4. Crystal Structure

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Solid State PhysicsIntroduction to

Crystal structureFree e Fermi gas

Energy bands

Magnetic properties

e confinement (quantum, nano devices)

Superconductivity



Physics?



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



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.



A protein molecule

1 2 3If we can write r r n a n b n c

, , fundamental translation vectorsa b c



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



Primitive cell (examples)



Wigner-Seitz cell



Wigner-Seitz primitive cell: 3D

f



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



Two Dimensional Lattices

Th Di i l L i T



Three Dimensional Lattice Types



Dimensions

F t B i L tti



Fourteen Bravais Lattices …

C bi l tti



Cubic space lattices

C bi l tti



Cubic lattices

BCC t t



BCC structure

P i iti t BCC



Primitive vectors BCC

El t ith BCC St t



Elements with BCC Structure

FCC l tti



FCC lattice

P i iti C ll FCC L tti



Primitive Cell: FCC Lattice



Structure

Crystal planes probing them



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



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



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



Indices of Planes: Cubic Crystal



indices

Some planes



Some planes

Use of Miller indices



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



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



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

4. Crystal Structure

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