3157:
3150:
3141:
3134:
3127:
3120:
3111:
3104:
3090:
3081:
3074:
3067:
3305:
3298:
3187:
3180:
3171:
3164:
3097:
3060:
3246:
3239:
3230:
3223:
3216:
3209:
3291:
3284:
3353:
3344:
3337:
3330:
3323:
3040:
3033:
3026:
3017:
3010:
3003:
2996:
2974:
2967:
2741:
2719:
2663:
2525:
2402:
2268:
2246:
2176:
2143:
2052:
1949:
1914:
1851:
2899:
2752:
2730:
2708:
2677:
2649:
2616:
2605:
2536:
2516:
2430:
2416:
2375:
2279:
2257:
2235:
2204:
2190:
2132:
2063:
2043:
1960:
1938:
1903:
1840:
1829:
2914:
2884:
2853:
2796:
2575:
2564:
2489:
2461:
2349:
2322:
2102:
2091:
2016:
1988:
1879:
1805:
1779:
1752:
2824:
2697:
2635:
2594:
2505:
2388:
2364:
2224:
2162:
2121:
2032:
1927:
1892:
2869:
2839:
2782:
2553:
2476:
2450:
2338:
2309:
2292:
2080:
2003:
1977:
1868:
1794:
1768:
1722:
2813:
2765:
1818:
1739:
111:
The magnetic space groups can be placed into three categories. First, the 230 colorless groups contain only spatial symmetry, and correspond to the crystallographic space groups. Then there are 230 grey groups, which are invariant under antisymmetry. Finally are the 1191 black-white groups, which
116:
and
Rosalia Guiccione) and Belov-Neronova-Smirnova. For colorless and grey groups, the conventions use the same names, but they treat the black-white groups differently. A full list of the magnetic space groups (in both conventions) can be found both in the original papers, and in several places
3695:
which are often used to classify artistic patterns. In that case, the 7 frieze groups with the addition of color reversal become 24 color-reversing frieze groups. Beyond the simple two-valued property, the idea has been extended further to three colors in three dimensions, and to even higher
2947:, but also contain additional symmetry elements. For black-white Bravais lattices, the number of black and white sites is always equal. There are 14 traditional Bravais lattices, 14 grey lattices, and 22 black-white Bravais lattices, for a total of 50 two-color lattices in three dimensions.
4548:
Phase transformations and evolution in materials : proceedings of a symposium sponsored by the Alloy Phase
Committee of the joint IMPMD/SMD of the Minerals, Metals, and Materials Society (TMS), held at the 2000 TMS Annual Meeting in Nashville, Tennessee, USA, March 12-16,
85:
gives a total of 122 magnetic point groups. However, although Heesch correctly laid out each of the magnetic point groups, his work remained obscure, and the point groups were later re-derived by Tavger and
Zaitsev. The concept was more fully explored by Shubnikov in terms of
3371:, but excluding the 14 gray lattices which look identical to the traditional lattices. The lattice symbols are those used for the traditional Bravais lattices. The suffix in the symbol indicates the mode of centering by the black (antisymmetry) points in the lattice, where
3661:. In the most general form, magnetic space groups can represent symmetries of any two valued lattice point property, such as positive/negative electrical charge or the alignment of electric dipole moments. The magnetic space groups place restrictions on the
3906:(1930-01-01). "Zur systematischen Strukturtheorie. IV - Über die Symmetrien zweiter Art in Kontinuen und Remidiskontinuen" [Systematic structure theory IV - On the symmetry of the second kind in continua and semicontinua].
1713:
of most of the magnetic point groups onto a flat surface. Not shown are the grey point groups, which look identical to the ordinary crystallographic point groups, except they are also invariant under the antisymmetry operation.
5145:
Xu, Yuanfeng; Elcoro, Luis; Song, Zhida; Wieder, Benjamin. J.; Vergniory, M. G.; Regnault, Nicolas; Chen, Yulin; Felser, Claudia; Bernevig, B. Andrei (2020). "High-throughput calculations of magnetic topological materials".
3860:(1930). "Zur systematischen Strukturtheorie. III - Über die vierdimensionalen Gruppen des dreidimensionalen Raumes" [Systematic Structure theory III - On the four-dimensional groups of three-dimensional space].
905:
in the following table. Here, the addition of an apostrophe to a symmetry operation indicates that the combination of the symmetry element and the antisymmetry operation is a symmetry of the structure. There are 32
891:
671:
464:
516:
3393:, and can be well-described by a magnetic space group. However, when this is not the case, the order does not correspond to any magnetic space group. These phases can instead be described by
3156:
3149:
3140:
3133:
3126:
3119:
3110:
3103:
3089:
3080:
3073:
3066:
46:, there is a so-called "antisymmetry" operation which turns all black lattice points white and all white lattice points black. Thus, the magnetic space groups serve as an extension to the
3304:
3297:
3186:
3179:
3170:
3163:
3096:
3059:
1678:
are colored red, and the magnetic point groups which are compatible with both ferromagnetism and ferroelectricity are purple. There are 31 magnetic point groups which are compatible with
3245:
3238:
3229:
3222:
3215:
3208:
3290:
3283:
3949:
Wills, Andrew S. (2017). "A historical introduction to the symmetries of magnetic structures. Part 1. Early quantum theory, neutron powder diffraction and the coloured space groups".
3039:
3032:
3025:
3016:
3009:
3002:
2995:
2973:
2966:
3352:
3343:
3336:
3329:
3322:
556:
298:
3637:, and a ferromagnetic or ferroelectric point group which is a subgroup of the prototypical group which can be reached by continuous motion of the atoms in the crystal structure.
708:
335:
238:
379:
753:
408:
90:. When applied to space groups, the number increases from the usual 230 three dimensional space groups to 1651 magnetic space groups, as found in the 1953 thesis of
3566:
3531:
3500:
3473:
3446:
807:
780:
57:. Compatibility with a material's symmetries, as described by the magnetic space group, is a necessary condition for a variety of material properties, including
596:
576:
355:
258:
112:
contain the more complex symmetries. There are two common conventions for giving names to the magnetic space groups. They are
Opechowski-Guiccione (named after
4708:
Perez-Mato, J M; Ribeiro, J L; Petricek, V; Aroyo, M I (2012-03-26). "Magnetic superspace groups and symmetry constraints in incommensurate magnetic phases".
81:, who first rigorously established the concept of antisymmetry as part of a series of papers in 1929 and 1930. Applying this antisymmetry operation to the 32
3645:
The main application of these space groups is to magnetic structure, where the black/white lattice points correspond to spin up/spin down configuration of
4545:
Laughlin, D. E.; Willard, M. A.; McHenry, M. E. (2000). "Magnetic
Ordering: Some Structural Aspects". In Gonis, Antonios; Turchi, Patrice E. A. (eds.).
5405:
3665:
of materials. Specifically, they place restrictions on the connectivity of the different electron bands, which in turn defines whether material has
5297:. Bridges: Mathematical Connections in Art, Music, and Science. The International Society of the Arts, Mathematics, and Architecture. p. 135.
5360:
4560:
4370:
4342:
94:. While the magnetic space groups were originally found using geometry, it was later shown the same magnetic space groups can be found using
3533:, and keeps only the symmetries which have not been broken during the phase transition. This can be tracked numerically by evolution of the
3666:
4979:
3389:
When the periodicity of the magnetic order coincides with the periodicity of crystallographic order, the magnetic phase is said to be
819:
4163:
3384:
1686:, leave at least one component of the spin invariant under operations of the point group. There are 31 point groups compatible with
5385:
901:
The following table lists all of the 122 possible three-dimensional magnetic point groups. This is given in the short version of
608:
91:
42:. To represent such a property, each lattice point is colored black or white, and in addition to the usual three-dimensional
416:
3741:"Black-and-White Symmetry, Magnetic Symmetry, Self-Duality and Antiprismatic Symmetry: The Common Mathematical Background"
902:
907:
82:
47:
29:
5245:
Rodríguez-Carvajal, Juan (1993). "Recent advances in magnetic structure determination by neutron powder diffraction".
