Laws of crystallography
Crystallography is the branch of science which deals with
the geometry, properties and structure of
crystals and crystalline substances. It is based on three fundamental laws.
The law of constancy
of interfacial angles: It states that the size of the faces or even
shapes of the crystals of a substance may vary widely with conditions of
crystallization yet the interfacial angles between any two corresponding be 90˚, irrespective of the size and shapes of faces. The
measurement of crystal angles is important in the study of crystals.
*Interfacial Angles- The crystals are
bound by plane faces. The angle between any two faces is called an interfacial
angle.
It
is indicated from the above figure that the shapes of crystals are different
but the angle between any two corresponding faces is 90˚.
2.
Law of rational indices (Hauy’s law): This law states that “the
ratio between the intercepts on the axes for the different faces of a crystal
can always be expressed by rational numbers”.
Let OX, OY and OZ be the three crystallographic axes and ABC
be the unit plane and LMN the plane of the crystal under study. The unit
intercepts are a fro OA, b for OB and c for OC. For the plane (face) LMN, the
intercepts will be OL (=2a),OM (=4b) and ON(=3c). Thus the intercepts for the
plane are in the ratio of 2a : 4b : 3c where 2,4,and 3 are simple intergral
whole number and are known as the are Weiss indies of the plane. It must be
noted that it is not necessary that the Weiss indices are always simple
integral whole number they may be fractional number or even infinity. Hence
these are replaced by Miller indices which are always simple intergral whole
numbers. These are represented by h, k and l corresponding to a, b and c,
Miller indices of a plane of a crystal are inversely
proportional to the intercepts of that plane on the various axes. Miller
indices for a plane may be obtained from Weiss indices (coefficients of the
unit lengths a, b, and c of the plane) by talking the reciprocals of the
coefficients of Weiss indices and multiplying throughout by the smallest number
on order to make all reciprocals as intergers.
For
illustration, consider a plane which in Weiss notation is given by ∞a: 2b: c.
The Miller indices of this plane may be calculated as below
Multiplying by 2 in order to get whole numbers =0,
1, 2
Thus the miller indices of the plane are 0, 1
and 2. Hence the plane is designated as the (012) plane, i.e. h=0, k=1 and l
=2.
Let
us now describe the planes in a cube. It should be remembered that if a plane
is parallel to an axis, its intercept with that axis would be infinity. Thus,
the miller index for that plane will be zero (1/∞
=0). For example, consider the shaded plane in following figure. It is clear
from the figure that the fractional intercepts of the shaded plane are: 1, ∞, ∞
Similarly, the miller indices of some planes in cubic crystals, shown shaded in following figure are as given below:
Fractional
intercepts 1 1 1
Reciprocal
of intercepts 1 1 1
Miller
indices 1
1 1
Ex : A crystal plane intercepts
the three crystallographic axes at a, 1/2b and 3/2c where a, b and c are the
unit lengths along X, Y and Z axes respectively. What are the miller indices of
this plane?
Solution. Form the given data the Weiss indices are 1, ½, 3/2, i.e.
A b c
1 ½ 3/2
Talking
reciproclas 1
2 2/3
Multiplying
by 3 3 6 2
Thus the miller indices of the
plane are 3, 6, 2 and hence the plane is designated as (362)plane.
Ex: Calculate the miller indices
of crystal plane which cut through the crystal axes at 2a, -3b, -3c.
Solution.
Given
a
b c
2 -3 -3
Talking
Reciprocals
1/2 -1/3 -1/3
Multiplying
by 6 3 -2 -2
Ex
: Calculate the miller indicies of
crystal plane which cut through the crystal axes at
(i) (2a, 3b, c)
(ii)(a, b, c)
(iii)(6a,
3b, 3c)
Solution
(i):
Following the procedure given above,
we prepare the table as follows:
a b c
2 3 1 Fractional intercepts
1/2 1/3
1 Reciprocals
3
2 6 Clear fractions
Hence, the miller indices are (326).
Solution(ii):
a b c
1 1 1 Fractional intercepts
1 1 1
Reciprocals of intercepts
1 1 1 Clear fractions
Hence, the miller indices are (111).
Solution
(iii):
a b c
6 3 3 Fractional intercepts
1/6 1/3 1/3 Reciprocals of the
intercepts
1
2 2 Clear
fractions
Hence, the miller indices are (122).
*Some
other planes represented by miller indice (h,k,l) are as follows-
Problems
for practice:
1. The Miller Indices of two parallel planes
in a crystal are-
(a)
Same (b) Different (c) May be same or different (d) None of these
2.
Calculate the miller indices of the crystal plane which cut through the crystal
axes at (2a, 3b, c)-
(a)
326 (b) 122 (c) 236 (d) 623
3.
