The document discusses different types of solids and their crystal structures. It defines crystalline and amorphous solids. Crystalline solids have an orderly arrangement of atoms and distinct crystal structures, while amorphous solids lack any orderly pattern. Common crystalline structures include face-centered cubic, body-centered cubic, and hexagonal close-packed. These structures differ in how atoms are arranged within the unit cell. The document also discusses properties like atomic radius, packing efficiency, and symmetry elements of different crystal structures.

1. MD MOYNUL
ISLAM
ID:20151107004

2. Solid: A solid is one of the three states of matter, in which atoms are closely packed as
compared to the other two states, e.g., gas and liquid. The atoms in the solid are not stationary
but are vibrating around fix points, giving rise to the orderly arrangement of crystal structures.
Solid may be classified as
 Crystalline solid, also called true solid,
 Amorphous solid and
 A combination of two.
Crystalline solids: In which solids, the atoms, molecules or ions are arranged in a systematic
and characteristic geometrical shape or pattern are called crystalline solid. In a crystalline solid,
the constituents are arranged in a regular, repeating three-dimensional pattern called the crystal
lattice.
Crystalline solid is two types. They are
 Metallic crystals: In the formation of meal crystal the ions are connected only
indirectly, through the free electrons surrounding them. Structures of some of the
more important metallic crystals are:
o Face centered cubic (FCC): 𝐶𝑢, 𝐴𝑔, 𝐴𝑙, 𝑁𝑖, 𝑃𝑏 etc.
o Body centered cubic (BCC): 𝑊, 𝑀𝑜, 𝐶𝑟 etc.
o Hexagonal close-packed (HCP): 𝑀𝑔, 𝑍𝑛, 𝑇𝑖, 𝐶𝑑, 𝑍𝑟 etc.
 Non-metallic crystals: Non-metallic crystals may have covalent or ionic bonding or
any of a great variety of mixed bonds. Atoms of two or more kinds are frequently
involved. Simple examples of non-metallic crystals are crystalline carbon and
crystallized polymers or plastics.

3. Amorphous solids: Amorphous solids (Gr amorphos = no form) have atoms, molecules and
ions arranged at random and in an irregular fashion or pattern and lacks the ordered crystalline
lattice. In amorphous materials, the elementary particles are mixed together in a disorderly
manner, their positions bearing no fixed relationship to those of their neighbor. The examples are
glass, silica, plastics, rubbers, polymers etc.
The difference between crystalline solids and amorphous solids are given below:
Properties Crystalline solid Amorphous solid
Geometrical shape
The crystals of every crystalline
solid have a defined geometrical
shape due to definite and orderly
arrangement of particles in three-
dimensional shape.
An amorphous solid does not have
any orderly pattern of arrangement
of particles and therefore, does not
have any definite geometrical
shape.
Melting point
The crystalline solids which directly
change into liquid state have
definite melting point.
Amorphous solids like glass do not
have definite melting point.
Cleavage planes
Crystalline solids have definite
cleavage planes which are inclined
to one another at a particular angle
for a given crystalline solid.
Amorphous solids do not have such
well-defined cleavage planes.
Anisotropic and
isotropic properties
As the orderly arranged particles in
crystalline solids are different in
different directions, these solids
show anisotropic properties.
As in amorphous solids, as those of
liquids and gases, the arrangement
of particles is random, these solids
show isotropic properties.
Symmetry
Crystalline solids have crystal
symmetry.
Amorphous solids do not have
symmetry.
Tensile strength
The tensile strength of crystalline
solid is very high.
The tensile strength of amorphous
solid is low.
Fracturing
The crystalline solids fracture in a
ductile manner.
The amorphous solids fracture in a
brittle manner.
Examples
Metals, 𝑁𝑎𝐶𝑙, 𝑍𝑛𝑂, 𝐾𝐶𝑙, 𝐶𝑠𝐶𝑙,
carbon etc.
Glass, plastics, polymers, rubbers,
silica etc.

