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Elements of Solid State Physics

J.P. Srivastava

Chapter 3

Reciprocal Lattice And Determination Of Crystalstructure - all with Video Answers

Educators

Chapter Questions

01:22

Problem 1

Calculate the energy carried by an e.m. wave of wavelength $1 \AA$ (i) in $\mathrm{eV}$ and (ii) in kelvin. Explain why no Bragg diffraction is observed for visible light.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:18

Problem 2

The Bragg angle for reflection from (111) planes in $\mathrm{Al}$ is $19.2^{\circ}$ for an $\mathrm{x}$ -ray beam of wavelength $1.54 \AA$. Calculate (a) the lattice constant of $\mathrm{Al}$ and (b) the interplanar spacing for these planes.

Ryan Hood
Ryan Hood
Numerade Educator
03:15

Problem 3

A two-dimensional direct lattice is formed by repeating a parallelogram of size $4 \mathrm{~cm} \times$ $3 \mathrm{~cm}$. If one of the angles of the parallelogram be $\pi / 3$, determine the primitive vectors of the reciprocal lattice.

Lauren Shelton
Lauren Shelton
Numerade Educator
06:08

Problem 4

Prove that the reciprocal lattice vector
$\mathrm{g}=h \mathrm{a}^{*}+k \mathrm{~b}^{*}+l \mathrm{c}^{*}$ is perpendicular to the plane $(h k l)$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:36

Problem 5

Calculate the angle between the reciprocal lattice vectors $\mathbf{g}_{100}$ and $\mathbf{g}_{111}$ of a simple cubic crystal. Find out the plane in the direct lattice to which $\left(\mathrm{g}_{100} \times \mathrm{g}_{111}\right)$ is perpendicular.

Ahmad Reda
Ahmad Reda
Numerade Educator
01:22

Problem 6

Show that for $\mathrm{BCC}$ structure with the basis of one atom, a scheme of assigning $1 / 8$ of each corner atom at its respective cell coordinate location to the unit cell plus the body centre atom in the central position leads to the same result for the geometrical structure factor for this lattice as the relation ( $3.35 \mathrm{~b}$ ).

Penny Riley
Penny Riley
Numerade Educator
01:31

Problem 7

Derive a relation for the geometrical structure factor of a crystal with $\mathrm{CsCl}$ structure. Would you expect (100) reflections to be present in the diffraction pattern?

David Collins
David Collins
Numerade Educator
04:05

Problem 8

Find the geometrical structure factor for an FCC crystal of monatomic basis. Which of the following reflections, $(100),(110),(111),(200),(220),(222),(211),(221)$ and (123) would be missing in the diffraction pattern?

Sanat Mukherjee
Sanat Mukherjee
Numerade Educator
03:56

Problem 9

Show that the structure factor for a monatomic HCP structure can have any of the six values
$f\left[1+\exp \left(\frac{i n \pi}{3}\right)\right]$ with $n=1, \ldots, 6$ as the reciprocal lattice vector ranges through the points of the simple hexagonal reciprocal lattice.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:16

Problem 10

Cuprous oxide has a cubic unit cell with oxygen atoms at the centre $(0,0,0)$ and at the corners $a(\pm 1, \pm 1, \pm 1) .$ The copper atoms are arranged in a tetrahedron around the central oxygen, at $1 / 2 a(1,1,1), 1 / 2 a(1,-1,-1), 1 / 2 a(-1,1,-1)$, and $1 / 2 a(-1,-1,1) .$ Calculate the structure factor and show that some reflections are determined only by the copper, others only by the oxygen atoms.

Catherine Lemar
Catherine Lemar
Numerade Educator
03:30

Problem 11

Powder diffraction patterns of three monatomic cubic crystals with $\mathrm{BCC}, \mathrm{FCC}$ and diamond structures are recorded using a Debye-Scherrer camera. The angles $2 \theta$ in degrees of the first four diffraction lines for the three samples marked as $A, B, C$ are as under:
$\begin{array}{ccc}A & B & C \\ 42.2 & 28.8 & 42.8 \\ 49.2 & 41.0 & 73.2 \\ 72.0 & 50.8 & 89.0 \\ 87.3 & 59.6 & 115.0\end{array}$
(a) Determine the structure type of each sample.
(b) What is the size of the cubic cell in each case? Take the wavelength of incident $x$ -rays as $1.5 \AA$.

Zachary Warner
Zachary Warner
Numerade Educator
01:43

Problem 12

For a cubic crystal the diffraction line from the planes with $\left(h^{2}+k^{2}+l^{2}\right)=8$ is observed at the angle of diffraction $10.23^{\circ}$. If only one line is observed at an angle lower than this, what is the crystal structure? Assuming the wavelength of $x$ -rays used as $0.71 \AA$, calculate the lattice parameters.

Aadit Sharma
Aadit Sharma
Numerade Educator
03:30

Problem 13

The $S$ values for the first three lines in the powder pattern of a cubic crystal are $34.88,40.36$ and $54.40 \mathrm{~mm}$ respectively. Given that the wavelength of $x$ -rays used is $0.71 \AA$ and the camera radius as $57.30 \mathrm{~mm}$, find out the crystal structure and the lattice parameter.

Zachary Warner
Zachary Warner
Numerade Educator
01:39

Problem 14

Rock salt (sodium chloride) has lattice parameter of $5.63 \AA$. When the $K_{\alpha 1}$ line from copper target $(\lambda=1.54 \AA)$ is used with a powder camera, what would be the first three $S$ values? Take the camera diameter as $57.30 \mathrm{~mm}$.

Anand Jangid
Anand Jangid
Numerade Educator