Introduction

Solids are of two types: Amorphous and crystalline. In amorphous solids, there is no order in the arrangement of their constituent atoms (molecules). Hence no definite structure could be assigned to them. A substance is said to be crystalline when the arrangement of the units (atoms, molecules or ions) of matter inside it is regular and periodic.

Space lattice

An array of points which describe the three dimensional arrangement of particles (atoms, molecules or ions) in a crystal structure is called space lattice. Here environment about each point should be identical.

Basis

A crystal structure is formed by associating with every lattice point a unit assembly of units or molecules identical in composition. This unit assembly is called basis.

A crystal structure is formed by the addition of a basis to every lattice point.

I.e., lattice + Basis = crystal structure.

Thus the crystal structure is real and the crystal lattice is imaginary.

Bravais lattice

For a crystal lattice, if each lattice point substitutes for an identical set of one or more atoms, then the lattice points become equivalent and the lattice is called Bravais lattice. On the other hand, if some of the lattice points are non-equivalent, then it is said to be a non-Bravais lattice.

Unit cell

The smallest portion of the crystal which can generate the complete crystal by repeating its own dimensions in various directions is called unit cell.

The position vector R for any lattice point in a space lattice can be written as

R= n₁a+n₂b+n₃c

Where a,b and c are the basis vector set. The angles between the vectors b and c, c and a, a and b are denoted asa,b and g and are called interfacial angles. The three basis vectors and the three interfacial angles, form a set of six parameters that define the unit cell, and are called lattice parameters.

Primitive cell

A primitive cell is a minimum volume unit cell. Consider a bravais lattice (in two dimensions) as shown below:

We can imagine two ways of identifying the unit cell in this structure. One is, with a₁ and b₁ as the basis vectors in which case, the unit cell will be a parallelogram. Here four lattice points are located at the vertices. This is a primitive cell. Other one is with the basis vectors a₂ and b₂ which would make a rectangle for the unit cell. Here in addition to the 4 points at the corners, one lattice point is at the centre. This is a nonprimitive cell.

crystal systems

Crystal systems

Bravais demonstrated mathematically that in 3-dimensions, there are only 14 different types of arrangements possible. These 14 Bravais lattices are classified into the seven crystal systems on the basis of relative lengths of basis vectors and interfacial angles.

Seven crystal systems are:

1. Cubic

2. Tetragonal

3. Orthorhombic

4. Monoclinic

5. Triclinic

6. Triogonal (Rhombohedral)

7. Hexagonal

14 Bravais lattices are

1. Simple cubic 2. Body centered cubic 3. Face centered cubic

4. Simple tetragonal 5. Body centered tetragonal

6. Simple orthorhombic 7. Base centered orthorhombic

8. Body centered orthorhombic 9. Face centered orthorhombic

10. Simple monoclinic 11. Base centered monoclinic

12. Triclinic

13. Triogonal( Rhombohedral)

14. Hexagonal

Planes

Direction and planes in a crystal

Many physical properties of crystalline solids are dependent on the direction of measurement or the planes across which the properties are studied. In order to specify directions in a lattice, we make use of lattice basis vectors a, b and c.

In general, any directional vector can be expressed as

R= n₁a+n₂b+n₃c

Where n₁, n₂ and n₃ are integers. The direction of the vector R is determined by these integers. If these numbers have common factors, they are removed and the direction of R is denoted as [n₁ n₂ n₃]. A similar set of three integers enclosed in a round bracket is used to designate planes in a crystal..

Lattice planes and Miller indices

The crystal lattice may be regarded as made up of a set of parallel, equidistant planes passing through the lattice points. These planes are known as lattice planes and may be represented by a set of three smallest possible integers. These numbers are called ‘Miller indices’ named after the crystallographer W.H.Miller.

Determination of Miller indices

Consider a crystal plane intersecting the crystal axes as shown:

The procedure adopted to find the miller indices for the plane is as follows:

1. Find the intercepts of the plane with the crystal axes along the basis vectors a, b and c. Let the intercepts be x, y and z respectively.

2. Express x, y and z as fractional multiples of the respective basis vectors. Then we obtain the fractions,

x/a, y/b, z/c.

3. Take the reciprocal of the three fractions to obtain

a/x, b/y, c/z

4. Find the least common multiple of the denominator, by which multiply the above three ratios. This operation reduces them to a set of 3 integers (h k l) called miller indices for the crystal plane.

For the plane given above,

1. x = 2a/3 y = 3b/2 z = 2c

2. (x/a y/b z/c) = (2/3 3/2 2)

3. Taking reciprocal, (3/2 2/3 1/2)

4. Multiplying throughout by the least common multiple 6 for the denominator, we have the miller indices, (9 4 3)

Expression for interplanar spacing

The interplanar spacing d_hkl is given by,

For a cubic lattice, a = b =c