A Tutorial Review on the Nonradiative Energy Transfer Rates between Lanthanide Ions

Albano N Carneiro Neto; Renaldo T Moura; Jorge A A Coelho; Mauro E Silva⁃Junior; Janderson L Costa; Oscar L Malta; Ricardo L Longo

doi:10.37188/CJL.EN20220007

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A Tutorial Review on the Nonradiative Energy Transfer Rates between Lanthanide Ions

Views on Basic Knowledge of Luminescence | Updated：2023-02-13

- A Tutorial Review on the Nonradiative Energy Transfer Rates between Lanthanide Ions Enhanced Publication
- Chinese Journal of Luminescence Vol.43, Issue 12, (2022)
- Affiliations：
  
  Physics Department and CICECO-Aveiro Institute of Materials， University of Aveiro， Aveiro,Portugal,3810-193;Department of Chemistry， Southern Methodist University,Dallas,United States,75275-0314;Department of Chemistry and Physics， Federal University of Paraíba， Areia， Brazil， 58397-000（permanent address）;Departamento de Química Fundamental， Universidade Federal de Pernambuco， Recife,Brazil,50740-560;Programa de Pós-Graduação em Ciência de Materiais， Universidade Federal de Pernambuco， Recife,Brazil,50670-901
- Author bio：
- Funds：
- DOI：10.37188/CJL.EN20220007 CLC： O482.31
- Published：05 December 2022，
  
  Received：20 May 2022，
  
  Revised：19 June 2022，
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Albano N Carneiro Neto,Renaldo T Moura Jr,Jorge A A Coelho等.镧系离子间无辐射能量传递速率的教程综述[J].发光学报,2022,43(12):1871-1891.

Neto Albano N Carneiro,Moura Renaldo T,Coelho Jorge A A,et al.A Tutorial Review on the Nonradiative Energy Transfer Rates between Lanthanide Ions[J].Chinese Journal of Luminescence,2022,43(12):1871-1891.
Albano N Carneiro Neto,Renaldo T Moura Jr,Jorge A A Coelho等.镧系离子间无辐射能量传递速率的教程综述[J].发光学报,2022,43(12):1871-1891. DOI： 10.37188/CJL.EN20220007.

Neto Albano N Carneiro,Moura Renaldo T,Coelho Jorge A A,et al.A Tutorial Review on the Nonradiative Energy Transfer Rates between Lanthanide Ions[J].Chinese Journal of Luminescence,2022,43(12):1871-1891. DOI： 10.37188/CJL.EN20220007.

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We emphasize the nature of the contributions which are different from those developed by Förster and Dexter theories because of the unique properties of the lanthanide ions. The expressions discussed here were based on Kushida's approach （electric multipolar mechanisms： dipole‑dipole（" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{d ‑ d}

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W_{e x}

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W_{m d ‑ m d}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818870&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818869&type=", "width": "7.45066643", "height": "3.21733332", "fontsize": "" } } } ] }, { "name": "text", "data": "）， and were improved to include shielding effects as well as an analytical expression for the " }, { "name": "italic", "data": [ { "name": "text", "data": "F" } ] }, { "name": "text", "data": "-factor （density of states in Fermi’s golden rule）. Similar to the original approach of Kushida， only the Forced Electric Dipole（FED） contributions to the Judd-Ofelt intensity parameters should be considered. A detailed discussion of selection rules and matrix elements calculations for the magnetic dipole‑magnetic dipole interaction is presented. In addition， step-by-step examples of Tb（Ⅲ）-Eu（Ⅲ） and Yb（Ⅲ）-Er（Ⅲ） energy transfer rates calculations are provided， with extensive Supporting Information， including scripts for calculations." } ] } ] }, { "lang": "zh", "data": [ { "name": "p", "data": [ { "name": "text", "data": "在本教程综述中，我们重新阐述和表达了镧系离子间无辐射能量传递（ET）速率的理论形式，并且强调了考虑镧系离子本身特异性所引发的与 Förster 和 Dexter 理论不同之处。所给出的表达式遵循了Judd‑Ofelt的4f‑4f跃迁理论框架之下的 Kushida 方法，并计入了如下电多极机制：偶极‑偶极（" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

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W_{q ‑ q}

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W_{e x}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818887&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818884&type=", "width": "4.23333359", "height": "3.21733332", "fontsize": "" } } } ] }, { "name": "text", "data": "）和磁偶极子⁃磁偶极子（" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{m d ⁃ m d}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818892&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818888&type=", "width": "7.45066643", "height": "3.21733332", "fontsize": "" } } } ] }, { "name": "text", "data": "）作用，并经过改进进一步包含了屏蔽效应以及给出了" }, { "name": "italic", "data": [ { "name": "text", "data": "F " } ] }, { "name": "text", "data": "因子（费米黄金规则中的态密度）的解析表达式。与 Kushida 的原始方法类似，我们只考虑了强制电偶极子（FED）对 Judd⁃Ofelt 强度参数的贡献，并细节性地讨论了磁偶极⁃磁偶极相互作用的选择定则以及相关矩阵元的计算。此外，我们还以Tb（Ⅲ）‑Eu（Ⅲ）和 Yb（Ⅲ）⁃Er（Ⅲ）的能量传递速率计算为例，逐步展示了计算过程、主要的计算支撑信息以及所使用的计算脚本。" } ] } ] } ], "keyword": [ { "lang": "zh", "data": [ [ { "name": "text", "data": "无辐射能量传递" } ], [ { "name": "text", "data": "镧系" } ], [ { "name": "text", "data": "理论计算" } ], [ { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": " 能量传递速率" } ], [ { "name": "text", "data": "选择定则" } ] ] }, { "lang": "en", "data": [ [ { "name": "text", "data": "nonradiative energy transfer" } ], [ { "name": "text", "data": "lanthanides" } ], [ { "name": "text", "data": "theoretical calculations" } ], [ { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": " energy transfer rates" } ], [ { "name": "text", "data": "selection rules" } ] ] } ], "highlights": [], "body": [ { "name": "sec", "data": [ { "name": "sectitle", "data": { "title": [ { "name": "text", "data": "1　Introduction" } ], "level": "1", "id": "s1" } }, { "name": "p", "data": [ { "name": "text", "data": "Energy transfer（ET） processes are ubiquitous in nature and are essential to life" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "1", "type": "bibr", "rid": "R1", "data": [ { "name": "text", "data": "1" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "2", "type": "bibr", "rid": "R2", "data": [ { "name": "text", "data": "2" } ] } } ], "rid": [ "R1", "R2" ], "text": "1-2", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". Despite being difficult to visualize, these ET processes are always present in our everyday life and obey the laws of thermodynamics（for macroscopic systems）. Many modern technologies are based on ET processes, for instance, those involving luminescence, where an excited sensitizer transfers energy to an activator that emits light（or electromagnetic radiation）. Most results of these ET processes are visible and quantifiable, but their detailed understanding and description, at the microscopic level, requires quantum mechanics. As a result, at least two concepts are important, namely, （quantum） states and transitions between these states." } ] }, { "name": "p", "data": [ { "name": "text", "data": "Photoluminescence involving lanthanide ions has some unique and remarkable features, which have been explored in many applications and new technologies, from biomedicine to engineering, and public health to public safety" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "3", "type": "bibr", "rid": "R3", "data": [ { "name": "text", "data": "3" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "7", "type": "bibr", "rid": "R7", "data": [ { "name": "text", "data": "7" } ] } } ], "rid": [ "R3", "R4", "R5", "R6", "R7" ], "text": "3-7", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". Energy transfer between lanthanide ions is particularly relevant to several processes such as long-range energy migration and energy upconversion. Resonances present in systems containing lanthanide ions of the same type, " }, { "name": "italic", "data": [ { "name": "text", "data": "e" } ] }, { "name": "text", "data": "." }, { "name": "italic", "data": [ { "name": "text", "data": "g" } ] }, { "name": "text", "data": "., Yb（Ⅲ）, allow the energy to be efficiently transferred between these ions, which can migrate for long distances depending on their concentration and the host matrix" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "8", "type": "bibr", "rid": "R8", "data": [ { "name": "text", "data": "8" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "11", "type": "bibr", "rid": "R11", "data": [ { "name": "text", "data": "11" } ] } } ], "rid": [ "R8", "R9", "R10", "R11" ], "text": "8-11", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". On the other hand, a lanthanide ion with relatively large oscillator strength, " }, { "name": "italic", "data": [ { "name": "text", "data": "e" } ] }, { "name": "text", "data": "." }, { "name": "italic", "data": [ { "name": "text", "data": "g" } ] }, { "name": "text", "data": "., Yb（Ⅲ） and Nd（Ⅲ）, can be employed as a sensitizer of another lanthanide ion, " }, { "name": "italic", "data": [ { "name": "text", "data": "e" } ] }, { "name": "text", "data": "." }, { "name": "italic", "data": [ { "name": "text", "data": "g" } ] }, { "name": "text", "data": "., Er（Ⅲ） and Tm（Ⅲ）, called an activator, which can be excited by sequential energy transfer processes, followed by emission of photons with higher energy than those absorbed photons; hence energy upconversion" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "10", "type": "bibr", "rid": "R10", "data": [ { "name": "text", "data": "10" } ] } }, { "name": "text", "data": "," }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "12", "type": "bibr", "rid": "R12", "data": [ { "name": "text", "data": "12" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "13", "type": "bibr", "rid": "R13", "data": [ { "name": "text", "data": "13" } ] } } ], "rid": [ "R12", "R13" ], "text": "12-13", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". In this context, energy transfer between lanthanide ions （" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": " ET） plays a fundamental and essential role in several processes that are explored and exploited in applications and new technologies" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "14", "type": "bibr", "rid": "R14", "data": [ { "name": "text", "data": "14" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "16", "type": "bibr", "rid": "R16", "data": [ { "name": "text", "data": "16" } ] } } ], "rid": [ "R14", "R15", "R16" ], "text": "14-16", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". Therefore, the focus of this contribution will be the quantitative description of the " }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": " ET processes." } ] }, { "name": "p", "data": [ { "name": "text", "data": "A common picture used for a general ET process employs two subsystems denoted as donor（D） and acceptor（A）. Initially, the donor is prepared in an excited state, " }, { "name": "italic", "data": [ { "name": "text", "data": "D" } ] }, { "name": "sup", "data": [ { "name": "text", "data": "*" } ] }, { "name": "text", "data": ", by absorbing energy（light, heat, and mechanical）, which may be represented as " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"i\"\"\" = \"\"\" separators=\"|\">, \"|\"D^{*} A\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819200&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819199&type=", "width": "17.10266685", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "text", "data": ". In other words, in the initial state, the energy absorbed by the system is localized at the donor subunit. Then, the system undergoes a transition to another state with the energy localized at the acceptor subsystem, represented as " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"f\"\"\" = \"\"\" separators=\"|\">, \"|\"D A^{*}\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818900&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818899&type=", "width": "18.28800011", "height": "5.33400011", "fontsize": "" } } } ] }, { "name": "text", "data": ". This general description requires some clarification and specification of different categories of ET processes as well as additional phenomena accompanying these processes. For instance, energy transfer can induce or be caused by electron（or charge） transfer between the subsystems. These coupled processes are important and relevant for many biological systems and applications, but they will not be considered here. The ET processes of interest are uncoupled energy transfer processes in the sense that are not caused nor can induce electron transfer. This is an important distinction that is not always clear in the literature and has caused some confusion." } ] }, { "name": "p", "data": [ { "name": "text", "data": "The transition mentioned above describes an ET process, represented by " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"D^{*} A\"\"\" \to \"\"\" separators=\"|\">, \"|\"D A^{*}\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818904&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818902&type=", "width": "23.53733444", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "text", "data": ", can be pictured as a deexcitation of the donor and an excitation of the acceptor, which may or may not involve photon（s）. When the deexcitation of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

D^{*}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818908&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818906&type=", "width": "3.55599999", "height": "2.96333337", "fontsize": "" } } } ] }, { "name": "text", "data": " occurs by photon emission（with energy " }, { "name": "italic", "data": [ { "name": "text", "data": "ћω=hν" } ] }, { "name": "text", "data": "）, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"D^{*}\"\"\" \to

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818911&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818909&type=", "width": "11.51466751", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"D\"\"\" + h ν

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818915&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818914&type=", "width": "13.37733269", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": ", and the emitted photon is absorbed by " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

A

\"\"\" separators=\"|\">, \"|\"A\"\"\" + h ν \to \"\"\" separators=\"|\">, \"|\"A^{*}\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818921&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818919&type=", "width": "25.39999962", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "text", "data": ", then the ET from " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

D^{*}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818924&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818923&type=", "width": "3.55599999", "height": "2.96333337", "fontsize": "" } } } ] }, { "name": "text", "data": " to " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

A

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818927&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818926&type=", "width": "2.28600001", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " is known as radiative energy transfer. This is also an important category of ET processes; however, they are usually not operative when the acceptor（or activator） is a lanthanide ion. This is because the oscillator strengths associated with 4f-4f transitions are very small（~10" }, { "name": "sup", "data": [ { "name": "text", "data": "-6" } ] }, { "name": "text", "data": "）" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "17", "type": "bibr", "rid": "R17", "data": [ { "name": "text", "data": "17" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "19", "type": "bibr", "rid": "R19", "data": [ { "name": "text", "data": "19" } ] } } ], "rid": [ "R17", "R18", "R19" ], "text": "17-19", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ", except for some transitions in Nd（Ⅲ） and Yb（Ⅲ）, which preclude radiative absorption. Thus, for those interested in lanthanide ions, the dominant ET processes are nonradiative（or radiationless）. Recently, the intramolecular ET processes in lanthanide complexes, namely, ligand-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "（Ⅲ） transfer rates, were revised and reviewed" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "20", "type": "bibr", "rid": "R20", "data": [ { "name": "text", "data": "20" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". Thus, the present review will focus on radiationless or nonradiative ET processes between lanthanide ions, " }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "." } ] }, { "name": "p", "data": [ { "name": "text", "data": "The main goal is to present a review for non-specialists, providing sufficient concepts to critically ascertain the approximations involved in the equations of the energy transfer rate mechanisms. Enough details will be provided for a qualitative assessment of the main factors affecting the ET processes in " }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": " systems as well as to perform estimations and calculations of these ET rates. The original equations were reorganized to ease their applications as well as their interpretations. In the Supporting Information, a derivation of Fermi's golden rule is provided together with the physical constants and unit conversions, details matrix elements calculations, and dimensional analysis of the main quantities." } ] }, { "name": "p", "data": [ { "name": "text", "data": "In the following sections, some concepts and equations involved in the quantum treatment of transitions will be present culminating in Fermi's golden rule for transition rates. This expression will then be particularized for nonradiative transitions between two lanthanide ions, namely, the " }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": " energy transfer rates. Detailed applications of these equations will be presented as tutorial examples for ET processes involved in the Yb（Ⅲ）-Er（Ⅲ） and Tb（Ⅲ）-Eu（Ⅲ） pairs." } ] } ] }, { "name": "sec", "data": [ { "name": "sectitle", "data": { "title": [ { "name": "text", "data": "2　Quantum Transitions" } ], "level": "1", "id": "s2" } }, { "name": "p", "data": [ { "name": "text", "data": "An energy transfer process can be regarded as a transition from a state representing the energy localized initially at the donor subsystem to another state characterized by the energy localized in the acceptor unit. This process can be described quantitatively by an energy transfer rate, that is, the probability of transition induced by a perturbation per unit of time, in units of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

s^{- 1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818931&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818929&type=", "width": "3.64066648", "height": "2.96333337", "fontsize": "" } } } ] }, { "name": "text", "data": ", because the probability is dimensionless. Thus, to comprehend the equations for the ET rates, representations of the initial and final states are required, which will be addressed in detail later, as well as the transition probability between these states per unit of time." } ] }, { "name": "p", "data": [ { "name": "text", "data": "For most transitions occurring at the molecular level, their rates can be quantitatively described by Fermi's golden rule. A detailed derivation of the expression for the transition rate, in particular Fermi's golden rule, can be found in many textbooks" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "21", "type": "bibr", "rid": "R21", "data": [ { "name": "text", "data": "21" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "22", "type": "bibr", "rid": "R22", "data": [ { "name": "text", "data": "22" } ] } } ], "rid": [ "R21", "R22" ], "text": "21-22", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " and is provided in the Supporting Information. Here we will present the main equations and the approximations and assumptions employed." } ] }, { "name": "p", "data": [ { "name": "text", "data": "Transitions at the molecular level are described by quantum mechanics. A system is described by its state represented by a （vector） state or wavefunction, which depends on the spatial and spin coordinates of its particles and on the time " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

t

{Ψ_{n} (t)}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818937&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818935&type=", "width": "12.02266598", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": " with corresponding energy eigenvalues " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{E_{n}}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818939&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818938&type=", "width": "6.77333355", "height": "4.40266657", "fontsize": "" } } } ] }, { "name": "text", "data": "." } ] }, { "name": "p", "data": [ { "name": "text", "data": "If the system is described by a time-independent Hamiltonian, " }, { "name": "italic", "data": [ { "name": "text", "data": "e" } ] }, { "name": "text", "data": "." }, { "name": "italic", "data": [ { "name": "text", "data": "g" } ] }, { "name": "text", "data": "., a molecule with contributions from the kinetic energy of the nuclei and electrons and Coulomb interactions between the charged particles （nuclei-nuclei, nuclei-electrons, and electrons-electrons）, the time-dependent Schrödinger equation can be separated and have solutions " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ψ_{n} (t) = ψ_{n} e^{- i E_{n} t / ћ}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818941&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818940&type=", "width": "24.55333328", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "text", "data": ", where " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ψ_{n}

Ψ_{m}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818946&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818945&type=", "width": "4.48733330", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": ", with energy " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

E_{m}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818948&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818947&type=", "width": "3.89466691", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": ", then it is stationary, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ψ_{m}^{*} (t) Ψ_{m} (t) = ψ_{m}^{*} e^{i E_{m} t / ћ} ψ_{m} e^{- i E_{m} t / ћ} = ψ_{m}^{*} ψ_{m} = {\"|\"ψ_{m}\"|\"}^{2} =

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818951&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818950&type=", "width": "74.42199707", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "text", "data": " constant, and it will remain in this state indefinitely. Thus, a perturbation " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\hat{V} (t)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818977&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818976&type=", "width": "6.77333355", "height": "4.23333359", "fontsize": "" } } } ] }, { "name": "text", "data": " is needed to promote a transition from this state to another state. This perturbation causes the temporal evolution of a non-stationary state " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ψ (t)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818981&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818980&type=", "width": "7.61999989", "height": "3.47133350", "fontsize": "" } } } ] }, { "name": "text", "data": ", which can still be represented as a linear combination of eigenfunctions " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{Ψ_{n} (t)}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818985&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818984&type=", "width": "12.02266598", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": " with coefficients " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{c_{n} (t)}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818987&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818986&type=", "width": "10.83733368", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": ". As a result, the probability of finding the system at time " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

t

k

{\"|\"c_{k} (t)\"|\"}^{2} = c_{k}^{*} (t) c_{k} (t)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818971&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818970&type=", "width": "29.97200012", "height": "4.99533367", "fontsize": "" } } } ] }, { "name": "text", "data": "." } ] }, { "name": "p", "data": [ { "name": "text", "data": "As a result of this perturbation, the system can be described by a Hamiltonian, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\hat{H}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818990&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818989&type=", "width": "2.70933342", "height": "3.64066648", "fontsize": "" } } } ] }, { "name": "text", "data": ", consisting of the time-independent Hamiltonian, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{\hat{H}}_{0}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818975&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818974&type=", "width": "3.72533321", "height": "4.74133301", "fontsize": "" } } } ] }, { "name": "text", "data": ", and the perturbation " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\hat{V} (t)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818977&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818976&type=", "width": "6.77333355", "height": "4.23333359", "fontsize": "" } } } ] }, { "name": "text", "data": ", " }, { "name": "italic", "data": [ { "name": "text", "data": "i" } ] }, { "name": "text", "data": "." }, { "name": "italic", "data": [ { "name": "text", "data": "e" } ] }, { "name": "text", "data": "., " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\hat{H} = {\hat{H}}_{0} + \hat{V} (t)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818979&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818978&type=", "width": "21.92866707", "height": "5.33400011", "fontsize": "" } } } ] }, { "name": "text", "data": ". A procedure developed by Dirac starts by replacing the wavefunction " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ψ (t)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818981&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818980&type=", "width": "7.61999989", "height": "3.47133350", "fontsize": "" } } } ] }, { "name": "text", "data": ", in the time-dependent Schrödinger equation employing " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\hat{H}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818990&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818989&type=", "width": "2.70933342", "height": "3.64066648", "fontsize": "" } } } ] }, { "name": "text", "data": ", by an expansion in terms of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{Ψ_{n} (t)}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818985&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818984&type=", "width": "12.02266598", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{c_{n} (t)}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818987&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818986&type=", "width": "10.83733368", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": ". After applying the time derivative and the full Hamiltonian " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\hat{H}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818990&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818989&type=", "width": "2.70933342", "height": "3.64066648", "fontsize": "" } } } ] }, { "name": "text", "data": " on this expansion and projecting the resulting equation onto state " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

k

\"<\"\"\"k\"|\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818994&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818993&type=", "width": "4.40266657", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": "）, an exact equation for the time evolution " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{\dot{c}}_{n} (t)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818996&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39818995&type=", "width": "7.28133297", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": " is obtained. By integrating the resulting equation from " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

t = 0

c_{n} (0)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819006&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819005&type=", "width": "8.04333401", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": ", to the time " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

t

c_{n} (t)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819004&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819003&type=", "width": "7.28133297", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": " is derived, so the probability could, in principle, be calculated. Thus, a uniquely define exact solution is obtained when the boundary conditions, namely, the initial coefficients " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

c_{n} (0)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819006&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819005&type=", "width": "8.04333401", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": " are known. These conditions are usually expressed as " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

c_{n} (t = 0) = δ_{n i}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819008&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819007&type=", "width": "21.25133324", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": ", where " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

i

c_{i} (t = 0) = 1

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819012&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819011&type=", "width": "19.38866615", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

c_{n \neq i} (t = 0) = 0

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819014&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819013&type=", "width": "22.77533340", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": "." } ] }, { "name": "p", "data": [ { "name": "text", "data": "However, the exact determination of the coefficient " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

c_{k} (t)

c_{k} (t)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819018&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819017&type=", "width": "7.19666624", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": ", " }, { "name": "italic", "data": [ { "name": "text", "data": "i" } ] }, { "name": "text", "data": "." }, { "name": "italic", "data": [ { "name": "text", "data": "e" } ] }, { "name": "text", "data": "., the exact solution is a set of coupled integral equations. So, approximations and assumptions are needed to obtain a computable expression." } ] }, { "name": "p", "data": [ { "name": "text", "data": "Suppose that the perturbation is so weak and applied during a time interval so short that all the coefficients remain close to their initial values and are approximated as " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

c_{n} (t) ≅ 0

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819020&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819019&type=", "width": "13.46199989", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": " for " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

n \neq i

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819022&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819021&type=", "width": "7.45066643", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

c_{i} (t) ≅ 1

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819024&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819023&type=", "width": "13.03866673", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": ". This is the so-called first-order perturbation or first-order approximation and yields the following expression for the coefficients （" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

k \neq i

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819026&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819025&type=", "width": "7.28133297", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": "） at time " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

t

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819028&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819027&type=", "width": "1.10066664", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "21", "type": "bibr", "rid": "R21", "data": [ { "name": "text", "data": "21" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "22", "type": "bibr", "rid": "R22", "data": [ { "name": "text", "data": "22" } ] } } ], "rid": [ "R21", "R22" ], "text": "21-22", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " :" } ] }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（1）" } ], "data": [ { "name": "math", "data": { "math": "

c_{k} (t) ≅ c_{k}^{(1)} = \frac{1}{i ћ} \int_{0}^{t} V_{k i} (t^{'}) e^{i ω_{k i} t^{'}} d t^{'}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819030&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819029&type=", "width": "46.65133286", "height": "7.53533268", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF1" } }, { "name": "p", "data": [ { "name": "text", "data": "where " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ω_{k i} = (E_{k} - E_{i}) / ћ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819032&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819031&type=", "width": "26.07733345", "height": "3.97933316", "fontsize": "" } } } ] }, { "name": "text", "data": " and the generic matrix element, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

