Gramin (ACS) Mahavidyalaya VasantNagar Kotgyal.

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DEPARTMENT OF PHYSICS

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B.Sc. Final Year

Solid State Physics

Crystal Structure

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Dr.Kendre D.K.

Head,Department of Physics

Gramin (ACS) Mahavidyalaya VasantNagar Kotgyal.

Crystal Structure

• Crystal Lattice and Translation Vectors
• Unit Cell
• Basis
• Symmetry Operations
• Point Group
• Space Group
• Types of Lattices
• Simple Crystal Structure ( HCP, FCC, SC)
• Structure of Diamond
• Structure of NaCl

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Introduction

Elements and their chemical compounds are found in three states 1. Solid 2. Liquid 3. Gases.

• If the atoms or molecules are arranged in some regular fashion then it is known as crystalline.
• When the atoms or molecules are arranged in an irregular fashion then it is known as amorphous.
• A crystal is a solid composed of periodic array of atoms .
• The study of Solid state physics aims to interpret the macroscopic properties in terms of the properties of the microscopic particles of which the solid is composed.
• The study of the geometric form and the other physical properties of crystalline solids by using X-rays , electron beams, etc constitute the science of Crystallography.

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• To describe the arrangement of imaginary points in space which has definite relationship with the atoms of crystal.
• The arrangement of infinite number of imaginary points in three dimensional space with each point having identical surroundings is known as a Point lattice or space lattice.
• The term identical surroundings means that the lattice has the same appearance when viewed from a point ‘r’ in the lattice as it has when viewed from any other point r’ w.r.t. some arbitrary origin. This is possible only if the lattice contains a small group of points called as pattern unit, which repeats itself in all directions by means of a Translation operator ‘T’.
• Translational operator is given by

T = n₁a +n₂b +n₃c

• For lattice to represent a crystal structure , we associate every lattice point with one or more atoms called the Basis or the pattern.
• When the basis is repeated with correct periodically in all directions gives the actual Crystal structure.

Thus

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Crystal Lattice and Translation Vectors

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• The parallelogram formed by the translational vectors may be regarded as building block for constructing the complete lattice and are known as Unit Cell of the lattice.
• The unit cell may be defined as the smallest unit of the lattice which on continuous repetition, generates the complete lattice.
• Both primitive and non primitive translational vectors may be used to construct a unit cell.
• Accordingly , a unit cell is named as primitive unit cell or non primitive unit cell.

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Unit Cell

A

C

M

D

E

F

G

H

P

NP

NP

K

L

B

N

• The space lattice is defined as an array of imaginary points which are so arranged in space that each point has identical surroundings.
• In order to obtain crystal structure , an atoms or a group of atoms must be placed on each lattice point in regular fashion. Such an atom or group of atoms is called as Basis.

Space Lattice + Basis = Crystal Structure

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Basis

�����Symmetry Operations������

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• A symmetry operation is that which transforms the crystals to itself i.e. a crystal remains invariant under a symmetry operation.
• Mainly Translation, Rotation, inversion, reflection operation.
• Translation operation applies only to lattice.
• All other operations apply to all objects and are collectively known as Point symmetry operation.
• Inversion operation is applicable only to three dimenssional crystals.

1. Translation Operation: The translation symmetry follows orderly arrangement of a lattice means that a lattice point r , under lattice translation vector operation T gives another point r^’ which is exactly identical to r i.e.

r^' = r + T

2. Rotation : A lattice is said to possess the rotation symmetry if its rotation by an angle ɵ about an axis transforms the lattice to itself.

The lattice always remains invarient by a rotation of 2Π , the angle 2Π must be integral multiple of ɵ i.e.

n ɵ = 2Π

ɵ = 2Π /n

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3. Reflection: A lattice is said to possess reflection symmetry if there exists a plane in the lattice which divides it into two identical halves which are mirror images of each other. Such a plane is represented by m.

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4. Inversion : Inversion is a point operation which is applicable to only three dimensional lattice. This symmetry element implies that each point located at r relative to a lattice point has an identical point located at r relative to same lattice point. It means that the lattice possess a centre of inversion denoted by i.

