Crystal Structures

Abstract

Although everyone has an intuitive idea of what a solid is, we will consider (in this book) only materials with a well defined crystal structure. What we mean by a well defined crystal structure is an arrangement of atoms in a lattice such that the atomic arrangement looks absolutely identical when viewed from two different points that are separated by a lattice translation vector. A few definitions are useful.

Appendices

Problems

1.1

Demonstrate that

• (a) the reciprocal lattice of a simple cubic lattice is simple cubic.

• (b) the reciprocal lattice of a body centered cubic lattice is a face centered cubic lattice.

• (c) the reciprocal lattice of a hexagonal lattice is hexagonal.

1.2

Determine the packing fraction of

• (a) a simple cubic lattice

• (b) a face centered lattice

• (c) a body centered lattice

• (d) the diamond structure

• (e) a hexagonal close packed lattice with an ideal $\frac{c}{a}$ ratio

1.3

Determine the separations between nearest neighbors, next nearest neighbors, $\ldots$ down to the 5th nearest neighbors for the lattices of the cubic system.

1.4

Work out the group multiplication table of the point group of an equilateral triangle.

1.5

The Bravais lattice of the diamond structure is fcc with two carbon atoms per primitive unit cell. If one of the two basis atoms is at (0, 0, 0), then the other is at $\left( \frac{1}{4},\frac{1}{4},\frac{1}{4}\right)$.

• (a) Illustrate that a reflection through the (100) plane followed by a non-primitive translation through $\left[ \frac{1}{4}, \frac{1}{4}, \frac{1}{4}\right]$ is a glide-plane operation for the diamond structure.

• (b) Illustrate that a 4-fold rotation about an axis in diamond parallel to the x axis passing through the point $(1, \frac{1}{4}, 0)$ (the screw axis) followed by the translation $[\frac{1}{4}, 0,0]$ parallel to the screw axis is a screw operation for the diamond structure.

1.6

A two dimensional hexagonal crystal has primitive translation vectors $\mathbf {a}_1=a\hat{x}$ and $\mathbf {a}_2=\frac{a}{2}\left( -\hat{x} +\sqrt{3}\hat{y}\right)$.

• (a) Show that the reciprocal lattice has primitive translation vectors $\mathbf {b}_1=\frac{b}{2}\left( \sqrt{3}\hat{x} +\hat{y}\right)$ and $\mathbf {b}_2=b\hat{y}$ with $b=\frac{4\pi }{\sqrt{3}a}$.

• (b) Draw the vectors from the origin to the nearest reciprocal lattice points in reciprocal space using $\mathbf {b}_1=\frac{b}{2}\left( \sqrt{3}\hat{x} +\hat{y}\right)$ and $\mathbf {b}_2=b\hat{y}$, and construct the first Brillouin zone.

• (c) An incident wave of wavevector $\mathbf {k}_0$ traveling in the $x-y$ plane is scattered by the two dimensional lattice into the direction of wavevector $\mathbf {k}$ in the $x-y$ plane. Find the values of $\mathbf {k}$, for which there are maxima in the diffraction pattern.

• (d) A graphene is a single layer of graphite of a hexagonal two dimensional lattice with two atoms per unit cell located at $\mathbf {r}_1=0$ and $\mathbf {r}_2=\frac{1}{3}\mathbf {a}_1 +\frac{2}{3}\mathbf {a}_2$. What is the two dimensional packing fraction of a graphene?

• (e) What is the structure factor $F(h_1, h_2)$ for X-ray scattering in a single layer of graphene? Take f as the atomic scattering factor of carbon.

1.7

CsCl can be thought of as a simple cubic lattice with two different atoms [at (0, 0, 0) and $\left( \frac{1}{2}, \frac{1}{2},\frac{1}{2}\right)$] in the cubic unit cell. Let $f_+$ and $f_-$ be the atomic scattering factors of the two constituents.

• (a) What is the structure amplitude $F(h_1,h_2,h_3)$ for this crystal?

• (b) An X-ray source has a continuous spectrum with wave numbers $\mathbf {k}$ satisfying: $\mathbf {k}$ is parallel to the [110] direction and $\frac{1}{\sqrt{2}}\left( \frac{2\pi }{a}\right) \le |\mathbf {k}|\le 3\times \sqrt{2}\left( \frac{2\pi }{a}\right)$, where a is the edge distance of the simple cube. Use the Ewald construction for a plane that contains the direction of incidence to show which reciprocal lattice vectors $\mathbf {K} (h_1,h_2,0)$ display diffraction maxima.

• (c) If $f_+=f_-$, which of these maxima disappear?

