PHYSICAL REVIEW 8 VOLUME 35, NUMBER 2 15 JANUARY 1987-I

Brief Reports

Brief Reports are short papers which report on completed research which, while meeting the usual Physical Review standards ofscientific quality, does not warrant a regular article (Ad.denda to papers previously published in the Physical Review by the sameauthors are included in Brief Reports )A. Brief Report may be no longer than 3& printed pages and must be accompanied by anabstract Th. e same publication schedule as for regular articles is followed, and page proofs are sent to authors

Symmetry and supersymmetry in crystals

L. H. BennettNational Bureau of Standards, Gaithersburg, Maryland 20899

R. E. WatsonBrookhauen National Laboratory, Upton, New York 11973

(Received 14 April 1986)

An examination of the disclination network associated with hexagon faces of Wigner-Seitzpolyhedra provides new insight into the supersymmetry associated with the three-dimensionalspace-group representation of centered crystals having nonunique asymmetric units in their descrip-tion.

The International Tables for Crystallography' list all ofthe 230 crystallographic space groups and, for each ofthese, the possible positions of the atoms. They also listthe asymmetric unit (in mathematics known as the funda-mental domain) for each space group. An asymmetricunit of space group is a part of space from which, by ap-plication of all symmetry operations of the space group,the whole of space is filled exactly. An asymmetric unit,together with its associated cluster of atomic positions,contains all the information necessary for the completeconstruction of the crystal structure. The asymmetricunit and its associated cluster of atoms is not alwaysuniquely determined by symmetry. In these cases, it issometimes useful to choose an asymmetric unit by lookingto its application; e.g., in order to describe a molecularcrystal, it is often convenient to select (if possible) atomicclusters that constitute one or several whole molecules.Alternatively, the authors of the International Tables forCrystallography have chosen the asymmetric units in sucha way that Fourier summations can be performed con-veniently. There is a certain arbitrariness in such choicesfor the fcc, bcc, and other centered crystals.

It is the purpose of this report to point out that thechoice of the asymmetric unit, for certain centered crys-tals, can be related to the symmetry in a higher-dimensional hyperspace from which the crystal structurecan be presumed to have been projected. Such hiddensymmetry, associated with the higher-dimensional space,is conventionally termed supersymmetry. Thissymmetry-related choice involves a criterion based on thetopology of the crystal lattice and is neither (i) for compu-tational convenience, nor (ii) does it make any reference to

35 845

chemical arguments. This choice is related to the factthat there is more information associated with the pack-ing of the atoms in the crystal than simply the crystalspace-group and point-group symmetries associated withthe site occupancies. This additional hidden symmetryinvolves the interpenetration of nets or lines of what maybe termed ligands or disclinations ' and gives new physi-cal insight into the chemical and other properties whichdepend on the crystal topology. We are not concernedhere with the projections of these nets from higher-dimensional space, but instead, with their consequences inthree dimensions.

A disclination is usually defined as a geometrical defectin an otherwise ordered crystal. Speaking of disclinationsmay thus seem odd when discussing perfectly orderedcrystal structures. However, three-dimensional crystalstructures may be understood in reference to crystallinesystems in higher than three-dimensional space. Somesuch structures may be best understood in terms of dis-clinations introduced in higher-order space which are thenprojected into three dimensions to yield crystalline struc-tures. ' While ligand or disclination lines can be impor-tant to the chemical, magnetic or s up erconductingproperties of a system, the point of the present report isthat, for some crystal structures, a reasonable and uniquechoice may be made of the asymmetric unit by letting itinvolve an unbroken net (or sets of nets) of disclinationlines.

As an example of such a choice, consider the a-manganese crystal, whose structure was determined tohave 58 atoms per unit cell, with space group Td (No. 217of the International Tables for Crystallography), and four

Work of the U. S. CgovernmentNot subject to U. S. copyright

846 BRIEF REPORTS 35

) ~

(a)

0Mn IQo

Q Mn Irt

—72 Disclinotion Lines&72 Disclination Lines

FIG. 1. The cluster of one Mn I, four Mn II, and twelve MnIII atoms about the (z, z, 2 ) central lattice position, (a) as

chosen by Bradley and Thewlis (Ref. 2), and (b) as suggested bythe disclination network. The twelve Mn IV atoms are notshown here since they are not involved in disclinations. Thelines in (a) are to guide the eye and are not intended to have anyphysical significance. The line joining the atoms in (b) are dis-clinations.