� Point Group , Space Group

• The crystals are classified on the basis of their symmetry which is compared with the symmetry of different point groups. Also the lattices consistent with the point group operations are limited. Such lattices are known as Bravais Lattice.
• The point symmetry of crystal structure as a whole is determined the point symmetry of the lattice as well as the basis. In order to determine the point symmetry of crystal structure, 1) a unit cell might show point symmetry at more than one location inside it. 2) The symmetry element comprising combined point and translation operation .
• The group of all symmetry elements of a crystal structure is called space group. It determines the symmetry of a crystal structure as a whole.

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Types of Lattices��

• The number of point groups in two and three dimension forms the basis for construction of different types of lattices. Only those lattices are permissible which are consistent with point group operation. Such lattices are called as Bravais Lattice.

Two Dimensional Lattice

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• a) Hexagonal a ≠ b, ɵ = 120^o
• b) Square a = b , ɵ =90^o

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• ^{C) Rectangular Centered}

a ≠ b ,ɵ =90^o

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• Three Dimensional Lattice : All the crystal system of three dimensional space and corresponding Bravais Lattice are as follows.

Crystal Structure ( HCP, FCC, SC)

• The basic crystal structure which are either monoatomic or contain simple basis includes closed packed structure like hexagonal close packed or face –centered cubic structure and loose packed structure like body centered cubic or simple cubic structure.
• A) Close packed Structure : Close packed Structure are mostly found in monoatomic crystals having non directional bonding, such as metallic bonding. This structure having coodination number of each atom is 12 i.e. each atom is surrounded by 12 similar and equal sized neighbors. Out of these 12 neighbors 6 lies in one plane , 3 are in adjacent parallel plane above this plane and 3 are in similar plane below it.

• A) Hexagonal close packed ( HCP ) Structure

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• This type of stacking is called as hcp stacking and structure is called as Hexagonal close packed ( HCP ) Structure. The name corresponds to the shape of conventional unit cell which is hexagonal as shown in figure. There are 12 atoms located at the corners , 2 at the centers of the basal planes, and 3 completely inside the hexagon forming a part of B- layer .

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• Face Centered Cubic Structure :

In this structure the stacking of first two layers is similar

to that of HCP structure . The

difference arises in third layer

which does not overlap the first

Layer . The conventional unit cell

is face centered cubic is shown in

figure . Examples of material having this type of structure are Cu, Ag, Au, Al etc. The packing fraction is given by

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= 0.74

• B. Loose packed Structure : A loose packed structure is that in which the coordination number of an atom is less than 12 or the packing fraction is less than 0.74. The most common and simplest form of loose packed structure are the body centered cubic (BCC) and simple cubic ( SC).
• a) Body centered cubic (BCC) structure:

The conventional unit cell of BCC structure is

non primitive which is shown in figure. It

Has cubical shape with atoms located at the

Corners and the body centre. The example

of BCC structure are Na, K , Mo, W etc.

The packing fraction is given by

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= 0.68

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• b) Simple cubic ( SC) Structure :

The conventional unit cell of SC structure is

Same as its primitive which is shown in figure.

The atoms are located at the corners only and

Touch one another along the cube edge.

Only polonium exhibits this type of structure at room temperature.

The packing fraction is given by

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= 0.52

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In a diamond structure in the conventional fcc lattice all atomic sites are occupied by

carbon atoms. Semiconductors such as

diamond (C), silicon(Si),germanium

The structure is a combination of two

identical inter penetrating fcc lattices.

One of the sub lattices is shifted along

the body diagonal of the cubic cell by

one quarter of the length of the diagonal.

The diamond structure is thus fcc with

a basis containing two identical atoms.

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Structure of Diamond

A simple method for constructing a diamond lattice considers it as a fcc structure with an extra atom placed at ¼a1, þ¼a2,þ¼a3 from each of the fcc atoms. The basic element of the structure is a tetrahedron where a C atom is at the center , and its four NNs are at the corners of the cube (or vice versa).Each atom forms type crystals form covalent bonding. The bonding energy is associated with the shared valence electrons between atoms and depends on the relative orientation of atoms

� Structure of NaCl

The structure of unit cell of NaCl is as shown in figure.

In NaCl structure the radii of Na & Cl

ions are such that each Na ion is

octahedrally coordinated to 6 Cl ions.

The unit cell is fcc with 4 Cl ions

occupying all four fcc and the four Na

ions occupying all the four positions octahedral voids. The NaCl structure can be viewed as two interpenetrating fcc sub lattices , one belongs to Na ions with its origin at the point (0,0,0) and other belongs to Cl ions with its origin at the point ( a/2,0,0).

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Thanks