3050:
3768:(1929-01-01). "Zur Strukturtheorie der ebenen Symmetriegruppen" [Structure theory of plane symmetry groups].
470:
5434:
3538:
3662:
3658:
3199:
2986:
95:
5429:
4215:
Kim, Shoon K. (1986). "The 38 assemblies of the general generator sets for 1421 magnetic double space groups".
3274:
3267:
3258:
2957:
1710:
382:
4863:
Aizu, Kêitsiro (1970-08-01). "Possible
Species of Ferromagnetic, Ferroelectric, and Ferroelastic Crystals".
4363:
The mathematical theory of symmetry in solids : representation theory for point groups and space groups
4085:
3591:
4394:
Litvin, Daniel B.; Kopský, Vojtěch (2011-05-26). "Seitz notation for symmetry operations of space groups".
4995:
Elcoro, Luis; Wieder, Benjamin J.; Song, Zhida; Xu, Yuanfeng; Bradlyn, Barry; Bernevig, B. Andrei (2021).
4448:
Teaching crystallographic andmagnetic point group symmetry using three-dimensional rendered visualizations
3697:
3650:
3623:
2940:
1699:
810:
711:
524:
266:
4971:
679:
306:
5439:
3670:
212:
66:
4656:
The mathematical theory of symmetry in solids: representation theory for point groups and space groups
4446:
4327:
Magnetic Group Tables: 1-, 2-, and 3-Dimensional
Magnetic Subperiodic Groups and Magnetic Space Groups
5317:
5254:
5211:
5085:
5018:
4915:
4872:
4829:
4786:
4727:
4682:
4593:
4503:
4403:
4224:
4116:
3968:
3598:
3313:
1698:. Similar symmetry arguments have been extended to other electromagnetic material properties such as
113:
87:
54:
360:
4820:
Dzyaloshinsky, I. (1958). "A thermodynamic theory of "weak" ferromagnetism of antiferromagnetics".
3685:
3681:
3677:
3605:
3576:
599:
20:
4258:
Opechowski, W.; Guccione, R. (1965). "Magnetic
Symmetry". In Rado, George T.; Suhl, Harry (eds.).
3421:
has been applied to magnetic phase transitions. The magnetic space group of disordered structure,
5181:
5155:
5127:
5075:
5008:
4759:
4717:
4527:
4039:
3992:
3958:
3931:
3885:
3839:
3814:(1930-01-01). "Zur systematischen Strukturtheorie. II" [Systematic structure theory II].
3793:
3714:
3616:
3571:
Important magnetic phase transitions include the paramagnetic to ferromagnetic transition at the
717:
168:
43:
4965:
4494:
Schmid, Hans (2008-10-09). "Some symmetry aspects of ferroics and single phase multiferroics".
4011:
3740:
38:
which classify the symmetries of a crystal both in space, and in a two-valued property such as
5356:
5333:
5270:
5227:
5173:
5119:
5101:
5044:
4975:
4939:
4931:
4888:
4845:
4802:
4777:
Dimmock, John O. (1963-05-15). "Use of
Symmetry in the Determination of Magnetic Structures".
4751:
4743:
4636:
4609:
4566:
4556:
4519:
4427:
4419:
4376:
4366:
4338:
4263:
4240:
4140:
4132:
4104:
4031:
3984:
3923:
3877:
3831:
3785:
3572:
1691:
209:
The types can be distinguished by their different construction. Type I magnetic space groups,
5289:
5325:
5262:
5219:
5165:
5109:
5093:
5034:
5026:
4923:
4880:
4837:
4794:
4735:
4690:
4601:
4546:
4511:
4411:
4361:
Bradley, C. J.; Cracknell, A. P. (2010). "The
Magnetic Groups and their corepresentations".
4330:
4232:
4124:
4023:
3976:
3915:
3869:
3823:
3777:
3680:
experiments. The resulting experimental profile can be matched to theoretical structures by
3418:
3368:
1703:
1687:
1675:
62:
4654:
Bradley, C. J.; Cracknell, A.P. (1972). "The magnetic groups and their corepresentations".
3544:
3509:
3478:
3451:
3424:
785:
758:
4961:
4906:
Litvin, D. B. (2008-02-19). "Ferroic classifications extended to ferrotoroidic crystals".
4670:
3903:
3857:
3811:
3765:
3580:
3534:
2944:
1695:
180:
78:
4739:
4515:
3669:. Thus, the magnetic space groups can be used to identify topological materials, such as
189:
Space groups, with additional anti-symmetry versions of half of the symmetry operations.
5321:
5258:
5215:
5089:
5022:
4919:
4876:
4833:
4790:
4731:
4686:
4597:
4507:
4407:
4228:
4120:
3972:
388:
5114:
5063:
5039:
4996:
1679:
1671:
581:
561:
340:
243:
58:
35:
5423:
5266:
5199:
5185:
4841:
4531:
4043:
3996:
3935:
3889:
3843:
3797:
3646:
3633:
which consist of a prototypical non-ferroic magnetic point group, the letter "F" for
3414:
39:
5131:
4763:
4282:
3719:
3692:
3654:
3402:
203:
Space groups, with additional combined spatial translation-time reversal symmetry.
5400:
3649:. More abstractly, the magnetic space groups are often thought of as representing
3367:
The table shows the 36 black-white Bravais lattices, including the 14 traditional
337:, are made up of all the symmetry operations of the crystallographic space group,
5202:(1969-06-02). "A profile refinement method for nuclear and magnetic structures".
4058:
3709:
150:
5291:
The Mathematics of Color-Reversing Decorative Friezes: Faaçdes of Pirgí, Greece
5030:
4472:
3919:
3873:
3827:
813:) pointing from a black colored point to a white colored point, or vice versa.
53:
The application of magnetic space groups to crystal structures is motivated by
5329:
5223:
5169:
4957:
4927:
4694:
4415:
4128:
3980:
3781:
3676:
Experimentally, the main source of information about magnetic space groups is
3622:
are purely antiferromagnetic. This theory developed into what is now known as
3401:
order. This is the same formalism often used to describe the ordering of some
5337:
5274:
5231:
5105:
5064:"Structure and topology of band structures in the 1651 magnetic space groups"
4935:
4892:
4849:
4806:
4798:
4747:
4640:
4613:
4523:
4423:
4380:
4334:
4244:
4136:
4035:
3988:
3927:
3881:
3835:
3789:
5381:"MAGNDATA: A collection of magnetic structures with portable cif-type files"
4570:
4303:
4267:
1694:. There are also 31 point groups compatible with the theoretically proposed
5177:
5123:
5097:
5074:(8). American Association for the Advancement of Science (AAAS): eaat8685.
5048:
4943:
4884:
4755:
4584:
Atoji, Masao (1965). "Graphical Representations of Magnetic Space Groups".
4431:
4144:
4627:
Belov, N.V.; Neronova, N. N.; Smirnova, T. S. (1957). "Shubnikov groups".
2740:
2718:
2662:
2524:
2401:
2267:
2245:
2175:
2142:
2051:
1948:
1913:
1850:
4970:. A Course of Theoretical Physics. Vol. 8. Pergamon Press. pp.
3634:
3503:
2898:
5380:
5308:
Harker, D. (1981). "The three-colored three-dimensional space groups".
4027:
4605:
2751:
2729:
2707:
2676:
2648:
2615:
2604:
2535:
2515:
2429:
2415:
2374:
2278:
2256:
2234:
2203:
2189:
2131:
2062:
2042:
1959:
1937:
1902:
1839:
1828:
1674:
are colored cyan, the magnetic point groups which are compatible with
4236:
357:, plus the product of those operations with time reversal operation,
2913:
2883:
2852:
2795:
2574:
2563:
2488:
2460:
2348:
2321:
2101:
2090:
2015:
1987:
1878:
1804:
1778:
1751:
5401:"Database of Magnetic Structures Determined by Neutron Diffraction"
5160:
5080:
5013:
4306:. University of the Basque Country - Bilbao Crystallographic Server
3963:
2823:
4722:
2696:
2634:
2593:
2504:
2387:
2363:
2223:
2161:
2120:
2031:
1926:
1891:
3448:, transitions to the magnetic space group of the ordered phase,
2868:
2838:
2781:
2552:
2475:
2449:
2337:
2308:
2291:
2079:
2002:
1976:
1867:
1793:
1767:
1721:
886:{\displaystyle {\mathcal {M}}_{IV}=G+{\mathcal {T}}\{E|t_{0}\}G}
167:
Space groups, with an additional anti-symmetry version of every
4592:(3). American Association of Physics Teachers (AAPT): 212–219.