4. Crystal: A crystal is a solid whose constituent atoms or molecules are arranged in a
systematic geometrical pattern.
The term ‘crystal’ of a material may be defined as a small body having a regular polyhedral
form bounded by smooth surfaces which are acquired under the action of its intermolecular
forces.
In other words, the three dimensional periodic arrangement of atoms in a solid is known as
crystal. An example for crystal is 𝑁𝑎𝐶𝑙.
Unit Cell: The unit cell is the simplest repeating unit in the crystal.
Crystal Lattice
The crystal lattice is used to describe the lattice of a real crystal. A lattice point in a crystal
lattice represents the position of a sodium ion or a chloride ion. The atoms arrange themselves in
a distinct pattern in space called a space lattice. In other words, the three dimensional network
of imaginary lines connecting the atoms in a crystalline solid is called the space lattice or crystal
lattice.
Bravais Lattice
Bravais lattices are more mathematical and abstract than crystal lattices. They are pretty much the same
as crystal lattices. Unlike the crystal lattice, however, lattice points in the Bravais lattice no longer
represent a position of a particular atom. Instead a lattice point represents a position in which an atom
can be placed. What matters in a Bravais lattice are not the points themselves, but how they are
arranged (i.e. symmetry). Ex-Na,Li

5. Symmetry
Crystal symmetry refers to the balanced pattern of the atomic structure which is reflected in the
external (crystal) shape. Different species vary in the symmetrical arrangement of faces. These
arrangements have certain ‘planes’ and ‘axes’ of symmetry. These form part of the definition of
the crystal system to which specific gemstones belong.
Plane of Symmetry
An imaginary plane dividing a body into two parts such that each is the reflected image of the
other. Crystals may have more than one plane of symmetry. i.e. a cube has nine planes of
symmetry.
Axis of Symmetry
An imaginary axis is placed through a perfect crystal so that during a single rotation about this
axis the outline of the crystal form appears identically more than once; 2, 3, 4 or 6 times.
Centre of Symmetry: (Centro-Symmetry)
Often present, it exists when every face of a perfect crystal is exactly opposite a similar face on
the other side of the crystal.

6. Thelatticeparameterrelationshipforthesevencrystalsystemisgivenbelow:
Crystalsystem Relationbetweenprimitives Interfaceangles
Cubic 𝑎 = 𝑏 = 𝑐 𝛼 = 𝛽 = 𝛾 = 90°
Monoclinic 𝑎 ≠ 𝑏 ≠ 𝑐 𝛼 = 𝛾 = 90°; 𝛽 ≠ 90°
Triclinic 𝑎 ≠ 𝑏 ≠ 𝑐 𝛼 ≠ 𝛽 ≠ 𝛾 ≠ 90°
Tetragonal 𝑎 = 𝑏 = 𝑐 𝛼 = 𝛽 = 𝛾 = 90°
Orthorhombicorrhombic 𝑎 ≠ 𝑏 ≠ 𝑐 𝛼 = 𝛽 = 𝛾 = 90°
Rhombohedralortrigonal 𝑎 = 𝑏 = 𝑐 𝛼 = 𝛽 = 𝛾 ≠ 90°
Hexagonal 𝑎 = 𝑏 ≠ 𝑐 𝛼 = 𝛽 = 90°; 𝛾 = 120°

7. Crystal structure: The structure of a compound in which the atoms are arranged in a regular
pattern or crystal form is known as crystal structure.
Crystals are two types according to the arrangement of atoms or molecules:
 Single crystal structure and
 Polycrystalline structure.
Single crystal structure: A single crystal is solid throughout which the atoms or molecules
are arranged in a regular repetitive pattern. Examples are 𝑁𝑎𝐶𝑙, diamond etc.
In the simplest form, the atomic pattern is continuous throughout the whole mass of the
metal. When this occurs the metal is said to be a single crystal. Such a state is uncommon.
Polycrystalline structure: A crystalline solid in which arrangements of many crystals are
separated by well-defined boundaries is called polycrystalline structure.
Usually the crystal pattern is discontinuous and changes its inclination from place to place.
Thus polycrystalline structure is formed.

