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Chapter 5: Thermal Properties of Crystals
The heat capacity at constant volume is defined
as CV =(¶U/¶T)V where U is the energy and T the
temperature.
The contribution of the phonons to the heat capacity
of a crystal is called the lattice heat capacity and
is denoted by Clat .
The total energy of the phonons at a temperature T
(=k T) in a crystal is the sum of the energy over all
B
phonon modes:
I. Phonon heat capacity
U = S S U = S S (<n >+1/2) ħ w ,
K p K,p K p K,p K,p
where <nK,p> is the thermal equilibrium occupancy
of phonons of wavevector K and polarization p.
The form of <nK,p> is given by the nck distri-
bution function:
w
<n >=1 exp( ) -1
k T
B
where the <…> denotes the average in thermal
equilibrium.
1. nck Distribution
分析对象: a set of identical harmonic oscillators in
thermal equilibrium.
基本思路:t=kB T
1. 第n个激发的量子态,其几率: N ∝exp(-nħw / t)
n
2. The ratio of the number of oscillators in their
(n+1)th quantum state of excitation to the
number in the nth quantum state is
N / N =exp(- ħ w/ t) (Boltzmann factor),
n+1 n
The fraction of the total number of oscillators in the nth
quantum state is:
N / S N = exp(-nħw / t)/ S p(-sħw / t)
n s s
1. nck Distribution
The average excitation quantum number of an
oscillator is (一个谐振子的平均量子数)
p( -sw/ k T )
s B 分母是一个等比数列,
<n >= exp(-sw/ k T ) 且比例小于1
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