2812:
2764:
755:, which is Seitz notation for null rotation and a translation,
1817:
1738:
5288:
David A. James; Loukas N. Kalisperis; Alice V. James (2003).
4914:(2). International Union of Crystallography (IUCr): 316–320.
4402:(4). International Union of Crystallography (IUCr): 415–418.
3691:
Adding the two-valued symmetry is also a useful concept for
3579:. Differences in the magnetic phase transitions explain why
3575:
and the paramagnetic to antiferromagnetic transition at the
851:
826:
686:
643:
615:
531:
477:
448:
423:
366:
313:
273:
219:
5210:(2). International Union of Crystallography (IUCr): 65–71.
5062:
Watanabe, Haruki; Po, Hoi Chun; Vishwanath, Ashvin (2018).
3604:
are weakly ferromagnetic, whereas the structurally similar
666:{\displaystyle {\mathcal {M}}_{III}=H+{\mathcal {T}}(G-H)}
5353:
General Results of Crystal Structure Analysis of Minerals
3908:
Zeitschrift für Kristallographie - Crystalline Materials
3862:
Zeitschrift für Kristallographie - Crystalline Materials
3816:
Zeitschrift für Kristallographie - Crystalline Materials
3770:
Zeitschrift für Kristallographie - Crystalline Materials
4681:(9). International Union of Crystallography: 543–548.
4365:. Oxford New York: Clarendon Press. pp. 569–681.
5355:. Springer Verlag Berlin Heidelberg. pp. 50–55.
3547:
3512:
3481:
3454:
3427:
822:
788:
761:
720:
682:
611:
584:
564:
527:
473:
459:{\displaystyle {\mathcal {M}}_{II}=G+{\mathcal {T}}G}
419:
391:
363:
343:
309:
269:
246:
215:
4658:. Oxford: Oxford University Press. pp. 586–587.
1690:; these are generalizations of the crystallographic
1670:
The magnetic point groups which are compatible with
4473:"On a magnetoelectric classification of materials"
3560:
3525:
3494:
3467:
3440:
2939:The black-white Bravais lattices characterize the
885:
801:
774:
747:
702:
665:
590:
570:
550:
510:
458:
402:
373:
349:
329:
292:
252:
232:
197:Black-White groups (black-white Bravais Lattices)
4785:(4). American Physical Society (APS): 1337–1344.
910:, 32 grey groups, and 58 magnetic point groups.
385:of an ordinary space group with the point group
4871:(3). American Physical Society (APS): 754–772.
4066:Journal of Experimental and Theoretical Physics
4016:Proceedings of the Indian Academy of Sciences A
4356:
4354:
4098:
4096:
511:{\displaystyle {\mathcal {M}}_{II}=G\times 1'}
5351:Koptsik, V. A. (1994). A. S. Marfunin (ed.).
4635:(3). American Institute of Physics: 311–322.
4105:"Comments on tables of magnetic space groups"
8:
877:
856:
742:
721:
240:are identical to the ordinary space groups,
4822:Journal of Physics and Chemistry of Solids
4329:. International Union of Crystallography.
4325:Litvin, D. B. (2013). Litvin, D. B (ed.).
4262:. Vol. 2A. New York: Academic Press.
3629:A related scheme is the classification of
5159:
5113:
5079:
5038:
5012:
4721:
4555:. Warrendale, Pa: TMS. pp. 121–137.
4010:Pantulu, P. V.; Radhakrishna, S. (1967).
3962:
3552:
3546:
3517:
3511:
3486:
3480:
3459:
3453:
3432:
3426:
871:
862:
850:
849:
831:
825:
824:
821:
793:
787:
766:
760:
736:
727:
719:
710:, are constructed with the use of a pure
691:
685:
684:
681:
642:
641:
620:
614:
613:
610:
583:
563:
536:
530:
529:
526:
482:
476:
475:
472:
447:
446:
428:
422:
421:
418:
390:
365:
364:
362:
342:
318:
312:
311:
308:
278:
272:
271:
268:
245:
224:
218:
217:
214:
5406:AGH University of Science and Technology
4997:"Magnetic Topological Quantum Chemistry"
4283:"ISO-MAG Table of Magnetic Space Groups"
4196:"Generalization of the Fedorov groups".
4177:"Generalization of the Fedorov groups".
2951:Magnetic (black-white) Bravais lattices
2949:
1716:
912:
381:. Equivalently, this can be seen as the
119:
4281:Harold T. Stokes; Branton J. Campbell.
4012:"A method of deriving shubnikov groups"
3731:
50:which describe spatial symmetry alone.
4083:A. V. Shubnikov; N. V. Belov (1964).
7:
4710:Journal of Physics: Condensed Matter
4496:Journal of Physics: Condensed Matter
4160:Generalization of the Fedorov groups
4057:Tavger, B.A.; Zaitsev, V.M. (1956).
3667:symmetry-protected topological order
551:{\displaystyle {\mathcal {M}}_{III}}
293:{\displaystyle {\mathcal {M}}_{I}=G}
4967:Electrodynamics of Continuous Media
703:{\displaystyle {\mathcal {M}}_{IV}}
330:{\displaystyle {\mathcal {M}}_{II}}
153:, without any additional symmetry.
5204:Journal of Applied Crystallography
4671:"Extensions of space-group theory"
4477:International Journal of Magnetism
2943:of the structure like the typical
233:{\displaystyle {\mathcal {M}}_{I}}
14:
3385:Superstructure (condensed matter)
1682:. These groups, sometimes called
5386:University of the Basque Country
5310:Acta Crystallographica Section A
4908:Acta Crystallographica Section A
4396:Acta Crystallographica Section A
4223:(5). AIP Publishing: 1484–1489.
4109:Acta Crystallographica Section A
3351:
3342:
3335:
3328:
3321:
3303:
3296:
3289:
3282:
3244:
3237:
3228:
3221:
3214:
3207:
3185:
3178:
3169:
3162:
3155:
3148:
3139:
3132:
3125:
3118:
3109:
3102:
3095:
3088:
3079:
3072:
3065:
3058:
3038:
3031:
3024:
3015:
3008:
3001:
2994:
2972:
2965:
2912:
2897:
2882:
2867:
2851:
2837:
2822:
2811:
2794:
2780:
2763:
2750:
2739:
2728:
2717:
2706:
2695:
2675:
2661:
2647:
2633:
2614:
2603:
2592:
2573:
2562:
2551:
2534:
2523:
2514:
2503:
2487:
2474:
2459:
2448:
2428:
2414:
2400:
2386:
2373:
2362:
2347:
2336:
2320:
2307:
2290:
2277:
2266:
2255:
2244:
2233:
2222:
2202:
2188:
2174:
2160:
2141:
2130:
2119:
2100:
2089:
2078:
2061:
2050:
2041:
2030:
2014:
2001:
1986:
1975:
1958:
1947:
1936:
1925:
1912:
1901:
1890:
1877:
1866:
1849:
1838:
1827:
1816:
1803:
1792:
1777:
1766:
1750:
1737:
1720:
1709:The following diagrams show the
558:, are constructed using a group
521:Type III magnetic space groups,
4217:Journal of Mathematical Physics
4059:"Magnetic Symmetry of Crystals"
676:Type IV magnetic space groups,
303:Type II magnetic space groups,
121:Types of magnetic space groups
4740:10.1088/0953-8984/24/16/163201
4716:(16). IOP Publishing: 163201.
4629:Soviet Physics Crystallography
4516:10.1088/0953-8984/20/43/434201
4502:(43). IOP Publishing: 434201.
