Neutron Scattering from Magnetic Materials part 2 doc

Determination of magnetic structures Most of the information on the nature of ordered magnetic phases or magnetic structurescomes from neutron diffraction experiments.. But neutron diffr

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[10] W Marshall and S.W Lovesey, Theory of Thermal Neutron Scattering, Oxford University Press (1971) [11] G.E Bacon, Neutron Diffraction, Third Edition, Clarendon, Oxford (1975).

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Magnetic Structures

Tapan Chatterji

Institut Laue–Langevin, B.P 156X, 38042 Grenoble cedex, France

E-mail: chatt@ill.fr

Contents

1 Introduction 27

2 Determination of magnetic structures 28

2.1 Polycrystalline samples 29

2.2 Single crystals 31

3 Ferromagnetic and simple antiferromagnetic structures 31

3.1 Cubic Bravais lattice 33

3.2 Hexagonal Bravais lattice 36

3.3 Tetragonal Bravais lattice 36

3.4 Single-k and multiple-k magnetic structures 36

4 Modulated magnetic structures 38

4.1 Sine-wave magnetic structures 39

4.2 Helimagnetic structures 39

5 Complex modulated structures 40

5.1 Magnetic structures of heavy rare-earth elements 40

5.2 Magnetic structures of light rare-earth elements 48

5.3 Spin density wave in chromium 49

5.4 Modulated magnetic structures in CeSb 52

5.5 Modulated magnetic structure of CeAl2 56

5.6 Modulated magnetic structures of EuAs3and Eu(As1−xPx )3 57

5.7 Modulated magnetic structures in MnP: Lifshitz point 62

5.8 Helimagnetic phase in CuO 64

5.9 Modulated magnetic structures in MnSi and FeGe 67

5.10 Microscopic origin of modulated magnetic structures 68

6 Magnetic structures of novel electronic materials 68

6.1 Magnetic structures of cuprates 68

6.2 Magnetic structures of manganites 79

7 Concluding remarks 87

Acknowledgment 88

References 88

NEUTRON SCATTERING FROM MAGNETIC MATERIALS

Edited by Tapan Chatterji

© 2006 Elsevier B.V All rights reserved

25

1 Introduction

Elements, alloys and chemical compounds containing atoms with incomplete d- and

f -shells exhibit unique properties characterized by their response to an applied magnetic

field or the magnetic susceptibility They are called magnetic materials Magnetic atomsare mainly situated in three rows of the periodic table, namely the transition elements, the

rare-earth and the actinide elements They have incomplete 3d- (4d-), 4f - and 5f -shells, respectively In Tables 1 and 2 we give 3d- and 4f -shell structures of the iron group and

rare-earth ions For an introduction to the magnetic properties of solids, the reader canconsult, for example, the textbooks of Kittel [1] and Ashcroft and Mermin [2] Due tothe exchange interaction between the magnetic atoms the magnetic moments usually order

at low temperatures Depending on the sign of the exchange the magnetic moments canorder to a ferro- or antiferromagnetic phase In the ferromagnetic phase in a single domainthe magnetic moments are all oriented parallel to each other whereas in a simple antifer-romagnetic phase magnetic moments group together in two sublattices In each sublatticethe moments are oriented parallel to each other but the orientation between the sublatticemoments is antiparallel In a true antiferromagnetic structure the magnetic moments intwo sublattices are equal and compensate each other If the magnetic moments of the twosublattices of the magnetic structure are not equal and therefore a net magnetic moment ex-ists, then the structure is called ferrimagnetic In more complex antiferromagnetic phasesthe magnetic moments can have noncollinear arrangements In another class of magneticmaterials, the magnetic moments are arranged in the form of a spiral or a helix Theseare called modulated magnetic structures In general the periodicities of these modulatedstructures are incommensurate with those of the crystal structures We already mentioned

in Chapter 1 that the first determination of a magnetic structure was done by Shull and

Table 1

Quantum numbers for spin S, orbital L, total angular momentum J , the ground-state terms

of iron group ions with the basic electron configuration 3d n Note that due to the quenching

of orbital angular momentum (L= 0) in a crystal field in the solid state, much better agreement with the experimental values of the magnetic moment is obtained by taking

J = S, than by taking J = |L ± S| appropriate to the free ion according to Hund’s rule

Table 2

Quantum numbers for spin S, orbital L, total angular momentum J , the ground-state term and the Landé factor g of the ground-state multiplet for rare earth with the basic electron configuration 4f n