different types of Mn sites. The Bravais lattice for thisspace group is body-centered cubic, with each lattice point[i.e., (0,0,0) or (0.5,0.5,0.5)] represented by a cluster of 29atoms. In their description of e-Mn, Bradley andThewlis chose a particular cluster, partially shown in Fig.1(a), to illustrate the structure, but they were careful tonote, "These clusters of atoms are in no way chemicalmolecules or even groups. An atom is no more related tothe other atoms within the cluster than to neighboringatoms outside the cluster. The cluster is in fact a meregeometrical conception, serving as an aid to the imagina-tion. " Other choices of the asymmetric unit lead to otherchoices of the atoms in a cluster. Now, we have alreadynoted that a-Mn is closely related to the Frank-Kaspertopologically close-packed (tcp) phases. These tcp phaseswere defined by Frank and Kasper to be made exclusivelyof structures composed of two or more types of fourWigner-Seitz (i.e., Voronoi) polyhedra involving 12 five-fold plus 0, 2, 3, and 4 sixfold faces. By conventional no-tation, ' these are termed (0,0, 12,0), (0,0, 12,2), (0,0, 12,3),and (0,0, 12,4), respectively, denoting the 12 fivefold andthe number of sixfold faces. The line joining an atom toits neighbor through a sixfold face was called a "majorligand. " If such a ligand network is constructed for a-Mn, it is found to have split into two disconnected netsextending to infinity in all directions. Such a split thendefines the asymmetric unit [except, of course, for thosesites having (0,0,12,0) topology, since these, by definitionhave no major ligands]. It is of interest to know that theparticular construction described happens to have selecteda different cluster than that shown in Fig. 1(a); this alter-nate cluster, involving one of the two disconnected nets, isshown in Fig. 1(b).

The above-mentioned major ligands have been shownby Nelson to be [

—72'I disclinations. The ligands in a-Mn are such [ —72'I disclinations, except for the minorligands between pairs of Mn III atoms near the surface ofthe unit cell. (The presence of these [+72') disclinations

involving fourfold faces between the Mn III pairs is whatprevents this structure from being a Frank-Kasper struc-ture. ) The second net is identical to the first, except thatthe cluster defining it is centered at the corners of thecube. While it is the crystal structure and not magnetismand chemical bonding which is of primary concern here,it should be noted that Mn sites connected by the [ —jdisclinations tend to lie close together: The ring of sixcommon nearest neighbors are forced radially outwards,thus allowing the pair to pull closer to each other. Strongchemical bonding (major ligands) is suggested both by thisclose proximity and by the sharing of more than the usualnumber of common nearest neighbors. Mn sites connect-ed by [ —

I disclinations appear to be important to magne-tism as well. The Mn I, II, and III sites depicted in Fig.1 have substantial aligned magnetic moments (which arecoupled antiferromagnetically along the [

—) disclination

lines within the cell and ferromagnetically across cellboundaries), whereas the Mn IV atoms, which are not in-volved in disclinations and are not shown in Fig. 1, haveessentially zero-valued moments. The magnetic momentson the second of the two nets, which are independent andinterpenetrating, are antiferromagnetically aligned to thefirst.

The correlation between magnetic-moment behaviorand the occurrence of disclinations is more general thanthe case of a-Mn above. Since the magnetic 3d elementsare small relative to the other alloy components, they fallinto sites without disclinations when they occur inFrank-Kasper structures. (Granted that such sites havefewer faces, they tend to be of smaller volume. ) Themagnetism in these Frank-Kasper systems tends to be ac-companied by only modest local magnetic moments. Incontrast, one Co site in the good hard magnet, SmCo5, has

[ —] (and also [+ } disclinations) and there is a sugges-tion that this site has a larger local moment than the otherCo site. Though they have lower Curie temperatures,Nd2Fe&7 and Nd2Fe&4B are still better hard magnets" andin these systems, the Fe sites labeled c and j2 in the twostructures, respectively, each with a (0,0,12,2) local envi-ronment, have particularly large local moments. There isthus the distinct suggestion that the occurrence of [

—I

disclinations encourages a large moment at an atomic sitein a magnetic system. It is further possible to speculatethat the magnetic coupling, involving as it likely does in-teractions via the common neighbors, bears an analogy tothe magnetic superexchange which is well known in oxidesystems.

Returning to the main issue of this article, other cen-tered crystal structures also have separated disclinationnetworks which may relate to supersymmetry. Considerthe low temperature form of uranium, a-U (Pearson sym-bol oC4, No. 63 of the International Tables for Crystallography, space group Cmcm) with an orthorhombic struc-ture having four atoms per unit cell. ' All atoms areequivalent and are associated with (0,8,0,4) Wigner-Seitzpolyhedra. The topology of these polyhedra, with no pen-tagonal faces but involving eight fourfold and four sixfoldfaces, ' are quite different than those found in n-Mn.Treating the four six-sided faces as propagating disclina-tions leads to two independent networks separated by

35 BRIEF REPORTS 847

( —, , —, ,0) in the A B-plane. Although the ct-Mn nets ex-tend to infinity in three dimensions, the fluted nets in a-Uare infinite in only two directions. The nets, obtained hereby geometry alone, are the same as those previously in-ferred from chemical insight in the crystallographic deter-mination.