4198:Soviet Physics Crystallography
916:Crystallographic point groups
863:
809:is a vector (usually given in
728:
660:
648:
374:{\displaystyle {\mathcal {T}}}
1:
908:Crystallographic point groups
179:Black-White groups (ordinary
151:crystallographic space groups
83:crystallographic point groups
77:A major step was the work of
48:crystallographic space groups
5267:10.1016/0921-4526(93)90108-i
5007:(1). Nature Research: 5965.
4842:10.1016/0022-3697(58)90076-3
4304:"Magnetic Space Groups List"
3537:, which belongs to a single
2935:Black-white Bravais lattices
5253:(1–2). Elsevier BV: 55–69.
5247:Physica B: Condensed Matter
4828:(4). Elsevier BV: 241–255.
4586:American Journal of Physics
3641:Applications and extensions
3275:Rhombohedral lattice system
3051:Orthorhombic lattice system
748:{\displaystyle \{E|t_{0}\}}
5456:
5031:10.1038/s41467-021-26241-8
4164:Leningrad State University
3920:10.1524/zkri.1930.73.1.346
3874:10.1524/zkri.1930.73.1.325
3828:10.1524/zkri.1930.72.1.177
3539:irreducible representation
3395:magnetic superspace groups
3382:
3379:Magnetic superspace groups
5330:10.1107/S0567739481000697
5224:10.1107/s0021889869006558
5170:10.1038/s41586-020-2837-0
4928:10.1107/s0108767307068262
4695:10.1107/S0365110X57001966
4416:10.1107/s010876731101378x
4158:Zamorzaev, A. M. (1953).
4129:10.1107/S0108767308039007
3981:10.1017/S0885715617000124
3782:10.1524/zkri.1929.71.1.95
3663:electronic band structure
3659:time translation symmetry
3653:. This is in contrast to
3311:
3272:
3265:
3256:
3200:Tetragonal lattice system
3197:
3048:
2987:Monoclinic lattice system
2984:
2955:
921:
578:, which is a subgroup of
4799:10.1103/physrev.130.1337
4335:10.1107/9780955360220001
3375:denotes edge centering.
3268:Hexagonal lattice system
3259:Hexagonal crystal family
2958:Triclinic lattice system
1711:stereographic projection
903:Hermann–Mauguin notation
5098:10.1126/sciadv.aat8685
4885:10.1103/physrevb.2.754
4675:Acta Crystallographica
4103:Grimmer, Hans (2009).
4089:. New York, Macmillan.
3671:topological insulators
3651:time reversal symmetry
3624:antisymmetric exchange
3562:
3527:
3496:
3469:
3442:
2941:translational symmetry
922:Magnetic point groups
887:
811:fractional coordinates
803:
776:
749:
704:
667:
592:
572:
552:
512:
460:
404:
375:
351:
331:
294:
254:
234:
67:topological insulation
5001:Nature Communications
4669:Mackay, A.L. (1957).
4471:Schmid, Hans (1973).
3657:, which instead have
3563:
3561:{\displaystyle G_{0}}
3528:
3526:{\displaystyle G_{0}}
3497:
3495:{\displaystyle G_{1}}
3470:
3468:{\displaystyle G_{1}}
3443:
3441:{\displaystyle G_{0}}
897:Magnetic point groups
888:
804:
802:{\displaystyle t_{0}}
777:
775:{\displaystyle t_{0}}
750:
705:
668:
593:
573:
553:
513:
461:
405:
376:
352:
332:
295:
255:
235:
107:Magnetic space groups
25:magnetic space groups
4162:(PhD) (in Russian).
3739:Gábor Gévay (2000).
3545:
3510:
3479:
3452:
3425:
3314:Cubic lattice system
820:
786:
759:
718:
680:
609:
582:
562:
525:
471:
417:
389:
361:
341:
307:
267:
244:
213:
114:Wladyslaw Opechowski
5322:1981AcCrA..37..286H
5259:1993PhyB..192...55R
5216:1969JApCr...2...65R
5090:2018SciA....4.8685W
5023:2021NatCo..12.5965E
4920:2008AcCrA..64..316L
4877:1970PhRvB...2..754A
4834:1958JPCS....4..241D
4791:1963PhRv..130.1337D
4732:2012JPCM...24p3201P
4687:1957AcCry..10..543M
4598:1965AmJPh..33..212A
4508:2008JPCM...20Q4201S
4408:2011AcCrA..67..415L
4229:1986JMP....27.1484K
4121:2009AcCrA..65..145G
3973:2017PDiff..32..148W
3686:simulated annealing
3682:Rietveld refinement
3678:neutron diffraction
2952:
122:
44:symmetry operations
21:solid state physics
4028:10.1007/BF03049452
3951:Powder Diffraction
3715:Magnetic structure
3558:
3523:
3492:
3465:
3438:
2950:
1700:magnetoelectricity
1692:polar point groups
919:Grey point groups
883:
799:
772:
745:
700:
663:
588:
568:
548:
508:
456:
403:{\displaystyle 1'}
400:
371:
347:
327:
290:
250:
230:
169:symmetry operation
120:
92:Alexandr Zamorzaev
16:Concept in physics
5435:Magnetic ordering
5362:978-3-642-78525-2
5154:(7831): 702–707.
4865:Physical Review B
4606:10.1119/1.1971375
4562:978-0-87339-468-0
4372:978-0-19-958258-7
4344:978-0-9553602-2-0
3573:Curie temperature
3419:phase transitions
3409:Phase transitions
3397:, which describe
3365:
3364:
2932:
2931:
1668:
1667:
591:{\displaystyle G}
571:{\displaystyle H}
350:{\displaystyle G}
253:{\displaystyle G}
207:
206:
143:Colorless groups
132:Number of groups
55:Curie's Principle
5447:
5416:
5414:
5413:
5396:
5394:
5393:
5367:
5366:
5348:
5342:
5341:
5305:
5299:
5298:
5296:
5285:
5279:
5278:
5242:
5236:
5235:
5196:
5190:
5189:
5163:
5142:
5136:
5135:
5117:
5083:
5068:Science Advances
5059:
5053:
5052:
5042:
5016:
4992:
4986:
4985:
4954:
4948:
4947:
4903:
4897:
4896:
4860:
4854:
4853:
4817:
4811:
4810:
4774:
4768:
4767:
4725:
4705:
4699:
4698:
4666:
4660:
4659:
4651:
4645:
4644:
4624:
4618:
4617:
4581:
4575:
4574:
4554:
4542:
4536:
4535:
4491:
4485:
4484:
4468:
4462:
4461:
4459:
4458:
4453:
4442:
4436:
4435:
4391:
4385:
4384:
4358:
4349:
4348:
4322:
4316:
4315:
4313:
4311:
4300:
4294:
4293:
4291:
4289:
4278:
4272:
4271:
4255:
4249:
4248:
4237:10.1063/1.527397
4212:
4206:
4205:
4193:
4187:
4186:
4179:Kristallografiya
4174:
4168:
4167:
4155:
4149:
4148:
4100:
4091:
4090:
4086:Colored Symmetry
4080:
4074:
4073:
4063:
4054:
4048:
4047:
4007:
4001:
4000:
3966:
3946:
3940:
3939:
3914:(1–6): 346–356.
3900:
3894:
3893:
3868:(1–6): 325–345.
3854:
3848:
3847:
3822:(1–6): 177–201.
