Chapter 6, PHONONS IN NON-METALS, THERMAL CONDUCTIVITY Video Solutions, The Solid State: An Introduction to the Physics of Crystals for Students of Physics, Materials Science, and Engineering

Problem 1

Estimate the mean free path of phonons in germanium at $300 \mathrm{~K}$ using the following information. Thermal conductrity $=80 \mathrm{~W} \mathrm{~m}^{-1} \mathrm{~K}^{-1}$, Debye $\theta=360 \mathrm{~K}$. atomic weight $=72.6$, density $=5500 \mathrm{~kg} \mathrm{~m}^{-3}$. mean velocity of sound $=4500 \mathrm{~m} \mathrm{~s}^{-1}$. (Assume that all the heat transport is by phonons.)

Chai Santi

Numerade Educator

03:21

Problem 2

The Debye $\theta$ of diamond is $2000 \mathrm{~K}$. Calculate the ratio of the thermal conductivity at $50 \mathrm{~K}$ to that at $4 \mathrm{~K}$, assuming that boundary scattering is dominant in both cases.

Suzanne W.

Numerade Educator

06:16

Problem 3

In a simple cubic lattice of spacing $0-2 \mathrm{~nm}$ a phonon travelling in the [100] direction with wave vector $(2 \pi, \lambda) 1.3 \times 10^{10} \mathrm{~m}^{-1}$ interacts and combines with another phonon of the same magnitude which is travellng in the $110^{-}$direction. Draw a diagram to show the magnitude and direction of the resultan. phonon and discuss the significance of the result.

Kai Chen

Princeton University

04:11

Problem 4

Estimate the relative importance of u-processes to the thermal resistivity at $100 \mathrm{~K}$ and $20 \mathrm{~K}$ for a crystal which has a Debye $\theta$ of $300 \mathrm{~K}$.

Ameer Said

Numerade Educator

01:39

Problem 5

Calculate the thermal conductivity at $\mathrm{l} \mathrm{K}$ of a rod of synthetic corundum $\left(\mathrm{Al}_2 \mathrm{O}_3\right)$ crystal of molecular weight 102 if its diameter is $3 \mathrm{~mm}$. The velocity of sound $=5000 \mathrm{~m} \mathrm{~s}^{-1}$, the density $=4000 \mathrm{~kg} \mathrm{~m}^{-3}$ and the Debye $\theta=1000 \mathrm{~K}$ (n b. eqn (5.25) is the expression for the specific heat of a quantity of material containing Avagadro's number of atoms).

Narayan Hari

Numerade Educator

01:01

Problem 6

The phonon dispersion relation in one dimension may be written in the form $\omega=C \mid \sin (a k / 2)$. Sketch the form of the curve for the range $-2 \pi^{\prime} a \leqslant k \leqslant 2 \pi / a$. If the velocity of sound in the material is $5000 \mathrm{~m} \mathrm{~s}^{-1}$ and $a$ is $0.3 \mathrm{~nm}$ what is the value of $C$ ? What are the phase and group velocities for $k=+\pi / 2 a,-\pi / 2 a$, $+\pi / a,+3 \pi / 2 a$ ? Which of these values of $k$ correspond to equivalent phonons?

Ajay Singhal

Numerade Educator