Magnetic Point Groups and Space Groups 219

gJ Lande´ factor Ms,0 magnetization at absolute saturation Happ applied magnetic field (vector) n molecular field coefficient Happ applied magnetic field (magnitude) N number of moments per unit volume Hm molecular field ND demagnetizing field coefficient J exchange integral NT total number of moments J total angular momentum S spin or spin angular momentum k0 propagation vector T temperature l orbital quantum number Tc critical temperature l orbital momentum TC Curie temperature L orbital momentum TN Neél temperature L Langevin function vat atom volume m magnetic quantum number x argument of the Langevin or Brillouin m0 measured magnetic moment at 0 K function m magnetic moment z number of first neighbors ml orbital moment w magnetic susceptibility ms spin moment mB Bohr magneton mz projection of m along Happ meff effective moment M magnetization m0 permittivity of vacuum Ms spontaneous magnetization

Introduction (a) Magnetic groups – also known as antisymmetry groups, Shubnikov groups, Heesch groups, Opechow- (b) ski–Guccione (OG) groups, as well as dichromatic, 2-color, or simply black-and-white groups – are the simplest extension to standard point group and space group theory. They allow directly to describe, clas- (c) sify, and study the consequences of the symmetry of crystals, characterized by having a certain property, Na Cl Na Cl Na Cl Na Cl Na Cl Na Cl associated with each crystal site, that can take one of (d) two possible values. This is done by introducing a single external operation of order two that interchanges the two possible values everywhere through- (e) out the crystal. This operation can be applied to the Figure 1 Several realizations of the simple 1D magnetic space crystal along with any of the standard point group group pp 1. Note that the number of symmetry translations is or space group operations, and is denoted by adding doubled if one introduces an operation e0 that interchanges the a prime to the original operation. Thus, any rota- two possible values of the property, associated with each crystal tion g followed by this external operation is denoted site. Also note that spatial inversion 1 can be performed without a 0 by g0. prime on each crystal site, and with a prime ð1 Þ between every two sites. (a) An abstract representation of the two possible val- To start with, a few typical examples of this two- ues as the two colors – black and white. The operation e0 is the valued property are given, some of which are illus- nontrivial permutation of the two colors. (b) A simple antifer- trated in Figure 1. In the section ‘‘Magnetic point romagnetic arrangement of spins. The operation e0 is time groups’’ the notion of a magnetic point group is dis- inversion which reverses the signs of the spins. (c) A ferromagne- 0 cussed, followed by a discussion on magnetic space tic arrangement of two types of spins, where e exchanges them as in the case of two colors. (d) A 1D version of salt. e0 ex- groups in the section ‘‘Magnetic space groups’’. The changes the chemical identities of the two atomic species. (e) A section ‘‘Extinctions in neutron diffraction of anti- function f(x) whose overall sign changes by application of the ferromagnetic crystals’’ describes one of the most operation e0. 220 Magnetic Point Groups and Space Groups direct consequences of having magnetic symmetry in applies equally to all the other two-valued properties. crystals which is the extinction of magnetic Bragg The magnetic crystal is described using a scalar spin peaks in neutron diffraction patterns. A conclusion density field S(r) whose magnitude gives the magnit- is given in the section ‘‘Generalizations of magnetic ude of an atomic magnetic moment, or some coarse- groups’’ by mentioning the generalization of magne- grained value of the magnetic moment, at position r. tic groups to cases where the property associated The sign of S(r) gives the direction of the spin – a with each crystal site is not limited to having only positive sign for up spins and a negative sign for one of two possible values. down spins. Thus, the function S(r) can be a discrete Consider first the structure of crystals of the set of delta functions defined on the crystal sites as in cesium chloride type. In these crystals the atoms Figure 1b, or a continuous spin density field as in are located on the sites of a body-centered cubic Figure 1e. (b.c.c.) lattice. Atoms of one type are located on the simple cubic lattice formed by the corners of the cu- Magnetic Point Groups bic cells, and atoms of the other type are located on the simple cubic lattice formed by the body centers of A d-dimensional magnetic point group GM is a the cubic cells. The parent b.c.c. lattice is partitioned subgroup of