6.
CRYSTALLGGRAPHY-I:
6.1
Introduction
6.2
Lattice point and space lattice
6.3
The basis and crystal structure
6.4
Unit cell and lattice parameter
6.5
Primitive cell
6.6
The seven crystal systems and Bravais space lattice
6.7
The twenty three symmetry elements in a cubic crystal
6.8
The unit cell characteristics
6.9
6.8.1
Number of atoms per unit cell
6.8.2
Atomic radius
6.8.3
Coordination number
6.8.4
Packing density
Relation between the density of crystal material and lattice constants in
cube lattice
7.
6.10
Lattice direction
6.11
Miller indices
6.12
Lattice planes
6.13
Separations between lattice planes in a cubic crystal
CRYSTALL8GRAPHY-II:
7.1
Introduction
7.2
X-ray diffraction
7.3
Laue method
7.4
Bragg's law
7.5
Bragg's x-ray spectrometer
7.6
Bragg's law and crystal structure
7.7
Crystallography by powder method
Crystallography-
I
Chapter 6
CRYSTALLOGRAPHY
-
I
tft.HNTRODUCTION:
1 •
•
d f
�JJy
�- That is about the
a complete definitiOn, yet it is a true description.
A
•
•
It may
e er�l
s of s 1· ds
m amorof
- are
u
N"
phous solids is limited to a few molecule distances. In polycrystalline materials, the
solid is made-up of grains which are highly ordered crystalline regions of irregular size
and orientation. Single crystals have long range order.
•
.,!'.,
N"
important properties of materials are found to depend on the structure of
crystals and on the electrons states with the crystals. At the beginning of the study of
c rystals it was their external form which was related to the physical properties.
The crystal structures are analyzed using x-ray diffraction technique invented by
Max. von Laue and extensively employed by W. H. Bragg and W. L. Bragg. The real
crystals have imperfections of different kinds. The study of crystal geometry helps us
to understand the diverse behavior of solids in their mechanical, metallurgical, electri­
cal, magnetic and optical properties. The imperfections in real crystals can be con­
trolled and suitably altered to improve the selected physical properties of the material.
�ATTICE POINTS AND SPACE LATTICE:
called� SJW�-· In a perfec�
periodici!Y i!
regular arrangemen�
i_n_�ace
is
generally varies in different
are called lattice points
these atoms are locatep. Such points in
about
atom� at
three-dimenThus
Lattice.
ais
v
dJlr.a
e
�E._l
ej
ic
tt
a
l
the
tile lattice points are identical,
Sloilal spae l�ttice may be defined as finite array of points in three-dimensions in
which every point has identical environment as any other point in the array.
129
Engineering Physics
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Let us now consider the case of a two­
dimensional array of points as shown in fig. 6.1. It
is obvious from the figure that environment about
any two points is the same and hence it represents
a lattice. This is explained as follows:
We chose any arbitrary point as origin
z
•
bL:
0
••
•
.
�n
1
•
a
•
•
d consider. the positiov vectors �
and
·r;
.
of.
any two lattice points by joining them to 0 as shown
•
in fig. 6.1. If the difference f of the two vectors
_
and
satisfies the following relations
Fig. 6.1
�
G
-->
T= n1a+n2b
where n1 and n2 are integrals� and
b
are fundamental translation vectors char­
acteristics of the array, then the array of points is a two dimensional lattice. For three
dimensional lattice:
f
n1
a + n2 6 + n3 c
Hence, it should be remembered that a crystal lattice refers to the geometry of
set of points in space whereas the structure of crystal refers to actual ordering of its
constituent ions, atoms, molecules in the space.
�THE BASIS AND CRYSTAL STRUCTURE:
0
0
0
0
0
0
0
0
•
0
�
0
0
0
0
Fig. 6.2
0
crystal structural is real, while the lattice is
imaginary. Thus
0
0
the basis or the pattern. When the basis is
tions, it gives the actual crystal structure. The
0
0
or more atoms (i.e. a unit assembly of atoms
repeated with correct periodicity in all direc-
0
0
0
0
0
0
0
ture, we associate every lattice point with one
or molecules identical in-composition) called
.0
0
0
0
0
0
0
0
For a lattice to represent a crystal struc-
0
lattice + basis
0
=
crystal structure.
Fig. 6.2 shows the basis or pattern representing each lattice point. It is observed from
130
Crystallography
-
I
the figure that a basis consists of three different atoms. It can also be observed that the
basis is identical in composition, arrangement and orientation.
l.k.41fNIT CELLS AND LATTICE PARAMETERS:
solids indicates that the small groups of atoms
The atomic order in
from a repetitive pattern. Thus, in describing crystal structures, it is often convenient
to subdivide the structure in to small repeat entities called
c
crystal some fundamental group­
unit cells, i.e. in
grouping of particles is
.
called a unit cel-·
Unit cells for most crystals are parallelopiped
or cubs having three sets of parallel phases. A unit cell
is chosen to represent the symmetry of the crystal struc­
ture, wherein all the atom positions in the crystal may
be generated by translations of the unit cell integral
Fig. 6.3
distances along each of its edges. Thus, the unit cell is
the basic structural unit or building block of the crystal
structure by virtue of its geometry and atomic positions within. Furthermore, more
than a single unit cell may be chosen for a particular crystal structure; however, we
generally use the unit cell having the highest geometrical symmetry. Unit cell may also
be regarded as the 'building blocks' that make up the crystal, each one indistinguish­
able from the next.
The unit cell is a parallelepiped formed by three non-coplanar vectors a, b and c.
The unit cell possesses all the structural properties of a bulk crystals. The geometry of
typical unit cell is shown in fig. 6.3. The primitive vectors a, band c defines the lengths
of three edges of the unit cell and represent the crystallographic axes. Each edge of
unit cell is the distance between the atoms of the same kinds and is often called the
formed
lattice constant. The
b and c is denoted by a , be­
betweel! a
tween a and c
y . The axial lengths a, b and c and the three
angles a, p �nd y are kn?wn lattice parameter of the unit celJ.
6.5 PRIMITIVE CELL:
In the literature, references to the unit cells and primitive cells are often made.
