A cubic crystal is modelled with two Einstein groups of modes

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

Solid State Physics, Ch Anharmonicity: expansion, phonon lifetimes and thermal conductivity

Problem

A cubic crystal is modelled with two Einstein groups of modes. Group A is one quarter of all modes, at

\hbar\omega_A=15\,\mathrm{meV}

, with mode Gruneisen parameter

\gamma_A=-1.3

. Group B is the remaining three quarters, at

\hbar\omega_B=48\,\mathrm{meV}

, with

\gamma_B=+0.95

. The molar heat capacity of a fraction

f_i

of the

3N

modes, each an Einstein oscillator at

\hbar\omega_i

, is

C_i=3Rf_i\dfrac{x_i^2e^{x_i}}{(e^{x_i}-1)^2}

with

x_i=\hbar\omega_i/k_BT

. Take

3B_TV_m=3.54\times10^6\,\mathrm{J\,mol^{-1}}

for this crystal. ((a)) Compute

C_A

and

C_B

T=90\,\mathrm{K}

. ((b)) Hence find the predicted linear thermal expansion coefficient

\alpha

90\,\mathrm{K}

and state its sign.

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