INTRODUCTION, Lattice points and space lattice, BASIS AND CRYSTAL STRUCTURE, UNIT CELLS AND LATTICE PARAMETERS, CRYSTAL SYSTEMS, CRYSTAL SYMMETRY

1. 1. Crystal Physics
1. INTRODUCTION: A crystal is a solid composed of atoms or
other microscopic particles arranged in an orderly repetitive array.
Three general types of solids:
1. Amorphous – order is limited to a few molecular distances.
2. Polycrystalline – made up of grains which are highly ordered
crystalline regions of irregular size and orientation.
3. Single crystal - have long range order.
Many important properties of materials are found to depend on the
structure of crystals and on the electron states within the crystals.
In addition to the physical properties, understanding regarding the
correlation of the structure of crystals and mechanical, thermal,
electrical and magnetic properties solids is developed. This is
primarily due to the advances in the bond theory of electron states
and in the theory of bonding in solids, which led to the
development of newer and better materials for electrical, electronic
and structural engineering. The study of crystal physics aims to
interpret the macroscopic properties in terms of properties of
microscopic particles of which the solid is composed.
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2. The study of geometric form and other physical properties of
crystalline solids by using X-rays, electron beams and neutron
beams constitutes the science of crystallography or crystal physics.
2. LATTICE POINTS AND SPACE LATTICE: The atomic
arrangement in a crystal is called crystal structure. In a perfect
crystal, there is a regular arrangement of atoms which is periodic in
space. The imaginary points in space about which these atoms are
located, are called lattice points and the totality of such points
forms a crystal lattice or space lattice.
If all the atoms at the lattice points are identical, the lattice is called
a Bravais lattice.
Thus, the three-dimensional space lattice may be defined as a finite
array of points in three-dimensions in which every point has identical
environment as any other point in the array.
Let us consider the case of two-dimensional array of points as shown
in Fig.1. The environment about any two points is the same and
hence it represents a space lattice, which is explained as follows:
We choose any arbitrary point as origin and consider the position
→ →
vectors r1 and r2 of any two lattice points by joining them to O.
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3. → → →
If the difference T of the two vectors r1 and r2 satisfies the following
relation:
→ → →
T = n1 a + n2 b
→ →
where n1 and n2 are integers and a and b are fundamental translation
vectors characteristics of the array, then the array of points is a two-
dimensional lattice.
Fig.1. Two-dimensional array of points
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4. For three-dimensional lattice:
→ → → →
T = n1 a + n2 b + n3 c
Hence, it should be remembered that a crystal lattice refers to the
geometry of a set of points in space whereas the structure of crystal
refers to the actual ordering of its constituent ions, atoms, molecules
in the space.
3. THE BASIS AND CRYSTAL STRUCTURE: For a lattice to
represent a crystal structure, we associate every lattice point with
one or more atoms (i.e. a unit assembly of atoms or molecules
identical in composition) called the basis or the pattern. When the
basis is repeated with correct periodicity in all directions, it gives the
actual crystal structure. The crystal structure is real, while the
lattice is imaginary. Thus,
Lattice + Basis = Crystal structure
Fig.2 shows a basis or pattern representing each lattice point. It is
observed from 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.
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5. In crystalline solids like copper and sodium, the basis is a single
atom, in NaCl and CsCl, the basis is diatomic one whereas in
crystals like CaF2, the basis is triatomic.
Fig.2. Basis or pattern
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6. 4. UNIT CELLS AND LATTICE PARAMETERS:
The atomic order in crystalline solids indicates that the small
groups of atoms (fundamental grouping of particles) form a repetitive
pattern called unit cell. Unit cells for most of the crystals are either
parallelopiped or cubes having three sets of parallel faces. A unit cell
is chosen to represent the symmetry of the crystal structure,
wherein all the atom positions in the crystal may be generated by
translations of the unit cell through integral distances along each of
the 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. Fig.3 shows a unit cell of a three-
dimension crystal lattice.
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Fig.3 Lattice parameters of a
unit cell

