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

International Tables for Crystallography (2016). Vol. A, ch. 3.6, pp. 854-856

Section 3.6.2.2.4. Opechowski–Guccione magnetic group type symbols and the standard set of coset representatives

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

3.6.2.2.4. Opechowski–Guccione magnetic group type symbols and the standard set of coset representatives

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The specification of the magnetic group type symbol and the standard set of coset representatives of the magnetic group type's representative group is based on the conventions introduced by Opechowski and Guccione (Opechowski & Guccione, 1965[link]; Opechowski, 1986[link]) for three-dimensional magnetic space groups. The specification was made in conjunction with Volume I of International Tables for X-ray Crystallography (1969[link]) (abbreviated here as ITXC I). One finds in ITXC I, for each group type [{\cal F}], a specification of the coordinate system used, and, in terms of that coordinate system, a specification of the subgroup of translations [\cal T] of the representative space group of that group type, and also indirectly a specification of a set of coset representatives of [\cal T] of that representative group of group type [{\cal F}]. These coset representatives are uniquely determined from the coordinate triplets of the explicitly printed general position of the space group. The symbol [{\cal F}] for the space group is taken to be the space-group symbol at the top of the page listing these coordinate triplets. The symbol for a group type [\ispecialfonts{\cal F}\!{\sfi 1}'] is that of the group type [{\cal F}] followed by [\ispecialfonts{\sfi 1}'], and the coset representatives of the representative group of the group type [\ispecialfonts{\cal F}\!{\sfi 1}'] consist of the set of coset representatives of [{\cal F}] and this set multiplied by [1'].

Example

In ITXC I, on the page for [{\cal F} = P2/m] one finds the following coordinate triplets of the general position:[x,y,z\semi \quad x,y, \overline z \semi\quad \overline x, \overline y, z\semi\quad \overline x, \overline y, \overline z ]determining the coset representative of the representative group [P2/m]:[\{1|0\}\semi\quad \{m_{001} |0\}\semi\quad \{2_{001}|0\}\semi\quad \{\overline 1 |0\}.]The coset representatives of the representative group [P2/m1'] are then:[\matrix{\{1|0\}\semi\hfill& \{m_{001} |0\}\semi\hfill& \{2_{001} |0\}\semi\hfill& \{\overline 1 |0\}\hfill\cr\{1|0\}'\semi\hfill& \{m_{001} |0\}'\semi\hfill& \{2_{001} |0\}'\semi\hfill& \{\overline 1|0\}'\hfill.}]

ITXC I has been replaced by IT A. Ones finds that, for some space groups, the set of coordinate triplets of the general positions explicitly printed in IT A differs from that explicitly printed in ITXC I. As a consequence, if one attempts to interpret the Opechowski–Guccione symbols (OG symbols) for magnetic groups using IT A, one will, in many cases misinterpret the meaning of the symbol (Litvin, 1997[link], 1998[link]). [It was suggested in these two papers that the original set of OG symbols should be modified so one could correctly interpret them using IT A instead of ITXC I. Adopting this ill-advised suggestion would have required in the future a new modification of the OG symbols whenever changes were made to the choices of coordinate triplets of the general position in IT A. Consequently, the meaning of the original OG symbols was specified by Litvin (2001[link]) by explicitly giving the coset representatives of the representative groups of each three-dimensional magnetic space group.]

Magnetic groups [{\cal M}_T]

The symbol for a magnetic group type [{\cal M}_T={\cal F}({\cal D}) =] [ {\cal D} \,\cup ({\cal F} - {\cal D})1'] and its representative group is based on the symbol for the group type [\cal F]. [\cal D] is an equi-translational subgroup of [{\cal F}], i.e. the translational subgroup [{\cal T}^{{\cal M}_T}] of the magnetic group [{\cal M}_T] is [\cal T], the translational subgroup of [{\cal F}]. The translational part of the group type symbol of an [{\cal M}_T] group is then the same as that of the group type [{\cal F}]. A number or letter in the remaining part of the symbol of [{\cal F}] appears unchanged in the symbol for [{\cal M}_T] if it is associated with a coset representative of the representative group [{\cal F}] that is also an element contained in the subgroup [{\cal D}] of [{\cal F}]. If not in [{\cal D}], i.e. if in [{\cal F}-{\cal D}], the number or letter appears in the symbol for [{\cal M}_T] with a prime to denote that the element in [{\cal M}_T] is coupled with [1'].

