A crystal of atoms in a volume (a cube of edge ) is treated by the Debye model:…

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

A crystal of

N

atoms in a volume

V

(a cube of edge

a

) is treated by the Debye model: standing sound waves with three independent polarisations, all travelling at one speed

c_s = 4.0\,\mathrm{km\,s^{-1}}

, with

\nu = \frac{c_s}{2a}\sqrt{n_x^2+n_y^2+n_z^2}

for positive integers

n_x,n_y,n_z

, and a sharp cutoff at

3N

modes. The atom number density is

n = N/V = 5.0\times10^{28}\,\mathrm{m^{-3}}

. Each mode is a quantised oscillator of mean energy

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

(zero-point energy dropped), and the crystal is weakly coupled to a reservoir at

T = 100\,\mathrm{K}

. ((a)) Derive the spectral energy density of the phonon gas,

u(\nu,T) = \frac{12\pi h\nu^3/c_s^3}{e^{h\nu/k_BT}-1}\qquad (\nu \le \nu_D),

tracing every factor to its step. ((b)) Evaluate

\nu_D

and confirm that

\nu = 2.0\,\mathrm{THz}

lies inside the allowed range. ((c)) Evaluate

u(\nu,T)

\nu = 2.0\,\mathrm{THz}

and

T = 100\,\mathrm{K}

. ((d)) What fraction of the classical share

k_BT

does one mode at

2.0\,\mathrm{THz}

hold at this temperature?

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