7. A space lattice is a regular distribution of points in space, in such a
manner that every point has identical surroundings. The lattice is
made up of a repetition of unit cells, and a unit cell can be
→ → →
completely described by the three vectors a, b and c when the length
of the vectors and the angles between them (, , ) are specified.
Taking any lattice point as the origin, all other points on the lattice
can be obtained by a repeated operation of the lattice vectors
→ → →
a, b and c . These lattice vectors and the above said interfacial
angles constitute the lattice parameters of the unit cell. If the values
of these intercepts and interfacial angles are known, we can easily
determine the form and actual size of the unit cell.
→ → →
The vectors a, b and c may or may not be equal. Also, the angles
, ,  may or may not be right angles. Based on these conditions,
there are seven different crystal systems. Thus, in a cubic system,
a = b = c and  =  =  = 90o. If atoms are existing only at the
corners of the unit cells, seven crystal systems will yield seven types
of lattice. More space lattices can be constructed by placing atoms
at the body centers of unit cells or at the centers of the faces.
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8. Bravais showed that the total number of different space lattice types
is only fourteen. Hence the term ‘Bravais Lattice’.
5. UNIT CELL VERSUS PRIMITIVE CELL: Primitive cell may be
defined as a 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
corners is known as ‘primitive cell’. The unit cell differs from the
primitive cell is that it is not restricted to being the equivalent of one
lattice point. In some cases, the two coincide. Thus, unit cells may be
primitive cells, but all the primitive cells need not be unit cells.
6. CRYSTAL SYSTEMS: There are thirty-two classes of crystal
systems based on the geometrical considerations (i.e. symmetry and
internal structure). But, it is a common practice to divide all the
crystal systems into seven groups or basic systems. These seven
basic crystal systems are distinguished from one another by the
angles between the three axes and the intercepts of the faces along
them. The basic crystal systems are: 1. Cubic (isometric),
2. Tetragonal, 3. Orthorhombic, 4. Monoclinic, 5. Triclinic, 6. Trigonal
(rhombohedral) and 7. Hexagonal. We shall discuss one by one.
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9. 1. Cubic Crystal System: a = b = c and  =  =  = 90o
The crystal axes are perpendicular to one another, and the repetitive
interval is the same along all the three axes. Cubic lattices may be
simple, body-centered or face-centered.
2. Tetragonal Crystal System: a = b  c and  =  =  = 90o
The crystal axes are perpendicular to one another. The repetitive
intervals along two axes are the same, but the interval along the
third axis is different. Tetragonal lattices may be simple or body-
centered.
3. Orthorhombic Crystal System: a  b  c and  =  =  = 90o
The crystal axes are perpendicular to one another, but the repetitive
intervals are different along all the three axes. Orthorhombic lattices
may be simple, base-centered, body-centered or face-centered.
4. Monoclinic Crystal System: a  b  c and  =  = 90o  
Two of the crystal axes are perpendicular to each other, but the third
is obliquely inclined. The repetitive intervals are different along all
the three axes. Monoclinic lattices may be simple or base-centered.
5. Triclinic Crystal System: a  b  c and       90o
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10. None of the crystal axes is perpendicular to any of the others, and
the repetitive intervals are different along all the three axes.
6. Trigonal Crystal System: a = b = c and  =  =   90o
The three axes are equal in length and are equally inclined to each
other at an angle other than 90o.
7. Hexagonal Crystal System: a = b  c and  =  = 90o ,  = 120o.
Two of the crystal axes are 60o apart while the third is perpendicular
to both of them. The repetitive intervals are the same along the axes
that are 60o apart, but the interval along the third axis is different.
The seven systems and their properties are given in Table-1.
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Seven
crystal
systems

11. 11/9/2023 DR. VBP 11
Seven crystal systems
(contd….)

12. 11/9/2023 DR. VBP 12
Sr.
No.
Crystal system Axial lengths of
unit cell (a, b and c)
Interaxial angles
(,  and )
Examples
1 Cubic a = b = c  =  =  = 90o NaCl, Au, Cu
2 Tetragonal a = b  c  =  =  = 90o TiO2 , SnO2
3 Orthorhombic a  b  c  =  =  = 90o KNO3, BaSO4
4 Monoclinic a  b  c  =  = 90o   Na2SO4 , FeSO4
5 Triclinic a  b  c       90o K2Cr2O7
6 Trigonal a = b = c  =  =   90o As, Sb, Bi, Calcite
7 Hexagonal a = b  c  =  = 90o ,  = 120o SiO2, Zn, Mg, Cd
Table-1: Seven basis crystal systems with their characteristics

