PX284 - K2 - debye model

assumptions

lattice vibrations are waves with dispersion relation:

ω = v_{s} q

where, $v_{s}$ is the speed of sound, and $q$ is the wavevector

thus, the DOS is given by:

g (q) d q = \frac{V q^{2}}{2 π^{2}} d q \times 3

the factor of $3$ corresponds to three polarizations of vibration - 2 transverse and a longitudinal

g (ω) d ω = \frac{3 V ω^{2}}{2 π^{2} v_{S}^{2}} d ω

debye also assumed a cut-off frequency, $ω_{D}$ , so as to enforce $3 N$ modes

\begin{aligned} \int_{0}^{ω_{D}} g (ω) d ω & = 3 N \\ \frac{V ω_{D}^{3}}{2 π^{2} v_{s}^{3}} & = 3 N \\ ω_{D} & = {(\frac{6 N π^{2} v_{s}^{3}}{V})}^{1 / 3} \\ or, g (ω) d ω & = \frac{9 N ω^{2}}{ω_{D}^{3}} d ω \end{aligned}

defining debye temperature, $θ_{D} = ℏ ω_{D} / k_{B}$

U = \int_{0}^{ω_{D}} g (ω) U_{S H O} d ω

ignoring ZPE:

U = \frac{9 N ℏ}{ω_{D}^{3}} \int_{0}^{ω_{D}} \frac{ω^{3}}{\exp (β ℏ ω) - 1} d ω

C_{V} = \frac{9 N ℏ}{ω_{D}} \int_{0}^{x_{D}} \frac{x^{4} \exp (x)}{(\exp (x) - 1)^{2}} d x

where $x = β ℏ ω$ and $x_{D} = β ℏ ω_{D} = θ_{D} / T$

$T \to 0 : x_{D} \to \infty$
integral is standard $= 4 π^{4} / 15$

C_{V} = \frac{12 π^{4}}{5} N k_{B} {(\frac{T}{θ_{D}})}^{3}

ie: $C_{V} \sim T^{4}$ at low temperatures

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