7.5: Low-temperature Heat Capacity
( \newcommand{\kernel}{\mathrm{null}\,}\)
If
G(ω)dω= number of normal modes with frequencies from ω to ω+dω
then
Ecrystal=∫∞0G(ω)eSHO(ω)dω and CcrystalV=∫∞0G(ω)cSHOV(ω)dω
and so forth.
Density of modes:
G(ω)dω=∑ branches [ vol. of shell in k -space ] (density of modes in k -space) =∑ branches [4π(kb(ω))2dkb](V8π3)
This formula holds for any isotropic dispersion relation kb(ω). For small values of ω the dispersion relation for each branch is linear (with sound speed cb) so
kb=ωcb and dkb=dωcb,
whence
G(ω)dω=∑ branches [4π(ωcb)2dωcb](V8π3)
=V2π2(3∑b=11c3b)ω2dω.
If we define the “average sound speed” cs through the so-called “harmonic cubed average”,
1c3s≡133∑b=11c3b,
then we have the small-ω density of modes
G(ω)dω=3V2π2ω2c3sdω.
At any temperature,
C crystal V=∫∞0G(ω)cSHOV(ω)dω.
Recall from equation (5.78) that
cSHOV(ω)=kB(ℏωkBT)2e−ℏω/kBT(1−e−hω/kBT)2,
and using the small-ω result (7.11), we have the low-temperature result
CcrystalV=3V2π21c3skB∫∞0ω2dω(ℏωkBT)2e−ℏω/kBT(1−e−ℏω/kBT)2.
For our first step, avoid despair — instead convert to the dimensionless variable
x=ℏωkBT
and find
C crystal V=3V2π21c3skB(kBTℏ)3∫∞0x4e−x(1−e−x)2dx
The integral is rather hard to do, but we don’t need to do it — the integral is just a number. We have achieved our aim, namely to show that at low temperatures, CV ∼ T3.
However, if you want to chase down the right numbers, after some fiddling you’ll find that
∫∞0x4e−x(1−e−x)2dx=4Γ(4)ζ(4)=415π4.
Thus, the low-temperature specific-heat of a solid due to a lattice vibration is
CcrystalV=kBV2π25(kBTℏcs)3.
7.3 How far do the atoms vibrate?
Consider a simplified classical Einstein model in which N atoms, each of mass m, move classically on a simple cubic lattice with nearest neighbor separation of a. Each atom is bound to its lattice site by a spring of spring constant K, and all the values of K are the same. At temperature T, what is the root mean square average distance of each atom from its equilibrium site? (Note: I am asking for an ensemble average, not a time average.)