2 Determination of magnetic structures

Most of the information on the nature of ordered magnetic phases or magnetic structurescomes from neutron diffraction experiments We have already shown that due to their wavenature thermal neutrons show interference phenomena Neutrons are diffracted from thenuclear structure yielding information on the spatial arrangements of the nuclei of theatoms in crystals In Chapter 1 we have shown that neutrons are also diffracted fromthe magnetic moments of the atoms In the present section we shall show how the po-sitions and the intensities of the diffracted neutron beams provide information about themagnetic structure Just as the crystal structure information is absolutely necessary for un-derstanding the physical properties of the crystals, the magnetic structure information is

a prerequisite to understanding of the magnetic properties of the materials Needless tosay to determine the magnetic structure one should know the crystal structure first Crystalstructures are usually determined by X-ray diffraction Only in special cases is neutron dif-

fraction needed But neutron diffraction is necessary to determine the magnetic structure.X-ray magnetic scattering can sometimes provide useful information about the details of amagnetic structure but is seldom employed to determine an unknown magnetic structure.

It is useful to stress that the determination of the magnetic structure is fundamentally ferent from conventional crystal structure determination which consists of determining theatomic positions The full determination of the magnetic structure consists of determin-ing both the magnitude and the direction of all magnetic atomic moments in the magneticunit cell The nuclear structure factor is a complex scalar quantity whereas the magneticstructure factor is a complex vector Since in a conventional diffraction experiment onlythe diffraction intensities can be measured the phase problem is aggravated in the mag-netic case However, because of the form of the dipole interaction between the neutronand the magnetization of the crystal, only those components of the magnetic structure fac-tor perpendicular to the scattering vector contribute to the scattered magnetic intensity It

dif-is thdif-is property which allows us to determine the moment direction Experimentally themagnetic structure is determined by using neutron diffraction intensities measured eitherfrom polycrystalline samples or from single crystals New solid state materials first becomeavailable in polycrystalline form and good single crystals become available (if at all) only

at a later stage, so magnetic structure determination is usually attempted first by the tron powder diffraction technique It must be pointed out that unambiguous determination

neu-of magnetic structure is in principle not possible in many cases by neutron diffraction frompolycrystalline samples Even neutron diffraction from a multidomain single crystal doesnot necessarily provide a unique magnetic structure solution In these cases only polarizedneutron diffraction on a single-domain crystal with three-dimensional polarization analysiscan provide a unique solution (see Chapter 5)

In Chapter 1 we have already derived the equations that form the basis of magneticstructure determination We are now concerned with the practical aspects of the determina-tion of the magnetic structure from neutron diffraction experiments This consists of fourimportant steps:

(1) identification of the propagation vector of the magnetic structure,

(2) determination of the coupling between the magnetic moments,

(3) determination of the moment directions, and

(4) determination of the moment values in Bohr magnetons

Before determining the magnetic structure of an ordered magnetic phase it is worthwhile

to characterize the magnetic ordering by bulk magnetic measurements like susceptibility

and magnetization This gives the ordering temperature TC or TN and also preliminaryinformation about the ordered magnetic phase The bulk magnetic measurements can showimmediately whether the ordered magnetic phase is ferro- or antiferromagnetic

2.1 Polycrystalline samples

We have already mentioned that since new magnetic materials are usually synthesized aspolycrystalline materials the magnetic structures are often determined from powder neu-tron diffraction If the ordered magnetic phase is ferromagnetic then we expect magneticreflections superimposed on the nuclear reflections The magnetic cell is the same as the

nuclear cell and the propagation vector k= 0 The magnetic structure determination is thenreduced to determining the moment direction and the magnitude of the ordered magneticmoment For this one needs to determine the neutron diffraction intensities from the poly-crystalline sample at least at two temperatures, one at the lowest attainable temperature

which must be much lower than the ordering temperature TCand the other at a temperature

higher than TC We assume, of course, that there are no more magnetic phase transitions

below TC The neutron powder diffraction pattern at the higher temperature is used to termine the nuclear contribution to the total intensity which can be subtracted from the lowtemperature intensity to give the magnetic intensity It is, however, desirable to measurethe powder diffraction pattern at several temperatures in order to characterize the ferro-

