Point factor group elements

Second, a multiplication table for the factor group is written down. The space group formed by the above symmetry elements is infinite, because of the translations. If we define the translations, which carry a point in one unit cell into the corresponding point in another unit cell, as equivalent to the identity operation, then the remaining symmetry elements form a group known as the factor, or unit cell, group. The factor... 

The set of PFs [gj gj] is called the factor system. Associativity (a) and the symmetry of [gi gf ] (d) are true for all factor systems. The standardization (b) and normalization (c) properties are conventions chosen by Altmann and Herzig (1994) in their standard work Point Group Theory Tables. Associativity (a) follows from the associativity property of the multiplication of group elements. For a spinor representation T of G, on introducing [/ j] as an abbreviation for [g, g ], ... 

If a crystal contains a center of symmetry among its space group elements, then for every atom at point xj = (xj, yj, zj), there is a corresponding atom at —xj = (—xj, —yj, —Zj). The structure factor equation for Fjt will therefore contain a term... 

Equation (2.59) is thus a representation of an arbitrary group element expressed as a left coset of this subgroup. Expressed in other words, the unitary transformation that is described by exp(i2) may alternatively be described by the unitary transformation exp(i/l )exp(iX ). It should be pointed out that there exist no simple relations between the parameters and the A ., and parameters. With the above factorization of the redundant part (A"), the unitary transformation of the reference state may be written as... 

Here the summation is performed over the repeating indexes. A is the transformation matrix with components Ay (ij= 1,2,3) and determinant det(A) = 1 the factor Ir denotes either the presence (tr=l) or the absence (tr = 0) of the time-reversal operation coupled to the space transformation Ay. For the case when the matrices A represent all the generating elements of the material point symmetry group (considered hereinafter) the identity = dY should be valid for nonzero components of the piezotensors. 

It was pointed out in Section IIC.5 that the understanding of the behavior of coordinates under symmetry operations is of great importance. Such an understanding is essential for the construction of symmetry coordinates and the choice of suitable coordinates is, indeed, often determined by such considerations. Schnepp and Ron (1969) therefore investigated the transformation of the Ug, under the operations of the factor group of a-Na, it- These transformations were found to be nonlinear and were summarized in Table 2 of their paper up to second order in the displacements. For the construction of symmetry coordinates it is sufficient to use the transformations to first order only since these coordinates must hold for arbitrarily small displacements. The transformations to second order are useful when it is desired to find relations between elements of the dynamical equation based on symmetry. The... 

If we now apply rotadonal nnmetxy (Factor II given in 2.2.1) to the 14 Bravais lattices, we obtain the 32 Point-Groups which have the factor of symmetry imposed upon the 14 Bravais lattices. The symmetry elements that have been used are ... 

If g is an element of the point group of the material meaning that p(r) and p(gr) are indistinguishable for all elements g in that group, corresponding Fourier components can differ only by a phase factor ... 

Thus the point group part of the operation works on the momentum coordinates and the translation part gives rise to a phase factor. We notice that this phase factor reduces to 1 in the diagonal elements, or in general when the difference between the the two arguments of N(p,p ) is a reciprocal lattice vector. 

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