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 [({\cal G})^*=(Fm\overline{3}m)^*], No. 225 of the arithmetic crystal class [m\overline{3}mI] there are the lines of k vectors Λ (α, α, α) and F [(\textstyle{1 \over 2}-\alpha, -\textstyle{1 \over 2}+3\alpha, \textstyle{1 \over 2}-\alpha)] of CDML, p. 41. By Figure 1.5.5.1[link] 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[link] the coefficients of [P=\textstyle{1 \over 4},\textstyle{1 \over 4},\textstyle{1 \over 4}] and [H=0,\textstyle{1 \over 2},0]. From these points or from the transformation listed at the top of Table 1.5.5.1[link] as `Parameter relations' the coefficients of the line F are obtained as [F=x,\textstyle{1 \over 2}-x,x\semi 0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}].

    The inspection of the symmetry diagram of [Fm\overline{3}m], No. 225, in IT A shows that a twofold rotation [\sf 2] (represented by the [{\sf{4}_2}] [\textstyle{1 \over 4},y,\textstyle{1 \over 4}] screw-rotation axis) leaves the point P invariant, whereas the point H is mapped onto the point R [\textstyle{1 \over 2}, \textstyle{1 \over 2}, \textstyle{1 \over 2}]. More formally: the rotation is described by [x,\textstyle{1 \over 2}-x,x \ \rightarrow] [\textstyle{1 \over 2}-x,\textstyle{1 \over 2}-x,\textstyle{1 \over 2}-x], where [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}]. The result is the line [F_1=[R\,P]]. It is uni-arm to the line Λ = x, x, x and the union [\Lambda \cup F_1] forms the Wintgen position [32\ f\ 3m]. An analoguous result is obtained for the same lines in the arithmetic crystal class [m\overline{3}I].

  • (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 [x,y,0] in the arithmetic crystal class [4/mmmI], see Fig. 1.5.5.4[link] and Table 1.5.5.4[link]. In CDML this plane is split into the parts [C=[\Gamma\, S_2\, R\, X]] and [D=[M\, S\, G]\sim [M_2\, S_2\, R]]. By the choice of the asymmetric unit the independent region of the Wintgen position is uni-arm: [[\Gamma M_2 X]=16\ l\ m..] [x,y,0]: [0 \,\lt \, x \,\lt \, y \,\lt \, \textstyle{1 \over 2}].

  • (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 [m\overline{3}mI] and [m\overline{3}I] the union [\Lambda\cup F_1] for the line [x,x,x], then [F_1=[P\,R]] forms a flagpole, whereas Λ forms an edge of the asymmetric unit, see Figs. 1.5.5.1[link] and 1.5.5.2[link].

    The same holds for the Wintgen position [96\ k\ ..m\quad x,x,z] of [m\overline{3}mI]. 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 [C\cup B_1\cup J_1] describes the independent part of that Wintgen position, see Fig. 1.5.5.1[link]. The part C forms a wall of the asymmetric unit; the part [B_1\cup J_1] forms a wing, see Fig. 1.5.5.1[link].








































to end of page
to top of page