PX284 - K1 - einstein model

considering lattice vibrations in a crystalline solid, eg: sound waves
thinking about normal modes of masses connected by springs
at high temperatures: $C_{V} \to 3 N k_{B}$

assumptions

each atom is an independent simple harmonic oscillator
- lattice vibrations are quantized, phonons
- in a crystal of $N$ atoms, there are $3 N$ independent modes
all atoms oscillate with the same frequency, $ω_{E}$
- making a very simple DOS
- thus, $Z = (Z_{S H O})^{3 N}$

U = 3 N U_{S H O} = 3 N ℏ ω_{E} (\frac{1}{2} + \frac{1}{\exp (β ℏ ω_{E}) - 1})

ignoring ZPE and replacing $ℏ ω_{E}$ with $k_{B} θ_{E}$ , where, $θ_{E}$ is the einstein temperature

U = \frac{3 N k_{B} θ_{E}}{\exp (β ℏ ω_{E}) - 1}

\begin{aligned} C_{V} & = {(\frac{\partial U}{\partial T})}_{V} \\ = 3 N k_{B} \frac{x^{2} e^{x}}{(e^{x} - 1)^{2}} \end{aligned}

where, $x = θ_{E} / T$

at low $T : x \to \infty, C_{V} \to 3 N k_{B} x^{2} e^{- x}$
at high $T : x \to 0, e^{x} \to 1 + x, C_{V} \to 3 N k_{B}$
this works really well at high and intermediate temperatures, but usually not great at low temperatures