8. Face centered cubic (FCC) structure: A FCC unit cell has an atom at each corner of the cube and
in addition, one atom at the intersection of the diagonals of each of the six faces of the cube
but none at the center of the cube. It is also known as cubic packed structure. It is more
common among metals than the BCC structure. 𝐴𝑙, 𝐶𝑢, 𝑃𝑏, 𝑁𝑖, 𝐶𝑜, 𝐴𝑔, 𝛾 𝐹𝑒 etc. posses FCC
arrangement. It is also formed in ceramic crystal.
We know total number of atoms in a cubic unit cell is
𝑁 =
𝑁𝑐
8
+
𝑁𝑓
2
+
𝑁𝑖
1
Here,
𝑁𝑐 = Total number of corner atom in unit cell
𝑁𝑓 = Total number of face atom in unit cell
𝑁𝑖 = Number of interior or center atom in unit cell
For FCC,
𝑁𝑐 = 8, 𝑁𝑓 = 6 and 𝑁𝑖 = 0
Therefore,
𝑁 =
8
8
+
6
2
+
0
1
= 1 + 3 + 0 = 4
Hexagonal close packed (HCP) structure: A HCP unit cell has one atom at each corner of the
hexagon, one atom at each center of the two hexagonal face (basal planes) and one atom at
each center of the line connecting the perpendiculars in case of three rhombuses, namely
𝐷𝐸𝐹𝐺, 𝐵𝐷𝐺𝐻 and 𝐵𝐾𝐸𝐷, which combine and form the HCP structure.
We know total number of atoms in a hexagonal unit cell is
𝑁 =
𝑁𝑐
6
+
𝑁𝑓
2
+
𝑁𝑖
1
Here,
𝑁𝑐 = Total number of corner atom in unit cell
𝑁𝑓 = Total number of face atom in unit cell
𝑁𝑖 = Number of interior or center atom in unit cell
For HCP,
𝑁𝑐 = 12, 𝑁𝑓 = 2 and 𝑁𝑖 = 3
Therefore,
𝑁 =
12
6
+
2
2
+
3
1
= 2 + 1 + 3 = 6

9. Atomic radius: Atomic radius may be defined as half the distance between the centers of two
neighboring atoms. It is possible to calculate the atomic radius by assuming that atoms are
spherical in shape and are in contact in a crystal.
Atomic radius of SC structure: We know that each simple cubic structure has one atom at each
of the corner of the cube. If ′𝑎′ is the lattice parameter (length of the cube edge) and ′𝑟′ is the
atomic radius, then
𝑎 = 2𝑟
⇒ 𝑟 =
𝑎
2
Atomic radius of BCC structure: we know a BCC unit cell has one atom in the center of the
cube and one atom each at all the corners. Let ′𝑎′ is the lattice parameter and ′𝑟′ is the atomic
radius.
From figure we see,
𝐴𝐺 = 𝑟 + 2𝑟 + 𝑟 = 4𝑟 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (1)
Again,
𝐸𝐺2
= 𝑎2
+ 𝑎2
= 2𝑎2
⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (2)
Then,
𝐴𝐺2
= 𝐸𝐺2
+ 𝐴𝐸2
= 2𝑎2
+ 𝑎2
= 3𝑎2
⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (3)
From equation (1) and (3) above,
𝐴𝐺2
= 4𝑟 2
= 16𝑟2
= 3𝑎2
⇒ 𝑟2
=
3𝑎2
16
⇒ 𝑟 =
3𝑎
4

10. Atomic radius of FCC structure: We know that a FCC unit cell has an atom at each corner of
the cube and in addition, one atom at the intersection of the diagonals of each of the six faces
of the cube but none at the center of the cube. Let ′𝑎′ is the lattice parameter and ′𝑟′ is the
atomic radius.
From figure we see,
𝐴𝐺 = 𝑟 + 2𝑟 + 𝑟 = 4𝑟 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (1)
Then considering ∆𝐴𝐵𝐺,
𝐴𝐺2
= 𝐴𝐵2
+ 𝐵𝐺2
⇒ 4𝑟 2
= 𝑎2
+ 𝑎2
⇒ 16𝑟2
= 2𝑎2
⇒ 𝑟2
=
2𝑎2
16
⇒ 𝑟 =
𝑎2
8
⇒ 𝑟 =
𝑎
2 2
⇒ 𝑟 =
2𝑎
4
Atomic packing factor or density of packing or packing efficiency: The packing of atoms in
a unit cell of the crystal structure of a material is known as atomic packing. It has been defined
as the ratio of the volume of the atoms per unit cell to the total volume of the unit cell. Then,
APF =
actual volume of atoms per unit cell
total volume of the unit cell
=
𝑣
𝑉
APF for SC structure: Let ′𝑎′ is the lattice parameter and ′𝑟′ is the atomic radius.
Volume of unit cell = 𝑎 × 𝑎 × 𝑎 = 𝑎3
.
Atoms per unit cell = 1.
Volume of one atom = 4
3 𝜋𝑟3
.
For SC structure, we know, 𝑎 = 2𝑟.
Therefore,
𝑣 =
4
3
𝜋
𝑎
2
3
=
4𝜋𝑎3
3 × 8
=
𝜋𝑎3
6
Hence,
𝐴𝑃𝐹 =
𝜋𝑎3
6
𝑎3
=
𝜋
6
= 0.52