Primitive cell may be defined as geometrical shape which, when repeated indefinitely
in three-dimensions, will fill all space and is the equivalent of one lattice point i. e. the
unit cell that contains one lattice point only at the comers is known as 'primitive cell'.
131
Engineering Physics
The unit cell differs from the primitive cell in that it is not restricted to being the
equivalent of one lattice point. In some cases, the two coincides. Thus, unit cells may
be primitive cells, but all the primitive cells need not be unit cells.
��HE SEVEN CRYSTAL SYSTEMS AND BRAVAIS SPACE LATTICE:
There are number of ways in which an actual crystal structures may be built. While
the possible crystal structures are unlimited, the possible schemes of space lattices are
however highly restricted. Each space lattice has some convenient set of axes which
Table 6.1 Crystal Systems and Bravais Lattices
Cubic
Bravais lattice
lengths and angles
System
Three equal axes at right angles
a
b
c, a
y
J3
90°
Simple
Body- centered
Face- centered
Tetragonal
Three axes at right angles,two equal
a
b * c,a
y
!3
90°
Simple
Example
Au,Cu,
NaCI, CaF2,
NaCJO"
Sn02, Ti02,
Body- centered
KnO,,
Three unequal axes at right angles,
a
Rhombohedral
or
b * c,
!3
a
y
90°
a b
Base -centered
MgS04
Face-
c, a
!3
As,Sb, Bi,
Simple
y * 90°
Two equal coplanar axes at 120°,
third axis at right angles
a
Monoclinic
BaS04,
Three equal axes,equally inclined
Trigonal
Hexagonal
*
b * c, a
!3
90°, y
Calcite
Si02,
Simple
Zn,Mg,
Cd,Agl
120°
Three unequal axes,one pair
CaS04 2H p
not at right angles
FeS04,
Na2S04
a
*
b
*
c, a
y
90°:t: !3
.
Simple
Base-- centered
Three unequal axes,unequally
Triclinic
,
Body- centered
inclined and none at right angles
a * b * c, a
*
!3 * y * 90°
132
Simple
K2Cr207
CuS04.5Hp
Crystallography- I
SIMPLE
CUBIC (P)
BODY-CENTERED
CUBIC (I)
FACE-CENTERED
CUBIC (F)
SIMPLE
TETRAGONAL
BODY-CENTERED
TETRAGONAL
SIMPLE
ORTHORHOMBIC
BODY -CENTERED
ORTHORHOMB_IC
(P)
(/)
(P)
(I)
BASE-CENTERED
ORTHORHOMBIC
FACE-CENTERED
ORTHORHOMBIC
RHOMBOHEDRAL
(C)
(F)
SIMPLE
MONOCLINIC
(P)
HEXAGONAL
·(R)
BASE-CENTERED
1\IONOCLINIC (C)
.t;.,(} .g!>I
133
(P)
TRICLINIC
(P)
Engineering Physics
need not be necessarily orthogonal. Further, the chosen units of length along the three
axes are not necessarily equal. The set of values that the six lattice parameters a, b, c,
ft,
p andy can take are limited to seven only and accordingly, all crystal
structures can
classified in to seven crystal systems.
If atoms are associated with only the comers of corresponding unit cells, there
could be only seven types of lattices. Such unit cells would be primitive cells. How­
ever, in addition to the seven primitive cells, there exist seven more non primitive cells
of three different types. The three types of non primitive cells are: body centered cell,
face centered cell and base centered cell. The four types of unit cells possible are thus:
i. simple, ii. body centered iii. face centered and iv. base centered cell.
In 1848, Auguste Bravais, the French crystallographer proved that there are
only fourteen space lattices in total which are required to describe all possible arrange­
ment of points in space subject to the condition that each lattice point has exactly
identical environment. The 14 space lattice are called Bravais lattices. The fourteen
Bravais lattices are described in table 6.1 and illustrated in fig. 6.4.
7 THE TWENTY THREE SYMMETRY ELEMENTS IN A CUBIC CRYS­
TALS:
Crystal possesses different external symmetries which are described by certain
operations. A symmetry operation is one that takes the crystal in to a configu­
ration identical to the initial configuration.
The crystal is said to possess a symmetry elements corresponding to an opera­
tion, if after performing the particular operation the crystal goes in to position indistin­
guishable from the initial position.
The main symmetry elements of crystalline solids are:
1.
Centre of symmetry or Inversion centre.
2.
Plane of symmetry or Reflection symmetry or Bilateral symmetry.
3.
Axis of symmetry or Rotation axis
6.7.1 Centre of Symmetry or Inversion Centre :
A centre of symmetry is such a point in the crystal that any straight line drawn
through this point intersects the crystal surface at equal distances in both directions
and joins identical points in the crystal. It is also called as 'inversion centre'.
134
Crystallography- I
Let us consider the cube shown in fig. 6.5. If
the body centre point and body diagonals are drawn
through it, each diagonal connects identical lattice points
located at equal distances and in opposite directions from
centre point. The centre point acts as a point mirror which
generates the second lattice point at an equal distance in
the opposite direction. Therefore, central point is the cen­
tre of symmetry or inversion point; which is located at
Fig. 6.5
the centre of the body.
6.7.2: Plane of symmetry or Reflection symmetry or Bilateral symmetry:
Highly regular crystal may be bilaterally symmetrical about several planes cut­
ting them in different directions. They may have several planes of symmetry.
There is one important characteristic,
however, about a crystallographic plane of
symmetry which differentiates it from our
ordinary conception of a plane geometrical
symmetry.
[he plane must be such that it
divides the crystal in to two equal portions,
but these two portions must be so situated
that they are mirror images of each other
Fig. 6.6
the
called plane of symmetry. There are three
�-·
��
planes of symmetry parallel to the faces
Fig 6. 7
of the cube.
The three straight planes of symmetry in a cube are shown in fig. 6.6.
Additionally there are six diagonal plane of symmetry which is shown in fig. 6. 7.
135
j
Physics
This plane " tormed by a pair of opposite parallel edges. Since there are six
'
such pair«
1:dges, the number of
diagonal planes of symmetry is six. Hence the
tlJtal number of diagonal planes of symmetry is six. Hence total number of planes of
symmetry is
6. 7.3:
(3+6)=9.
