Chapter 10, Vibrations of a One-Dimensional Diatomic Chain Video Solutions, The Oxford Solid State Basics

Problem 1

Normal modes of a One-Dimensional Diatomic Chain
(a) What is the difference between an acoustic mode and an optical mode.
$\triangleright$ Describe how particles move in each case.
(b) Derive the dispersion relation for the longitudinal oscillations of a one-dimensional diatomic massand-spring crystal where the unit cell is of length $a$ and each unit cell contains one atom of mass $m_{1}$ and one atom of mass $m_{2}$ connected together by springs with spring constant $\kappa$, as shown in the figure (all springs are the same, and motion of particles is in one dimension only).
(c) Determine the frequencies of the acoustic and optical modes at $k=0$ as well as at the Brillouin zone boundary.
$\triangleright$ Describe the motion of the masses in each case (see margin note 4 of this chapter!).
$\triangleright$ Determine the sound velocity and show that the group velocity is zero at the zone boundary.
$\triangleright$ Show that the sound velocity is also given by $v_{s}=\sqrt{\beta^{-1} / \rho}$ where $\rho$ is the chain density and $\beta$ is the compressibility.
(d) Sketch the dispersion in both reduced and extended zone scheme.
$\triangleright$ If there are $N$ unit cells, how many different normal modes are there?
D How many branches of excitations are there? I.e., in reduced zone scheme, how many modes are there there at each $k ?$
(e) What happens when $m_{1}=m_{2}$ ?

Khoobchandra Agrawal

Numerade Educator

04:55

Problem 2

Decaying Waves
Consider the alternating diatomic chain dispersion as discussed in the text Eq. $10.6$ and shown in Fig. 10.6. For frequencies above $\omega_{+}(k=0)$ there are no propagating wave modes, and similarly for frequencies between $\omega_{-}(k=\pi / a)$ and $\omega_{+}(k=\pi / a)$ there are no propagating wave modes. As in Exercise $9.4$, if this chain is driven at a frequency $\omega$ for which there are no propagating wave modes, then there will be a decaying, or evanescent, wave instead. By solving Eq. $10.6$ for a complex $k$, find the length scale of this decaying wave.

Ameer Said

Numerade Educator

25:04

Problem 3

General Diatomic Chain*
Consider a general diatomic chain as shown in Fig. $10.1$ with two different masses $m_{1}$ and $m_{2}$ as well as two different spring constants $\kappa_{1}$ and $\kappa_{2}$ and lattice constant $a$.
(a) Calculate the dispersion relation for this system.
(b) Calculate the acoustic mode velocity and compare it to $v_{s}=\sqrt{\beta^{-1} / \rho}$ where $\rho$ is the chain density and $\beta$ is the compressibility.

Dr. Rajveer Singh

Numerade Educator

25:04

Problem 4

Second Neighbor Diatomic Chain*
Consider the diatomic chain from Exercise 10.1. In addition to the spring constant $\kappa$ between neighboring masses, suppose that there is also a next nearest-neighbor coupling with spring constant $\kappa^{\prime}$ connecting equivalent masses in adjacent unit cells. Determine the dispersion relation for this system. What happens if $\kappa^{\prime} \gg \kappa ?$

Dr. Rajveer Singh

Numerade Educator

05:15

Problem 5

Triatomic Chain*
Consider a mass-and-spring model with three different masses and three different springs per unit cell as shown in this diagram.

As usual, assume that the masses move only in one dimension.
(a) At $k=0$ how many optical modes are there? Calculate the energies of these modes. Hint: You will get a cubic equation. However, you already know one of the roots since it is the energy of the acoustic mode at $k=0$
$(b)^{*}$ If all the masses are the same and $\kappa_{1}=\kappa_{2}$ determine the frequencies of all three modes at the zone boundary $k=\pi / a$. You will have a cubic equation, but you should be able to guess one root which corresponds to a particularly simple normal mode.
(c) $^{*}$ If all three spring constants are the same, and $m_{1}=m_{2}$ determine the frequencies of all three modes at the zone boundary $k=\pi / a$. Again you should be able to guess one of the roots.

Khoobchandra Agrawal

Numerade Educator