11. APF for BCC structure: Let ′𝑎′ is the lattice parameter and ′𝑟′ is the atomic radius.
Volume of unit cell = 𝑎 × 𝑎 × 𝑎 = 𝑎3
.
Atoms per unit cell = 2.
Volume of one atom = 2 × 4
3 𝜋𝑟3
.
For BCC structure, we know, 𝑟 = 3𝑎
4.
Therefore,
𝑣 =
8
3
𝜋
3𝑎
4
3
=
24 3𝜋𝑎3
3 × 64
=
3𝜋𝑎3
8
Hence,
𝐴𝑃𝐹 =
3𝜋𝑎3
8
𝑎3
=
3𝜋
8
= 0.68
APF for FCC structure: Let ′𝑎′ is the lattice parameter and ′𝑟′ is the atomic radius.
Volume of unit cell = 𝑎 × 𝑎 × 𝑎 = 𝑎3
.
Atoms per unit cell = 4.
Volume of one atom = 4 × 4
3 𝜋𝑟3
.
For FCC structure, we know, 𝑟 = 2𝑎
4.
Therefore,
𝑣 =
16
3
𝜋
2𝑎
4
3
=
32 2𝜋𝑎3
3 × 64
=
2𝜋𝑎3
6
Hence,
𝐴𝑃𝐹 =
2𝜋𝑎3
6
𝑎3
=
2𝜋
6
= 0.74

12. AFP for HCP structure: Let ′𝑎′ is the lattice parameter, ′𝑐′ is the distance between two basal
planes and ′𝑟′ is the atomic radius.
Then from ∆𝐻𝐺𝐷, area,
𝐴 =
1
2
× 𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡
⇒ 𝐴 =
1
2
× 𝑎 × 𝑎 sin 60°
⇒ 𝐴 =
1
2
𝑎2
sin 60°
Therefore total area,
𝐻𝐺𝐹𝐸𝐾𝐵 = 6 × 𝐴
⇒ 𝐻𝐺𝐹𝐸𝐾𝐵 = 6 ×
1
2
𝑎2
sin 60°
⇒ 𝐻𝐺𝐹𝐸𝐾𝐵 = 3𝑎2
sin 60°
Hence the volume of a HCP unit cell,
𝑉 = 3𝑎2
sin 60° × 𝑐
⇒ 𝑉 = 3𝑎2
𝑐 ×
3
2
Now, atoms per unit cell = 6.
Volume of six atoms = 6 × 4
3 𝜋𝑟3
= 8𝜋𝑟3
.
For HCP structure, 𝑟 = 𝑎
2.
So, volume of six atoms, 𝑣 = 6 × 4
3 𝜋 𝑎
2
3
= 𝜋𝑎3
.
Hence,
𝐴𝑃𝐹 =
𝜋𝑎3
3𝑎2 𝑐 ×
3
2
=
2𝜋𝑎
3 3𝑐
=
2𝜋
3 3
𝑐
𝑎
=
2𝜋
3 3 × 1.633
= 0.74
𝑐
𝑎
= 1.633

13. Derive an equation to calculate the density of crystal materials.
Let,
𝑛 = Number of atoms or molecules per unit cell
𝑀 = Molecular weight of the crystalline materials
𝑁 = Avogadro’s number = 6.023 × 1023
Now,
Weight of 𝑁 molecules or atoms = 𝑀 𝑔
Weight of 𝑛 molecules or atoms = 𝑀 × 𝑛
𝑁 𝑔 = weight of the unit cell
We know that weight of unit cell is 𝜌𝑎3
.
Where,
𝜌 = Density of crystal materials
𝑎 = Lattice parameter
Therefore,
𝜌𝑎3
=
𝑀 × 𝑛
𝑁
⇒ 𝜌 =
𝑀 × 𝑛
𝑎3 × 𝑁