3808:
3802:
3801:
3762:
3756:
3755:
3745:
3736:
3577:Néel temperature
3567:
3565:
3564:
3559:
3557:
3556:
3532:
3530:
3529:
3524:
3522:
3521:
3501:
3499:
3498:
3493:
3491:
3490:
3474:
3472:
3471:
3466:
3464:
3463:
3447:
3445:
3444:
3439:
3437:
3436:
3417:of second-order
3369:Bravais lattices
3355:
3346:
3339:
3332:
3325:
3307:
3300:
3293:
3286:
3248:
3241:
3232:
3225:
3218:
3211:
3189:
3182:
3173:
3166:
3159:
3152:
3143:
3136:
3129:
3122:
3113:
3106:
3099:
3092:
3083:
3076:
3069:
3062:
3042:
3035:
3028:
3019:
3012:
3005:
2998:
2976:
2969:
2953:
2945:Bravais lattices
2922:
2916:
2907:
2901:
2892:
2886:
2877:
2871:
2860:
2855:
2846:
2841:
2826:
2815:
2804:
2798:
2790:
2784:
2767:
2754:
2743:
2732:
2721:
2710:
2699:
2684:
2679:
2670:
2665:
2656:
2651:
2642:
2637:
2618:
2607:
2596:
2577:
2566:
2555:
2538:
2527:
2518:
2507:
2496:
2491:
2483:
2478:
2463:
2452:
2437:
2432:
2423:
2418:
2409:
2404:
2395:
2390:
2377:
2366:
2351:
2340:
2329:
2324:
2316:
2311:
2294:
2281:
2270:
2259:
2248:
2237:
2226:
2211:
2206:
2197:
2192:
2183:
2178:
2169:
2164:
2145:
2134:
2123:
2104:
2093:
2082:
2065:
2054:
2045:
2034:
2023:
2018:
2010:
2005:
1990:
1979:
1962:
1951:
1940:
1929:
1916:
1905:
1894:
1881:
1870:
1853:
1842:
1831:
1820:
1807:
1796:
1781:
1770:
1759:
1754:
1746:
1741:
1724:
1717:
1704:piezoelectricity
1688:ferroelectricity
1676:ferroelectricity
1659:
1652:
1645:
1638:
1631:
1576:
1569:
1563:
1509:
1503:
1497:
1491:
1485:
1408:
1402:
1397:
1366:
1360:
1354:
1348:
1342:
1288:
1282:
1277:
1224:
1218:
1212:
1206:
1200:
1123:
1117:
1112:
958:
952:
947:
913:
892:
890:
889:
884:
876:
875:
866:
855:
854:
839:
838:
830:
829:
808:
806:
805:
800:
798:
797:
781:
779:
778:
773:
771:
770:
754:
752:
751:
746:
741:
740:
731:
709:
707:
706:
701:
699:
698:
690:
689:
672:
670:
669:
664:
647:
646:
631:
630:
619:
618:
597:
595:
594:
589:
577:
575:
574:
569:
557:
555:
554:
549:
547:
546:
535:
534:
517:
515:
514:
509:
507:
490:
489:
481:
480:
465:
463:
462:
457:
452:
451:
436:
435:
427:
426:
409:
407:
406:
401:
399:
380:
378:
377:
372:
370:
369:
356:
354:
353:
348:
336:
334:
333:
328:
326:
325:
317:
316:
299:
297:
296:
291:
283:
282:
277:
276:
259:
257:
256:
251:
239:
237:
236:
231:
229:
228:
223:
222:
181:Bravais lattices
123:
63:ferroelectricity
5455:
5454:
5450:
5449:
5448:
5446:
5445:
5444:
5430:Crystallography
5420:
5419:
5411:
5409:
5399:
5391:
5389:
5379:
5376:
5371:
5370:
5363:
5350:
5349:
5345:
5307:
5306:
5302:
5294:
5287:
5286:
5282:
5244:
5243:
5239:
5200:Rietveld, H. M.
5198:
5197:
5193:
5144:
5143:
5139:
5061:
5060:
5056:
4994:
4993:
4989:
4982:
4962:Evgeny Lifshitz
4956:
4955:
4951:
4905:
4904:
4900:
4862:
4861:
4857:
4819:
4818:
4814:
4779:Physical Review
4776:
4775:
4771:
4707:
4706:
4702:
4668:
4667:
4663:
4653:
4652:
4648:
4626:
4625:
4621:
4583:
4582:
4578:
4563:
4552:
4544:
4543:
4539:
4493:
4492:
4488:
4470:
4469:
4465:
4456:
4454:
4451:
4445:DeGraef, Marc.
4444:
4443:
4439:
4393:
4392:
4388:
4373:
4360:
4359:
4352:
4345:
4324:
4323:
4319:
4309:
4307:
4302:
4301:
4297:
4287:
4285:
4280:
4279:
4275:
4257:
4256:
4252:
4214:
4213:
4209:
4195:
4194:
4190:
4176:
4175:
4171:
4157:
4156:
4152:
4102:
4101:
4094:
4082:
4081:
4077:
4061:
4056:
4055:
4051:
4009:
4008:
4004:
3948:
3947:
3943:
3902:
3901:
3897:
3856:
3855:
3851:
3810:
3809:
3805:
3776:(1–6): 95–102.
3764:
3763:
3759:
3743:
3738:
3737:
3733:
3728:
3706:
3696:dimensions and
3643:
3620:
3613:
3609:
3602:
3595:
3588:
3584:
3548:
3543:
3542:
3535:order parameter
3513:
3508:
3507:
3482:
3477:
3476:
3455:
3450:
3449:
3428:
3423:
3422:
3411:
3387:
3381:
2937:
2924:
2920:
2917:
2909:
2905:
2902:
2894:
2890:
2887:
2879:
2875:
2872:
2862:
2858:
2856:
2848:
2844:
2842:
2830:
2827:
2819:
2816:
2806:
2802:
2799:
2791:
2788:
2785:
2771:
2768:
2758:
2755:
2747:
2744:
2736:
2733:
2725:
2722:
2714:
2711:
2703:
2700:
2686:
2682:
2680:
2672:
2668:
2666:
2658:
2654:
2652:
2644:
2640:
2638:
2622:
2619:
2611:
2608:
2600:
2597:
2581:
2578:
2570:
2567:
2559:
2556:
2542:
2539:
2531:
2528:
2519:
2511:
2508:
2498:
2494:
2492:
2484:
2481:
2479:
2467:
2464:
2456:
2453:
2439:
2435:
2433:
2425:
2421:
2419:
2411:
2407:
2405:
2397:
2393:
2391:
2381:
2378:
2370:
2367:
2355:
2352:
2344:
2341:
2331:
2327:
2325:
2317:
2314:
2312:
2298:
2295:
2285:
2282:
2274:
2271:
2263:
2260:
2252:
2249:
2241:
2238:
2230:
2227:
2213:
2209:
2207:
2199:
2195:
2193:
2185:
2181:
2179:
2171:
2167:
2165:
2149:
2146:
2138:
2135:
2127:
2124:
2108:
2105:
2097:
2094:
2086:
2083:
2069:
2066:
2058:
2055:
2046:
2038:
2035:
2025:
2021:
2019:
2011:
2008:
2006:
1994:
1991:
1983:
1980:
1966:
1963:
1955:
1952:
1944:
1941:
1933:
1930:
1920:
1917:
1909:
1906:
1898:
1895:
1885:
1882:
1874:
1871:
1857:
1854:
1846:
1843:
1835:
1832:
1824:
1821:
1811:
1808:
1800:
1797:
1785:
1782:
1774:
1771:
1761:
1757:
1755:
1747:
1744:
1742:
1728:
1725:
1696:ferrotorodicity
1657:
1650:
1643:
1636:
1629:
1574:
1567:
1561:
1507:
1501:
1495:
1489:
1483:
1406:
1400:
1395:
1364:
1358:
1352:
1346:
1340:
1286:
1280:
1275:
1222:
1216:
1210:
1204:
1198:
1121:
1115:
1110:
956:
950:
945:
899:
867:
823:
818:
817:
789:
784:
783:
762:
757:
756:
732:
716:
715:
683:
678:
677:
612:
607:
606:
580:
579:
560:
559:
528:
523:
522:
500:
474:
469:
468:
420:
415:
414:
392:
387:
386:
359:
358:
339:
338:
310:
305:
304:
270:
265:
264:
242:
241:
216:
211:
210:
109:
104:
96:generating sets
79:Heinrich Heesch
75:
36:symmetry groups
17:
12:
11:
5:
5453:
5451:
5443:
5442:
5437:
5432:
5422:
5421:
5418:
5417:
5397:
5375:
5374:External links
5372:
5369:
5368:
5361:
5343:
5316:(3): 286–292.
5300:
5280:
5237:
5191:
5137:
5054:
4987:
4981:978-0750626347
4980:
4949:
4898:
4855:
4812:
4769:
4700:
4661:
4646:
4619:
4576:
4561:
4537:
4486:
4463:
4437:
4386:
4371:
4350:
4343:
4317:
4295:
4273:
4250:
4207:
4188:
4185:: 15–20. 1957.
