Phonons

Thermal Properties

Phonon Heat Capacity

References

• Kittel, “Introduction to Solid State Physics”, Chapter 5

Using the more fundamental heat capacity, with respect to constant volume $C_V=(\frac{\partial U}{\partial T}{)}_V$ Contributions of phonons to heat capacity of a crystal is called lattice heat capacity.

The total energy of phonons at a temperature $\tau =k_{b}T$ may be written as a sum of energy over all phonon modes, eg. a superposition of wave modes. Must sum over allowed values of K and polarizations (to define dimensionality).

$$U_{lat}=\sum _K\sum _p{U}_{K,p}=\sum _k\sum _p⟨n_{K,p}⟩\hslash {\omega }_{K,p}$$

$⟨n_{K,p}⟩$ is the average occupation number of a certain phonon polarization and k-value in thermal equilibrium. It is given by the Planck distribution

$$⟨n⟩=\frac{1}{exp(\hslash \omega /\tau )-1}$$

Normal Mode Enumeration

$U_{lat}={\sum }_K{\sum }_p{U}_{K,p}={\sum }_k{\sum }_p\frac{\hslash {\omega }_{K,p}}{\exp (\hslash {\omega }_{K,p}/\tau )-1}$ Replace summation over K by integral. Crystal has $D_p(\omega )d\omega$ modes of a given polarization in the frequency range $\omega$ to $\omega +d\omega$.

$$U_{lat}=\sum _p\int d{\omega }_p{D}_p(\omega )\frac{\hslash \omega }{\exp (\hslash \omega /\tau )-1}$$

Lattice heat capacity found by differentiating above expression with respect to T. Define $x=\hslash \omega /\tau =\hslash \omega /k_{B}T$

$$C_{lat}=k_B\sum _p\int d{\omega }_p{D}_p(\omega )\frac{x^2\exp (x)}{(\exp (x)-1{)}^2}$$

$D(\omega )$ is often called density of states

Density of States in One Dimension

Boundary value problem for vibrations of a one-dimensional line of length $L$ carrying $N+1$ particles of separation $a$. Particles at $s=0$ and $s=N$, first and last, are fixed in place. Each normal vibrational mode of certain polarization has form of standing wave.

$$u_s=u(0)e^{-i{\omega }_{K,p}t}\sin (Ksa)$$

Boundary conditions require $sin(Ksa)=0$ for $s=N$ and $s=0$. The condition is satisfied for $Ksa=n\pi$ so that $K=n\pi /Na$ where $Na=L$. Of the N+1 atoms only N-1 are allowed to move.

$$K=\frac{n\pi }{L},n=1\cdots N-1$$

In one dimension there is one mode per $\Delta K=\pi /L$ so that the number of modes per unit range of K is,

$$\text{modes}=L/\pi \cdots \mathrm{for}n\le N\mathrm{and}K\le \pi /a$$

In one dimension there is 3 polarizations for each K, one longitudinal and two transverse. The number of states in one dimension is

$$D_1(\omega )d\omega =\frac{L}{\pi }\frac{dK}{d\omega }d\omega =\frac{L}{\pi }\frac{d\omega }{d\omega /dK}$$