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
Honeycomb net: Bravais lattice
with two point basis
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
Summary: Bravais Lattices (Nets)
in Two Dimensions
Escher loved two dimensional
structures too
Summary: Fourteen Bravais
Lattices in Three Dimensions
Fourteen Bravais Lattices …
FCC Structure
FCC lattice
Primitive Cell: FCC Lattice
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.
Elements That Have FCC Structure
Simple Hexagonal Bravais Lattice
Primitive Cell: Hexagonal System
HCP Crystal
Hexagonal Close Packing
HexagonalClosePacked
HCP lattice is not a Bravais lattice, because orientation of the environment
Of a point varies from layer to layer along the c-axis.
HCP: Simple Hexagonal Bravais
With Basis of Two Atoms Per Point
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:
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
CsCl Basis
CsCl Basis
CeCl Crystals
Diamond Crystal Structure
ZincBlende structure
Symmetry planes
The End: Chapter 1
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
Close Packing 2
Close Packing 3
Close Packing 4
Close Packing 5
NaCl Basis
Close Packing of Spheres