Symmetry analysis in neutron diffraction studies of magnetic structures

Journal of Magnetism and Magnetic Materials 12 (1979) 239-248 0 North-Holland Publishing Company

SYMMETRY ANALYSIS IN NEUTRON DIFFRACTION

STUDIES OF MAGNETIC STRUCTURES

1. A phase transition concept to describe magnetic structures in crystals Yu.A. IZYUMOV and V.E. NAISH Institute of Metal Physics, Ural Science Research Center of the USSR Academy

of Sciences, Sverdlovsk, USSR

Received 27 December 1978

On the basis of phase transition symmetry theory there has been developed an efficient method of calculating the possible magnetic structures liable to arise from the paramagnetic phase of the crystal. Every magnetic structure is described by the superposition of the basis functions of the irreducible representation incorporated in the magnetic representation of the crystal symmetry group. Convenient formulas are given to calculate the basis functions and composition of the magnetic representation. These formulas permit calculation by referring only to the tables of space group irreducible representations. The technique is illustrated on the example of magnetic structures in crystals with the symmetry group D$.

where tn is the vector of translation from the O-cell into the n-cell, i is the number of an atom in the cell (what is implied here is cells and translations in the paramagnetic phase of the crystal). It is just this relation that determines the wave vector of a magnetic structure. Since we have found it at the first stage of decoding, the problem has now reduced to determining u magnetic atomic vectors of a primitive cell. The problem involved is usually solved by a variational method, by prescribing a trial magnetic structure and comparing the calculated diffraction picture with the observed value. 30 variables are subject to variation, and the difficulties enhance as u increases. The advantage of symmetry analysis is that it permits a drastic reduction in the number of variables varied. This reduction is facilitated by deliberately selecting only those magnetic structures which are allowed by the symmetry of the initial crystal. The symmetry approach in the theory of magnetic structures rests on the mathematical idea, first used by Dzialoshinsky [ 1,2], that any magnetic structure with a prescribed K may be expanded in basis functions of irreducible representations of the space group.of the crystal with this K:

1. Introduction Every magnetic structure may be regarded as resulting from a phase transition from a certain initial phase of the crystal. It is the purpose of this series of papers to demonstrate how the concept of Landau phase transition symmetry theory can be applied to the problem of enumerating the magnetic structures liable to occur in a given crystal. The above problem is highly instrumental in neutron diffraction analysis of crystals. The deciphering of a magnetic structure involving the use of a neutron diffraction pattern is known to consist of two stages. The first stage is to determine the wave vector K of the magnetic structure using the system of magnetic reflections (broadly speaking, there may well be several vectors). This enables one to find automatically the magnetic lattice of the crystal. The second stage is aimed at determining the orientation of atomic magnetic moments in the primitive cell by the intensity of reflections. If we know the wave vector of a magnetic structure, we can express the magnetic moments (spins) of atoms in any cell of the crystal in terms of the magnetic moments of atoms in the zero cell by means of the equation S,,,. = @*tnSgi

,

(2)

(1) 239

240

Yu.A. Izyumov,

V.E. Naish /Neutron

SK here is a column vector comprising atomic spin vectors that prescribes the magnetic structure of the crystal with a wave vector K, and $I? is a column of symmetrical combinations of atomic vectors transforming according to the v irreducible representation. These functions are considered to be known since in the theory there is a method to compute them for any K. Ci are the arbitrary mixing coefficients to be determined. Eq. (2) is absolutely exact if one disregards the crystal lattice distortions arising from the occurrence of magnetic order and proceeds from the fact any function invariant under group CD may be series-expanded in basis functions of irreducible representations of group G which is its supergroup. Such a relation underlies Landau’s theory of phase transitions in which SK occurs as the density of a certain quantity specifying the crystal upon transition to a dissymmetric phase. Landau’s constructive idea is based on the premise that the phase transition always goes over an irreducible representation of the initial-crystal symmetry group and thus in (2) there remains one term from the sums over V:

