Materials Science and Engineering 294–296 (2000) 508–511

Magnetic quasicrystals: what can we expect to seein their neutron diffraction data?

Ron Lifshitz∗School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel

Received 1 September 1999; accepted 29 March 2000

Abstract

The theory of magnetic symmetry in quasicrystals is used to characterize the nature of magnetic peaks expected in elastic neutrondiffraction experiments. It has been established that there is no symmetry-based argument which forbids the existence of quasiperiodiclong-range magnetic order. Suggestions are offered as to where one should look for the simplest kinds of antiferromagnetic quasicrystals.© 2000 Elsevier Science B.V. All rights reserved.

Keywords:Quasicrystals; Magnetism; Quasiperiodicity; Long-range order; Neutron diffraction

1. Introduction

In recent years, we have witnessed a careful experimentalinvestigation on the question of long-range magnetic orderin rare-earth based icosahedral quasicrystals [1–3]. Never-theless, discussions of this matter have been somewhat un-clear as to the actual nature of the magnetic order one wouldexpect to see in antiferromagnetic (AF) quasicrystals, if theywere to exist. A partial answer to this question can be ob-tained from a theory of the symmetry of magnetically or-dered quasicrystals [4]. I intend to show here that such a the-ory not only provides a valuable tool for analyzing neutrondiffraction data, but also helps to narrow down the possiblemagnetic ordering one would expect to see in the classesof quasicrystals that are known to exist today. I hope thatthis will help in guiding the continuing search for new qua-sicrystals with this unique physical property.

2. The spin density field and its symmetry

A magnetically-ordered crystal, whether periodic or ape-riodic, is most directly described by its spin density fieldS(r ). This field is a three-component real-valued function,transforming like an axial vector under O(3) and changingsign under time inversion. One may think of this function asdefining a set of classical magnetic moments, or spins, onthe atomic sites of the material. For quasiperiodic crystals,

∗ Tel.: +972-3-640-5145; fax:+972-3-642-2979.E-mail address:ronlif@post.tau.ac.il (R. Lifshitz).

the spin density field may be expressed as a Fourier sumwith a countable infinity of wave vectors

S(r ) =∑k∈L

S(k) eik·r . (1)

The setL of all integral linear combinations of the wavevectors in Eq. (1) is called themagnetic lattice. Its rank D isthe smallest number of wave vectors needed to generate it byintegral linear combinations. For quasiperiodic crystals, bydefinition, the rank is finite. For the special case of periodiccrystals the rank is equal to the dimensiond of physicalspace.

In elastic neutron scattering experiments, every wave vec-tor k in L is a candidate for a magnetic Bragg peak whoseintensity is given by

I (k) ∝ |S(k)|2 − |k̂ · S(k)|2, (2)

wherek is the scattering wave vector andk̂ a unit vector inits direction. I have shown elsewhere [5] that under genericcircumstances there can be only three reasons for not ob-serving a magnetic Bragg peak atk, even thoughk is in L:(a) the intensityI(k)6=0, but is too weak to be detected in theactual experiment; (b) the intensityI(k)=0 becauseS(k) isparallel tok; and (c) the intensityI(k)=0 because magneticsymmetry requires the Fourier coefficientS(k) to vanish. Ishall explain below exactly how this symmetry requirement,or ‘selection rule’, comes about.

The theory of magnetic symmetry in quasiperiodic crys-tals, which is described in more detail in Ref. [4], is a refor-mulation of Litvin and Opechowski’s theory of spin space

0921-5093/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved.PII: S0921-5093(00)01071-6

R. Lifshitz / Materials Science and Engineering 294–296 (2000) 508–511 509

groups [6–8]. Their theory, which is applicable to periodiccrystals, is extended to quasiperiodic crystals by followingthe ideas of Rokhsar, Wright, and Mermin’s ‘Fourier-spaceapproach’ to crystallography [9]. At the heart of this ap-proach is a redefinition of the concept of point-group sym-metry which enables one to treat quasicrystals directly inphysical space [10]. The key to this redefinition is the obser-vation that point-group rotations (proper or improper), whenapplied to a quasiperiodic crystal, do not leave the crystalinvariant but rather take it into one that contains the samespatial distributions of bounded structures of arbitrary size.