OðdÞ10, where O(d) is the group of in this way into two identical sublattices, related by a d-dimensional rotations, and 10 is the time inversion body-diagonal translation. One may describe the group containing the identity e and the time invers- symmetry of the cesium chloride structure as having ion operation e0. Note that the term ‘‘rotation’’ is a simple cubic lattice of ordinary translations, with a used to refer to proper as well as improper rotations basis containing one cesium and one chlorine atom. such as mirrors. Alternatively, one may describe the symmetry of the Three cases exist: (1) all rotations in GM appear crystal as having twice as many translations, forming with and without time inversion; (2) half of the ro- a b.c.c. lattice, half of which are primed to indicate tations in GM are followed by time inversion and the that they are followed by the external operation that other half are not; and (3) no rotation in GM is fol- exchanges the chemical identities of the two types of lowed by time inversion. More formally, if G is any atoms. A similar situation occurs in crystals whose subgroup of O(d), it can be used to form at most structure is of the sodium chloride type, in which a three types of magnetic point groups, as follows: simple cubic lattice is partitioned into two identical 0 face-centered cubic (f.c.c.) sublattices. 1. GM ¼ G 1 . Thus, each rotation in G appears in Another typical example is the orientational or- GM once by itself, and once followed by time 0 dering of magnetic moments (spins), electric dipole inversion. Note that in this case e AGM. 0 moments, or any other tensorial property associated 2. GM ¼ H þ g H; where G ¼ H þ gH; and geH: with each site in a crystal. Adopting the language of Thus, exactly half of the rotations in G, which spins, if these can take only one of two possible belong to its subgroup H, appear in GM by them- orientations – ‘‘up’’ and ‘‘down’’ – as in simple an- selves, and the other half, belonging to the coset tiferromagnets, the same situation as above is seen, gH, are followed by time inversion. Note that in 0 with the two spin orientations replacing the two this case e eGM. chemical species of atoms. In the case of spins, the 3. GM ¼ G: Thus, GM contains rotations, none of external prime operation is a so-called antisymmetry which are followed by time inversion. operation that reverses the signs of the spins. Phys- ically, one may think of using the operation of time Enumeration of magnetic point groups is thus inversion to reverse the directions of all the spins straightforward. Any ordinary point group G (i.e., without affecting anything else in the crystal. any subgroup, usually finite, of O(d)) is trivially a Finally, consider a periodic or quasiperiodic scalar magnetic point group of type 3, and along with the function of space f(r), as in Figure 1e, whose average time inversion group gives another magnetic point value is zero. It might be possible to extend the point group of type 1, denoted as G10: One then lists all group or the space group describing the symmetry of distinct subgroups H of index 2 in G, if there are any, the function f(r) by introducing an external prime to construct additional magnetic point groups of type operation that changes the sign of f(r). 2. These are denoted either by the group–subgroup For the sake of clarity, the picture of up-and-down pair G(H), or by using the International (Hermann– spins, and the use of time inversion to exchange them Mauguin) symbol for G and adding a prime to those shall be adopted for the remaining of the discussion. elements in the symbol that do not belong to It should be emphasized, though, that most of what the subgroup H (and are therefore followed by time is said here (except for the discussion of extinctions) inversion). The set of magnetic groups (here magnetic Magnetic Point Groups and Space Groups 221

point groups GM), formed in this way from a single is of type 1 if time inversion by itself leaves the crys- ordinary group (here an ordinary point group G), tal invariant to within a translation. Recall that in is sometimes called the magnetic superfamily of the this case, any rotation in the magnetic point group group G. For example, the orthorhombic point group can be performed either with or without time invers- G ¼ mm2 ¼fe; mx; my; 2zg hasÈÉ three subgroups of ion. If time inversion cannot be combined with a index 2 ðH1 ¼ fge; mx , H2 ¼ e; my , and H3 ¼ translation to leave the crystal invariant, the magne- fgÞe; 2z , of which the first two are equivalent through tic point group is of type 2, in which case half the a relabeling of the x and y axes. This yields a total of rotations are performed without time inversion and four magnetic point groups: mm2, mm210, m0m20 the