diffraction of magnetic structures I

soft mode mechanism. This is so because the paramagnetic phase is not ordered and does not contain normal oscillations (spin waves) such as those in the magnetically ordered phase. However, when approaching the point of transition, there develop magnetic order fluctuations in the paramagnetic crystal. These fluctuations may be classified by irreducible representations of the initial-phase group. Accordingly, those fluctuations whose intrinsic energy goes to zero earlier (in terms of temperature)sort of “freeze” into the crystal, forming a relevant magnetic structure. This reasoning provides ample grounds for the hypothesis that the magnetic structure results also from one space group irreducible representation. Then formula (3) should be applicable. The hypotehsis under consideration enables one to use effectively the symmetry in decoding neutron diffraction patterns at the second stage of the investigation (after the wave vector of the magnetic structure has already been determined). In fact, the magnetic reflection intensity corresponding to the neutron scattering vector 4 is determined by the relation [3] I4 = {I F(q)I’

- le. F(q)12},

(4)

(3) This is fairly easy to understand if we refer to the example of structural transitions in distortion-type crystals resulting from minor displacements of crystal atoms from the equilibrium positions. The possible types of atomic displacements in the initial crystal (normal oscillations or phonons) are classified with due regard for the irreducible representations of its space group. If there is a soft mode in the phonon spectrum and the energy of the mode goes to zero at some temperature T, , then at a temperature below T,,, the number of such phonons begins to increase anomalously, and their condensation already forms static displacements of the same symmetry as that of soft mode oscillations. The soft mode is said to freeze into the crystal, thereby diminishing its symmetry. In this sense it is said, too, that the structural phase transition goes over an irreducible representation describing the symmetry of the soft mode. As far as the magnetic structure of the crystal arising from the paramagnetic phase as a result of a second-order phase transition is concerned, we are not in a position to say that it is produced through the

where e is the unit vector of scattering and.F(q) structural amplitude of magnetic scattering: F(q) = F

the

e- @-+Si

(5)

with Xi and Si being the coordinate and spin of the ith atom of the cell, respectively. The spin vector of any atom of the crystal may be represented as a superposition of basis functions of the form Si=F

C{S(;Yli),

(6)

where S(y I i) is the vector from the aNdimensional column of the basis function $J?. The amplitude F(q) may also be represented as a superposition of amplitudes F(q)=?

GiITe-

iq-“iS(y

I i)) = F

C{@(q). (7)

We now see that instead of using formulas (4) and (5) and varying 30 variables SF it is possible to employ formulas (4) and (7) and vary the I, (I, is the dimen-

Yu.A. Izyumov, V.E. Naish /Neutron diffraction of magnetic structures I

sion of the v irreducible representation) variables Ci only. The procedure is to take in succession, one by one, all V’Ssince the quantities F?(q) can be computed in advance. The problem has thus reduced to calculating the quantities S(y I i) that form the basis functions of the irreducible representations of the space group of the crystal. Bertaut [4,5] is the first to have applied the theory of space group representations to determine the possible magnetic structures in the crystal in a neutron diffraction analysis. He has given a multitude of examples of using symmetry analysis [6-81. The technique developed by Bertaut has been utilized by a number of other authors (see, for instance, ref. [9]) in practical neutron diffraction studies. Since, in the concept of phase transitions, the magnetic structure is described by basis functions of irreducible representations of the initial group of the crystal, the method we propose here has a number of points in common with Bertaut’s representative analysis. A detailed comparison of these methods will be given in the last paper of this series. For the time being it suffices to note that in our approach, which is based on the physical ideas of the phase transition theory, great emphasis is laid on the mixing of basis functions of irreducible representations. Here we also focus our attention on the important role played by the notion of a primitive (but not a unit) cell of the initial crystal. The series submitted incorporates four papers. The first paper contains a formulation of the general approach to the problem. Apart from this, a scheme is worked out to compute the basis functions of magnetic representation of crystal which are directly connected with the possible magnetic structures. Simple working formulas are obtained that permit calculation of these functions using space group irreducible-representation tables only. The second paper is dedicated to a study of the translational properties of magnetic structures, i.e., a deals with magnetic lattices and techniques of determining them from the system of magnetic reflections of the neutron diffraction pattern. The third paper considers in detail an example of the magnetic structure of spinels in which the possibilities proposed method are illustrated. The final paper is devoted to an investigation into the symmetry properties of the exchange Hamiltonian and to ascertaining what information on possible magnetic structures can be derived

241

from this study. We hope these papers contain an exhaustive description of the magnetic-structure symmetry analysis method which is requisite for the problem of magnetic neutron diffraction analysis.

