Although a discussion of photons is not a traditional part of an introductory solid-state physics course, bosons play an important role in solids. Phonons, plasmons, magnons, and Cooper pairs are bosons, so it would make sense to introduce the noninteracting boson gas in the introductory course in much the same way that the noninteracting fermi gas for electrons is introduced. There is not much additional effort required to do this because the counting of the modes is the same. From the photon density of states, you can directly derive some famous results such as the Planck radiation curve, the Stefan-Boltzmann law, and Wien's law.
A discussion of photons is found in the section on the quantization of the electromagnetic field. First, the photon modes are found. Photons are put in a box with a size $L_x\times L_y \times L_z$ and periodic boundary conditions are applied. To satisfy the boundary conditions, the photons must have wave vectors where $k_x = 2\pi n_x/L_x$, $k_y = 2\pi n_y/L_y$, $k_z = 2\pi n_z/L_z$ where $n_x,n_y,n_z$ are integers. The length of the wave vector is $|\vec{k}| = 2\pi \sqrt{\left(\frac{n_x}{L_x}\right)^2+\left(\frac{n_y}{L_y}\right)^2+\left(\frac{n_z}{L_z}\right)^2}$. The wavelength of this mode is $\lambda = 2\pi /|\vec{k}|$ and the energy of this mode is $E=hc/\lambda$. Here $h$ is Planck's constant and $c$ is the speed of light.
Once all of the modes are known, we can calculate the occupation of the modes using the Bose-Einstein factor, \[ \begin{equation} f_{BE}(E)=\frac{1}{\exp\left(\frac{E}{k_BT}\right)-1}. \end{equation} \]$f_{BE}$ is the mean number of photons in a mode with energy $E$. Here $k_B$ is Boltzmann's constant, and $T$ is the temperature in Kelvin. By summing over all modes we can calculate the total energy in the collection of photons. Then by using some relations from thermodynamics, it is possible to calculate other properties like the specific heat and the radiation pressure.
The density of states $D(k)$ in 1, 2, and 3 dimensions is the same for light, sound, and electrons. The calculation of the thermodynamic properties is almost the same in all three cases. The results are tabulated for a system of noninteracting photons and a system of noninteracting electrons.
There are also more advanced discussions of the thermodynamic properties of noninteracting photons based on statistical physics and the thermodynamic properties of noninteracting electrons based on statistical physics.
Calculate the density of states $D(k)$ in one dimension. From the density of states and the dispersion relation for photons, $\omega = ck$, determine the wavelength spectral density in one dimension $u(\lambda )$.
(a) Photons are confined to a box $20\,\mu\text{m} \times 30\,\mu\text{m} \times 40\,\mu \text{m}$. How many allowed $k$-states are there in the visible range ($\lambda =$ 390 nm - 700 nm)?
(b) For which wavelength in the visible range is the density of states $D(\lambda )$ the lowest?
(c) What are the units of the density of states $D(E)$ and $D(\lambda )$ in three dimensions?
(a) 1401658
When light travels through a periodic material where the speed of light is a periodic function of space, we proceed the same way as before, first calculating the allowed modes and then using the Bose-Einstein factor to determine the occupation of the modes. The calculation of the modes for a periodic material is difficult. For a layered material, the solutions can be calculated analytically. For other cases, the plane wave method can be used to find the modes. A result that is found is that there are certain frequency bands where there are no propagating photon modes. These forbidden frequency bands are called photonic band gaps. Given the photon density of states, you should be able to calculate thermodynamic properties like the specific heat. Links to programs that calculate these thermodynamic properties numerically are found in the left column of the table of photonic crystals.
(a) Show that $\vec{k}$ vectors on the Brilloun zone boundaries satisfy the diffraction condition $\Delta \vec{k} = \vec{G}$. Derive Bragg's law $2d\sin\theta = n\lambda$ from the diffraction condition.
(b) A silicon crystal with a diamond crystal structure (fcc) is etched so that it forms a periodic structure with a hexagonal Bravais lattice. The period of the etched structure is 1000 times the period of the lattice constant of silicon. Is the fcc Bravais lattice relevant for calculating the photonic bandgap of the crystal? Explain your answer.
The photon density of states of a one-dimensional photonic crystal is given below. What is the photon contribution to the specific heat at 300 K?
$c_v = 1.05 \times 10^{-17}$ J m-1 K-1.