Axis of symmetry or Rotation axis:
The
axis of symmetry may thus be defined as a line such that the crystal
assumes a congruent position for every rotation of
[ r.
The value of n decides the
fold of the axis.
If
n =
If n
=
This rotation axis is known as axis of symmetry.
1, crystal has to rotate through 360° to achieve self coincidence. Such an axis is
known as identity axis.
2, crystal has to rotate through 180° to achieve self coincidence. Such an axis is
known as
If n
=
3, crystal has to rotate through 120° to achieve self coincidence. Such an axis is
known as
If n
=
=
triad axis.
4, crystal has to rotate through 90° to achieve self coincidence. Such an axis is
known as
If n
diad axis.
tetrad axis.
6, crystal has to rotate through 60° to achieve self coincidence. Such an axis is
known as hexad axis.
Cubic crystal show only one, two, three, four and six fold symmetry. Cubic
crystals do not show five fold and more than six fold symmetry. Cube possesses only
three axis of four fold symmetry.
6.7.3.1:
Tetrad:
If cube is rotated about a line perpendicular to
one of its faces at the mid-point, it will tum in to a congru­
ent position by every rotation of 90° i. e. 4 positions during
compete revolution. This normal is thus an axis of four-fold
symmetry i. e. tetrad axis which is normal
to
each of the
three pairs of parallel faces. The three tetrad axes of cube
are shown in fig. 6.8.
Fig. 6.8
136
Crystallography
6.7.3.2:
-
I
Triad:
Let the cube be now rotated about solid diago­
nal (body diagonal) through 120° to get congruence,
and such a line, is therefore triad axis. The various po­
sitions (four) are shown in fig. 6.9.
6.7.3.3:
Diad:
Let the cube be
now rotated about an
axes joining the midpoints of a pair of op-
t;
Fig. 6.9
posite parallel edges
proves to be diad axis. There are six such axes present
in cube as shown in fig. 6.10.
�
Fig. 6. 10
The total numbers of crystallographic symme­
try elements of the cubic system are summarized as follows:
1.
Centre of symmetry
2.
Plane of symmetry
1
Straight plane
3
Diagonal plane
6
9
3.
Axes of symmetry
Tetrd axes
3
Triad axes
4
Diad axes
6
13
23
==
Thus there are
TWENTY THREE �ymmetry elements in cubic crystals.
�HE UNIT CELL CHARACTERISTICS:
The unit cell is characterized by a number of atoms per unit cell, atomic
radius, coordination number, packing factor etc. These parameters can be computed
for simple cube (sc), body centered cube (bee) and face centered cube (fcc).
137
Engineering Physics
6.8.1
Number of Atoms per Unit Cell:
6.8.1.1 Simple cube (sc):
A simple cube has eight lattice points at its eight
comers, which are occupied by eight atoms. In the three
dimensional arrangement, each comer atom is linked to
eight surrounding unit cells. It therefore contributes
equally its volume and mass to the eight adjacent cells.
An isolated simple cube cell indicating the actual con­
tribution of each comer atom is to the unit cell is shown
in fig. 6.11. It is seen that each comer in effect contrib­
Fig. 6.11
utes only 1/81h of its content to a unit cell.
Total no. of atoms in sc = 1/8 atoms per comer X 8 comers of unit cell
= 1 atom I unit cell.
6.8.1.2 Body Centered Cube (bee):
A bee has eight atoms at eight comers of the cube
and one atom within the volume of the cell. An isolated
bee cell is shown in fig. 6.12 which indicates the actual
contribution from the atoms. The atom at the centre of the
body of the cell cannot be shared by the adjacent cell and
therefore contributes fully its volume and mass of unit cell
in which it is located. The atom at the comer contribute
each 1 /8
th
Fig. 6.12
share.
:. Total no. of atoms/bee cell= 1 X body centre atom/unit cell+ 1 /8 atoms/comer
X 8 comers of unit cell
= 1+ 1
= 2 atoms I unit cell.
6.8.1.3 Face Centered Cube (fcc):
The face centered cubic unit cell is non primitive cell having six atoms at the
centre, six faces and eight atoms at the eight comers of the cube. An isolated fcc cell is
shown in fig. 6.13. Each face of the cell is common to two adjacent cells. Therefore,
each face centered atom contributes only half of its volume. Each comer atom contrib-
138
Crystallography
utes 1/8
111
-
l
of its content.
Total no. of Atoms in fcc crystal=
=
112 atom/face X 6faces/cell +
1/8atoms/corner X Scorners/cell
=3+1
=4 atoms I unit cell
6.8.2 Atomic Radius (r) and Nearest Neighbor Dis­
Fig.6.13
tance (2r):
The atomic radius (r) is defined as half the distance between nearest neigh­
bors in the crystal of pure element. Generally it is expressed in terms of cube edge 'a'.
The distance between the centres of two nearest neighboring atoms is called
'nearest neighbor distance'. It will be 2r, ifr is the radius of the atom.
6.8.2.1 Simple cube (sc):
I
I
'
'
'
'
Nearest neighbor distance=2r=a
Atomic radius= r= a/2
'
I
I
'
I
/
6.8.2.2 Body Centered cube (bee):
'
,,
'
' ----� /
:
I
We can calculate atomic radius of bcc using fig. 6.15
(AC)2 = a2
+
a2
Fig. No. 6.14
2a2
(CD)2
(AC)2 + (AD)l
2a2 + a2
3a2
:. CD= .J3 a
:. 4r =
.J3
a
Nearest neighbor distance i.e.2r =
139
Engineering Physics
J3a
2
}\ tom1c rad.ms
r
.
J3a
4
:
6.8.2.3
.
a
One side of unit cell
r
J3
Face Centered cube (fcc):
We can calculate atomic radius of fcc us­
ing fig. 6.16
(BD)2
(CDf + (BCf
a2 + a2
2a2
BD
4r
J2a
J2a
· hbor d.1stance 1.