14. 1. Aluminium has FCC structure and a density of 𝟐. 𝟔𝟗𝟗 × 𝟏𝟎 𝟑
𝒌𝒈/𝒎 𝟑
. Calculate its atomic
radius. Atomic weight of aluminium is 𝟐𝟔. 𝟗𝟖 𝒈/𝒎𝒐𝒍𝒆 and 𝑵 = 𝟔. 𝟎𝟐𝟑 × 𝟏𝟎 𝟐𝟑
𝒂𝒕𝒐𝒎𝒔/
𝒎𝒐𝒍𝒆.
We know,
𝜌 =
𝑛𝑀
𝑎3 𝑁
⇒ 𝜌 =
𝑛𝑀
4𝑟
2
3
𝑁
⇒ 𝜌 =
𝑛𝑀 × 2 2
16𝑟3 𝑁
⇒ 𝑟3
=
𝑛𝑀 × 2 2
16𝜌𝑁
⇒ 𝑟3
=
4 × 26.98 × 2 2
16 × 2.7 × 6.023 × 1023
⇒ 𝑟3
= 2.93 × 10−24
⇒ 𝑟 = 1.43 × 10−8
∴ 𝑟 = 1.43 × 10−8
𝑐𝑚
Given,
𝑛 = 4 for FCC
𝑀 = 26.98 𝑔/𝑚𝑜𝑙𝑒
𝑎 =
4𝑟
2
𝜌 = 2.699 × 103
𝑘𝑔/𝑚3
= 2.7𝑔/𝑐𝑚3
𝑁 = 6.023 × 1023
𝑎𝑡𝑜𝑚𝑠/𝑚𝑜𝑙𝑒
𝑟 =?

15. Bragg’s equation: In 1913 the father-and-son, William Lawrence Bragg and William Henry Bragg
worked out a mathematical relation to determine interatomic distances from X-ray diffraction
patterns. This relation is called the Bragg’s equation. They showed that:
i. The X-ray diffracted from atoms in crystal planes obeys the laws of reflection.
ii.The two rays reflected by successive planes will be in phase if the extra distance travelled by
the second ray is an integral number of wavelengths.
The equation is
𝑛𝜆 = 2𝑑 sin 𝜃
Where,
𝜃 = diffraction angle
𝜆 = wave length of the X-ray
𝑑 = distance between the planes
Derivation: Let a beam of X-rays is falling on the crystal surface. Two successive atomic planes of
the crystal are shown separated by a distance 𝑑. Let the X-rays of wavelength 𝜆 strikes the planes at an
angle 𝜃. For the first plane, 𝐴𝐵 is incident ray and 𝐵𝐶 is reflected ray. For the second plane, 𝐷𝐸 is
incident ray and 𝐸𝐹 is reflected ray. Now we draw perpendicular lines 𝐵𝐺 and 𝐵𝐻 on 𝐷𝐸 and 𝐸𝐹
respectively.
After falling on first plane some of the rays will be reflected at the same angle. Some of the rays
will penetrate and get reflected from the second plane. These rays will reinforce those reflected from
the first plane if the extra distance travelled by them (𝐺𝐸 + 𝐸𝐻) is equal to integral number, 𝑛, of
wavelengths. That is,
𝑛𝜆 = 𝐺𝐸 + 𝐸𝐻 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (1)
Geometry shows that
𝐺𝐸 = 𝐸𝐻 = 𝐵𝐸 sin 𝜃
From equation (1), we get,
𝑛𝜆 = 𝐵𝐸 sin 𝜃 + 𝐵𝐸 sin 𝜃
⇒ 𝑛𝜆 = 𝑑 sin 𝜃 + 𝑑 sin 𝜃
⇒ 𝑛𝜆 = 2𝑑 sin 𝜃
This is Bragg’s equation.
[Derived]

16. Types Of Bonds
The properties of a solid can usually be predicted from the valence and bonding preferences of
its constituent atoms. Four main bonding types are discussed here: ionic, covalent, metallic, and
molecular. Hydrogen-bonded solids, such as ice, make up another category that is important in
a few crystals. There are many examples of solids that have a single bonding type, while other
solids have a mixture of types, such as covalent and metallic or covalent and ionic.