V_{k n}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819034&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819033&type=", "width": "4.14866638", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": ", of the perturbation operator " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\hat{V}

V_{k n} (t) = \int ψ_{k}^{*} \hat{V} (t) ψ_{n} d τ \equiv \"<\"\" separators=\"|\">, k | \hat{V} (t) | n\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819038&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819037&type=", "width": "51.90066528", "height": "4.99533367", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF2" } }, { "name": "p", "data": [ { "name": "text", "data": "with the respect to the stationary eigenfunctions " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ψ_{k}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819040&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819039&type=", "width": "3.47133350", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ψ_{n}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819042&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819041&type=", "width": "3.55599999", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": ". Notice that the first equality employs the integral representation of the matrix element, which is equivalent to that in the ‘bra-ket’（or Dirac） notation." } ] } ] }, { "name": "sec", "data": [ { "name": "sectitle", "data": { "title": [ { "name": "text", "data": "3　Fermi’s Golden Rule for Radiationless（or Nonradiative） Transitions" } ], "level": "1", "id": "s3" } }, { "name": "p", "data": [ { "name": "text", "data": "Several processes are induced by a coupling between subsystems or between states and radiation is not required, representing the so-called radiationless or nonradiative transitions. For example, in a time " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

t = 0

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819062&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819061&type=", "width": "7.45066643", "height": "2.70933342", "fontsize": "" } } } ] }, { "name": "text", "data": " the system （instantaneously） absorbs energy （" }, { "name": "italic", "data": [ { "name": "text", "data": "e" } ] }, { "name": "text", "data": "." }, { "name": "italic", "data": [ { "name": "text", "data": "g" } ] }, { "name": "text", "data": "., light absorption, mechanical or thermal excitations） and is promoted to an excited state, thus initiating a coupling or interaction. This is the initial condition of the system, and this excited state is its initial state " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

i

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819727&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819726&type=", "width": "1.10066664", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": ". The nonadiabatic electronic coupling or the spin-orbit coupling between the initial state " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

i

k

i

k

i \to k

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819056&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819055&type=", "width": "8.89000034", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": ". Another example is a system of two uncoupled subunits, " }, { "name": "italic", "data": [ { "name": "text", "data": "e" } ] }, { "name": "text", "data": "." }, { "name": "italic", "data": [ { "name": "text", "data": "g" } ] }, { "name": "text", "data": "., donor and acceptor " }, { "name": "italic", "data": [ { "name": "text", "data": "D" } ] }, { "name": "text", "data": "-" }, { "name": "italic", "data": [ { "name": "text", "data": "A" } ] }, { "name": "text", "data": ". Upon excitation, at " }, { "name": "italic", "data": [ { "name": "text", "data": "t" } ] }, { "name": "text", "data": "=0, localized on the donor subsystem, the system can be represented as " }, { "name": "italic", "data": [ { "name": "text", "data": "D" } ] }, { "name": "sup", "data": [ { "name": "text", "data": "*" } ] }, { "name": "text", "data": "-" }, { "name": "italic", "data": [ { "name": "text", "data": "A " } ] }, { "name": "text", "data": "and a coupling or interaction between " }, { "name": "italic", "data": [ { "name": "text", "data": "D" } ] }, { "name": "text", "data": " and " }, { "name": "italic", "data": [ { "name": "text", "data": "A" } ] }, { "name": "text", "data": " appears and promotes a nonradiative transition to another state that can lead to energy transfer or electron transfer, depending on the nature of the donor and acceptor, and on the initial and final states. In these radiationless transitions, the coupling or interaction initiated at " }, { "name": "italic", "data": [ { "name": "text", "data": "t" } ] }, { "name": "text", "data": "=0 does not depend explicitly on time, so it is constant in time: " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\hat{V} (t) \equiv \hat{V} =

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819058&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819057&type=", "width": "16.34066582", "height": "4.23333359", "fontsize": "" } } } ] }, { "name": "text", "data": " constant." } ] }, { "name": "p", "data": [ { "name": "text", "data": "Thus, for nonradiative energy transfer processes, a constant perturbation " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\hat{V}

t = 0

t = τ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819064&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819063&type=", "width": "7.36600018", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": ". For this perturbation, the coefficient describing state " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

k

t \geq τ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819068&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819067&type=", "width": "7.36600018", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " becomes" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "21", "type": "bibr", "rid": "R21", "data": [ { "name": "text", "data": "21" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " :" } ] }, { "name": "dispformula", "data": { "label": [], "data": [ { "name": "math", "data": { "math": "

c_{k} = - \frac{i}{ћ} V_{k i} \int_{0}^{τ} e^{i ω_{k i} t} d t,

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819070&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819069&type=", "width": "30.05666542", "height": "7.53533268", "fontsize": "" } } } ], "id": "DF3" } }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（3）" } ], "data": [ { "name": "math", "data": { "math": "

V_{k i} = \int ψ_{k}^{*} \hat{V} ψ_{i} d τ \equiv \"<\"\" separators=\"|\">, k | \hat{V} | i\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819072&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819071&type=", "width": "44.61933517", "height": "6.43466663", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF4" } }, { "name": "p", "data": [ { "name": "text", "data": "where it was considered that the matrix element " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

V_{k i}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819132&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819131&type=", "width": "3.72533321", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " has a weak dependence on the quasi-continuum states around " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

k

τ

τ

k

P_{k}^{i} = {\"|\"V_{k i}\"|\"}^{2} F_{k}^{i} (ω_{k i}, τ),

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819084&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819083&type=", "width": "32.34266663", "height": "4.99533367", "fontsize": "" } } } ], "id": "DF5" } }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（4）" } ], "data": [ { "name": "math", "data": { "math": "

F_{k}^{i} (ω_{k i}, τ) = \frac{1}{ћ^{2}} \frac{s i n^{2} (ω_{k i} τ / 2)}{{(ω_{k i} / 2)}^{2}} .

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819086&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819085&type=", "width": "44.70399857", "height": "12.44600010", "fontsize": "" } } }, { "name": "text", "data": " " } ], "id": "DF6" } }, { "name": "p", "data": [ { "name": "text", "data": "This expression yields the same probability for the reverse transition: " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

P_{k}^{i} = P_{i}^{k}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819088&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819087&type=", "width": "11.17599964", "height": "4.06400013", "fontsize": "" } } } ] }, { "name": "text", "data": ", because " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{\"|\"V_{k i}\"|\"}^{2} = {\"|\"V_{i k}\"|\"}^{2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819090&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819089&type=", "width": "18.79599953", "height": "4.99533367", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F_{k}^{i} = F_{i}^{k}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819092&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819091&type=", "width": "11.43000031", "height": "4.06400013", "fontsize": "" } } } ] }, { "name": "text", "data": "." } ] }, { "name": "p", "data": [ { "name": "text", "data": "For " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

τ ≳ 2 π ℏ / E_{k}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819094&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819093&type=", "width": "17.44133186", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": ", the function " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F_{k}^{i} (ω_{k i}, τ)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819162&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819161&type=", "width": "14.73200035", "height": "4.82600021", "fontsize": "" } } } ] }, { "name": "text", "data": " becomes sharply peaked, with the height of the peak around " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

E_{k}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819138&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819137&type=", "width": "3.38666677", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " increasing as " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

τ^{2} / ћ^{2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819100&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819099&type=", "width": "7.45066643", "height": "3.38666677", "fontsize": "" } } } ] }, { "name": "text", "data": " and width decreasing as " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

2 π ћ / τ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819102&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819101&type=", "width": "9.48266602", "height": "3.21733332", "fontsize": "" } } } ] }, { "name": "text", "data": ", so the area of this peak is approximately " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

2 π τ / ћ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819104&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819103&type=", "width": "9.48266602", "height": "3.21733332", "fontsize": "" } } } ] }, { "name": "text", "data": ", which grows linearly with " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

τ

τ

δ E = 2 π ℏ / τ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819110&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819109&type=", "width": "17.94933319", "height": "3.13266683", "fontsize": "" } } } ] }, { "name": "text", "data": ", which is consistent with the energy-time uncertainty principle because " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

δ E τ = 2 π ћ > ћ / 2

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819112&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819111&type=", "width": "26.24666595", "height": "3.21733332", "fontsize": "" } } } ] }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "21", "type": "bibr", "rid": "R21", "data": [ { "name": "text", "data": "21" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". So, the final states after the perturbation are better described as nearly a continuum of energy values with a narrow distribution represented by a density of states, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ρ (E)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819124&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819123&type=", "width": "7.78933382", "height": "3.55599999", "fontsize": "" } } } ] }, { "name": "text", "data": ", which is the number of states per energy unit around energy " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

E

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819120&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819119&type=", "width": "2.28600001", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ρ (E) d E

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819118&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819117&type=", "width": "12.02266598", "height": "3.55599999", "fontsize": "" } } } ] }, { "name": "text", "data": " is the number of （continuum） states in the range " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

E

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819120&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819119&type=", "width": "2.28600001", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " to " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

E + d E

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819122&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819121&type=", "width": "11.43000031", "height": "2.70933342", "fontsize": "" } } } ] }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "21", "type": "bibr", "rid": "R21", "data": [ { "name": "text", "data": "21" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "22", "type": "bibr", "rid": "R22", "data": [ { "name": "text", "data": "22" } ] } } ], "rid": [ "R21", "R22" ], "text": "21-22", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". In other words, the perturbation, represented by the interaction （or coupling）, has generated many final states closely spaced in energy. For a molecular system, other mechanisms of energy dispersion will also be present, such as a distribution of closely spaced rotational‑vibrational energy levels. Thus, if the final states have a density of states " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ρ (E)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819124&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819123&type=", "width": "7.78933382", "height": "3.55599999", "fontsize": "" } } } ] }, { "name": "text", "data": ", then the total probability of transition to state " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

k

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819126&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819125&type=", "width": "1.69333339", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " becomes an integral of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

P_{k}^{i} ρ (E) d E

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819128&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819127&type=", "width": "15.66333294", "height": "4.99533367", "fontsize": "" } } } ] }, { "name": "text", "data": " over all final states accessible under the influence of the perturbation:" } ] }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（5）" } ], "data": [ { "name": "math", "data": { "math": "

P_{\bar{k}}^{i} = \int {\"|\"V_{k i}\"|\"}^{2} F_{k}^{i} (ω_{k i}, τ) ρ (E_{k}) d E_{k} ≅ {\"|\"V_{k i}\"|\"}^{2} ρ (E_{k}) \int F_{k}^{i} (ω_{k i}, τ) d E_{k}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819130&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819129&type=", "width": "124.20600128", "height": "6.43466663", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF7" } }, { "name": "p", "data": [ { "name": "text", "data": "where the last approximation considers that the matrix element " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

V_{k i}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819132&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819131&type=", "width": "3.72533321", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " and the density of state " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ρ (E_{k})

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819166&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819165&type=", "width": "8.80533314", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " are constant in the integration range. Because the function " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F_{k}^{i} (ω_{k i}, τ)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819162&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819161&type=", "width": "14.73200035", "height": "4.82600021", "fontsize": "" } } } ] }, { "name": "text", "data": " is sharply peaked, the integrand has significant values only around " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

E_{k}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819138&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819137&type=", "width": "3.38666677", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": ", which justifies the last approximation. This property of the function " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F_{k}^{i} (ω_{k i}, τ)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819162&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819161&type=", "width": "14.73200035", "height": "4.82600021", "fontsize": "" } } } ] }, { "name": "text", "data": " also allows for extending the integration from " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

- \infty

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819142&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819141&type=", "width": "4.74133301", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " to " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

+ \infty

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819144&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819143&type=", "width": "4.74133301", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": ", so the last integral is just " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

2 π τ / ℏ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819146&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819145&type=", "width": "9.48266602", "height": "3.13266683", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

P_{\bar{k}}^{i}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819148&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819147&type=", "width": "3.38666677", "height": "4.06400013", "fontsize": "" } } } ] }, { "name": "text", "data": " has a linear dependence on " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

τ

\bar{k}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819152&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819151&type=", "width": "1.69333339", "height": "3.13266683", "fontsize": "" } } } ] }, { "name": "text", "data": " represents the summation over over the quasi-continuum （" }, { "name": "italic", "data": [ { "name": "text", "data": "e.g" } ] }, { "name": "text", "data": "., rovibrational） contributions.. As a result, the transition rate " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{i \to k} \equiv W_{k i}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819154&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819153&type=", "width": "16.76399994", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": " from an initial state " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

i

k

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819725&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819724&type=", "width": "1.69333339", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": ", which is the change of the probability per unit of time, can be written as" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "21", "type": "bibr", "rid": "R21", "data": [ { "name": "text", "data": "21" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "22", "type": "bibr", "rid": "R22", "data": [ { "name": "text", "data": "22" } ] } } ], "rid": [ "R21", "R22" ], "text": "21-22", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": "：" } ] }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（6）" } ], "data": [ { "name": "math", "data": { "math": "

W_{i \to k} \equiv W_{k i} = \frac{d P_{k}^{i}}{d t} ≅ \frac{P_{\bar{k}}^{i}}{τ} = \frac{2 π}{ћ} {\"|\"V_{k i}\"|\"}^{2} ρ (E_{k}) .

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819160&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819159&type=", "width": "64.17733002", "height": "7.78933382", "fontsize": "" } } }, { "name": "text", "data": " " } ], "id": "DF8" } }, { "name": "p", "data": [ { "name": "text", "data": "This is called Fermi’s golden rule and provides the rate of transition." } ] }, { "name": "p", "data": [ { "name": "text", "data": "Despite this derivation being performed for constant perturbation, the same expression is obtained for periodic potentials that describe, for instance, the electromagnetic radiation." } ] }, { "name": "p", "data": [ { "name": "text", "data": "For a proper application of this expression for the transition rate, it is important to present the approximations and assumptions as well as their consequences （see Supporting Information）." } ] }, { "name": "p", "data": [ { "name": "text", "data": "The task of applying " }, { "name": "xref", "data": { "text": "Eq.（6）", "type": "disp-formula", "rid": "DF8", "data": [ { "name": "text", "data": "Eq.（6）" } ] } }, { "name": "text", "data": " to a given process becomes （ⅰ） the proper representation of the initial and final states of the system, （ⅱ） defining and establishing the perturbation, and （ⅲ） determining the density of states." } ] }, { "name": "p", "data": [ { "name": "text", "data": "The sharply peaked function " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F_{k}^{i} (ω_{k i}, τ)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819162&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819161&type=", "width": "14.73200035", "height": "4.82600021", "fontsize": "" } } } ] }, { "name": "text", "data": " appearing in " }, { "name": "xref", "data": { "text": "Eq.（5）", "type": "disp-formula", "rid": "DF7", "data": [ { "name": "text", "data": "Eq.（5）" } ] } }, { "name": "text", "data": " becomes a Dirac delta function at the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

τ \to \infty

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819164&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819163&type=", "width": "10.58333302", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " limit. As a result, the density of states " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ρ (E_{k})

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819166&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819165&type=", "width": "8.80533314", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " in " }, { "name": "xref", "data": { "text": "Eq.（6）", "type": "disp-formula", "rid": "DF8", "data": [ { "name": "text", "data": "Eq.（6）" } ] } }, { "name": "text", "data": " can be replaced by the delta function " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

δ (E_{k} - E_{i})

τ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819170&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819169&type=", "width": "1.77800000", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "21", "type": "bibr", "rid": "R21", "data": [ { "name": "text", "data": "21" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "22", "type": "bibr", "rid": "R22", "data": [ { "name": "text", "data": "22" } ] } } ], "rid": [ "R21", "R22" ], "text": "21-22", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " :" } ] }, { "name": "dispformula", "data": { "label": [], "data": [ { "name": "math", "data": { "math": "

w_{k i} ≅ \frac{2 π}{ћ} {\"|\"V_{k i}\"|\"}^{2} δ (E_{k} - E_{i}),

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819172&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819171&type=", "width": "40.38600159", "height": "7.36600018", "fontsize": "" } } } ], "id": "DF9" } }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（7）" } ], "data": [ { "name": "math", "data": { "math": "

W_{k i} = \int w_{k i} ρ (E_{k}) d E_{k} .

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819174&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819173&type=", "width": "32.25799942", "height": "6.43466663", "fontsize": "" } } }, { "name": "text", "data": " " } ], "id": "DF10" } }, { "name": "p", "data": [ { "name": "text", "data": "This expression is known as state-to-state Fermi's golden rule. The delta function " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

δ (E_{k} - E_{i})

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819176&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819175&type=", "width": "15.40933418", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": " expresses the condition that at the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

τ \to \infty

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819178&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819177&type=", "width": "9.90600014", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " limit, only transitions that satisfy energy conservation can be caused by a secular（time-independent） interaction. The presence of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

δ (E_{k} - E_{i})

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819180&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819179&type=", "width": "16.00200081", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": " in " }, { "name": "xref", "data": { "text": "Eq.（7）", "type": "disp-formula", "rid": "DF10", "data": [ { "name": "text", "data": "Eq.（7）" } ] } }, { "name": "text", "data": " does not violate the energy-time uncertainty principle, because as " }, { "name": "italic", "data": [ { "name": "text", "data": "τ→∞" } ] }, { "name": "text", "data": ", the energy dispersion " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

δ E = 2 π ћ / τ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819182&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819181&type=", "width": "17.94933319", "height": "3.21733332", "fontsize": "" } } } ] }, { "name": "text", "data": " vanishes." } ] } ] }, { "name": "sec", "data": [ { "name": "sectitle", "data": { "title": [ { "name": "text", "data": "4　Rates of Energy Transfer Processes" } ], "level": "1", "id": "s4" } }, { "name": "p", "data": [ { "name": "text", "data": "As mentioned, the application of " }, { "name": "xref", "data": { "text": "Eq.（6）", "type": "disp-formula", "rid": "DF8", "data": [ { "name": "text", "data": "Eq.（6）" } ] } }, { "name": "text", "data": " for an energy transfer process between a donor and acceptor D-A requires the proper representation of the initial and final states. Usually, the subsystems D and A are considered uncoupled or weakly coupled, so that the state of the system can be represented by a （tensor） product of wavefunctions describing the subunits, namely, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ψ \equiv Ψ_{D A} ≅ Ψ_{D} Ψ_{A}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819184&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819183&type=", "width": "25.23066711", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": " or " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"Ψ\"\"\" \equiv \"\"\" separators=\"|\">, \"|\"D A\"\"\" ≅

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819186&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819185&type=", "width": "19.81200027", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"D\"\"\" \"\"\" separators=\"|\">, \"|\"A\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819188&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819187&type=", "width": "10.92199993", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": ". In this representation, the wavefunction " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ψ_{A}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819190&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819189&type=", "width": "4.40266657", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": " depends on the spatial and spin coordinates of the nuclei and electrons constituting the acceptor and similarly " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ψ_{D}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819192&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819191&type=", "width": "4.40266657", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": " the donor. Despite this simple notation, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ψ_{D}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819194&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819193&type=", "width": "4.40266657", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ψ_{A}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819196&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819195&type=", "width": "4.40266657", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": " represent multielectronic states that may require more than one determinant （multideterminantal） for an adequate description. Employing a single particle or monoelectronic representation may lead to improper descriptions and an unreliable picture of the energy transfer process." } ] }, { "name": "p", "data": [ { "name": "text", "data": "The initial state " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"i\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819198&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819197&type=", "width": "3.72533321", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": " of an energy transfer process can be described by the donor at a higher or excited energy level and the acceptor at a lower or ground level denoted as " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"i\"\"\" = \"\"\" separators=\"|\">, \"|\"D^{*} A\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819200&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819199&type=", "width": "17.10266685", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "text", "data": ". This represents the additional energy to be localized at the donor subsystem. Whereas the final state " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"f\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819202&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819201&type=", "width": "5.33400011", "height": "5.33400011", "fontsize": "" } } } ] }, { "name": "text", "data": " would have this additional energy localized at the acceptor subunit and is represented as " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"f\"\"\" = \"\"\" separators=\"|\">, \"|\"D A^{*}\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819204&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819203&type=", "width": "18.28800011", "height": "5.33400011", "fontsize": "" } } } ] }, { "name": "text", "data": ". As a result, the energy transfer process " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"i\"\"\" \to \"\"\" separators=\"|\">, \"|\"f\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819206&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819205&type=", "width": "15.15533447", "height": "5.33400011", "fontsize": "" } } } ] }, { "name": "text", "data": " can be pictured as a simultaneous deexcitation of the donor and excitation of the acceptor " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"D^{*} A\"\"\" \to \"\"\" separators=\"|\">, \"|\"D A^{*}\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819208&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819207&type=", "width": "23.53733444", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "text", "data": ", without the presence of any photon or charge transferred. The ET process can then be represented as " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"D^{*}\"\"\" \"\"\" separators=\"|\">, \"|\"A\"\"\" \to \"\"\" separators=\"|\">, \"|\"D\"\"\" \"\"\" separators=\"|\">, \"|\"A^{*}\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819210&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819209&type=", "width": "30.56466866", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "text", "data": " to emphasize the independence of the subsystems. In addition, this representation allows for selection rules of the ET process. In other words, depending on the perturbation " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\hat{V}

