Section 1.5.4.3. Wintgen positions

In order to avoid confusion, in the following the analogues to the Wyckoff positions of will be called Wintgen positions of ; the coordinates of the Wyckoff position are replaced by the k-vector coefficients of the Wintgen position, the Wyckoff letter will be called the Wintgen letter, and the symbols for the site symmetries of are to be read as the symbols for the little co-groups of the k vectors in . The multiplicity of a Wyckoff position is retained in the Wintgen symbol in order to facilitate the use of IT A for the description of symmetry in k space. However, it is equal to the multiplicity of the star of k only in the case of primitive lattices .

In analogy to a Wyckoff position, a Wintgen position is a set of orbits of k vectors. Each orbit as well as each star of k can be represented by any one of its k vectors. The zero, one, two or three parameters in the k-vector coefficients define points, lines, planes or the full parameter space. The different stars of a Wintgen position are obtained by changing the parameters.

Remark. Because reciprocal space is a vector space, there is no origin choice and the Wintgen letters are unique (in contrast to the Wyckoff letters, which may depend on the origin choice). Therefore, the introduction of Wintgen sets in analogy to the Wyckoff sets of IT A, Section 8.3.2 is not necessary.

It may be advantageous to describe the different stars belonging to a Wintgen position in a uniform way. For this purpose one can define:

For non-holosymmetric space groups the representation domain Φ is a multiple of the basic domain Ω. CDML introduced new letters for stars of k vectors in those parts of Φ which do not belong to Ω. If one can make a new k vector uni-arm to some k vector of the basic domain Ω by an appropriate choice of Φ and Ω, one can extend the parameter range of this k vector of Ω to Φ instead of introducing new letters. It turns out that indeed most of these new letters are unnecessary. This restricts the introduction of new types of k vectors to the few cases where it is indispensible. Extension of the parameter range for k means that the corresponding representations can also be obtained by parameter variation. Such representations can be considered to belong to the same type. In this way a large number of superfluous k-vector names, which pretend a greater variety of types of irreps than really exists, can be avoided (Boyle, 1986). For examples see Section 1.5.5.1.