other half with time inversion. (equivalently expressed as mm020), and m0m02. Figure 2a shows an example of a magnetically or- Magnetic point groups of type 3 are equivalent to dered crystal whose magnetic point group is of type 1. ordinary nonmagnetic point groups and are listed This is a square crystal with magnetic point group 0 here only for the sake of completeness. Strictly GM ¼ 4mm1 where time inversion can be followed speaking, they can be said to describe the symmetry by a translation to leave the crystal invariant. Figure 2b of ferromagnetic structures with no spin–orbit coup- shows an example of a crystal whose magnetic point ling. Groups of this type are not considered here group is of type 2. This is a hexagonal crystal with 0 0 any further. Magnetic point groups of type 2 can be point group GM ¼ 6 mm . Note that all right-side-up used to describe the point symmetry of finite objects, triangles contain a blue circle (spin up), and all as described in the next section, as well as that up-side-down triangles contain a green circle (spin of infinite crystals, as described in the section down). Time inversion, exchanging the two types of ‘‘Magnetic point groups of crystals.’’ Magnetic point spins, cannot be combined with a translation to groups of type 1 can be used to describe the point recover the original crystal. Time inversion must be symmetry of infinite crystals, but because they in- combined with an operation, such as the sixfold rota- clude e0 as a symmetry element they cannot be used tion or the horizontal mirror that interchanges the to describe the symmetry of finite objects, as is now two types of triangles, to recover the original crystal. defined. Note that the vertical mirror (the first m in the International symbol) leaves the crystal invariant Magnetic Point Groups of Finite Objects without requiring time inversion, and is therefore unprimed in the symbol. The magnetic symmetry of a finite object, such as a More generally, the magnetic point group of a d- molecule or a cluster containing an equal number of dimensional magnetically ordered ‘‘quasiperiodic two types of spins, can be described by a magnetic crystal (quasicrystal),’’ is defined as the set of primed point group. One can say that a primed or unprimed or unprimed rotations from OdðÞ10 that leave rotation from OðdÞ10 is in the magnetic point the crystal ‘‘indistinguishable.’’ This means that the group G of a finite object in d dimensions, if it M rotated and unrotated crystals contain the same spa- leaves the object invariant. Clearly, only magnetic tial distributions of finite clusters of spins of arbitrary point groups of type 2, listed above, can describe the size. The two are statistically the same though not symmetry of finite magnetically ordered structures. necessarily identical. For the special case of perio- This is because time inversion e0 changes the orientat- dic crystals, the requirement of indistinguishability ions of all the spins, and therefore cannot leave reduces to the requirement of invariance to within a the object invariant unless it is accompanied by a translation. nontrivial rotation. It should be mentioned that in Figure 3 shows two quasiperiodic examples, ana- this context, point groups of type 1 are sometimes logous to the two periodic examples of Figure 2. called ‘‘gray’’ groups, as they describe the symmetry Figure 3a shows an octagonal crystal with magnetic of ‘‘gray’’ objects which stay invariant under the point group G ¼ 8mm10. One can see that time exchange of black and white. M inversion rearranges the spin clusters in the crystal, but they all still appear in the crystal with the same Magnetic Point Groups of Crystals spatial distribution. This is because any finite spin The magnetic point group of a d-dimensional magne- cluster and its time-reversed image appear in the tically ordered ‘‘periodic crystal’’, containing an crystal with the same spatial distribution. Figure 3b equal number of up and down spins, is defined as shows a decagonal crystal with magnetic point group 0 0 the set of primed or unprimed rotations from OdðÞ GM ¼ 10 m m. In this case, time inversion does not 10 that leave the crystal ‘‘invariant to within a trans- leave the crystal indistinguishable. It must be com- lation’’. The magnetic point group of a crystal can be bined either with odd powers of the tenfold rotation, either of the first or of the second type listed above. It or with mirrors of the vertical type. 222 Magnetic Point Groups and Space Groups

(a) (b)

0 Figure 2 Periodic antiferromagnets. (a) A square crystal with magnetic point group GM ¼ 4mm1 and magnetic space group pp 4mm 0 0 0 0 (also denoted pc 4mm), (b) A hexagonal crystal with magnetic point group GM ¼ 6 mm and magnetic space group p6 mm .