13. 7. CRYSTAL SYMMETRY: The definite ordered arrangement of the
faces and edges of a crystal is known as crystal symmetry. A sense
of symmetry is a powerful tool for the study of internal structure of
crystals.
Crystals possess different symmetries or symmetry elements. They
are described by certain operations. A symmetry operation is one
that leaves the crystal and its environment invariant.
Symmetry operations performed about a point or a line are called
point group symmetry operations and symmetry operation
performed by translations as well as rotations are called space
group symmetry operations. Crystals exhibit both types of
symmetries independently and in compatible combinations. A
symmetry operation is a transformation performed on the body which
leaves it unchanged or invariant. i.e. after performing an operation
on the body, if the body becomes indistinguishable from its initial
configuration, the body is said to possess a symmetry element
corresponding to that particular operation. The following are the
different point group symmetry elements exhibited by crystals:
1.Centre of symmetry or inversion center, 2. Reflection symmetry and
3. Rotation symmetry.
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14. 8. THE BRAVAIS SPACE LATTICE: There are number of ways in
which an actual crystal may be built up and atoms piled together
resulting in a great many crystal structures. But each of the
structures consists of some fundamental patterns repeated at each
point of a space lattice. The scheme of repetitions of a space lattice
are very limited in number while the possible crystal structures are
almost unlimited. But in 1848, Bravais showed that there are only 14
different arrays or networks of lattice in which points can be
arranged in space so that each point has identical surroundings.
These are known as ‘Bravais space lattices’.
To specify a given arrangement of points in a space lattice of atoms in
a structure, a set of coordinate axes with an origin at one of the
lattice points is chosen. Cubic crystals, e.g. are referred to a cubic set
of axes, i.e. three axes of equal length that stand perpendicular to
one another and form the three edges of the cube. The seven different
systems used in crystallography posses certain characteristics like
equality of angles and equality of lengths. These seven
crystallographic axes generate 14 space lattices. Most of the common
metals and some important alkali halides have cubic structures. In
cubic crystals, atoms are packed in a cubic pattern with three
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15. different types of repetitions: Simple Cubic (sc), Body Centered Cubic
(bcc) and Face Centered Cubic (fcc). The 14 Bravais space lattices are
shown in Fig.4. Table-2 gives seven crystal systems and the fourteen
lattices.
Fig.4 Fourteen space lattices
and the seven crystal
systems.
(Contd…)
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16. 11/9/2023 DR. VBP 16
Fig.4 Fourteen
space lattices
and the seven
crystal
systems.
(Contd…)

17. 11/9/2023 DR. VBP 17
Fig.4 Fourteen
space lattices
and the seven
crystal
systems.
(Contd…)

18. 11/9/2023 DR. VBP 18
Sr.
No.
System No. of
lattices
Space (Bravais)
lattices
Lattice
symbol
Nature of unit cell
1 Cubic 3 Simple P a = b = c
Body centered I  =  =  = 90o
Face centered F
2 Tetragonal 2 Simple P a = b  c
Body centered I  =  =  = 90o
3 Orthorho 4 Simple P a  b  c
Base centered C  =  =  = 90o
Body centered I
Face centered F
4 Monoclinic 2 Simple P a  b  c
Base centered C  =  = 90o  
5 Triclinic 1 Simple P a  b  c,       90o
6 Trigonal 1 Simple P a = b = c,  =  = 90o  
7 Hexagonal 1 Simple P a = b  c,  =  = 90o ,  = 120o
Table-2: The seven crystal systems and the fourteen Bravais lattices

