A model crystal of atoms fills a cube of edge (volume ) and supports only…

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 model crystal of

N

atoms fills a cube of edge

a

(volume

V = a^3

) and supports only longitudinal sound waves: there is one polarisation, and every such wave travels at the same speed

c_L = 5.0\,\mathrm{km\,s^{-1}}

(a given number). A standing wave has

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

with

n_x, n_y, n_z

positive integers. The atom number density is

n = N/V = 4.0\times10^{28}\,\mathrm{m^{-3}}

. Because there is a single branch, the count of modes is cut off at exactly

N

modes. The crystal is weakly coupled to a reservoir at temperature

T

; each mode is a quantised oscillator of mean energy

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

, with the zero-point energy separated off and dropped. Any contribution from conduction electrons is ignored. ((a)) Derive the density of modes

g(\nu)

for this crystal, naming the origin of every factor. ((b)) Derive the cutoff frequency

\nu_D

from the condition that there are

N

modes in all, and evaluate

\nu_D

and

\theta_D = h\nu_D/k_B

. ((c)) Derive

C_V

for

T \ll \theta_D

in the form (number)

\,\times Nk_B(T/\theta_D)^3

, using the stated integral

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

, and evaluate it per mole at

T = 6.0\,\mathrm{K}

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