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. 863

Section 3.6.4. Comparison of OG and BNS magnetic group type symbols

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

3.6.4. Comparison of OG and BNS magnetic group type symbols

| top | pdf |

There are other notations for magnetic group type symbols than the notations of Opechowski & Guccione (1965[link]) and Belov, Neronova & Smirnova (1957[link]): for example for the three-dimensional magnetic group 55.10.450 P2cbam the Shubnikov notation is [\hbox to 11pt{\hrulefill}\llap{\rm III}_{58}^{403}] (Koptsik, 1966[link]; Shubnikov & Koptsik, 1974[link]) or [{\rm Sh}_{58}^{403}] (Ozerov, 1969a[link],b[link]) (see also Zamorzaev, 1976[link]). There are also the variations of the Opechowski & Guccione notation put forward by Grimmer (2009[link], 2010[link]). We shall limit ourselves here to a detailed comparison of the Opechowski & Guccione and Belov, Neronova & Smirnova notations.

For all group types in the reduced magnetic superfamily of [{\cal F}], the Opechowski & Guccione (1965[link]) magnetic group type symbols (OG symbols) are based on the symbol of the group [{\cal F}]. Belov, Neronova & Smirnova (1957[link]) also base their symbols (BNS symbols) on the symbol of the group [{\cal F}], but only for magnetic groups of the type [{\cal F}], [\ispecialfonts{\cal F}\!{\sfi 1}'] and [{\cal M}_T]. For magnetic groups [{\cal M}_R = {\cal F}({\cal D}) = {\cal D}\, \cup({\cal F} - {\cal D})1'], where [{\cal D}] is an equi-class subgroup of [{\cal F}], the BNS symbol is based on the symbol of the group [{\cal D}], the non-primed subgroup of index 2. A magnetic group [{\cal M}_R] can be written as [\ispecialfonts{\cal M}_R = {\cal F}({\cal D}) = {\cal D} \,\cup {\sfi t}_{\alpha}{}'{\cal D}], where [\ispecialfonts{\sfi t}_{\alpha}] is a translation of [{\cal F}] not in [{\cal D}]. The BNS symbol for a magnetic group of the type [{\cal M}_R] is the symbol for the group type [{\cal D}] with a subindex inserted on the symbol for the translational subgroup of [{\cal D}] to denote the translation [\ispecialfonts{\sfi t}_{\alpha}{}'].

Example

The representative three-dimensional space group [{\cal F} = Pmm2] has a translational subgroup generated by the three translations {1∣1, 0, 0}, {1∣0, 1, 0} and {1∣0, 0, 1} and the standard set of coset representatives[\{1|0\}\quad \{m_{100}|0\}\quad \{m_{010}|0\}\quad \{2_{001}|0\}.]The three-dimensional magnetic space group 25.10.165 [{\cal F}({\cal D})] = [Pmm2(Pcc2)] has a subgroup [{\cal D}] with a translational sub­group generated by the three translations {1∣1, 0, 0}, {1∣0, 1, 0} and {1∣0, 0, 2}, [\ispecialfonts{\sfi t}_\alpha{}'] = {1∣0, 0, 1}′, and a set of coset representatives[\{1|0\}\quad \{m_{100}|0,0,1\}\quad \{m_{010}|0,0,1\}\quad \{2_{001}|0\}.]The OG magnetic group type symbol is, see Section 3.6.2.2.4[link], P2cmm′2, i.e. based on the symbol for the group type [{\cal F}] = [Pmm2]. The BNS symbol is Pccc2, i.e. based on the symbol for the subgroup [{\cal D} = Pcc2] of [{\cal F}], with a subindex `c' attached to `P' to denote the translation [\ispecialfonts{\sfi t}_\alpha{}'] = {1∣0, 0, 1}′ in [{\cal M}_R] = [\ispecialfonts{\cal F}({\cal D}) = {\cal D} \,\cup {\sfi t}_{\alpha}{}'{\cal D}].

A side-by-side comparison of OG magnetic group type symbols and BNS symbols is given in Litvin (2013[link]). As the OG and BNS symbols are the same for magnetic groups [\cal F], [\ispecialfonts{\cal F}\!{\sfi 1}'] and [{\cal M}_T], BNS symbols are explicitly listed only for groups of type [{\cal M}_R]. Examples of this comparison are given in Table 3.6.4.1[link].

Table 3.6.4.1| top | pdf |
Comparisons of three-dimensional OG and BNS magnetic group type symbols

Serial No.OGBNS[{\cal F}({\cal D})]
44.1.324 Imm2    
44.2.325 Imm21′    
44.3.326 Imm2′    
44.4.327 Imm′2    
44.5.328 IPmm2 PImm2 Imm2(Pmm2)
44.6.329 IPmm′2′ PImm21 Imm2(Pmm21)
44.7.330 IPmm′2 PInn2 Imm2(Pnn2)

References

Belov, N. V., Neronova, N. N. & Smirnova, T. S. (1957). 1651 Shubnikov groups. Sov. Phys. Crystallogr. 1, 487–488. See also (1955) Trudy Inst. Krist. Acad. SSSR, 11, 33–67 (in Russian), English translation in Shubnikov, A. V, Belov, N. V. & others (1964). Colored Symmetry. London: Pergamon Press.
Grimmer, H. (2009). Comments on tables of magnetic space groups. Acta Cryst. A65, 145–155.
Grimmer, H. (2010). Opechowski–Guccione-like symbols labelling magnetic space groups independent of tabulated (0, 0, 0)+ sets. Acta Cryst. A66, 284–291.
Koptsik, V. A. (1966). Shubnikov Groups. Handbook on the Symmetry and Physical Properties of Crystal Structures. Izd. MGU (in Russian). English translation of text: Kopecky, J. & Loopstra, B. O. (1971). Fysica Memo 175. Stichting, Reactor Centrum Nederland.
Litvin, D. B. (2013). Magnetic Group Tables, 1-, 2- and 3-Dimensional Magnetic Subperiodic Groups and Space Groups. Chester: International Union of Crystallography. Freely available from http://www.iucr.org/publ/978–0–9553602–2–0 .
Opechowski, W. & Guccione, R. (1965). Magnetic symmetry. In Magnetism, edited by G. T. Rado & H. Suhl, Vol. 2A, ch. 3. New York: Academic Press.
Ozerov, R. P. (1969a). The design of tables of Shubnikov space groups of dichroic symmetry. Kristallografiya, 14, 393–403.
Ozerov, R. P. (1969b). The design of tables of Shubnikov space groups of dichroic symmetry. Sov. Phys. Crystallogr. 14, 323–332.
Shubnikov, A. V. & Koptsik, V. A. (1974). Symmetry in Science and Art. New York: Plenum Press.
Zamorzaev, A. M. (1976). The Theory of Simple and Multiple Antisymmetry. Kishinev: Shtiintsa. (In Russian.)








































to end of page
to top of page