Gold has a Debye temperature (a given number)

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Lattice dynamics and thermal properties of solids · Lattice dynamics and thermal properties of solids

Thermal Physics, Ch Photons and phonons: blackbody radiation and the heat capacity of solids

Problem

Gold has a Debye temperature

\theta_D = 165\,\mathrm{K}

(a given number). Model the metal as

N

atoms in a volume

V

carrying standing sound waves with three polarisations and a single speed

c_s

, with

\nu = (c_s/2a)\sqrt{n_x^2+n_y^2+n_z^2}

in a cube of edge

a

and a sharp cutoff at

3N

modes; each mode is a quantised oscillator of mean energy

h\nu/(e^{h\nu/k_BT}-1)

with the zero-point energy dropped. The conduction electrons are ignored throughout. ((a)) Starting from the mode count, derive the low-temperature heat capacity

C_V = \frac{12\pi^4}{5}Nk_B\left(\frac{T}{\theta_D}\right)^3 \qquad (T \ll \theta_D),

using the stated integral

\int_0^\infty x^3/(e^x-1)\,\mathrm{d}x = \pi^4/15

, and say at which step the cutoff

\nu_D

stops acting as the upper limit of the integral, even though it survives in the answer through

\theta_D

. ((b)) Evaluate

C_V

for one mole of gold at

T = 8.0\,\mathrm{K}

. ((c)) At what temperature does the lattice heat capacity of gold reach one per cent of its classical value

3Nk_B

? Comment on whether the

T^3

law may be trusted there.

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