\"\"\" separators=\"|\">, \"|\"D^{*}\"\"\" \to \"\"\" separators=\"|\">, \"|\"D\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819214&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819213&type=", "width": "18.03400040", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "text", "data": " or excitation " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"A\"\"\" \to \"\"\" separators=\"|\">, \"|\"A^{*}\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819216&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819215&type=", "width": "17.27199936", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "text", "data": " may not be allowed, so this energy transfer pathway is forbidden. However, it is important to remember that the selection rules will also depend on the nature of the donor and acceptor subsystems." } ] }, { "name": "p", "data": [ { "name": "text", "data": "Because the D-A subunits are considered independent, the perturbation can be promptly and easily pictured as the interaction between these subsystems. So, the perturbation " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\hat{V}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819302&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819301&type=", "width": "2.37066650", "height": "3.64066648", "fontsize": "" } } } ] }, { "name": "text", "data": " can be represented by the Coulomb （or electrostatic） interactions between electron densities and nuclear charges of the donor and the acceptor. This perturbation leads to a mechanism known as multipolar" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "23", "type": "bibr", "rid": "R23", "data": [ { "name": "text", "data": "23" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " or, more restricted, dipolar or Förster" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "24", "type": "bibr", "rid": "R24", "data": [ { "name": "text", "data": "24" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". On the other hand, approximating the wavefunction of the system by a （tensor） product causes the electrons of the donor to be distinguishable from the electrons of the acceptor, which violates the indistinguishability and antisymmetry principles of quantum mechanics. So, the exchanges between the electrons of the donor and the acceptor can be regarded as a perturbation of the D-A subunits. This perturbation can be described by an exchange operator and originates a mechanism known as exchange or Dexter" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "25", "type": "bibr", "rid": "R25", "data": [ { "name": "text", "data": "25" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " ." } ] }, { "name": "p", "data": [ { "name": "text", "data": "A step further in the development of a workable expression for the energy transfer rate can be taken by realizing that, usually, the time scale of the nuclear motion is much longer than of the electronic motion. As a result, the molecular wavefunction can be approximately factored into a product of electronic and nuclear wavefunctions. This is the well-known Born-Oppenheimer approximation, which has been used since the dawn of quantum mechanics and its successes and limitations are well-established" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "26", "type": "bibr", "rid": "R26", "data": [ { "name": "text", "data": "26" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "27", "type": "bibr", "rid": "R27", "data": [ { "name": "text", "data": "27" } ] } } ], "rid": [ "R26", "R27" ], "text": "26-27", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". So, the wavefunctions describing the initial state with the energy localized at the donor, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"i\"\"\" = \"\"\" separators=\"|\">, \"|\"D^{*} A\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819220&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819219&type=", "width": "17.44133186", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "text", "data": ", and the final state related to the energy localized at the acceptor, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"f\"\"\" = \"\"\" separators=\"|\">, \"|\"D A^{*}\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819222&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819221&type=", "width": "18.54199982", "height": "5.33400011", "fontsize": "" } } } ] }, { "name": "text", "data": ", can be approximated as" } ] }, { "name": "dispformula", "data": { "label": [], "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"i\"\"\" ≅ {\"\"\" separators=\"|\">, \"|\"D^{*} A\"\"\"}_{e} {\"\"\" separators=\"|\">, \"|\"D^{*} A\"\"\"}_{n u c}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819224&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819223&type=", "width": "31.24200058", "height": "5.67266655", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF11" } }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（8）" } ], "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"f\"\"\" ≅ {\"\"\" separators=\"|\">, \"|\"D A^{*}\"\"\"}_{e} {\"\"\" separators=\"|\">, \"|\"D A^{*}\"\"\"}_{n u c}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819226&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819225&type=", "width": "31.91933441", "height": "5.75733376", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF12" } }, { "name": "p", "data": [ { "name": "text", "data": "where " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{\"\"\" separators=\"|\">, \"|\"D^{*} A\"\"\"}_{e} \equiv ψ_{e} (D^{*} A)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819228&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819227&type=", "width": "25.90800095", "height": "5.67266655", "fontsize": "" } } } ] }, { "name": "text", "data": " is the electronic wavefunction of the donor in an excited state and the acceptor in the ground state, while " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{\"\"\" separators=\"|\">, \"|\"D A^{*}\"\"\"}_{e} \equiv ψ_{e} (D A^{*})

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819230&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819229&type=", "width": "27.00866890", "height": "5.67266655", "fontsize": "" } } } ] }, { "name": "text", "data": " is the electronic wavefunction of the donor in the electronic ground state and the acceptor in an excited state. These electronic states have corresponding nuclear wavefunctions represented as " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{\"\"\" separators=\"|\">, \"|\"D^{*} A\"\"\"}_{n u c} \equiv ψ_{n u c} (D^{*} A)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819232&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819231&type=", "width": "31.91933441", "height": "5.67266655", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{\"\"\" separators=\"|\">, \"|\"D A^{*}\"\"\"}_{n u c} \equiv ψ_{n u c} (D A^{*})

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819234&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819233&type=", "width": "30.90333366", "height": "5.67266655", "fontsize": "" } } } ] }, { "name": "text", "data": ", which are usually described as vibrational wavefunctions. It is important to emphasize that the electronic wavefunction " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{\"\"\" separators=\"|\">, \"|\"X\"\"\"}_{e} \equiv ψ_{e} (X)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819236&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819235&type=", "width": "20.91266632", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "text", "data": " is a parametric function of the nuclear positions " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

R_{α}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819274&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819273&type=", "width": "3.72533321", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": ", with " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

α

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819240&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819239&type=", "width": "2.20133328", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " being the label of each nucleus in the D-A system. As a result, any expectation value concerning the electronic wavefunction will be a function of the set " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{R_{α}}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819242&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819241&type=", "width": "7.02733374", "height": "4.40266657", "fontsize": "" } } } ] }, { "name": "text", "data": ", namely, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

A ({R_{α}}) = \"<\"\" separators=\"|\">, \hat{A}\" = \int ψ_{e}^{*} (X) \hat{A} ψ_{e} (X) d τ \equiv {\"<\"\" separators=\"|\">, X | \hat{A} | X\"}_{e}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819244&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819243&type=", "width": "71.62799835", "height": "6.43466663", "fontsize": "" } } } ] }, { "name": "text", "data": ".Substitution of these factored representations of the initial and final states in " }, { "name": "xref", "data": { "text": "Eq.（6）", "type": "disp-formula", "rid": "DF8", "data": [ { "name": "text", "data": "Eq.（6）" } ] } }, { "name": "text", "data": " yields the following expression for the energy transfer rate" } ] }, { "name": "dispformula", "data": { "label": [], "data": [ { "name": "math", "data": { "math": "

W_{E T} ≅ \frac{2 π}{ћ} {\"|\"V_{f i}\"|\"}^{2} F

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819246&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819245&type=", "width": "25.14599991", "height": "7.36600018", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF13" } }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（9）" } ], "data": [ { "name": "math", "data": { "math": "

V_{f i} = \int ψ_{e}^{*} (D A^{*}) \hat{V} ψ_{e} (D^{*} A) d τ \equiv {\"<\"\" separators=\"|\">, D A^{*} | \hat{V} | D^{*} A\"}_{e}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819248&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819247&type=", "width": "66.71733856", "height": "6.43466663", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF14" } }, { "name": "p", "data": [ { "name": "text", "data": "where " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

V_{f i}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819272&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819271&type=", "width": "3.38666677", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " is the electronic matrix element due to the time-independent part " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\hat{V}

F

\int ψ_{n u c}^{*} (D A^{*})

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819256&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819255&type=", "width": "16.84866714", "height": "6.43466663", "fontsize": "" } } } ] }, { "name": "text", "data": "·" } ] }, { "name": "p", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ψ_{n u c} D^{*} A) d τ \equiv

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819258&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819257&type=", "width": "21.50533485", "height": "4.99533367", "fontsize": "" } } } ] }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819260&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819259&type=", "width": "0.84666669", "height": "0.93133330", "fontsize": "" } } } ] }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

(\"<\"\" separators=\"|\">, D^{*} A, D A^{*}\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819262&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819261&type=", "width": "17.35666656", "height": "3.89466691", "fontsize": "" } } } ] }, { "name": "text", "data": ", and from the density of states " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ρ (E)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819264&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819263&type=", "width": "7.78933382", "height": "3.55599999", "fontsize": "" } } } ] }, { "name": "text", "data": ". This separation into an electronic contribution " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

V_{f i}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819272&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819271&type=", "width": "3.38666677", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " and a " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F

\hat{V}

V_{f i}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819272&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819271&type=", "width": "3.38666677", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " is a function of the nuclear positions " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

R_{α}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819274&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819273&type=", "width": "3.72533321", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": ", " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

V_{f i} \equiv V_{f i} (\"{\"R_{α}\"}\")

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819276&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819275&type=", "width": "22.43666649", "height": "4.40266657", "fontsize": "" } } } ] }, { "name": "text", "data": ". Notice that in the wavefunction representation of the matrix element, the notation " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ψ_{e}^{*}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819278&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819277&type=", "width": "3.55599999", "height": "4.74133301", "fontsize": "" } } } ] }, { "name": "text", "data": " represents the complex conjugate of the function " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ψ_{e}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819280&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819279&type=", "width": "3.47133350", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": ", whereas " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

D A^{*}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819282&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819281&type=", "width": "5.50333309", "height": "2.96333337", "fontsize": "" } } } ] }, { "name": "text", "data": " or " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

D^{*} A

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819284&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819283&type=", "width": "6.18066692", "height": "2.96333337", "fontsize": "" } } } ] }, { "name": "text", "data": " represents the acceptor or the donor in an excited electronic state. This confusion does not arise in the ‘bra-ket’ notation." } ] }, { "name": "p", "data": [ { "name": "text", "data": "So far, the proposed model, expressed in " }, { "name": "xref", "data": { "text": "Eq.（9）", "type": "disp-formula", "rid": "DF14", "data": [ { "name": "text", "data": "Eq.（9）" } ] } }, { "name": "text", "data": ", to describe energy transfer processes is general and can be applied to intermolecular D…A organic and/or inorganic pairs" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "28", "type": "bibr", "rid": "R28", "data": [ { "name": "text", "data": "28" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "29", "type": "bibr", "rid": "R29", "data": [ { "name": "text", "data": "29" } ] } } ], "rid": [ "R28", "R29" ], "text": "28-29", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " as well as energy transfer in intramolecular D-A pair such as lanthanide complexes L-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "（ligand-lanthanide ion）" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "20", "type": "bibr", "rid": "R20", "data": [ { "name": "text", "data": "20" } ] } }, { "name": "text", "data": "," }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "30", "type": "bibr", "rid": "R30", "data": [ { "name": "text", "data": "30" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "33", "type": "bibr", "rid": "R33", "data": [ { "name": "text", "data": "33" } ] } } ], "rid": [ "R30", "R31", "R32", "R33" ], "text": "30-33", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " and lanthanide-lanthanide ion pairs Ln" }, { "name": "sub", "data": [ { "name": "text", "data": "D" } ] }, { "name": "text", "data": "-Ln" }, { "name": "sub", "data": [ { "name": "text", "data": "A" } ] }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "34", "type": "bibr", "rid": "R34", "data": [ { "name": "text", "data": "34" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "37", "type": "bibr", "rid": "R37", "data": [ { "name": "text", "data": "37" } ] } } ], "rid": [ "R34", "R35", "R36", "R37" ], "text": "34-37", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". For the latter case, it is important to distinguish two situations: " }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "sub", "data": [ { "name": "text", "data": "D" } ] }, { "name": "text", "data": "-L-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "sub", "data": [ { "name": "text", "data": "A" } ] }, { "name": "text", "data": ", when there are bridging ligands between the lanthanide ions; and " }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "sub", "data": [ { "name": "text", "data": "D" } ] }, { "name": "text", "data": "-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "sub", "data": [ { "name": "text", "data": "A" } ] }, { "name": "text", "data": ", when no ligands bridge the ions. The case of " }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "sub", "data": [ { "name": "text", "data": "D" } ] }, { "name": "text", "data": "-L-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "sub", "data": [ { "name": "text", "data": "A" } ] }, { "name": "text", "data": " will not be considered here because the direct energy transfer pathways between the lanthanide ions are very inefficient compared to the ligand-lanthanide energy transfer pathways. So, most likely this case involves nonradiative energy transfer processes between the ligands and the lanthanide ions. Recently, the theoretical modeling of intramolecular ligand-lanthanide energy transfer was reviewed" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "20", "type": "bibr", "rid": "R20", "data": [ { "name": "text", "data": "20" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". So, here the focus will be on the direct energy transfer in " }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "sub", "data": [ { "name": "text", "data": "D" } ] }, { "name": "text", "data": "-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "sub", "data": [ { "name": "text", "data": "A" } ] }, { "name": "text", "data": " pairs with no bridging ligands will be described." } ] } ] }, { "name": "sec", "data": [ { "name": "sectitle", "data": { "title": [ { "name": "text", "data": "5　Energy Transfer Rates Between Lanthanide Ions" } ], "level": "1", "id": "s5" } }, { "name": "p", "data": [ { "name": "text", "data": "A quantitative description of the nonradiative energy transfer between lanthanide ions （" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": " ET） was developed in the early 1970s" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "38", "type": "bibr", "rid": "R38", "data": [ { "name": "text", "data": "38" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ", after the success of the Judd-Ofelt theory for the 4f-4f transition intensities" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "39", "type": "bibr", "rid": "R39", "data": [ { "name": "text", "data": "39" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "40", "type": "bibr", "rid": "R40", "data": [ { "name": "text", "data": "40" } ] } } ], "rid": [ "R39", "R40" ], "text": "39-40", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " and way before the development of the ligand-lanthanide ET model" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "20", "type": "bibr", "rid": "R20", "data": [ { "name": "text", "data": "20" } ] } }, { "name": "text", "data": "," }, { "name": "xref", "data": { "text": "32", "type": "bibr", "rid": "R32", "data": [ { "name": "text", "data": "32" } ] } }, { "name": "text", "data": "," }, { "name": "xref", "data": { "text": "41", "type": "bibr", "rid": "R41", "data": [ { "name": "text", "data": "41" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". This sequence of events may be explained by the very similar nature of the donor and acceptor states in the " }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": " ET, in contrast to the very dissimilar nature of the states involved in ligand-lanthanide ET. Lanthanide ions have a unique feature related to the shielding of the " }, { "name": "italic", "data": [ { "name": "text", "data": "N" } ] }, { "name": "text", "data": " electrons in the 4f subshell, 4f" }, { "name": "italic", "data": [ { "name": "sup", "data": [ { "name": "text", "data": "N" } ] } ] }, { "name": "text", "data": ". Because the filled subshells 5s" }, { "name": "sup", "data": [ { "name": "text", "data": "2" } ] }, { "name": "text", "data": "5p" }, { "name": "sup", "data": [ { "name": "text", "data": "6" } ] }, { "name": "text", "data": " are more radially external than the 4f" }, { "name": "italic", "data": [ { "name": "sup", "data": [ { "name": "text", "data": "N" } ] } ] }, { "name": "text", "data": " subshell, the electronic density of 5s" }, { "name": "sup", "data": [ { "name": "text", "data": "2" } ] }, { "name": "text", "data": "5p" }, { "name": "sup", "data": [ { "name": "text", "data": "6" } ] }, { "name": "text", "data": " shields the 4f electrons from the interactions with the surroundings （" }, { "name": "italic", "data": [ { "name": "text", "data": "e" } ] }, { "name": "text", "data": "." }, { "name": "italic", "data": [ { "name": "text", "data": "g" } ] }, { "name": "text", "data": "., ligating atoms, ligands, ions, lattice, solvent, " }, { "name": "italic", "data": [ { "name": "text", "data": "etc" } ] }, { "name": "text", "data": ".） of the lanthanide. As a result, the 4f states are practically those of the free-ion, where the effects of the surroundings can be treated as a small perturbation, if necessary. This feature was explored in the development of the Judd-Ofelt theory, and the successes of this formulation validate this picture of the 4f states. Therefore, donor and acceptor states in the energy transfer processes between 4f levels of lanthanide ions are well-defined and several theoretical tools developed previously could be employed. These " }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": " ET processes can be pictured as an excited state within the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

4 f^{N_{D}}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819286&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819285&type=", "width": "5.50333309", "height": "2.87866688", "fontsize": "" } } } ] }, { "name": "text", "data": " configuration of the donor lanthanide being deexcited to a lower level （usually the ground state） and its energy is transferred to the acceptor lanthanide causing an excitation within its " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

4 f^{N}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819288&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819287&type=", "width": "4.57200003", "height": "2.87866688", "fontsize": "" } } } ] }, { "name": "text", "data": " configuration. Because the 4f donor and acceptor states are well localized on each lanthanide, the donor-acceptor distance is taken as the distance between the lanthanide ions. This feature is not present in a ligand-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": " system, because the states of the ligand are usually delocalized, the donor-acceptor distance is much more difficult to define and determine." } ] }, { "name": "p", "data": [ { "name": "text", "data": "Because of the localized nature of the 4f states, the donor and acceptor electronic wavefunctions can be approximately treated as independent or uncoupled, that is," } ] }, { "name": "dispformula", "data": { "label": [], "data": [ { "name": "math", "data": { "math": "

{\"\"\" separators=\"|\">, \"|\"D^{*} A\"\"\"}_{e} \equiv ψ_{e}^{*} (D A^{*}) ≅ \"\"\" separators=\"|\">, \"|\"D^{*}\"\"\" \"\"\" separators=\"|\">, \"|\"A\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819290&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819289&type=", "width": "44.02666473", "height": "5.67266655", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF15" } }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（10）" } ], "data": [ { "name": "math", "data": { "math": "

{\"\"\" separators=\"|\">, \"|\"D A^{*}\"\"\"}_{e} \equiv ψ_{e}^{*} (D A^{*}) ≅ \"\"\" separators=\"|\">, \"|\"D\"\"\" \"\"\" separators=\"|\">, \"|\"A^{*}\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819292&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819291&type=", "width": "44.28066635", "height": "5.67266655", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF16" } }, { "name": "p", "data": [ { "name": "text", "data": "where " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"D^{*}\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819294&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819293&type=", "width": "6.51933336", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"D\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819296&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819295&type=", "width": "5.33400011", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": " represent the electronic wavefunction of the isolated donor in the excited and ground states, respectively, whereas " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"A\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819298&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819297&type=", "width": "4.91066647", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"A^{*}\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819300&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819299&type=", "width": "6.18066692", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "text", "data": " of the isolated acceptor in the ground and excited states. As a result, the perturbation operator " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\hat{V}

\"\"\" separators=\"|\">, \"|\"D^{*}\"\"\" \"\"\" separators=\"|\">, \"|\"A\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819336&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819335&type=", "width": "12.19200039", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"D\"\"\" \"\"\" separators=\"|\">, \"|\"A^{*}\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819338&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819337&type=", "width": "12.19200039", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "text", "data": ". This has a classical analogue related to Coulomb interactions between the charge densities and can be represented by the following operator" } ] }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（11）" } ], "data": [ { "name": "math", "data": { "math": "

{\hat{V}}_{C} = \sum_{i \in D} \sum_{j \in A} \frac{e^{2}}{\"|\"r_{i} - r_{j}\"|\"} \equiv \sum_{i \in D} \sum_{j \in A} \frac{e^{2}}{r_{i j}}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819308&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819307&type=", "width": "45.21200180", "height": "10.15999985", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF17" } }, { "name": "p", "data": [ { "name": "text", "data": "where " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

r_{i}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819310&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819309&type=", "width": "2.11666679", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

r_{j}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819312&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819311&type=", "width": "1.94733346", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " are the position vectors of electrons " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

i

j

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819330&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819329&type=", "width": "1.52400005", "height": "3.47133350", "fontsize": "" } } } ] }, { "name": "text", "data": " （belonging to the acceptor）, respectively." } ] }, { "name": "p", "data": [ { "name": "text", "data": "Another more subtle interaction comes from the principles of quantum mechanics, which has no classical analogue. In the approximation expressed by " }, { "name": "xref", "data": { "text": "Eq.（10）", "type": "disp-formula", "rid": "DF16", "data": [ { "name": "text", "data": "Eq.（10）" } ] } }, { "name": "text", "data": ", it is implicitly assumed that the donor has a fixed number of electrons, " }, { "name": "italic", "data": [ { "name": "text", "data": "e" } ] }, { "name": "text", "data": "." }, { "name": "italic", "data": [ { "name": "text", "data": "g" } ] }, { "name": "text", "data": "., " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

N_{D}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819318&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819317&type=", "width": "3.97933316", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": ", as well as the acceptor has " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

N_{A}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819320&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819319&type=", "width": "3.97933316", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": "-electrons. However, when these subsystems interact with each other, the electrons of the donor must become indistinguishable from the electrons of the acceptor, as a principle of quantum mechanics. This can be accomplished by an exchange operator that permutes the indices （or labels） of each electron of the donor with each electron of the acceptor, which can be represented by the following operator" } ] }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（12）" } ], "data": [ { "name": "math", "data": { "math": "

{\hat{V}}_{e x} = \sum_{i \in D} \sum_{j \in A} \frac{e^{2}}{r_{i j}} P_{i j}, P_{i j} = \frac{1}{2} + {\hat{s}}_{i} \cdot {\hat{s}}_{j}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819322&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819321&type=", "width": "48.68333054", "height": "8.80533314", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF18" } }, { "name": "p", "data": [ { "name": "text", "data": "where " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{\hat{s}}_{i}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819324&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819323&type=", "width": "2.11666679", "height": "3.89466691", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{\hat{s}}_{j}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819326&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819325&type=", "width": "2.03200006", "height": "3.89466691", "fontsize": "" } } } ] }, { "name": "text", "data": " are the spin operators of electrons " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

i

j

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819330&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819329&type=", "width": "1.52400005", "height": "3.47133350", "fontsize": "" } } } ] }, { "name": "text", "data": ". It is important to realize that the wavefunctions of the isolated donor and acceptor are normalized and antisymmetrized, so they individually satisfy the principles of quantum mechanics." } ] }, { "name": "p", "data": [ { "name": "text", "data": "The electronic wavefunctions of 4f states of an isolated lanthanide ion are accurately described by the total electronic angular momentum quantum number " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

J

ψ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819334&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819333&type=", "width": "2.37066650", "height": "3.38666677", "fontsize": "" } } } ] }, { "name": "text", "data": ". Thus, the initial " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"D^{*}\"\"\" \"\"\" separators=\"|\">, \"|\"A\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819336&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819335&type=", "width": "12.19200039", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "text", "data": " and final " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"D\"\"\" \"\"\" separators=\"|\">, \"|\"A^{*}\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819338&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819337&type=", "width": "12.19200039", "height": "5.07999992", "fontsize": "" } } } ] }, { "name": "text", "data": " states, " }, { "name": "xref", "data": { "text": "Eq. （10）", "type": "disp-formula", "rid": "DF16", "data": [ { "name": "text", "data": "Eq. （10）" } ] } }, { "name": "text", "data": ", in the ET process are represented as " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"ψ_{D}^{*} J_{D}^{*}\"\"\" \"\"\" separators=\"|\">, \"|\"ψ_{A} J_{A}\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819340&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819339&type=", "width": "21.42066574", "height": "5.24933338", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"\"\" separators=\"|\">, \"|\"ψ_{D} J_{D}\"\"\" \"\"\" separators=\"|\">, \"|\"ψ_{A}^{*} J_{A}^{*}\"\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819342&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819341&type=", "width": "21.42066574", "height": "5.24933338", "fontsize": "" } } } ] }, { "name": "text", "data": ". Notice that the projection " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