17. Lattice energy of crystal system
The stability of an ionic solid is measured in terms of its lattice energy (U0). It is defined as the energy
released when on mole of an ionic crystal is formed from one mole of gaseous positive and one mole of gaseous
negative ions, when these are separated from each other y infinite distance. Greater the value of lattice
energy, more stable, more stable is the ionic solid. The lattice energy of sodium chloride is -
786 kj/mole. The negative sign indicates that the process is exothermic and that the energy of
the system is lowered as the solid is formed. The high value suggests the stability of the solid.
Factors Affecting Lattice Energy
 as the charge of the ions increases, the lattice energy increases
 as the size of the ions increases, the lattice energy decreases
Lattice energies are also important in predicting the solubility of ionic solids in H2O. Ionic
compounds with smaller lattice energies tend to be more soluble in H2O.
Cohesive energy: The amount of energy evolved when a crystalline solid is formed from infinitely
separated atoms or the amount of energy required to separate atoms in a crystalline solid to
infinite distance is known as cohesive or binding energy of the crystal. The interatomic forces
which hold the atoms together in a solid are responsible for crystal formation. So, the energy of
the crystal is lower than that of the free atoms by an amount equal to the energy required to pull
the atoms to an infinite distance. This energy is called the binding energy of the crystal.

18. What are defects and imperfections in crystals? Describe them with neat sketches.
Defects and imperfections in crystals: A crystal is a solid composed of atoms, ions or molecules arranged
in a pattern which is repetitive in three dimensions. In an ideal crystal, the atomic arrangement is perfectly
regular and continuous throughout. But, real crystals are never perfect; lattice distortion and various
imperfections, irregularities or defects are generally present in them. If the species in crystal are arranged in
irregular or random pattern, it is said to be defect or imperfection in crystal.
All defects and imperfections in crystals can be conveniently classed under four main divisions, namely,
a) Point defects or imperfections or zero dimensional imperfections
i. Vacancies
ii. Interstitialcies
iii. Impurities
iv. Substitutional
v. Schottky defect
b) Line defects or imperfections
i. Edge dislocation
ii. Screw dislocation
c) Planar, surface, interfacial or grain boundaries defects or imperfections
i. Grain boundaries
ii. Tilt boundaries
iii. Twin boundaries
d) Volume defects or imperfections

19. Point defects: The imperfections which take place due to imperfect packing of atoms during crystallization
are known as point imperfections. The point imperfections also take place due to vibrations of atoms at high
temperature.
i. Vacancies: Whenever one or more atoms are missing from a normally occupied position within a
crystal lattice is known as vacancy.
ii. Interstitialcies: An interstitial defect arises when an atom occupies a definite position in the lattice that
is not normally occupied in the perfect crystal.
iii. Impurities: Impurities may be small particles (such as slag inclusions in metals) embedded in the
structure or foreign (metal) atoms in the lattice. Impurities give rise to compositional defects.
iv. Electronic defects or Schottky defects: Electronic defects are the result of errors in charge distribution
in solids.

20. Line defects: The most important two-dimensional or line defect is the dislocation. A dislocation may be
defined as a disturbed region between two substantially perfect parts of a crystal.
i. Edge dislocation: An edge dislocation lies perpendicular to its Burger’s vector. It moves (in its slip plane) in
the direction of the Burger’s vector (slip direction)
ii. Screw dislocation: An screw dislocation lies parallel to its Burger’s vector. It moves (in the slip plane) in a
direction perpendicular to the Burger’s vector (slip direction).
Figure 1: Edge dislocation

21. Surface or planar defects: The defects which are take place on the surface of a material are known as
surface defects.
i. Grain boundaries: Grain boundaries are those planar imperfects ions in polycrystalline materials that
separate crystals (grains) of different orientation.
ii. Tilt boundaries: Till boundary may be regarded as an array of edge dislocations (⊥). Actually tilt
boundary is a class of low angle boundaries.

22. i. Twin boundaries: A twin boundary separates two parts of crystal having the same orientation and they
look like mirror image of each other.
Volume defects: Volume defects such as cracks or stacking faults may arise where there is only small
dissimilarity (electrostatically) between the stacking sequence of closed packed planes in FCC and HCP metals.
It is possible for one atom layer to be out of sequence relative to the atoms of the layers above and below,
giving a mistake or defect.

