UNIT - II
1.CRYSTALLOGRPHY
Introduction
Solids are aggregates of atoms. These atoms are arranged throughout the volume of the
solids. Solids are classified into two categories on the basis of the arrangement of atoms. Those
are
1) Crystalline solids and
2) Amorphous solids
Crystalline solids
In crystalline solid the atoms are arranged in periodic manner in all three
Examples: Iron, copper, silver, sulphur etc. are some elements which form crystalline solids.
Potassium chloride, sodium nitrate etc are some of the compounds, which are crystalline.
Amorphous solids
In Amorphous solids the arrangement of atoms is random. They have no directional
properties and therefore they are called isotropic substances. They do not possess any regular
shape and they have wide range of melting point.
The Basis and Crystal Structure:
Associating with every lattice point a unit assembly of atoms or molecules identical in
composition, arrangement and orientation forms the crystal structure. This unit assembly is
called the basis. 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
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Lattice
Basis
Crystal Structure
Above figures show crystal structure from Space lattice and Basis. It is observed from the figure
that the basis consists of two atoms. The number of atoms in the basis may vary from a single
atom to many atoms. For example in sodium and copper, the basis is single atom and in NaCl
and CsCl, the basis is diatomic.
Unit Cell:
The atomic order in crystalline solids indicates that the small groups of atoms form a
repetitive pattern. Thus in describing crystal structures, it is often convenient to subdivide the
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structure into small repeat entities called unit cells. A unit cell is chosen to represent the
symmetry of the crystal structure, where in all the atomic positions in the crystal may be
generated by translation of the unit cell integral distances along each of its edges. Thus the unit
cell is the basic building block of the crystal structure.
Unit cell for most crystals are parallelepiped or cubes having three sets of parallel faces.
The choice of a unit cell is not unique but can be constructed in a number of ways as shown in
the figure. The unit cell should be chosen in such a way that it gives the symmetry of crystal
lattice.
The figure above shows a unit cell of a three-dimensional crystal lattice.
Lattice Parameters of a unit cell:
In order to define lattice parameters, we should first define axes. The crystallographic
axes are those lines drawn parallel to the lines of intersection of any three faces of the unit cell
which do not lie in the same plane. The angles between the three crystallographic axes are
known as interfacial angles and are denoted by , and . The intercepts a, b and c on
crystallographic axes (x, y and z ) define the dimensions of unit cell and are known as primitives
or characteristic intercepts on the axes. These primitives and interfacial angles constitute the
lattice parameters of the unit cell. It is thus obvious that if the values of these intercepts and
interfacial angles are known, we can easily determine the form and actual size of the unit cell.
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Primitive Cell: Primitive cell is defined as the unit cell that contains one lattice point at corners.
Thus the number of lattice points per unit cell is one. The unit cell differs from the primitive cell
in that it is not restricted to begin the equivalent of one lattice point.
Carbon crystallized polymers and plastics are examples of nonmetallic crystals or amorphous
solids.
Difference between amorphous and crystalline solids
Crystalline Solids
1.They have long range order of atoms
2.Crystals have sharp melting points
3.Because they have regular arrangement, their properties vary with direction, hence they are
anisotropic
4. Cleavage of crystal occurs along certain planes.
Amorphous Solids:
1. The arrangement of atoms is random.
2. The don’t have sharp melting point i.e. wide range of melting point.
3. They have no directional property and therefore they are isotropic
4. They give a rough surface on cutting along any direction.
Lattice Points:
The atomic arrangement in a crystal is called crystal structure. In a perfect crystal there is
a regular arrangement of atoms. It is very convenient to imagine points in space about which
these atoms are located. Such points in space are called “Lattice Points” and totality of such
points forms a crystal lattice or space lattice. If the arrangement is in three dimensions, that
arrangement is called 3-D space lattice.
Bravais Lattices:
A Three dimensional space lattice is generated by repeated translation of three non
coplanar vectors a,b and c. There are only fourteen distinguishable ways of arranging of points in
three dimensional space. These 14 space lattices are known as Bravais Lattices. These lattices
are classified into seven crystal systems. Each system is characterized by the values of lattice
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parameters a,b ,c and α,β,γ. The classification of these systems is shown in the below table.
S.No.
Crystal
System
Number of Bravais lattice
lattices in
the system
Unit cell characteristics
1
Cubic
3
a=b=c
= ==
90o
2
Tetragonal
2
a=bc
3
Orthorhombic
4
1. Simple
2. Body centered
3. Face centered
1. Simple
2. Body centered
1. Simple
2. Base centered
3. Body centered
4. Face centered
1. Simple
= ==
90o
= ==
90o
1. Simple
a=bc
1. Simple
2. Base centered
1. Simple
abc
4
5
Rhombohedral 1
(Trigonal)
Hexagonal
1
6
Monoclinic
2
7
Triclinic
1
abc
a=b=c
abc
= =
90o
 =  = 90o
= 120o
 = = 90o
 90o
 