Example

The orthorhombic space-group type [{\cal F} = Pca2_1] has the magnetic space-group type number 29.1.198. The representative group is defined by a orthorhombic translational subgroup [\cal T] denoted by the letter P in Pca21 and the standard set of coset representatives[\{1|0\} \quad\{m_{100}|{\textstyle{1\over 2}},0,{\textstyle{1\over 2}}\}\quad \{m_{010}|{\textstyle{1\over 2}},0,0\}\quad \{2_{001}|0,0,{\textstyle{1\over 2}}\}.]The magnetic space-group type 29.5.202 is a group [{\cal M}_T] whose symbol is [Pc'a'2_1]. In this case we have [Pc'a'2_1 =] [P2_1 \,\cup (Pca2_1 - P2_1)1'], i.e. [{\cal F} = Pca2_1] and [{\cal D} = P2_1]. The symbol `21' in the symbol for [{\cal F} = Pca2_1] refers to the coset representative {2001∣0, 0, ½}, an element in [{\cal D} = P2_1]. Consequently, the symbol `21' appears unprimed in the symbol for [{\cal M}_T] ([Pc'a'2_1]) and the coset representative {2001∣0, 0, ½} appears as an unprimed coset representative in the standard set of coset representatives of [{\cal M}_T]. The symbols `c' and `a' in [{\cal F} = Pca2_1] refer to the coset representatives {m100∣½, 0, ½} and {m010∣½, 0, 0}, respectively, neither of which are contained in [{\cal D}]. Consequently, both symbols appear primed in the symbol [Pc'a'2_1] for [{\cal M}_T] and the coset representatives {m100∣½, 0, ½} and {m010∣½, 0, 0} appear as primed coset representatives in the standard set of coset representatives of [{\cal M}_T]. The representative magnetic space group [Pc'a'2_1] then has the orthorhombic translational subgroup [\cal T] denoted by the letter P and the standard set of coset representatives[\{1|0\} \quad \{m_{100}|{\textstyle{1\over 2}},0,{\textstyle{1\over 2}}\}' \quad \{m_{010}|{\textstyle{1\over 2}},0,0\}'\quad \{2_{001}|0,0,{\textstyle{1\over 2}}\}.]

Magnetic groups [{\cal M}_R]

The symbol for a group type [{\cal M}_R = {\cal F}({\cal D}) = {\cal D} \,\cup ({\cal F} - {\cal D})1'] and its representative group is also based on the symbol for the group [{\cal F}]. [This is in contradistinction to the BNS symbols of [{\cal M}_R] groups (Belov et al., 1957[link]), where the symbol for an [{\cal M}_R] group type is based on the symbol for the group [{\cal D}], see Section 3.6.4[link].] As this is an equi-class magnetic group, half the translations of [{\cal F}] are now coupled with [1'] in [{\cal M}_R] and half the translations remain unprimed in [{\cal M}_R]. The unprimed translations constitute the translational subgroup [{\cal T}^{\cal D}] of [{\cal D}]. We can then write the coset decomposition of the translational subgroup [{\cal T}] of [{\cal F}] with respect to the translational subgroup [{\cal T}^{\cal D}] of [{\cal D}] as[\ispecialfonts{\cal T} = {\cal T}^{\cal D} \cup {\sfi t}_{\alpha} {\cal T}^{\cal D},]where [\ispecialfonts{\sfi t}_\alpha] denotes a chosen coset representative, a translation of [{\cal F}] which appears primed (coupled with [1']) in [{\cal M}_R]. The translational subgroup [{\cal T}^{{\cal M}_R}] of [{\cal M}_R] can then be written as[\ispecialfonts{\cal T}^{{\cal M}_R} = {\cal T}^{\cal D} \cup {\sfi t}_{\alpha}{}' {\cal T}^{\cal D}.]Symbols for the translational groups [{\cal T}], the translational subgroups [{\cal T}^{\cal D}] of [{\cal T}], the translational groups [{\cal T}^{{\cal M}_R}] of [{\cal M}_R] and the choice of the translations [\ispecialfonts{\sfi t}_{\alpha}] are given in Fig. 1 of Litvin (2013[link]).

