Problem 2: CaTiO₃

CaTiO₃ has a cubic perovskite crystal structure. The lattice contants are given in the figure.

(a) A crystal of CaTiO₃ is  mm ×  mm ×  mm. How many $k$ states are there in the first Brillouin zone?

Number of $k$ states =

(b) How many optical branches of there of the phonon dispersion of CaTiO₃ ?

Number of optical branches =

(c) How many phonon modes does this crystal have?

phonon modes =

(d) The speed of sound at low frequencies is $c=$  m/s. What is the the wavelength of a phonon mode with a frequency of $\omega = $ 0000 rad/s travelling in the [111] direction?

$\lambda =$ m

(e)At  K, what is the mean number of phonons in a phonon state with a frequency of $\omega = $  $\times 10^{12}$ rad/s?

phonons =

(f) What is the length of the reciprocal lattice vector $\vec{G}$?

$|\vec{G}|=$ m^-1

(g) A electron is in $k$-state $\vec{k}= $  $\hat{k}_x$ m^-1. The wavefunction at point $\vec{r}$ is $\psi (\vec{r}) = \psi_0$. What is the wavefunction at $\vec{r}+a\,\hat{x}$? Here $a$ is the lattice constant.

$\psi (\vec{r}+a\,\hat{x}) =$ $\psi_0$ + $i$ $\psi_0$