de-magnetic phase transition and determine the ordering temperature TCfrom neutron fraction If the magnetic moment is small then the powder neutron diffraction technique isnot very helpful because separation of the weak magnetic intensity from the strong nuclearintensity involves large inaccuracy In this case, polarized neutron diffraction from a singlecrystal is recommended (see Chapters 4 and 5) It is possible to have an antiferromagnetic

dif-structure with the propagation vector k= 0 In this case, the magnetic reflections in eral are superimposed on the nuclear reflections as in the case of ferromagnetic structuresdiscussed above If the magnetic structure is antiferromagnetic with a propagation vector

gen-k = 0 then magnetic Bragg scattering is present at positions which are different from those

of nuclear Bragg reflections and the determination of magnetic intensities by the method

of subtraction becomes more accurate Once the positions of the magnetic intensities areobtained, the next important task is to determine the periodicity of the magnetic unit cell

It is convenient to use the nuclear or paramagnetic unit cell and express the magnetic

pe-riodicity in terms of the propagation vector k If k is commensurate with the nuclear cell, i.e., when for example, k= (1

2, 0, 0) (the magnetic cell is 2a × b × c), or k = (1

2,12, 0)

(the magnetic cell is 2a × 2b × c) or k = (1

2,12,12) (the magnetic cell is 2a × 2b × 2c),

then the determination of the propagation vector from the powder diffraction diagram isnot a very difficult task A simple graphical method described by Rossat-Mignod [4] is

adequate to determine k But if the magnetic cell is incommensurate with the nuclear cell,

i.e., the components of the propagation vector cannot be expressed by simple fractions, the

determination of k becomes more difficult In such case, neutron diffraction investigation

with a single crystal, if available, is recommended Otherwise a computer program oped by Wilkinson et al [5] can be used to index the magnetic reflections of the neutronpowder diffraction pattern Wilkinson et al [5] illustrate their method by giving several ex-amples Once the propagation vector of the magnetic structure (step 1) is determined onetries to determine the coupling (step 2) between the Fourier components of the magneticmoments Sometimes group theoretical methods are employed for this purpose Chapter 3gives the details of this method In addition the readers are referred to the relevant litera-ture [4,6,7] The spin density Patterson function, which is analogous to the crystallographicPatterson function, is also employed to determine the magnetic coupling from the magneticintensity data [8–10] This method works best with a good single crystal data set, but canalso be employed by the calculation of the magnetic ion radial distribution function usingthe magnetic intensity obtained from a powder diffraction diagram [11] The third step

devel-of the magnetic structure determination is to find the moment directions The intensities

of the magnetic reflections, especially the absent reflections, provide useful information

on the spin directions Otherwise one uses a least-squares method in which the momentdirections are continuously varied to get the best agreement between the observed and cal-culated magnetic structure factors [12] If the magnetic structure model is correct then theleast-squares refinement of the magnetic structure factors provides the information on themoment directions (step 3) and moment values in Bohr magnetons (step 4) Lastly oneshould point out that a complete structure determination is often not possible with powderdiffraction data [13,14].

2.2 Single crystals

If a single crystal is available then the determination of the magnetic structure is easier,first of all because the propagation vector can be determined unambiguously Secondly,the intensities of the magnetic reflections can be determined without the problem of over-lapping reflections For the calculation of magnetic structure factors and refinement of themagnetic structure using single neutron diffraction data the reader is referred to the Cam-bridge Crystallography Subroutine Library developed by Brown and Matthewman [15].However, one must always remember that the intensities are affected more by the extinc-tion effect in the single crystal case This is especially so for crystals which tend to growwith a high degree of perfection Also the correction for the absorption effects is morecomplex Historically magnetic single crystal experiments have normally been performedwith a single detector and there is always a danger of missing some information, espe-cially in the cases where several propagation vectors coexist So even if single crystalsare available it is always worthwhile to carry out a powder diffraction experiment first, or

to perform the single crystal experiment using a large multidetector As we have alreadypointed out the measurement of neutron diffraction intensities from a single crystal is notalways enough for the unique determination of the magnetic structure especially in the highsymmetry cases One needs to produce a single domain by applying symmetry breakingperturbations like magnetic field or uniaxial stress Sometimes unpolarized neutron dif-fraction from a monodomain single crystal is also not enough to determine the magneticstructure unambiguously In this case, it is necessary to use polarized neutrons with three-dimensional polarization analysis for a unique determination of the magnetic structure.This will be discussed in Chapter 5