2. Indispensable information representation theory

from space group

Since the method is based on the theory of representation of space groups of crystals, it is reasonable to remind one of its fundamental notions. The space group C contains the following elements of symmetry rotation elements {h lrh} (h is the symbol of rotation operation and rh the accompanying translation), pure translation {hr I t} (h 1 is the unit element, t the translation vector), and mixed elements (h I sh + t}. The set of elements {h 1 1t} for all t’s form a translation group which is a subgroup of group G. Its irreducible representations #(g) are one-dimensional and are of the form

where K is the index of representation tor from the first Brillouin zone

- the wave vec-

b,, bz and b3 here are the main vectors of the reciprocal lattice which are expressed in terms of fundamental translations (11,UZ, a3 of the crystal by means of the known equations b = 2n [a2 x 431 1

v

b = 2n 143 x 41 ’

2

v

’

b3 = 21T bl x a21 v ’

where v is the unit cell volume. In expression (9), Nr , N2 and N3 are the dimensions of the main region of the crystal, and pi are integers. All in all, there are N = Nr N2N3 possible values of the vector K, where N is the number of unit cells in the crystal. The irreducible representations of space group G are specified by the wave vector star {K} which is a set of nonequivalent wave vectors KL (L = 1,2, . . .. Ix) obtained from one vector by the action of all the elements of group G. The individual vectors from set

242

Yu.A. Izyumov, V.E. Naish /Neutron diffraction of magnetic structures I

{K} are called star arms. The number of star arms ZK depends on the position a vector from {K} occupies, i.e., it depends on how symmetric or general this position is. The number of stars qualitatively differing in symmetry is usually small, and all the various types thereof are enumerated and analyzed in reference manuals on representations of Fedorov groups. Let us consider some wave vector K. If the position it occupies is not general, then part of the group G elements leaves it invariant or transform it into an equivalent vector differing from that involved by an arbitrary vector b of the reciprocal lattice. The set of all such elements forms a certain group GK which is a subgroup of group G. The latter is called the wave vector group. In order to include an element g = {h It} from group G into group GK it is necessary that the condition hK=K+b

(11)

be satisfied. Wave vector group is an important notion primarily because its irreducible representations are relatively easy to find. They are labeled by symbols dK” where v is the number of a representation. The matrices of these representations de(g) are tabulated for elements g = {h 1zh} of the so-called zero block of group GK, i.e., for the set of elements containing no integral translations. For an arbitrary element of group GK, the relation

(12) holds which reduces everything to zero block elements. Hence X, /J = 1 , .... I,, where 1, is the dimension of the vth irreducible representation of group GK. Every irreducible representation of group GK generates an irreducible representation of the entire space group G. This representation is specified by vector star {K} and number v and is usually labeled by symbol D{K b. The group G irreducible representation matrix is related to the group GK irreducible representation matrix by the induction formula [IO]:

The indexes L and M here number the star {K} arms, and K is one of the arms of that star. Elements gL and

gM are group G elements which group GK does not contain. It is convenient to choose the so-called representatives of expansion of group G in group GK (or expansion of point group G” in point group GE) in the capacity of the above elements ‘K

G=5L=l

a, GK

or Go=LGl hLGk.

It is noteworthy that the star arms arise from the first arm K1 by the action of the rotational part of the representative element gL = { hL I thL }: KL =hLK,.