1.8

A simple cubic structure is constructed in which two planes of A atoms followed by two planes of B atoms alternate in the [100] direction.

• (a) What is the crystal structure (viewed as a non-Bravais lattice with four atoms per unit cell)?

• (b) What are the primitive translation vectors of the reciprocal lattice?

• (c) Determine the structure amplitude $F(h_1,h_2,h_3)$ for this non-Bravais lattice.

1.9

Powder patterns of three cubic crystals are found to have their first four diffraction rings at the values of the scattering angles $\phi _i$ given below (Table 1.5):

 

Table 1.5 Scattering angles of the samples

The crystals are monatomic, and the observer believes that one is body centered, one face centered, and one is a diamond structure.

• (a) What structures are the crystals A, B, and C?

• (b) The wave length $\lambda$ of the incident X-ray is $0.95\,\AA$. What is the length of the cube edge for the cubic unit cell in A, B, and C, respectively?

1.10

Determine the ground state atomic configurations of C(6), O(8), Al(13), Si(14), Zn(30), Ga(31), and Sb(51).

1.11

Consider 2N ions in a linear chain with alternating $\pm e$ charges and a repulsive potential $AR^{-n}$ between nearest neighbors.

• (a) Show that the internal energy becomes

  $$ U(R) = 2\ln 2 \frac{Ne^2}{R} \left[ \frac{1}{n}\left( \frac{R_0}{R}\right) ^{n-1}-1\right] , $$

  where $R_0$ is the equilibrium separation of the ions.

• (b) Let the crystal be compressed such that $R_0 \rightarrow R_0 -\delta$. Show that the work done in compressing the crystal of a unit length can be written as $\frac{1}{2} C\delta ^2$, and determine the expression for C.

Summary

In this chapter first we have introduced basic geometrical concepts useful in describing periodic arrays of objects and crystal structures both in real and reciprocal spaces assuming that the atoms sit at lattice sites.

A lattice is an infinite array of points obtained from three primitive translation vectors ${\mathbf a}_1$, ${\mathbf a}_2$, ${\mathbf a}_3$. Any point on the lattice is given by

$$ {\mathbf n} = n_1{\mathbf a}_1+n_2{\mathbf a}_2+n_3{\mathbf a}_3. $$

Any pair of lattice points can be connected by a vector of the form

$$ {\mathbf T}_{n_1n_2n_3} = n_1{\mathbf a}_1+n_2{\mathbf a}_2+n_3{\mathbf a}_3. $$

Well defined crystal structure is an arrangement of atoms in a lattice such that the atomic arrangement looks absolutely identical when viewed from two different points that are separated by a lattice translation vector. Allowed types of Bravais lattices are discussed in terms of symmetry operations both in two and three dimensions. Because of the requirement of translational invariance under operations of the lattice translation, the rotations of $60, 90, 120, 180, \text{ and } 360^{\circ }$ are allowed.

If there is only one atom associated with each lattice point, the lattice of the crystal structure is called Bravais lattice. If more than one atom are associated with each lattice point, the lattice is called a lattice with a basis. If ${\mathbf a}_1, {\mathbf a}_2, {\mathbf a}_3$ are the primitive translations of some lattice, one can define a set of primitive translation vectors ${\mathbf b}_1, {\mathbf b}_2, {\mathbf b}_3$ by the condition

$$ {\mathbf a}_i \cdot {\mathbf b}_j = 2 \pi \delta _{ij}, $$

where $\delta _{ij} =0$ if i is not equal to j and $\delta _{ii} = 1$. It is easy to see that

$$ {\mathbf b}_i = 2\pi \frac{{\mathbf a}_j \times {\mathbf a}_k}{\mathbf a}_i \cdot \left( {\mathbf a}_j \times {\mathbf a}_k\right) , $$

where i, j, and k are different. The lattice formed by the primitive translation vectors ${\mathbf b}_1, {\mathbf b}_2, {\mathbf b}_3$ is called the reciprocal lattice (reciprocal to the lattice formed by ${\mathbf a}_1, {\mathbf a}_2, {\mathbf a}_3$), and a reciprocal lattice vector is given by

$$ {\mathbf G}_{h_1 h_2 h_3} = h_1 {\mathbf b}_1 + h_2 {\mathbf b}_2 +h_3 {\mathbf b}_3. $$

Simple crystal structures and principles of commonly used experimental methods of wave diffraction are also reviewed briefly. Connection of Laue equations and Bragg’s law is shown. Classification of crystalline solids are then discussed according to configuration of valence electrons of the elements forming the solid.