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
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