. Nearest ne1g
·
e. 2r
J\tomic radius
J2a
r
One side of unit cell
6.8.3
J2a
-2
4
a
4
r
J2
Coordination Number (N):
The coordination number is defined as the num­
ber of equidistant nearest neighbors that an atom has in
the given structure. Greater is the coordination number,
more closely packed up will be the structure.
Fig. 6.17
140
Crystallography
-
I
6.8.3.1 Simple cube (sc):
any atom 'A' in sc cell, there would be six equally spaced nearest neigh­
bor atom each at a distance 'a' from that atom. Four atoms lie in the plane of atom,
while one is vertically above it and one vertically below as shown in fig. 6.17. Any
other atom in the lattice would be at a distance greater than 'a' from the atom umk1
consideration.
:. Coordination number of sc
N
6.
6.8.3.2 Body centered cube (bee):
In bee the comer atoms do not touch each other. How­
ever, each comer atom is in contact with the bod): cen­
tered atom. As there are eight unit cells around each
corner, the ato m located at each comer would be si­
Fig. 6.18
multaneously touching the eight body centered atoms
around it. The same is the case for a body centered atom
as shown in fig. 6.18. It is in contact with all the eight
comer atoms.
Coordination number of bee
N
8.
6.8.3.3. Face centered cube (fcc):
z
X
(a)
)-.y
Fig. 6.19
(b)
In fcc each comer atom is in contact with the face centered atom. It would be
simultaneously touching four atoms in the xy plane, four atoms in yz p lan e and four
atoms in zx plane, making up of 12 atoms.
141
Engineering Physics
Coordination number of fcc=N= 12.
fcc cell has maximum value for the coordination number.
�:8.4 Packing Density or Packing Fraction or Packing Factor:
It is defined as the fraction of the space occupied by atom in a unit cell. or It is ratio
of volume occupied by atom in a unit cell (v) to the total volume of unit cell (V)
I.e.
P=­
f
v
v
6.8.4.1 Simple cube (sc):
No. of atoms per unit cell = 1
4
Volume of ONE atom
=V =
One side of unit cell
=a= 2r
Volume of unit cell
=V= a3
Packing density
-
3
1U
3
-
'
4
�
Jtr "
=3
6
0.52
52%.
6.8.4.2 Body centered cube (bee):
No. of atoms per unit cell =2
Volume ofTWO atoms
=v= 2X
One side of unit cell
=a = J3
142
4r
4
-n:r
3
3
Crystallography
=V=
Volume of unit cell
2x
Packing density
a
4
3
3
nr
3
a
J3n
8
=0.68
= 68%.
6.8.4.3
Face centered cube (fcc):
No. of atoms per unit cell=4
Volume of FOUR atom
=v =4 X
One side of unit cell
=a=
Volume ofunit cell
=V= a3
4
3
4r
J2
4
3
4x-nr
3
:. Packing density
a3
=
.fin
6
= 0.74
74%.
143
nr3
-
I
Engineering Physics
The characteristics of these three types of cubic unit cell are summarized
in Table· 6.2
Table: 6.2
I
Sr.
No.
1
2
Properties
sc
bee
fcc
Volume of unit cell (V)
a3
a3
aJ
I
2
4
No. of atoms
unit cell
1
No. of atoms per unit
4
5
6
7
8
2
-
volume
a3
Coordination number ( N )
Nearest neighbor distance
(2r)
a3
aJ
8
12
a
--
aJ3
--
2
2
aJ3
--
-
--
2
Lattice constant (a)
7r
-
6
a-Jl
a-Jl
4
4
4r
4r
J3
J3Jr
--
J2
J2Jr
--
-
2r
Atomic packing factor
-
6
a
Atomic radius ( r )
4
-
8
6
0.52
0.68
0.74
52%
68%
74%
6.9 RELATION BETWEEN THE DENSITIES OF CRYSTAL MATERIAL &
LATTICE CONSTANT IN CUBE LATTICE:
The dimensions of unit cell or the inter-atomic distance in a crystal lattice can
be computed from knowledge of:
1. Molecular weight of crystalline compound (M)
2. Avogadro's number (N)
3. Density of material (p)
4. Its crystalline form.
Consider cube crystal of lattice constant
Let
n - number of atoms per unit cell.
p - the density of crystal material.
A - atomic weight of the material
N- Avogadro's number
Alp - the material will contain n atoms.
144
=
a
Crystallography -
n atoms in a unit cell will occupy volume=
Thus
aJ=
[ )(
nA
pN
a
where
I
J = nA
p
N
a - is lattice parameter
p - density of crystal
n - no. of atoms per unit cell
A - Atomic weight of crystal
Based on the above formula the lattice parameter for bee and fcc can be calculated as:
6.9.1
Body centered cubic lattice (bee):
e. g. a - iron crystal
Atomic weight of a-Iron crystal= 55.85
Density of a- iron= 7.86 gm./cm3.
Mass of each molecule=
Atomic Weight
Avogadro'sNo.
55.85
6 .0 2x i 023 gm.
Number of atoms per unit cell=
Mass of TWO atoms=
2 X 55.85
.
6.02 X 102'
2
(1)
------------------
Length of unit cell = a
Volume of unit cell= a3
Mass of unit cell = a3 X 7.86 gm.
--------------------(2)
Equating equation ( 1 ) and (2)
a3x?.S 6=
2x55.85
6.02xl023
a= 2.87
A
{ FormulaUsed.is ( a3p= }whereA-AtomicWeight}
1 45
Engineering Physics
6.9.2
··--· -··- -·-·
Face centered cubic lattice (fcc):
e. g. NaCl
Molecular weight ofNaCl = 23
=
Density ofNaCI
=
35.5
+
58.5
2.18 gm./cm3.
Mass of each molecule =
M olecularW eight
A vagadro'sN umber
58.5
g m.
,
.c
Number of molecules in unit cell= 4
Mass of 4 Molecules=
Fig. 6.20
4x58.5
---------(1)
6_02x 1023 gm.
Length of unit cell= a
Volume of unit cell= a3.