(a) (b)

0 Figure 3 Quasiperiodic antiferromagnets. (a) An octagonal quasicrystal with magnetic point group GM ¼ 8mm1 and magnetic space group pp8mm, ﬁrst appeared in an article by Niizeki (1990) Journal of Physics A: Mathematical and General 23: 5011. (b) A decagonal 0 0 0 0 quasicrystal with magnetic point group GM ¼ 10 m m and magnetic space group p10 m m based on the tiling of Li, Dubois, and Kuo (1994) Philosophical Magazine Letters 69: 93. Magnetic Point Groups and Space Groups 223

Magnetic Space Groups crystal class’’ (i.e., have the same magnetic point group), but to two distinct ‘‘magnetic arithmetic crys- The full symmetry of a magnetically ordered crystal, tal classes.’’ Translation-equivalent magnetic space described by a scalar spin density field S( ), is given by r groups are denoted by taking the International sym- its magnetic space group G . It was mentioned earlier M bol for G, and adding a prime to those elements in the that the magnetic point group G is the set of primed M symbol that do not belong to H and are therefore or unprimed rotations that leave a periodic crystal followed by time inversion. invariant to within a translation, or more generally, Enumeration of class-equivalent magnetic space leave a quasiperiodic crystal indistinguishable. One group types can proceed in two alternative ways. still needs to specify the distinct sets of translations tg 0 Given a space group G, one may consider all distinct or t 0 that can accompany the rotations g or g in a g sublattices of index 2 of the Bravais lattice of G. periodic crystal to leave it invariant, or provide the Alternatively, given a space group H, one may con- more general characterization of the nature of the sider all distinct superlattices of index 2 of the indistinguishability in the case of quasicrystals. Bravais lattice of H. It is also to be noted that if one Periodic Crystals prefers to look at the reciprocal lattices in Fourier space, instead of the lattices of real-space transla- As in the case of point groups, one can take any space tions, the roles of superlattice and sublattice are group G of a periodic crystal and form its magnetic exchanged. Although the lattice of G contains twice superfamily, consisting of one trivial group GM ¼ G, as many points as the lattice of H, the reciprocal 0 one gray group GM ¼ G1 , and possibly nontrivial lattice of G contains exactly half of the points of the 0 magnetic groups of the form GM ¼ H þðg tg0 ÞH, reciprocal lattice of H. where H is a subgroup of index 2 of G. Only the Because of the different enumeration methods, latter, nontrivial groups are relevant as magnetic there are conflicting notations in the literature for space groups consisting of operations that leave the class-equivalent magnetic space groups. The OG no- magnetically ordered crystal ‘‘invariant.’’ These non- tation follows the first approach (sublattices of G), trivial magnetic space groups are divided into two taking the International symbol for the space group types depending on their magnetic point groups GM: G with a subscript on the Bravais class symbol denoting the Bravais lattice of H. The Belov notation 1. Class-equivalent magnetic space groups. The follows the second approach (superlattices of H), magnetic point group GM is of type 1, and there- taking the International symbol for the space group fore all rotations in G appear with and without H with a subscript on the Bravais class symbol time inversion. In this case the point groups of G denoting one or more of the primed translations te0 in 0 and H are the same, but the Bravais lattice of H the coset ðÞe jte0 H. The Lifshitz notation, which fol- contains exactly half of the translations that are in lows a third approach (sublattices of H in Fourier the Bravais lattice of G. space), resembles that of Belov but generalizes more 2. Translation-equivalent magnetic space groups. easily to quasiperiodic crystals. The magnetic point group GM is of type 2. Here There is an additional discrepancy in the different the Bravais lattices of G and H are the same, but notations which has been the cause of some confu- the point group H of H is a subgroup of index 2 sion over the years. It is due to the fact that when 0 of the point group G of G. e AGM, ordinary mirrors or rotations when unprimed may become glide planes or screw axes when Enumeration of translation-equivalent magnetic primed, and vice versa. The only consistent way to space group types is very simple. Given a space group avoid this confusion is to use only unprimed elements