International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.5, p. 190   | 1 | 2 |

## Section 1.5.5.4.2. Splitting of k-vector types

M. I. Aroyoa* and H. Wondratschekb

aDepartamento de Fisíca de la Materia Condensada, Facultad de Cienca y Technología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain , and bInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail:  wmpararm@lg.ehu.es

#### 1.5.5.4.2. Splitting of k-vector types

| top | pdf |

The Brillouin zone as well as the unit cell are always convex bodies; the same holds for the representation domain of CDML and for the choice of the asymmetric unit. It is thus sometimes unavoidable that the k-vector types are split and that the different parts belong to different arms and to different stars of k vectors. Sometimes this splitting of k-vector types may be avoided by an appropriate choice of the asymmetric unit; sometimes the introduction of flagpoles and wings is necessary to make the k-vector types uni-arm.

#### Examples

 (1) In the reciprocal-space group , No. 225 of the arithmetic crystal class there are the lines of k vectors Λ (α, α, α) and F of CDML, p. 41. By Figure 1.5.5.1 one sees that the line Λ connects the points Γ and P, the line F connects the points P and H. One takes from the corresponding Table 1.5.5.1 the coefficients of and . From these points or from the transformation listed at the top of Table 1.5.5.1 as `Parameter relations' the coefficients of the line F are obtained as . The inspection of the symmetry diagram of , No. 225, in IT A shows that a twofold rotation (represented by the  screw-rotation axis) leaves the point P invariant, whereas the point H is mapped onto the point R . More formally: the rotation is described by  , where . The result is the line . It is uni-arm to the line Λ = x, x, x and the union forms the Wintgen position . An analoguous result is obtained for the same lines in the arithmetic crystal class . (2) In the following example the splitting of a Wintgen position happens if a representation domain of the Brillouin zone is chosen. The splitting can be avoided by the choice of the asymmetric unit. We consider the plane in the arithmetic crystal class , see Fig. 1.5.5.4 and Table 1.5.5.4 . In CDML this plane is split into the parts and . By the choice of the asymmetric unit the independent region of the Wintgen position is uni-arm:  : . (3) The splitting of a Wintgen position can be avoided if flagpoles and wings are admitted, i.e. if the minimal domain is described by a non-convex body. If one chooses in the first example of the arithmetic crystal classes and the union for the line , then forms a flagpole, whereas Λ forms an edge of the asymmetric unit, see Figs. 1.5.5.1 and 1.5.5.2 . The same holds for the Wintgen position of . In the representation domain which is simultaneously the asymmetric unit, this Wintgen position is split into three parts B, C and J, which form three of the four walls of the (tetrahedral) minimal domain. By proper symmetry operations these three parts can be made uni-arm to the part C, such that their union describes the independent part of that Wintgen position, see Fig. 1.5.5.1 . The part C forms a wall of the asymmetric unit; the part forms a wing, see Fig. 1.5.5.1 .