Mass of unit cell = a3 X 2.18 gm.-------------(2)
Equating equations (1) and (2)
3
a x2.18
a=
6.02x
5.63
A
The distance between adjacent atom i.e. Na and Cl = d
d
{
(
6
5. 3
2
= 2.815
A
3
Formula Used is a p=
M-Molecular Weight of Crystal
}
6.10 LATTICE DIRECTION:
The direction of any line in a lattice may be described by first drawing a line
through the origin parallel to the given line and then giving the coordinates of any
point on the line through the origin.
146
Crystallography
Let the line pass through the origin of the unit cell and any
-
I
having coor­
dinates u, v, w where, these numbers are not necessarily integral. Then uvw is written
in square bracket as [uvw]. These are the indices of the direction of the line. They are
.tlso the indices of any line parallel to the given line.
Whatever, the value of u, v, w they are
[233)
always converted to a set of smallest in­
tegers by
or division
throughout: e.g.
f?24]
[Yz Yz 1], [ 112]
[112] is the
c
and
all represent the same direction,
form. Nega­
tive indices are written with a bar over
the number. e.g.
[ uvw ] .
Direction in­
dices are illustrated in fig. 6.21.
[lZOJ
Fig.
Note
how one can mentally shift the origin, to
avoid using adjacent unit cell, in finding
6.21
a direction like
Directions related by summary are
[ 12o] .
directions of a form, and a set of
these are represented by the indices of one of them enclosed in angular bracket i.e.<>,
e.g. the four
diagonals of a cube
sented by the symbol
[111 ] , [ i 1 ], [iiI]
and
<111>.
[ ll 1 ]
may all be repre­
6.11 MILLER INDICES:
Miller indices are defined
the reciprocal ofthe fractional intercepts which
the plane makes with the crystallographic
Miller
of a crystal face or plane is given by three smallest figures which
are inversely proportional to the
Miller evoJ
p arameter s of the fa c e
a method to desi gn ate a plane in a
.
by three numbers
(hld) known as Miller
The
in the determination of
the
147
indices of a
are illustrated with
Engineering Physics
z
3C
2C
y
Fig.
6.22
X
i) Determine the coordinates of the intercepts made by the plane along the three
crystallographic axes (x, y,
z
axes).
y
z
2a
3b
c
pa
qb
rc
X
ii)
(p
=
2, q
=
3 and r
=
1 )
Express the intercepts as multiples of the unit cell dimensions, or lattice pa-
rameters along the axes, i. e.
2a
3b
c
a
b
c
2
3
iii) Determine the reciprocals of these numbers:
2
iv)
3
Reduce these reciprocals to the smallest set of integral numbers and enclose
them in brackets:
1
6x2
( 3
1
6x3
2
148
6xl
6)
Crystallography
-
I
In general it is denoted by (hkl). We also notice that:
1
-
:
1
-
q
p
:
1
r
-
=h :k :l
I 1 I
-·-·
.
2 3 I
.
3·2·6
.
.
i.e. ( 3 2
6)
Thus Miller indices may be defined as the reciprocals of the intercepts made
by the plane on the crystallographic axis, when reduced to smallest numbers.
6.11.1 Important Features of Miller Indies of Crystal Planes:
i)
All the parallel equidistant planes have the same Miller indices. Thus the Miller
indices define set of parallel planes.
ii)
A plane parallel to one of the coordinate axes has an intercept of infinity.
iii)
If the Miller indices of two planes have
same ratio ( i.e. 844 and 422 or
211 ), then planes are parallel to each other.
iv)
If (hkl) are the miller indices of a plane, then the plane cuts the axes into h, k
and 1 equal segments respectively.
v)
A plane passing th�ough the origin is defined in terms of a parallel plane
having non-zero intercepts.
vi)
The angle 8
between two directions [ u1 v1 w1] and [ u2 v2 w2] can be
basically calculated from the expression:
cos 8 =
u1u2+v1v7+w1w2
2 )112
2
( u12+v12+w12 )112 ( u2+v;+w2
-
1
vii) The normal to the plane with index number (hkl) is the direction [hkl].
This fact can be utilized to find the angle between two planes by using the
above relation.
viii) The distance d between adjacent planes of a set of parallel planes of the
indices (hkl) is given by:
d=
a
,
h2+k2+1"
149
where a is edge of cube.
l:!-ngineering Ph\·,ics
PLAI'�ES:
T}
or entation of planes in a lattice may also be represented symbolically ac-
:ording to a system polularized the English crystalographer Miller. In general case, the
given plane will be fitted
with
respecti v e
the
crystalographic axes and
ct
since
(Ilk/)
the s e
co nvinetnt
frame
from
of
referance, we might de­
].,\
scribe the orentation of the
u
plane by
(a)
axs
Fig. 6.23
giving the actual
distance measured from the
lh\
origin at which it intercepts
the t hree axes one of the typical plane is shown in fig. 6.23.
We may determine the Miller indices of the plane shown in fig. 6.23 as follows:
4A
1A
Axial lengths
Intercept lengths
Fractional Intercepts
Yt
4
Miller indices
sA
4A
3A
3A
Y2
1
2
1
:. Plane is (421)
As mentioned earlier, if a plane is parallel to a given axis, its fractional inter­
cept on that axis is taken as infinity and the corresponding Miller indices is zero. If
plane cuts a negative axis, the corresponding index is negative and written with bar
over it. Planes whose indices are the negatives of one another are parallel and lie on
opposite sides of the origin e. g. (21 0)
and (210) . The planes ( nh nk nl) are parallel
to the planes (hkl) and have 1/n th spacing . The same plane may belong to two different
sets, the Miller indices of one set being multiplies of those of the other; thus same
plane belongs to the (21 0) set and the ( 420) set and in fact, the planes (21 0) set from
every second plane in the ( 420) set.
In cubic system it is convenient to remember that the direction
perpendicular to the plane
(hkl)
[hld]
is always
of the same indices, but this is not generally true in
other systems. Further fa miliarity with Miller indices can be gained from the study of
fig. ?.24, 6.25 and 6.26.
150
15; l
i(I(J_l.i
((1()1;\
!IIJJ
((JO])
UO!):aJ!O
<0 �D
z
(ZOI)
(OII)
Coo<:)
(OOI)
Engineering Physics
6.13 SEPARATIO � BETWEEN LATTICE PLANES IN A CUBIC CRYSTAL:
The cube edge is a. Let (hkl) be the Miller indices of the plane A B C. This
plane belongs to a family of planes whose Miller indices are (hkl) because Miller indi­
ces
represent a set of planes.
