Download - Crystal Lattice
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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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