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CHAPTER-06

Elements of Crystallography

Crystallography

• The word crystallography is the combination of two words: crystal and graphein, where crystal has its usual meaning while the Greek word graphein means to write. Hence, crystallography means writing about the complete crystal.

• In a single crystal, the position and orientation of atoms or molecules are uniform and continuous throughout the entire crystal as shown in Fig. 6.1. The examples of single crystal are diamond, ruby, etc.

CRYSTAL STRUCTURE

• There are two types of single crystals known as elemental and ionic. The examples of elemental single crystals are Al, Fe, Cu, etc., while the examples of ionic single crystals are AgCl, CuSO4, etc.

• There is another class of material known as polycrystalline in which the whole crystal is made up of smaller crystallites, as shown in Fig. 6.2. Each constituent of the small crystallite is called grain.

SPACE LATTICE

• To describe the arrangement of atoms in a crystal, three-dimensional arrangement of imaginary points in space that have identical points in its surroundings is known as space lattice.

BASIS

• An atom or a group of atoms placed on each lattice point to obtain a crystal structure, is called the basis, and it acts as a building unit, or a structural unit, for the complete crystal structure.

Space lattice + Basis = Crystal structure

• The generation of a crystal structure from a two-dimensional lattice and a basis is shown in the figure. The basis consists of two atoms, represented by and .

UNIT CELL

• A unit cell may be defined as the smallest unit of the lattice which, on continuous repetition, generates the complete lattice.

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• In following Fig. the parallelogram ABCD represents a two-dimensional primitive cell, whereas the parallelograms EFGH and KLMN represent non-primitive cells. The smallest volume is known as primitive unit cell.

LATTICE PARAMETER

• In the following figure vectors and along x-, y-, and z-axes, respectively, are called crystallographic axes. The angles between these axes are called interfacial angles, which are measured as α, β, and γ . Primitive vectors and the interfacial angles α, β, and γ together are known as lattice parameter of a crystal.

SEVEN CRYSTAL SYSTEMS

Simple sketch of all the seven crystal systems

BRAVAIS SPACE LATTICE

Bravais suggested that there are only 14 ways of arranging points in space.

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• In the case of cubic system, there are three types of possible Bravais lattices

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• In the case of tetragonal system, there are following two types of Bravais lattices

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• In the case of orthorhombic system, there are following four types of possible Bravais lattices:

• In the case of rhombohedral or trigonal system, there is only one possible Bravais lattice

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• In the case of hexagonal system, there is only one possible Bravais lattice known as simple hexagonal

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• Monoclinic system has two possible bravais lattices

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• Triclinic crystal system has only one possible lattice known

SYMMETRY ELEMENT OF A CRYSTALLINE SOLID

Axis of Symmetry

• 3 axes have four-fold symmetry known as tetrad. These axes pass through opposite face centres.
• 4 axes have three-fold symmetry known as triad. These axes pass through diagonally opposite corners.
• 6 axes have two-fold symmetry known as diad. These axes pass through the centres of opposite edges.

Planes of Symmetry

• If a plane is able to cut a crystal into two halves in such a way that one half becomes the mirror image of the other half, then that plane is known as the plane of symmetry.

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• There are 9 planes—3 are parallel to the face of the cube and 6 are diagonal planes passing through diagonally opposite parallel edges.

Centre of Symmetry

• It is a point in a crystal such that if a line is drawn from any point on the crystal through this point and an equal distance is produced on the other side of this central point, then it meets an identical point.

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• There is only one centre of symmetry for cubic system.

COORDINATES OF LATTICE POINT

• The coordinates of the corner lattice points for simple cube, body-centred cube, and face-centred cube systems may be considered as (0,0,0).

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• Coordinates of corner lattice points, coordinates of body-centred lattice points, and coordinates of face-centred lattice points are as following:

(a) Coordinates of corner lattice points, (b)coordinates of body-centred lattice points, and (c) coordinates of face-centred lattice points

NUMBER OF ATOMS PER UNIT CELL

• In the simple cubic crystal, total number of atoms per unit cell is 1.