This generalized notion of symmetry, termed ‘indistingui-shability’, is captured by requiring that any symme-try operation of the magnetic crystal leave invariant allspatially-averaged autocorrelation functions of its spindensity fieldS(r ) for any order n and for any choice ofcomponentsαi∈{x,y,z},

C(n)α1...αn

(r1, . . . , rn)

= limV →∞

1

V

∫V

dr Sα1(r1 − r ) · · · Sαn(rn − r ). (3)

I have shown in the Appendix of Ref. [11] that an equiva-lent statement for the indistinguishability of any two quasi-periodic spin density fields,S(r ) and S′(r ), is that theirFourier coefficients are related as

S′(k) = e2πiχ(k)S(k), (4)

where χ , called agauge function, is a real-valued scalarfunction which is linear (modulo integers) onL. Only in thecase of periodic crystals can one replace 2πχ (k) by k·d,reducing indistinguishability to the requirement that the twocrystals differ at most by a translationd.

With this in mind, we define thepoint group Gof themagnetic crystal to be the set of operationsg from O(3)that leave it indistinguishable to within rotationsγ in spinspace, possibly combined with time inversion. Accordingly,for every pair (g,γ ) there exists a gauge function,Φ

γg (k),

called aphase function, which satisfies

S(gk) = e2πiΦγg (k)γ S(k). (5)

SinceS([gh]k)=S(g[hk]), one can easily establish that thetransformationsγ in spin space form a groupΓ and that thepairs (g,γ ) satisfying the point-group condition (5) form asubgroup ofG×0 which we call thespin point group GS.The corresponding phase functions, one for each pair inGS,must satisfy thegroup compatibility condition,

∀(g, γ ), (h, η) ∈ GS : Φγη

gh (k) ≡ Φγg (hk) + Φ

ηh(k), (6)

where ‘≡’ denotes equality modulo integers. Aspin spacegroup, describing the symmetry of a magnetic crystal,whether periodic or aperiodic, is thus given by a magneticlatticeL, a spin point groupGS, and a set of phase functionsΦ

γg (k), satisfying the group compatibility condition (6).

3. The diffraction pattern: a thinned-out magneticlattice or a shifted nuclear lattice?

I said earlier that every wave vector in the magnetic latticeis a candidate for a diffraction peak unless symmetry forbidsit. We are now in a position to understand how this happens.Given a wave vectork∈L, we examine all spin point-groupoperations (g,γ ) for which gk=k. These elements form asubgroup of the spin point group, called thelittle spin groupof k, Gk

S. For elements (g,γ ) of GkS, the point-group condi-

tion (5) can be rewritten as

γ S(k) = e−2πiΦγg (k)S(k). (7)

This implies that the Fourier coefficientS(k) is requiredto be a simultaneous eigenvector of all spin transforma-tions γ in the little spin group ofk, with the eigenvaluesgiven by the corresponding phase functions. If a non-trivialthree-dimensional axial vector satisfying Eq. (7) does notexist thenS(k) will necessarily vanish. If such an eigenvec-tor does exist its form might still be constrained to lie in aparticular subspace of spin space.

Of particular interest are spin transformationsγ that leavethe spin density field indistinguishable without requiring anyrotation in physical space. These transformations are pairedin the spin point group with the identity rotatione and forma normal and Abelian subgroup ofΓ called thelattice spingroup Γe. In the special case of periodic crystals, the ele-ments ofΓe are spin transformations that, when combinedwith translations, leave the magnetic crystal invariant.

The lattice spin group plays a key role in determining theoutcome of elastic neutron scattering, for if a magnetic crys-tal has a non-trivial lattice spin groupΓe then{e}×Γe⊆Gk

Sfor every k in the magnetic lattice, restricting the form ofall the S(k)’s. This may result in a substantial thinning-outof the magnetic lattice, whereby only a fraction of the wavevectors give rise to actual magnetic Bragg peaks. Becausethis thinning of the magnetic lattice is often quite exten-sive, it is common practice to describe the magnetic peaksnot as a thinned-out magnetic lattice, but rather in termsof the nuclear latticeL0 (the one observed above the mag-netic ordering temperature) which is shifted by the so-called‘magnetic propagation vectors’. These two descriptions arein fact equivalent and with some care can be used inter-changeably.

4. Where should we look?

In the past I have tabulated all the decagonal spin spacegroups [12], as well as all the lattice spin groups for icosa-hedral quasicrystals [4]. In the latter case I also listed ex-plicitly, for every wave vectork in the magnetic lattice,whether through Eq. (7) symmetry requiresS(k) to vanishor to take any special form. In a future publication I planto provide complete tables of spin space groups and the

510 R. Lifshitz / Materials Science and Engineering 294–296 (2000) 508–511

requirements which they impose on neutron scattering ex-periments for all the relevant quasiperiodic crystal systems(octagonal, decagonal, dodecagonal, and icosahedral).

