PHY.K02UF Molecular and Solid State Physics
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PHY.K02UF Molecular and Solid State Physics | ||||
Two important many-particle quantum systems are noninteracting fermions and noninteracting bosons. Noninteracting fermions are used as a simple model for electrons in metals which will be discussed in the section on electrons. Noninteracting bosons are used to describe photons (particles of light), phonons (particles of sound), plasmons (particles of plasma waves), and other bosons that inhabit solids. We will start with a description of photons in solids because photons are very nearly noninteracting and they are well described by a system on noninteracting particles.
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 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.
You should know how to calculate the density of states $D(k)$ in 1, 2 and 3 dimensions. You should know how to calculate the macroscopic properties (like the specific heat) of a gas of noninteracting photons. You should be able to explain how to calculate the thermodynamic quantities listed in the table summarizing the results of the quantization of the wave equation.
6.1 (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?
6.2 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 $u(\lambda )$.
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.
6.x (a) Show that $\vec{k}$ vectors on the Brilloun zone boundaries satisfy the diffraction condition $\Delta \vec{k} = \vec{G}$. The reciprocal lattice has inversion symmetry so if $\vec{k}'-\vec{k}$ is a reciprocal lattice vector so is $\vec{k}-\vec{k}'$. Draw $\vec{k}-\vec{k}'=\vec{G}$ with $\vec{k}$ and $\vec{G}$ starting from the origin. Then construct the Brillouin zone boundary that cooresponds to the reciprocal lattice vector.
(b) A silicon crystal with a diamond crystal structure (fcc) is etched so that it forms a periodic structure with a hexagonal Bravais lattice. A 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.
6.3 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?