G, all that one needs to do is to consider all for the symbols of class-equivalent magnetic space subgroups H of index 2 of the point group G of G, group types. This is the approach adopted by both as explained earlier. The only additional considera- the Belov and the Lifshitz notations, but unfortu- tion is that with an underlying Bravais lattice, there nately not by the OG notation, having introduced may be more than one way to orient the subgroup H errors into their list of magnetic space group types. In relative to the lattice. For example, consider the any case, there is no need to leave the 10 at the end of magnetic point group m0m20 which, as noted earlier, the symbol, as it is clear from the existence of a sub- is equivalent to mm020. If the lattice is an A-centered script on the lattice symbol that the magnetic point 0 orthorhombic lattice, then the x- and y-axes are no group GM contains e . longer equivalent, and one needs to consider both Consider, for example, all the class-equivalent options as distinct groups. It can be said that Am0m20 magnetic space group types with point group 432. and Amm020 belong to the same ‘‘magnetic geometric In the cubic system there are two lattice–sublattice 224 Magnetic Point Groups and Space Groups pairs: (1) The simple cubic lattice P has a face-cen- vectors needed to generate L by integral linear com- tered sublattice F of index 2, and (2) The b.c.c. lattice binations. If D is finite the crystal is quasiperiodic. If, I has a simple cubic sublattice P of index 2. Recall in particular, D is equal to the number d of spatial that the reciprocal of F is a b.c.c. lattice in Fourier dimensions, and L extends in all d directions, the space, denoted by I, and the reciprocal of I is an crystal is periodic. In this special case, the magnetic f.c.c. lattice, denoted by F. In the Belov notation one lattice L is reciprocal to a lattice of translations in real has Fs432, Fs4j32, PI432, and PI4j32 with j ¼ 1; 2; 3: space that leave the magnetic crystal invariant. The The subscript s is for ‘‘simple’’. In the Lifshitz nota- reciprocal lattices L of magnetic crystals are classified tion, using Fourier space lattices, these groups be- into Bravais classes, just like those of ordinary non- come IP432, IP4j32, PF 432, and PF 4j32. In the OG magnetic crystals. notation these groups should become PF432, PF4j32, IP432, and IP4j32. Instead, they list PF432, PF4232, Phase functions and space group types The precise 0 0 0 0 IP432, IP4132; IP4 32 ,andIP4132 , using primes mathematical statement of indistinguishability, used inconsistently and clearly missing two groups. It earlier to define the magnetic point group, is the is therefore recommended that one refrains from requirement that any symmetry operation of the using primes when denoting class-equivalent magne- magnetic crystal leaves invariant all spatially aver- tic space groups. aged nth-order autocorrelation functions of its spin- The reader is referred to the ‘‘Further reading’’ density field, section for complete lists of magnetic space group CðÞn ðr ; y; r Þ types. However, some statistics are summarized. 1 nZ Starting from the 17 space group types in two dimen- 1 ¼ lim drSðr rÞ?Sðr rÞ½2 - 1 n sions, also known as the 17 plane groups, one can V N V V form a total of 80 magnetic space group types as One can prove that two quasiperiodic spin density follows: 17 trivial groups, 17 gray groups, and 46 fields S(r) and SˆðrÞ are indistinguishable in this way, nontrivial groups of which 18 are class-equivalent if their Fourier coefficients, defined in eqn [1], are and 28 are translation equivalent. Starting from the related by 230 space group types in three dimensions, one can 2piwðkÞ form a total of 1651 magnetic space group types as SˆðkÞ¼e SðkÞ½3 follows: 230 trivial groups, 230 gray groups, and where w is a real-valued linear function (modulo 1191 nontrivial groups of which 517 are class- integers) on L, called a ‘‘gauge function’’. By this it is equivalent and 674 are translation equivalent. These meant that for every pair of wave vectors k and k in numbers have no particular significance other than 1 2 the magnetic lattice L, the fact that they are surprisingly large. wðk1 þ k2Þwðk1Þþwðk2Þ½4 Quasiperiodic Crystals where ‘‘’’ means equal to within adding an integer. Lattices and Bravais classes Magnetically-ordered Thus, for each element g or g0 in the magnetic point quasiperiodic crystals, in general, possess no lattices group GM of the crystal, that by definition leaves the of real space translations that