The symbol for a magnetic group type [{\cal M}_R = {\cal F}({\cal D})] and its representative group is based on the symbol of the group type [{\cal F}], and is also a symbol for the subgroup [{\cal D}] of unprimed elements: the translational part of the symbol of [{\cal F}] is replaced by the symbol for the translational subgroup [{\cal T}^{\cal D}] of [{\cal D}]. If a coset representative {Rτ(R)} of [{\cal T}] in [{\cal F}] appears as the coset representative {Rτ(R) + tα} of [{\cal T}^{\cal D}] in [{\cal D}], then the number or letter corresponding to {Rτ(R)} in the symbol for [{\cal F}] is primed. If {Rτ(R)} appears unchanged as a coset representative of [{\cal T}^{\cal D}] in [{\cal D}], then the number or letter corresponding to {Rτ(R)} in the symbol for [{\cal F}] is unchanged (Opechowski & Litvin, 1977[link]). The resulting symbol is a symbol for [{\cal D}] based on the symbol for [{\cal F}] and is also a symbol for the magnetic group [{\cal M}_R = {\cal F}({\cal D})]: the symbol specifies not only [{\cal D}] but also [{\cal F}]; by deleting the subindex on the translational part of the symbol and the primes on the rotational part, one obtains the symbol specifying [{\cal F}]. Having specified [{\cal D}] and [{\cal F}], one has specified the group [{\cal M}_R = {\cal F}({\cal D})].

Example

Consider again the group 29.1.198, [{\cal F} = Pca2_1], where[{\cal F} = \{1|0\}{\cal T} \cup \{m_{100}|{\textstyle{1\over 2}},0,{\textstyle{1\over 2}}\}{\cal T} \cup\{m_{010}|{\textstyle{1\over 2}},0,0\}{\cal T} \cup \{2_{001}|0,0,{\textstyle{1\over 2}}\}{\cal T}.]The symbol for the [{\cal M}_R = {\cal F}({\cal D})] group type 29.7.204 is [P_{2b}c'a'2_1] and is based on the symbol for [{\cal F} = Pca2_1]. The translational subgroup [{\cal T}^{\cal D}] of [{\cal D}] is given by the symbol P2b where [\ispecialfonts{\sfi t}_\alpha] = {1∣0, 1, 0}, i.e. [{\cal T}^{\cal D}] is generated by the three translations {1∣1, 0, 0}, {1∣0, 2, 0} and {1∣0, 0, 1} of [{\cal T}], and the translational subgroup [{\cal T}^{{\cal M}_R}] of [P_{2b}c'a'2_1] is given by [\ispecialfonts{\cal T}^{{\cal M}_R} = {\cal T}^{\cal D} \cup {\sfi t}_\alpha{}'{\cal T}^{\cal D}]. The two primed symbols [c'] and [a'] in [P_{2b}c'a'2_1] refer to the fact that the two coset representatives {m100∣½, 0, ½} and {m010∣½, 0, 0} that appear in the set of standard coset representatives of [{\cal T}] in [{\cal F}] appear as the coset representatives {m100∣½, 1, ½} and {m010∣½, 1, 0} in the set of standard coset representatives of [{\cal T}^{\cal D}] in [{\cal D}]. As the symbol 21 in [P_{2b}c'a'2_1] is not primed, the coset representative {2001∣0, 0, ½} of [{\cal T}] in [{\cal F}] remains unchanged as a coset representative of [{\cal T}^{\cal D}] in [{\cal D}]. We have then the subgroup[\eqalign{{\cal D} &=\{1|0\} {\cal T}^{\cal D} \cup \{m_{100}|{\textstyle{1\over 2}},1,{\textstyle{1\over 2}}\}{\cal T}^{\cal D} \cup\{m_{010}|{\textstyle{1\over 2}},1,0\}{\cal T}^{\cal D} \cr&\quad\cup \{2_{001}|0,0,{\textstyle{1\over 2}}\}{\cal T}^{\cal D}.