M_{J}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819344&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819343&type=", "width": "3.97933316", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " of the angular momentum " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

J

S

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819348&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819347&type=", "width": "1.86266661", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " and orbital " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

L

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819350&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819349&type=", "width": "1.94733346", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " angular momenta （and their projections, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

M_{S}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819352&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819351&type=", "width": "4.14866638", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

M_{L}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819354&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819353&type=", "width": "4.14866638", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": "） basis as the so-called intermediate coupling scheme. This transformation is quite convenient because it allows the choice of a diagonal matrix representation of the operator of interest." } ] }, { "name": "p", "data": [ { "name": "text", "data": "Once the wavefunctions describing the initial and final states have been properly represented, the matrix element involving the perturbation operator " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\hat{V} = {\hat{V}}_{C} + {\hat{V}}_{e x}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819356&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819355&type=", "width": "18.54199982", "height": "4.74133301", "fontsize": "" } } } ] }, { "name": "text", "data": " can be determined. Notice that this operator involves the two-electron interaction " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

e^{2} / r_{12}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819416&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819415&type=", "width": "7.28133297", "height": "4.06400013", "fontsize": "" } } } ] }, { "name": "text", "data": ", which can be expanded in terms of the spherical harmonics or equivalently in terms of Racah's irreducible spherical tensor operators" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "23", "type": "bibr", "rid": "R23", "data": [ { "name": "text", "data": "23" } ] } }, { "name": "text", "data": "," }, { "name": "xref", "data": { "text": "42", "type": "bibr", "rid": "R42", "data": [ { "name": "text", "data": "42" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". This allows the factorization of the matrix element into radial and angular parts, where this latter part can be calculated analytically. As a result, the following equations for the energy transfer rates between lanthanide ions can be obtained" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "16", "type": "bibr", "rid": "R16", "data": [ { "name": "text", "data": "16" } ] } }, { "name": "text", "data": "," }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "34", "type": "bibr", "rid": "R34", "data": [ { "name": "text", "data": "34" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "37", "type": "bibr", "rid": "R37", "data": [ { "name": "text", "data": "37" } ] } } ], "rid": [ "R34", "R35", "R36", "R37" ], "text": "34-37", "type": "bibr" } }, { "name": "text", "data": "," }, { "name": "xref", "data": { "text": "43", "type": "bibr", "rid": "R43", "data": [ { "name": "text", "data": "43" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " and classified the dipole⁃dipole," }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{d ⁃ d}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819360&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819359&type=", "width": "5.67266655", "height": "3.72533321", "fontsize": "" } } } ] } ] }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（13）" } ], "data": [ { "name": "math", "data": { "math": "

W_{d ⁃ d} = \frac{4 π}{3 ћ} \frac{S_{d}^{D} S_{d}^{A}}{R_{D A}^{6}} F

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819362&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819361&type=", "width": "26.75466537", "height": "8.55133343", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF19" } }, { "name": "p", "data": [ { "name": "text", "data": "the dipole⁃quadrupole and quadrupole⁃dipole," } ] }, { "name": "dispformula", "data": { "label": [], "data": [ { "name": "math", "data": { "math": "

W_{d q ⁃ q d}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819364&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819363&type=", "width": "7.61999989", "height": "3.72533321", "fontsize": "" } } }, { "name": "text", "data": " ," } ], "id": "DF20" } }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（14）" } ], "data": [ { "name": "math", "data": { "math": "

W_{d q ⁃ q d} = \frac{π}{ћ} \frac{(S_{d}^{D} S_{q}^{A} + S_{q}^{D} S_{d}^{A})}{R_{D A}^{8}} F

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819366&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819365&type=", "width": "41.31733322", "height": "8.80533314", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF21" } }, { "name": "p", "data": [ { "name": "text", "data": "the quadrupole⁃quadrupole, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{q ⁃ q}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819802&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819801&type=", "width": "5.67266655", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": "," } ] }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（15）" } ], "data": [ { "name": "math", "data": { "math": "

W_{q ⁃ q} = \frac{28 π}{5 ћ} \frac{S_{q}^{D} S_{q}^{A}}{R_{D A}^{10}} F

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819370&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819369&type=", "width": "28.61733437", "height": "8.80533314", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF22" } }, { "name": "p", "data": [ { "name": "text", "data": "the magnetic dipole⁃magnetic dipole " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{m d ⁃ m d}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819372&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819371&type=", "width": "8.55133343", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": "," } ] }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（16）" } ], "data": [ { "name": "math", "data": { "math": "

W_{m d ⁃ m d} = \frac{4 π}{3 ћ} \frac{S_{m d}^{D} S_{m d}^{A}}{R_{D A}^{6}} F

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819374&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819373&type=", "width": "31.75000000", "height": "8.55133343", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF23" } }, { "name": "p", "data": [ { "name": "text", "data": "and the exchange, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{e x}

W_{e x} = \frac{2 π}{ћ} {\"[\"(\frac{e^{2}}{R_{D A}}) ρ_{f f}^{2}\"]\"}^{2} F

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819378&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819377&type=", "width": "35.22133255", "height": "10.41399956", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF24" } }, { "name": "p", "data": [ { "name": "text", "data": "where " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

S_{d}

e r g \cdot c m^{3}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819382&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819381&type=", "width": "11.76866722", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": "） is the modified forced electric dipole strength" } ] }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（18）" } ], "data": [ { "name": "math", "data": { "math": "

S_{d}^{x} = \frac{e^{2} {(1 - σ_{1}^{x})}^{2}}{2 J_{x} + 1} \sum_{K = 2,4, 6} Ω_{K}^{x} \"<\"ψ_{x}^{*} J_{x}^{*}\"\" \"‖\"U^{(K)}\"‖\" {\"\"\" separators=\"|\">, ψ_{x} J_{x}\"}^{2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819384&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819383&type=", "width": "68.83399963", "height": "9.90600014", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF25" } }, { "name": "p", "data": [ { "name": "text", "data": "while " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

S_{q}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819386&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819385&type=", "width": "3.04800010", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " （in " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

e r g \cdot c m^{5}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819388&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819387&type=", "width": "11.76866722", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": "） is the modified quadrupole strength" } ] }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（19）" } ], "data": [ { "name": "math", "data": { "math": "

S_{q}^{x} = \frac{e^{2} (1 - σ_{2}^{x})^{2}}{2 J_{x} + 1} \"<\"f\"\" \"‖\"C^{(2)}\"‖\" {\"\"\" separators=\"|\">, f\"}^{2} \times {\"<\"\" separators=\"|\">, r^{2}\"}_{x}^{2} \"<\"ψ_{x}^{*} J_{x}^{*}\"\" \"‖\"U^{(2)}\"‖\" {\"\"\" separators=\"|\">, ψ_{x} J_{x}\"}^{2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819390&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819389&type=", "width": "109.89733887", "height": "8.29733276", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF26" } }, { "name": "p", "data": [ { "name": "text", "data": "whereas " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

S_{m d}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819392&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819391&type=", "width": "4.48733330", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " （in " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

e r g \cdot c m^{3}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819394&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819393&type=", "width": "11.76866722", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": "） is the magnetic dipole strength analogue" } ] }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（20）" } ], "data": [ { "name": "math", "data": { "math": "

\begin{array}{c} S_{m d}^{x} = \frac{μ_{B}^{2} (1 - σ_{m d}^{x})^{2}}{(2 J_{x} + 1)} \"<\"ψ_{x}^{*} J_{x}^{*}\"\" \"‖\"\hat{M}\"‖\" {\"\"\" separators=\"|\">, ψ_{x} J_{x}\"}^{2}, \\ \hat{M} \equiv \hat{L} + g_{S} \hat{S} \end{array}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819396&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819395&type=", "width": "66.71733856", "height": "15.83266735", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF27" } }, { "name": "p", "data": [ { "name": "text", "data": "with " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

x = A

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819398&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819397&type=", "width": "8.72066593", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " for the acceptor and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

x = D

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819400&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819399&type=", "width": "8.72066593", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " for the donor, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

J_{x} = J_{D}^{*}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819402&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819401&type=", "width": "10.41399956", "height": "4.40266657", "fontsize": "" } } } ] }, { "name": "text", "data": " being the total angular momentum quantum number of the excited state of the donor or " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

J_{x} = J_{A}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819404&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819403&type=", "width": "10.41399956", "height": "4.14866638", "fontsize": "" } } } ] }, { "name": "text", "data": " of the ground state of the acceptor, and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

g_{S} =

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819406&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819405&type=", "width": "5.92666674", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": " 2.002 319 3 being the electron spin " }, { "name": "italic", "data": [ { "name": "text", "data": "g" } ] }, { "name": "text", "data": "-factor" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "44", "type": "bibr", "rid": "R44", "data": [ { "name": "text", "data": "44" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". See the Supporting Information for dimensional analysis and unit conversion." } ] }, { "name": "p", "data": [ { "name": "text", "data": "The total energy transfer rate will be a summation of each contribution in Eqs.（" }, { "name": "xref", "data": { "text": "13", "type": "disp-formula", "rid": "DF19", "data": [ { "name": "text", "data": "13" } ] } }, { "name": "text", "data": "）-（" }, { "name": "xref", "data": { "text": "17", "type": "disp-formula", "rid": "DF24", "data": [ { "name": "text", "data": "17" } ] } }, { "name": "text", "data": "）. Usually, there is one contribution that dominates this sum, however, there are exceptions, and the relative contributions change strongly with the donor-acceptor distance, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

R_{D A}

S_{d}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819848&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819847&type=", "width": "3.04800010", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " is denoted as " }, { "name": "italic", "data": [ { "name": "text", "data": "forced" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "electric" } ] }, { "name": "text", "data": " dipole strength analogue because " }, { "name": "italic", "data": [ { "name": "text", "data": "only" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "the" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "forced" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "electric" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "dipole" } ] }, { "name": "text", "data": " （FED） mechanism contributes to the calculation of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ω_{K}

Ω_{K}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819414&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819413&type=", "width": "4.06400013", "height": "3.89466691", "fontsize": "" } } } ] }, { "name": "text", "data": ", like in the Judd-Ofelt theory" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "38", "type": "bibr", "rid": "R38", "data": [ { "name": "text", "data": "38" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": "." } ] }, { "name": "p", "data": [ { "name": "text", "data": "These equations are the same as those published in the revisited versions of the energy transfer rates between lanthanide ions" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "34", "type": "bibr", "rid": "R34", "data": [ { "name": "text", "data": "34" } ] } }, { "name": "text", "data": "," }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "36", "type": "bibr", "rid": "R36", "data": [ { "name": "text", "data": "36" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "37", "type": "bibr", "rid": "R37", "data": [ { "name": "text", "data": "37" } ] } } ], "rid": [ "R36", "R37" ], "text": "36-37", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". However, those involving multipole and magnetic dipole mechanisms were rearranged and separated into more symmetrical and easily interpretable forms. In addition, these separations allow a stepwise calculation of the contributing terms, which can assist the analysis of the transfer rates and determine if there are dominant contributions from either donor or acceptor." } ] }, { "name": "p", "data": [ { "name": "text", "data": "The energy transfer rates are expressed in terms of well-defined and calculable quantities. It must be emphasized that the choice of the two-electron interaction as " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

e^{2} / r_{12}

ℏ = h / (2 π)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819418&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819417&type=", "width": "16.93333435", "height": "3.89466691", "fontsize": "" } } } ] }, { "name": "text", "data": " is the reduced Planck constant （see the Supporting Information for the values of the physical constants and unit conversion）," } ] }, { "name": "p", "data": [ { "name": "text", "data": "⋅ " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

e

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819420&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819419&type=", "width": "1.43933344", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " is the elementary charge in " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

e s u

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819422&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819421&type=", "width": "4.99533367", "height": "2.70933342", "fontsize": "" } } } ] }, { "name": "text", "data": "," } ] }, { "name": "p", "data": [ { "name": "text", "data": "⋅ " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

μ_{B} = e ћ / (2 m_{e} c)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819424&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819423&type=", "width": "23.79133224", "height": "3.97933316", "fontsize": "" } } } ] }, { "name": "text", "data": " is the Bohr magneton," } ] }, { "name": "p", "data": [ { "name": "text", "data": "⋅ " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"f\"\" \"‖\"C^{(2)}\"‖\" \"\"\" separators=\"|\">, f\" = - 1.366

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819426&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819425&type=", "width": "33.61266708", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " is the one-electron doubly reduced spherical tensor matrix element （dimensionless）" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "23", "type": "bibr", "rid": "R23", "data": [ { "name": "text", "data": "23" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " ," }, { "name": "italic", "data": [ { "name": "text", "data": "on" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "intrinsic" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "quantities" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "of" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "the" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "lanthanide" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "ion" } ] }, { "name": "text", "data": ":" } ] }, { "name": "p", "data": [ { "name": "text", "data": "⋅ " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

J_{D}^{*}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819428&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819427&type=", "width": "3.13266683", "height": "4.40266657", "fontsize": "" } } } ] }, { "name": "text", "data": " is the total angular momentum quantum number of the excited state of the donor and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

J_{A}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819430&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819429&type=", "width": "3.13266683", "height": "4.06400013", "fontsize": "" } } } ] }, { "name": "text", "data": " of the ground state of the acceptor," } ] }, { "name": "p", "data": [ { "name": "text", "data": "⋅ " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"ψ^{*} J^{*}\"\" \"‖\"U^{(K)}\"‖\" \"\"\" separators=\"|\">, ψ J\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819432&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819431&type=", "width": "24.72266769", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " （dimensionless） are the doubly reduced matrix elements of the unit irreducible tensor operator " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

U^{(K)}

K

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819782&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819781&type=", "width": "2.53999996", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " evaluated in the intermediate coupling scheme. Their square, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"ψ^{*} J^{*}\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819438&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819437&type=", "width": "9.31333351", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": " " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"‖\"U^{(K)}\"‖\" {\"\"\" separators=\"|\">, ψ J\"}^{2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819440&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819439&type=", "width": "17.01799965", "height": "5.16466665", "fontsize": "" } } } ] }, { "name": "text", "data": ", are often abbreviated as " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

U^{(K)}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819442&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819441&type=", "width": "5.75733376", "height": "3.04800010", "fontsize": "" } } } ] }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "45", "type": "bibr", "rid": "R45", "data": [ { "name": "text", "data": "45" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " or " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

U (K)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819444&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819443&type=", "width": "8.63599968", "height": "3.47133350", "fontsize": "" } } } ] }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "46", "type": "bibr", "rid": "R46", "data": [ { "name": "text", "data": "46" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " and tabulated values can be found in reference " }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "47", "type": "bibr", "rid": "R47", "data": [ { "name": "text", "data": "47" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ", " }, { "name": "italic", "data": [ { "name": "text", "data": "e" } ] }, { "name": "text", "data": "." }, { "name": "italic", "data": [ { "name": "text", "data": "g" } ] }, { "name": "text", "data": "., Eu（Ⅲ） has " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"{}^{7} F_{2}\"\" \"‖\"U^{(2)}\"‖\" {\"\"\" separators=\"|\">, {}^{5} D_{0}\"}^{2} =

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819446&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819445&type=", "width": "28.70199966", "height": "5.16466665", "fontsize": "" } } } ] }, { "name": "text", "data": "0.003 2 and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"{}^{7} F_{4}\"\" \"‖\"U^{(4)}\"‖\" {\"\"\" separators=\"|\">, {}^{5} D_{0}\"}^{2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819448&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819447&type=", "width": "24.63800049", "height": "5.16466665", "fontsize": "" } } } ] }, { "name": "text", "data": "=0.002 3," } ] }, { "name": "p", "data": [ { "name": "text", "data": "⋅ " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"ψ^{*} J^{*}\"\" \"‖\"\hat{M}\"‖\" \"\"\" separators=\"|\">, ψ J\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819450&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819449&type=", "width": "22.35199928", "height": "5.75733376", "fontsize": "" } } } ] }, { "name": "text", "data": " （dimensionless）, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\hat{M} \equiv \hat{L} + g_{S} \hat{S} ≅

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819452&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819451&type=", "width": "20.65866661", "height": "5.58799982", "fontsize": "" } } } ] } ] }, { "name": "p", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\hat{L} + 2 \hat{S}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819454&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819453&type=", "width": "10.15999985", "height": "3.64066648", "fontsize": "" } } } ] }, { "name": "text", "data": " is the doubly reduced matrix element of the angular " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\hat{L}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819456&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819455&type=", "width": "1.94733346", "height": "3.64066648", "fontsize": "" } } } ] }, { "name": "text", "data": " and spin " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\hat{S}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819458&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819457&type=", "width": "1.86266661", "height": "3.64066648", "fontsize": "" } } } ] }, { "name": "text", "data": " operators calculated in the intermediate coupling scheme" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "48", "type": "bibr", "rid": "R48", "data": [ { "name": "text", "data": "48" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "50", "type": "bibr", "rid": "R50", "data": [ { "name": "text", "data": "50" } ] } } ], "rid": [ "R48", "R49", "R50" ], "text": "48-50", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ", have typical values of 0.1-0.6 for states with different " }, { "name": "italic", "data": [ { "name": "text", "data": "L" } ] }, { "name": "text", "data": " and " }, { "name": "italic", "data": [ { "name": "text", "data": "S" } ] }, { "name": "text", "data": " terms" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "37", "type": "bibr", "rid": "R37", "data": [ { "name": "text", "data": "37" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " and assume high values into the same " }, { "name": "italic", "data": [ { "name": "text", "data": "L" } ] }, { "name": "text", "data": " and " }, { "name": "italic", "data": [ { "name": "text", "data": "S" } ] }, { "name": "text", "data": " terms （" }, { "name": "italic", "data": [ { "name": "text", "data": "e" } ] }, { "name": "text", "data": "." }, { "name": "italic", "data": [ { "name": "text", "data": "g" } ] }, { "name": "text", "data": "., Tb（Ⅲ） has " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"|\"\"<\"{}^{7} F_{5}\"\" \"‖\"\hat{M}\"‖\" \"\"\" separators=\"|\">, {}^{7} F_{6}\"\"|\" = 4.38

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819460&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819459&type=", "width": "34.96733475", "height": "6.51933336", "fontsize": "" } } } ] }, { "name": "text", "data": "）" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "51", "type": "bibr", "rid": "R51", "data": [ { "name": "text", "data": "51" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". It is important to emphasize that this matrix element is also expressed in units of " }, { "name": "italic", "data": [ { "name": "text", "data": "ћ" } ] }, { "name": "text", "data": ", in which case, the Bohr magneton " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

μ_{B}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819462&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819461&type=", "width": "3.80999994", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": " cannot include " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ћ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819464&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819463&type=", "width": "1.94733346", "height": "2.70933342", "fontsize": "" } } } ] }, { "name": "text", "data": "," } ] }, { "name": "p", "data": [ { "name": "text", "data": "⋅ " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"\" separators=\"|\">, r^{t}\" \equiv \"<\"\" separators=\"|\">, 4 f | r^{t} | 4 f\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819466&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819465&type=", "width": "22.69066620", "height": "3.64066648", "fontsize": "" } } } ] }, { "name": "text", "data": " is the expectation value （integral） of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

r^{t}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819468&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819467&type=", "width": "2.03200006", "height": "2.96333337", "fontsize": "" } } } ] }, { "name": "text", "data": " with respect to the 4f radial function and the values for " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

t =

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819470&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819469&type=", "width": "4.48733330", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": "2, 4, and 6 can be found in Ref.[" }, { "name": "xref", "data": { "text": "52", "type": "bibr", "rid": "R52", "data": [ { "name": "text", "data": "52" } ] } }, { "name": "text", "data": "], " }, { "name": "italic", "data": [ { "name": "text", "data": "e" } ] }, { "name": "text", "data": "." }, { "name": "italic", "data": [ { "name": "text", "data": "g" } ] }, { "name": "text", "data": "., " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"\" separators=\"|\">, r^{2}\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819472&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819471&type=", "width": "5.75733376", "height": "3.64066648", "fontsize": "" } } } ] }, { "name": "text", "data": " are 0.997, 0.893, 0.710, 0.773 a.u. for Eu（Ⅲ）, Tb（Ⅲ）, Yb（Ⅲ）, and Er（Ⅲ）, respectively" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "52", "type": "bibr", "rid": "R52", "data": [ { "name": "text", "data": "52" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": "," } ] }, { "name": "p", "data": [ { "name": "text", "data": "⋅ " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ω_{K} \equiv Ω_{K} (F E D)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819474&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819473&type=", "width": "22.69066620", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": " is the forced electric dipole contribution to the intensity parameter of rank " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

K

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819782&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819781&type=", "width": "2.53999996", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": ", in units of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

10^{- 20} c m^{2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819478&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819477&type=", "width": "13.03866673", "height": "3.04800010", "fontsize": "" } } } ] }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "20", "type": "bibr", "rid": "R20", "data": [ { "name": "text", "data": "20" } ] } }, { "name": "text", "data": "," }, { "name": "xref", "data": { "text": "34", "type": "bibr", "rid": "R34", "data": [ { "name": "text", "data": "34" } ] } }, { "name": "text", "data": "," }, { "name": "xref", "data": { "text": "36", "type": "bibr", "rid": "R36", "data": [ { "name": "text", "data": "36" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " ," }, { "name": "italic", "data": [ { "name": "text", "data": "on" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "the" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "surroundings" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "of" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "the" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "lanthanide" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "ion" } ] }, { "name": "text", "data": ":" } ] }, { "name": "p", "data": [ { "name": "text", "data": "⋅ " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

(1 - σ_{k})

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819480&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819479&type=", "width": "12.53066730", "height": "4.31799984", "fontsize": "" } } } ] }, { "name": "text", "data": ", with " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

k = 1

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819482&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819481&type=", "width": "8.04333401", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

2

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819484&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819483&type=", "width": "1.86266661", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": ", is the shielding factor for donor and acceptor approximated as" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "34", "type": "bibr", "rid": "R34", "data": [ { "name": "text", "data": "34" } ] } }, { "name": "text", "data": "," }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "53", "type": "bibr", "rid": "R53", "data": [ { "name": "text", "data": "53" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "54", "type": "bibr", "rid": "R54", "data": [ { "name": "text", "data": "54" } ] } } ], "rid": [ "R53", "R54" ], "text": "53-54", "type": "bibr" } }, { "name": "text", "data": "]" } ] } ] }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（21）" } ], "data": [ { "name": "math", "data": { "math": "