Let ON= d1 be the perpendicular distance of the planeAB C from the origin.
'
Let a , J3' and y' (different from the interfacial angles a, J3 and
between coordinate axes X, Y, Z respectively and ON.
z
z
Fig. No. 6.27
152
y) be the angles
I
The intercepts of the plane on
a
=h
From fig.
6.27(a),
I
,
three axes are:
)B a
a
( = k amiOC , = �-----------------(
1)
we have:
dl
cosa=- cos
OA'
From fig. 6.27(b),
dl ------(2
dl
[J' -an
d cosy=--=
)
OC
OB
+l+ z2
d
a' + i� cos2 fJ' +
) (cos� r·)]
[ (cos� ) (
(ON)2 =d:'- = [dl:: (cos2a' ) +dl2 (cos2 fJ' ) + di� (cos2 r· )J' :'
dl = dl [cos2 a'+ cos'- fJ'+cos2 rl'"
dl2
Le.
= dl2
cos2 a'+cos1 fJ'+cos::
Thus
y'= 1--------------(3)
this in equation (2), we get:
d
l
]:' [ ]:: [ dl ]:'
[
[ r
OA
dl
+
d11
OB
"
0
Let OM
first plane ABC.
=1
�I
=1
a�
d( = "
=
OC
[0)
+k- +1-
0
a-
dI
+
+
-,
h
h
+k- +1.
,
a
------- ------(4)
-
,
+k- +!o
•
plane PQR parallel to
distance of
The intercepts of this plane on the three crystallographic axes are:
=
d� be the perpendicular
OA'=
and
x2
(ONi
2a
h
OB'=
cosa'=
d,
OA''
2a
k
andOC'=
2a
L
d,
andcosY'=-OB'
I
OC'
d,
cosfJ'=-
[d; cos2 a'+d; cos2 [J'+d� cos2 r]
; = dg [cos" a'+ cos'- fJ'+cos'- r']
(OJ\,1)' =
d
153
-------------(5)
the
Engineering Physics
i.e. cos� a'+cos2 fJ'+cos2 r'= l
substuting this in equation
(5) we get:
[ r[ r[
+
+
d;·
--=:;4a
0
4a2
(
d,
1
(h""+k"+ l')
" =l
Co -
-
=
1
o
h-+k +l
2a
o
+k" +12
-
Thus the interplaner spacing between the two adjacent parallel plane s of
Miller indices (hkl) is given by:
******
Obj e ctiv es Type Questions
1.
Question Bank
Explain in brief: lattice point, space lattice, basis, crystal structure, primitive
4 each
2
cell.
3
Explain in brief: Unit cell and lattice parameters.
4
radius, coordination number, packing density.
5
Marks
Calculate following unit cell parameters in sc, bee & fcc crystals: atomic
IdentifY I show various lattice directions and planes in given diagram.
4 each
2 each
List out various features of Miller indices
Descriptive Type Questions or Notes
1
2
3
4
5
Write note on: the seven crystal system and Bravais space lattice.
Write note on: twenty three symmetry elements of cube.
Derive relation between the density of crystal material and lattice constant in
cube lattice. Illustrate your answer using bee and fcc crystal lattice.
Write note on Miller Indices.
Derive relation to determine separation between
Numericals will be asked based on articles such as: the unit cell
relation between the density of crystal material and lattice constant, separation
154
5
8
8
8
8
planes in cubic lattice. 8
Numericals
between lattice planes in cubic crystal etc.
4
Crystallography II
-
Chapter 7
CRYSTALLOGRAPHY
•
II
7.1 INTRODUCTION:
This chapter deals with the study of the phenomenon ofx-ray diffraction. There
was little knowledge of interior structure of the various crystals known. These crys­
tals are built up by periodic repetition of atoms or molecules and these are situated 1
to 5 A0 apart. On the other hand, there were indications, that x-ray might be electro­
magnetic waves about 1 or 2 A0 in wavelength. In addition, the phenomenon of dif­
fraction was well understood, and it was known that diffraction, as of visible light by
ruled grating occurred whenever wave motion encountered a set of regularly spaced
scattering objects, provide that wavelength of wave motion was of the same order of
magnitude as the repeat distance between the scattering centers.
German physicist von Laue (1879-1960) in 1912 came to conclusion that, if
crystals are composed of regularly spaced atoms which might act as scattering centers
for x-rays and if x-rays were electromagnetic waves of wavelength of the order of
inter-atomic distance in crystal, then it should be possible to diffract x-rays by means
of crystals. He has successfully recorded diffraction pattern on photographic plate, in
year 1912. The diffraction pattern recorded, further called as Laue spots.
7.2 X-RAY DIFFRACTION:
X-rays are electromagnetic waves like ordinary light; therefore they should ex­
hibit interference and diffraction. The wavelength ofx-rays is of the order of 1A0, so
the ordinary devices such as ruled diffraction grating do not produce
observable
effects with x-rays.
In 1912, German physicist Laue suggested that a crystal which considered of
three dimensional array of regularly spaced atoms could serve the purpose of a grat­
ing. Crystal consists of parallel equidistant planes passing through these lattice points,
which are known as lattice planes. The orientation of these planes lies in various (all)
directions as shown in Fig. 7 .1. Here the crystal differs from the ordinary grating in
the sense that the diffracting center in the crystal are not in one plane. Hence, crystal
act as space grating rather than a plane grating.
155
Engineering Physics
On the suggestion of
'
l
(b)
Laue, his associates, W. Friedrich
�
and P. Knipping in diffracting x­
•
I
rays by passing them through a
thin crystal of zinc blend. The
•
diffraction pattern obtained con­
sists of central spot and series of
•
(a;
spots arranged in definite pattern
�
•
____
•///
•
/
,
/•
//
\C)
about the central spot. This symmetrical pattern of spots is known
•
as Laue pattern or Laue spots
and proves that x-rays are elec­
Fig 7.1
tromagnetic radiations. Von Laue
was awarded with the Nobel Prize in Physics in 1914 for this work.