(15)

It is evident from formula (13) that if the dimension of the vth irreducible representation of the wave vector group is l,, then the dimension of the same representation of the space group is 1, X 1~. The form of the representation matrices D {K)v depends on the choice of the initial star arm. But if another arm of the star is taken to be initial, one obtains an equivalent representation (it should be remembered that two representations whose characters coincide are called equivalent representations). For practical calculations connected with the application of space groups representation theory we will refer to Kovalev’s book [ 111. It contains a list of all the stars for all the 230 Fedorov groups and specifies the elements of the wave vector groups. The matrices of irreducible representations are written out for every group GK. The information furnished suffices to construct basis functions of irreducible representations of the wave vector group on the space of atomic spins. This procedure is carried out in the next section of the paper.

3. Magnetic representation and its basis functions Examine a crystal containing N primitive cells with u magnetic atoms in each of them. At the outset we will consider only a system of crystallographically equivalent atoms. Let some magnetic structure be realized in the crystal. In other words, let the vector of the atomic magnetic moment be specified for each of the atoms. We thus have some 3uN-dimensional state vector which is transformed by the action of elements g of space group G of the crystal into another

Yu.A. Izyumov, V.E. Naish /Neutron diffraction of magnetic structures 1

state vector from the same space. Such a transformation brings about some representation of group G which we shall call a magnetic one. To begin with, let us consider the case when the coherent structure is described by one harmonic, i.e., by one wave vector K.Then the magnetic representation can be readily constructed. The functions $# are convenient to choose as unit vectors of magnetic representation. These functions may be written in the form of 3uNdimensional columns (16) where the direct sum is taken over all N primitive cells of the crystal which are connected with the zero cell by translations f,, , and ai0 is a 3o-dimensional column in which all the components are equal to zero, except for one component. The latter component which is equal to unity corresponds to the magnetic atomj of the zero cell and to the spin direction p (p = x, y, z). The functions I$ are translation operator functions, therefore the evident relation T(t,) J/g = e-iK’tn@

(17)

holds true. Let us take one of the elements g = {h i zh} of wave vector group GK and consider its action on the atom of the crystal. It must be noted that in the general case the atom from the zero cell with the number j and coordinate xi becomes an atom with the number i of another cell: gXj=hwi+th=Xj+Up(g,j). Symbolically manner MO

(18)

this can also be written in the following

+ (r$) *

(19)

With T(g) operating on the function J/g, the atomic number transforms according to (19) and there occurs a factor e-iK’aP according to (17), as well as a rotation transformation of the axial vector components. The rotation transformation is described by a rotation matrix 6,R&, where Rtp is a matrix corresponding to the rotation of the polar vector in three-dimensional space by the action of point transformation h, with 6, = 1 for usual rotations and with a,, = -1 for inversion rotations. The wave vector K

243

remains since g E GK. Thus, the result of the T(g) operation on function rjJi is

(20) where the number i on the right-hand side is determined according to (19). This equation may be written as

(21) which defines the explicit form of the magnetic representation matrix of wave vector group GK: (22)

with the dimensions of these matrices being 30 X 30. The magnetic representation is reducible and may be expanded into irreducible representations dK” of the group GK: d$ = c

V

n,dKV .

(23)

The numbers 5 are found according to the usual group theory formula which, upon summation over all the translations of the crystal, reduces to the following form (24) Here XKu(g) is the character of irreducible tion dKV of group GK, and

X$(g)=~,spRh~-iK’ap(8’i)Si,~

representa-

(25)

is the character of magnetic representation *. Summation in formula (24) is now taken only over elements {h I th} of the zero block of group GK or, which is the same, over the elements of the group of directions of the wave vector group; n(G$ is the number of elements in Gk. Thus, to ascertain the composi-

* It may be shown (see the fourth paper of this series) that the goup GK magnetic representation we have introduced is equivalent to Bertaut’s magnetic representation [S] $ X V, where $ is a permutation representation of group CR, and V is the representation according to which the axial vector is transformed.