4169:
4150:
4115:(2): 145–155.
4092:
4075:
4049:
4022:(2): 107–111.
4002:
3957:(2): 148–155.
3941:
3895:
3849:
3803:
3757:
3730:
3729:
3727:
3724:
3723:
3722:
3717:
3712:
3705:
3702:
3642:
3639:
3618:
3611:
3607:
3600:
3593:
3586:
3582:
3555:
3551:
3520:
3516:
3489:
3485:
3462:
3458:
3435:
3431:
3410:
3407:
3399:incommensurate
3380:
3377:
3363:
3362:
3360:
3358:
3356:
3348:
3347:
3340:
3333:
3326:
3318:
3317:
3309:
3308:
3301:
3294:
3287:
3279:
3278:
3271:
3263:
3262:
3254:
3253:
3251:
3249:
3242:
3234:
3233:
3226:
3219:
3212:
3204:
3203:
3195:
3194:
3192:
3190:
3183:
3175:
3174:
3167:
3160:
3153:
3145:
3144:
3137:
3130:
3123:
3115:
3114:
3107:
3100:
3093:
3085:
3084:
3077:
3070:
3063:
3055:
3054:
3046:
3045:
3043:
3036:
3029:
3021:
3020:
3013:
3006:
2999:
2991:
2990:
2982:
2981:
2979:
2977:
2970:
2962:
2961:
2936:
2933:
2930:
2929:
2927:
2925:
2918:
2910:
2903:
2895:
2888:
2880:
2873:
2864:
2863:
2857:
2849:
2843:
2835:
2833:
2831:
2828:
2820:
2817:
2808:
2807:
2800:
2792:
2786:
2778:
2776:
2774:
2772:
2769:
2760:
2759:
2756:
2748:
2745:
2737:
2734:
2726:
2723:
2715:
2712:
2704:
2701:
2692:
2691:
2689:
2687:
2681:
2673:
2667:
2659:
2653:
2645:
2639:
2630:
2629:
2627:
2625:
2623:
2620:
2612:
2609:
2601:
2598:
2589:
2588:
2586:
2584:
2582:
2579:
2571:
2568:
2560:
2557:
2548:
2547:
2545:
2543:
2540:
2532:
2529:
2521:
2512:
2509:
2500:
2499:
2493:
2485:
2480:
2472:
2470:
2468:
2465:
2457:
2454:
2445:
2444:
2442:
2440:
2434:
2426:
2420:
2412:
2406:
2398:
2392:
2383:
2382:
2379:
2371:
2368:
2360:
2358:
2356:
2353:
2345:
2342:
2333:
2332:
2326:
2318:
2313:
2305:
2303:
2301:
2299:
2296:
2287:
2286:
2283:
2275:
2272:
2264:
2261:
2253:
2250:
2242:
2239:
2231:
2228:
2219:
2218:
2216:
2214:
2208:
2200:
2194:
2186:
2180:
2172:
2166:
2157:
2156:
2154:
2152:
2150:
2147:
2139:
2136:
2128:
2125:
2116:
2115:
2113:
2111:
2109:
2106:
2098:
2095:
2087:
2084:
2075:
2074:
2072:
2070:
2067:
2059:
2056:
2048:
2039:
2036:
2027:
2026:
2020:
2012:
2007:
1999:
1997:
1995:
1992:
1984:
1981:
1972:
1971:
1969:
1967:
1964:
1956:
1953:
1945:
1942:
1934:
1931:
1922:
1921:
1918:
1910:
1907:
1899:
1896:
1888:
1886:
1883:
1875:
1872:
1863:
1862:
1860:
1858:
1855:
1847:
1844:
1836:
1833:
1825:
1822:
1813:
1812:
1809:
1801:
1798:
1790:
1788:
1786:
1783:
1775:
1772:
1763:
1762:
1756:
1748:
1743:
1735:
1733:
1731:
1729:
1726:
1680:ferromagnetism
1672:ferromagnetism
1666:
1665:
1663:
1661:
1654:
1647:
1640:
1633:
1625:
1624:
1622:
1620:
1618:
1616:
1613:
1610:
1606:
1605:
1603:
1601:
1599:
1597:
1594:
1591:
1587:
1586:
1584:
1582:
1580:
1578:
1571:
1564:
1557:
1556:
1554:
1552:
1550:
1548:
1546:
1543:
1539:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1516:
1515:
1513:
1511:
1505:
1499:
1493:
1487:
1480:
1479:
1477:
1475:
1473:
1470:
1467:
1464:
1460:
1459:
1457:
1455:
1453:
1450:
1447:
1444:
1440:
1439:
1437:
1435:
1432:
1429:
1426:
1423:
1419:
1418:
1416:
1414:
1412:
1410:
1404:
1398:
1392:
1391:
1389:
1387:
1385:
1383:
1380:
1377:
1373:
1372:
1370:
1368:
1362:
1356:
1350:
1344:
1337:
1336:
1334:
1332:
1330:
1328:
1325:
1322:
1318:
1317:
1315:
1313:
1311:
1309:
1306:
1303:
1299:
1298:
1296:
1294:
1292:
1290:
1284:
1278:
1272:
1271:
1269:
1267:
1265:
1263:
1261:
1258:
1254:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1231:
1230:
1228:
1226:
1220:
1214:
1208:
1202:
1195:
1194:
1192:
1190:
1188:
1185:
1182:
1179:
1175:
1174:
1172:
1170:
1168:
1165:
1162:
1159:
1155:
1154:
1152:
1150:
1147:
1144:
1141:
1138:
1134:
1133:
1131:
1129:
1127:
1125:
1119:
1113:
1107:
1106:
1104:
1102:
1100:
1098:
1095:
1092:
1088:
1087:
1085:
1083:
1080:
1077:
1074:
1071:
1067:
1066:
1064:
1062:
1060:
1057:
1054:
1051:
1047:
1046:
1044:
1042:
1040:
1038:
1035:
1032:
1028:
1027:
1025:
1023:
1020:
1017:
1014:
1011:
1007:
1006:
1004:
1002:
1000:
998:
995:
992:
988:
987:
985:
983:
981:
979:
976:
973:
969:
968:
966:
964:
962:
960:
954:
948:
942:
941:
939:
937:
935:
933:
931:
928:
924:
923:
920:
917:
898:
895:
894:
893:
882:
879:
874:
870:
865:
861:
858:
853:
848:
845:
842:
837:
834:
828:
796:
792:
769:
765:
744:
739:
735:
730:
726:
723:
697:
694:
688:
674:
673:
662:
659:
656:
653:
650:
645:
640:
637:
634:
629:
626:
623:
617:
587:
567:
545:
542:
539:
533:
519:
518:
506:
503:
499:
496:
493:
488:
485:
479:
466:
455:
450:
445:
442:
439:
434:
431:
425:
398:
395:
383:direct product
368:
346:
324:
321:
315:
301:
300:
289:
286:
281:
275:
249:
227:
221:
205:
204:
201:
198:
195:
191:
190:
187:
184:
177:
173:
172:
165:
162:
159:
155:
154:
147:
144:
141:
137:
136:
133:
130:
127:
108:
105:
103:
100:
88:color symmetry
74:
71:
59:ferromagnetism
15:
13:
10:
9:
6:
4:
3:
2:
5452:
5441:
5438:
5436:
5433:
5431:
5428:
5427:
5425:
5408:
5407:
5402:
5398:
5388:
5387:
5382:
5378:
5377:
5373:
5364:
5358:
5354:
5347:
5344:
5339:
5335:
5331:
5327:
5323:
5319:
5315:
5311:
5304:
5301:
5293:
5292:
5284:
5281:
5276:
5272:
5268:
5264:
5260:
5256:
5252:
5248:
5241:
5238:
5233:
5229:
5225:
5221:
5217:
5213:
5209:
5205:
5201:
5195:
5192:
5187:
5183:
5179:
5175:
5171:
5167:
5162:
5157:
5153:
5149:
5141:
5138:
5133:
5129:
5125:
5121:
5116:
5111:
5107:
5103:
5099:
5095:
5091:
5087:
5082:
5077:
5073:
5069:
5065:
5058:
5055:
5050:
5046:
5041:
5036:
5032:
5028:
5024:
5020:
5015:
5010:
5006:
5002:
4998:
4991:
4988:
4983:
4977:
4973:
4969:
4968:
4963:
4959:
4953:
4950:
4945:
4941:
4937:
4933:
4929:
4925:
4921:
4917:
4913:
4909:
4902:
4899:
4894:
4890:
4886:
4882:
4878:
4874:
4870:
4866:
4859:
4856:
4851:
4847:
4843:
4839:
4835:
4831:
4827:
4823:
4816:
4813:
4808:
4804:
4800:
4796:
4792:
4788:
4784:
4780:
4773:
4770:
4765:
4761:
4757:
4753:
4749:
4745:
4741:
4737:
4733:
4729:
4724:
4719:
4715:
4711:
4704:
4701:
4696:
4692:
4688:
4684:
4680:
4676:
4672:
4665:
4662:
4657:
4650:
4647:
4642:
4638:
4634:
4630:
4623:
4620:
4615:
4611:
4607:
4603:
4599:
4595:
4591:
4587:
4580:
4577:
4572:
4568:
4564:
4558:
4551:
4550:
4541:
4538:
4533:
4529:
4525:
4521:
4517:
4513:
4509:
4505:
4501:
4497:
4490:
4487:
4483:(4): 337–361.