A simple interpretation of diffraction pattern was given by W.L. Bragg Accord­
ing to him the spots are produced due to the reflection of some of the incident x-rays
from the various sets of parallel crystal planes (called Bragg's planes ) which contain
a large number of atoms.
7.3 LAUE METHOD:
X-rays produced by an x-ray tube are defined in to a narrow beam by a set of
lead screens S1 and S2 having pin holes at their centers. A thin crystal Cis mounted in
the path of x-ray beam and a photographic film is positioned beyond it, as illustrated
in fig. 7.2
Photographic
plate
•
•
•
•
••
•
•
Undiffracted
beam
(a)
Fig. 7.2
156
••
•
(b)
•
•
•
Crystallography-
II
As the ray beam penetrates the crystal C some of the rays are scattered or de­
flected away by atoms from their initial direction. The scattered x-rays emerge from
the crystal in specific direction as highly narrow beams and they are intercepted by the
photographic film. On developing the exposed film a pattern of bright spots corre­
sponding to maximum intensity are observed. They are more commonly referred
as
Laue spots.
The pattern of Laue spots is uniquely characteristics of the crystal C. The cen­
tral bright patch on the film corresponds to the main unscattered beam. A hole is often
cut in the film so that the central spot is not recorded. In the experiment conducted by
W. Friedrich and P. Knipping, the wavelength of x-rays was determined from the
measurement of angular position of Laue spots and from the knowledge of the separa­
tion of the atoms in the crystal. This method has been subsequently used to determine
the crystal structure using x-rays of known wavelength.
7.4 BRAGG'S LAW:
William Henry Bragg (1862-1942) and William Lawrence Bragg (1890-1971),
the father and son team of British
physicists, derived in 1913 a simple equation
wavelength of x-rays to the angular
relating the
p
position of the scattered beams and
R
Q
s
Plane 0
Plane
Plane
crystal. They were awarded by Nobel
P rize in physics in 1915 for their ser­
t
1
d
Plane
the separation of atomic planes in the
t
11
vices in analysis of crystal structure
by means of x-rays.
A crystal may be regarded as a
stack of parallel planes of atoms. The
111
atomic planes are often called Bragg
Fig. 7.3
planes. Every crystal has several sets
of Bragg planes oriented in different directions and each plane in a given set has the
same distribution of atom. Two different sets of Bragg planes are shown in fig. 7.3
Consider a ray PA refracted at atom A in the direction AR from plane I and
another ray QB refracted at another atom B in the direction BS.
157
Engineering Physics
Now, from another atom A, draw two perpendiculars AC and AD on QB and BS
respective1y. The two refracted rays will be in phase or out of phase depending on the
path daTerence.
If the path difference (CB+BD) is wavelength A or multiple of wavelength A i.e.
nA., then two rays will reinforce each other and produce an intense spot.
Thus condition of reinforcement can be written as BC+BD=nA
From fig. 7.3
Consider !! ABC
.
sm
B
BC
=-
AB
BC=AB. sine
BC =d sine
Similarly consider !! ABD
. 8
sm=
BD
AB
BD =AB. sine
BD=d sine
where e is the angle between the incident ray and the planes of refraction
( i. e. glancing angle)
2d sine = n A ----------- ( 1)
where d - the interplaner spacing and
n= 1, 2, 3, ........... etc. stands for P', 2nd, 3rct order maxima respectively.
The above equation
(1)
is known as Bragg's Law.
Different directions in which intense refraction will be produced can be ob­
tained by giving values toe i.e.
158
Crystallography
For P' maxima,
sin 81
For 2nd maxima,
sin 82
For 3n1 maxima,
sin 83
=
=
=
-
II
'A I 2 d
2f. I 2 d
3f.l 2 d
......... and so on.
It should be remembered that the intensity goes on decreasing as the order of
spectrum increases.
Note: Generally Braggs scattering is regarded as Bragg's reflection and hence
are known as Braggs planes. At certain glancing angles, reflections from these sets of
parallel planes are in phase with each other and hence they reinforce each other to
produce maximum intensity. For other angles the reflection from different planes are
out of phase and hence they reinforce to produce either zero intensity or extremely
feeble intensity.
�RAGG'S X-RAY SPECTROMETER:
Bragg devised an apparatus known as x-ray crystal spectrometer where the crystal
was used not as transmission grating, but as a reflection grating. The schematic ar­
rangement of Bragg's spectrometer is shown in fig. 7.4
c
s1 s2
Fig. 7.4
159
Engi>1eering
X-rays fred
obtam
a narrow
X-ray tube were allowed to pass
slits S1 and S2 so as to
beam which is then allow to strike a single crystal D mounted on a
The crystal is rotated by means of the turn-table so as to increase the glancangle at which X-rays are incident at the exposed face of the crystal. The photo­
t:;raphic plate or ionization chamber is used for measuring the intensities of the re­
flected rays. The angles, for which reflection intensities are maximum, give the val­
ues of 8 of equation nA
crystal,
n =
occurs n
=
=
2d sin8. The process is carried out for each plane of the
1, creates the condition for the lower angle at which the lowest reflection
2, creates the condition for next higher angle at which maximum reflection
occurs and so on.
Thus,
for n
for n
for n
=
=
=
1,
A
2,
2A
3,
3A
=
2d sin 81
=
=
2d sin 82
2d sin 83 , etc.
where 81' 82 and 83 are glancing angles for n
=
1, 2 and 3 respectively.
Now,
Hence, by measuring glancing angles at which reflection occurs, we can deter­
mine the interp1aner spacing knowing the wavelength of x-rays. If the above propor­
tionality is verified the assumption ofBragg's
theory that x-rays are reflected like ordinary
light, gets proved.
..
c
Col
From the graph of glancing angle 8 and
''-
ionization current as shown in fig. 7 .5, the
:J
u
.
glancing angle 81,82, 83 or first, second and
c
0
third order reflections are measured. It can
�
..
0
N
·c:
.Q
n=
I
I
be seen that sin 81: sin82: sin 83
I
I
I
I
1
,2
incident
angle
Fig. 7.5
=
1: 2: 3. This
shows that the assumption that x-rays gets re­
flected like ordinary light is justified. Here
we have assumed that the x-ray beam is
monochromatic.