23. 1. Whatisunderstoodbytheterm‘solidsolution’?
Solid solution: A homogeneous mixture of two or more crystalline metals in varying proportion is known
as solid solution. The metal which is present in a small quantity in a solid solution is called solute while the
metalwhich ispresentinlargequantityiscalled solute.Solid solutionhasonlyonephase.
Solid solution forms more rapidly when the solvent and solute atoms have similar size and comparable
electron structure. For example brass is a solid solution of 𝐶𝑢 and 𝑍𝑛; a typical composition of 𝐶𝑢 64% and
𝑍𝑛 36%,where 𝐶𝑢 atomsare solvent and 𝑍𝑛 atomsare solute.
Therearetwotypesofsolid solution:
 Interstitialsolidsolutionand
 Substitutionalsolid solution.
1. What are the conditions for the formation of solid solution?
Three conditions must be fulfilled to form a solid solution. Such as –
 Two or more metals that are involved to form a solid solution should have similar crystal structure.
 Metals that are involved to form a solid solution should be of same group in the periodic table and
should have same oxidation number.
 Metals that are involved to form a solid solution should have nearly same atomic structure or atomic
size.

24. 1. Explain with neat sketches the substitutional solid solution and interstitial solid solution.
Substitutional solid solution: A solid solution in which solute atoms directly replace some solvent atoms so
that they lie at normal atoms site of the crystal structure then this type of solid solution is called substitutional
solid solution.
Substitutional solid solution forms when the solute and solvent atoms posses equal or approximately
equal diameter. For example atomic diameter of 𝐶𝑢 is 2.522Å and 𝑁𝑖 is 2.487Å and these two form
substitutional solid solution.
Substitutional solid solutions are two types:
 Random or disordered substitutional solid solution and
 Ordered substitutional solid solution.
Random or disordered substitutional solid solution: A solid solution in which the solute atoms replace the
solvent atoms randomly is called random substitutional solid solution and this type of substitution is called
random substitution. Following is the structure of random substitutional solid solution.
Ordered substitutional solid solution: A substitutional solid solution in which the solute atoms replace by
the solvent atoms orderly is called ordered substitutional solid solution and this type of substitution is known
as ordered substitution. 𝐶𝑢 − 𝑍𝑛, 𝐴𝑢 − 𝐶𝑢, 𝐶𝑢2 𝑀𝑛𝐴𝑙 are some examples of ordered structure. Following is
the structure of ordered substitutional solid solution.
Interstitial solid solution: A solid solution in which solute atoms are located in the interstices of the
solvent lattice is called interstitial solid solution.
Interstitial solid solution can form only when the solute are small enough to fit into the space between the
solvent atoms. They are unable to substitute solvent atoms and can only to fit interstices or spaces in the
crystal lattice of solvent atoms. For example FCC 𝐹𝑒 is the solvent for 𝐶.

25. Miller indices: Miller indices are a system of notation that denotes the orientation of the faces of a crystal
and the planes and directions of atoms within that crystal.
Some General Principles:
 If a Miller index is zero, the plane is parallel to that axis.
 The smaller a Miller index, the more nearly parallel the plane is to the axis.
 The larger a Miller index, the more nearly perpendicular a plane is to that axis.
 Multiplying or dividing a Miller index by a constant has no effect on the orientation of the plane
 Miller indices are almost always small.
Determination of Miller indices: Following are the rules for determination of Miller indices:
a) At first we should have to determine the intercepts of the plane on the axes 𝑥, 𝑦, 𝑧 in terms of lattice
constant.
b) Now we have to take the reciprocal terms of the intercepts.
c) If the results are fractions, then they are changed to the small integers having the same ratio i.e.,
multiplied each reciprocal or fraction by the same number such as the common denominator.
d) At last the result is enclosed in parenthesis (ℎ 𝑘 𝑙).
Example:
𝑥 𝑦 𝑧
Intercept 1 ∞ 1
Reciprocal
1
1
= 1
1
∞
= 0
1
1
= 1
Miller indices ℎ 𝑘 𝑙 = (1 0 1)
𝑥 𝑦 𝑧
Intercept 1 3 2
Reciprocal
1
1
=
1
1
× 6 =
6
1
3
=
1
3
× 6 =
2
1
2
=
1
2
× 6 =
3
Miller indices ℎ 𝑘 𝑙 = (6 2 3)