}]We note that these same coset representatives of [{\cal T}^{\cal D}] in [{\cal D}] are also the coset representatives of the standard set of coset representatives of [{\cal T}^{{\cal M}_R}] in [{\cal M}_R].[\eqalign{{\cal M}_R &= \{1|0\}{\cal T}^{{\cal M}_R} \cup \{m_{100}|{\textstyle{1\over 2}},1,{\textstyle{1\over 2}}\}{\cal T}^{{\cal M}_R} \cup\{m_{010}|{\textstyle{1\over 2}},1,0\}{\cal T}^{{\cal M}_R} \cr&\quad\cup \{2_{001}|0,0,{\textstyle{1\over 2}}\}{\cal T}^{{\cal M}_R}}]and the standard set of coset representatives of [P_{2b}c'a'2_1] listed in the tables is the same set of coset representatives:[\{1|0\} \quad\{m_{100}|{\textstyle{1\over 2}},1,{\textstyle{1\over 2}}\}\quad \{m_{010}|{\textstyle{1\over 2}},1,0\}\quad \{2_{001}|0,0,{\textstyle{1\over 2}}\}.]Since [\ispecialfonts{\cal T}^{{\cal M}_R} = {\cal T}^{\cal D} \cup {\sfi t}_{\alpha}{}' {\cal T}^{\cal D} ], it follows that [{\cal M}_R ={\cal D}\ \cup ] [({\cal F}-{\cal D})1'] and[\eqalign{{\cal M}_R &= \{1|0\} {\cal T}^{\cal D} \cup \{m_{100}|{\textstyle{1\over 2}},1,{\textstyle{1\over 2}}\} {\cal T}^{\cal D} \cup\{m_{010}|{\textstyle{1\over 2}},1,0\} {\cal T}^{\cal D} \cr&\quad\cup \{2_{001}|0,0,{\textstyle{1\over 2}}\} {\cal T}^{\cal D} \cup \{1|0,1,0\}' {\cal T}^{\cal D} \cup \{m_{100}|{\textstyle{1\over 2}},0,{\textstyle{1\over 2}}\}' {\cal T}^{\cal D} \cr&\quad\cup\{m_{010}|{\textstyle{1\over 2}},0,0\}' {\cal T}^{\cal D} \cup \{2_{001}|0,1,{\textstyle{1\over 2}}\}' {\cal T}^{\cal D}.}]Consequently, a primed number or letter in the symbol for [{\cal M}_R] (which is also a symbol for [{\cal D}]) denotes that the corresponding coset representative appears in [{\cal D}] coupled with [\ispecialfonts{\sfi t}_\alpha] and primed in [({\cal F}-{\cal D})1'], e.g. [a'] in [P_{2b}c'a'2_1] denotes that the coset {m100∣½, 0, ½} appears as {m100∣½, 1, ½} in [{\cal D}] and as {m100∣½, 0, ½}′ in [({\cal F}-{\cal D})1']. An unprimed number or letter in the symbol for [{\cal M}_R] (which is also a symbol for [{\cal D}]) denotes that the corresponding element appears unchanged in [{\cal D}] and coupled with [\ispecialfonts{\sfi t}_\alpha] and primed in [({\cal F}-{\cal D})1'], [e.g.] the symbol 21 in [P_{2b}c'a'2_1] denotes that {2001∣0, 0, ½} is in [{\cal D}] and {2001∣1, 0, ½}′ is in [({\cal F}-{\cal D})1'].

