p882dh3 - Brief Summary of a Few Central Concepts in...

Brief Summary of a Few Central Concepts in Condensed Matter
Physics
Physics 880.06: Winter, 2010-11
David G. Stroud
In these notes I give a brief survey of a few central concepts in condensed
matter physics. More details can be found in a number of standard sources,
e. g. Ashcroft and Mermin. These notes are a work in progress: I hope to
update them from time to time.
1. Bravais Lattice.
A Bravais lattice is a network of points in d-dimensional space (usually
we take d = 3) defined by the equation
R =
d
summationdisplay
i =1
n
i
a
i
, (1)
where the n
i
’s are positive or negative integers. The vectors a
i
are called
basis vectors , and they must all be linearly independent.
For example, in a cubic lattice in d = 3, the conventional choice of the
three basis vectors is a
i
= a ˆ x
i
, where ˆ x
i
is a unit vector in the i th direction.
2. Unit Cell; Primitive Cell.
1

A unit cell for a crystal structure is a volume which, if translated in turn
by some subset of the Bravais lattice vectors, will fill all space. The smallest
possible unit cell is called the primitive cell . It is a volume which, if translated
in turn by all of the Bravais lattice vectors, will fill all space. The choice
of primitive cell (and of unit cell) is not unique, and may be made in any
convenient way. One common choice of primitive cell is the Wigner-Seitz cell ;
this is that volume which is closer to a given Bravais lattice point than to
any other Bravais lattice point. For example, in a simple cubic lattice, the
Wigner-Seitz cell is also a cube; in a body-centered-cubic (bcc) lattice, it is a
fourteen-sided polyhedron consisting of six square and eight hexagonal faces
(see picture in Ashcroft and Mermin).
The conventional unit cell is the unit cell which is generally chosen. For
example, in the bcc lattice, the conventional unit cell is the cube of edge a,
with lattice points on the corners and one lattice point in the center of the
cube. This actually has the volume of two primitive cells.
3. Crystal Structure.
A crystal structure is defined by a Bravais lattice and a basis of atoms.
The basis of s atoms is defined by the s basis vectors and by the identities of
the atoms occupying the basis sites.
Many crystal structures have a basis of just one atom, in which case there
is only one basis vector which may be chosen to be zero. Some well-known
2

crystal structures with a basis of one atom are the face-centered cubic (fcc),
body-centered cubic (bcc) and simple cubic (sc) structures.
One common crystal structure is the diamond structure (this has an fcc
Bravais lattice with a basis of two identical atoms; well-known examples
include C in its diamond form, Si, and Ge). The zinc-blende structure is like
the diamond structure, except that the two atoms forming the basis are not
identical. Examples include GaAs and InSb. The hexagonal close-packed
(hcp) crystal structure has a simple hexagonal Bravais lattice with a basis
of two identical atoms; examples include Mg, Zn, and Cd. The honeycomb