(1 - σ_{k}) = ρ {(2 β)}^{k + 1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819486&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819485&type=", "width": "33.69733429", "height": "5.07999992", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF28" } }, { "name": "p", "data": [ { "name": "text", "data": "where " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ρ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819488&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819487&type=", "width": "2.03200006", "height": "3.38666677", "fontsize": "" } } } ] }, { "name": "text", "data": " is the overlap integral between the valence subshells of the ligating atom and the 4f subshell of the lanthanide ion and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

β = 1 / (1 \pm ρ)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819490&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819489&type=", "width": "21.84399986", "height": "4.91066647", "fontsize": "" } } } ] }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "53", "type": "bibr", "rid": "R53", "data": [ { "name": "text", "data": "53" } ] } }, { "name": "text", "data": "," }, { "name": "xref", "data": { "text": "55", "type": "bibr", "rid": "R55", "data": [ { "name": "text", "data": "55" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " ," } ] }, { "name": "p", "data": [ { "name": "text", "data": "⋅ " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

σ_{m d}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819492&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819491&type=", "width": "4.99533367", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " is the shielding factor, for donor and acceptor, which shields the magnetic dipole interactions," } ] }, { "name": "p", "data": [ { "name": "italic", "data": [ { "name": "text", "data": "on" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "the" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "lanthanide" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "pair" } ] }, { "name": "text", "data": ":" } ] }, { "name": "p", "data": [ { "name": "text", "data": "⋅ " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

R_{D A}

R_{D A} = \sqrt[]{(x_{D} - x_{A})^{2} + (y_{D} - y_{A})^{2} + (z_{D} - z_{A})^{2}}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819496&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819495&type=", "width": "65.78600311", "height": "5.41866684", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF29" } }, { "name": "p", "data": [ { "name": "text", "data": "with " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

(x_{D}, y_{D}, z_{D})

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819498&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819497&type=", "width": "15.74800014", "height": "4.74133301", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

(x_{A}, y_{A}, z_{A})

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819500&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819499&type=", "width": "15.74800014", "height": "4.74133301", "fontsize": "" } } } ] }, { "name": "text", "data": " being the Cartesian coordinates of the donor and acceptor centers, respectively," } ] }, { "name": "p", "data": [ { "name": "text", "data": "⋅ " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ρ_{f f}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819745&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819744&type=", "width": "3.30200005", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": " is the overlap integral between the 4f subshells of the donor and acceptor lanthanide ions" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "53", "type": "bibr", "rid": "R53", "data": [ { "name": "text", "data": "53" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": "." } ] }, { "name": "p", "data": [ { "name": "text", "data": "The last quantity to determine the ET transfer rates is the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F

ρ_{D} (E)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819506&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819505&type=", "width": "9.56733322", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " and of the acceptor " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ρ_{A} (E)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819508&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819507&type=", "width": "9.56733322", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": ". In the original model for the nonradiative energy transfer between organic species " }, { "name": "italic", "data": [ { "name": "text", "data": "via" } ] }, { "name": "text", "data": " the dipole-dipole mechanism" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "24", "type": "bibr", "rid": "R24", "data": [ { "name": "text", "data": "24" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ", this factor was assigned to and determined as the spectral overlap between the absorption band of the acceptor and the emission band of the donor. As a result, the emission spectrum of the donor and the absorption spectrum of the acceptor are recorded separately and properly rescaled to determine the area under which these spectra overlap. However, this requires the spectra of the isolated donor and acceptor, which might not be viable, as well as a delicate procedure for determining the area of the spectral overlap. This would also preclude the development of a quantitative and predictive model. Considering that the density of states " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ρ_{D} (E)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819548&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819547&type=", "width": "9.56733322", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ρ_{A} (E)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819542&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819541&type=", "width": "9.56733322", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " are Gaussian functions, then the area of their overlap can be expressed as a sum of two error functions that depend on the widths of the Gaussian functions, their energy maxima, and the point of their intersection. Despite this area being calculable, there is not an analytical expression for it, which requires numerical modeling." } ] }, { "name": "p", "data": [ { "name": "text", "data": "It should be noticed that a typo occurred in the expression of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{m d ⁃ m d}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819828&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819827&type=", "width": "8.55133343", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " in previous publications" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "16", "type": "bibr", "rid": "R16", "data": [ { "name": "text", "data": "16" } ] } }, { "name": "text", "data": "," }, { "name": "xref", "data": { "text": "37", "type": "bibr", "rid": "R37", "data": [ { "name": "text", "data": "37" } ] } }, { "name": "text", "data": "," }, { "name": "xref", "data": { "text": "56", "type": "bibr", "rid": "R56", "data": [ { "name": "text", "data": "56" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ",where " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

σ_{1}^{x}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819516&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819515&type=", "width": "3.64066648", "height": "4.06400013", "fontsize": "" } } } ] }, { "name": "text", "data": " was used instead of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

σ_{m d}^{x}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819530&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819529&type=", "width": "4.99533367", "height": "4.06400013", "fontsize": "" } } } ] }, { "name": "text", "data": " as shown in " }, { "name": "xref", "data": { "text": "Eq.（20）", "type": "disp-formula", "rid": "DF27", "data": [ { "name": "text", "data": "Eq.（20）" } ] } }, { "name": "text", "data": ". Indeed, because the electric dipole interactions, which are shielded by electron densities represented as " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

σ_{1}^{x}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819520&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819519&type=", "width": "3.64066648", "height": "4.06400013", "fontsize": "" } } } ] }, { "name": "text", "data": ", have different nature than the magnetic dipole interactions, the shielding of these interactions, represented as " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

σ_{m d}^{x}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819522&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819521&type=", "width": "4.99533367", "height": "4.06400013", "fontsize": "" } } } ] }, { "name": "text", "data": ", should not be the same. Hence, the introduction of the new notation " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

σ_{m d}^{x}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819524&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819523&type=", "width": "4.99533367", "height": "4.06400013", "fontsize": "" } } } ] }, { "name": "text", "data": ". However, because the shielding of the 4f electrons by the remaining electrons of the lanthanide ion and of the ligands involves closed-shells （" }, { "name": "italic", "data": [ { "name": "text", "data": "e" } ] }, { "name": "text", "data": "." }, { "name": "italic", "data": [ { "name": "text", "data": "g" } ] }, { "name": "text", "data": "., 5s" }, { "name": "sup", "data": [ { "name": "text", "data": "2" } ] }, { "name": "text", "data": "5p" }, { "name": "sup", "data": [ { "name": "text", "data": "6" } ] }, { "name": "text", "data": "）, then it is expected that the magnetic shielding " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

σ_{m d}^{x}

σ_{m d}^{x}

σ_{m d}^{x}

W_{m d ⁃ m d}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819532&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819531&type=", "width": "8.55133343", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " " }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "57", "type": "bibr", "rid": "R57", "data": [ { "name": "text", "data": "57" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "58", "type": "bibr", "rid": "R58", "data": [ { "name": "text", "data": "58" } ] } } ], "rid": [ "R57", "R58" ], "text": "57-58", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " or to consider " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

σ_{m d}^{x} \approx 0

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819534&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819533&type=", "width": "11.43000031", "height": "4.06400013", "fontsize": "" } } } ] }, { "name": "text", "data": " in the calculations of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{m d ⁃ m d}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819828&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819827&type=", "width": "8.55133343", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "16", "type": "bibr", "rid": "R16", "data": [ { "name": "text", "data": "16" } ] } }, { "name": "text", "data": "," }, { "name": "xref", "data": { "text": "37", "type": "bibr", "rid": "R37", "data": [ { "name": "text", "data": "37" } ] } }, { "name": "text", "data": "," }, { "name": "xref", "data": { "text": "56", "type": "bibr", "rid": "R56", "data": [ { "name": "text", "data": "56" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ", which will cause a slight overestimation of the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{m d ⁃ m d}

ρ_{D} (E)

ρ_{A} (E)

F

F

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type=", "width": "2.37066650", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": "-factor, which becomes a function of the temperature by assuming a Boltzmann distribution for the vibrational states of the initial and final electronic states" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "20", "type": "bibr", "rid": "R20", "data": [ { "name": "text", "data": "20" } ] } }, { "name": "text", "data": "," }, { "name": "xref", "data": { "text": "34", "type": "bibr", "rid": "R34", "data": [ { "name": "text", "data": "34" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". This formulation is relevant because it explicitly accounts for the creation of phonons when the donor state has energy higher than the acceptor. It also accounts for the annihilation of phonons when the donor state is below in energy relative to the acceptor state because the energy mismatch must be compensated by phonons" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "20", "type": "bibr", "rid": "R20", "data": [ { "name": "text", "data": "20" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". By considering the bands of the donor and acceptor, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ρ_{D} (E)

ρ_{A} (E^{'})

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819550&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819549&type=", "width": "10.32933331", "height": "4.99533367", "fontsize": "" } } } ] }, { "name": "text", "data": ", have Gaussian profiles, an analytical expression for the integral of a product of two Gaussian functions can be obtained" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "60", "type": "bibr", "rid": "R60", "data": [ { "name": "text", "data": "60" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". This comes from the fact that a product of two Gaussian functions is a well-defined Gaussian function, whose define integral has analytical expression and the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F

e r g^{- 1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819554&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819553&type=", "width": "6.77333355", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": "）" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "34", "type": "bibr", "rid": "R34", "data": [ { "name": "text", "data": "34" } ] } }, { "name": "text", "data": "]" } ] } ] }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（23）" } ], "data": [ { "name": "math", "data": { "math": "

F = \frac{Υ}{{(γ_{D}^{2} + γ_{A}^{2})}^{1 / 2}} \times e^{- {(Δ / γ_{D})}^{2} Γ} G (Δ, T)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819556&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819555&type=", "width": "53.42466354", "height": "9.82133293", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF30" } }, { "name": "p", "data": [ { "name": "text", "data": "where" } ] }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（24）" } ], "data": [ { "name": "math", "data": { "math": "

Γ = \"[\"1 - \frac{1}{1 + {(γ_{D} / γ_{A})}^{2}}\"]\" l n 2

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819558&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819557&type=", "width": "42.84133148", "height": "14.39333248", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF31" } }, { "name": "dispformula", "data": { "label": [], "data": [ { "name": "math", "data": { "math": "

Υ = {(\frac{l n 2}{π})}^{1 / 2} \frac{1}{c h}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819560&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819559&type=", "width": "23.96066666", "height": "9.05933285", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF32" } }, { "name": "p", "data": [ { "name": "text", "data": "with " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

γ_{D}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819719&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819718&type=", "width": "3.55599999", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

γ_{A}

c m^{- 1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819574&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819573&type=", "width": "6.51933336", "height": "2.96333337", "fontsize": "" } } } ] }, { "name": "text", "data": "） of the donor and acceptor, respectively, so " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

2 π c ћ γ_{x}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819568&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819567&type=", "width": "10.75266743", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": " has units of energy （in " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

e r g

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819570&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819569&type=", "width": "4.57200003", "height": "3.47133350", "fontsize": "" } } } ] }, { "name": "text", "data": "）, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Δ

c m^{- 1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819574&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819573&type=", "width": "6.51933336", "height": "2.96333337", "fontsize": "" } } } ] }, { "name": "text", "data": "） being the energy difference （mismatch） between the donor and acceptor states, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Δ = E_{D} - E_{A}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819576&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819575&type=", "width": "18.96533203", "height": "3.89466691", "fontsize": "" } } } ] }, { "name": "text", "data": ", and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Υ = 2.3646184 \times 10^{15} c m^{- 1} \cdot e r g^{- 1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819578&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819577&type=", "width": "49.52999878", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": ". The function " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

G (Δ, T)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819677&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819676&type=", "width": "11.43000031", "height": "3.47133350", "fontsize": "" } } } ] }, { "name": "text", "data": " in " }, { "name": "xref", "data": { "text": "Eq.（23）", "type": "disp-formula", "rid": "DF30", "data": [ { "name": "text", "data": "Eq.（23）" } ] } }, { "name": "text", "data": " collects contributions from the overlaps of the vibrational wavefunctions, which can be expressed in terms of the Huang-Rhys parameters with dependence on the average vibrational quantum number and on the number of phonons required for energy conservation. An approximated expression for this function is" } ] }, { "name": "dispformula", "data": { "label": [ { "name": "text", "data": "（25）" } ], "data": [ { "name": "math", "data": { "math": "

G (Δ, T) = \"{\"\begin{matrix} 1, Δ > 0 \\ e^{Δ / (k_{B} T)}, Δ < 0 \end{matrix}\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819582&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819581&type=", "width": "41.14799881", "height": "9.90600014", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF33" } }, { "name": "p", "data": [ { "name": "text", "data": "which represents an approximation to the multiphonon mechanism used to compensate for the energy conservation, with " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

k_{B}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819584&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819583&type=", "width": "3.04800010", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " being the Boltzmann constant." } ] }, { "name": "p", "data": [ { "name": "text", "data": "This reorganized form of the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F

Δ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819588&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819587&type=", "width": "2.45533323", "height": "2.79399991", "fontsize": "" } } } ] }, { "name": "text", "data": " and the bandwidths " }, { "name": "italic", "data": [ { "name": "text", "data": "ћ" } ] }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

γ_{D}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819590&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819589&type=", "width": "3.55599999", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ћ γ_{A}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819592&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819591&type=", "width": "5.41866684", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": ". For instance, in the case of " }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": " energy transfer, the bandwidths are narrow and practically the same, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

γ_{D} ≅ γ_{A} \equiv γ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819594&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819593&type=", "width": "18.11866570", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": ", so " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F = (Υ / γ \sqrt[]{2}) e^{{- (Δ / γ)}^{2} (l n 2) / 2} G (Δ)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819596&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819595&type=", "width": "46.39733505", "height": "5.16466665", "fontsize": "" } } } ] }, { "name": "text", "data": ". For narrow bands, the restriction on the energy mismatch " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Δ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819598&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819597&type=", "width": "2.45533323", "height": "2.79399991", "fontsize": "" } } } ] }, { "name": "text", "data": " is quite severe. For instance, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

γ ≅ 200 c m^{- 1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819600&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819599&type=", "width": "19.47333336", "height": "3.64066648", "fontsize": "" } } } ] }, { "name": "text", "data": " gives " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F ≅ 8 \times 10^{12} e^{{- 0.35 (Δ / γ)}^{2}} e r g^{- 1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819602&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819601&type=", "width": "39.11600113", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": ", for " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Δ > 0

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819604&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819603&type=", "width": "8.55133343", "height": "2.79399991", "fontsize": "" } } } ] }, { "name": "text", "data": ". If " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Δ ≅ 1 000 c m^{- 1} = 5 γ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819606&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819605&type=", "width": "29.46400070", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": ",then " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

e^{- 0.3466 \cdot 5^{2}} ~ 10^{- 4}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819608&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819607&type=", "width": "19.13466644", "height": "3.55599999", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F ~ 10^{9} e r g^{- 1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819610&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819609&type=", "width": "16.59466553", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": "; however, if " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Δ ≅ 2 000 c m^{- 1} = 10 γ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819612&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819611&type=", "width": "31.41133499", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": ", then " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

e^{- 0.3466 \cdot 10^{2}} ~ 10^{- 15}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819614&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819613&type=", "width": "21.16666603", "height": "3.55599999", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F ~ 10^{- 2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819616&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819615&type=", "width": "11.00666618", "height": "3.04800010", "fontsize": "" } } } ] } ] }, { "name": "p", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

e r g^{- 1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819618&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819617&type=", "width": "6.77333355", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": ". So, a twofold increase in " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Δ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819642&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819641&type=", "width": "2.45533323", "height": "2.79399991", "fontsize": "" } } } ] }, { "name": "text", "data": " leads to a " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

10^{11}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819622&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819621&type=", "width": "5.75733376", "height": "2.96333337", "fontsize": "" } } } ] }, { "name": "text", "data": " decrease of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F

Δ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819642&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819641&type=", "width": "2.45533323", "height": "2.79399991", "fontsize": "" } } } ] }, { "name": "text", "data": " in the argument of the exponential. This situation is quite distinct from that involving a sensitizer with a broad band, where " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

γ_{D} ≫ γ_{A}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819628&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819627&type=", "width": "12.19200039", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": ". In this case, the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819630&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819629&type=", "width": "2.37066650", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": "-factor becomes " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F = (Υ / γ_{D}) e^{{- (Δ / γ_{D})}^{2} l n 2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819632&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819631&type=", "width": "28.78666496", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

G (Δ)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819634&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819633&type=", "width": "8.04333401", "height": "3.47133350", "fontsize": "" } } } ] }, { "name": "text", "data": " .So, for a donor bandwidth of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

γ_{D} ≅ 2 000 c m^{- 1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819636&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819635&type=", "width": "23.53733444", "height": "4.74133301", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Δ ≅ 6 000 c m^{- 1} = 3 γ_{D}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819638&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819637&type=", "width": "31.07266808", "height": "4.91066647", "fontsize": "" } } } ] }, { "name": "text", "data": ", then " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F ~ 2 \times 10^{9} e r g^{- 1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819640&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819639&type=", "width": "22.35199928", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": ". Despite this energy mismatch " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Δ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819642&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819641&type=", "width": "2.45533323", "height": "2.79399991", "fontsize": "" } } } ] }, { "name": "text", "data": " being six times larger than that of narrow bands, the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F

F

F

G (Δ, T)

Δ < 0

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819652&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819651&type=", "width": "8.80533314", "height": "2.79399991", "fontsize": "" } } } ] }, { "name": "text", "data": " or " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Δ > 0

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819654&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819653&type=", "width": "8.80533314", "height": "2.79399991", "fontsize": "" } } } ] }, { "name": "text", "data": "（" }, { "name": "xref", "data": { "text": "Eq.25", "type": "disp-formula", "rid": "DF33", "data": [ { "name": "text", "data": "Eq.25" } ] } }, { "name": "text", "data": "）. For very narrow bandwidths values （" }, { "name": "italic", "data": [ { "name": "text", "data": "e" } ] }, { "name": "text", "data": "." }, { "name": "italic", "data": [ { "name": "text", "data": "g" } ] }, { "name": "text", "data": "., " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

γ_{D} = γ_{A} \approx 100

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819656&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819655&type=", "width": "21.59000015", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": " cm" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": "）, the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F

Δ

F

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819708&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819707&type=", "width": "2.03200006", "height": "2.28600001", "fontsize": "" } } } ] }, { "name": "text", "data": "-factor on the energy mismatch " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Δ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819664&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819663&type=", "width": "2.11666679", "height": "2.37066650", "fontsize": "" } } } ] }, { "name": "text", "data": " and the bandwidth " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

γ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819666&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819665&type=", "width": "1.77800000", "height": "2.87866688", "fontsize": "" } } } ] }, { "name": "text", "data": "， for " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

γ_{D} = γ_{A} = γ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819668&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819667&type=", "width": "15.57866764", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": " and temperature " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

T =

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819670&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819669&type=", "width": "4.48733330", "height": "2.28600001", "fontsize": "" } } } ] }, { "name": "text", "data": " 300 K. Graph in linear scale in Fig.S1." } ] } ], "subcaption": [], "note": [], "graphics": [ { "print": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819671&type=", "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819673&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819672&type=", "width": "76.03378296", "height": "69.81262207", "fontsize": "" } ] } }, { "name": "p", "data": [ { "name": "text", "data": "Temperature effects on the energy transfer rates can be rationalized from the changes of the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F

G (Δ, T)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819677&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819676&type=", "width": "11.43000031", "height": "3.47133350", "fontsize": "" } } } ] }, { "name": "text", "data": " when " }, { "name": "italic", "data": [ { "name": "text", "data": "Δ" } ] }, { "name": "text", "data": "<" }, { "name": "text", "data": "0（direct） and the widening of the bandwidths due to the temperature rising （indirect）. " }, { "name": "xref", "data": { "text": "Fig.2", "type": "fig", "rid": "F2", "data": [ { "name": "text", "data": "Fig.2" } ] } }, { "name": "text", "data": " shows the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F

Δ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819681&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819680&type=", "width": "2.45533323", "height": "2.79399991", "fontsize": "" } } } ] }, { "name": "text", "data": ", for fixed values of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

γ_{D} = γ_{A} \approx 400

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819683&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819682&type=", "width": "21.59000015", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": " cm" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": ". For the temperature effects on the bandwidth broadening, consider the following hypothetical case: a co-doped Nd（Ⅲ）,Yb（Ⅲ）-containing material at an initial temperature " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

T_{i}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819685&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819684&type=", "width": "2.79399991", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " presents " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

γ_{D} (T_{i}) = γ_{A} (T_{i}) \approx 300

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819687&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819686&type=", "width": "33.69733429", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " cm" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": " for a pair of donor and acceptor transitions （" }, { "name": "italic", "data": [ { "name": "text", "data": "e" } ] }, { "name": "text", "data": "." }, { "name": "italic", "data": [ { "name": "text", "data": "g" } ] }, { "name": "text", "data": "., Nd（Ⅲ） " }, { "name": "sup", "data": [ { "name": "text", "data": "4" } ] }, { "name": "text", "data": "G" }, { "name": "sub", "data": [ { "name": "text", "data": "5/2" } ] }, { "name": "text", "data": "→" }, { "name": "sup", "data": [ { "name": "text", "data": "4" } ] }, { "name": "text", "data": "I" }, { "name": "sub", "data": [ { "name": "text", "data": "15/2" } ] }, { "name": "text", "data": " as donor and Yb（Ⅲ） " }, { "name": "sup", "data": [ { "name": "text", "data": "2" } ] }, { "name": "text", "data": "F" }, { "name": "sub", "data": [ { "name": "text", "data": "7/2" } ] }, { "name": "text", "data": "→" }, { "name": "sup", "data": [ { "name": "text", "data": "2" } ] }, { "name": "text", "data": "F" }, { "name": "sub", "data": [ { "name": "text", "data": "5/2" } ] }, { "name": "text", "data": " as acceptor, with " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Δ ≅

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819689&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819688&type=", "width": "5.58799982", "height": "2.79399991", "fontsize": "" } } } ] }, { "name": "text", "data": "1 100 cm" }, { "name": "sup", "data": [ { "name": "text", "data": "-1[" }, { "name": "xref", "data": { "text": "47", "type": "bibr", "rid": "R47", "data": [ { "name": "text", "data": "47" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": "）, and the sample is uniformly heated to a final temperature " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

T_{f}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819691&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819690&type=", "width": "2.79399991", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " and both bandwidths increase by 50 cm" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": ", so （" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

γ_{D} (T_{f}) = γ_{A} (T_{f}) \approx 350

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819693&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819692&type=", "width": "33.69733429", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " cm" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": "）. According to " }, { "name": "xref", "data": { "text": "Eq.（23）", "type": "disp-formula", "rid": "DF30", "data": [ { "name": "text", "data": "Eq.（23）" } ] } }, { "name": "text", "data": " and approximating " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Δ ≅

