Parameter ranges

Aroyo, M. I.; Wondratschek, H.

doi:10.1107/97809553602060000553

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

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International Tables for Crystallography (2006). Vol. B. ch. 1.5, pp. 171-172 | 1 | 2 |

Section 1.5.5.3. Parameter ranges

M. I. Aroyo^a ^* and H. Wondratschek^b

^a Faculty of Physics, University of Sofia, bulv. J. Boucher 5, 1164 Sofia, Bulgaria , and ^bInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: aroyo@phys.uni-sofia.bg

1.5.5.3. Parameter ranges

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For the uni-arm description of a Wintgen position it is easy to check whether the parameter ranges for the general or special constituents of the representation domain or asymmetric unit have been stated correctly. For this purpose one may define the field of k as the parameter space (point, line, plane or space) of a Wintgen position. For the check, one determines that part of the field of k which is inside the unit cell. The order of the little co-group $[\bar{{\cal G}} ^{{\bf k}}]$ ( $[\bar{{\cal G}} ^{{\bf k}}]$ represents those operations which leave the field of k fixed pointwise) is divided by the order of the stabilizer [which is the set of all symmetry operations (modulo integer translations) that leave the field invariant as a whole]. The result gives the independent fraction of the above-determined volume of the unit cell or the area of the plane or length of the line.

If the description is not uni-arm, the uni-arm parameter range will be split into the parameter ranges of the different arms. The parameter ranges of the different arms are not necessarily equal; see the second of the following examples.

Examples

(1) Line $[\Lambda \cup F_{1}]$ : In $[(Fm\bar{3}m)^*]$ the line x, x, x has stabilizer $[\bar{3}m]$ and little co-group $[\bar{{\cal G}} ^{{\bf k}} = 3m]$ . Therefore, the divisor is 2 and x runs from 0 to $[{1 \over 2}]$ in $[0 \lt x \lt 1]$ .
(2) Plane $[B \cup C \cup J]$ : In $[(Fm\bar{3}m)^*]$ , the stabilizer of x, x, z is generated by m.mm and the centring translation $[t({1 \over 2}, {1 \over 2}, 0)]$ modulo integer translations $[(\hbox{mod }{\bf T}_{\rm int})]$ . They generate a group of order 16; $[\bar{{\cal G}} ^{{\bf k}}]$ is ..m of order 2. The fraction of the plane is $[{2 \over 16} = {1 \over 8}]$ of the area $[2^{1/2} a^{*2}]$ , as expressed by the parameter ranges $[0 \lt x \lt {1 \over 4}]$ , $[0 \lt z \lt {1 \over 2}]$ . There are six arms of the star of x, x, z: x, x, z; $[\bar{x}]$ , x, z; x, y, x; x, y, $[\bar{x}]$ ; x, y, y; x, $[\bar{y}]$ , y. Three of them are represented in the boundary of the representation domain: , $[C = \Gamma NP]$ and $[J = \Gamma HP]$ ; see Fig. 1.5.5.1. The areas of their parameter ranges are $[{1 \over 32}, {1 \over 32}]$ and $[{1 \over 16}]$ , respectively; the sum is $[{1 \over 8}]$ .

