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 a method for the determination of the parameter ranges was described. A few examples shall display the procedure.

 (1) Arithmetic crystal class , line : In the reciprocal-space group of the arithmetic crystal class , the line has stabilizer and little co-group . Therefore, the divisor is 12:6 = 2 and is running from 0 to . The same result holds for the line in the reciprocal-space group of the arithmetic crystal class : the stabilizer generated by is of order 6, , the quotient is again , the parameter range is the same as for . (2) Arithmetic crystal class , plane : In , the stabilizer of is generated by and the centring translation mod (integer translations). They generate a group of order 16; is of order 2. The fraction of the plane is of the area in the (centred) unit cell, as expressed by the parameter ranges . There are six arms of the star of : ; ; ; ; ; . Three of them (, and ) 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. The areas of their parameter ranges are and , respectively; the sum is . Arithmetic crystal class , the same result holds in the reciprocal-space group . The stabilizer generated by and by the centring translation mod (integer translations) forms a group of order 8; the order of the little co-group . The quotient is again , the parameter range is the same as for but the plane belongs to the general position GP because the little co-group is trivial. (3) Arithmetic crystal class , reciprocal-space group , plane : the stabilizer of the plane A is generated by and , order 32, (site-symmetry group) , order 2. Consequently, [Γ H N] is of the unit square : . In , the stabilizer of , here , is and , order 16, with the same group of order 2. Therefore, [Γ H2 H] is of the unit square in ; . (4) Arithmetic crystal class , line : In the stabilizer is generated by and mod (integer translations), order 16, is of order 4. The divisor is 4 and thus . In the stabilizer is generated by and mod (integer translations), order 8, and , order 2; the divisor is 4 again and 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 and , i.e. and , line : The points ; (and ) are special points; the parameter ranges are open: , . (2) Arithmetic crystal class , plane : In all corners Γ, N, N2, H2 and all edges are special points or lines. Therefore, the parameter ranges are open: : , where the lines : and : are special lines and thus excepted. (3) Arithmetic crystal classes and , plane : In both reciprocal-space groups, and , and holds. The line is a special line, its k vectors have little co-groups of higher order than that of the planes and the boundaries of both planes are open. The same holds for the boundary for . The k vectors of the lines and , 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 . Therefore, for the arithmetic crystal class , the plane is open at its boundaries and in the range . In the asymmetric unit of the lines : and : belong to the plane, and the boundary of the plane A is here closed. The boundary line : of the plane AA is equivalent to the range of the part A and thus does not belong to the asymmetric unit; here the boundary of the plane is open.