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819695&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819694&type=", "width": "5.58799982", "height": "2.79399991", "fontsize": "" } } } ] }, { "name": "text", "data": " 1 000 cm" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": " （as illustrated in " }, { "name": "xref", "data": { "text": "Fig.3", "type": "fig", "rid": "F3", "data": [ { "name": "text", "data": "Fig.3" } ] } }, { "name": "text", "data": "）, the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F

F

Δ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819701&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819700&type=", "width": "2.11666679", "height": "2.37066650", "fontsize": "" } } } ] }, { "name": "text", "data": " and the temperature. " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

γ_{D} = γ_{A} = 400

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819703&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819702&type=", "width": "18.54199982", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": " cm" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": ". Graph in linear scale in Fig.S2." } ] } ], "subcaption": [], "note": [], "graphics": [ { "print": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819704&type=", "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819706&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819705&type=", "width": "74.02540588", "height": "69.37142944", "fontsize": "" } ] } }, { "name": "fig", "data": { "id": "F3", "caption": [ { "lang": "en", "label": [ { "name": "text", "data": "Fig.3" } ], "title": [ { "name": "text", "data": "Dependence of the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819708&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819707&type=", "width": "2.03200006", "height": "2.28600001", "fontsize": "" } } } ] }, { "name": "text", "data": "-factor on the energy bandwidths of the donor （" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

γ_{D}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819710&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819709&type=", "width": "3.04800010", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": "） and the acceptor （" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

γ_{A}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819712&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819711&type=", "width": "3.04800010", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": "）. " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Δ = 1 000

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819714&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819713&type=", "width": "12.95400047", "height": "2.45533323", "fontsize": "" } } } ] }, { "name": "text", "data": " cm" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": ". Graph in linear scale in Fig.S3." } ] } ], "subcaption": [], "note": [], "graphics": [ { "print": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819715&type=", "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819717&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819716&type=", "width": "75.41313171", "height": "67.54042053", "fontsize": "" } ] } }, { "name": "p", "data": [ { "name": "text", "data": "Notice that the graph in " }, { "name": "xref", "data": { "text": "Fig.3", "type": "fig", "rid": "F3", "data": [ { "name": "text", "data": "Fig.3" } ] } }, { "name": "text", "data": " presents symmetry with respect to the interchange of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

γ_{D}

γ_{A}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819721&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819720&type=", "width": "3.55599999", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": " values, which is not evident in Eqs.（" }, { "name": "xref", "data": { "text": "23", "type": "disp-formula", "rid": "DF30", "data": [ { "name": "text", "data": "23" } ] } }, { "name": "text", "data": "） and （" }, { "name": "xref", "data": { "text": "24", "type": "disp-formula", "rid": "DF31", "data": [ { "name": "text", "data": "24" } ] } }, { "name": "text", "data": "） because of the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

e^{- {(Δ / γ_{D})}^{2} Γ}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819723&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819722&type=", "width": "11.17599964", "height": "3.47133350", "fontsize": "" } } } ] }, { "name": "text", "data": " term. However, it makes sense once the Dirac delta function acting in " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

k

i

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819727&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819726&type=", "width": "1.10066664", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " states should be the same when the order of states is reversed, " }, { "name": "italic", "data": [ { "name": "text", "data": "i" } ] }, { "name": "text", "data": "." }, { "name": "italic", "data": [ { "name": "text", "data": "e" } ] }, { "name": "text", "data": "., " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

δ (E_{k} - E_{i}) = δ (E_{i} - E_{k})

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819729&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819728&type=", "width": "37.08399963", "height": "4.31799984", "fontsize": "" } } } ] }, { "name": "text", "data": " in " }, { "name": "xref", "data": { "text": "Eq.（7）", "type": "disp-formula", "rid": "DF10", "data": [ { "name": "text", "data": "Eq.（7）" } ] } }, { "name": "text", "data": "." } ] } ] }, { "name": "sec", "data": [ { "name": "sectitle", "data": { "title": [ { "name": "text", "data": "6　Selection Rules for Energy Transfer Rates" } ], "level": "1", "id": "s6" } }, { "name": "p", "data": [ { "name": "text", "data": "Lanthanide ions have a wealthy number of levels in the optical region that can be active in energy transfer processes. So, the number of ET pathways between two lanthanide ions can be substantially large and the selection rules are important to exclude those pathways that are null. In fact, the selection rules of the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

J

Δ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819733&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819732&type=", "width": "2.45533323", "height": "2.79399991", "fontsize": "" } } } ] }, { "name": "text", "data": ", in " }, { "name": "xref", "data": { "text": "Eq.（23）", "type": "disp-formula", "rid": "DF30", "data": [ { "name": "text", "data": "Eq.（23）" } ] } }, { "name": "text", "data": ", are very useful to select the donor and acceptor states for adequate ET pathways." } ] }, { "name": "p", "data": [ { "name": "text", "data": "The exchange contribution, " }, { "name": "xref", "data": { "text": "Eq.（17）", "type": "disp-formula", "rid": "DF24", "data": [ { "name": "text", "data": "Eq.（17）" } ] } }, { "name": "text", "data": ", to the ET rate " }, { "name": "italic", "data": [ { "name": "text", "data": "does" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "not" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "have" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "selection" } ] }, { "name": "text", "data": " " }, { "name": "italic", "data": [ { "name": "text", "data": "rules" } ] }, { "name": "text", "data": " for the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

J

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819735&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819734&type=", "width": "1.77800000", "height": "2.96333337", "fontsize": "" } } } ] }, { "name": "text", "data": " quantum number, that is, it always present, regardless of the donor or acceptor states. This is particular to lanthanide-lanthanide ET processes because in this case, the isotropic contribution does not vanish, so no dependence on the angular momentum appears. For the case of ligand-lanthanide ET processes, because the ligand states are usually restricted to singlets and triplets, with a singlet ground state, the contribution from the spin operator vanishes and only transitions with " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"|\"∆ J\"|\" = 0

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819737&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819736&type=", "width": "11.76866722", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": " or 1 are allowed （nonvanishing）" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "20", "type": "bibr", "rid": "R20", "data": [ { "name": "text", "data": "20" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ".Despite this lack of selection rules, the exchange contribution is strongly dependent on the lanthanide-lanthanide distance, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

R_{D A}

R_{D A}^{- 2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819741&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819740&type=", "width": "5.16466665", "height": "4.06400013", "fontsize": "" } } } ] }, { "name": "text", "data": " in " }, { "name": "xref", "data": { "text": "Eq. （17）", "type": "disp-formula", "rid": "DF24", "data": [ { "name": "text", "data": "Eq. （17）" } ] } }, { "name": "text", "data": ", and, mainly of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ρ_{f f}^{4}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819743&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819742&type=", "width": "3.30200005", "height": "4.74133301", "fontsize": "" } } } ] }, { "name": "text", "data": " dependence. Indeed, the overlap integral between 4f-4f radial functions, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ρ_{f f}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819745&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819744&type=", "width": "3.30200005", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": ", decreases rapidly with the lanthanide-lanthanide distance （" }, { "name": "xref", "data": { "text": "Fig.4", "type": "fig", "rid": "F4", "data": [ { "name": "text", "data": "Fig.4" } ] } }, { "name": "text", "data": "）" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "53", "type": "bibr", "rid": "R53", "data": [ { "name": "text", "data": "53" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". As a result, depending on the value of the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F

W_{e x}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819749&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819748&type=", "width": "4.82600021", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": ", can be promptly ruled out and would be considered mostly when the other contributions are null." } ] }, { "name": "fig", "data": { "id": "F4", "caption": [ { "lang": "en", "label": [ { "name": "text", "data": "Fig.4" } ], "title": [ { "name": "text", "data": "4f-4f overlap integrals （lg scale of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

ρ_{f f}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819751&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819750&type=", "width": "2.87866688", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": "） as a function of " }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": " distance （" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

R_{D A}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819753&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819752&type=", "width": "4.48733330", "height": "3.21733332", "fontsize": "" } } } ] }, { "name": "text", "data": " in Å） for the Tb（Ⅲ）-Eu（Ⅲ）， Yb（Ⅲ）-Er（Ⅲ）， and Nd（Ⅲ）-Yb（Ⅲ） pairs. DFT calculations with BP86 functional" }, { "name": "sup", "data": [ { "name": "text", "data": "［" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "61", "type": "bibr", "rid": "R61", "data": [ { "name": "text", "data": "61" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "62", "type": "bibr", "rid": "R62", "data": [ { "name": "text", "data": "62" } ] } } ], "rid": [ "R61", "R62" ], "text": "61-62", "type": "bibr" } }, { "name": "text", "data": "］" } ] }, { "name": "text", "data": "， TZ2P basis set" }, { "name": "sup", "data": [ { "name": "text", "data": "［" }, { "name": "xref", "data": { "text": "63", "type": "bibr", "rid": "R63", "data": [ { "name": "text", "data": "63" } ] } }, { "name": "text", "data": "］" } ] }, { "name": "text", "data": "，and the inclusion of ZORA scalar relativistic effects" }, { "name": "sup", "data": [ { "name": "text", "data": "［" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "64", "type": "bibr", "rid": "R64", "data": [ { "name": "text", "data": "64" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "65", "type": "bibr", "rid": "R65", "data": [ { "name": "text", "data": "65" } ] } } ], "rid": [ "R64", "R65" ], "text": "64-65", "type": "bibr" } }, { "name": "text", "data": "］" } ] }, { "name": "text", "data": " were performed in the ADF program" }, { "name": "sup", "data": [ { "name": "text", "data": "［" }, { "name": "xref", "data": { "text": "66", "type": "bibr", "rid": "R66", "data": [ { "name": "text", "data": "66" } ] } }, { "name": "text", "data": "］" } ] }, { "name": "text", "data": ". Parametric functions derived from these calculations for any " }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "-" }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": " pair can be found in reference" }, { "name": "sup", "data": [ { "name": "text", "data": "［" }, { "name": "xref", "data": { "text": "53", "type": "bibr", "rid": "R53", "data": [ { "name": "text", "data": "53" } ] } }, { "name": "text", "data": "］" } ] }, { "name": "text", "data": ". Graph in linear scale in Fig.S4." } ] } ], "subcaption": [], "note": [], "graphics": [ { "print": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819754&type=", "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819756&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819755&type=", "width": "71.07788086", "height": "43.91893005", "fontsize": "" } ] } }, { "name": "p", "data": [ { "name": "text", "data": "In the case of the multipolar mechanisms, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{d ⁃ d}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819758&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819757&type=", "width": "5.67266655", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": ", " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{d q ⁃ q d}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819800&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819799&type=", "width": "7.61999989", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": ", and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{q ⁃ q}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819802&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819801&type=", "width": "5.67266655", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": ", the selection rules are: " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"|\"J^{*} - J\"|\" \leq K \leq J^{*} + J

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819764&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819763&type=", "width": "30.98800087", "height": "4.31799984", "fontsize": "" } } } ] }, { "name": "text", "data": ".These come from the fact that these rates will vanish when the dipole " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

S_{d}^{x}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819766&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819765&type=", "width": "3.04800010", "height": "4.06400013", "fontsize": "" } } } ] }, { "name": "text", "data": " or quadruple " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

S_{q}^{x}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819768&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819767&type=", "width": "3.04800010", "height": "4.06400013", "fontsize": "" } } } ] }, { "name": "text", "data": " strengths are null. On the other hand, these quantities will be null when all reduced matrix elements " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"ψ_{x}^{*} J_{x}^{*}\"\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819770&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819769&type=", "width": "8.80533314", "height": "4.48733330", "fontsize": "" } } } ] }, { "name": "text", "data": " " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"‖\"U^{(K)}\"‖\" \"\"\" separators=\"|\">, ψ_{x} J_{x}\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819772&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819771&type=", "width": "17.86466599", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " are zero, hence these selection rules.For instance, consider the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{5} D_{4} \to {}^{7}F F_{5}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819774&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819773&type=", "width": "14.73200035", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " transition of Tb（Ⅲ） as a donor state, then " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"|\"J_{D}^{*} - J_{D}\"|\" = 1

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819776&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819775&type=", "width": "17.01799965", "height": "4.31799984", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

J_{D}^{*} + J_{D} = 9

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819778&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819777&type=", "width": "17.18733406", "height": "4.40266657", "fontsize": "" } } } ] }, { "name": "text", "data": ", so " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

1 \leq K \leq 9

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819780&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819779&type=", "width": "15.07066536", "height": "2.70933342", "fontsize": "" } } } ] }, { "name": "text", "data": " which covers all possible values of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

K

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819782&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819781&type=", "width": "2.53999996", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " and increases the chance of having nonzero matrix elements. On the other hand, if the acceptor state is the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{7} F_{0} \to {}^{5}D D_{1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819784&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819783&type=", "width": "14.73200035", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " transition of Eu（Ⅲ）, then " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"|\"J_{A}^{*} - J_{A}\"|\" = 1

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819786&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819785&type=", "width": "19.38866615", "height": "4.31799984", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

J_{A}^{*} + J_{A} = 1

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819788&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819787&type=", "width": "17.18733406", "height": "4.40266657", "fontsize": "" } } } ] }, { "name": "text", "data": ", so " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

K = 1

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819790&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819789&type=", "width": "8.72066593", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " and the transition is forbidden because matrix elements " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"ψ_{x}^{*} J_{x}^{*}\"\" \"‖\"U^{(1)}\"‖\" \"\"\" separators=\"|\">, ψ_{x} J_{x}\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819792&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819791&type=", "width": "26.58533096", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " are zero. However, these selection rules may be slightly relaxed by the " }, { "name": "italic", "data": [ { "name": "text", "data": "J" } ] }, { "name": "text", "data": "-mixing effect, providing very small values of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"ψ_{x}^{*} J_{x}^{*}\"\" \"‖\"U^{(2)}\"‖\" {\"\"\" separators=\"|\">, ψ_{x} J_{x}\"}^{2} \approx 0.0032 \times 0 . 05^{2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819794&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819793&type=", "width": "56.98066711", "height": "5.16466665", "fontsize": "" } } } ] }, { "name": "text", "data": " for the specific cases involving the Eu（Ⅲ） " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{7} F_{0}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819978&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819977&type=", "width": "4.65666676", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " level. Therefore, this effect is negligible in the energy transfer between lanthanide ions." } ] }, { "name": "p", "data": [ { "name": "text", "data": "So, the contributions from multipolar mechanisms, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{d ⁃ d}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819854&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819853&type=", "width": "5.67266655", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": ", " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{d q ⁃ q d}

W_{q ⁃ q}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819802&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819801&type=", "width": "5.67266655", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": ", to the ET rate " }, { "name": "italic", "data": [ { "name": "text", "data": "via" } ] }, { "name": "text", "data": " the pathway Tb（Ⅲ） " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{5} D_{4} \to {}^{7}F F_{5}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819880&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819879&type=", "width": "14.39333248", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " to Eu（Ⅲ） " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{7} F_{0} \to {}^{5}D D_{1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819806&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819805&type=", "width": "14.73200035", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " are null. However, this pathway is allowed by the exchange mechanism and might be allowed by the magnetic dipole mechanism. Hence, for the multipolar mechanisms, another acceptor state of Eu（Ⅲ） needs to be considered, " }, { "name": "italic", "data": [ { "name": "text", "data": "e" } ] }, { "name": "text", "data": "." }, { "name": "italic", "data": [ { "name": "text", "data": "g" } ] }, { "name": "text", "data": "., " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{7} F_{1} \to {}^{5}D D_{1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819808&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819807&type=", "width": "15.83266735", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": ". In this case, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"|\"J_{A}^{*} - J_{A}\"|\" = 0

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819810&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819809&type=", "width": "19.38866615", "height": "4.31799984", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

J_{A}^{*} + J_{A} = 2

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819812&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819811&type=", "width": "17.18733406", "height": "4.40266657", "fontsize": "" } } } ] }, { "name": "text", "data": ", so " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

K = 2

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819814&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819813&type=", "width": "8.72066593", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": " and the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{7} F_{1} \to {}^{5}D D_{1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819816&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819815&type=", "width": "14.73200035", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " transition is allowed because matrix element " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

U^{(2)}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819818&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819817&type=", "width": "5.33400011", "height": "3.13266683", "fontsize": "" } } } ] }, { "name": "text", "data": " of Eu（Ⅲ） is nonzero." } ] }, { "name": "p", "data": [ { "name": "text", "data": "The remaining mechanism for ET between lanthanide ions is the magnetic dipole⁃magnetic dipole, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{m d ⁃ m d}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819828&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819827&type=", "width": "8.55133343", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": ", in Eq.（" }, { "name": "xref", "data": { "text": "16", "type": "disp-formula", "rid": "DF23", "data": [ { "name": "text", "data": "16" } ] } }, { "name": "text", "data": "） and （" }, { "name": "xref", "data": { "text": "20", "type": "disp-formula", "rid": "DF27", "data": [ { "name": "text", "data": "20" } ] } }, { "name": "text", "data": "）. Thus, the selection rules are given by the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"ψ^{*} J^{*}\"\" \"‖\"\hat{L} + g_{S} \hat{S}\"‖\" \"\"\" separators=\"|\">, ψ J\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819822&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819821&type=", "width": "30.90333366", "height": "5.75733376", "fontsize": "" } } } ] }, { "name": "text", "data": " matrix elements, which will be zero unless " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

∆ J = 0, \pm 1

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819824&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819823&type=", "width": "17.44133186", "height": "3.21733332", "fontsize": "" } } } ] }, { "name": "text", "data": ", except " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

J^{*} = J = 0

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819826&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819825&type=", "width": "15.49400043", "height": "3.30200005", "fontsize": "" } } } ] }, { "name": "text", "data": ". As a result, the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{m d ⁃ m d}

{}^{5} D_{4} \to {}^{7}F F_{5}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819830&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819829&type=", "width": "14.39333248", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " to Eu（Ⅲ） " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{7} F_{0} \to {}^{5}D D_{1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819832&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819831&type=", "width": "14.13933277", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " pathway will be nonzero because " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

∆ J = \pm 1

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819834&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819833&type=", "width": "11.51466751", "height": "3.04800010", "fontsize": "" } } } ] }, { "name": "text", "data": " for both transitions, which satisfy the selection rules." } ] }, { "name": "p", "data": [ { "name": "text", "data": "Notice that for Eu（Ⅲ）, the first excited state " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{7} F_{1}

{}^{7} F_{0}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819978&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819977&type=", "width": "4.65666676", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": ", so the populations of these states will depend on the temperature " }, { "name": "italic", "data": [ { "name": "text", "data": "via" } ] }, { "name": "text", "data": " the Boltzmann factor. When comparing the relative contributions of each mechanism to the total ET rate, it is common to weigh those rates involving the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{7} F_{0}

{}^{7} F_{1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819936&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819935&type=", "width": "4.65666676", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " levels by ca. 0.65 and 0.33 （at 300 K）, respectively. However, care must be exercised when the ET rates are going to be employed in the rate equations. In this case, the thermal effects are already included in the initial populations when solving the rate equations, so the weighting procedure should not be overcounted." } ] } ] }, { "name": "sec", "data": [ { "name": "sectitle", "data": { "title": [ { "name": "text", "data": "7　Forced Electric Dipole Intensity Parameter， " }, { "name": "italic", "data": [ { "name": "text", "data": "Ω" }, { "name": "sub", "data": [ { "name": "text", "data": "K" } ] } ] }, { "name": "text", "data": "（FED）" } ], "level": "1", "id": "s7" } }, { "name": "p", "data": [ { "name": "text", "data": "The forced electric dipole （FED） contribution to the intensity parameter " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ω_{K}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819844&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819843&type=", "width": "4.06400013", "height": "3.89466691", "fontsize": "" } } } ] }, { "name": "text", "data": " （with " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

K =

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819846&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819845&type=", "width": "5.41866684", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": "2, 4, and 6）, that appears in the （forced） electric dipole strength " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