Clearly, the theory of spin space groups provides a help-ful tool for analyzing neutron diffraction experiments. Itlists the patterns of magnetic Bragg peaks, compatible witheach symmetry class, which can then be directly comparedwith experiment. But, on a more basic level, this theory an-swers one of the fundamental questions that have been de-bated in recent years, which is whether it is even possible tohave long-range quasiperiodic magnetic order. It establishesthat even though symmetry may impose constraints on thepossible forms of magnetic order one can have in a givenquasicrystal, it clearly does not forbid the existence of suchorder. Thus, there is no symmetry-based argument whichdisallows long-range magnetic order in quasicrystals.

Why is it then, that we have not yet observed unequivocallong-range magnetic order in a quasicrystal? It might bebecauseenergeticconsiderations lead to local frustration andspin-glass ordering; it might be due to some otherphysicalargument; or — it might be simply because we have notfound it yet. If this is the case, then a more practical questionto ask of a theory of magnetic symmetry is whether it canoffer any suggestions as to where to look for such order.Indeed, symmetry considerations may assist us in decidingin which quasicrystal systems to look first for thesimplestkind of non-trivial magnetic ordering. Such ordering wouldbe the quasiperiodic analog of a simple AF periodic crystalwhere half the spins are pointing ‘up’ and the other half arepointing ‘down’. Symmetry arguments can guide us to thosesystems where such ordering is possible.

I, therefore, close this essay with a short discussion ofwhat this quasiperiodic AF order looks like, followed bythe list of systems which are compatible with such order. Itwould then be up to the metallurgists and material scientiststo find the right chemical systems which can sustain localmagnetic moments and at the same time are likely to havestable phases in these crystal systems.

5. The quasiperiodic antiferromagnet

The simple AF crystal, whether periodic or aperiodic, hasa lattice spin groupΓe containing only two elements: theidentity operationε and time inversionτ . In the case of timeinversion, the selection rule (7) becomes

τS(k) ≡ −S(k) = e−2πiΦτe (k)S(k), (8)

which requiresS(k) to vanish unlessφτe (k)≡1/2. On the

other hand, application of the group compatibility condition(6) to (e,τ )2=(e,ε) gives two possible values for this phase,

φτe (k) ≡ 0 or 1

2. (9)

It is not too difficult to show that exactly half of the wavevectors in the magnetic latticeL haveφτ

e (k)≡0 and will,

therefore, not appear in the neutron diffraction pattern. Thesewave vectors constitute a sublatticeL0 of index 2 inL. Onecan then describe the set of wave vectors appearing in thediffraction diagram either as the magnetic latticeL withoutall the wave vectors inL0, or asL0 shifted byq, whereq, a‘magnetic propagation vector’, is any vector inL which isnot in the sublatticeL0. In the simplest scenarioL0 is alsothe nuclear lattice, but this is not necessarily the case.

Consider a one-dimensional spin chain with this latticespin group. If the chain is periodic, then its (Fourier) mag-netic lattice is given by all integral multiples of a singlewave vectorb∗ (I will keep the superscript * as a reminderthat we are in Fourier space). Because phase functions arelinear, it suffices to specify the value ofφτ

e on b∗ and thatwill determine its value on any wave vector in the lattice. Ofthe two possible values (see Eq. (9)) the first,φτ

e (b∗)≡0 willresult through the selection rule (8) inS(k) being zero everywhere and, therefore,S(r )=0 as well. The only non-trivialassignment is thereforeφτ

e (b∗)≡1/2 which through the se-lection rule (8) implies that all lattice wave vectors that areeven multiples ofb∗ will be missing, or ‘extinct’, from thediffraction pattern.

If the spin chain is quasiperiodic, say having a rank of 2,then its magnetic lattice will be given by all integral linearcombinations of two wave vectors,b∗

1 andb∗2, whose mag-

nitudes are incommensurate. In this case the phase functionφτ

e is fully determined by specifying its two independent val-ues onb∗

1 andb∗2. At first glance, it would seem as if there

are three distinct non-trivial assignments of values given by

(φτe (b∗

1), φτe (b∗

2)) ≡ (0, 12) or (1

2, 0) or (12, 1

2). (10)

It turns out that these three assignments are equivalent,leading to the same spin space group, due to the fact that fora quasiperiodic chain one has the added freedom of changingthe basis of the magnetic lattice. A basis transformationfrom (b∗

1, b∗2) to (b∗

1+b∗2, b∗

2) or to (b∗1, b∗

1+b∗2) takes one,

respectively, from the first or second assignment in Eq. (10)to the third. Thus, the diffraction pattern of a quasiperiodicAF spin chain can always be described as a magnetic latticegiven by wave vectors of the formk=n1b∗