160
Crystallography
-
II
�BRAGG'S LAW AND CRYSTAL STRUCTURE:
We have derived Bragg's law in article number 7.4. According to which
2d sin 8
=
nA
where d - interplaner distance
8 - glancing angle
n - 1. 2. 3
. .
.. etc. for 1st 2nd, 3rct
,
. .
.
.. . .
.
order respectively.
A -wavelength ofx-rays incident on it.
This formula was derived by Bragg pair in 1913. This can be used for determi­
nation of crystal structure.
This structure of rock salt (NaCl) crystal was studied by using Bragg's ioniza­
tion spectrometer and intensity of ionization current was determined for different glanc­
ing angles. If a graph is plotted between intensity of ionization current and glancing
angle, the intensity order of reflection diagram is obtained as shown in fig. 7.5.
The experimental results from Bragg's method have shown that if x-ray from a
palladium anticathode are used, the first order reflection maxima occurred at 5.9°,
8.4° and 5.2° for
(100), (110) and (111) planes respectively.
According to Bragg's law
2d sin 8
=
nA
For 1st order reflection n
=
1 and hence
2sinB
I.e.
doc
1
sin e
Therefore
d100 : d110: d111
d1oo:
d11o:diii=
1
1
1
sinB1
sin e2
sin e3
sin5.9°
161
sin8A-
sin 5.2°
Engineering Physics
9.73: 6.84: 11.04
1:0.703: 1.14
1
d!OO: dl!O: dill
1
1
1
=
The table
(100), (110)
7.1
and
2
} : .J2 : .J3
=
l :.J2:
.J3
2
gives the values of ratios between interplaner distance of planes
(Il l ) for the cubic structure.
Thus it can be confirmed that the so­
dium chloride crystal has a face centered cubic structure.
In case ofKCl crystal, Bragg's obtained maxima of reflected x-rays at glancing
angles
5.22°, 7.30° and 9.05° respectively using three different reflecting planes.
We have
d : d2: d3
l
di : d2: d3
doc
I.e.
1
s in O
1
1
1
sin01
sin02
sin03
1
=
°
sin 5.22
°
sin 7.30
=
10.99 : 7.87 : 6.35
=
1 : 0.716 :0.577
1
d : d 2: d3
l
=
1:
.J2 :
sin 9.05
°
1
.J3
=
l : .J2
:
.fj
This result shows thatKCl is a simple cubic crystal as represented in table
162
7.1.
Crystallography - II
Table 7.1 : Ratios between interplaner distances
I
j
1
1
1
Sr.No.
Type of cubic structure
I
Simple cube
2
Body centered cube
1:
3
Face centered cube
l:.fi: J3
-
d,
-
-
d3
dz
I:.fi: .J3
:
.fi
2
7.7 CRYSTALLOGR APHY BY POWDER METHOD:
Investigation of crystal structure using Laue method is possible only when the
material is available in the form of single crystals of reasonable size. There are many
materials for which it is impossible to obtain single crystals of required size. For such
Exil for
X-rays
Film
Collimalor
(a)
Fig. 7.6
(b)
materials powder photography is highly suitable. One form of powder photography is
known as Debye-Scherrer method invented by P. Debye and Scherrer. In this method
the material under investigation is crushed into a fine grain powder and compressed
into a thin rod or packed into capillary tube.
163
Engineering Physics
Fig. 7.6 shows the principle of Debye-Scherrer powder method. A strip of
is mounted round the inside of a cylindri­
photography film wrapped in opaque
cal drum. The specimen is positioned vertically at the center of the drum. A narrow
beam of monochromatic x-rays enter and leave the drum through the aperature on
opposite
of the drum. The principle of the technique is that the powder consists
of millions of tiny crystals oriented at random in all possible direction. Each crystal­
lite has the some system of atomic planes. Some of the crystallites are bound to lie
with their planes at glancing angle 8 to the incident ray such that Bragg's equation is
satisfied. Each such crystal will produce a spot on photographic plate. Reflections
will be produced by all such crystallites whose normal to the planes from a cone, as
illustrated in fig. 7.7 & 7.8.
Consequently the re­
flected rays will lie in the cone
of semi angle28. The inter sec­
tion of cone formed by the re­
flected rays with the photo­
graphic plate yields a circle as
shown in
7.7.
Such reflection occur
(a)
different sets of crystallites ly­
pointwh�
incident beam
enters (�& = 180")
ing at different angle to the in­
cident beam. Further higher or­
der reflection at 48, 88 etc. also
(b)
occur.
Fig. 7.7
2S - 180"
29
I
The various possible val­
..
OCI
T
ues of28 can be calculated from
Cu
the positions of the arcs and the
radius of the camera drum shows
a
typical pattern obtained in this
w
method.
If xis the distance at which
Zn
a reflected beam strikes the film
from the center then
Fig. 7.8
164
Oystallography
28 =
8 =
-
II
where r is radius of drum
Jr
900 XI
1Cf'
Let xi, x2, x be the distance between symmetrical arcs on the stretched photo­
3
graphic film then 28 I
=
180°x1
Similarly
1[[
82
Le. 8 1
90° X2
=
and
90o xl
1Cf'
83 =
90° x,
1Cf'
Using the value of 8 in to the Bragg's equation, the interplaner distance
ing) d can be
Such a diffraction pattern helps us to
crystalline materials. Amorphous material does not have
amorphous materials from
planes. Therefore
diffraction rings are not produced on the film. However, they may produce smeared
ring as there is some kind of short range order in the arrangement of its molecules.
information may also be obtained from x-ray diffraction regarding the
structure of metals; polymers etc. e.g.
1.
the diffraction rings consists of separate small spots, it means that the
of large size crystallites.
2.
The x-ray
shows
the crystallites are arranged haphazardly or are
oriented along an axis or plane, as occurs in wires and sheets of metals, as in the
case
vegetable fibers, in polymer strands etc. In place of rings, the films show
arcs. If the orientation is of high degree, only tiny spots are obtained in place of
arcs.
*****
165