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**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 in the set {+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 involved are:
**

Type | Point Groups | Space Groups |
---|---|---|

Nonmagnetic | 32 | 230 |

Magnetic | 58 | 1321 |

All | 90 | 1651 |

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 = {V_{k}, 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 T_{i}and T_{j}are symmetry transformations of the crystal structure then their composition T_{j}T_{i}is also a symmetry transformation because T_{j}(T_{i}S)=T_{i}S=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.

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).

(Three Versions)

- Any type of symmetry exhibited by the point
group of a crystal is possessed by every physical property of the crystal.
- The tensor describing a physical property must be
invariant under the symmetry operations of the crystal's point group.
- No asymmetry can appear in a physical effect that does not
exist in the crystal or in the external influences upon the crystal.

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 = {V_{k}, k ε K}. A
symmetry transformation for S is a transformation T
such that T(S)=S. Thus the transformations is a set
{T_{i}, such that T_{i}S=S for i ε I}.

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

Suppose the property P is expressed as a matrix M.
If the transformation T_{ij} has the effect
of interchanging the x_{i} and x_{j}
axes then T_{ij}(M) = M means that
m_{ij} = m_{ji}. If the set of symmetry operations
includes all possible interchanges of axes then
m_{ij} = m_{ji} for all i and j and thus
M is a symmetric matrix.

On the other hand, if T_{ij}(M) has the effect
of reversing the sign of an element when the axes are
interchanged then m_{ij} = -m_{ji} and
M is an antisymmetric matrix and thus m_{ii} = 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:

- A.R. Billings,
*Tensor Properties of Materials*, Wiley-Interscience, 1969. - 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. - Barry M. McCoy and Tai Tsun Wu,
*The Two Dimensional Ising Model*Harvard University Press, 1973. - J.F. Nye,
*Physical Properties of Crystals: Their Representation by Tensors and Matrices* - 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. - Joe Rosen,
*A Symmetry Primer for Scientists* - 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.