Chapter 5, Thermal Properties Of Solids Video Solutions, Elements of Solid State Physics

Problem 1

The thermal conductivity maximum of a synthetic sapphire sample of $3 \mathrm{~mm}$ diameter is observed at $30 \mathrm{~K}$. The measured maximum value is $2.7 \times 10^{4} \mathrm{~W} \mathrm{~m}^{-1} \mathrm{~K}^{-1}$. If the speed of sound in sapphire is $10^{4} \mathrm{~m} \mathrm{~s}^{-1}$, calculate the heat capacity of the sapphire at $30 \mathrm{~K}$.

Suzanne W.

Numerade Educator

09:44

Problem 2

Obtain expressions for the thermal energy and the phonon heat capacity of
(a) a system of two harmonic oscillators, and
(b) a system with two energy levels. Explain the difference in the two results.

Tianyu Li

Numerade Educator

01:46

Problem 3

Derive an expression for the temperature at which the thermal lattice energy is equal to the zero point energy in the Einstein model. Write down the corresponding condition in the Debye model.

Hunza Gilgit

Numerade Educator

18:19

Problem 4

Using the Debye approximation for a one-dimensional monatomic lattice with atomic spacing $a$ and sound speed $v$, show that
$$
\omega_{\mathrm{D}}=\frac{\pi v}{a} \quad \text { and } \quad \theta_{\mathrm{D}}=\frac{\hbar \omega_{\mathrm{D}}}{k_{\mathrm{B}}}
$$

Dr. Rajveer Singh

Numerade Educator

08:37

Problem 5

Derive the integral expressions for the thermal energy and phonon heat capacity. Show that
$C_{V}=\frac{\pi^{2} k_{\mathrm{B}}}{3 a}\left(\frac{T}{\theta_{\mathrm{D}}}\right)$ per unit length at low temperatures
$=\frac{k_{\mathrm{B}}}{a}$ per unit length at high temperatures.

Crystal Wang

Numerade Educator

02:04

Problem 6

Solve the above problem for a two-dimensional crystal and show that at low temperatures $C_{V} \propto T^{2}$

Manik Pulyani

Numerade Educator

07:12

Problem 7

Consider the dispersion relation (4.4) for a one-dimensional monatomic lattice of $N$ atoms. Show that the density of normal modes is given by
$$
D(\omega)=\frac{2 N}{\pi} \frac{1}{\left(\omega_{m}^{2}-\omega^{2}\right)^{1 / 2}}
$$

Khoobchandra Agrawal

Numerade Educator

01:14

Problem 8

Show that for temperatures well below $\theta_{\mathrm{D}}$, the mean free path of a phonon can be expressed as
$$
\Lambda=\frac{f K_{\mathrm{ph}} \theta_{\mathrm{D}}^{3}}{v T^{3}}
$$
where $v$ is the speed of sound and $f$ is defined in $(5.40)$.

Ajay Singhal

Numerade Educator

08:37

Problem 9

In the approximation that the heat capacity is temperature independent, prove the following for the variation of thermal expansion coefficient $\alpha$ :
$$
\frac{d \alpha}{d T}=-\frac{9 g f C_{V}^{2}}{8 v^{4} x_{0}}
$$
where $x_{0}$ is the equilibrium interatomic separation.

Crystal Wang

Numerade Educator

06:05

Problem 10

Show that $u=\sum_{n=1}^{\infty} a_{n} \exp ($ in $\omega t)$ is an approximate solution multiples of the harmonic
frequency $\omega=\sqrt{\frac{f}{M}}$, to the equation of motion
$$
M \ddot{\mathbf{u}}+f \mathbf{u}-\frac{1}{2} g \mathbf{u}^{2}=0
$$
for an anharmonic oscillator.

Ameer Said

Numerade Educator

07:12

Problem 11

Show that the phonon density of states $D(\omega)$ for a diatomic linear chain diverges at the maximum frequency and at frequencies on either side of the gap while it tends to be a constant as $\omega \rightarrow 0$.

Khoobchandra Agrawal

Numerade Educator