Post on 03-Apr-2018
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Solid State PhysicsIntroduction to
Crystal structureFree e Fermi gas
Energy bands
Magnetic properties
e confinement (quantum, nano devices)
Superconductivity
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Physics?
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Basics
Crystal Periodic array of atoms
Lattice Periodic array of points in space
Basis Group of atoms attached to each lattice
point or each elementary
parallelepiped
Lattice + Basis Crystal structure
Platinum (STM image)NaClInsulin
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Bravais Lattice
An infinite array of discrete points with anarrangement and orientation that appearsexactly the same, from any of the points thearray is viewed from.
A three dimensional Bravais lattice consists of all points with position vectors R that can bewritten as a linear combination of primitivevectors. The expansion coefficients must beintegers.
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A protein molecule
1 2 3If we can write r r n a n b n c
, , fundamental translation vectorsa b c
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Primitive Unit Cell
A primitive cell or primitive unit cell is a volume of
space that when translated through all the vectors in aBravais lattice just fills all of space without either
overlapping itself or leaving voids.
A primitive cell must contain precisely one lattice point.
Primitive lattice cell
Parallelepiped defined by a, b, c
= unit cell = minimum volume cell
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Primitive cell (examples)
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Wigner-Seitz cell
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Wigner-Seitz primitive cell: 3D
f
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Fundamental types of lattices
Crystal lattices can be mapped intothemselves by the lattice translations T and byvarious other symmetry operations.
A typical symmetry operation is that of rotation
about an axis that passes through a latticepoint. Allowed rotations of : 2 π, 2π/2,2π/3,2π/4, 2π/6
(Note: lattices do not have rotation axes for 1/5, 1/7 …) times 2π
T Di i l L i
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Two Dimensional Lattices
Th Di i l L i T
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Three Dimensional Lattice Types
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Dimensions
F t B i L tti
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Fourteen Bravais Lattices …
C bi l tti
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Cubic space lattices
C bi l tti
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Cubic lattices
BCC t t
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BCC structure
P i iti t BCC
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Primitive vectors BCC
El t ith BCC St t
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Elements with BCC Structure
FCC l tti
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FCC lattice
P i iti C ll FCC L tti
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Primitive Cell: FCC Lattice
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Structure
Crystal planes probing them
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Crystal planes – probing them
XY
1
2
a
Coherent incident light Diffracted light
Path difference XYbetween diffracted
beams 1 and 2:
sin = XY/a
XY = a sin
For 1 and 2 to be in phase and give constructiveinterference, XY = , 2, 3, 4…..n
so a sin = n where n is the order of diffraction
Constructive Interference
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Beam 2 lags beam 1 by XYZ = 2d sin
so 2d sin = n Bragg’s Law
X
Y
Z
d
Incident radiation “Reflected” radiation
Transmitted radiation
1
2
Constructive Interference
Miller indices of lattice plane
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Miller indices of lattice plane The indices of a crystal plane (h,k,l) are defined to be
a set of integers with no common factors, inversely
proportional to the intercepts of the crystal plane alongthe crystal axes:
Indices of Planes: Cubic Crystal
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Indices of Planes: Cubic Crystal
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indices
Some planes
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Some planes
Use of Miller indices
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Where does a protein crystallographer see the Miller
indices?
• Common crystal faces areparallel to lattice planes
• Each diffraction spot can be
regarded as a X-ray beam
reflected from a lattice plane,
and therefore has a unique
Miller index.
Use of Miller indices
Simple Crystal Structures
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Simple Crystal Structures
There are several crystal structures of common
interest: sodium chloride, cesium chloride, hexagonalclose-packed, diamond and cubic zinc sulfide.
Each of these structures have many different
realizations.
References
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References
A. Beiser – “Concepts of Modern Physics”, 6 Ed., Tata
McGraw-Hill (New Delhi, 2003)
Charles Kittel – “Introduction to Solid State Physics”, 7
Ed., John Wiley and Sons (New York, 1996)
www.wikipedia.org