The same result holds for $[(Fm\bar{3})^*]$ : the stabilizer is generated by and $[t({1 \over 2}, {1 \over 2}, 0)]$ mod $[{\bf T}_{\rm int}]$ and is of order 8, $[|\bar{{\cal G}} ^{{\bf k}}| = | \{{\bf 1}\}| = 1]$ , the quotient is again $[{1 \over 8}]$ , the parameter range is the same as for $[(Fm\bar{3}m)^*]$ . The planes $[H\Gamma_{1}P]$ and $[N\Gamma_{1}P]$ are equivalent to $[J = \Gamma HP]$ and $[C = \Gamma NP]$ , and do not contribute to the parameter ranges.
(3) Plane x, y, 0: In $[(Fm\bar{3}m)^*]$ the stabilizer of plane A is generated by and $[t({1 \over 2}, {1 \over 2}, 0)]$ , order 32, $[\bar{{\cal G}} ^{{\bf k}}]$ (site-symmetry group) m.., order 2. Consequently, ΓHN is $[{1 \over 16}]$ of the unit square $[a^{*2}: 0 \lt y \lt x \lt {1 \over 2} - y]$ . In $[(Fm\bar{3})^*]$ , the stabilizer of $[A \cup AA]$ is mmm. plus $[t({1 \over 2}, {1 \over 2}, 0)]$ , order 16, with the same group $[\bar{{\cal G}} ^{{\bf k}}]$ . Therefore, $[\Gamma H \Gamma_{1}]$ is $[{1 \over 8}]$ of the unit square $[a^{*2}]$ in $[(Fm\bar{3})^{*}: 0 \lt y \lt x \lt {1 \over 2}]$ .
(4) Line x, x, 0: In $[(Fm\bar{3}m)^*]$ the stabilizer is generated by m.mm and $[t ({1 \over 2}, {1 \over 2}, 0)\hbox{ mod } {\bf T}_{\rm int}]$ , order 16, $[\bar{{\cal G}} ^{{\bf k}}]$ is m.2m of order 4. The divisor is 4 and thus $[0 \lt x \lt {1 \over 4}]$ . In $[(Fm\bar{3})^*]$ the stabilizer is generated by and $[t ({1 \over 2}, {1 \over 2}, 0)\hbox{ mod } {\bf T}_{\rm int}]$ , order 8, and $[\bar{{\cal G}} ^{{\bf k}} = m..]$ , order 2; the divisor is 4 again and $[0 \lt x \lt {1 \over 4}]$ is restricted to the same range.⁶

Data for the independent parameter ranges are essential to make sure that exactly one k vector per orbit is represented in the representation domain Φ or in the asymmetric unit. Such data are much more difficult to calculate for the representation domains and cannot be found in the cited tables of irreps.

In the way just described the inner parameter range can be fixed. In addition, the boundaries of the parameter range must be determined:
(5) Line x, x, x: In ( $[Fm\bar{3}m]$ )* and ( $[Fm\bar{3}]$ )* the points 0, 0, 0; $[{1 \over 2}, {1 \over 2}, {1 \over 2}]$ (and $[{1 \over 4}, {1 \over 4}, {1 \over 4}]$ ) are special points; the parameter ranges are open: $[0 \lt x \lt {1 \over 4}, {1 \over 4} \lt x \lt {1 \over 2}]$ .
(6) Plane x, x, z: In $[(Fm\bar{3}m)^*]$ all corners Γ, N, $[N_{1}]$ , $[H_{1}]$ and all edges are either special points or special lines. Therefore, the parameter ranges are open: x, x, z: $[0 \lt x \lt {1 \over 4}]$ , $[0 \lt z \lt {1 \over 2}]$ , where the lines x, x, x: $[0 \leq x \leq {1 \over 4}]$ and $[x, x, {1 \over 2} - x]$ : $[0 \leq x \leq {1 \over 4}]$ are special lines and thus excepted.
(7) Plane x, y, 0: In both $[(Fm\bar{3}m)^*]$ and $[(Fm\bar{3})^*]$ , $[0 \lt x]$ and $[0 \lt y]$ holds. The k vectors of line x, x, 0 have little co-groups of higher order and belong to another Wintgen position in the representation domain (or asymmetric unit) of $[(Fm\bar{3}m)^*]$ . Therefore, x, y, 0 is open at its boundary x, x, 0 in the range $[0 \lt x \lt {1 \over 4}]$ . In the asymmetric unit of $[(Fm\bar{3})^*]$ the line x, x, 0: $[0 \lt x \lt {1 \over 4}]$ belongs to the plane, in this range the boundary of plane A is closed. The other range x, x, 0: $[{1 \over 4} \lt x \lt {1 \over 2}]$ is equivalent to the range $[0 \lt x \lt {1 \over 4}]$ and thus does not belong to the asymmetric unit; here the boundary of AA is open.