S_{d}

Ω_{K}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819850&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819849&type=", "width": "4.06400013", "height": "3.89466691", "fontsize": "" } } } ] }, { "name": "text", "data": " contains contributions from both FED and dynamic coupling （DC） mechanisms. Usually, the FED contribution to " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ω_{K}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819852&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819851&type=", "width": "4.06400013", "height": "3.89466691", "fontsize": "" } } } ] }, { "name": "text", "data": " is very small; however, it depends on the lanthanide ion and its environment, so no trends have been observed" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "37", "type": "bibr", "rid": "R37", "data": [ { "name": "text", "data": "37" } ] } }, { "name": "text", "data": "," }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "67", "type": "bibr", "rid": "R67", "data": [ { "name": "text", "data": "67" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "69", "type": "bibr", "rid": "R69", "data": [ { "name": "text", "data": "69" } ] } } ], "rid": [ "R67", "R68", "R69" ], "text": "67-69", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". It cannot be neglected, because despite it being small, the contributions of the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{d ⁃ d}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819854&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819853&type=", "width": "5.67266655", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{d q ⁃ q d}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819856&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819855&type=", "width": "7.61999989", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " rates can be relevant or even dominant for several ET pathways. In addition, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ω_{K} (F E D)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819864&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819863&type=", "width": "14.22399998", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": " contains effects from the environment of the lanthanide ion, which can be used to design systems with more efficient ET pathways. Therefore, values of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ω_{K} (F E D)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819860&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819859&type=", "width": "14.22399998", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": " have to be obtained from quantitative models. For instance, the simple overlap model （SOM）" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "55", "type": "bibr", "rid": "R55", "data": [ { "name": "text", "data": "55" } ] } }, { "name": "text", "data": "," }, { "name": "xref", "data": { "text": "70", "type": "bibr", "rid": "R70", "data": [ { "name": "text", "data": "70" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " yields quantitative expressions for " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ω_{K} (F E D)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819924&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819923&type=", "width": "14.22399998", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": ", which can be evaluated using the JOYSpectra web platform" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "71", "type": "bibr", "rid": "R71", "data": [ { "name": "text", "data": "71" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". For Eu（Ⅲ） and Tb（Ⅲ） in a unique nine-coordination site of the doped material Sr" }, { "name": "sub", "data": [ { "name": "text", "data": "3" } ] }, { "name": "text", "data": "Tb" }, { "name": "sub", "data": [ { "name": "text", "data": "0.90" } ] }, { "name": "text", "data": "Eu" }, { "name": "sub", "data": [ { "name": "text", "data": "0.10" } ] }, { "name": "text", "data": "⁃" } ] }, { "name": "p", "data": [ { "name": "text", "data": "（PO" }, { "name": "sub", "data": [ { "name": "text", "data": "4" } ] }, { "name": "text", "data": "）" }, { "name": "sub", "data": [ { "name": "text", "data": "3" } ] }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "36", "type": "bibr", "rid": "R36", "data": [ { "name": "text", "data": "36" } ] } }, { "name": "text", "data": "," }, { "name": "xref", "data": { "text": "71", "type": "bibr", "rid": "R71", "data": [ { "name": "text", "data": "71" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ", two sets of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ω_{K} (F E D)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819864&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819863&type=", "width": "14.22399998", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": " " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"{\"Ω_{2}; Ω_{4}; Ω_{6}\"}\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819866&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819865&type=", "width": "20.15066719", "height": "4.40266657", "fontsize": "" } } } ] }, { "name": "text", "data": " （in units of 10" }, { "name": "sup", "data": [ { "name": "text", "data": "-20" } ] }, { "name": "text", "data": " cm" }, { "name": "sup", "data": [ { "name": "text", "data": "2" } ] }, { "name": "text", "data": "） are obtained: {0.60; 0.21; 0.26} for Tb（Ⅲ） and {0.82; 0.38; 0.52} for Eu（Ⅲ）." } ] }, { "name": "p", "data": [ { "name": "text", "data": "These calculations were performed using the values of the force constant, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

k_{L n ⁃ L}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819868&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819867&type=", "width": "5.58799982", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": ", of the " }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "-Ligating atom bond obtained from density functional theory （DFT） calculations and, therefore, the values of the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

g

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819874&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819873&type=", "width": "1.94733346", "height": "3.47133350", "fontsize": "" } } } ] }, { "name": "text", "data": "-charge factors were determined for all " }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "⁃O pairs with Eq.S80" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "36", "type": "bibr", "rid": "R36", "data": [ { "name": "text", "data": "36" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ".In the specific case of Sr" }, { "name": "sub", "data": [ { "name": "text", "data": "3" } ] }, { "name": "text", "data": "Tb" }, { "name": "sub", "data": [ { "name": "text", "data": "0.90" } ] }, { "name": "text", "data": "Eu" }, { "name": "sub", "data": [ { "name": "text", "data": "0.10" } ] }, { "name": "text", "data": "⁃" } ] }, { "name": "p", "data": [ { "name": "text", "data": "（PO" }, { "name": "sub", "data": [ { "name": "text", "data": "4" } ] }, { "name": "text", "data": "）" }, { "name": "sub", "data": [ { "name": "text", "data": "3" } ] }, { "name": "sup", "data": [ { "name": "text", "data": " " } ] }, { "name": "text", "data": "materials, the calculated " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

k_{L n ⁃ L}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819872&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819871&type=", "width": "5.67266655", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " is in the range of 0.38-0.42 mdyn/Å and it provides " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

g

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819874&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819873&type=", "width": "1.94733346", "height": "3.47133350", "fontsize": "" } } } ] }, { "name": "text", "data": "-charge factors around 1.29. For further details on applying the SOM model, see Supporting Information and references [20, 35, 55, 67, 70-76]." } ] } ] }, { "name": "sec", "data": [ { "name": "sectitle", "data": { "title": [ { "name": "text", "data": "8　Illustrative Examples of How to Calculate Energy Transfer Rates" } ], "level": "1", "id": "s8" } }, { "name": "p", "data": [ { "name": "text", "data": "Here we will illustrate step-by-step examples of some energy transfer pathways between " }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "（Ⅲ） ions. The first example （" }, { "name": "bold", "data": [ { "name": "text", "data": "E1" } ] }, { "name": "text", "data": "） involves the Tb（Ⅲ） [" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{5} D_{4} \to {}^{7}F F_{5}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819946&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819945&type=", "width": "14.73200035", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": "] → Eu（Ⅲ） [" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{7} F_{1} \to {}^{5}D D_{1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819878&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819877&type=", "width": "13.97000027", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": "] energy transfer pathway which is the dominant one for Tb-Eu processes" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "36", "type": "bibr", "rid": "R36", "data": [ { "name": "text", "data": "36" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "37", "type": "bibr", "rid": "R37", "data": [ { "name": "text", "data": "37" } ] } } ], "rid": [ "R36", "R37" ], "text": "36-37", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". As an exercise of a direct application of selection rules on " }, { "name": "italic", "data": [ { "name": "text", "data": "J" } ] }, { "name": "text", "data": ", the Tb（Ⅲ）[" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{5} D_{4} \to {}^{7}F F_{5}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819880&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819879&type=", "width": "14.39333248", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": "]→Eu（Ⅲ）[" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{7} F_{0} \to {}^{5}D D_{1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819882&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819881&type=", "width": "14.39333248", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": "] （" }, { "name": "bold", "data": [ { "name": "text", "data": "E2" } ] }, { "name": "text", "data": "） will also be addressed. These two pathways are illustrated in " }, { "name": "xref", "data": { "text": "Fig.5", "type": "fig", "rid": "F5", "data": [ { "name": "text", "data": "Fig.5" } ] } }, { "name": "text", "data": "." } ] }, { "name": "fig", "data": { "id": "F5", "caption": [ { "lang": "en", "label": [ { "name": "text", "data": "Fig.5" } ], "title": [ { "name": "text", "data": "Energy level diagram showing Eu（Ⅲ） and Tb（Ⅲ） levels. The green arrow denotes the Tb（Ⅲ） " }, { "name": "sup", "data": [ { "name": "text", "data": "5" } ] }, { "name": "text", "data": "D" }, { "name": "sub", "data": [ { "name": "text", "data": "4" } ] }, { "name": "text", "data": "→" }, { "name": "sup", "data": [ { "name": "text", "data": "7" } ] }, { "name": "text", "data": "F" }, { "name": "sub", "data": [ { "name": "text", "data": "5" } ] }, { "name": "text", "data": " transition（donor） while the blue lines represent Eu（Ⅲ） " }, { "name": "sup", "data": [ { "name": "text", "data": "7" } ] }, { "name": "text", "data": "F" }, { "name": "sub", "data": [ { "name": "text", "data": "1，0" } ] }, { "name": "text", "data": "→" }, { "name": "sup", "data": [ { "name": "text", "data": "5" } ] }, { "name": "text", "data": "D" }, { "name": "sub", "data": [ { "name": "text", "data": "1" } ] }, { "name": "text", "data": " transitions （acceptors）. The levels involved in " }, { "name": "bold", "data": [ { "name": "text", "data": "E1" } ] }, { "name": "text", "data": " and " }, { "name": "bold", "data": [ { "name": "text", "data": "E2" } ] }, { "name": "text", "data": " are highlighted in red." } ] } ], "subcaption": [], "note": [], "graphics": [ { "print": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819883&type=", "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819885&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819884&type=", "width": "73.15191650", "height": "76.03033447", "fontsize": "" } ] } }, { "name": "p", "data": [ { "name": "text", "data": "Since the discovery of the upconversion process by F. Auzel" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "77", "type": "bibr", "rid": "R77", "data": [ { "name": "text", "data": "77" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "78", "type": "bibr", "rid": "R78", "data": [ { "name": "text", "data": "78" } ] } } ], "rid": [ "R77", "R78" ], "text": "77-78", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ", ET between lanthanide ions has become an important issue regarding luminescent materials based on " }, { "name": "italic", "data": [ { "name": "text", "data": "Ln" } ] }, { "name": "text", "data": "（Ⅲ） ions" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "blockXref", "data": { "data": [ { "name": "xref", "data": { "text": "79", "type": "bibr", "rid": "R79", "data": [ { "name": "text", "data": "79" } ] } }, { "name": "text", "data": "-" }, { "name": "xref", "data": { "text": "84", "type": "bibr", "rid": "R84", "data": [ { "name": "text", "data": "84" } ] } } ], "rid": [ "R79", "R80", "R81", "R82", "R83", "R84" ], "text": "79-84", "type": "bibr" } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". Thus, we will also show how to calculate two energy transfer steps that are in good energy resonance conditions: Yb（Ⅲ） [" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{2} F_{5 / 2} \to {}^{2}F F_{7 / 2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819887&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819886&type=", "width": "18.20333290", "height": "5.58799982", "fontsize": "" } } } ] }, { "name": "text", "data": "] → Er（Ⅲ） [" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{4} I_{15 / 2} \to {}^{4}I I_{11 / 2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819889&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819888&type=", "width": "17.78000069", "height": "5.58799982", "fontsize": "" } } } ] }, { "name": "text", "data": "] （" }, { "name": "bold", "data": [ { "name": "text", "data": "E3" } ] }, { "name": "text", "data": "） and Yb（Ⅲ）[" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{2} F_{5 / 2} \to {}^{2}F F_{7 / 2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819891&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819890&type=", "width": "17.78000069", "height": "5.58799982", "fontsize": "" } } } ] }, { "name": "text", "data": "] → Er（Ⅲ） [" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{4} I_{11 / 2} \to {}^{4}I I_{7 / 2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820058&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820057&type=", "width": "16.76399994", "height": "5.58799982", "fontsize": "" } } } ] }, { "name": "text", "data": "] （" }, { "name": "bold", "data": [ { "name": "text", "data": "E4" } ] }, { "name": "text", "data": "） pathways" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "16", "type": "bibr", "rid": "R16", "data": [ { "name": "text", "data": "16" } ] } }, { "name": "text", "data": "," }, { "name": "xref", "data": { "text": "35", "type": "bibr", "rid": "R35", "data": [ { "name": "text", "data": "35" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". " }, { "name": "xref", "data": { "text": "Fig.6", "type": "fig", "rid": "F6", "data": [ { "name": "text", "data": "Fig.6" } ] } }, { "name": "text", "data": " illustrates these pathways in " }, { "name": "bold", "data": [ { "name": "text", "data": "E3" } ] }, { "name": "text", "data": " and " }, { "name": "bold", "data": [ { "name": "text", "data": "E4" } ] }, { "name": "text", "data": "." } ] }, { "name": "fig", "data": { "id": "F6", "caption": [ { "lang": "en", "label": [ { "name": "text", "data": "Fig.6" } ], "title": [ { "name": "text", "data": "Energy level diagram showing Er（Ⅲ） and Yb（Ⅲ） levels. The green arrow denotes the Yb（Ⅲ） " }, { "name": "sup", "data": [ { "name": "text", "data": "2" } ] }, { "name": "text", "data": "F" }, { "name": "sub", "data": [ { "name": "text", "data": "5/2" } ] }, { "name": "text", "data": "→" }, { "name": "sup", "data": [ { "name": "text", "data": "2" } ] }, { "name": "text", "data": "F" }, { "name": "sub", "data": [ { "name": "text", "data": "7/2" } ] }, { "name": "text", "data": " transition （donor） while the blue lines represent the two upconverting transitions （acceptors） considered in " }, { "name": "bold", "data": [ { "name": "text", "data": "E3" } ] }, { "name": "text", "data": " and " }, { "name": "bold", "data": [ { "name": "text", "data": "E4" } ] }, { "name": "text", "data": ". All levels involved in the energy transfer examples are highlighted in red." } ] } ], "subcaption": [], "note": [], "graphics": [ { "print": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819894&type=", "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819896&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819895&type=", "width": "69.99908447", "height": "76.91113281", "fontsize": "" } ] } }, { "name": "p", "data": [ { "name": "text", "data": "⋅ " }, { "name": "bold", "data": [ { "name": "text", "data": "E1" } ] } ] }, { "name": "p", "data": [ { "name": "xref", "data": { "text": "Tab.1", "type": "table", "rid": "T1", "data": [ { "name": "text", "data": "Tab.1" } ] } }, { "name": "text", "data": " contains the input data considering Tb（Ⅲ） [" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{5} D_{4} \to {}^{7}F F_{5}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819898&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819897&type=", "width": "14.81666565", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": "] and Eu（Ⅲ） [" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{7} F_{1} \to {}^{5}D D_{1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819900&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819899&type=", "width": "14.81666565", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": "] transitions as donor and acceptor, respectively." } ] }, { "name": "table", "data": { "id": "T1", "caption": [ { "lang": "en", "label": [ { "name": "text", "data": "Tab.1" } ], "title": [ { "name": "text", "data": "Input data used to calculate the Tb（Ⅲ）［" }, { "name": "sup", "data": [ { "name": "text", "data": "5" } ] }, { "name": "text", "data": "D" }, { "name": "sub", "data": [ { "name": "text", "data": "4" } ] }, { "name": "text", "data": "→ " }, { "name": "sup", "data": [ { "name": "text", "data": "7" } ] }, { "name": "text", "data": "F" }, { "name": "sub", "data": [ { "name": "text", "data": "5" } ] }, { "name": "text", "data": "］Eu（Ⅲ）［" }, { "name": "sup", "data": [ { "name": "text", "data": "7" } ] }, { "name": "text", "data": "F" }, { "name": "sub", "data": [ { "name": "text", "data": "1" } ] }, { "name": "text", "data": "→" }, { "name": "sup", "data": [ { "name": "text", "data": "5" } ] }, { "name": "text", "data": "D" }, { "name": "sub", "data": [ { "name": "text", "data": "1" } ] }, { "name": "text", "data": "］ in E1. The number of the equation where the input data enters is also indicated" } ] } ], "note": [], "table": [ { "head": [ [ { "align": "center", "style": "border-top:solid;border-bottom:solid;", "data": [] }, { "align": "center", "style": "border-top:solid;border-bottom:solid;", "data": [ { "name": "text", "data": "Eq." } ] }, { "align": "center", "style": "border-top:solid;border-bottom:solid;", "data": [ { "name": "text", "data": "Donor" } ] }, { "align": "center", "style": "border-top:solid;border-bottom:solid;", "data": [ { "name": "text", "data": "Acceptor" } ] } ] ], "body": [ [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

J

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820042&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820041&type=", "width": "1.35466671", "height": "2.28600001", "fontsize": "" } } } ] } ] }, { "align": "center", "data": [ { "name": "text", "data": "18⁃20" } ] }, { "align": "center", "data": [ { "name": "text", "data": "4" } ] }, { "align": "center", "data": [ { "name": "text", "data": "1" } ] } ], [ { "align": "center", "data": [ { "name": "italic", "data": [ { "name": "text", "data": "ρ" } ] } ] }, { "align": "center", "data": [ { "name": "text", "data": "21" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0.057" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0.057" } ] } ], [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"\"\" \"‖\"U^{(2)}\"‖\" {\"\"\" separators=\"|\">\"}^{2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820044&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820043&type=", "width": "12.95400047", "height": "4.06400013", "fontsize": "" } } } ] } ] }, { "align": "center", "data": [ { "name": "text", "data": "18⁃19" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0.014 2" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0.002 5" } ] } ], [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"\"\" \"‖\"U^{(4)}\"‖\" {\"\"\" separators=\"|\">\"}^{2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820046&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820045&type=", "width": "12.95400047", "height": "4.06400013", "fontsize": "" } } } ] } ] }, { "align": "center", "data": [ { "name": "text", "data": "18" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0.001 3" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0" } ] } ], [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"\"\" \"‖\"U^{(6)}\"‖\" {\"\"\" separators=\"|\">\"}^{2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820048&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820047&type=", "width": "12.95400047", "height": "4.06400013", "fontsize": "" } } } ] } ] }, { "align": "center", "data": [ { "name": "text", "data": "18" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0.002 2" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0" } ] } ], [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"\" separators=\"|\">, r^{2}\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820008&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820007&type=", "width": "4.57200003", "height": "3.04800010", "fontsize": "" } } } ] }, { "name": "text", "data": " （10" }, { "name": "sup", "data": [ { "name": "text", "data": "-17" } ] }, { "name": "text", "data": " cm" }, { "name": "sup", "data": [ { "name": "text", "data": "2" } ] }, { "name": "text", "data": "）" } ] }, { "align": "center", "data": [ { "name": "text", "data": "19" } ] }, { "align": "center", "data": [ { "name": "text", "data": "2.30" } ] }, { "align": "center", "data": [ { "name": "text", "data": "2.57" } ] } ], [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"\"\" \"‖\"L + 2 S\"‖\" {\"\"\" separators=\"|\">\"}^{2}

γ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820012&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820011&type=", "width": "1.60866666", "height": "2.53999996", "fontsize": "" } } } ] }, { "name": "text", "data": " （cm" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": "）" } ] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "23⁃24" } ] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "350" } ] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "350" } ] } ], [ { "align": "left", "style": "border-bottom:solid;", "data": [] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "Eq." } ] }, { "colspan": "2", "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "Donor⁃Acceptor" } ] } ], [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

R_{D A} /

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819916&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819915&type=", "width": "5.33400011", "height": "2.87866688", "fontsize": "" } } } ] }, { "name": "text", "data": "cm" } ] }, { "align": "center", "data": [ { "name": "text", "data": "22" } ] }, { "colspan": "2", "align": "center", "data": [ { "name": "text", "data": "3.97×10" }, { "name": "sup", "data": [ { "name": "text", "data": "-8" } ] } ] } ], [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Δ

ρ_{f f}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819920&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819919&type=", "width": "2.70933342", "height": "3.38666677", "fontsize": "" } } } ] } ] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "17" } ] }, { "colspan": "2", "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "6.70×10" }, { "name": "sup", "data": [ { "name": "text", "data": "-4" } ] }, { "name": "text", "data": " at 3.97 Å" } ] } ] ], "foot": [] } ], "graphics": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819922&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819921&type=", "width": "77.69583130", "height": "67.04952240", "fontsize": "" } } }, { "name": "p", "data": [ { "name": "text", "data": "Thus, when the data in " }, { "name": "xref", "data": { "text": "Tab.1", "type": "table", "rid": "T1", "data": [ { "name": "text", "data": "Tab.1" } ] } }, { "name": "text", "data": " is applied to the indicated equations with the sets of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ω_{K} (F E D)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819924&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819923&type=", "width": "14.22399998", "height": "4.57200003", "fontsize": "" } } } ] }, { "name": "text", "data": ", mentioned previously for the case of Sr" }, { "name": "sub", "data": [ { "name": "text", "data": "3" } ] }, { "name": "text", "data": "Tb" }, { "name": "sub", "data": [ { "name": "text", "data": "0.90" } ] }, { "name": "text", "data": "Eu" }, { "name": "sub", "data": [ { "name": "text", "data": "0.10" } ] }, { "name": "text", "data": "（PO" }, { "name": "sub", "data": [ { "name": "text", "data": "4" } ] }, { "name": "text", "data": "）" }, { "name": "sub", "data": [ { "name": "text", "data": "3" } ] }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "36", "type": "bibr", "rid": "R36", "data": [ { "name": "text", "data": "36" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ", we obtain the following rates for each mechanism:" } ] }, { "name": "dispformula", "data": { "label": [], "data": [ { "name": "math", "data": { "math": "

W_{d ⁃ d} = 6.6 \times 10^{- 2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819926&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819925&type=", "width": "25.23066711", "height": "4.06400013", "fontsize": "" } } }, { "name": "text", "data": " s" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": "，" } ], "id": "DF34" } }, { "name": "dispformula", "data": { "label": [], "data": [ { "name": "math", "data": { "math": "

W_{d q ⁃ q d} = 4.2 \times 10^{1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819928&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819927&type=", "width": "26.07733345", "height": "4.06400013", "fontsize": "" } } }, { "name": "text", "data": " s" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": "，" } ], "id": "DF35" } }, { "name": "dispformula", "data": { "label": [], "data": [ { "name": "math", "data": { "math": "

W_{q ⁃ q} = 5.1 \times 10^{4}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819930&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819929&type=", "width": "24.12999916", "height": "4.06400013", "fontsize": "" } } }, { "name": "text", "data": " s" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": "，" } ], "id": "DF36" } }, { "name": "dispformula", "data": { "label": [], "data": [ { "name": "math", "data": { "math": "

W_{m d ⁃ m d} = 6.8 \times 10^{- 1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819932&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819931&type=", "width": "28.02466965", "height": "4.06400013", "fontsize": "" } } }, { "name": "text", "data": " s" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": "，" } ], "id": "DF37" } }, { "name": "dispformula", "data": { "label": [], "data": [ { "name": "math", "data": { "math": "

W_{e x} = 1.6 \times 10^{5}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819934&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819933&type=", "width": "23.28333473", "height": "4.06400013", "fontsize": "" } } }, { "name": "text", "data": " s" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": "." } ], "id": "DF38" } }, { "name": "p", "data": [ { "name": "text", "data": "The total energy transfer rate is given by the sum of all these contributions and multiplied, in this case, by the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{7} F_{1}

\begin{array}{c} W_{T O T A L} = 0 . 33 \times (W_{d ⁃ d} + W_{d q ⁃ q d} + W_{q ⁃ q} + W_{m d ⁃ m d} + W_{e x}) \\ = 7.0 \times 10^{4} \end{array}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819938&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819937&type=", "width": "75.01467133", "height": "9.82133293", "fontsize": "" } } }, { "name": "text", "data": " s" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": "." } ], "id": "DF39" } }, { "name": "p", "data": [ { "name": "text", "data": "These rates are in agreement with those presented in reference" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "36", "type": "bibr", "rid": "R36", "data": [ { "name": "text", "data": "36" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": " for the same pathway at the same " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

R_{D A}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819940&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819939&type=", "width": "5.16466665", "height": "3.72533321", "fontsize": "" } } } ] }, { "name": "text", "data": " distance. A slight deviation in the quadrupole-quadrupole mechanism from reference" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "36", "type": "bibr", "rid": "R36", "data": [ { "name": "text", "data": "36" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": "（" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{q ⁃ q} = 5.67 \times 10^{4}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819942&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819941&type=", "width": "26.92399979", "height": "4.06400013", "fontsize": "" } } } ] }, { "name": "text", "data": " s" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": "） was observed. This is due to the use of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"\" separators=\"|\">, r^{2}\" = 2.57 \times 10^{- 17}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819944&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819943&type=", "width": "29.04066467", "height": "3.64066648", "fontsize": "" } } } ] }, { "name": "text", "data": " for the Tb（Ⅲ）." } ] }, { "name": "p", "data": [ { "name": "text", "data": "⋅ " }, { "name": "bold", "data": [ { "name": "text", "data": "E2" } ] } ] }, { "name": "p", "data": [ { "name": "text", "data": "To estimate each contribution to the rate of the Tb（Ⅲ） [" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{5} D_{4} \to {}^{7}F F_{5}