3 Ferromagnetic and simple antiferromagnetic structures

In the last section we discussed the methods of magnetic structure determination In thepresent section we will describe the different types of magnetic ordering that have beenfound by neutron diffraction As previously mentioned most of our knowledge of theordered magnetic phases and their structures originate directly from neutron diffractiondata We will not attempt to describe exhaustively all types of magnetic structures deter-mined Interested readers can consult [16] and [17] for magnetic structures However, manymagnetic structures have been solved since those publications Rossat-Mignod [4] has de-scribed some selected interesting magnetic structures Examples of magnetic structures are

given in the book by Bacon [18] For modulated magnetic structures readers are referred

to [19,20] Here we will discuss some of the most common magnetic structures tered in practice The most frequently encountered structure of the ordered magnetic phase

encoun-is a simple ferromagnetic or an antiferromagnetic structure A ferromagnetic structure can

be described by a propagation vector k= 0 which means that the magnetic unit cell isidentical to the chemical unit cell Magnetic reflections are observed in the same posi-tions as nuclear reflections in neutron diffraction experiments Examples of ferromagnetic

structures among 3d elements are body centered cubic Fe, hexagonal Co, face centered

cubic Ni The low temperature magnetic structures of several hexagonal rare-earth metalsare also ferromagnetic There are many examples of ferromagnetic structures in transitionmetal, rare-earth and actinide compounds A simple antiferromagnetic structure is defined

by a propagation vector which corresponds to a symmetry point of the Brillouin zone, i.e.,

k= H/2, where H is a reciprocal lattice vector In Table 3 we give the symmetry points

of the Brillouin zones of the 14 Bravais lattices The propagation vectors corresponding

to each of these symmetry points define distinct antiferromagnetic structures The gation vector corresponding to the symmetry points which are marked by a star are those

propa-which keep the full symmetry of the paramagnetic group Gp, i.e., Gk = Gp The

corre-sponding magnetic structures do not have S or rotational domains.

Table 3

Symmetry points of the Brillouin zones of the 14 Bravais lattices (after Rossat-Mignod [4]) The symmetry

points, k= H/2, are associated with an antiferromagnetic structure Those marked by an asterisk (∗) keep the

full symmetry of the group Gk= Gp

Bravais lattice Symmetry points, k= H/2

3.1 Cubic Bravais lattice

From Table 3 we notice that for the face centered cubic (f.c.c.) lattice there are two try points and two possible propagation vectors for simple antiferromagnetic ordering with

symme-k= H/2: (0, 0, 1) and (1

2,12,12) corresponding to the so-called type-I and type-II

antifer-romagnetic ordering, respectively Note that the type-III ordering with k= (1,1

2, 0) should

not be called antiferromagnetic ordering since k = H/2 Rossat-Mignod [4] preferred

to classify this magnetic structure as a commensurate structure For type-III structures

k= (1,1

2, 0)=1

4(4, 2, 0) = H/4 Similarly the commonly encountered magnetic structure

type-IA with the wave vector k= (0, 0,1

2) = H/4 is a commensurate structure rather than

an antiferromagnetic structure The stability condition of the three types of ordering in

the f.c.c lattice has been given by Villain [21] on the basis of nearest neighbor J1 and

next nearest neighbor J2Heisenberg exchange interaction The stability condition for the

type-I structure is J1< 0 < J2and those for type-II and type-III are J2<12J1< −J2and1

2J1< J2< 0, respectively Furthermore, the stability condition for a ferromagnetic

struc-ture is J1+ J2> 0 Type-I and type-II magnetic structures are quite common for many

transition metal and rare-earth binary compounds which crystallize with the NaCl-typecrystal structure Transition metal oxides MnO, FeO, CoO and NiO and the chalcogenides