19. 9. METALLIC CRYSTAL STRUCTURE: Since atomic bonding in this
group of materials is metallic, within the hard sphere model for the
representation of a crystals structure, each sphere represents an ion
core. Three relatively simple crystal structures are found for most of
the common metals: Body centered cubic, face centered cubic and
hexagonal closed packed. An important characteristic of most of the
metals is that they are crystalline; i.e. the atoms are arranged in
some regular repeatable pattern indefinitely in space.
1. Simple Cubic (sc) Structure: The simplest and easiest structure
to describe is the simple cubic structure. Fig.5 shows the unit cell of
a simple cubic structure.
Fig.5 Simple cubic structure
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20. There is only one lattice point at each of the eight corners of the unit
cell. If we take an atom at one corner as the center, it is observed
that this atom is surrounded by six equidistant nearest neighbors
and hence the coordination number of a simple cubic lattice is six. In
this structure, each corner atom is shared by eight unit cells. Hence
the share of each corner atom to a unit cell is one-eighth of an
atom. In this way, the total number of atoms in one unit cell will be
1/8 x 8 = 1 atom. In other words, the effective no. of lattice points in
a simple cubic cell is one. Thus SC is a primitive cell. In this
structure, the atoms tough each other along the edges. Hence the
nearest neighbor distance is 2r = a. SC structure has the following
features:
i. Coordination number, N = 6
ii. Nearest neighbor distance, 2r = a
iii. Lattice distance, a = 2r
iv. Number of atoms/unit cell, n = 1/8 x 8 = 1
v. Number of lattice points = 1
vi. Volume of all atoms in a unit cell, v = 1 x 4r3/3
vii. Volume of unit cell, V = a3 = (2r)3
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21. Hence the packing factor or density of packing of this structure is:
PF = v/V = 4r3/3a3
PF = 4r3/3(2r)3 = /6 = 0.52 or 52%
Only ONE element polonium at a certain temp region exhibits this
structure; SC structure is a loosely packed structure.
Some important crystal structure terms are defined below:
Coordination number (N): The coordination number is defined as
the no. of equidistant nearest neighbors that an atom has in the
given structure. Greater is the coordination number, the more closely
packed up will be the structure.
Nearest neighbor distance (2r): The distance between the centers of
two nearest neighboring atoms is called nearest neighbor distance. It
will be 2r if r is the radius of the atom.
Atomic radius (r): Atomic radius is defined as half the distance
between nearest neighbors in a crystalline solid without impurity.
Atomic packing factor (APF): The fraction of space occupied by atoms
in a unit cell is known as atomic packing factor or simply packing
factor. It is the ratio of vol. of atoms occupying unit cell to the vol. of
unit cell relating to that structure.
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22. 2. Body Centered Cubic (bcc) Structure: Elements that exhibit this
structure are Li, Na, K and Cr. In this structure, in a unit cell, there
are 8 atoms at the eight corners and another atom at the body
center. As the body center is contained entirely within the unit cell
and is not shared by any surrounding unit cell, the number of
atoms/unit cell in a bcc structure is 1/8 x 8 + 1 = 2. The corner
atoms do not touch each other, but each corner atom touches the
body center atom along the body diagonal. Hence, the coordination
no. of this structure is 8.
Fig.6 Body centered cubic structure
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23. The calculation of the lattice constant can be made with the help of
Fig.6.
(AC)2 = a2 + a2 = 2a2
(FC)2 = (AC)2 + (AF)2
(FC)2 = 2a2 + a2 = 3a2
(4r)2 = 3a2, a2 = (4r)2/3, a = 4r/3 or 2r = a3/2.
We shall now calculate the packing factor from the following data:
i. Coordination number, N = 8
ii. Nearest neighbor distance, 2r = a3/2
iii. Lattice distance, a = 4r/3
iv. Number of atoms/unit cell, n = 1/8 x 8 + 1 = 2
v. Volume of all atoms in a unit cell, v = 2 x 4r3/3
vi. Volume of unit cell, V = a3 = 64r3/33.
Therefore, atomic packing factor,
APF = v/V = 8r3 33/3 x 64r3 = 3 /8 = 0.68 or 68%.
bcc structure is a closely packed structure.
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24. Closest packing: In metallic crystals in which all the atoms are
identical, there are two forms of closest packing: face centered cubic
(fcc) and hexagonal close packed (hcp).
A closest packing is a way of arranging equi-dimensional objects in
space so that the available space is filled more efficiently. Such an
arrangement is achieved when each object is in contact with the
maximum no. of like objects.
Fig.7 Closest packing of spheres in 2-dimensional and in 3- dimensions
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25. 3. Face Centered Cubic (fcc) Structure: In this lattice, there are
eight atoms at the eight corners of the unit cell and six atoms at the
centers of six faces. Considering the atom at the face center as origin,
it can be observed that the face is common to two unit cells and
there are twelve points surrounding it situated at a distance equal to
half the face diagonal of the unit cell. Thus, the coordination no. of
fcc lattice is twelve. The atoms touch each other along the face
diagonal. Each corner atom is shared by 8 surrounding unit cells
and each of the face centered atom is shared by 2 surrounding unit
cells.
Thus, the total no. of
Atoms in fcc structure
is (1/8 x 8 + 6 x ½) = 4.
Thus we note in fcc
(e.g. Cu, Al, Pb and Ag)
structure that:
Fig.8 Face centered
cubic structure
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26. i. Coordination number, N = 12
ii. Nearest neighbor distance, 2r = a2/2
iii. Lattice distance, a = 4r/2
iv. Number of atoms/unit cell, n = 1/8 x 8 + 6 x ½ = 4
v. Volume of all atoms in a unit cell, v = 4 x 4r3/3
vi. Volume of unit cell, V = a3 = 64r3/22.
Therefore, atomic packing factor of fcc structure is calculated:
APF = v/V = 16r3 22/3 x 64r3 = 2/6 = /32 = 0.74 or 74%.
4. Hexagonal Close Packed (hcp) Structure: The specific
hexagonal structure formed by Mg is shown in Fig.9. The unit cell
contains one atom at each corner, one atom each at the center of
the hexagonal faces and three more atoms within the body of the
cell. Each atom touches three atoms in the layer below its plane,
six atoms in its own plane, and three atoms in the layer above.
Hence the coordination no. of this structure is 12. Further, the
atoms touch each other along the edge of the hexagon.
Thus, a = 2r.
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27. 11/9/2023 DR. VBP 27
The top layer contains seven atoms.
Each corner atoms is shared by 6
Surrounding hexagonal cells and
the center atom is shared by 2
surrounding cells. The three atoms
within the body of the cell are fully
contributing to the cell.
Thus, the total no. of atoms in a
unit cell is 3/2 + 3/2 + 3 = 6.
Calculation of [c/a] ratio for an ideal
hexagonal close packed structure:
Fig.9 Ideal hexagonal close packed structure
Fig.9 a. Bottom layer of hcp
structure