90o
Crystal Systems:
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 where the length of vectors and the angles between
them (,, ) are completely specified. The vectors a , bc may or may not be equal. Also,
the angles , , and  may or may not be right angles. Based on these conditions, there are seven
different crystal systems. If atoms exist only at the corners of the unit cells, the seven crystal
systems will yield seven types of lattices. 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 structures are:
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(1) Cubic system:
In this system all the three vectors are equal in magnitude and are perpendicular to each other
i.e. a= b = c and  =  =  = 90o. These crystals have three types of lattices depending upon
arrangement of atoms. They are simple cubic, body centered and face centered cubic.
Examples: Au, Cu and NaCl
(2) Tetragonal system:
In this system all the two vectors are equal in magnitude and are perpendicular to each other
i.e. a = b  c and  =  =  = 90o. There are two lattices in this system, simple and body
centered.
Examples: TiO2, SnO2 and NiSO4
(3) Orthorhombic system:
In This system, the crystal axes are perpendicular to one another, but the repetitive
interval are different along the three axes i.e. a  b  c and  =  =  = 90o. Orthorhombic
may be simple, body centered, base centered and face centered.
Examples: KNO3, BaSO4 and PbCO3
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(4) Monoclinic system:
In this system three vectors are unequal in magnitude and two of the crystal axes are
perpendicular to each other but the third is obliquely inclined i.e.a  b  c and  =  =90o 
These crystals have two types of lattices depending upon the arrangement of atoms. Simple
and base centered systems.
Examples: CaSO4.2H2O, FeSO4 and Na2SO4
(5) Triclinic system
In this system all the three vectors are unequal in magnitude and no crystal axis is
perpendicular to each other i.e. a  b  c and       90o . The try clinic lattice is only
one lattice.
Examples: K2Cr2O7 and CuSO4.5H2O
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(6) Trigonal (or Rhombohedral) system:
In this system the The three axes are equal in length and are equally inclined to each other at
an angle other than 90o. i.e.a = b = c and  =  =   90o
Examples: As, Sb, Bi and Calcite
(7) Hexagonal system:
In this system a = b  c and  =  = 90o  = 120o.Two axes of the unit cell are equal in length
in one plane at 120o with each other and third axes is perpendicular to this plane.
Examples: SiO2, Zn, Mg and Cd
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Properties of Unit Cell:
Coordination number:
Coordination number is defined as the number of equi-distant nearest neighbors that an
atom has in the given structure.
Nearest neighbor distance:
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 Packing Factors:
The fraction of the space occupied by atoms in a unit cell is known as atomic packing
factor. It is the ratio of the volume of the atoms occupying the unit cell to the volume of the unit
cell relating to that structure.
Atomic packing factor =
PACKING FRACTIONS:
Simple Cubic Structure (SC):
Simple cubic structure is shown in the figure. In a simple cubic lattice there is one lattice
point at each of the eight corners of the unit cell. Each atom in this lattice is surrounded by six
equidistant nearest neighbors hence the coordination number is six. In this structure, eight unit
cells share each corner atom. Hence share of each corner atom to a unit cell is 1/8 of an atom.
Therefore the total number of atoms (or effective number of atoms) in one unit cell will be
1
x 8 = 1. Atom
8
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Thus SC is a primitive cell. The atoms in this structure touch each other along the cube
edge. Hence nearest neighbor distance (2r) = a , where ‘a’ is the side of the cube and ‘r’ is radius
of atoms.
Volume of the atoms in unit cell v =
4 3
r
3
Volume of unit cell V = a 3 = (2r)3
4 3
r
v 3

Atomic packing factor for simple cubic (SC) structure is = 
 = 0.52 or 52%
3
V (2r )
6
Hence SC is a loosely packed structure. The element polonium at a certain temperature region
exhibits this structure.
Body centered Cubic Structure (BCC):
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In this structure, a unit cell contains 8 atoms at the eight corners and another atom at the
body center. Eight unit cells share each corner atom. Therefore the number of atoms per unit cell
in a BCC structure is
(
1
x 8) + 1 = 2 atoms
8
The corner atoms don’t touch each other, but each corner atom touches the body center atom
along body diagonal. Hence coordination number = 8.
The side of the cube is ‘ a ’ and radius of atom is ‘r’. From the figure,the nearest neighbor
distance 2r =
3
a
2
Volume of all the atoms in a unit cell v = 2 x
Volume of unit cell = a 3 = (
4r
3
4 3
r
3
)3
4
2   r3
v
3

Atomic packing factor =
= 0.68 or 68%
3
V
 4r 


 3
Packing factor is 68%.
Li, Na, K and Cr exhibit this structure.
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Face centered Cubic Structure (FCC):
In the case of FCC lattice, there are eight atoms at the eight corners of the unit cell and
six atoms at the centers of six faces shown in figure. The unit cell 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 coordination number of FCC is 12. Each of the six face centered
atoms is shared by two adjoining cubes. 8 surrounding unit cells share each of the corner atoms.
1
1
Total number of atoms per unit cell = (  8 ) + (  6 ) = 4 atoms
8
2
From the figure (4r) = 2a
Nearest neighbor distance 2r =
2
a
2
where ‘r’ is radius of atom and ‘ a ’ is side of the cube.
Volume of all the atoms in unit cell (v) = 4 x
4 3
r
3
Volume of unit cell (V) = a 3
4
4   r3
v
3
Atomic packing factor =
=
= 0.74 or 74%
3
V
 4r 


 2
Therefore FCC is a close packed Structure. Copper, aluminum, lead and silver have this
structure.
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Relationship between lattice constant and density of a crystal: (cubic only)
Let ‘’ be the density of the material. The volume of a unit cell for a cubic lattice is ‘a3’.Hence
mass of the unit cell = a3.
Let ‘n’ be the number of atoms or molecules in a unit cell.
Mass of an atom or molecule =
M
; where M is the atomic or molecular weight and NA is
NA
the Avogadro number.
 a 3 
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nM
NA
or
a3
nM
N A