α-MnS and MnSe have type-II antiferromagnetic structures Rare-earth monopnictides

and monochalcogenides RX (R= rare-earth, X = N, P, As, Sb, Bi, S, Se, Te) also haveNaCl-type crystal structures These compounds order usually with type-II antiferromag-netic structures, the exceptions being cerium and neodymium monopnictides which havetype-I magnetic structures Type IA ordering has been found in CeSb and CeBi at lowtemperatures [4] Type-III ordering has been found in MnS2 which crystallizes with the

pyrite-type crystal structure (space group Pa¯3) [18] Although the Mn atoms are situated

on an f.c.c Bravais lattice, strictly speaking there are four Bravais lattices This type of

ordering has also been found [18] in β-MnS, which crystallizes with the cubic zincblende

structure The Mn atoms are situated on an f.c.c sublattice It is to be noted that the f.c.c.lattice is inherently frustrated with respect to antiferromagnetic ordering Topological frus-tration can be explained most simply by considering an equilateral triangle of magneticmoments If any of the magnetic moments are aligned antiparallel, then the third momentcannot be simultaneously aligned antiparallel to both, so the third magnetic moment isfrustrated and is unable to decide where to point An f.c.c lattice has many such equilat-eral triangles (two nearest neighbors of a given atom can themselves be nearest neighbors)and therefore many magnetic moments are frustrated Because of this frustration the f.c.c.lattice leads to a wealth of observed magnetic phases A different magnetic structure ob-

served in CrN is called a type-IV structure with the wave vector k= (1

2,12, 0) Again this

structure should be considered as a commensurate structure rather than an netic structure Another type of magnetic structure been found in MnSe2 [18] in which

antiferromag-Mn atoms lie on an f.c.c Bravais lattice but the crystal structure is primitive cubic (Pa¯3)

and therefore strictly speaking there are actually four Bravais lattices The magnetic

struc-ture is a commensurate square-wave type with wave vector k= 1

3, 0, 1  and 3k = 0, 1, 0.

The four common modes of the magnetic ordering of the f.c.c lattice have been illustrated

in Figure 1

Fig 1 Four different ordering modes for face centered cubic lattices The black and white spheres represent oppositely oriented spins The figure shows two fundamental unit cells in each case Also indicated are the prop-

agation vectors (after Brown [10]).

For a body centered cubic (b.c.c.) lattice there are two symmetry points with k= H/2

corresponding to the wave vectors k = (0, 0, 1) and k = (1

2,12, 0) Neglecting the

long-period sine-wave modulation (spin-density-wave) [22], the magnetic structure of

b.c.c Cr can be described approximately by the wave vector k= (0, 0, 1) The magnetic

moments of the Cr atoms situated on the corners of the unit cell have parallel tion and are antiparallel to the direction of the magnetic moments of the body centered

orienta-Cr atoms The magnetic moments are parallel to one of the cubic direction perpendicular

to the wave vector However, the actual magnetic structure of Cr is a long-period density-wave structure

spin-Fig 2 Different ordering modes for simple cubic lattices The black and white spheres represent oppositely oriented spins The figure shows two fundamental unit cells in each case Also indicated are the propagation

vectors (after Brown [10]).

For a primitive cubic lattice there are three symmetry points with k= H/2 viz.,

k= (0, 0,1

2), k = (1

2,12, 0) and k = (1

2,12,12) The corresponding three different types

of ordering are illustrated in Figure 2 along with the ferromagnetic ordering The and the Cu3Au-type crystal structures with the magnetic rare-earth atoms on a simple cu-bic Bravais lattice are quite common in rare-earth actinide intermetallics The equiatomic

CsCl-rare-earth intermetallic compounds RM (R= rare-earth, M = Cu, Ag, Au, Zn, Mg, Rh,etc.) have the CsCl-type crystal structure RZn compounds are ferromagnets RCu and

RAg compounds order antiferromagnetically with the wave vector k= (1

struc-3.2 Hexagonal Bravais lattice

There exist five symmetry points in a hexagonal lattice defining three kinds of

mag-netic structures: (1) hexagonal (k= 0, k = (0, 0,1

3,13,12)) Note that the triangular

structures are not strictly antiferromagnetic

3.3 Tetragonal Bravais lattice

For a body centered tetragonal lattice there are the following symmetry points with

k= H/2: k = (0, 0, 1) (c > a), k = (1, 0, 0) (c < a), k = (1

2,12, 0) and k = (1

2, 0,12).

