ANALYSE DES STRUCTURES MAGNÉTIQUES ET THÉORIE DES GROUPESON TWO CLASSIFICATIONS OF MAGNETIC STRUCTURES

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ANALYSE DES STRUCTURES MAGNÉTIQUES ET THÉORIE DES GROUPESON TWO

CLASSIFICATIONS OF MAGNETIC STRUCTURES

W. Opechowski

To cite this version:

W. Opechowski. ANALYSE DES STRUCTURES MAGNÉTIQUES ET THÉORIE DES GROUPE-

SON TWO CLASSIFICATIONS OF MAGNETIC STRUCTURES. Journal de Physique Colloques,

1971, 32 (C1), pp.C1-457-C1-461. �10.1051/jphyscol:19711155�. �jpa-00213975�

ANAL YSE DES STRUCTURES MA GNETIQUES ET THEORIE DES GRBUPES

ON TWO CLASSIFICATIONS OF MAGNETIC STRUCTURES

by W. OPECHOWSKI

Department of Physics, University of British Columbia, Vancouver 8, Canada

R6sum6. - Une formulation gknerale de la classification (appelBe ici C2) des structures magnetiques, baste sur l'emploi de reprksentations des groupes d'espace, est donnke. On montre que cette classification est Bquivalente, dans un sens exactement defini au point de vue mathematique, a la classification (appelke ici Cl') basee sur l'emploi des groupes magnetiques, pourvu qu'on impose au crystal les conditions cycliques usuelles. Si l'on n'impose pas de telles conditions, la classification C2 m$ne dans certains cas (par exemple, structures helicoidales) B des difficultks mathkmatiques, qui sont likes B. la necessite d'utiliser des representations infinies, tandis que la classification Cl' reste valable sans aucune modi- fication.

Abstract. - A general formulation of the classification (called here C2) of magnetic structures, based on the use of representations of space groups is given. It is shown that this classification is, in a precisely defined mathematical sense, equivalent to the classification (called here Cl') based on the use of magnetic groups, provided the usual cyclic boundary conditions are imposed on the crystal. If such conditions are not imposed Classification C2 meets in some cases (for example, helical structures) with mathematical difficulties, due to the necessity of using infinite-dimensional representa- tions, while Classification Cl' remains valid without any modifications.

I. Introduction. - In this paper an attempt is made to clarify some rather essential mathematical points concerning the relation of two classifications of magnetic structures. The paper has thus a purely mathematical character ; it is a somewhat complicated exercise in elementary group theory. What these mathematical points are should become apparent towards the end of this Introduction.

As is well-known, the generally accepted classifi- cation of all possible crystals is based on the use of space groups. In that classification each (perfect) crystal is assigned a classification label which in the simplest case can be written as

where F is a space group, and r(') is the position of any of the identical atoms of mass m which form the crystal. The positions of all other atoms are then Fr"), where each element F of the space group F is taken in turn. Instead of specifying the label (I. 1) we could thus simply write F#), at least when we are not interested in the value of m ; Fr") is then the set of all positions of the atoms of the crystal. A crystal describable by a label of the form (I. 1) will be called a simple crystal. A simple crystal is thus a crystal whose atoms occupy one set of equivalent positions. If a crystal is not simple but composite then its classifica- tion label is of the form

[F ; rC1), . . . , dk) ; m . . ^{m ]} . (I. 2) The set of the positions of all the atoms of a composite crystal thus consists of the k disjoint subsets Fr('), ...,

Fr(,), the atoms of different subsets having in general different masses m,, ... m,.

Since space groups have always a finite number of generators, classification labels such as (I. 1) or (1.2) always consist of a finite number of real num- bers, which determine a crystal completely and uni- quely.

The problem of establishing a classification of all magnetic structures amounts to defining an analogous kind of label for magnetic structures. Here it is perhaps

useful to say explicitly that from a purely geometrical point of view which is adopted in this paper a magnetic structure is nothing else but an axial vector function defined on a crystal, and having the property of changing sign under time inversion. I will use in this paper the term spin arrangement for such function, to avoid possible dynamical connotations of the term magnetic structure.

