Space group elements

Corresponding relations for arbitrary space group elements show that if the orbitals r) which make up the density transform asthe irreducible representations of the space group, the density is invariant under all the operations of that group. 

If (III.24) holds we get the corresponding result for arbitrary space group elements [Rim]... 

The law of binary composition in (k) is the Herring multiplication rule, which is the Seitz rule for the multiplication of space-group elements supplemented by the additional condition that ( t) is to be replaced by ( 0) whenever exp(—ik t) = + 1. 

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... 

The set of fractional translations v in the space-group elements g = t R depends on the choice of origin (with respect to which the space-group elements are written) and on the labehng of axes (choice of setting) [19]. 

An important special case is obtained when we specialize the space group element S v(S) + t(m) to the operation E t(m), where E is the 3x3 unit matrix this is the operation of displacing the crystal through the lattice translation t(m). In this case g = + m, 9k = k and (F.12b) yields... 

If we consider a space group element of the form g = S 0 where S is a rotation about the x-axis or a reflection through the xz-plane (Fig.F.2), then gti = ty For S = C2x = rotation by an angle 180 about the x-axis we have... 

The nonunitary magnetic space group of MnF2, P42/mnm contains, in addition to the elements of Pnnrn, the following anti-elements... 

The BaBPOs compound was first prepared and structurally characterized by Bauer [12]. Figure 21.2 shows the crystal structure of BaBPOs. Its structure is similar to all stillwellite-like compounds with the space group P322. Its main structural elements are spiral tetrahedral chains [001] built of three-membered rings. The contact between the BO4 tetrahedra that form the central part of these chains are reinforced by PO4 tetrahedra and thus [BPO5] heterotetrahedral chain complexes are produced. 

A crystal characterised by a space group G has an electron density p(r) which is invariant under all elements Rim ofG ... 

If the elements of the number density matrix in position space are invariant under all operations of the space group, i.e. if... 

We notice that neither the momentum distribution nor the reciprocal form factor seems to carry any information about the translational part of the space group. The non diagonal elements of the number density matrix in momentum space, on the other hand, transform under the elements of the space group in a way which brings in the translational parts explicitly. 

In the case of a perfect crystal the Hamiltonian commutes with the elements of a certain space group and the wave functions therefore transform under the space group operations accorc g to the irreducible representations of the space group. Primarily this means that the wave functions are Bloch functions labeled by a wave vector k in the first Brillouin zone. Under pure translations they transform as follows... 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

Space lattices and crystal systems provide only a partial description of the crystal structure of a crystalline material. If the structure is to be fully specified, it is also necessary to take into account the symmetry elements and ultimately determine the pertinent space group. There are in all two hundred and thirty space groups. When the space group as well as the interatomic distances are known, the crystal structure is completely determined. 

The 230 space-group types are listed in full in International Tables for Crystallography, Volume A [48], Whenever crystal symmetry is to be considered, this fundamental tabular work should be consulted. It includes figures that show the relative positions of the symmetry elements as well as details concerning all possible sites in the unit cell (cf. next section). 

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