Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Phymcs 251

Solution:
Graphite is an insulator and its specific heat is contributed entirely by
the crystal lattice. When the temperature T increases from zero, the vi-
brational modes corresponding to the motion between layers is first excited
since the coupling between the carbon atoms in different layers is much
weaker. By the Debye model, we have

w=ck.

The number of longitudinal waves in the interval k to k+dk is (L/2~)~2~kdk,

where L is the length of the graphite crystal. From this, we obtain the

number of the longitudinal waves in the interval w to w + dw, L2wdw/2xci,

where CII is the velocity of longitudinal waves. Similarly, the number of

transversal waves in the interval w to w + dw is ----,

L2wdw

TC?

Therefore, the Debye frequency spectrum is given by

w < WD (Debye frequency).

where

hW

z=-- XD = __ twD , kB being Boltzmann’s constant.

kBT’ kB T

At low temperatures, ~WD >> kBT, i.e., XD >> 1, then,

O0 x3ez

dx