Recently

there present many works to discuss the phase transition-like behaviour of the polaron in polar crystals. However

all these works didn't consider the incorporate effect of the temperature and electron-phonon Frohlich Coupling constant r. In this paper

starting from the Bogolyubov inequality for the free energy

we discuss this incorporate effect on the phase transition-like behaviour of the polaron.The Hamiltonian of a system consisting of the electrons and phonons is shown as (2.1) in the text. And assume that the Bogolyubov ineqality between the real free energy

F(H)

and the model free energy

F

mod

(H)

also holds for the polaron. The Bogolyubov ineqality has the form (2.4) in the text. In this representation

H

can be regarded as the sum of

H

0

and a perturbed term (

H-H

0

).

H

0

is represented [by (2.8) which is called the model Hamiltonian. In the course of calculation

we introduce two unitary transformations U

1

and U

2

. Then

applying these transformations to (2.4)

we finally obtain the expression (2.27) from which the free energy can be determinded. Lagrange multiplier

u

in (2.27) has the meaning of the translation velocity of the polaron. The effective mass of the polaron and the mean numbers of phonons N in the cloud around the electron are

respectively

represented by (3.5) and (3.6). The expectation value of the polaron energy with respect to the ground state |0

>

at the different temperature is expressed by (3.4).