For two-dimensional magnetic space groups with square and hexagonal lattices, three-dimensional magnetic space groups with tetragonal, hexagonal, rhombohedral and cubic lattices, and magnetic layer and rod groups with tetragonal, trigonal or hexagonal lattices, each letter or number in the group type symbol may represent not a single symmetry direction but a set of symmetry directions, see Table 1.3 in Litvin (2013[link]), Table 2.1.3.1[link] in the present volume and Table 1.2.4.1[link] in IT E. Stokes & Campbell (2009[link]) have pointed out that this can lead to not being able to determine the standard set of coset representatives in [{\cal M}_R] groups from the Opechowski–Guccione magnetic group symbol. Consequently, we introduce the convention that in determining the standard set of coset representatives, each letter or number in the group type symbol in these groups refers to the first symmetry direction of each set of symmetry directions listed in the aforementioned tables.

Example

The standard set of coset representatives of 94.1.786 P42212 is[\matrix{\{1|0\}\semi\hfill & \{4_{001} |{\textstyle{1\over 2}},{\textstyle{1\over 2}},{\textstyle{1\over 2}}\}\semi\hfill& \{2_{001}|0\}\semi\hfill& \{4_{001}|{\textstyle{1\over 2}},{\textstyle{1\over 2}},{\textstyle{1\over 2}}\}\hfill\cr  \{2_{100}|{\textstyle{1\over 2}},{\textstyle{1\over 2}},{\textstyle{1\over 2}}\}\semi\hfill& \{2_{010} |{\textstyle{1\over 2}},{\textstyle{1\over 2}},{\textstyle{1\over 2}}\}\semi\hfill&\{2_{110} |0\}\semi\hfill& \{{2_{1\bar10}} |0\}.\hfill}]For the three-dimensional magnetic group 94.7.792 [P_{2c}4_2'2_1'2], the standard set of coset representatives is[\matrix{\{1|0\}\semi\hfill& \{4_{001} |{\textstyle{1\over 2}},{\textstyle{1\over 2}},{\textstyle{3\over 2}}\}\semi\hfill& \{2_{001} |0,0,1\}\semi\hfill& \{4_{001}|{\textstyle{1\over 2}},{\textstyle{1\over 2}},{\textstyle{1\over 2}}\}\cr  \{2_{100}| {\textstyle{1\over 2}},{\textstyle{1\over 2}},{\textstyle{3\over 2}}\}\semi\hfill& \{2_{010} |{\textstyle{1\over 2}},{\textstyle{1\over 2}},{\textstyle{1\over 2}}\}\semi\hfill& \{2_{110} |0,0,1\}\semi\hfill& \{{2_{1\bar10}} |0\}.\hfill}]The secondary position in the Hermann–Mauguin symbol [P_{2c}4_2'2_1'2] denotes the set of symmetry directions {[100], [010]}. With this convention, the primed symbol [2_1'] denotes that the corresponding coset representative {2100∣½, ½, ½} of P42212 appears in the coset representatives of [P_{2c}4_2'2_1'2] coupled with the translation [\ispecialfonts{\sfi t}_\alpha] = {1∣0, 0, 1}. The third position in the Hermann–Mauguin symbol [P_{2c}4_2'2_1'2] denotes the set of symmetry directions [\{ [1 \overline 10],[110]\}]. The unprimed symbol 2 denotes that the coset representative [\{{2_{1\bar 10}} |0\}] of P42212 appears unchanged in the coset representatives of [P_{2c}4_2'2_1'2].

The impact of this convention is that nine Opechowski–Guccione symbols of three-dimensional magnetic space groups need to be changed:

  Old symbol New symbol
93.6.781 P2c4222 P2c4222′
93.8.783 PI4222 PI4222′
93.9.784 P2c42′22′ P2c42′22
     
153.4.1270 P2c3112 P2c3112′
154.4.1274 P2c3121 P2c312′1
     
180.6.1401 P2c6222 P2c6222′
180.7.1402 P2c62′22′ P2c62′22
     
181.6.1408 P2c6422 P2c6422′
181.7.1409 P2c64′2′2 P2c64′2′2′

To have all [{\cal M}_R] group symbols represent subgroups [\cal D], six symbols for three-dimensional magnetic space groups were based (Opechowski & Guccione, 1965[link]) on the symbol of the subgroup [{\cal D}] instead of the symbol for [{\cal F}]. These are the groups 144.3.1236 P2c32, 145.3.1239 P2c31, 151.4.1262 P2c3212, 152.4.1266 P2c3221, 153.4.1270 P2c3112 and 154.4.1274 P2c3121. Additional groups are the rod groups 43.3.231 [{\scr p}_{2c}3_2], 44.3.234 [{\scr p}_{2c}3_1], 47.4.246 [{\scr p}_{2c}3_212] and 48.4.250 [{\scr p}_{2c}3_112].

References

International Tables for X-ray Crystallography (1969). Vol. I, 3rd ed., edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Earlier editions 1952, 1965.]
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.
Litvin, D. B. (1997). Changes in the Opechowski–Guccione symbols for magnetic space groups due to changes in the International Tables of Crystallography. Ferroelectrics, 204, 211–215.
Litvin, D. B. (1998). On Opechowski–Guccione magnetic space-group symbols. Acta Cryst. A54, 257–261.
Litvin, D. B. (2001). Magnetic space-group types. Acta Cryst. A57, 729–730.
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. (1986). Crystallographic and Metacrystallographic Groups. Amsterdam: North Holland.
Opechowski, W. & Guccione, R. (1965). Magnetic symmetry. In Magnetism, edited by G. T. Rado & H. Suhl, Vol. 2A, ch. 3. New York: Academic Press.
Opechowski, W. & Litvin, D. B. (1977). Error corrections corrected: A remark on Bertaut's article, Simple derivation of magnetic space groups. Ann. Phys. 2, 121–125.
Stokes, H. T. & Campbell, B. J. (2009). Table of magnetic space groups, http://stokes.byu.edu/magneticspacegroupshelp.html.








































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