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SMES 3203 SOLID STATE PHYSICS (3 CREDITS)

References:

1. Introduction to Solid State Physics C. Kittel (John Wiley)

2. Elementary Solid State Physics M.A. Omar (Addison Wesley)

3. Solid State Physics J.S. Blakemore (Saunders)

4. Fundamental of Solid State Physics J. Richard Christman (John Wiley)

5. Waves, Atoms & Solids D.A. Davies (Longman)

6. Solid State Physics C.M. Kachhava (McGraw Hill)

7. Elements of Solid State Physics M.N. Rudden and J. Wilson (John Wiley)

1. Crystal Structure

1.1 Periodicity of Crystal

Solid material classified into 2 basic groups: crystalline and amorphous

amorphous shows short range ordering in its nearest neighbour bonds

eg. polymerized plastics, carbon blacks

crystalline shows long range ordering

atomic arrangement regularly repeated

position is exactly periodic

eg sodium chloride, diamond, silicon

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perfect crystal is periodic from - to + along the x and y directions

if crystal is translated by any vector R, the crystal appears exactly the same

as it did before the translation ie crystal remains invariant under any

translation

R : translational vector

Rn = n1a + n2b 2-D

Rn = n1a + n2b + n3c 3-D

n1, n2, n3 - arbitrary integers

a, b, c - basis vector, form 3 adjacent edges of a parallelepiped

- not necessary orthogonal

crystal lattice

geometrical pattern which represents the positions of every atoms

divided into 2 classes

Bravais lattice

non-Bravais lattice

Bravais lattice

all lattice points equivalent

that is all atoms in the crystal of the same type

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non-Bravais lattice

mixture of 2 or more interpenetrating Bravais lattices

A and A are not equivalent since lattice is not invariant (variant) under

translation by AA although A and A are of the same kind

eg. : A and A, B and B, C and C

non-Bravais lattice also referred to as a lattice with a basis

regarded as a combination of 2 or more interpenetrating Bravais lattices with

fixed orientations relative to each other

example: A, B, C form one Bravais lattice and A, B, C .. form

another Bravais lattice

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basis vector

position vector of any lattice point : Rn = n1a + n2b

D : (0,2) B : (1,0) F : (0,-1)

a and b form a set of basis vectors for the lattice

positions of all lattice points can be expressed by Rn = n1a + n2b

set of all vectors expressed by Rn = n1a + n2b called lattice vectors

choice of basis vectors is by convenience

a and b (=a + b) can be chosen as a basis

unit cell

2-D : area of parallelogram whose sides are basis vectors a and b

S= a x b

: area S of parallelogram whose sides are vectors a and b

S= a x (a +b) = a x b = S

3-D : volume of parallelepiped whose sides are basis vectors a, b and c

V= a . b x c

primitive unit cell

same area/volume although different shape

contains 1 lattice point

minimum area/volum

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non-primitive unit cell

area is multiple of area of primitive unit cell

S1 1 lattice point

S2 2 lattice points

area of S2 = 2 x area of S1

use of non-primitive cell S2 shows rectangular symmetry

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Wigner-Seitz primitive cell

(i) draw lines to connect a given lattice point to all nearby lattice points

(ii) at the midpoint and normal to these lines, draw new lines or planes

lines 2D

planes 3D

(iii) smallest area/volume enclosed Wigner-Seitz primitive cell

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1.2 Crystal Symmetry

inversion centre

cell has an inversion centre if there is a point at which the cell remains

invariant when a mathematical transformation r -r is performed on it

for every lattice vector Rn = n1a + n2b + n3c there is an associated lattice

vector Rn = -n1a - n2b - n3c

all Bravais lattices have an inversion centre

non-Bravais lattices may or may not have an inversion centre depending on

the symmetry of the basis

reflection plane

plane in a cell such that when a mirror reflection in this plane is performed,

the cell remains invariant

example:

cubic - 9 reflection planes : 3 parallel to the faces, 6 each of which passes

through 2 opposite edges

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rotation axis

axis such that if the cell is rotated around it through some angle, the cell

remains invariant

axis called n-fold if the rotation angle is n

2

example:

cubic - has three 4-fold axes normal to the faces : A1 becomes A2

- has four 3-fold axes each passing through two opposite corners :

A1 becomes A3

- has six 2-fold axes joining the centres of opposite edges : A1

becomes A4

rotation-reflection axes, glide planes etc complicated elements of

symmetry

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1.3 Lattices

7 crystal system

can be divided into 14 Bravais lattices

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