The miller indices of the shaded plane shown in the figure below are-
(a)
(001) (b) (010) (c) (011) (d) (100)
4.
(hkl) represents –
(a)
Lattice parameters (b) Crystal
faces (c) Crystal Systems (d) Miller Indices
5.
If a plane in a cubic crystal intersects the a, b and c axes at 1, 2 and
Infinity, respectively. Its Miller Indices will be –
(IISc-Integrated PhD 2010)
(a)
(120) (b) (210) (c) (12∞) (d) (21∞)
6.
If weiss indices of face of a crystal are 1, ∞, ∞, then its miller indices will
be:
(a)
100 (b) 011 (c) 010 (d) 101 (BHU 2016)
7.
In any crystal, ratio of weiss indices of the face is 2:4:3, then the miller
indices would be-
(a)
634 (b) 346 (c) 436 (d) 643 (BHU 2016)
8.
The miller indices of the face having intercept (a/2, 2b, ∞c) where a, b, c are
the unit cell axes, is:
(University of Hyderabad 2015)
(a)
(410) (b) (220) (c) (210) (d) (012)
9.
The miller indices of the planes with intercepts 4a, 6b, ∞c where a, b, c are
the unit cell edge lengths are-
(BHU 2015)
(a)
(320) (b) (230) (c) (023) (d) (46∞)
10.
A crystal plane is parallel to the x-axis and y-axis and makes an intercept of
½ on the z-axis. The miller indices of the planes are-
(IISc 2012)
(a)
112 (b) 220 (c) 001 (d) 002
Answers:
1
(a) 2. (a) 3. (d) 4. (d) 5. (b) 6. (a)
7.
(a) 8. (a) 9. (a) 10. (d)
3.
The law of symmetry : it states that
all crystals of a substance possess the same elements* of symmetry. The three
important elements of symmetry are (i) plane of symmetry, (ii) axis of symmetry,
(iii) centre of symmetry.
(i)
Plane of symmetry : it is an
imaginary plane which can divide the crystal into two halves such that one is
the mirror image of the other. A cubic crystal ,e.g., has two types of plane of
symmetry-
(a)
Rectangular planes of symmetry : These are
the planes situated midway and parallel to the two opposite faces. Since a
cubic crystal has six faces, i.e. three pairs of opposite faces, it has three
rectangular plane of symmetry as shown in following figure-
(b)
Diagonal plane of Symmetry: These are
the plane touching the opposite edges. Since there are 12 edges or six pairs of
edges. So there are 6 diagonal plane of symmetry.As they lie like the diagonal
that is why these are called as diagonal plane of symmetry.
(ii) Axis or Line of Symmetry: It is an imaginary axes (line)
about which the crystal may be rotated so that it gives the equivalent
orientation more than once in a complete rotation through 360˚. It is donated
by Cn (n is an integral number)
On the basis of n, these are of
following types-
(a)
Axes of 2-fold Symmetry (C2):
On rotation of 180˚, we can get equivalent orientation. It pass through the
diagonally opposite edges. Total edges are 12 so that 6 two fold axes, are
present in a cube.
(b)
Axes of 3-fold symmetry (C3):
On rotation of 120˚ through the diagonally opposite corners of the cube. We can
get equivalent orientations. Total corners are eight. So 4 such type of
symmetry elements are there i.e. 4 C3 axes.
(c) Axes of 4
fold of symmetry: On rotation of 90˚
through two opposite faces, we will get equivalent orientations. Total faces
are 6, so 3 4-fold elements of symmetry are present in a cube.
Hence
Total number of elements of symmetry in a cubic crystal are 6 C2 + 3
C4 + 4 C3
(iii)
Centre of Symmetry: It is a such point
in the crystal that any line drawn through it intersects the surface of the
crystal at equal distances on either side. A crystalmay have one or more axes
of symmetry or plane of symmetry but it never has more than one centre of
symmetry.
So total number of elements of symmetry in a
cubic crystal are 23.
Plane of Symmetry = 3 + 6 = 9
Axes of Symmetry = 3 + 4 +6 = 13
Centre of symmetry = 1
Problems for
practice:
1.
The total number of elements of symmetry in a cubic crystal is-
(a)
9 (b) 23 (c) 13 (d) 1
2.
A crystal may have one or more planes of symmetry as well as one or more axes
of symmetry but it has-
(a)
Two centres of symmetry (b) no centre of symmetry
(c)
one centre of symmetry (d) four
centres of symmetry
3. Total number of axes of three-fold symmetry in a cubic crystal are-
(a) 3 (b) 4 (c) 6 (d) 13
Answer:
1. (b) 2. (c) 3. (b)
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