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

International Tables for Crystallography (2010). Vol. B, ch. 1.5, pp. 190-191   | 1 | 2 |

Section 1.5.5.4.4. Ranges of independent parameters

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

1.5.5.4.4. Ranges of independent parameters

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In Section 1.5.4.3[link] a method for the determination of the parameter ranges was described. A few examples shall display the procedure.

  • (1) Arithmetic crystal class [m\overline{3}mI], line [\Lambda \cup F_1]: In the reciprocal-space group [(Fm\overline{3}m)^*] of the arithmetic crystal class [m\overline{3}mI], the line [x,x,x] has stabilizer [\overline{3}m] and little co-group [\overline{{\cal G}}^{\bf k} = 3m]. Therefore, the divisor is 12:6 = 2 and [x] is running from 0 to [\textstyle{1 \over 2}].

    The same result holds for the line [\Lambda \cup F_1] in the reciprocal-space group [(Fm\overline{3})^*] of the arithmetic crystal class [m\overline{3}I]: the stabilizer generated by [\overline{\sf 3}] is of order 6, [|\overline{{\cal G}}\,^{\bf k}|= |\{{\sf 3}\}| = 3], the quotient is again [\textstyle{1 \over 2}], the parameter range is the same as for [(Fm\overline{3}m)^*].

  • (2) Arithmetic crystal class [m\overline{3}mI], plane [B_1\cup C \cup J_1]: In [(Fm\overline{3}m)^*], the stabilizer of [x,x,z] is generated by [m.mm] and the centring translation [t(\textstyle{1 \over 2},\textstyle{1 \over 2},0)] mod (integer translations). They generate a group of order 16; [\overline{{\cal G}}\,^{\bf k}] is [..m] of order 2. The fraction of the plane is [\textstyle{2 \over 16} = \textstyle{1 \over 8}] of the area [\sqrt{2}a^{*2}] in the (centred) unit cell, as expressed by the parameter ranges [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}, 0 \,\lt \, z \,\lt \, \textstyle{1 \over 2}]. There are six arms of the star of [x,x,z]: [x,x,z]; [\overline{x},x,z]; [x,y,x]; [x,y,\overline{x}]; [x,y,y]; [x,\overline{y},y]. Three of them ([x,x,z], [\overline{x},x,z] and [x,y,x]) are represented in the boundaries of the representation domain: C = [Γ N P], B = [H N P] and J = [Γ H P], see Fig. 1.5.5.1[link]. The areas of their parameter ranges are [\textstyle{1\over 32}, \textstyle{1 \over 32}] and [\textstyle{1\over 16}], respectively; the sum is [\textstyle{1 \over 8}].

    Arithmetic crystal class [m\overline{3}I], the same result holds in the reciprocal-space group [(Fm\overline{3})^*]. The stabilizer generated by [2/m..] and by the centring translation [t(\textstyle{1 \over 2},\textstyle{1 \over 2},0)] mod (integer translations) forms a group of order 8; the order of the little co-group [|\overline{{\cal G}}\,^{\bf k}|= |\{{\sf 1}\}| = 1]. The quotient is again [\textstyle{1\over 8}], the parameter range is the same as for [(Fm\overline{3}m)^*] but the plane belongs to the general position GP because the little co-group is trivial.

  • (3) Arithmetic crystal class [m\overline{3}mI], reciprocal-space group [(Fm\overline{3}m)^*], plane [x,y,0]: the stabilizer of the plane A is generated by [4/mmm] and [t(\textstyle{1 \over 2},\textstyle{1 \over 2},0)], order 32, [\overline{{\cal G}}\,^{\bf k}] (site-symmetry group) [m..], order 2. Consequently, [Γ H N] is [\textstyle{1\over 16}] of the unit square [a^{*2}]: [0 \,\lt \, x \,\lt \, y \,\lt \, \textstyle{1 \over 2}-x]. In [(Fm\overline{3})^*], the stabilizer of [x,y,0], here [A \cup AA], is [mmm.] and [t(\textstyle{1 \over 2},\textstyle{1 \over 2},0)], order 16, with the same group [\overline{{\cal G}}\,^{\bf k}=m..] of order 2. Therefore, [Γ H2 H] is [\textstyle{1 \over 8}] of the unit square [a^{*2}] in [(Fm\overline{3})^*]; [0 \,\lt \, y \,\lt \, \textstyle{1 \over 2} - x \,\lt \, \textstyle{1 \over 2}].

