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Physics LibreTexts

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(3b=11c3b)ω2dω.

If we define the “average sound speed” cs through the so-called “harmonic cubed average”,

1c3s133b=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(ωkBT)2eω/kBT(1ehω/kBT)2,

and using the small-ω result (7.11), we have the low-temperature result

CcrystalV=3V2π21c3skB0ω2dω(ωkBT)2eω/kBT(1eω/kBT)2.

For our first step, avoid despair — instead convert to the dimensionless variable

x=ωkBT

and find

C crystal V=3V2π21c3skB(kBT)30x4ex(1ex)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, CVT3.

However, if you want to chase down the right numbers, after some fiddling you’ll find that

0x4ex(1ex)2dx=4Γ(4)ζ(4)=415π4.

Thus, the low-temperature specific-heat of a solid due to a lattice vibration is

CcrystalV=kBV2π25(kBTcs)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.)


This page titled 7.5: Low-temperature Heat Capacity is shared under a CC BY-SA license and was authored, remixed, and/or curated by Daniel F. Styer.

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