28. Let ‘c’ be the height of the unit cell and ‘a’ be its edge. The three body
atoms lie in a horizontal plane at c/2 from the orthocenters of
alternate equilateral triangles at the top or base of the hexagonal cell.
These three atoms just rest on the three atoms at the corners of the
triangles. In triangle ABY,
cos 30o = AY/AB , AY = a cos 30o = a3/2 and
(AZ)2 = (AX)2 + (ZX)2 and in triangle, AXZ,
AX = 2/3 AY = 2 a3/3x2 = a/3. Substituting this in
equation (1), we get,
a2 = a2/3 + c2/4, c2/4 = a2 – a2/3 = 2a2/3,
c2/a2 = 8/3 or c/a = [8/3]1/2
Volume of the unit cell:
Area of the base = Six times the area of the triangle AOB,
Area of the triangle, AOB = ½ (BO) (AY) = ½ a x a3/2
Thus, area of the base = 6 a23/2 x 2 = 33 a2/2
Volume of the cell, V = 33 a2c/2
Calculation of packing factor of hcp:
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29. i. Coordination number, N = 12
ii. Nearest neighbor distance, 2r = a
iii. Lattice distance, a = 2r
iv. Number of atoms/unit cell, n = 3/2 + 3/2 + 3 = 6
v. Volume of all atoms in a unit cell, v = 6 x 4r3/3
= 24a3/3x(2)3 = a3
vi. Volume of unit cell, V = a3 = 33a2c/2 with c/a = [8/3]1/2
Therefore, atomic packing factor (v/V) of hcp structure is calculated:
APF = (a3)/(33 a2c/2) = 2a/33c = (2/33) [3/8]1/2
= 2/6 = /32 = 0.74 or 74%
Mg, Zn and Cd crystallise in this structure. The packing factor of fcc
and hcp are 74% and hence they are known as closest packing
structures.
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30. 10. RELATION BETWEEN THE DENSITY OF CRYSTAL MATERIAL
AND LATTICE CONSTANT IN A CUBIC LATTICE:
Consider a cubic crystal of lattice constant a. Let the no. of atoms
per unit cell be n and  be the density of a crystal material. The
atomic weight of the material is MA. Let NA be the Avogadro’s
number. Thus [MA/]m3 of the material will contain NA atoms. Hence,
n atoms in a unit cell will occupy a volume [MA n/NA].
Thus, a3 = MA n/NA ; or  = MA n/ NA a3 ; a = [MA n/NA]1/3.
11. DIRECTIONS, PLANES AND MILLER INDICES: In a crystal
there exists directions and planes which contain a large
concentration of atoms. It is necessary to locate these directions and
planes for crystal analysis. In Fig.10, two directions are shown by
arrows in 2-Ds. These directions pass through the origin O and end
at A and B respectively.
The directions are described by giving the coordinates of first whole
numbered point (x,y) through which each of the directions passes.
For direction OA, it is (1,1) and for OB it is (3,1). In 3-Ds, the
directions are described by the coordinates of the first whole
numbered point (x,y,z).
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31. Fig.10 Crystal directions
Generally, square brackets are used to indicate a direction. A few
directions are shown below:
OA [110], OB [010], OC [111], OD [102] and OE [112]. The crystal
lattice may be regarded as made-up of an aggregate of a set of
parallel equidistant planes, passing through the lattice points, which
are known as lattice planes. For a given lattice, the lattice planes
can be chosen in different ways as shown in Fig.11. The problem is
that how to designate these planes in a crystal.
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32. Fig.11 Different
crystal planes
Miller evolved a method to designate a set of a parallel planes in a
crystal by three numbers (h k l) known as Miller indices.
The steps in the determination of Miller indices of a set of parallel
planes are illustrated in Fig.12.
(i) Determine the coordinates of the intercepts made by the plane
along the three crystallographic axes (x,y,z).
x y z
2a 3b c
pa qb rc (p = 2, q = 3 and r = 1)
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33. (ii) Express the intercepts as
multiples of the unit dimensions,
or lattice parameters along the
axes.
2a/a 3b/b c/c
2 3 1
(iii) Determine the reciprocals of
these numbers:
1/2 1/3 1/1
(iv) Reduce these reciprocals to
the smallest set of integral
numbers and enclose them in
brackets:
6 x 1/2 6 x 1/3 6 x 1
(3 2 6)
In general it is denoted by (hkl).
We also notice that:
1/p : 1/q : 1/r = h : k : l
1/2 : 1/3 : 1/1 = 3 : 2 : 6
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Fig.12 Miller indices of planes

34. Thus, Miller indices may be defined as the reciprocals of the
intercepts made by the planes on the crystallographic axes when
reduced to smallest numbers.
12. IMPORTANT FEATURES OF MILLER INDICES:
1. All the parallel equidistant planes have the same Miller indices.
Thus, the Miller indices define a set of parallel planes.
2. A plane parallel to one of the coordinate axes has an intercept of
infinity.
3. If the Miller indices of two planes have the same ratio, i.e. (8 4 4)
and (4 2 2) or (2 1 1), then the planes are parallel to each other.
4. If (hkl) are the Miller indices of a plane, then the plane cuts the
axes into h, k and l equal segments respectively.
13. IMPORTANT PLANES ND DIRECTIONS IN A CUBIC CRYSTAL:
In Fig.13(a), the plane cuts the Y-axis at  and Z-axis at .
Therefore, the coordinates of the intercepts of the plane are , a and
. i.e. , 1 and  are the intercepts as the multiples of the unit cell
dimensions. Take the reciprocals. They are 1/ , 1 and 1/, or 0,1
and 0. Hence (010) are the Miller indices of the plane and (010) plane
is perpendicular to [010] direction, (110) and (111) planes are shown
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35. in Fig.13(b) and (c). We also see [110] and [111] directions
respectively.
Fig.13 Planes and
directions in
a cubic crystal
11/9/2023 DR. VBP 35