leave them invariant, crystal indistinguishable, there exists such a gauge but they do have reciprocal lattices (or Fourier mod- function FgðkÞ or Fg0 ðkÞ, called a ‘‘phase function’’, ules) L that can be inferred directly from their neu- satisfying tron diffraction diagrams. To be a bit more specific, ( 2piFgðkÞ consider spin density fields with well-defined Fourier e SðkÞ; gAGM SðgkÞ¼ ½5 transforms 2piFg0 ðkÞ 0A X e SðkÞ; g GM ik . r SðrÞ¼ SðkÞe ½1 Since for any g,hAG, Sð½ghkÞ¼Sðg½hkÞ, the cor- kAL responding phase functions for elements in GM, in which the set L contains, at most, a countable whether primed or not, must satisfy the ‘‘group com- infinity of plane waves. In fact, the ‘‘reciprocal lat- patibility condition’’, tice’’ L of a magnetic crystal can be defined as the set Fg hwðkÞFgðhkÞþFhwðkÞ½6 of all integral linear combinations of wave vectors k determined from its neutron diffraction diagram, as where the asterisk and the dagger denote optional one expects to see a magnetic Bragg peak at every primes. A ‘‘magnetic space group,’’ describing the linear combination of observed peaks, unless it is symmetry of a magnetic crystal, whether periodic or forbidden by symmetry, as explained in the next aperiodic, is thus given by a lattice L, a magnetic section. The ‘‘rank’’ D of L is the smallest number of point group GM, and a set of phase functions FgðkÞ Magnetic Point Groups and Space Groups 225

and Fg0 ðkÞ, satisfying the group compatibility condi- referred to references in the ‘‘Further reading’’ section tion [6]. for details regarding the enumeration of magnetic Magnetic space groups are classified in this for- space groups of quasiperiodic crystals and for com- malism into magnetic space group types by organ- plete lists of such groups. izing sets of phase functions satisfying the group compatibility condition [6] into equivalence classes. Two such sets, F and F# , are equivalent if: (1) they Extinctions in Neutron Diffraction of describe indistinguishable spin density fields related Anti-Ferromagnetic Crystals as in [3] by a gauge function w; or (2) they correspond It was said earlier that every wave vector k in the to alternative descriptions of the same crystal that lattice L of a magnetic crystal is a candidate for a differ by their choices of absolute length scales and # diffraction peak unless symmetry forbids it. One can spatial orientations. In case (1), F and F are related now understand exactly how this happens. Given by a ‘‘gauge transformation,’’ a wave vector kAL, all magnetic point-group opera- # tions g or g0 for which gk ¼ k are examined. For FgðkÞFgðkÞþwðgk kÞ½7 such elements, the point-group condition [5] can be where, again, the asterisk denotes an optional prime. rewritten as ( In the special case that the crystal is periodic 2piFgðkÞ e SðkÞ; gAGM ðD ¼ dÞ, it is possible to replace each gauge function SðkÞ¼ ½9 2piF 0 ðkÞ 0 by a corresponding d-dimensional translation t, sat- e g SðkÞ; g AGM isfying 2pwðkÞ¼k t, thereby reducing the requirement of indistinguishability to that of invariance to requiring S(k) to vanish unless FgðkÞ0, or within a translation. Only then does the point group Fg0 ðkÞ1=2, or unless both conditions are satisfied 0 condition [5] become when both g and g are in GM. It should be noted that ( the phase values in eqn [9], determining the extinc- Sðr tgÞ; gAGM tion of S(k), are independent of the choice of gauge SðgrÞ¼ 0 ½8 0 A Sðr tg Þ; g GM [7], and are therefore uniquely determined by the magnetic space-group type of the crystal. the gauge transformation [7] becomes a mere shift of Particularly striking are the extinctions when the the origin, and the whole description reduces to that magnetic point group is of type 1, containing time given in the section ‘‘Periodic crystals.’’ The reader is inversion e0. The relation ðe0Þ2 ¼ e implies – through

(a) (b) Figure 4 Extinctions of magnetic Bragg peaks in crystals with magnetic point groups of type 1, containing time inversion e0. Both ﬁgures show the positions of expected magnetic Bragg peaks indexed by integers ranging from –4 to 4. Filled circles indicate observed peaks and open circles indicate positions of extinct peaks. The origin and the D vectors used to generate the patterns are indicated by an additional large circle. (a) corresponds to the square crystal in Figure 2a with magnetic space group pp4mm, and (b) corresponds to the octagonal crystal in Figure 3a with magnetic space group pp8mm. 226 Magnetic Point Groups and Space Groups the group compatibility condition [6] and the