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The Symmetry Group Structures of Crystals

A crystal's symmetries can be described in terms of the geometric operations which produce identical configurations. The set of symmetry operations and results of their combinations define a mathematical structure called a group. The symmetry operations which involve only rotations, reflections and inversion define the point group of the crystal.

The point group does not include the translations of the crystal lattice which produce identical configurations. When translations as well as rotations, reflections and inversions are taken into account the group defined is called the space group of the crystal.

Some crystal structures possess not only spatial symmetry but also symmetries involving spin direction at the lattice points. If the basis of a crystal involves spins there may be symmetry operations which involve not only translations, rotations, reflections and inversions but also systematic changes in the spins. The spin coordinate must be either +1, 0, -1. The groups which involve the changes in the spin coordinate are called magnetic groups.

There are 32 different point groups for three dimensional crystal lattices and 230 space groups. The ordinary point and space groups may be considered as special cases of the magnetic point and space groups. The numbers invoved are:

TypePoint GroupsSpace Groups
Nonmagnetic32230
Magnetic581321
All901651

The Mathematical Formulation of Symmetry Groups

Let V be a vector, three dimensional in the nonmagnetic case or four dimensional in the magnetic case where the fourth component is limited to the values {+1, -1}. Let T be a linear transformation that is defined on the vector space of the V's.

The geometry of the crystals is given as a set S of V's; i.e., S = {Vk, k ε K}. A symmetry transformation for S is a transformation T such that T(S)=S.

Note that the composition of any two symmetry transformations for S is also a symmetry transformation for S; i.e.,

If Ti and Tj are symmetry transformations of the crystal structure then their composition TjTi is also a symmetry transformation because Tj(TiS)=TiS=S.

In order for the symmetry transformation of a crystal to form a mathematical group it is necessary, in addition to the above property for the composition of transformations, that there exist an identity transformation and an inverse transformation for each transformation element. The identity element is just the transformation that changes nothing. The inverse of each transformation can be determined. Thus the symmetry transformations of a crystal form a mathematical group.

How Do the Symmetries of a Crystal
Affect its Physical Properties?

The answer to this question is given in terms of a principle formulated by Franz Neumann (1795-1898) at the University of Königsberg in the late nineteenth century (1885).

Neumann's Principle:
(Three Versions)

Consider now the definition of the symmetry transformations of a crystal structure. The geometry of the crystals is given as a set S of vectors; i.e., S = {Vk, k ε K}. A symmetry transformation for S is a transformation T such that T(S)=S. Thus the transformations is a set {Ti, such that TiS=S for i ε I}.

If P is the representation of some physical property of a crystal then by Neumann's Principle:

Ti(P) = P for all i in I

Suppose the property P is expressed as a matrix M. If the transformation Tij has the effect of interchanging the xi and xj axes then Tij(M) = M means that mij = mji. If the set of symmetry operations includes all possible interchanges of axes then mij = mji for all i and j and thus M is a symmetric matrix.

On the other hand, if Tij(M) has the effect of reversing the sign of an element when the axes are interchanged then mij = -mji and M is an antisymmetric matrix and thus mii = 0.

The magnetic symmetry of such crystals determines the structure of the tensors representing their magnetic properties. The simplest versions of such structures are the Ising Models (one, two and three dimensional).

(To be continued.)

References:

  1. A.R. Billings, Tensor Properties of Materials, Wiley-Interscience, 1969.
  2. Robert R. Birss, Symmetry and Magnetism (North-Holland Publishing Co.: Amsterdam, 1964). Melvin Lax, Symmetry Principles in Solid State and Molecular Physics, John wiley and Sons, 1974.
  3. Barry M. McCoy and Tai Tsun Wu, The Two Dimensional Ising Model Harvard University Press, 1973.
  4. J.F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices
  5. W. Opechowski and Rosalia Guccione, "Magnetic Symmetry," in George T. Rado and Harry Suhl (eds.), Magnetism, vol. IIA (Academic Press: New York, 1965), pp. 105-165.
  6. Joe Rosen, A Symmetry Primer for Scientists
  7. K.W.H. Stevens, "Spin Hamiltonians," in George T. Rado and Harry Suhl (eds.), Magnetism, vol. I (Academic Press: New York, 1965), pp. 1-23.