We want thus to assign to each spin arrangement a classification label defined in the same way for all possible spin arrangements, and consisting of a finite number of real numbers which determine a spin arrangement completely and uniquely.

Let us forget for a moment that spin arrangements change sign under time inversion. If we apply an element of the space group F of the crystal on which a spin arrangement is defined to that spin arrangement then the spin arrangement will either remain unchan- ged or be transformed into another spin arrangement defined on the same crystal. This trivial observation suggests two ways of classifying all possible spin arrangements :

1) by assigning to a given spin arrangement the subgroup H of F, which consists of all those elements of F which leave it unchanged

we shall call this Classification C1 ;

2) by assigning to a given spin arrangement all those distinct spin arrangements which arise from it by applying all the elements of F ; a group-theory argument shows that this is equivalent to assigning to the spin arrangement a representation of F which is contained in that permutation representation of F according to which those distinct spin arrangements transform

we shall call this Classification C2.

Of course a classification label C1 or C2 will consists of more than just a symbol of a subgroup of F or a symbol of a representation of F. However before discussing the question, what these additional data in the classification labels are, we have to modifiy Classi- fication C1 by taking into account time inversion which we have disregarded so far. Classification C1

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711155

C 1

-

so modified will be called Cl'. In Cl' we assign to a

given spin arrangement a subgroup K of F

A, where A is the time-inversion group, which consists of the identity E and time inversion E' ; namely the subgroup K that consists of all those elements of F

A which leave the spin arrangement unchanged.

As is well-known, such a group K is a trivial or non- trivial magnetic group, but not necessarily a 3-dimen- sional magnetic space group (also called a Shubnikov group). Trivial means consisting of elements which are all of the form (F, E), and non-trivial means not containing the element (E, E'), where (F, A) is the gene- ral symbol for an element of F

A. (Terminology and notation used in this paper are, if not defined explicitly, those of Opechowski and Guccione [l], with one exception : an element (F, A) is there deno- ted by FA.)

Since the representations of F O A are direct pro- ducts of the representations of F and those of A, there is no point in introducing a Classification C2' which would be in the same relation to C2 as Cl' is to Cl.

From the way I have introduced here the idea of the Classifications Cl' and C2 it appears plausible that they are equivalent in some precise mathematical sense. However this conjecture can possibly become a theorem only after the Cl' and C2 labels have been defined completely. So far we may only conclude that a classification label of a spin arrangement defined on a simple crystal Fr will have the form

[Fr ; K ; ...l in C l ' , and

[Fr ; r(G,), r(G,), ..., r(Gq) ; ...l ^{in C2} .

Here T(Gj), j

1, 2, ... g, is the matrix which corres- ponds to the generating element G j of F in a representa- tion r ^of F. The dots

... ^>> indicate the additional data and conventions necessary and sufficient to determine a spin arrangement completely and uni- quely from a Cl' or C2 label. Evidently each spin arrangement can be assigned a Cl' label, but there are spin arrangements to which no C2 label corres- ponds if the representation Tis to be finite-dimensional.

However it turns out that by introducing the usual cyclic boundary conditions this incompleteness of Classification C2 can be removed.

What the dots stand for in the case of Cl' is known from the general discussion given by Opechowski and Guccione [l]. No equally general discussion of C2 has been published so far. In this paper I will indicate what the dots in a C2 label stand for. This amounts to giving a general formulation of Classifi- cation C2. I will also state in which precise mathema- tical sense the two classifications, Cl' and C2, are equivalent. Because of space limitations I shall have to omit the proofs and discussion of illustrative exam- ples ; all that is given in a paper by Opechowski and Dreyfus p]. For the same reason I will only consider the case of simple crystals. The generalization to the case of spin arrangements defined on composite crys- tals is straightforward.

References to earlier papers in which what are called here Classifications Cl' and C2 were discussed more or less explicitly, or used can be found in the

two papers, [l] and [2], mentioned above. Here I will only say that the idea of C2 was outlined and illustrated with examples in a paper by Bertaut [3], presented to the 1967 International Conference on Magnetism. What is called here Classification C2 is called by Bertaut representation analysis of magnetic structures rather than a classification, probably because he was interested in something more than merely a classification. However Bertaut did not formulate the principles of C2 in a sufficiently general way to make possible a correct statement of the mathematical relation between C2 and Cl'. Despite that, Bertaut did repeatedly express an opinion about this relation.