Yu.A. Izyumov, V.E. Naish /Neutron diffraction of magnetic structures I

244

tion of the magnetic representation it suffices to find the table of atomic permutations given by the action of the group GK zero-block elements according to eq. (18) and to resort to space group irreducible-representations tables. We now will give cut-and-dried instructions on how to construct the basis functions of group GK irreducible representations contained in the magnetic representation. To this end, we avail ourselves of the known projection operator formula $P

=igEGK

&g)

F(g) ti ,

(26)

where $ is some starting function and dg is the matrix of the representation selected. The index p should be fiied here. Then formula (26) determines I, pieces of basis functions of the Z,-dimensional representation. We shall take function @ (16) to be the starting function J/ and make use of formula (21). In the expression obtained we then take summation over all the translations and, with due regard for equation (22), find

Inserting expression (16) for I/.$, we present the final result for the 3uN vector as the direct sum of 3udimensional vectors e: +p

= C.e,iK*r,, n

(28)

where the @’ represent, in turn, the direct sum over u of the usual axial three-dimensional vectors Se(y I i) determined for every single magnetic atom i of the zero cell of the crystal @ with

= g’So(yIi)

(29)

As is seen from (28), the atomic vector in the nth cell of the crystal is related to the vector in the zeroth cell by the equation S (KvIi)= n h

eiK”nSO(~Ii).

(31)

When calculating according to formula (30), it is necessary to fix the indexes cc,j and 13which are enclosed in the square brackets. This enables determination of the start for projection of basis functions on to this space. Passing over to other sets of indexes leads either to other independent sets of basis functions (there are just as many basis functions as times the irreducible representation is contained in the magnetic one) or identically to zero. Let us now discuss the problem about the physical sense of the magnetic representation basis functions. First of all we note the condition of orthogonality which is easy to obtain using formula (30) and the general properties inherent to the irreducible representations of any group

It should be borne in mind that at the moment we are considering a system of crystallographically equivalent atoms occupying only one position in group G. Summation over i is here taken over the magnetic atoms bound with one another by the group transformations. It follows from (32) that the sum of the squares of the modulus of vectors Sc mains on the primitive cell atoms FlS~(~li)12

= const.

(33)

This result provides no in rmation concerning the 5a normalization of vectors S,-,( A’ I i) belonging to an individual atom. A direct calculation by use of expression (30) yields (34) so in the general case the modulus of vector Se for separate atoms of the crystal is not conserved in the irreducible-representation basis function. For onedimensional representations it follows from (34) that the magnitude of the atomic spin is conserved. For multidimensional representations the magnetic struc-

Yu.A. Izyumov, V.E. Naish /Neutron diffraction of magnetic structures I

ture corresponds to a superposition of basis functions. It is worth noting that the necessity of conserving atomic spins should impose certain restrictions on the mixing coefficients. Formula (30) gives an expression for basis functions on a star arm, for instance Kr. It is no difficult matter to show that the basis functions on the other arm KL , are related to it by a simple equation

where gLXi = hLXi + thL = xi t up, i.e., it is necessary to affect the atomic spin of the initial-arm basis function by the representative element of the new arm. Finally, we should explain why it is the crystal space group irreducible representations that we were resorting to in the foregoing. Why is this so despite the fact that the magnetic structure is considered to arise as a result of a phase transition from the initial paramagnetic state and the real symmetry group of the crystal is group G X R, where R is the spin inversion operation? It is from the property of the direct product that it follows that the irreducible representations of group G X R are obtained by simply multiplying those of group G into a doubled set of representations, even and odd with respect to R. When calculating the magnetic representation characters d(g) accordin to formula (29, it is only in M(gR) will differ for all the sign that A(g) and J& group G X R elements: &(gR) = -d(g).

(36)

Let us now trave the calculation of the entry coefficients n, according to formula (24) for even d!!” and odd d!!” representations of group G XR, taking into account that #(gR> = e(g), x!?‘(gR) = --x!?‘(g) and that the group G$- X R order has now become I twice as large:

245

netic group contains no even representations at all and the occurrence of the magnetic structure is characterized by its odd representations. The role of even representations is different, viz. they pertain to purely structural phase transitions within the paramagnetic phase. Formula (37) for the entry number of irreducible representations coincides with (24), so only the space group G is involved therein. The formulas for the basis functions of irreducible representations coincide exactly in the same way. Therefore we can characterize the magnetic structure by space group irreducible representations.

