Specific heat and thermal expansion

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```					                 Specific heat and thermal expansion
The energy of lattice vibrations is quantized (phonons). Phonons are thermally excited
thermal phonons. Their energy can be calculated based on the harmonic oscillator model.
1
En = (n + )ℏω with n=0,1,2,3…. To get the total energy, one has to sum up N oscillators.
2

How many oscillators are excited at a given temperature and what is the mean energy at this
temperature?          Calculate the mean value of the occupancies <n>. One obtains the mean
thermal equilibrium occupancy of a certain state n as function of the thermal energy kT:
1
< n >=                   which is the Bose-Einstein distribution, valid for all quanta with an
ℏω
exp( ) − 1
kt
integer spin. Therefore the total energy of excited phonons is given by
1
E = ∑ (< n > + )ℏω (q ) .(*)
q             2
The number of allowed values of frequencies per unit volume of q space, for each branch for
a system with propagating waves within a crystal of size L³=(Na)³ is
∆ω (q ) = ( L / 2π )³ = const. . The sum of spectral number densities Sj(ω) over all the branches
ω max             3p     ω max

j within the phonon dispersion    ∫      S (ω )dω = ∑      ∫ S (ω )dω = 3 pN
j            depends on the number
0               j =1     0

of atoms per cell p and number of unit cells N. In 3D space the spectral density of states
L        dF
(DOS) Sj(ω) is given by S j (ω ) = ( )³∫∫             where dF is an area element with ω(q) =
const. and gradqω the derivative dω/dq in 3D. In general case, the last formula has no
analytic solution. Therefore two models exit:

A: Einstein model, where the optical branch is approximated by a constant
 0   ω ≠ ωE
S j (ω ) =              where ωE is the Einstein frequency.
3 pN ω = ω E
B: Debye model, where the acoustic branches are approximated by ω=vs q with vs=const. as
3N
the sound velocity up to a certain qmax. Here S j (ω ) =    ω ² (for p=1) and ω < ωD, with ωD
ωD
1
 3 Nvs3  3
being the Debye frequency given by : ωD =    ( L / 2π )²4π  .

               
Both approximations are helpful to calculate the specific heat contribution of phonons. Here
one needs to calculate the temperature dependent part of the equ.(*) which is:
ℏω
exp( )
∂E                ∞        ℏω         kT
cv = ( )V = const = k ∫ dωS (ω )( )²                  .
∂T               0         kT (exp( ℏω ) − 1)²
kT
Using Einstein model for S(ω) one gets cv = 3k for the case of high T, i.e. ℏω << kT , but a
wrong result for low temperatures. Alternatively the Debye model provides the correct
T 3 x D x 4e x
relation for the low T case: cV = 3Nk ( 3 ) ∫           where ΘD is the Debye temperature
Θ D 0 (e x − 1)²
(Temperature where all possible phonon modes are exited) and xD=ΘD/T.

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