  • (4) Arithmetic crystal class [m\overline{3}mI], line [x,x,0]: In [(Fm\overline{3}m)^*] the stabilizer is generated by [m.mm] and [t(\textstyle{1 \over 2},\textstyle{1 \over 2},0)] mod (integer translations), order 16, [\overline{{\cal G}}\,^{\bf k}] is [m.2m] of order 4. The divisor is 4 and thus [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}]. In [(Fm \overline{3})^*] the stabilizer is generated by [2/m..] and [t(\textstyle{1 \over 2},\textstyle{1 \over 2},0)] mod (integer translations), order 8, and [\overline{{\cal G}}\,^{\bf k} = m.. ], order 2; the divisor is 4 again and [0 \,\lt \, x \,\lt\, \textstyle{1 \over 4}] is restricted to the same range.

In the way just described the inner part of the parameter range can be fixed. The boundaries of the parameter range must be determined in addition:

  • (1) Arithmetic crystal classes [m\overline{3}mI] and [m\overline{3}I], i.e. [(Fm \overline{3} m)^*] and [(Fm \overline{3})^*], line [x,x,x]: The points [0,0,0]; [\textstyle{1 \over 2},\textstyle{1 \over 2},\textstyle{1 \over 2}] (and [\textstyle{1 \over 4},\textstyle{1 \over 4},\textstyle{1 \over 4}]) are special points; the parameter ranges are open: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}], [\textstyle{1 \over 4} \,\lt \, x \,\lt \, \textstyle{1 \over 2}].

  • (2) Arithmetic crystal class [m\overline{3}mI], plane [x,x,z]: In [(Fm\overline{3}m)^*] all corners Γ, N, N2, H2 and all edges are special points or lines. Therefore, the parameter ranges are open: [x,x,z]: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}, 0 \,\lt \, z \,\lt \, \textstyle{1 \over 2}], where the lines [x,x,x]: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}] and [x, x, \textstyle{1 \over 2} -x]: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}] are special lines and thus excepted.

  • (3) Arithmetic crystal classes [m\overline{3}mI] and [m\overline{3}I], plane [x,y,0]: In both reciprocal-space groups, [(Fm\overline{3}m)^*] and [(Fm\overline{3})^*], [0 \,\lt \, x] and [0 \,\lt \, y] holds. The line [0,y,0 = \Delta] is a special line, its k vectors have little co-groups of higher order than that of the planes [x,y,0] and the boundaries of both planes are open. The same holds for the boundary [x,0,0 \sim 0,y,0] for [(Fm\overline{3})^*]. The k vectors of the lines [x,x,0] and [x,\textstyle{1 \over 2}-x,0], Σ and G, also have little co-groups of higher order and belong to other Wintgen positions in the representation domain (or asymmetric unit) of [(Fm\overline{3}m)^*]. Therefore, for the arithmetic crystal class [m\overline{3}mI], the plane [A = x,y,0] is open at its boundaries [x,x,0] and [x,\textstyle{1 \over 2}-x,0] in the range [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}]. In the asymmetric unit of [(Fm\overline{3})^*] the lines [x,x,0]: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}] and [x,\textstyle{1 \over 2}-x,0]: [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}] belong to the plane, and the boundary of the plane A is here closed. The boundary line [x,\textstyle{1 \over 2}-x,0]: [\textstyle{1 \over 4} \,\lt \, x \,\lt \, \textstyle{1 \over 2}] of the plane AA is equivalent to the range [0 \,\lt \, x \,\lt \, \textstyle{1 \over 4}] of the part A and thus does not belong to the asymmetric unit; here the boundary of the plane [A\cup AA] is open.








































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