1+n2b∗2 where

all the vectors withn1+n2 are extinct. Equivalently, it maybe described as a latticeL0, generated by the wave vectorsb∗

1+b∗2 andb∗

1−b∗2, and shifted by the vectorb∗

1.Knowing the different possibilities in Fourier space al-

lows us to immediately construct simple direct-space exam-ples of AF spin chains having these symmetries. Fig. 1a andb show two periodic AF spin chains in which the ‘magneticunit cell’ is twice or four-times as large as the ‘nuclear unitcell’. Both of these chains will exhibit the same magneticdiffraction peaks, the only way to distinguish them being adirect comparison with the nuclear diffraction pattern, whichcan be obtained above the magnetic ordering temperature.Fig. 1c–e show three AF Fibonacci chains, obtained by set-ting the ratiob∗

1/b∗2 to the golden mean (1+√

5)/2, and usingthe three different assignments of the phase function valuesgiven in Eq. (10). Again, as discussed above, all three are

R. Lifshitz / Materials Science and Engineering 294–296 (2000) 508–511 511

Fig. 1. Examples of antiferromagnetic spin chains: (a) and (b) are peri-odic with different magnetic unit cells; (c)–(e) the AF Fibonacci chains,obtained as described in the text by using a modified grid method. In (c)spins separated by a long (L) segment are anti-parallel and those sepa-rated by a short (S) segment are parallel; in (d) the opposite occurs; andin (e) all nearest-neighbor pairs are anti-parallel.

expected to have the same magnetic diffraction peaks andthe only way to distinguish them is a comparison with thenuclear diffraction pattern.

Which of the actual quasicrystal systems that are knownto exist today allow a simple AF order? Axial quasicrys-tals admit two kinds of simple AF order. Since they are allquasiperiodic in the plane normal to then-fold axis and pe-riodic along this axis, it is always possible to have periodicAF order along then-fold axis. This would give an AF qua-sicrystal, but not in the true sense that we are interestedin. Only whenn is a power of 2 is it possible to have truequasiperiodic AF order in the plane normal to then-fold axis[11]. Thus, among the known axial quasicrystals one shouldconcentrate the search for simple AF order in the octagonalcrystal system.

Only two of the three Bravais classes in the icosahedralsystem admit simple AF order [4]. Such order is possible

if the nuclear lattice is either simple (giving a magneticlattice which is body-centered in Fourier space) or if thenuclear lattice is face-centered in Fourier space (giving asimple icosahedral magnetic lattice). Unfortunately, most ofthe known icosahedral quasicrystals, including the rare-earthbased ones, are face-centered in direct space and, there-fore, do not allow simple AF order. Furthermore, icosahedralquasicrystals which are body-centered in direct space arenot yet known to exist.Thus, in the icosahedral system,one should look for simple AF order in crystals that have asimple icosahedral nuclear lattice.

References

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[2] Z. Islam, I.R. Fisher, J. Zarestky, P.C. Canfield, C. Stassis, A.I.Goldman, Phys. Rev. B 57 (1998) R11047.

[3] T.J. Sato, H. Takakura, A.-P. Tsai, K. Shibata, Phys. Rev. Lett. 81(1998) 2364.

[4] R. Lifshitz, Phys. Rev. Lett. 80 (1998) 2717.[5] R. Lifshitz (unpublished). For a sketch of the argument, see W.

Sikora (Ed.), Extended Abstracts of the Workshop on Applicationof Symmetry Analysis to Diffraction Investigation. University ofMining and Metallurgy, Kraków, Poland, 1996, p. 70.

[6] D.B. Litvin, W. Opechowski, Physica 76 (1974) 538.[7] D.B. Litvin, Acta Cryst. A 29 (1973) 651.[8] D.B. Litvin, Acta Cryst. A 33 (1977) 279.[9] For a review, see N.D. Mermin, Rev. Mod. Phys. 64 (1992) 3. For

an elementary introduction, see R. Lifshitz, Physica A 232 (1996)633.

[10] As opposed to the alternative superspace approach, a review ofwhich is provided by T. Janssen, A. Janner, A. Looijenga-Vos,P.M. de Wolff, in: A.J.C. Wilson (Ed.), International Tables forCrystallography, Vol. C, Kluwer Academic, Dordrecht, 1992, p. 797.

[11] R. Lifshitz, Rev. Mod. Phys. 69 (1997) 1181.[12] R. Lifshitz, in: C. Janot, R. Mosseri (Eds.), Proceedings of the

5th International Conference on Quasicrystals, World Scientific,Singapore, 1995, p. 43.