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Why Study Solid State Physics?

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Page 1: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Why Study Solid State Physics?

Page 2: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Ideal Crystal

• An ideal crystal is a periodic array of structural units, such as atoms or molecules.

• It can be constructed by the infinite repetition of these identical structural units in space.

• Structure can be described in terms of a lattice, with a group of atoms attached to each lattice point. The group of atoms is the basis.

Page 3: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Bravais Lattice

• An infinite array of discrete points with an arrangement and orientation that appears exactly the same, from any of the points the array is viewed from.

• A three dimensional Bravais lattice consists of all points with position vectors R that can be written as a linear combination of primitive vectors. The expansion coefficients must be integers.

Page 4: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Crystal lattice: Proteins

Page 5: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Crystal Structure

Page 6: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Honeycomb: NOT Bravais

Page 7: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Honeycomb net: Bravais lattice with two point basis

Page 8: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Crystal structure: basis

Page 9: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Translation Vector T

Page 10: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Translation(a1,a2), Nontranslation Vectors(a1’’’,a2’’’)

Page 11: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Primitive Unit Cell

• A primitive cell or primitive unit cell is a volume of space that when translated through all the vectors in a Bravais lattice just fills all of space without either overlapping itself or leaving voids.

• A primitive cell must contain precisely one lattice point.

Page 12: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed
Page 13: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Fundamental Types of Lattices

• Crystal lattices can be mapped into themselves by the lattice translations T and by various other symmetry operations.

• A typical symmetry operation is that of rotation about an axis that passes through a lattice point. 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

Page 14: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Five fold axis of symmetry cannot exist

Page 15: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Two Dimensional Lattices

• There is an unlimited number of possible lattices, since there is no restriction on the lengths of the lattice translation vectors or on the angle between them. An oblique lattice has arbitrary a1 and a2 and is invariant only under rotation of π and 2 π about any lattice point.

Page 16: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Oblique lattice: invariant only under rotation of pi and 2 pi

Page 17: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Two Dimensional Lattices

Page 18: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Three Dimensional Lattice Types

Page 19: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Wigner-Seitz Primitive Cell: Full symmetry of Bravais Lattice

Page 20: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Conventional Cells

Page 21: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Cubic space lattices

Page 22: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Cubic lattices

Page 23: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

BCC Structure

Page 24: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

BCC Crystal

Page 25: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

BCC Lattice

Page 26: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Primitive vectors BCC

Page 27: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Elements with BCC Structure

Page 28: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Summary: Bravais Lattices (Nets) in Two Dimensions

Page 29: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Escher loved two dimensional structures too

Page 30: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Summary: Fourteen Bravais Lattices in Three Dimensions

Page 31: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Fourteen Bravais Lattices …

Page 32: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

FCC Structure

Page 33: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

FCC lattice

Page 34: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Primitive Cell: FCC Lattice

Page 35: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

FCC: Conventional Cell With Basis

• We can also view the FCC lattice in terms of a conventional unit cell with a four point basis.

• Similarly, we can view the BCC lattice in terms of a conventional unit cell with a two point basis.

Page 36: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Elements That Have FCC Structure

Page 37: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Simple Hexagonal Bravais Lattice

Page 38: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Primitive Cell: Hexagonal System

Page 39: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

HCP Crystal

Page 40: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Hexagonal Close Packing

Page 41: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

HexagonalClosePacked

HCP lattice is not a Bravais lattice, because orientation of the environmentOf a point varies from layer to layer along the c-axis.

Page 42: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

HCP: Simple Hexagonal Bravais With Basis of Two Atoms Per Point

Page 43: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

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 along the crystal axes:

Page 44: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Indices of Crystal Plane

Page 45: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Indices of Planes: Cubic Crystal

Page 46: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

001 Plane

Page 47: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

110 Planes

Page 48: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

111 Planes

Page 49: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Simple Crystal Structures

• There are several crystal structures of common interest: sodium chloride, cesium chloride, hexagonal close-packed, diamond and cubic zinc sulfide.

• Each of these structures have many different realizations.

Page 50: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

NaCl Structure

Page 51: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

NaCl Basis

Page 52: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

NaCl Type Elements

Page 53: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

CsCl Structure

Page 54: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

CsCl Basis

Page 55: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

CsCl Basis

Page 56: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

CeCl Crystals

Page 57: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Diamond Crystal Structure

Page 58: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

ZincBlende structure

Page 59: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Symmetry planes

Page 60: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

The End: Chapter 1

Page 61: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed
Page 62: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Bravais Lattice: Two Definitions

The expansion coefficients n1, n2, n3 must be integers. The vectors a1,a2,a3 are primitive vectors and span the lattice.

Page 63: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

HCP Close Packing

Page 64: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

HCP Close Packing

Page 65: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Close Packing 2

Page 66: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Close Packing 3

Page 67: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Close Packing 4

Page 68: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Close Packing 5

Page 69: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

NaCl Basis

Page 70: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed

Close Packing of Spheres

Page 71: Why Study Solid State Physics?. Ideal Crystal An ideal crystal is a periodic array of structural units, such as atoms or molecules. It can be constructed