In a very recent paper (Bertaut [4]) he says this :

... surprisingly enough, this new point of view

- that is C2 -

offers a wider frame for the descrip- tion and analysis of magnetic structures than invariance under Shubnikov groups. It can be shown indeed that Shubnikov groups can only describe magnetic structures which belong to one-dimensional real representations of the 230 space groups. In our theory >>

-that is C2

there is no limitation either on the nature, real or complex, nor on the dimension of the irreducible representation

In other words, Bertaut asserts that Clf and C2 are not equivalent, C2 being applicable to many more magnetic structures than Cl', and that this is surprising. The conclusion of the pre- sent paper is that the two classifications are equiva- lent if the usual cyclic boundary conditions are impo- sed on crystals (otherwise C2 would have to be gene- ralized to the case of infinite-dimensional representa- tions). There is thus nothing to be surprised about.

Since I want to describe the relation between C l f and C2 (this is done in Section IV), I have to indicate (in Section 11) the form of a complete Cl' label, although this has already been done in an earlier publication (see [l]). The form of a complete C2 label is indicated in Section I11 and the conditions are formulated under which a spin arrangement corres- ponding to a C2 label may exist ; that is, does not vanish identically.

11. Classification Cl'. - First a few preliminary remarks. If we write an element F of a space group F as F

(R ( v), where R is a proper or improper rota- tion matrix and v is a column matrix representing a primitive or non-primitive translation, then applying F to a spin arrangement S(r) means replacing S(r) by another spin arrangement, denoted by [F] S(r) and defined by

3

[F] ~ ( ~ ' ( r )

C ^{6, Rij} s ( ~ ) ( F - r) ,

j= 1

6,

det R , i

1,2, 3 . (11.1) Similarly applying an element m

(F, A) of a magnetic group m that is a subgroup of F C 3 A to S(r) means replacing S(r) by

[m] S(r)

[F, A ] S(r)

[F] S(r) , (11.2) where

E , = + l i f A = E , and & , = - l i f ~ = ~ f

(in Reference [l] the notation FS and mS was used

instead of [F] S and [m] S).

Consider a simple crystal Fr,, an arbitrary sub- group L of F, an a magnetic group m,(H) which belongs to the family of L and whose unprimed elements form the group H. If

F = L + ^LF, + ^LF, + ..., (11.3) then

Fr,

Lr, + LF,r, + L F 3 r , + -., (II.3A) and the sets of atoms located a t LF,r and LFp r are either identical or have no atom in common. Hence, if the index of L in F is l' the number of such distinct sets LF, r, is I" 5 I'. Of course I' and E" may be infi- nite. The Cl' label of a spin arrangement defined on Fr and whose symmetry group is m,(M) has then the form

[F ; m ; S S , . S ( (11.4) where

r y = F y r , y = 1 , 2 ,..., I". (11.5) Since a spin arrangement is a single-valued vector function, the l" spin vectors must satisfy certain admissibility conditions

whose formulation is quite straightforward but rather long, and for that reason will not be repeated here. If the admissibility conditions cannot be satisfied at all then no spin arran- gement with the symmetry group m,(H) can exists in the crystal Fr,.

If L

F then the magnetic space group M,(H) and the choice of one admissible spin vector deter- mines the whole spin arrangement uniquely ; the Cl' label is then

[ F ; %(H); S(rt)], (11.6) and one obtains the whole spin arrangement from it by applying to S(rl) each element of M,(H) in turn.

If L is a 3-dimensional space group contained in F as a proper subgroup then 2" is finite, and the spin arrangement is obtained from its Cl' label by applying each element of ML(H) in turn to the I" admissible spin vectors.

If L is a 2-dimensional space group the number of admissible spin vectors in a Cl' lable would in general be infinite. However in the case of helical spin arran- gements it is sufficient to specify several or even only one admissible spin vector together with an infinte countable (but not discrete) group of rotations with one generator, or perhaps several such groups (for details, see References [I] and [2]). Such additional symmetries of the admissible spin vectors appearing in the Cl' labels are closely related t o the spin-space groups introduced by Brinkman and Elliott [5].

