(3) Convert these intercepts as the
multiplication of lattice parameter and a constant as p, q and r for OA, OB and
OC respectively.
OA = p.a, OB = q.a, OC = r.a
Here we consider unit cell as cubic,
so lattice parameter = ‘a’.
(4) Now find 1/p, 1/q and 1/r find the LCM of p, q and r.
(5) Multiply the LCM of p, q and r to the
values 1/p, 1/q and 1/r to get the values of h, k and l respectively.
The value of (h k l) represents the Miller indices.
Note :
(1) If a plane occurs on the left hand side of
the origin then it will be represented by negative Miller indices. Its notation
can be done by making a ‘bar’ on the numeric value of Miller indices as
indicates a negative Miller indices with
negative ‘h’.
indicates a negative Miller indices with
negative ‘h’.
(2) For representing the planes and directions
of a hexagonal unit cell we use Miller bravis indices which contains four axis
system and it is represented by (h k i l).
Q.8 Discuss the features of miller indices.
Ans. Salient Features of Miller Indices: -
Some important features of Miller indices
are as follows:
(i) Miller indices of equally
spaced parallel planes are the same.
(ii) A plane parallel to
one of the coordinate axes has an intercept at infinity and so the Miller index
of the plane is zero for that axis.
(iii) The plane passing
through the origin is defined in terms of a parallel plane having non-zero
intercept. Alternately, if the plane passes through the origin, the origin has
to be shifted for indexing the plane.
(iv) Any two planes
having miller indices (h1 k1 l1) and (h2 k2 l2) will be
perpendicular to each other if
h1 h2 + k1 k2 + l1 l2 = 0
(v) When Miller indices
contain integers of more than one digit, the indices are separated by a comma
for clarity, e.g. (3, 4,12) or (4, 11, 17).
(vi) The direction [h k l] is normal to
the plane having Miller index (h k l) in a cubic system. This is generally not
true for non-cubical crystal systems.
(vii) Planes with low index numbers have wide interplanar
spacing as compared with those having high index
numbers.
(viii) All members of family of planes are not necessarily
parallel to one another. Similar is the case of crystal directions of a family.
Q.9 Write short notes on the following :
(a)
Interplaner spacing, (b) Linear Density, (c) Planer Density.
Ans. (a)
Interplaner Spacing: -
The spacing between a plane (hkl) and the other
parallel plane passing through the origin is called interplaner spacing. It is
denoted by d[hkl]. It is measured at right angles to the
planes. As an example, the interplaner spacing of (100) planes in cubic crystal
is equal to the lattice constant a. For (200) planes, this value is (a/2).
Interplaner spacing may be obtained from the following relations in cubic and
tetragonal crystals.
For cubic
unit cell, 
..........(1)
..........(1)
And for
tetragonal unit cell
..........(2)
..........(2)
Members of
a family of planes {hkl} have the same interplaner spacing.
(b)
Linear Demsity: -
Linear density may be defined as the number of effective
atoms NeL per unit length on certain
length L along any direction in a unit cell. It can be expressed as
(c)
Planer Density: - (AKTU. 2005 - 06)
The number of atoms per unit area of
a crystal plane is known as planer density. This density infact expresses the
packing of atoms on a plane. The rate-of plastic deformation is significantly
influenced by it. Hence calculation of number of atoms per unit area (generally
per square millimeter) becomes essential.
The planer density can be expressed as
where Ne is the effective number of atoms on the plane
whose area is A.
Q.10. Briefly describe the crystel structure of
diamod.
Ans. Diamond Cubic (DC) Structure:-
Carbon exists in two forms viz.
diamond and graphite. Both have quite different characteristics and
properties. Diamond has (sp3 )
hybrid covalent bond. Each of its atoms has four bonds. The bonds are
directional in nature. The bonds are primary in nature and extend in a
three-dimensional network.