{}^{7} F_{0} \to {}^{5}D D_{1}

{}^{7} F_{0} \to {}^{5}D D_{1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819950&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819949&type=", "width": "13.97000027", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " transition （" }, { "name": "xref", "data": { "text": "Tab.2", "type": "table", "rid": "T2", "data": [ { "name": "text", "data": "Tab.2" } ] } }, { "name": "text", "data": "）." } ] }, { "name": "table", "data": { "id": "T2", "caption": [ { "lang": "en", "label": [ { "name": "text", "data": "Tab.2" } ], "title": [ { "name": "text", "data": "Input data used to calculate the Tb（Ⅲ）［" }, { "name": "sup", "data": [ { "name": "text", "data": "5" } ] }, { "name": "text", "data": "D" }, { "name": "sub", "data": [ { "name": "text", "data": "4" } ] }, { "name": "text", "data": "→ " }, { "name": "sup", "data": [ { "name": "text", "data": "7" } ] }, { "name": "text", "data": "F" }, { "name": "sub", "data": [ { "name": "text", "data": "5" } ] }, { "name": "text", "data": "］→Eu（Ⅲ）［" }, { "name": "sup", "data": [ { "name": "text", "data": "7" } ] }, { "name": "text", "data": "F" }, { "name": "sub", "data": [ { "name": "text", "data": "0" } ] }, { "name": "text", "data": "→" }, { "name": "sup", "data": [ { "name": "text", "data": "5" } ] }, { "name": "text", "data": "D" }, { "name": "sub", "data": [ { "name": "text", "data": "1" } ] }, { "name": "text", "data": "］ in E2. The number of the equation where the input data enters is also indicated. The other quantities that are not displayed are still the same as presented in Tab.1" } ] } ], "note": [], "table": [ { "head": [ [ { "align": "center", "style": "border-top:solid;border-bottom:solid;", "data": [] }, { "align": "center", "style": "border-top:solid;border-bottom:solid;", "data": [ { "name": "text", "data": "Eq." } ] }, { "align": "center", "style": "border-top:solid;border-bottom:solid;", "data": [ { "name": "text", "data": "Donor" } ] }, { "align": "center", "style": "border-top:solid;border-bottom:solid;", "data": [ { "name": "text", "data": "Acceptor" } ] } ] ], "body": [ [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

J

\"<\"\"\" \"‖\"U^{(2)}\"‖\" {\"\"\" separators=\"|\">\"}^{2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820044&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820043&type=", "width": "12.95400047", "height": "4.06400013", "fontsize": "" } } } ] } ] }, { "align": "center", "data": [ { "name": "text", "data": "18⁃19" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0.014 2" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0" } ] } ], [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"\"\" \"‖\"U^{(4)}\"‖\" {\"\"\" separators=\"|\">\"}^{2}

\"<\"\"\" \"‖\"U^{(6)}\"‖\" {\"\"\" separators=\"|\">\"}^{2}

\"<\"\"\" \"‖\"L + 2 S\"‖\" {\"\"\" separators=\"|\">\"}^{2}

γ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820012&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820011&type=", "width": "1.60866666", "height": "2.53999996", "fontsize": "" } } } ] }, { "name": "text", "data": " （cm" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": "）" } ] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "23⁃24" } ] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "350" } ] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "106" } ] } ], [ { "align": "left", "style": "border-bottom:solid;", "data": [] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "Eq." } ] }, { "colspan": "2", "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "Donor⁃Acceptor" } ] } ], [ { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Δ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820016&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820015&type=", "width": "1.86266661", "height": "2.11666679", "fontsize": "" } } } ] }, { "name": "text", "data": " /cm" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] } ] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "23， 25" } ] }, { "colspan": "2", "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "-631" } ] } ] ], "foot": [] } ], "graphics": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819966&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819965&type=", "width": "77.69583130", "height": "55.38971710", "fontsize": "" } } }, { "name": "p", "data": [ { "name": "text", "data": "It can be noticed that for this pathway, according to the selection rules, the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{7} F_{0} \to {}^{5}D D_{1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819968&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819967&type=", "width": "14.73200035", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": " transition has all " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"\"\" \"‖\"U^{(K)}\"‖\" {\"\"\" separators=\"|\">\"}^{2} = 0

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819970&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819969&type=", "width": "23.19866753", "height": "4.65666676", "fontsize": "" } } } ] }, { "name": "text", "data": ". Therefore, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{d ⁃ d} = W_{d q ⁃ q d} = W_{q ⁃ q} = 0

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819972&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819971&type=", "width": "34.37466812", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": " and this pathway has only the contributions of the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{e x} = 3.3 \times 10^{3} s^{- 1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819974&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819973&type=", "width": "27.85533524", "height": "4.14866638", "fontsize": "" } } } ] }, { "name": "text", "data": " and " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{m d ⁃ m d} = 1.3 \times 10^{1} s^{- 1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819976&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819975&type=", "width": "31.58066559", "height": "4.14866638", "fontsize": "" } } } ] }, { "name": "text", "data": ", in accordance with those in reference" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "36", "type": "bibr", "rid": "R36", "data": [ { "name": "text", "data": "36" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": ". Adding these rates, considering the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{7} F_{0}

G (Δ, T)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819980&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819979&type=", "width": "11.76866722", "height": "4.14866638", "fontsize": "" } } } ] }, { "name": "text", "data": ", yields the total rate:" } ] }, { "name": "dispformula", "data": { "label": [], "data": [ { "name": "math", "data": { "math": "

\begin{array}{c} W_{T O T A L} = 0.64 \times G (Δ, T) \times (W_{m d ⁃ m d} + W_{e x}) \\ = 1.0 \times 10^{3} \end{array}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819982&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819981&type=", "width": "72.22066498", "height": "8.97466660", "fontsize": "" } } }, { "name": "text", "data": " s" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": "." } ], "id": "DF40" } }, { "name": "p", "data": [ { "name": "text", "data": "It is important to pay attention that, in this case, " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

G (Δ, T)

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819984&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819983&type=", "width": "12.53066730", "height": "4.14866638", "fontsize": "" } } } ] }, { "name": "text", "data": " was considered outside of the " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

F

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819986&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819985&type=", "width": "2.37066650", "height": "2.62466669", "fontsize": "" } } } ] }, { "name": "text", "data": "-factor just to match with the data in the literature. But it must be considered for each case appropriately." } ] }, { "name": "p", "data": [ { "name": "text", "data": "⋅ " }, { "name": "bold", "data": [ { "name": "text", "data": "E3" } ] } ] }, { "name": "p", "data": [ { "name": "text", "data": "The first step in the Yb-Er upconversion process is given by the Yb（Ⅲ）[" }, { "name": "sup", "data": [ { "name": "text", "data": "2" } ] }, { "name": "text", "data": "F" }, { "name": "sub", "data": [ { "name": "text", "data": "5/2" } ] }, { "name": "text", "data": "→" }, { "name": "sup", "data": [ { "name": "text", "data": "2" } ] }, { "name": "text", "data": "F" }, { "name": "sub", "data": [ { "name": "text", "data": "7/2" } ] }, { "name": "text", "data": "]→Er（Ⅲ） [" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{4} I_{15 / 2} \to {}^{4}I I_{11 / 2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819988&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819987&type=", "width": "18.54199982", "height": "5.58799982", "fontsize": "" } } } ] }, { "name": "text", "data": "] pathway. " }, { "name": "xref", "data": { "text": "Tab.3", "type": "table", "rid": "T3", "data": [ { "name": "text", "data": "Tab.3" } ] } }, { "name": "text", "data": " contains the data for calculating the pairwise energy transfer for the shorter Yb⁃Er distance （" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

R_{D A} = 3.5

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819990&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819989&type=", "width": "15.15533447", "height": "3.80999994", "fontsize": "" } } } ] }, { "name": "text", "data": " Å） in NaYF" }, { "name": "sub", "data": [ { "name": "text", "data": "4" } ] }, { "name": "text", "data": "∶Yb,Er material." } ] }, { "name": "table", "data": { "id": "T3", "caption": [ { "lang": "en", "label": [ { "name": "text", "data": "Tab.3" } ], "title": [ { "name": "text", "data": "Input data used to calculate Yb（Ⅲ）［" }, { "name": "sup", "data": [ { "name": "text", "data": "2" } ] }, { "name": "text", "data": "F" }, { "name": "sub", "data": [ { "name": "text", "data": "5/2" } ] }, { "name": "text", "data": "→" }, { "name": "sup", "data": [ { "name": "text", "data": "2" } ] }, { "name": "text", "data": "F" }, { "name": "sub", "data": [ { "name": "text", "data": "7/2" } ] }, { "name": "text", "data": "］→ Er（Ⅲ）［" }, { "name": "sup", "data": [ { "name": "text", "data": "4" } ] }, { "name": "text", "data": "I" }, { "name": "sub", "data": [ { "name": "text", "data": "15/2" } ] }, { "name": "text", "data": "→" }, { "name": "sup", "data": [ { "name": "text", "data": "4" } ] }, { "name": "text", "data": "I" }, { "name": "sub", "data": [ { "name": "text", "data": "11/2" } ] }, { "name": "text", "data": "］ in E3. The number of the equation where the input data enters is also indicated. The values of " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Ω_{K}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819992&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39819991&type=", "width": "3.64066648", "height": "3.30200005", "fontsize": "" } } } ] }, { "name": "text", "data": " represent those obtained only with the FED contribution" } ] } ], "note": [], "table": [ { "head": [ [ { "align": "center", "style": "border-top:solid;border-bottom:solid;", "data": [] }, { "align": "center", "style": "border-top:solid;border-bottom:solid;", "data": [ { "name": "text", "data": "Eq." } ] }, { "align": "center", "style": "border-top:solid;border-bottom:solid;", "data": [ { "name": "text", "data": "Donor" } ] }, { "align": "center", "style": "border-top:solid;border-bottom:solid;", "data": [ { "name": "text", "data": "Acceptor" } ] } ] ], "body": [ [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

J

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820042&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820041&type=", "width": "1.35466671", "height": "2.28600001", "fontsize": "" } } } ] } ] }, { "align": "center", "data": [ { "name": "text", "data": "18⁃20" } ] }, { "align": "center", "data": [ { "name": "text", "data": "5/2" } ] }, { "align": "center", "data": [ { "name": "text", "data": "15/2" } ] } ], [ { "align": "center", "data": [ { "name": "italic", "data": [ { "name": "text", "data": "ρ" } ] } ] }, { "align": "center", "data": [ { "name": "text", "data": "21" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0.065" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0.065" } ] } ], [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"\"\" \"‖\"U^{(2)}\"‖\" {\"\"\" separators=\"|\">\"}^{2}

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\"<\"\"\" \"‖\"U^{(6)}\"‖\" {\"\"\" separators=\"|\">\"}^{2}

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Ω_{2}

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Ω_{4}

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Ω_{6}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820006&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820005&type=", "width": "3.04800010", "height": "2.96333337", "fontsize": "" } } } ] }, { "name": "text", "data": " （10" }, { "name": "sup", "data": [ { "name": "text", "data": "-20" } ] }, { "name": "text", "data": " cm" }, { "name": "sup", "data": [ { "name": "text", "data": "2" } ] }, { "name": "text", "data": "）" } ] }, { "align": "center", "data": [ { "name": "text", "data": "18" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0.551" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0.637" } ] } ], [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"\" separators=\"|\">, r^{2}\"

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820008&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820007&type=", "width": "4.57200003", "height": "3.04800010", "fontsize": "" } } } ] }, { "name": "text", "data": " （10" }, { "name": "sup", "data": [ { "name": "text", "data": "-17" } ] }, { "name": "text", "data": " cm" }, { "name": "sup", "data": [ { "name": "text", "data": "2" } ] }, { "name": "text", "data": "）" } ] }, { "align": "center", "data": [ { "name": "text", "data": "19" } ] }, { "align": "center", "data": [ { "name": "text", "data": "1.82" } ] }, { "align": "center", "data": [ { "name": "text", "data": "1.99" } ] } ], [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"\"\" \"‖\"L + 2 S\"‖\" {\"\"\" separators=\"|\">\"}^{2}

γ

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820012&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820011&type=", "width": "1.60866666", "height": "2.53999996", "fontsize": "" } } } ] }, { "name": "text", "data": " （cm" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": "）" } ] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "23⁃24" } ] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "400" } ] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "450" } ] } ], [ { "align": "left", "style": "border-bottom:solid;", "data": [] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "Eq." } ] }, { "colspan": "2", "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "Donor⁃Acceptor" } ] } ], [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

R_{D A}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820014&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820013&type=", "width": "4.23333359", "height": "2.87866688", "fontsize": "" } } } ] }, { "name": "text", "data": " /cm" } ] }, { "align": "center", "data": [ { "name": "text", "data": "22" } ] }, { "colspan": "2", "align": "center", "data": [ { "name": "text", "data": "3.50×10" }, { "name": "sup", "data": [ { "name": "text", "data": "-8" } ] } ] } ], [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Δ

ρ_{f f}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820018&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820017&type=", "width": "2.70933342", "height": "3.38666677", "fontsize": "" } } } ] } ] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "17" } ] }, { "colspan": "2", "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "1.38×10" }, { "name": "sup", "data": [ { "name": "text", "data": "-3" } ] }, { "name": "text", "data": " at 3.50 Å" } ] } ] ], "foot": [] } ], "graphics": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820020&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820019&type=", "width": "77.69583130", "height": "77.24517822", "fontsize": "" } } }, { "name": "p", "data": [ { "name": "text", "data": "Applying the data in " }, { "name": "xref", "data": { "text": "Tab.3", "type": "table", "rid": "T3", "data": [ { "name": "text", "data": "Tab.3" } ] } }, { "name": "text", "data": " leads to:" } ] }, { "name": "dispformula", "data": { "label": [], "data": [ { "name": "math", "data": { "math": "

W_{d ⁃ d} = 5.2 \times 10^{2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820022&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820021&type=", "width": "24.12999916", "height": "4.06400013", "fontsize": "" } } }, { "name": "text", "data": " s" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": "，" } ], "id": "DF41" } }, { "name": "dispformula", "data": { "label": [], "data": [ { "name": "math", "data": { "math": "

W_{d q ⁃ q d} = 3.1 \times 10^{4}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820024&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820023&type=", "width": "26.07733345", "height": "4.06400013", "fontsize": "" } } }, { "name": "text", "data": " s" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": "，" } ], "id": "DF42" } }, { "name": "dispformula", "data": { "label": [], "data": [ { "name": "math", "data": { "math": "

W_{q ⁃ q} = 3.2 \times 10^{6}

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W_{m d ⁃ m d} = 0

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820028&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820027&type=", "width": "14.90133286", "height": "3.80999994", "fontsize": "" } } }, { "name": "text", "data": "，" } ], "id": "DF44" } }, { "name": "dispformula", "data": { "label": [], "data": [ { "name": "math", "data": { "math": "

W_{e x} = 3.6 \times 10^{6}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820030&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820029&type=", "width": "23.28333473", "height": "4.06400013", "fontsize": "" } } }, { "name": "text", "data": " s" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] }, { "name": "text", "data": "." } ], "id": "DF45" } }, { "name": "p", "data": [ { "name": "text", "data": "The total pairwise energy transfer Yb（Ⅲ） [" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{2} F_{5 / 2} \to {}^{2}F F_{7 / 2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820032&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820031&type=", "width": "18.20333290", "height": "5.58799982", "fontsize": "" } } } ] }, { "name": "text", "data": "] → Er（Ⅲ） [" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{4} I_{15 / 2} \to {}^{4}I I_{11 / 2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820034&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820033&type=", "width": "18.11866570", "height": "5.58799982", "fontsize": "" } } } ] }, { "name": "text", "data": "] pathway is " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

W_{T O T A L} = 6.9 \times 10^{6} s^{- 1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820036&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820035&type=", "width": "32.59666443", "height": "4.14866638", "fontsize": "" } } } ] }, { "name": "text", "data": ", which also reproduces the value for the shortest Yb⁃Er distance for the first upconverting step from reference" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "16", "type": "bibr", "rid": "R16", "data": [ { "name": "text", "data": "16" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": "." } ] }, { "name": "p", "data": [ { "name": "text", "data": "⋅ " }, { "name": "bold", "data": [ { "name": "text", "data": "E4" } ] } ] }, { "name": "p", "data": [ { "name": "text", "data": "Repeating the same procedure as " }, { "name": "bold", "data": [ { "name": "text", "data": "E3" } ] }, { "name": "text", "data": " but updating the acceptor transition from " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{4} I_{15 / 2} \to {}^{4}I I_{11 / 2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820038&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820037&type=", "width": "17.78000069", "height": "5.58799982", "fontsize": "" } } } ] }, { "name": "text", "data": " to " }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{4} I_{11 / 2} \to {}^{4}I I_{7 / 2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820040&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820039&type=", "width": "17.18733406", "height": "5.58799982", "fontsize": "" } } } ] }, { "name": "text", "data": ", we obtain the input data in " }, { "name": "xref", "data": { "text": "Tab.4", "type": "table", "rid": "T4", "data": [ { "name": "text", "data": "Tab.4" } ] } }, { "name": "text", "data": "." } ] }, { "name": "table", "data": { "id": "T4", "caption": [ { "lang": "en", "label": [ { "name": "text", "data": "Tab.4" } ], "title": [ { "name": "text", "data": "Input data used to calculate the Yb（Ⅲ）［" }, { "name": "sup", "data": [ { "name": "text", "data": "2" } ] }, { "name": "text", "data": "F" }, { "name": "sub", "data": [ { "name": "text", "data": "5/2" } ] }, { "name": "text", "data": "→ " }, { "name": "sup", "data": [ { "name": "text", "data": "2" } ] }, { "name": "text", "data": "F" }, { "name": "sub", "data": [ { "name": "text", "data": "7/2" } ] }, { "name": "text", "data": "］→Er（Ⅲ）［" }, { "name": "sup", "data": [ { "name": "text", "data": "4" } ] }, { "name": "text", "data": "I" }, { "name": "sub", "data": [ { "name": "text", "data": "11/2" } ] }, { "name": "text", "data": "→" }, { "name": "sup", "data": [ { "name": "text", "data": "4" } ] }, { "name": "text", "data": "I" }, { "name": "sub", "data": [ { "name": "text", "data": "7/2" } ] }, { "name": "text", "data": "］ in " }, { "name": "bold", "data": [ { "name": "text", "data": "E4" } ] }, { "name": "text", "data": ". The number of the equation where the input data enters is also indicated" } ] } ], "note": [], "table": [ { "head": [ [ { "align": "center", "style": "border-top:solid;border-bottom:solid;", "data": [] }, { "align": "center", "style": "border-top:solid;border-bottom:solid;", "data": [ { "name": "text", "data": "Eq." } ] }, { "align": "center", "style": "border-top:solid;border-bottom:solid;", "data": [ { "name": "text", "data": "Donor" } ] }, { "align": "center", "style": "border-top:solid;border-bottom:solid;", "data": [ { "name": "text", "data": "Acceptor" } ] } ] ], "body": [ [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

J

\"<\"\"\" \"‖\"U^{(2)}\"‖\" {\"\"\" separators=\"|\">\"}^{2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820044&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820043&type=", "width": "12.95400047", "height": "4.06400013", "fontsize": "" } } } ] } ] }, { "align": "center", "data": [ { "name": "text", "data": "18⁃19" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0.122 5" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0.032 0" } ] } ], [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"\"\" \"‖\"U^{(4)}\"‖\" {\"\"\" separators=\"|\">\"}^{2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820046&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820045&type=", "width": "12.95400047", "height": "4.06400013", "fontsize": "" } } } ] } ] }, { "align": "center", "data": [ { "name": "text", "data": "18" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0.409 6" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0.265 3" } ] } ], [ { "align": "center", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"\"\" \"‖\"U^{(6)}\"‖\" {\"\"\" separators=\"|\">\"}^{2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820048&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820047&type=", "width": "12.95400047", "height": "4.06400013", "fontsize": "" } } } ] } ] }, { "align": "center", "data": [ { "name": "text", "data": "18" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0.857 5" } ] }, { "align": "center", "data": [ { "name": "text", "data": "0.154 5" } ] } ], [ { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

\"<\"\"\" \"‖\"L + 2 S\"‖\" {\"\"\" separators=\"|\">\"}^{2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820050&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820049&type=", "width": "16.51000023", "height": "3.30200005", "fontsize": "" } } } ] } ] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "20" } ] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "30.86" } ] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "0" } ] } ], [ { "align": "left", "style": "border-bottom:solid;", "data": [] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "Eq." } ] }, { "colspan": "2", "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "Donor⁃Acceptor" } ] } ], [ { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

Δ /

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820052&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820051&type=", "width": "2.70933342", "height": "2.11666679", "fontsize": "" } } } ] }, { "name": "text", "data": "cm" }, { "name": "sup", "data": [ { "name": "text", "data": "-1" } ] } ] }, { "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "23， 25" } ] }, { "colspan": "2", "align": "center", "style": "border-bottom:solid;", "data": [ { "name": "text", "data": "-111" } ] } ] ], "foot": [] } ], "graphics": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820054&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820053&type=", "width": "77.69583893", "height": "42.71717072", "fontsize": "" } } }, { "name": "p", "data": [ { "name": "text", "data": "In this case, the total pairwise energy transfer of the Yb（Ⅲ） [" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{2} F_{5 / 2} \to {}^{2}F F_{7 / 2}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820056&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820055&type=", "width": "17.44133186", "height": "5.58799982", "fontsize": "" } } } ] }, { "name": "text", "data": "] → Er（Ⅲ） [" }, { "name": "inlineformula", "data": [ { "name": "math", "data": { "math": "

{}^{4} I_{11 / 2} \to {}^{4}I I_{7 / 2}

W_{T O T A L} = 5.1 \times 10^{6} s^{- 1}

", "graphicsData": { "small": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820060&type=", "big": "http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39820059&type=", "width": "32.59666443", "height": "4.14866638", "fontsize": "" } } } ] }, { "name": "text", "data": ", in agreement with the value for the shortest Yb⁃Er distance for the second upconverting step from reference" }, { "name": "sup", "data": [ { "name": "text", "data": "[" }, { "name": "xref", "data": { "text": "16", "type": "bibr", "rid": "R16", "data": [ { "name": "text", "data": "16" } ] } }, { "name": "text", "data": "]" } ] }, { "name": "text", "data": "." } ] }, { "name": "p", "data": [ { "name": "text", "data": "All these examples are worked out as separated Mathcad® script files available in the Supporting Information （Examples_Mathcad.zip）." } ] } ] }, { "name": "sec", "data": [ { "name": "sectitle", "data": { "title": [ { "name": "text", "data": "9　Conclusions" } ], "level": "1", "id": "s9" } }, { "name": "p", "data": [ { "name": "text", "data": "In this tutorial review, we presented the equations for the energy transfer rates between lanthanide ions in an alternative form, which aimed at an easier interpretation and a simpler step-by-step calculation. Four examples were provided to illustrate this procedure for relevant lanthanide pairs. We expect that the present work will guide students and stimulate the research fn the field of energy transfer between lanthanide ions." } ] } ] } ], "footnote": [], "reflist": { "title": [ { "name": "text", "data": "References" } ], "data": [ { "id": "R1", "label": "1", "citation": [ { "lang": "en", "text": [ { "name": "text", "data": "JAHNKE T" }, { "name": "text", "data": "， " }, { "name": "text", "data": "SANN H" }, { "name": "text", "data": "， " }, { "name": "text", "data": "HAVERMEIER T" }, { "name": "text", "data": "， " }, { "name": "text", "data": "et al" }, { "name": "text", "data": ". " }, { "name": "text", "data": "Ultrafast energy transfer between water molecules" }, { "name": "text", "data": " ［J］. " }, { "name": "text", "data": "Nat. 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A Tutorial Review on the Nonradiative Energy Transfer Rates between Lanthanide Ions Enhanced Publication