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PH0101 UNIT 4 LECTURE 4RECIPROCAL LATTICE

FEATURES OF RECIPROCAL LATTICE

GRAPHICAL REPRESENTATION

GENERAL PROCEDURE

SIMPLE CUBIC STRUCTURE

BODY CENTERED CUBIC STRUCTURE

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RECIPROCAL LATTICE

The concept of ‘reciprocal lattice’ provides a device for tabulating both the slopes and the interplanar spacings of the planes of a crystal lattice.

In a crystal, there exist many sets of planes with different orientations and spacing. These planes can cause diffraction.

If we draw normals to all sets of planes, from a common origin, then the end points of normals form a lattice which is called as ‘reciprocal lattice’.

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FEATURES OF RECIPROCAL LATTICE

Each point in the reciprocal lattice preserves the characteristics of the set of planes which it represents.

Its direction with respect to the origin represents the orientation of the planes.

Its distance from the origin represents the interplanar spacing of the planes.

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GRAPHICAL REPRESENTATION

Consider all the planes belonging to a single zone.

Since all the planes to be considered are parallel to a

common line, known as zone line, the normals to these

planes lie in the same plane (normal to the zone axis).

In this way, these planes can be represented by a two

dimensional reciprocal lattice.

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FORMATION OF RECIPROCAL LATTICE

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The figure shows the unit cell of a monoclinic

crystal looking along its unique axis designated by C.

The cell edges are ‘a’ and ‘b’.

The example also shows the edge view of four (h k l)

planes, namely (100), (110), (120) and (010).

RECIPROCAL LATTICE

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GENERAL PROCEDURE FOR LOCATING THE RECIPROCAL LATTICE

A lattice point is taken as common origin.

From the common origin, draw a normal to each plane.

Place a point on the normal to each plane (h k l) at a distance from the origin equal to .

Such points form a periodic array called reciprocal lattice.

hkld1

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CRYSTAL STRUCTURE OF MATERIALS

NUMBER OF ATOMS PER UNIT CELL (n) The total number of atoms present in an unit cell is

known as number of atoms per unit cell.COORDINATION NUMBER (CN)

It is the number of nearest neighboring atoms to a particular atom.

ATOMIC RADIUS (r) It is the radius of an atom. It is also defined as half the

distance between two nearest neighboring atoms in a crystal.

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CRYSTAL STRUCTURE OF MATERIALS

ATOMIC PACKING FACTOR (APF) It is the ratio of volume occupied by the atoms or molecules

in an unit cell (v) to the total volume of the unit cell (V).

APF =

APF = No. of atoms present in the unit cell x Volume of the atom

Volume of the unit cell

Volume occupied by the atoms in an unit cellVolume of the unit cell

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SIMPLE CUBIC STRUCTURE

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SIMPLE CUBIC STRUCTURE

Each and every corner atom is shared by eight adjacent unit cells. The contribution of each and every corner atom to one unit cell is 1/8.

The total number of atoms present in a unit cell =1/8 x 8 =1.

CORDINATION NUMBER (CN)

For SCC atom, there are four nearest neighbours in its own plane.

There is another nearest neighbour in a plane which lies just above this atom and yet another nearest neighbour in another plane which lies just below this atom.

Therefore the co-ordination number is 6.

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SIMPLE CUBIC STRUCTURE

ATOMIC RADIUS (R) Since the atoms touch along cube edges, the atomic radius for a

simple cubic unit cell is, r =

(where a = 2r, is the lattice constant)

ATOMIC PACKING FACTOR (APF)

APF =

v = 1 4/3 r3

a2

vV

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SIMPLE CUBIC STRUCTURE

V = a3

APF =

APF =

Substituting r = , we get,

APF =

3

3

1 4 3 ra

3

3

4 r3a

a2

3

3

3 3

a44 a2

63a 8 3a

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SIMPLE CUBIC STRUCTURE

Therefore packing density = /6 = 0.5236

APF = 0.52

Thus 52 percent of the volume of the simple cubic unit cell is occupied by atoms and the remaining 48 percent volume of the unit cell is vacant or void space.

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BODY CENTERED CUBIC STRUCTURE

A body centred cubic structure has eight comer

atoms and one body centred atom.

The atom at the centre touches all the eight

corner atoms.

The BCC structure is shown in the figure.

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BODY CENTERED CUBIC STRUCTURE

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BODY CENTERED CUBIC STRUCTURE

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BODY CENTERED CUBIC STRUCTURE

In BCC unit cell, each and every corner atom is shared by eight adjacent unit cells. So, the total number of atoms contributed by the corner atoms is × 8 = 1.

A BCC unit cell has one full atom at the centre of the unit cell.

The total number of atoms present in a BCC unit cell = 1+1 = 2.

18

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BODY CENTERED CUBIC STRUCTURE

CO-ORDINATION NUMBER (CN) Let us consider a body centred atom. The nearest neighbour for a

body centred atom is a corner atom. A body centred atom is surrounded by eight corner atoms.

Therefore, the co-ordination number of a BCC unit cell = 8.

ATOMIC RADIUS (R) For a BCC unit cell, the atomic radius can be calculated

From figure AB = BC = AD = ‘a’ and CD = 4r.

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BODY CENTERED CUBIC STRUCTURE

From the triangle, ACD,

CD2 = AC2 + AD2

CD2 = 2a2 + a2

(4r)2 = 3a2

16r2 = 3a2

i.e. r2 = 23a

16

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BODY CENTERED CUBIC STRUCTURE

atomic radius r = a

ATOMIC PACKING FACTOR (APF)

APF =

The number of atoms present in an unit cell = 2

v = 2 x 4/3 r3

V = a3

34

vV

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BODY CENTERED CUBIC STRUCTURE

APF =

Substituting r = we get

APF =

APF = = 0.68

3

3

2 4 / 3 ra

3 a4

3

3

3 a2 4 / 3

4

a

38

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BODY CENTERED CUBIC STRUCTURE

Thus 68 percent of the volume of the BCC unit cell is occupied by atoms and the remaining 32 percent volume of the unit cell is vacant or void space.

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