4. Example: magnetic structure of crystals with space group Did

To illustrate the method developed we will perform a symmetry analysis of magnetic structures in ilmenitetype crystals having a space group D$ -R%. The compounds CoCOa and MnCOs belonging to this structural type have antiferromagnetic structures with weak ferromagnetism [ 121. The magnetic atoms in these compounds occupy the positions 2(b): l(0 0 ),.2(0

0 4)

(38)

(the coordinates of the atoms are written in a system shown in fig. 1 which corresponds to a hexagonal arrangement). Group D& contains 12 elements in the principal block. These elements, written in table 1 [l 11,include elements h3 and h, which represent rotations about a triad axis; h8, hl,, and hlz representing rotations \ about two-fold axes (fig. 2), and h 13 which represents

(37)

-Lx ‘b- - h(Gjt)

XMKM

x’ K”(g)

-

h

Thus, the magnetic representation of the paramag-

Fig. 1. Hexagonal arrangement of system of coordinates for g~oup D$-J (according to Kovalev [Ill).

Yu.A. Izyumov,

246

V.E. Naish /Neutron

diffmction

of magnetic structures I

Table 1 Transition of Co atoms in COCOS crystal by the action of group DQd elements [positions 2(b)] Elements

Atoms 1

2

{hl 1000)

1

2

{h31000}

1

2

{hglOOO}

1

2

{h81000}

2

1

{hlo1000)

2

1

{h121000)

2

1

{hulOOf)

1

2

b100$)

1

2

{h1,100!)

1

2

{hz,lOO&)

2

1

{hz,lOO$)

2

1

{hz,lOOd)

2

1

Fig. 2. Unit cell of COCOS crystal. Only magnetic atoms are shown [positions 2(b)].

found in [ 111. In table 2 are given the matrices of these representations. For two-dimensional representations 75 and 76, these matrices have been made real preliminarily with the help of some unitary transformation. Beferring to table 2 we find the characters d(g) of the magnetic representation of group D$. Only two kinds, viz. x&(/z,) = &(hrs) = 6 are other than zero. It is now easy to obtain the expansion of the magnetic representation into irreducible ones &$=o = 71 +

inversion *. The other elements are products of the above rotations by inversion. All the inversion elements contain an accompanying translation by half the period along the triad axis, z = (0 0 f }. Table 1 also shows permutations of atoms by the action of the group elements. AlI the returning translations up are absent here, because they are not necessary in the case K_= 0. The magnetic structures of compounds such as COCOS converse the chemical unit cell, that is why to describe them it is necessary to consider the irreducible representations of the group D&r with K = 0. This star has six irreducible representations which can be

* In Kovalev’s tables of which we make extensive use, the rotational elements of the group are labeled by the symbol ht, where i runs from 1 to 24 in the case of hexagonal symmetry of the highest class. They form the set of rotational elements of all the less symmetric groups. Element h13 denotes inversion, and element h12+t represents the product of hi by inversion. All the cubic symmetry elements are also labeled by hi, where i runs from 1 to 48, with h25 corresponding to inversion and with element h24+t being equal to the product of hi by inversion.

73 +

27s.

The results of the calculations of the tions for these representations are given One-dimensional representations r1 and antiferro- and ferromagnetism along the tal axis. The basis functions of repeated

(39) basis funcin ‘table 3. r3 describe principal crysrepresenta-

tions rs and 7; describe ferro- and antiferromagnetic ordering in the basis plane. For pure basis functions, the orientation of the moment is determined by the ratio of the quantities u/u, and a superposition of two basis functions, r5 or T;, gives the position of the moments in an arbitrary direction in the plane. In experiments it is just the magnetic structures corresponding to the calculated basis functions (table 4) that are found in ilmenite-type crystals. In a number of compounds, specified in table 4, weak ferromagnetism is observed in the antiferromagnetic structure with the vector of antiferromagnetism lying in the basis plane. It is noteworthy that ferromagnetism occurs in the same plane. Such a structure corresponds to the superposition of the basis functions of the irreducible representation r5 which occurs twice. There is one more class of CoF3 type compounds which have the same space group D$. The magnetic