Among the body centered tetragonal systems, K2NiF4- and ThCr2Si2-type structures have

been widely investigated The former type of structures are found for many 3d compounds

which form quasi-two-dimensional magnetic systems, whereas the latter type of structure

is common for heavy fermion superconductors Another family which has been studiedrecently is the parent compounds R2CuO4(R= Pr, Nd, Gd, etc.) of the electron-dopedhigh temperature superconductors We wish to point out that it is possible to have an anti-

ferromagnetic structure with k= 0 In this case, the magnetic unit cell is identical with thechemical cell (primitive) The positive and negative sublattices are related to one another

by one of the symmetry elements of the space group of the chemical structure, rather than

by a lattice translation The transition metal difluorides MnF2, FeF2, CoF2and NiF2with

rutile type crystal structure (P 42/mnm) illustrate this situation [10] The magnetic

struc-tures of these compounds can be described by the wave vector k= (0, 0, 0) The magnetic

unit cell is the same as the nuclear unit cell The two magnetic sublattices characterized

by positions (000) and (12,12,12) are related to one another by the screw tetrad of the space

group The magnetic structure is illustrated in Figure 3

3.4 Single-k and multiple-k magnetic structures

The problem of distinguishing between single-k and multiple-k magnetic structures is a

general problem of magnetic structure determination [4] and is relevant for both surate and incommensurate structures This problem is acute for high symmetry crystal

commen-Fig 3 The antiferromagnetic structure adopted by 3d transition metal diflourides with the rutile-type crystal

structure.

structures like f.c.c structures but is also important for tetragonal and hexagonal

sys-tems For example, for type-II magnetic ordering (k= 1

2,12,12) of the magnetic atoms

on an f.c.c lattice the magnetic reflections may originate because of the presence of eral modulations (propagation vectors) in the crystal or from several domains with a sin-gle modulation (propagation vector) There is no way to distinguish between these casesfrom the diffraction patterns, which look alike One may be able to distinguish betweenthese magnetic structures only by applying magnetic fields or uniaxial stresses along one

sev-or several crystallographic directions The examples of the most symmetrical multiple-k structures associated with a wave vector k= 1

2,12,12 or k = 0, 0, 1 for the f.c.c

Bra-vais lattice is shown in Figure 4(a) For k= 1

2,12,12 there are four possible magnetic

structures: single-k, double-k, triple-k and quadruple-k magnetic structures Note that

the actual moment directions of these magnetic structures are quite different, along100

or110 or along 111 For k = 0, 0, 1 there exist three possibilities (most symmetrical):

single-k, double-k and triple-k In Figure 4(b) we illustrate similarly the most symmetrical multiple-k magnetic structures associated with a wave vector k= 0, 0,1

2 or k = 1

2,12, 0for the primitive cubic Bravais lattice For both the wave vectors the most symmetrical

structures are single-k, double-k and triple-k Note that in the case of primitive cubic

lat-tice, the difference between the three structures is not only in moment directions but also

in the size of the magnetic unit cell

Fig 4 (a) Typical examples of the most symmetrical multiple-k structures associated with the wave vector

k=  1 1 and k = 001 for the face centered cubic lattice (b) Typical examples of the most symmetrical multiple-k structures associated with the wave vector k= 00 1

2 and k = 1

2 1

2 0  for the primitive cubic lattice (after Rossat-Mignod [4]).

4 Modulated magnetic structures

A modulated magnetic structure is a superstructure of the crystal structure [19] for whichthe magnetic periodicity, caused by the spin modulation, is large compared to the periodic-ity of the crystalline lattice The translational properties of the modulated structure are fixed

by the propagation or the wave vector Ferromagnetic structures are described by the wave

vector k= 0 and therefore have the same periodicity as the nuclear structure A simpleantiferromagnetic structure is a special case in which the wave vector components are 12

For example, for the type-II antiferromagnet MnO the wave vector k= (1

2,12,12) A

modu-lated structure is described by a wave vector which has in general nonrational components

A modulated magnetic structure can often be considered to be a modulation of the initialferromagnetic or antiferromagnetic structure and is described by the wave vector

where the modulation vector δ is usually small, k0 is either the initial ferromagnetic

(k0= 0) or the antiferromagnetic wave vector (for example, k0= (1

2,12,12)) The

modu-lated magnetic structure is characterized by the appearance of satellite magnetic reflectionsclose to the initial ferromagnetic (superimposed on the nuclear reflection) or antiferromag-

2< /small>,12< /sub>,12< /sub>) (the magnetic cell is 2a × 2b × 2c),

then the determination of the propagation vector from the... 0, 0) (the magnetic cell is 2a × b × c), or k = (1

2< /small>,12< /sub>, 0)

(the magnetic cell is 2a × 2b × c) or... transition metal difluorides MnF2, FeF2, CoF2and NiF2with

rutile type crystal structure (P 42< /sub>/mnm) illustrate this situation [10] The magnetic

struc-tures