4482:
4478:
4474:
4467:
4464:
4450:
4449:
4441:
4438:
4433:
4429:
4425:
4421:
4417:
4413:
4409:
4405:
4401:
4397:
4390:
4387:
4382:
4378:
4374:
4368:
4364:
4357:
4355:
4351:
4346:
4340:
4336:
4332:
4328:
4321:
4318:
4305:
4299:
4296:
4284:
4277:
4274:
4269:
4265:
4261:
4254:
4251:
4246:
4242:
4238:
4234:
4230:
4226:
4222:
4218:
4211:
4208:
4203:
4199:
4192:
4189:
4184:
4180:
4173:
4170:
4165:
4161:
4154:
4151:
4146:
4142:
4138:
4134:
4130:
4126:
4122:
4118:
4114:
4110:
4106:
4099:
4097:
4093:
4088:
4087:
4079:
4076:
4071:
4067:
4060:
4053:
4050:
4045:
4041:
4037:
4033:
4029:
4025:
4021:
4017:
4013:
4006:
4003:
3998:
3994:
3990:
3986:
3982:
3978:
3974:
3970:
3965:
3960:
3956:
3952:
3945:
3942:
3937:
3933:
3929:
3925:
3921:
3917:
3913:
3910:(in German).
3909:
3905:
3899:
3896:
3891:
3887:
3883:
3879:
3875:
3871:
3867:
3864:(in German).
3863:
3859:
3853:
3850:
3845:
3841:
3837:
3833:
3829:
3825:
3821:
3818:(in German).
3817:
3813:
3807:
3804:
3799:
3795:
3791:
3787:
3783:
3779:
3775:
3772:(in German).
3771:
3767:
3761:
3758:
3753:
3749:
3742:
3735:
3732:
3725:
3721:
3718:
3716:
3713:
3711:
3708:
3707:
3703:
3701:
3699:
3694:
3693:frieze groups
3689:
3687:
3683:
3679:
3674:
3672:
3668:
3664:
3660:
3656:
3655:time crystals
3652:
3648:
3647:electron spin
3640:
3638:
3636:
3632:
3627:
3625:
3621:
3614:
3603:
3596:
3589:
3578:
3574:
3569:
3553:
3549:
3540:
3536:
3518:
3514:
3505:
3487:
3483:
3460:
3456:
3433:
3429:
3420:
3416:
3415:Landau theory
3408:
3406:
3404:
3403:quasicrystals
3400:
3396:
3392:
3386:
3378:
3376:
3374:
3370:
3361:
3359:
3357:
3354:
3350:
3349:
3345:
3341:
3338:
3334:
3331:
3327:
3324:
3320:
3319:
3316:
3315:
3310:
3306:
3302:
3299:
3295:
3292:
3288:
3285:
3281:
3280:
3277:
3276:
3270:
3269:
3264:
3261:
3260:
3255:
3252:
3250:
3247:
3243:
3240:
3236:
3235:
3231:
3227:
3224:
3220:
3217:
3213:
3210:
3206:
3205:
3202:
3201:
3196:
3193:
3191:
3188:
3184:
3181:
3177:
3176:
3172:
3168:
3165:
3161:
3158:
3154:
3151:
3147:
3146:
3142:
3138:
3135:
3131:
3128:
3124:
3121:
3117:
3116:
3112:
3108:
3105:
3101:
3098:
3094:
3091:
3087:
3086:
3082:
3078:
3075:
3071:
3068:
3064:
3061:
3057:
3056:
3053:
3052:
3047:
3044:
3041:
3037:
3034:
3030:
3027:
3023:
3022:
3018:
3014:
3011:
3007:
3004:
3000:
2997:
2993:
2992:
2989:
2988:
2983:
2980:
2978:
2975:
2971:
2968:
2964:
2963:
2960:
2959:
2954:
2948:
2946:
2942:
2934:
2928:
2926:
2915:
2911:
2900:
2896:
2885:
2881:
2870:
2866:
2865:
2854:
2850:
2840:
2836:
2834:
2832:
2825:
2821:
2814:
2810:
2809:
2797:
2793:
2783:
2779:
2777:
2775:
2773:
2766:
2762:
2761:
2753:
2749:
2742:
2738:
2731:
2727:
2720:
2716:
2709:
2705:
2698:
2694:
2693:
2690:
2688:
2678:
2674:
2664:
2660:
2650:
2646:
2636:
2632:
2631:
2628:
2626:
2624:
2617:
2613:
2606:
2602:
2595:
2591:
2590:
2587:
2585:
2583:
2576:
2572:
2565:
2561:
2554:
2550:
2549:
2546:
2544:
2537:
2533:
2526:
2522:
2517:
2513:
2506:
2502:
2501:
2490:
2486:
2477:
2473:
2471:
2469:
2462:
2458:
2451:
2447:
2446:
2443:
2441:
2431:
2427:
2417:
2413:
2403:
2399:
2389:
2385:
2384:
2376:
2372:
2365:
2361:
2359:
2357:
2350:
2346:
2339:
2335:
2334:
2323:
2319:
2310:
2306:
2304:
2302:
2300:
2293:
2289:
2288:
2280:
2276:
2269:
2265:
2258:
2254:
2247:
2243:
2236:
2232:
2225:
2221:
2220:
2217:
2215:
2205:
2201:
2191:
2187:
2177:
2173:
2163:
2159:
2158:
2155:
2153:
2151:
2144:
2140:
2133:
2129:
2122:
2118:
2117:
2114:
2112:
2110:
2103:
2099:
2092:
2088:
2081:
2077:
2076:
2073:
2071:
2064:
2060:
2053:
2049:
2044:
2040:
2033:
2029:
2028:
2017:
2013:
2004:
2000:
1998:
1996:
1989:
1985:
1978:
1974:
1973:
1970:
1968:
1961:
1957:
1950:
1946:
1939:
1935:
1928:
1924:
1923:
1915:
1911:
1904:
1900:
1893:
1889:
1887:
1880:
1876:
1869:
1865:
1864:
1861:
1859:
1852:
1848:
1841:
1837:
1830:
1826:
1819:
1815:
1814:
1806:
1802:
1795:
1791:
1789:
1787:
1780:
1776:
1769:
1765:
1764:
1753:
1749:
1740:
1736:
1734:
1732:
1730:
1723:
1719:
1718:
1715:
1712:
1707:
1705:
1701:
1697:
1693:
1689:
1685:
1681:
1677:
1673:
1664:
1662:
1655:
1648:
1641:
1634:
1627:
1626:
1623:
1621:
1619:
1617:
1614:
1611:
1608:
1607:
1604:
1602:
1600:
1598:
1595:
1592:
1589:
1588:
1585:
1583:
1581:
1579:
1572:
1565:
1559:
1558:
1555:
1553:
1551:
1549:
1547:
1544:
1541:
1540:
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1517:
1514:
1512:
1506:
1500:
1494:
1488:
1482:
1481:
1478:
1476:
1474:
1471:
1468:
1465:
1462:
1461:
1458:
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41:
40:electron spin
37:
33:
31:
26:
22:
5440:Group theory
5410:. Retrieved
5404:
5390:. Retrieved
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3720:Group theory
3690:
3675:
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3631:Aizu species
3630:
3628:
3570:
3412:
3398:
3394:
3391:commensurate
3390:
3388:
3372:
3366:
3312:
3273:
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3049:
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675:
520:
302:
208:
161:Grey groups
135:Description
110:
76:
52:
28:
24:
18:
3710:Space group
3698:more colors
782:. Here the
712:translation
102:Description
5424:Categories
5412:2020-01-22
5392:2020-01-22
5161:2003.00012
5081:1707.01903
5014:2010.00598
4958:Lev Landau
4457:2020-01-17
3964:1609.09666
3904:Heesch, H.