36. 14. SEPARATION BETWEEN LATTICE PLANES IN A CUBIC
CRYSTAL: The cube edge is a. Let (hkl) be the Miller indices of the
plane ABC. This plane belongs to a family of planes whose Miller
indices are (hkl) because Miller indices represent a set of planes. Let
ON = d1 be the perpendicular distance of the plane ABC from the
origin. Let ’, ’ and ’ (different from interfacial angles ,  and  ) be
the angles between coordinates axes X,Y,Z respectively and ON.
The intercepts of the plane on the three axes are:
OA = a/h , OB = a/k and OC = a/l … (2)
From Fig.14(a), we have:
cos ’ = d1/OA, cos ’ = d1/OB and cos ’ = d1/OC … (3)
From Fig.14(b), we have:
(ON)2 = x2 + y2 + z2 = d1
2 = [d1
2 (cos2’) + d1
2 (cos2’) + d1
2 (cos2’)]
d1 = d1 [cos2’ + cos2’ + cos2’ ]1/2
Thus,
cos2’ + cos2’ + cos2’ = 1 … (4)
Substituting the value cos ’, cos ’, cos ’ of from eq (3) to eq (4), we
get,
11/9/2023 DR. VBP 36

37. 11/9/2023 DR. VBP 37
Fig.14 Interplanar spacing

38. [d1/OA]2 +[d1/OB]2 +[d1/OC]2 = 1 = [d1h/a]2 + [d1k/a]2 + [d1l/a]2 = 1
d1
2 [h2 + k2 + l2]/a2 = 1  d1
2 = a2 /(h2 + k2 + l2)
d1 = a / (h2 + k2 + l2) … (5)
Let OM = d2 be the perpendicular distance of the next plane PQR
parallel to the first plane ABC. The intercepts of this plane on the
three crystallographic axes are:
OA’ = 2a/h , OB’ = 2a/k and OC’ = 2a/l and
cos ’ = d2/OA’, cos ’ = d2/OB’ and cos ’ = d2/OC’ … (6)
(OM)2 = d2
2 = [d2
2 cos2’ + d2
2 cos2’ + d2
2 cos2’]
d2
2 = d2
2 [cos2’ + cos2’ + cos2’]  cos2’ + cos2’ + cos2’ = 1
Substituting this in equation (6), we get:
[d2h/2a]2 + [d2k/2a]2 + [d2l/2a]2 = 1
d2
2/[h2 + k2 + l2]/4a2 = 1  d2
2 = 4a2 /(h2 + k2 + l2)
d2 = 2a / (h2 + k2 + l2) … (7)
Thus the inter-planar spacing between two adjacent parallel planes
of Miller indices (h k l) in a cubic lattice is given by:
d = d2 - d1 = a / (h2 + k2 + l2) … (8)
11/9/2023 DR. VBP 38

39. 15. RECIPROCAL LATTICE: Physical properties of crystalline solids
are different in different directions in them. i.e. crystalline solids are
anisotropic w.r.t. many of the physical properties. Hence a method of
representation of planes and directions has become inevitable in the
studies of crystal structure. Miller introduced the usage of a set of
three integers, Miller indices, to represent a set of parallel planes and
these indices when put inside square brackets are also used to
indicate the directions. Miller indices of different sets of parallel
planes have already discussed.
Let us consider one such plane of Miller indices (h k l) nearest to the
origin within the unit cell of a cubic crystal. Here h k l are related to
the orientation of the plane with reference to the crystal axes. The
fractional coordinates of intercepts of the plane (h k l) on a, b, c axes
are 1/h, 1/k, 1/l respectively.
The equation of this plane may be written as,
hx + ky + lz = 1 … (9)
where (x,y,z) are the coordinates of a point lying on the plane.
From eq (9), one can have two possible interpretations:
1. h k l may be considered fixed and x, y, z are variables and
11/9/2023 DR. VBP 39

40. 2. x, y, z may be considered fixed and h, k, l are variable.
In first case, eq (9) represents a set of points lying on the and in the
second case the same equation represents a set of planes passing
through a fixed point. Thus, a plane and the set of points on it or a
point and the set of planes through it are said to be dual to each
other.
After X-rays were discovered, many physicists tried to characterize
them. One of the problems they were facing was their inability to
correlate the measured X-ray intensities to their energies. It was
Bragg who established a relationship between the angle at which X-
rays are deflected and their wavelength, , considering a crystal as
made up of parallel planes of atoms periodically repeated by lattice
translations and spaced by a distance d from each other. It is now
familiar as Bragg’s law and for practicizing crystallographers, it is
represented by,
 = 2dhkl sin … (10)
or = 2d sinhkl … (11)
or = 2dhkl sinhkl … (12)
This eqn diffraction indicates that the diffraction is by the planes
11/9/2023 DR. VBP 40

