With the development of the solid science and experimental technology

the properties of the surface polaron in crystals have been of considerable interest.Many investigators studied the properties of the effective mass of the surface polaron by the method of a perturbation

a strong-coupling variational wavefunction

a linear-combination operator and a Green-function et al.However

the research of these methods only was restricted to the case of crystals having only one mode of the longitudinal optical(LO)phonon.However

a large number of polar crystals

with several atoms per unit cell

have more than one LO phonon branch.For example

in cuprite such as CuO

2

there are two LO phonon modes and a large number of perovskites (SrTiO

3

BaTiO

3

LiNbO

3

…)have more than two modes.In recent years the polaron problem with many LO phonon branches has been studied.The properties of polaron in polyatomic polar crystals are discussed by the method of a linear combination operator and a simple unitary transformation by one of the authors.However

the influences of the electron interaction with both bulk longitudinal optical phonons and surface optical(SO) phonons on the properties of the surface polaron in polyatomic crystals has not been investigated so far.In this paper

the effective mass of the surface polaron in polyatomic semi-infinite crystals

weak coupling with both bulk longitudinal optical and surface optical phonons

are studied using a linear combination operator and Lagrange multiplier method.