Chapter 13, Reciprocal Lattice, Brillouin Zone, Waves in Crystals Video Solutions, The Oxford Solid State Basics

Problem 1

Reciprocal Lattice
Show that the reciprocal lattice of a fcc (facecentered cubic) lattice is a bcc (body-centered cubic) lattice. Correspondingly, show that the reciprocal lattice of a bcc lattice is an fcc lattice. If an fcc lattice has conventional unit cell with lat-

Penny Riley

Numerade Educator

02:08

Problem 2

Lattice Planes
Consider the crystal shown in Exercise 12.3. Copy this figure and indicate the [210] direction and the (210) family of lattice planes.

Chai Santi

Numerade Educator

00:49

Problem 3

Directions and Spacings of Crystal Planes D. texplain briefly what is meant by the terms "crystal planes" and "Miller indices".
D. Show that the general direction $[h k l]$ in a cubic crystal is normal to the planes with Miller indices $(h k l)$.
D Is the same true in general for an orthorhombic crystal?
D. Show that the spacing $d$ of the $(h k l)$ set of planes in a cubic crystal with lattice parameter $a$ is
$$
d=\frac{a}{\sqrt{h^{2}+k^{2}+l^{2}}}
$$

Chai Santi

Numerade Educator

07:02

Problem 4

Reciprocal Lattice (a) Define the term Reciprocal Lattice. (b) Show that if a lattice in $3 d$ has primitive lattice vectors a $_{1}$, a $_{2}$ and as then primitive lattice vectors for the reciprocal lattice can be taken as $$ b_{1}=2 \pi \frac{a_{2} \times a_{3}}{a_{1} \cdot\left(a_{2} \times a_{3}\right)} $$ $b_{2}=2 \pi \frac{a_{3} \times a_{1}}{a_{1} \cdot\left(a_{2} \times a_{3}\right)}$
D. What is the general an orthorhombic crystal? †Reciprocal Lattice
$$
b_{3}=2 \pi \frac{a_{1} \times a_{2}}{a_{1} \cdot\left(a_{2} \times a_{3}\right)}
$$
For an orthorhombic lattice, show that $|\mathbf{b j}|=$ $2 \pi /|\mathbf{a j}|$. Hence, show that the length of the reciprocal lattice vector $\mathbf{G}=h \mathbf{b}_{1}+k \mathbf{b}_{2}+l \mathbf{b}_{3}$ is equal to $2 \pi / d$, where $d$ is the spacing of the $(h k l)$ planes (see question 13.3)

Khoobchandra Agrawal

Numerade Educator

06:08

Problem 5

More Reciprocal Lattice
A two-dimensional rectangular crystal has a unit cell with sides $a_{1}=0.468 \mathrm{~nm}$ and $a_{2}=0.342 \mathrm{~nm}$.
(a) Draw to scale a diagram of the reciprocal lattice.

D. Label the reciprocal lattice points for indices in the range $0 \leqslant h \leqslant 3$ and $0 \leqslant k \leqslant 3$.
(b) Draw the first and second Brillouin zones using the Wigner-Seitz construction.

Khoobchandra Agrawal

Numerade Educator

08:26

Problem 6

Brillouin Zones
(a) Consider a cubic lattice with lattice constant
a. Describe the first Brillouin zone. Given an arbitrary wavevector $k$, write an expression for an equivalent wavevector within the first Brillouin zone (there are several possible expressions you can write).
(b) Consider a triangular lattice in two dimensions (primitive lattice vectors given by Eqs. 12.3). Find the first Brillouin zone. Given an arbitrary wavevector $\mathbf{k}$ (in two dimensions), write an expression for an equivalent wavevector within the first Brillouin zone (again there are several possible expressions you can write).

Hafiz Shahzaib

Numerade Educator

10:43

Problem 7

Number of States in the Brillouin Zone A specimen in the form of a cube of side $L$ has a primitive cubic lattice whose mutually orthogonal fundamental translation vectors (primitive lattice vectors) have length $a$. Show that the number of different allowed $\mathrm{k}$-states within the first Brillouin zone equals the number of primitive unit cells forming the specimen. (One may assume periodic boundary conditions, although it is worth thinking about whether this still holds for hard-wall boundary conditions as well.)

Hafiz Shahzaib

Numerade Educator

12:16

Problem 8

Calculating Dispersions in $d>1^{*}$
(a) In Exercises $9.8$ and $11.9$ we discussed dispersion relations of systems in two dimensions (if you have not already solved those exercises, you should do so now).

D. In Exercise 11.9, describe the Brillouin zone (you may assume perpendicular lattice vectors with length $a_{1}$ and $a_{2}$ ). Show that the tight-binding dispersion is periodic in the Brillouin zone. Show that the dispersion curve is always flat crossing a zone boundary.

D. In Exercise $9.8$, describe the Brillouin zone. Show that the phonon dispersion is periodic in the Brillouin zone. Show that the dispersion curve is always flat crossing a zone boundary.
(b) Consider a tight binding model on a threedimensional foc lattice where there are hopping matrix elements $-t$ from each site to each of the nearest-neighbor sites. Determine the energy spectrum $E(\mathbf{k})$ of this model. Show that near $\mathbf{k}=\mathbf{0}$ the dispersion is parabolic.

Dr. Rajveer Singh

Numerade Educator