41. Specified (h k l) or at the particular angle sinhkl , often called the
Bragg angle of the (h k l). When the wavelength of , of X-rays is
known eqn (10) will be helpful in relating the measured angle  to the
inter-planar spacing dhkl as in the case of diffraction expts. The form
of Bragg’s law as given by eqn (11) is useful in determining the
unknown wavelength of X-ray by using measured angle  as in
spectroscopic expts. But in real practice there are many a set of
planes, with a variety of orientations and spacings that can diffract a
beam of X-rays. Hence one has to deal with planes of several slopes
and it is very difficulty to bring into visualization several slopes of the
planes. Also the Miller indices of a plane do not define a plane
uniquely since all parallel planes have a common normal and hence
identical Miller indices. Hence it is clear that a simplified
representation of the planes in a crystal would be helpful in the
understanding of X-ray diffraction phenomenon. It is here the
geometrical concept of reciprocal lattice is handy.
Concept of Reciprocal Lattice: Whenever it is necessary to consider
sets of planes in a crystal we can think in terms of their normals
since geometrically we have an advantage. The planes are of two
dimensions while normals are lines of one dimensional nature.
11/9/2023 DR. VBP 41

42. Simply normals to a set of planes may be viewed as a set of planes
rather than the planes themselves. But it is not enough we consider
the orientations of planes alone to study the diffraction of X-rays by
crystals but it is also required to be known the interplanar specings
d as only they determine the reflection angles . These interplanar
spacings may also be represented in the normals to the planes by
appropriately limiting their lengths.
From the discussion, it is clear that one can indicate the orientation
of a set of parallel planes by their common normal and the
interplanar spacing by restricting the length of the normals
proportionately.
Now, if the normals are drawn, to each set of parallel planes, from a
common origin and their lengths are made proportional to reciprocal
of the respective interplanar spacings, the points at the end of these
normals form a lattice array. Since the distances in this array are
reciprocal to distances in the crystal, the array is called the
‘reciprocal lattice’ of the crystal.
The points in the reciprocal lattice are called reciprocal lattice points.
These points in 3-d space form the ‘reciprocal lattice space’. This
reciprocal lattice space is also called the k-space. From the concept
11/9/2023 DR. VBP 42

43. of reciprocal lattice it may be understood that the ‘coordinates of
points’ in the reciprocal lattice space are defined by (h k l), the Miller
indices.
Geometrical Construction: The general rules for constructing the
reciprocal lattice may be formulated by the following steps:
1. Fix up some point in the direct lattice as a common origin.
2. From this common origin draw normals to each and every set of
parallel planes in the direct lattice.
3. Fix the length of each normal equal to the reciprocal of the
interplanar spacing (1/dhkl ) of the set of parallel planes (h k l) it
represents.
4. Put a point at the end of each normal.
The collection of all these points in space is the reciprocal lattice
space. It should be noted that the direction of each of the points from
the origin preserves the orientation of the set of parallel planes it
represents and the distance of each of the points from the origin
preserves the interplanar spacing of that stack of parallel planes. It is
well known that the d-spacing of parallel planes in crystals of few Ao.
Hence the distance of the reciprocal lattice point on an normal from
11/9/2023 DR. VBP 43

44. the origin (1/dhkl ) will be too large to be accommodated within the
frame work of the figure representing the reciprocal lattice. It is
because of this reason one may have to introduce a suitable scale
factor while picturing a reciprocal lattice. Thus, for convenience, let
us define the length of each normal (hkl ) explained in step 3 of the
procedure to be,
hkl = K (1/dhkl ) … (13)
where, is the scale factor chosen for the construction of the
reciprocal lattice. With all the rules specified above and eqn (13) one
can make an attempt to construct a reciprocal lattice in 2-d space as
detailed below:
Let us consider, as an example for construction of reciprocal lattice,
the unit cell of a monoclinic crystal (a  b  c;  =  = 90o  ) looking
along its unique axis, namely, c-axis. To make the construction
further simplified, one may consider planes belonging to single zone.
This makes the normal to all parallel planes belonging to the zone all
to lie in the same plane, normal to the zone axis, i.e. normal to the c-
axis. In Fig.15 the zone axis lies perpendicular to the plane of the
diagram and hence all normals to the parallel planes of the family of
11/9/2023 DR. VBP 44