fact inversion. Here as well, elements of the form ðe; gÞ that FeðkÞ0 – that Fe0 ðkÞ0or1=2. It then fol- play a central role in the theory, especially in deter- lows from the linearity of the phase function that on mining the symmetry constraints imposed by the g exactly half of the wave vectors in L Fe0 ðkÞ1=2, corresponding phase functions FeðkÞ on the patterns and on the remaining half, which form a sublattice of magnetic Bragg peaks, observed in elastic neutron L0 of index 2 in L, Fe0 ðkÞ0. Thus, in magnetic diffraction experiments. crystals with magnetic point groups of type 1 at least half of the diffraction peaks are missing. Figure 4 shows the positions of the expected magnetic Bragg Acknowledgments peaks corresponding to the square and octagonal The author would like to thank David Mermin for magnetic structures shown in Figures 2a and 3a, teaching him the symmetry of crystals, and Shahar illustrating this phenomenon. Even-Dar Mandel for joining him in his investigation of the magnetic symmetry of quasicrystals. The au- Generalizations of Magnetic Groups thor’s research in these matters is funded by the Israel Science Foundation through grant No. 278/00. There are two natural generalizations of magnetic groups. One is to color groups with more than two See also: Crystal Symmetry; Disordered Magnetic Sys- colors, and the other is to spin groups where the tems; Magnetic Materials and Applications; Magnetoelas- spins are viewed as classical axial vectors free to ticity; Paramagnetism; Periodicity and Lattices; Point rotate continuously in any direction. Groups; Quasicrystals; Space Groups. An n-color point group GC is a subgroup of OðdÞSn, where Sn is the permutation group of n PACS: 61.50.Ah; 61.12.Bt; 75.25. þ z; 61.44.Br; colors. Elements of the color point group are pairs 75.50.Kj; 61.68. þ n; 02.20.Bb; 75.10. b; 75.50.Ee ðg; gÞ where g is a d-dimensional (proper or improper) rotation and g is a permutation of the n colors. As before, for ðg; gÞ to be in the color point group of a Further Reading ﬁnite object it must leave it invariant, and for ðg; gÞ to Lifshitz R (1996) The symmetry of quasiperiodic crystals. Physica be in the color point group of a crystal it must leave it A 232: 633–647. indistinguishable, which in the special case of a Lifshitz R (1997) Theory of color symmetry for periodic and qua- periodic crystal reduces to invariance to within a siperiodic crystals. Reviews of Modern Physics 69: 1181. Lifshitz R and Even-Dar Mandel S (2004) Magnetically-ordered translation. To each element ðg; gÞAGC corresponds g quasicrystals: Enumeration of spin groups and calculation of a phase function FgðkÞ, satisfying a generalized vers- magnetic selection rules. Acta Crystallographica A 60: 167–178. ion of the group compatibility condition [6]. The Litvin DB (1973) Spin translation groups and neutron diffraction color point group contains an important subgroup of analysis. Acta Crystallographica A 29: 651–660. elements of the form ðe; gÞ containing all the color Litvin DB (1977) Spin point groups. Acta Crystallographica A 33: permutations that leave the crystal indistinguishable 279–287. Litvin DB and Opechowski W (1974) Spin groups. Physica 76: without requiring any rotation g. 538–554. A spin point group GS is a subgroup of OðdÞ Mermin ND (1992) The space groups of icosahedral quasicrystals 0 SOðdsÞ1 , where SO(ds) is the group of ds-dimen- and cubic, orthorhombic, monoclinic, and triclinic crystals. sional proper rotations operating on the spins, and 10 Reviews of Modern Physics 64: 3–49. is the time inversion group as before. Note that the Opechowski W (1986) Crystallographic and Metacrystallographic Groups. Amsterdam: North-Holland. dimension of the spins need not be equal to the di- Opechowski W and Guccione R (1965) Magnetic symmetry. In: mension of space (e.g., one may consider a planar Rado GT and Suhl H (eds.) Magnetism, vol. 2A, chap. 3. New arrangement of 3D spins). Also note that because the York: Academic Press. spins are axial vectors there is no loss of generality by Schwarzenberger RLE (1984) Colour symmetry. Bulletin of the restricting their rotations to being proper. Elements London Mathematical Society 16: 209–240. Senechal M (1990) Crystalline Symmetries: An Informal Mathe- of the spin point group are pairs ðg; gÞ, where g is a matical Introduction. Bristol: Adam Hilger. d-dimensional (proper or improper) rotation and g Shubnikov AV, Belov NV, and others (1964) Colored Symmetry. is a spin-space rotation possibly followed by time New York: Pergamon Press.