Yu.A. Izyumov,

V.E. Naish /Neutron

diffraction of magnetic structures 1

247

Table 2 Irreducible representation of group D$, for K = 0 (in the case of representations 73.74 and 76 odd with respect to inversion the matrices for the elements specified at the bottom of the table should be multiplied by -1) _ _~_~-_.___--~ _--ht

h3

hs

ha

71

1

1

1

1

72

1

1

1

1

73

1

1

1

-1

-1

74

1

11

1

-1

-1

1

$1

75

01 01

-2

_G -12

-$

fi -12

qf

_+

-“$

76

1 0

-t

_Js

rt

8

0

1

_T

;

hl3

hts

0 -1

hI7

ho

Table 3 Basis functions of the group DQd irreducible representation withK=O Representation

Atom * 1

2

-___71

75 7;

* Ufv = 2 - J5

ooi

001 001 UVO vu0 uvo

73

GO .--

001 UVO vu0 iXl

vu0

---

for 75, u/v = 2 + fl

for 75.

Table 4 Magnetic structure of the crystals with the space group Did and with the positions 2(b) for magnetic atoms Representation

Magnetic structure

Crystals

2(OOW) l(OOw); l(OOw); 2(OOw) 1(uv0); 2(iiiIO) 1 (uv0); 2(uvol 1(uv0); 2(zixl) additional mode

CoF3, FeC03 CrBr3 CrF3 FeF3

l(u1qO);

2@1qO)

l(uvw); Z(lTi+ additional mode l(ul~lwl);

2041v1w1)

1 1

1

-10 -10

-1

hlz

hlo ___.___1

0

-1

_d

L&1 ;

_$

hzz

i2

1 2

h24

atoms also occupy positions 2(b) as in the case of CoC03 type compounds. Consequently, the results contained in table 4 also hold true for these crystals. In the compounds under consideration there occur magnetic structures which correspond to each of the irreducible representations. In the latter case, the magnetic structures correspond to two irreducible representations, rl and r5. The symmetry analysis of many experimentally determined magnetic structures carried out in the book [ 121 shows that in the overwhelming majority of cases the magnetic structure is characterized by a single space group irreducible representation. This is what permits us to use effectively symmetry analysis when decoding an unknown magnetic structure. However, there are cases, which also take place in the example we have given, of transition over several irreducible representations. The problem of transition to a magnetically ordered state over several irreducible representations will be treated in detail in the final paper of this series.

Acknowledgements

FeB03, MnC03 1 CoCO3, NiC03

The authors wish to express their sincere appreciation to S.V. Vonsovsky, R.P. Ozerov and V.P. Plakhtij for a helpful discussion of the problems touched upon in the paper.

248

Yu.A. Izyumov, V.E. Naish /Neutron diffraction of magnetic structures 1

References [l] I.E. Dzialoshinsky, JETP 32 (1957) 1547. [ 21 I.E. Dzialoshinsky, JETP 46 (1964) 1420. [ 31 Yu.A. Izyumov and R.P. Ozerov, Magnetic Neutron Diffraction (Plenum, New York, 1970). [4] E.F. Bertaut, ActaCryst. A24 (1968) 217. [S] E.F. Bertaut, J. de Phys. 32 (1971) C1-462. [6] E.F. Bertaut and .I. Dulac, Acta Cryst. A28 (1972) 580. (71 E.F. Bertaut, Acta Cryst. A28 (1972) 477.

[ 81 A. Kallel, H. Boller and E.F. Bertaut, J. Phys. Chem. Solids 35 (1974) 1139. [9] W. Prandle, Z. Krist. 144 (1976) 198. [lo] G.Ya. Lyubarski, Tjeorija grupp i jeje primjenjenije v fizikje (Gostekhizdat, 1957). [ 1 l] O.V. Kovalev, Irreducible Representations of the Space Groups (Gordon and Breach, New York, 1965). [ 121 A. Oles, F. Kajzar, M. Kucab and W. Sikora, Magnetic Structures by Neutron Diffraction (Warszawa, Krakow, 1976).