3858:Heesch, H.
3812:Heesch, H.
3766:Heesch, H.
3726:References
3383:See also:
1684:admissible
34:, are the
5338:0567-7394
5275:0921-4526
5232:0021-8898
5186:226036258
5106:2375-2548
4936:0108-7673
4893:0556-2805
4850:0022-3697
4807:0031-899X
4748:0953-8984
4723:1107.2358
4641:0038-5638
4614:0002-9505
4532:120569385
4524:0953-8984
4424:0108-7673
4381:859155300
4260:Magnetism
4245:0022-2488
4137:0108-7673
4072:(3): 430.
4044:118874086
4036:0370-0089
3997:118533941
3989:0885-7156
3936:102161512
3928:2196-7105
3890:102161514
3882:2196-7105
3844:101972126
3836:2196-7105
3798:102004261
3790:2196-7105
1531:6/m'm'm'
1525:6'/m'mm'
1252:4'/m'm'm
1246:4/m'm'm'
655:−
498:×
176:Type III
149:Ordinary
30:Shubnikov
5178:33116291
5132:51910083
5124:30083612
5049:34645841
4964:(1960).
4944:18285626
4764:11738423
4756:22447842
4571:44883836
4432:21694481
4268:31184704
4204:: 10–15.
4145:19225196
3754:: 57–60.
3704:See also
3504:subgroup
2735:6/m'm'm'
2724:6'/m'mm'
2273:4'/m'm'm
2240:4/m'm'm'
1537:6'/mmm'
1528:6/mm'm'
1522:6/mmm1'
1243:4/mm'm'
1240:4'/mmm'
1237:4/mmm1'
505:′
397:′
194:Type IV
158:Type II
117:online.
5318:Bibcode
5255:Bibcode
5212:Bibcode
5115:6070365
5086:Bibcode
5040:8514474
5019:Bibcode
4916:Bibcode
4873:Bibcode
4830:Bibcode
4787:Bibcode
4728:Bibcode
4683:Bibcode
4594:Bibcode
4504:Bibcode
4404:Bibcode
4225:Bibcode
4117:Bibcode
3969:Bibcode
3635:ferroic
2757:6/mm'm'
2713:6'/mmm'
2284:4/mm'm'
2262:4'/mmm'
1534:6/m'mm
1249:4/m'mm
1079:m'm'm'
140:Type I
73:History
5359:
5336:
5273:
5230:
5184:
5176:
5148:Nature
5130:
5122:
5112:
5104:
5047:
5037:
4978:
4974:–119.
4942:
4934:
4891:
4848:
4805:
4762:
4754:
4746:
4639:
4612:
4569:
4559:
4530:
4522:
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4379:
4369:
4341:
4310:14 Apr
4288:14 Apr
4266:
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4143:
4135:
4042:
4034:
3995:
3987:
3934:
3926:
3888:
3880:
3842:
3834:
3796:
3788:
3597:, and
2746:6/m'mm
2251:4/m'mm
1943:m'm'm'
1856:2'/m'
1615:4'3m'
1612:43m1'
1596:4'32'
1593:4321'
1519:6/mmm
1472:6m'm'
1469:6'mm'
1466:6mm1'
1452:62'2'
1449:6'22'
1446:6221'
1428:6'/m'
1425:6/m1'
1234:4/mmm
1187:4m'm'
1184:4'mm'
1181:4mm1'
1167:42'2'
1164:4'22'
1161:4221'
1149:4'/m'
1140:4/m1'
1076:mm'm'
1073:mmm1'
1059:2'm'm
1056:m'm'2
1053:mm21'
1037:2'2'2
1034:2221'
1016:2'/m'
1013:2/m1'
32:groups
23:, the
5295:(PDF)
5182:S2CID
5156:arXiv
5128:S2CID
5076:arXiv
5009:arXiv
4760:S2CID
4718:arXiv
4553:(PDF)
4528:S2CID
4452:(PDF)
4062:(PDF)
4040:S2CID
3993:S2CID
3959:arXiv
3932:S2CID
3886:S2CID
3840:S2CID
3794:S2CID
3748:Forma
3744:(PDF)
3502:is a
2829:4'32'
2702:6/mmm
2685:'m'2
2671:'m2'
2657:m'2'
2621:6'mm'
2610:6m'm'
2580:6'2'2
2569:62'2'
2530:6'/m'
2229:4/mmm
2212:'2'm
2198:'2m'
2148:4'mm'
2137:4m'm'
2107:42'2'
2096:4'22'
2057:4'/m'
1965:m'm'm
1919:mm'2'
1908:m'm'2
1884:2'2'2
1845:2'/m
1834:2/m'
1545:231'
1510:m'2'
1504:'m2'
1498:'2m'
1492:m21'
1434:6'/m
1431:6/m'
1324:3m1'
1305:321'
1225:2'm'
1219:'m2'
1213:'2m'
1207:2m1'
1146:4/m'
1143:4'/m
1082:mmm'
1022:2'/m
1019:2/m'
600:index
598:with
129:Name
126:Type
27:, or
5357:ISBN
5334:ISSN
5271:ISSN
5228:ISSN
5174:PMID
5120:PMID
5102:ISSN
5045:PMID
4976:ISBN
4940:PMID
4932:ISSN
4889:ISSN
4846:ISSN
4803:ISSN
4752:PMID
4744:ISSN
4637:ISSN
4610:ISSN
4567:OCLC
4557:ISBN
4549:2000
4520:ISSN
4428:PMID
4420:ISSN
4377:OCLC
4367:ISBN
4339:ISBN
4312:2019
4290:2019
4264:OCLC
4241:ISSN
4141:PMID
4133:ISSN
4032:ISSN
3985:ISSN
3924:ISSN
3878:ISSN
3832:ISSN
3786:ISSN
3617:FeCO
3615:and
3599:CoCO
3592:MnCO
3413:The
2861:'3m'
2541:6/m'
2520:6/m'
2184:2'm'
2068:4/m'
2047:4/m'
1954:mmm'
1873:222
1823:2/m
1653:'m'
1639:m1'
1609:43m
1590:432
1463:6mm
1443:622
1422:6/m
1379:61'
1361:'m'
1349:m1'
1327:3m'
1308:32'
1260:31'
1178:4mm
1158:422
1137:4/m
1094:41'
1070:mmm
1050:mm2
1031:222
1010:2/m
994:m1'
975:21'
200:517
186:674
164:230
146:230
5326:doi
5263:doi
5251:192
5220:doi
5166:doi
5152:586
5110:PMC
5094:doi
5035:PMC
5027:doi
4972:116
4924:doi
4881:doi
4838:doi
4795:doi
4783:130
4736:doi
4691:doi
4602:doi
4512:doi
4412:doi
4331:doi
4233:doi
4125:doi
4024:doi
3977:doi
3916:doi
3870:doi
3824:doi
3778:doi
3684:or
3541:of
3506:of
2893:'m'
2818:432
2643:m2
2599:6mm
2558:622
2510:6/m
2424:'m'
2380:3m'
2354:32'
2170:2m
2126:4mm
2085:422
2037:4/m
1932:mmm
1897:mm2
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1702:or
1660:'m
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1570:1'
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