An example of an intermetallic compound is Ti2Ni(Pearson symbol cF96, No. 227 of the InternationalTables for Crystallography, space group Fd 3m), an fccstructure with 96 atoms per cell. ' There is one type of Nisite, with (0,0,12,0) polyhedra, and two Ti sites, one with(0,0, 12,0) and the other with (0,4, 10,0). The only disclina-tions are the four I + I disclinations (from the four-sidedfaces) associated with the second Ti site. There are fourindependent nets, extending in three dimensions to infini-ty, again providing a unique choice for the asymmetricunit.

There are crystal structures where independent disclina-tion networks are obtained unrelated to choices of theasymmetric unit, but with hidden symmetry having physi-cal significance. An outstanding example is in theFrank-Kasper structure associated with A 15 high-temperature superconductors, such as Nb3Sn (Pearsonsymbol cP 8, No. 223 of the International Tables for Crys

tallography, space group Pm 3n). The Sn atoms are(0,0, 12,0) with no disclinations; the Nb are (0,0,12,2) withtwo disclinations. The disclination structure is a set ofthree independent lines parallel to the cube axes. Theselinear chains of atoms have traditionally been assumed'to be essential to the superconductivity and perhaps to thelow-temperature lattice phase transition.

Much as other areas of physics have involved higher-dimensional group theory, it appears that we are enteringa new era in which it may be useful to invoke a crystallog-raphy in higher dimensions. ' We have shown here that,in some cases, there is a unique geometrical definition ofthe asymmetric unit intrinsic to the structure, related todisclinations which may be viewed as arising from projec-tions from higher-dimensional space, but extractable fromthe three-dimensional geometry alone.

Thanks are due to Mr. M. Hartmann, Dr. M. Melamud,and Dr. C. Witzagall for their aid in writing and revisingthe computer program. Brookhaven Laboratory is sup-ported by the Division of Materials Sciences, U.S. Depart-ment of Energy, under Contract No. DE-AC02-76CH00016.

'International Tables for Crystallography, Vol. 4, Space Group-Symmetry, edited by T. Hahn (Reidel, Dordrecht, Holland,1983).

This observation has been recognized, e.g. , by A. L. Loeb, J.Solid State Chem. ], 237 (1970); and W. B. Pearson, Comput-er Modeling of Phase Diagrams (The Metallurgical Society,Warrendale, PA, 1986), p. 163.

F. C. Frank and J. S. Kasper, Acta Crystallogr. 11, 184 (1958);12, 483 (1959).

4J. F. Sadoc, J. Non-Cryst. Solids 44, 1 (1981); J. Phys. (Paris)44, L707 (1983).

5D. R. Nelson, Phys. Rev. B 28, 5155 (1983).P. Kramer and R. Neri, Acta Crystallogr. Sect. A 40, 580

(1984)~

7L. H. Bennett, R. E. Watson, and W. B. Pearson, J. Magn.Magn. Mater. 54-57, 1537 (1986).

A. J. Bradley and J. Thewlis, Proc. R. Soc. London, Ser. A 115,456 (1927).

R. E. Watson and L. H. Bennett, Scripta Metall. 19, 535

(1985). a-Mn differs from the Frank-Kasper phases in thatthe Mn III-site has the topology (0,1,10,3). The 1, whichrefers to the number of four-sided faces, can be considered aminor ligand. Inclusion of it in the disclination structurecomplicates the discussion, but does not change any of thestated conclusions.Notation (a, b, c,d, . . . ) indicates a polyhedron having a faceswith three edges, b with four, c with five, d with six, etc.

' J. J. Croat, J. F. Herbst, R. W. Lee, and F. E. Pinkerton,Appl. Phys. Lett. 44, 148 (1984).

' C. S. Barrett, M. H. Mueller, and R. L. Hitterman, Phys. Rev.129, 625 (1963).

P. Duwez and J. L. Taylor, Trans. AIME 188, 1173 (1950).' For example, see S. V. Vonkovsky, Yu. A. Izyumov, and E. Z.

Kumaev, Superconductiuity of Transition Metals, Their Alloysand Compounds (Springer-Verlag, Berlin, 1982).For example, S. Sachdev and D. R. Nelson, Phys. Rev. 32,14&0 (1985); P. Bak, Phys. Rev. Lett. 54, 1517 (1985); J. F.Sethna, Phys. Rev. B 31, 6278 (1985); see also Refs. 3—5.