111. Classification CZ. - The complete C2 label of any spin arrangement defined on a simple crystal Fr, and characterized by the d-dimensional representa- tion r ^of F has the form

where the d spin vectors c,, a,, ..., o, are of equal length and must satisfy certain admissibility condi- tions which arise from the single-valuedness of spin arrangements. These admissibility conditions are of course quite different from those satisfied by the admis-

sible spin vectors appearing in the Cl' labels ; they are given explicitly by equations (111.4) below. If these conditions cannot be satisfied no spin arrange- ments defined on Fr, and transforming according to r

can exist.

A C2 label such as (111.1) determines uniquely not just one spin arrangement but d spin arrangements, and a special convention must be introduced if d > l in order to make a C2 label to characterize one spin arrangement.

To state the procedure by means of which a C2 label is used to determine the spin arrangement charac- terized by it, one first defines a 3 d-dimensional repre- sentation D of F as follows : D consists of matrices each of which is the direct product of the matrix T(F) and a matrix 6, R, and which correspond to the elements of F as determined by the homomor- phism

F

(R I

D(F)

6, R o r(l;).

Next one defines d spin arrangements by assigning to the atom at r

Fr, a spin vector Sa(r) whose com- ponents are

These d spin arrangements can be shown to transform according to T, that is,

[F] sa(r)

C

(111.3)

B

provided the d vectors a5 satisfy the admissibility conditions

= C C ~(R(rl)):a;ip of) (111.4)

i B

for all those matrices D(R(~,)) of D which corres- pond to the elements of the site point group R(r,) of r,. When deriving the above equations the assump- tion was made that the matrices R are orthogonal and the matrices T(F) are unitary. This assumption does not constitute any restriction on the generality of the treatment.

The d spin vectors are clearly those assigned to the atom at r, in the d spin arrangements defined by (111.2).

Since we want the C2 label to characterize one single spin arrangement it is necessary in the case of d > 1 to introduce a convention according to which the C2 label is, for example, the label of that spin arrangement in which the spin vector assigned to the atom at r, is a,, that is, S,(r). This particular conven- tion is adopted in the remainder of this paper.

It should be emphasized that the representation r

need not be irreducible. In fact, restricting the choice of r to irreducible representations would make Classi- fication C2 very much incomplete.

Finally it must be remembered that only those C2

labels have a physical meaning for which the spin

arrangement S,(r) as defined by (111.2) is a real vector

function. For example, a C2 label in which r is a

one-dimensional complex representation has no phy-

sical meaning. On the other hand, a rqresentation

C 1

-

reducible into two complexe-conjugate one-dimen-

sional representations may very well occur in a C2 label. In fact, this is the case of the spin arrangement in DyCrO,, as established by Bertaut and Mares- chal [6] ; for details, see Reference [2], Example 3.

IV. Relation between Cl' and C2. - 1. DBTER-

MINING THE

Cl'

C2

Let

suppose that a C2 label is given. To determine the Cl' label of the spin arrangement S(r) characterized by that C2 label it is not necessary first to construct S(r) from the C2 label. One can construct the symmetry group of S(r) directly from the matrices of the repre- sentation T b y means of the following theorem (remem- ber the convention adopted at the end of Section 111 !) :

The magnetic group

mL(H)

H + H L ' , L'

(L,E1), L

F , (IV.1) is the symmetry group of the spin arrangement Sl(r) if and only if the matrices of the restriction of the representation r of F to the subgroup L, and no other matrices of r, have the property that for all H ele- ments of H

r(H)pl

dpl (IV .2)

and

~ ( H L ' ) ~ ~

6p1 (IV .3) where p

1,2, ..., d, and d is the dimension of r.

If equations (IV. 3) are not satisfied but (IV. 2) are then the symmetry group of S,(r) is the trivial magnetic group H.

Once the symmetry group of Sl(r) is found, one determines the positions defined by (11.5) from (11.3) and (II.3A) and the spin vectors of the atoms located at these positions from (111.2).