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• In the body-centred cubic crystal, , total number of atoms per unit cell is 2.

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• In the body-centred cubic crystal, , total number of atoms per unit cell is 4.

(a) Number of atoms per unit cell in simple cubic system, (b) number of atoms per unit

cell in body-centred cubic system, and (c) number of atoms per unit cell in face-centred cubic system

PACKING DENSITY OR ATOMIC PACKING FACTOR

MILLER INDICES

Miller indices are defined as three smallest possible integers which have the same ratios as the reciprocal of intercepts of the plane concerned on the three axes.

Orientation of a plane with different axial intercepts

Procedure for Finding Miller Indices

1. Choose a system of three coordinate axes, preferably along the crystallographic axis.

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2. Find the intercepts of the plane along the coordinate axes x, y, z.

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3. Express these intercepts in the terms of axial unit.

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4. Take the reciprocal of these intercepts.

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5. Taking LCM of numerical values of the above reciprocals, reduce them to the smallest three integers which have the same ratio.

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6. Enclose the result obtained in step 5 in parentheses as (hkl), which are known as Miller indices of the crystal plane.

DIFFERENT CRYSTAL PLANES WITH MILLER INDICES

Three planes BCGF, ADGF, and AHF of a cubic crystal having Miller indices (100), (101), and (111), are shown in following figures

(a) BCGF plane with Miller indices (100), (b) ADGF plane with Miller indices (101), and (c) AHF plane with Miller indices (111)

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IMPORTANT FEATURES OF MILLER INDICES

• Equally spaced parallel planes will have the same index numbers.

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• If a plane is parallel to any one of the three coordinate axes, its intercept on that axis is infinity. Hence, the Miller index for that direction is zero.

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• The Miller indices are used to define a set of parallel planes and not a particular plane.

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• A plane passing through the origin is defined in terms of a parallel plane having nonzero intercepts.

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• For a cubic crystal system, the distance d between the adjacent planes of a set of parallel planes of the indices (hkl) is given as

• For a cubic crystal system

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(a) the distance d between the adjacent planes of a set of parallel planes of the indices (hkl) is given as

where a is the edge of the cube.

(b) the angle ѳ between two crystallographic directions [hkl] and [h’ k’ l’ ] can be calculated as

INTERPLANAR SPACING

The perpendicular distance between the corresponding parallel planes of a family having Miller indices (hkl) is known as interplanar spacing, which is usually denoted by d_hkl.

ATOMIC RADIUS IN A CUBIC SYSTEM

• The radius of the atoms is known as atomic radius and is equal to half of the distance between the centres of two adjacent atoms placed symmetrically in the cube.

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• The atomic radius of SC structure, r = a/2
• Simple cubic structure and its atomic radius is shown in the figure.

Atomic Radius of Face-Centred Cubic Structure

• The atomic radius of Face centered cubic structure,

• Face Centered Cubic structure and its atomic radius is shown in the following figure.

Atomic Radius of Body-Centred Cubic Structure

• The atomic radius of body centered cubic structure is

• Body Centered Cubic structure and its atomic radius is shown in the following figure.

DIMENSIONS OF UNIT CELL

• The dimensions of a unit cell, can be given as:

COORDINATION NUMBER

It is defined as the number of atoms directly surrounding a given atom.

Where n is the numbers of atom per unit cell and N is the Avogadro number

INTERPLANAR SPACING IN CUBIC SYSTEM

Simple Cubic Crystal: Corresponding to different Miller indices the interplanar spacing of simple cubic crystal can be given as:

Face-Centred Cubic Crystal

Face Centered Cubic Crystal: Corresponding to different Miller indices the interplanar spacing of Face Centered cubic crystal are shown in following figure.

Body-Centered Cubic Crystal

For the BCC crystal, (100), (110), and (111) planes are shown in Figs. 6.27 (a), (b), and (c), respectively.