Fundamentals of the Physics of Solids / 16-back-matter

are introduced with arbitrary coe cients, the following formulas are valid:

where the fl(r ) are spherical Bessel functions of the first or second kind. The same function can be expanded around a di erent origin (displaced by r with respect to the first), using the spherical coordinates r , θ , ϕ of the vector r = r + r . The obtained functions fl(r ) are related to the fl(r ) via the Wigner 3j symbols:

where r, θ, ϕ are the spherical coordinates of r.

References

1.G. B. Arfken and H.-J. Weber, Mathematical Methods for Physicists, Sixth Edition, Elsevier/Academic Press, Amsterdam (2005).

2.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Edited by M. Abramowitz and I. A. Stegun, 10th printing, Dover Publications, Inc., New York (1972). It is available online at www.math.sfu.ca/ cbm/aands.

3.I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Sixth Edition, Edited by A. Je rey and D. Zwillinger, Academic Press, Inc., Boston (2000).

632 C Mathematical Formulas

4.A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, Vol. 1. Elementary Functions, Gordon and Breach Science Publishers, New York (1986).

5.A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, Vol. 2. Special Functions, Gordon and Breach Science Publishers, New York (1990).

D

Fundamentals of Group Theory

As mentioned in Chapter 6, crystalline materials occupy a special place among solids not in the least because the symmetries of their structure manifest themselves in macroscopic properties, which highly facilitates the theoretical description of their behavior. Symmetry operations form a group in the mathematical sense, therefore statements of general validity can often be based on group-theoretical considerations. In this appendix we shall review some basic notions and relationships of group theory, then present the irreducible representations of the Oh group, and finally list the group theoretical theorems related to the quantum mechanical eigenvalue problem.

D.1 Basic Notions of Group Theory

There exists a whole wealth of group theoretical textbooks written by physicists for physicists. A few of them are listed among the references. These contain plenty of solid-state physics applications. Therefore below we shall discuss only a few basic notions and theorems.

D.1.1 Definition of Groups

The set of elements R, S, T, . . . are said to form a group G if a binary operation called multiplication is defined among group elements, which satisfies the following requirements:

1.For any two elements R and S the product, denoted as R S, is an element T of the group:

The operation of group multiplication does not need to be commutative: in general the order of the elements is not immaterial. A group is called commutative or Abelian if group multiplication satisfies the commutative law, i.e., R S = S R for any two elements R and S.

634 D Fundamentals of Group Theory

2.Multiplication satisfies the associative law, i.e., for any elements R, S, and T

where, naturally, multiplication within the parentheses has to be performed first.

3. There exists an identity element (also called unit element) E in the set. When any element R of the group is multiplied by E, it remains unchanged:

4.For any element R of the group there exists a unique element S in the group such that

This element S is called the inverse of R, and is denoted by S ≡ R−1.

The number g of elements in the set is called the order of the group. The number of elements can be either finite or infinite; these correspond to finite and infinite groups. In solid-state physics applications we deal mostly with finite groups.

When the group is finite, the elements in the sequence R, R2, R3, . . .

repeat themselves from a certain point onward. Consequently one of these elements must be the identity element. The order of an element R in the group is the least positive integer n such that Rn = E.

A subset G of the group elements is called a subgroup of G if it satisfies in itself the previous requirements that define a group – i.e., it contains the identity element, for every element in G its inverse is also in G , and the product of any two elements in G is also in G .

A subgroup G of the group G is an invariant subgroup or normal subgroup if R S R−1 G for every R G and every S G .

If R1, S1, T1, . . . are the elements of a group G1 of order g1, and R2, S2, T2, . . . are the elements of a group G2 of order g2, then the direct product (or Kronecker product ) of the two groups, denoted by G1 G2 is the group made up of the elements (R1, R2), (R1, S2), . . . , (S1, R2), (S1, S2), . . . , with the group multiplication defined as

(R1, S2)(T1, U2) = (R1T1, S2U2) . (D.1.5)

This group is of order g1 g2.

If G1 and G2 are such subgroups of G that 1.) the elements in G1 commute with the elements in G2; 2.) they have a single element in common, the identity element; and 3.) each element in G can be written as the product of an element in G1 and an element in G2, then the group G is isomorphic to the direct product group G1 G2.

The group G is the semidirect product of its subgroups G1 and G2 (G = G1 G2), if 1.) the elements in G1 do not commute with the elements in G2

636 D Fundamentals of Group Theory

If the transformation between r and r is a one-to-one correspondence, then the inverse of the transformation O exists,

and this mapping can be given by the inverse of the matrix D:

Consider a group G, and associate a linear operator O(R) and the corresponding matrix Dij (R) with each element R. If the same multiplication rule applies to the operators as to the group elements, i.e., for RS = T

The identity element E is associated with the identity operator and the unit matrix, while the inverse of the element R is associated with the inverse of the operator O(R) and the inverse of the matrix D(R). These matrices provide a representation of the group. The representation is faithful if the mapping is not only homomorphic but also isomorphic, i.e., there exists a one-to-one correspondence.

Two representations of a group are equivalent if there exists a unitary matrix U U † = U −1 that transforms the matrices D(1)(R) and D(2)(R) belonging to the two representations into each other for each element of the group:

For every representation of a finite group there exists an equivalent representation in which the matrices are unitary. In what follows we shall only deal with unitary representations.

Being an invariant expression of the matrix elements, the diagonal sum (trace) of the matrices in the representation of the group plays an important