Crystallography
Category : JEE Main & Advanced
“The branch of science that deals with the study of structure, geometry and properties of crystals is called crystallography”.
(1) Symmetry in Crystal : A crystal possess following three types of symmetry,
(i) Plane of symmetry : It is an imaginary plane which passes through the centre of a crystal can divides it into two equal portions which are exactly the mirror images of each other.
(ii) Axis of symmetry : An axis of symmetry or axis of rotation is an imaginary line, passing through the crystal such that when the crystal is rotated about this line, it presents the same appearance more than once in one complete revolution i.e., in a rotation through 360°. Suppose, the same appearance of crystal is repeated, on rotating it through an angle of 360°/n, around an imaginary axis, is called an n-fold axis where, n is known as the order of axis. By order is meant the value of n in \[2\pi /n\] so that rotation through \[2\pi /n,\] gives an equivalent configuration. 
(iii) Centre of symmetry : It is an imaginary point in the crystal that any line drawn through it intersects the surface of the crystal at equal distance on either side.
Only simple cubic system have one centre of symmetry. Other system do not have centre of symmetry.
The total number of planes, axes and centre of symmetries possessed by a crystal is termed as elements of symmetry.
A cubic crystal possesses total 23 elements of symmetry.
\[\frac{\begin{align} & Plane\,\,of\,\,symmetry(3+6)=9 \\ & Plane\,\,of\,\,symmetry(3+4+6)=13 \\ & Axis\,\,of\,\,symmetry(1)=1 \\ \end{align}}{Total\,\,elements\,\,of\,symmetry\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,23}\]
(2) Laws of crystallography : Crystallography is based on three fundamental laws.
(i) Law of constancy of interfacial angles : This law states that angle between adjacent corresponding faces is inter facial angles of the crystal of a particular substance is always constant inspite of different shapes and sizes and mode of growth of crystal. The size and shape of crystal depend upon the conditions of crystallisation. This law is also known as Steno's Law.
(ii) Law of rational indices : This law states that the ratio of intercepts of different faces of a crystal with the three axes are constant and can be expressed by rational numbers that the intercepts of any face of a crystal along the crystallographic axes are either equal to unit intercepts (i.e., intercepts made by unit cell) a, b, c or some simple whole number multiples of them e.g., na, n' b, n''c, where n, n' and n'' are simple whole numbers. The whole numbers n, n' and n'' are called Weiss indices. This law was given by Hauy.
(iii) Law of constancy of symmetry : According to this law, all crystals of a substance have the same elements of symmetry is plane of symmetry, axis of symmetry and centre of symmetry.
Miller indices : Planes in crystals are described by a set of integers (h, k and l) known as Miller indices. Miller indices of a plane are the reciprocals of the fractional intercepts of that plane on the various crystallographic axes. For calculating Miller indices, a reference plane, known as parametral plane, is selected having intercepts a, b and c along x, y and z-axes, respectively. Then, the intercepts of the unknown plane are given with respect to a, b and c of the parametral plane.
Thus, the Miller indices are :
\[h=\frac{a}{\text{intercept of the plane along }x\text{-axis}}\]
\[k=\frac{b}{\text{intercept of the plane along }y\text{-axis}}\]
\[l=\frac{c}{\text{intercept of the plane along }z\text{-axis}}\]
The distance between the parallel planes in crystals are designated as \[{{d}_{hkl}}\]. For different cubic lattices these interplanar spacing are given by the general formula,
\[{{d}_{(hkl)}}=\frac{a}{\sqrt{{{h}^{2}}+{{k}^{2}}+{{l}^{2}}}}\]
Where a is the length of cube side while h, k and l are the Miller indices of the plane. When a plane is parallel to an axis, its intercept with that axis is taken as infinite and the Miller will be zero. Negative signs in the Miller indices is indicated by placing a bar on the intercept. All parallel planes have same Miller indices.
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