Random frequency modulations exist in many systems

such as magnetic system and glasses at low temperature

dye-solution. In magnetic systems

spin flips of doped ions or nuclei in the host lattice can cause the magnetic fluctuations at the impurity site

and hence the stochastic change of the transition frequency of the impurity ions. It is the main source of doped ions’ optical dephasing at low temperature. Because the interactions have various time scales and strengths

they can lead to a time-dependent broadening spectral line (heterogeneous line)

the process is socalled spectral diffusion. Several theoretical methods have been used to explain this phenomenon. They all divided the interactions into only two time scales

the fast one results from spin flipping of the bulk nuclei

and the slow one comes from the contributions of the nearby nuclear spins in a frozen core. However

to describe the complete dynamic process in practical systems

these two time scales are obviously not enough. In addition

spectral diffusion is a typical frequency domain process

but almost no theoretical work started from the pure frequency domain point.In this paper

the spectral diffusion in paramagnetic ions doped system at high field

in which has the modulation-frequency-dependent jump rates

is studied in both frequency and time domains. In general

the spectral diffusion goes as [1-exp(-

WT

w

)]

x

. Without frozen core

the values of

x

are 1/2 and 1 for frequency and time domains

respectively. Considering frozen core effect

the spectral diffusions are slow down by the factor of (ω

eff

/ω

0

)

2

where ω

0

is the frequency characterized frozen core

and ω

eff

are ω

max

and

H

∞

for frequency and time domains

respectively. When the frozen core is large enough

the spectral diffusions in two domains tend to have the same form as

x

≈0.22.