PHY.K02UF Molecular and Solid State Physics

Phonon dispersion of an fcc crystal

The eigen value problem that must be solved to determine the fcc phonon dispersion relation is, (show derivation)

\begin{equation} \left[ \begin{array}{c} 4 - \cos\left(\frac{k_xa}{2} + \frac{k_ya}{2}\right) - \cos\left(\frac{k_xa}{2} + \frac{k_za}{2}\right) - \cos\left(\frac{k_xa}{2} - \frac{k_ya}{2}\right) - \cos\left(\frac{k_xa}{2} - \frac{k_za}{2}\right) & - \cos\left(\frac{k_xa}{2} + \frac{k_ya}{2}\right) + \cos\left(\frac{k_xa}{2} - \frac{k_ya}{2}\right) & - \cos\left(\frac{k_xa}{2} + \frac{k_za}{2}\right) + \cos\left(\frac{k_xa}{2} - \frac{k_za}{2}\right) \\ - \cos\left(\frac{k_xa}{2} + \frac{k_ya}{2}\right) + \cos\left(\frac{k_xa}{2} - \frac{k_ya}{2}\right) & 4 - \cos\left(\frac{k_xa}{2} + \frac{k_ya}{2}\right) - \cos\left(\frac{k_ya}{2} + \frac{k_za}{2}\right) - \cos\left(\frac{k_xa}{2} - \frac{k_ya}{2}\right) - \cos\left(\frac{k_ya}{2} - \frac{k_za}{2}\right) & - \cos\left(\frac{k_ya}{2} + \frac{k_za}{2}\right) + \cos\left(\frac{k_ya}{2} - \frac{k_za}{2}\right) \\ - \cos\left(\frac{k_xa}{2} + \frac{k_za}{2}\right) + \cos\left(\frac{k_xa}{2} - \frac{k_za}{2}\right) & - \cos\left(\frac{k_ya}{2} + \frac{k_za}{2}\right) + \cos\left(\frac{k_ya}{2} - \frac{k_za}{2}\right) & 4 - \cos\left(\frac{k_xa}{2} + \frac{k_za}{2}\right) - \cos\left(\frac{k_ya}{2} + \frac{k_za}{2}\right) - \cos\left(\frac{k_xa}{2} - \frac{k_za}{2}\right) - \cos\left(\frac{k_ya}{2} - \frac{k_za}{2}\right) \end{array} \right] \left[ \begin{array}{b} u_{\vec{k}}^x \\ u_{\vec{k}}^y \\ u_{\vec{k}}^z \end{array} \right]=\frac{M\omega^2}{C} \left[ \begin{array}{b} u_{\vec{k}}^x \\ u_{\vec{k}}^y \\ u_{\vec{k}}^z \end{array} \right] \end{equation}

The eigen values $\lambda = \frac{M\omega^2}{C}$ can be found by setting the determinant of the matrix to zero.

\begin{equation} \left| \begin{array}{c} 4 - \cos\left(\frac{k_xa}{2} + \frac{k_ya}{2}\right) - \cos\left(\frac{k_xa}{2} + \frac{k_za}{2}\right) - \cos\left(\frac{k_xa}{2} - \frac{k_ya}{2}\right) - \cos\left(\frac{k_xa}{2} - \frac{k_za}{2}\right) -\lambda & - \cos\left(\frac{k_xa}{2} + \frac{k_ya}{2}\right) + \cos\left(\frac{k_xa}{2} - \frac{k_ya}{2}\right) & - \cos\left(\frac{k_xa}{2} + \frac{k_za}{2}\right) + \cos\left(\frac{k_xa}{2} - \frac{k_za}{2}\right) \\ - \cos\left(\frac{k_xa}{2} + \frac{k_ya}{2}\right) + \cos\left(\frac{k_xa}{2} - \frac{k_ya}{2}\right) & 4 - \cos\left(\frac{k_xa}{2} + \frac{k_ya}{2}\right) - \cos\left(\frac{k_ya}{2} + \frac{k_za}{2}\right) - \cos\left(\frac{k_xa}{2} - \frac{k_ya}{2}\right) - \cos\left(\frac{k_ya}{2} - \frac{k_za}{2}\right) -\lambda & - \cos\left(\frac{k_ya}{2} + \frac{k_za}{2}\right) + \cos\left(\frac{k_ya}{2} - \frac{k_za}{2}\right) \\ - \cos\left(\frac{k_xa}{2} + \frac{k_za}{2}\right) + \cos\left(\frac{k_xa}{2} - \frac{k_za}{2}\right) & - \cos\left(\frac{k_ya}{2} + \frac{k_za}{2}\right) + \cos\left(\frac{k_ya}{2} - \frac{k_za}{2}\right) & 4 - \cos\left(\frac{k_xa}{2} + \frac{k_za}{2}\right) - \cos\left(\frac{k_ya}{2} + \frac{k_za}{2}\right) - \cos\left(\frac{k_xa}{2} - \frac{k_za}{2}\right) - \cos\left(\frac{k_ya}{2} - \frac{k_za}{2}\right) -\lambda \end{array} \right|=0 \end{equation}

The matrix is symmetric with six independent elements. It has the form,

\begin{equation} \left| \begin{array}{c} m_{11} -\lambda & m_{12} & m_{13} \\ m_{12} & m_{22} -\lambda & m_{23} \\ m_{13} & m_{23} & m_{33} -\lambda \end{array} \right|=0 \end{equation}

This can be written as a cubic equation in $\lambda$.

$$-\lambda^3 +(m_{11}+m_{22}+m_{33})\lambda^2+(m_{12}^2+m_{13}^2+m_{23}^2 - m_{11}m_{22} - m_{11}m_{33} - m_{22}m_{33})\lambda + m_{11}m_{22}m_{33} +2m_{12}m_{13}m_{23} - m_{12}^2m_{33} - m_{13}^2m_{22}-m_{23}^2m_{11} =0$$

Cubic equations can be solved using Cardano's formula. The standard form of a cubic equation is,

$$a\lambda^3 + b\lambda^2 + c\lambda + d =0$$

Code that will calculate the coefficients $a,b,c,d$ and the matrix elements from the $\vec{k}$-vector is,

$k_xa=$  $k_ya=$  $k_za=$

$\sqrt{\frac{M}{C}}\omega = $



$\sqrt{M/C}\omega_1$

 $\sqrt{M/C}\omega_2$ 

$\sqrt{M/C}\omega_3$

 $\Gamma: 0,0,0$ 

0

0

0

$X: \frac{2\pi}{a},0,0$

$2$

$2$

$2\sqrt{2}$

$L: \frac{\pi}{a},\frac{\pi}{a},\frac{\pi}{a}$

$\sqrt{2}$

$\sqrt{2}$

$2\sqrt{2}$

$W: \frac{2\pi}{a},\frac{\pi}{a},0$

$2$

$\sqrt{6}$

$\sqrt{6}$

$U: \frac{2\pi}{a},\frac{\pi}{a},\frac{\pi}{a}$

$\sqrt{2}$

$2$

$\sqrt{6}$

$K: \frac{3\pi}{2a},\frac{3\pi}{2a},0$

$1.8477$

$2.3268$

$2.6131$