International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.6, p. 861

Section 3.6.3.5. Asymmetric unit

D. B. Litvina*

aDepartment of Physics, The Eberly College of Science, Penn State – Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA
Correspondence e-mail: u3c@psu.edu

3.6.3.5. Asymmetric unit

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An asymmetric unit is a simply connected smallest part of space which, by application of all symmetry operations of the magnetic group, exactly fills the whole space. For subperiodic groups, because the translational symmetry is of a lower dimension than that of the space, the asymmetric unit is infinite in size. The asymmetric unit for subperiodic groups is defined by setting the limits on the coordinates of points contained in the asymmetric unit. For example, the asymmetric unit for the magnetic layer group 32.3.199 pm′21n′ is[{\bf Asymmetric\ unit}\quad 0\,\lt\, x \,\lt\, {\textstyle{1\over 2}}\semi\quad 0 \,\lt\, y \,\lt\, 1\semi\quad 0 \,\lt\, z]Since the translational symmetry of a magnetic space group is of the same dimension as that of the space, the asymmetric unit is a finite part of space. The asymmetric unit is defined, as above, by setting the limits on the coordinates of points contained in the asymmetric unit. For example, for the magnetic space group 140.3.1198 I4/mcm one has[{\bf Asymmetric\ unit}\quad 0\,\lt\, x \,\lt\, {\textstyle{1\over 2}}\semi\quad 0 \,\lt\, y \,\lt\, {\textstyle{1\over 2}}\semi\quad 0 \,\lt\, z \,\lt\, {\textstyle{1\over 4}}\semi\quad y \,\lt\, {\textstyle{1\over 2}} - x]Drawings showing the boundary planes occurring in the tetragonal, trigonal and hexagonal systems, together with their algebraic equations, are given in Fig. 2.1.3.11[link] . Drawings of asymmetric units for cubic groups have been published by Koch & Fischer (1974[link]). The asymmetric units have complicated shapes in the trigonal, hexagonal and cubic crystal systems, and consequently are also specified by giving the vertices of the asymmetric unit. For example, for the magnetic space group 176.1.1374 P63/m one finds

Asymmetric unit 0 < x < 2/3; 0 < y < 2/3; 0 < z < 1/4;  
    x < (1 + y)/2; y < min(1 − x, (1 + x)/2)  
Vertices 0, 0, 0 1/2, 0, 0 2/3, 1/3, 0 1/3, 2/3, 0 0, 1/2, 0
  0, 0, 1/4 1/2, 0, 1/4 2/3, 1/3, 1/4 1/3, 2/3, 1/4 0, 1/2, 1/4

Because the asymmetric unit is invariant under time inversion, all magnetic space groups [\cal F], [\ispecialfonts{\cal F}\!{\sfi 1}'] and [{\cal F}({\cal D})] of the magnetic superfamily of type [{\cal F}] have identical asymmetric units, the asymmetric unit of the group [{\cal F}] (as in the present volume).

References

Koch, E. & Fischer, W. (1974). Zur Bestimmung asymmetrischer Einheiten kubischer Raumgruppen mit Hilfe von Wirkungsbereichen. Acta Cryst. A30, 490–496.








































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