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 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.

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.

Crystal lattice: Proteins

Crystal Structure

Honeycomb: NOT Bravais

Crystal structure: basis

Translation Vector T

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

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.

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

Five fold axis of symmetry cannot exist

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.

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

Two Dimensional Lattices

Three Dimensional Lattice Types

Wigner-Seitz Primitive Cell: Full symmetry of Bravais Lattice

Conventional Cells

Cubic space lattices

Cubic lattices

BCC Structure

BCC Crystal

BCC Lattice

Primitive vectors BCC

Elements with BCC Structure

FCC Structure

FCC lattice

Primitive Cell: FCC Lattice

Elements That Have FCC Structure

HCP Crystal

Hexagonal Close Packed

HCP

Primitive Cell: Hexagonal System

Miller indices of lattice plan 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:

Indices of Crystal Plane

Indices of Planes: Cubic Crystal

001 Plane ?

110 Planes

111 Planes

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.

NaCl Structure

NaCl Basis

NaCl Type Elements

CsCl Structure

CeCl Basis

CeCl Basis

CeCl Crystals

ZincBlende structure

Diamond Crystal Structure

Symmetry planes

The End: Chapter 1

Bragg Diffraction

Periodic Table

Lanthanoids and Actinoids

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.

HCP Close Packing

HCP Close Packing

NaCl Basis

lattice