2. DETERMINING

C2

Cl'

- This is much more complicated, and I will only sketch the procedure. If the magnetic group occurring in the Cl' label of a spin arrangement S(r) is m,(H), one first decomposes F into cosets of H :

(We use the symbols;G,, ..., G,, for the cosets represen- tatives of F in this case to avoid confusion with the coset representatives of F used in eq. (II.3).) Next one shows that the h' spin arrangements

transform according to the finite (if h' is finite) or infinite (if h' is not finite) transitive permutation repre- sentation P, (dimension h') of F generated by the sub- group H of F. The case of h' infinite presents mathe- matical difficulties, and has not been investigated.

However if one introduces the usual cyclic boun- dary conditions h' will always be finite.

If the h' spin arrangements (IV. 5) are linearly inde- pendent then PH is the representation r occurring in the C2 label corresponding to the given Cl' label.

If they are not then a direct sum rH of a certain num-

ber of the irreducible representations into which P, can be decomposed will appear in the C2 label (P, is never irreducible, except in the trivial case where h'

1). One can always choose the basis in the carrier space of that direct sum such that one of the basis vectors is the spin arrangement characterized by the given Cl' label. The spin vectors

occurring in the C2 label are then determined by taking the (vector) values of the spin arrangements forming the basis at r,. The C2 label corresponding to a given Cl' label is thus determined by the latter only up to an equivalence transformation which leaves S,@) unchan- ged.

If the Cl' label of a spin arrangement has the simple form (11.6) the corresponding C2 label is very easily obtained by the above sketched procedure. The permutation representation P, of F is then 2-dimen- sional (since h'

21, and is given by

where G, is a fixed element of F not belonging to H ; furthermore

By reducing P, one finds

the direct sum r13 becomes an (one-dimensional) alternating representation of F. The C2 label corres- ponding to the Cl' label (11.6) is thus

[Frl; H + + l , H G 2 - + - 1 ;

o l

S(rl)] . (IV. 6) A comparison of the label (11.6) and (IV. 6) illus- trates the well-known one-to-one correspondence between the alternating representations of a group and its subgroups of index 2.

3. CONCLUSIONS.

If one imposes on the crysta the usual periodic boundary conditions Classi- fications Cl' and C2 are equivalent in the sense defined in Subsections IV. 1 and IV.2. If no such conditions are adopted Classification Cl' remains valid in all its generality while Classification C2 can no longer be used as it stands ; it would have to be adapted to the case of infinite-dimensional represell- tations.

Classification Cl' is simpler than Classification C2 because a Cl' label contains only the minimum neces- sary and sufficient information for determining the spin arrangement labdled by it, while a C2 label contains a good deal of superfluous information, as we have seen in Szction 111. That is also why the procedure to assign the classification label to a given spin arrangement requires in general fewer steps in the case of Cl' than in the case of C2. However the superfluous information contained in C2 labels may of course be useful for some other than classification purposes.

On the other hand it is clear that both these classi-

fications are of doubtful value for labelling those spin arrangements which have partially random charac- ter.

Finally, I would like to make a remark that goes beyond the purely mathematical question discussed in this paper. The group L which appears in a Cl' label is, for many experimentally determined magnetic structures, a proper subgroup of the space group of

the magnetically ordered crystal, that is, of the space group F as experimentally determined by the X- ray methods. It seems to me that the following conjec- ture is physically very plausible : if the precision of the determination of F was sufficiently high one would always find that L

F ; in other'words, all magnetic structures would turn out to have a Cl' label of the simple form (11.6).

References

[l] OPECHOWSKI (W.)

GUCCIONE (R.), Magnetism [4] BERTAUT (E. F.),

Phys., 1969,

1592.

(Rado and Suhl, editors), Vol. 11, Ch.

Academic

BRINKMAN (W. F.)

ELLIOTT (R. J.), PYOC. Roy.

Press, 1965. Soc. London, 1966, A

i[21 OpECHoWsK1

publication in (W.)

Acta Cryst.

submitted for 161 BERTAUT (E. F.)

MARESCHAL (J.), J. Physique,

131 BERTAUT (E. F.), Acta Cryst., 1968, A

217. 1968, 29, 67.