45. the zone will lie in the plane of the diagram. Fig.15 shows the edge
views of four (h k l) planes, viz. (100), (110), (120) and (010) all
belonging to the [001] zone. It must be clear that the normals to the
family of planes (100), (200), (300) etc. are parallel. It is also known
that d100 = 2d200 = 3d300 = 4d400 etc. Hence while 100 = K (1/d100),
200 = 2100 , 300 = 3100 and so on.
Fig.15 Graphical
construction
of reciprocal lattice.
11/9/2023 DR. VBP 45

46. Therefore, the distance of the reciprocal lattice point representing the
set of parallel planes (h 0 0) from the origin will be n times greater
than the distance of the reciprocal lattice point representing the set
of parallel planes (h 0 0 or simply the distance of the reciprocal
lattice point from the origin representing the set of parallel planes (3
0 0) will be at a distance from the origin equal to three times the
distance of the reciprocal lattice point representing the set of parallel
planes (100). In reciprocal lattice, ‘it is the convention to represent
the reciprocal lattice point representing the set of parallel planes with
Miller indices (h k l) by the same Miller indices without the
parenthesis’. Thus in Fig.15, it may be seen that the reciprocal lattice
points representing the set of parallel planes (100), (200), (300) etc.
are indicated by 100, 200, 300 etc. Since all the set of parallel planes
(100), (200), (300) etc. have common normal, all the reciprocal lattice
points 100, 200, 300 etc. lie along the same straight line. As we have
taken the planes belonging to [001] zone, soon after the reciprocal
lattice points of all the (h k 0) planes are plotted we get a 2-d lattice
as shown in Fig.15. Buerger has proved that all these reciprocal
lattice points must lie on a 2-d lattice. A vector-algebraic proof
establishing that a 3-d reciprocal lattice is similarly formed in a real
11/9/2023 DR. VBP 46

47. crystal was given by Ewald.
16. X-RAY DIFFRACTION: Bragg’s Law: The concept of reciprocal
lattice may now be applied to understand the actual situation under
which X-rays are diffracted by crystal planes and thereby the Bragg’s
law. Bragg’s law given by eqn (12) may now be represented, to suit
our convenience by
sin hkl = (1/dhkl )/(2/) … (31)
11/9/2023 DR. VBP 47
Now let us consider a circle of radius 1/
and construct a right angled triangle with
its diameter as hypotenuse and one of the
sides of length equal to 1/dhkl . This is
shown in Fig.18. The angle opposite to the
side of length equal to 1/dhkl will be equal
to  as evidenced by eq (31). Fig.18 explains
Bragg’s law graphically.
We will try to understand the Bragg’s law from Fig.18 by making the
latter resemble to a real practical situation. Let us suppose the
direction of incident X-ray beam to coincide with the diameter AO in
figure. Since PAO is equal to hkl , the Bragg angle, the side AP of the

48. triangle APO should have the same slope as those of the set of
parallel planes which diffract X-rays.
Let us go more closer to the real situation by modifying Fig.18 to
Fig.19 where one may imagine one of the set of parallel planes of
Miller indices (h k l ), diffracting X-rays being put at the center
making an angle  with AO and AO is the direction of the incident X-
ray beam. In Fig.19, OP is normal both to the crystal plane (when
extended) and AP. Hence OCP is equal to
2hkl making CP the direction of the diffracted X-ray beam. Hence
Fig.19 is a graphical representation of Bragg’s law in terms of the
reciprocal lattice vector (since OP is equal to 1/dhkl it represents the
reciprocal lattice vector). Thus, the following conditions may be
emphasized to explain Bragg’s law with the aid of Fig.19.
1. The crystal has to be thought of being placed at the center of a
circle of radius 1/. In real cases it should be imagined to be at
the center of a sphere of radius 1/.
2. Origin of the reciprocal lattice vector is the point O, where the X-
ray beam leaves the circle (sphere of three dimensions).
3. The point P has to the position of the reciprocal lattice point
11/9/2023 DR. VBP 48

49. (at the end of the reciprocal lattice vector hkl or G of length 1/dhkl ) or
in other words it is sufficient to understand that whenever a
reciprocal lattice point lies on the circle (sphere in 3-d) Bragg’s law
will be satisfied and the diffracted X-ray beam passes along CP. In 3-
ds this sphere of radius 1/ is also called Ewald’s sphere of reflection
or simply sphere of reflection or simply Ewald’s sphere. Thus
reciprocal lattice may be used in conjunction with the sphere of
reflection to explain X-ray diffraction experiment.
11/9/2023 DR. VBP 49
As it is easy to imagine the reciprocal
lattice point centered at the point of
emergence of the X-ray beam from the
sphere of reflection, a construction
like Fig.19 can be used to analyse the
different ways in which a crystal can
be made to satisfy Bragg’s condition
for its various planes.
It is important to note that the X-ray diffraction takes place only
when a reciprocal lattice point intersects or lies on the sphere of
reflection. Hence a photographic record of the diffracted X-ray beams
is nothing but a photographic record of the reciprocal lattice.