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Bragg diffraction from magnetic materials
Physica B 318 (2002) 251–260
Bragg diffraction from magnetic materials Bente Lebech Department of Materials Research, Ris National Laboratory, DK-4000 Roskilde, Denmark
Abstract Neutrons form a penetrating neutral probe, which makes it possible to use neutrons scattering techniques to study bulk materials, localise both light and heavy atoms and to distinguish between isotopes (e.g. hydrogen and deuterium). These properties make neutron scattering complementary to X-ray scattering when studying crystalline properties of matter. In addition, neutrons possess magnetic moments of the same order of magnitude as the atomic magnetic moments in elements and this makes neutrons highly suited for studies of the order and interactions between the magnetic moments in magnetic materials. This will be illustrated by reviewing the ordered magnetic structures found in some simple elements and in some chemically more complex systems containing several magnetic elements. The different scattering techniques (two- and three-axis neutron scattering, small angle neutron scattering, polarised neutron scattering (polarisation analysis) and magnetic X-ray scattering) and their relevance for the particular study will be elucidated. r 2002 Elsevier Science B.V. All rights reserved. PACS: 75.10.b; 75.25.+z; 75.30.m Keywords: Neutron scattering; Magnetic structures; Rare earths; Transition metals; Actinides
1. Introduction Neutrons form a penetrating neutral probe, which makes it possible to use neutron scattering techniques to study bulk material, localise both light and heavy atoms and to distinguish between isotopes (hydrogen and deuterium). This feature is caused by the fundamental properties of the neutron: It has a mass similar to hydrogen, energies from room temperature and below and wavelengths comparable with distances in solid matter. These are properties which make neutron scattering complementary to X-ray scattering when studying crystalline properties of matter and for many years neutron scattering has been E-mail address: [email protected] (B. Lebech).
used to locate hydrogen in organic compounds. In addition, neutrons possess magnetic moments of the same order of magnitude as the atomic magnetic moment in compounds containing magnetic elements. This makes neutrons superbly suited for studies of the order and interactions between the magnetic moments caused by the unpaired electrons in magnetic elements. With a quotation from the Nobel poster [1] in connection with award of the 1994 Nobel prize in physics to Clifford G. Shull and Bertram B. Brockhouse ‘Neutrons show where atoms are and what atoms do.’ Since Shull [2] and collaborators solved the first magnetic structure in 1949 a kaleidoscopic array of exotic magnetic structures has been revealed. Magnetic structures such as helices, cycloids, cones, spin-slips, multi-q and magnetic
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 0 7 8 4 - 6
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important to note that for neutrons the coherent nuclear scattering amplitudes vary irregularly with atom number and that the coherent magnetic scattering amplitudes is of the same order of magnitude as the nuclear scattering amplitudes. The neutron coherent nuclear scattering amplitudes arise from the nucleus only and are independent of sin y=l: This contrasts the neutron coherent magnetic scattering amplitudes which arise because of unpaired electrons in the electron cloud and therefore depend on sin y=l in a similar way as for X-ray or electron scattering. Most of the magnetic structures are periodic with magnetic unit cells that repeat themselves over distances ranging from a few times the extension of the chemical unit cell to more than hundred times the extension of the chemical unit cell. The very long period structures are rare and exist mainly in compounds having crystal structures, which lack a centre of inversion. In these symmetries a special term in the free energy, the so-called Dzyaloshinskii-Moriya term may cause a
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NEUTRONS
Magnetic scattering amplitudes 5f - experiments on mono pnictides and chalcogenides 4f - elements, calculated 3d - elements, calculated Nuclear scattering amplitudes Naturally occuring elements Isotopes
125
100
Ag
Au
X-RAYS
sin θ /λ = 0
75
sinθ /λ= 0.5
ELECTRONS
sin θ /λ = 0
Cu
sinθ /λ= 0.5
Al
50 58
Ni
25
0
-13
SCATTERING AMPLITUDE -9 cm for neutrons and x-rays) (10 cm for electrons)
flux line lattices exist in magnetic materials ranging from the the chemically simplest to very complex compounds. Many compounds that are vital for technological applications like high Tc super-conducting materials and permanent magnets order magnetically at low temperatures and the understanding of the magnetic sub-lattice interactions in these compounds on a microscopic level is essential. For this, neutron diffraction is the best-suited experimental technique but X-ray diffraction plays an important complementary role when it comes to elucidating certain aspects or predictions based on the neutron scattering results. In order to illustrate the complementarity between X-ray and neutron diffraction, Fig. 1 summarises the coherent neutron scattering amplitude for all atoms and isotopes [3] and indicates the scattering amplitudes of X-rays and electrons [4]. The magnetic scattering amplitudes calculated as described in Ref. [3] for some compounds containing 3d (8), 4f (X) and 5f (M) elements are also included in Fig. 1. It is
(10
62
Ni
0
20
40
60
80
100
ATOM NUMBER Fig. 1. Neutron coherent scattering amplitudes (nuclear and magnetic) for all the elements. Also shown are some corresponding values for X-rays and electrons. Note that the ordinate for electrons differ from the ordinate used for neutrons and X-rays.
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slowly varying instability of the ferromagnetic order. The most well known examples are cubic MnSi, FeGe and their alloys [5] with periods of ( B600 A ( up to B1800 A. ( A more recent B180 A, example is CsCuCl3 [6]. Many magnetic structures may conveniently be described by a sum of cosine functions including one fundamental harmonic and its higher harmonics. These are called modulated magnetic structures. They include helices, cones, helifans, cycloids and single-q or multi-q structures, some of which will be described below. Additional details may be found for instance in Ref. [7]. These structures can be changed, sometimes dramatically, with changed external conditions such as temperature, applied magnetic fields and applied pressure, which affect the magnetic interaction parameters. Fig. 2 shows an artists impression of how the magnetic order of the rare-earth element Ho changes when an applied magnetic field is increased. The crystal structure of Ho is hexagonal close packed. All the magnetic moments in planes normal to the hexagonal c-axis are parallel, but the direction of the moments in the different planes rotate. In Fig. 1, each layer is represented by a step in a staircase and the arrow and the step points in the direction of the magnetic moments in that layer. The zero field magnetic structure is called a helix structure (left) and is observed in heavy rare-earth elements close to the Ne! el temperature [8]. At lower temperatures, regular faults or missing steps occur and the helix changes to the so-called spinslip structure [9]. The fan (centre) is the precursor of the complete ferromagnetic alignment that
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occurs when very high fields are applied to a compound with the helix structure at zero fields. The helifan (right), composed of helix- and fanlike sections, may occur at intermediate fields. The latter structure was suggested in Ref. [10] and used to describe the magnetic structure in Ho at 50 K in an applied magnetic field of 1.2 T. In Sections 2 and 3 we describe briefly some experimental techniques for determination of magnetic structures and remark on some of the features mentioned above. In Section 4 follows a summary of the result of a recent magnetic structure study which resolves long-term controversial interpretations of the magnetic structure of REFe4Al8 compounds and illustrates the complementarity of the use of conventional neutron diffraction, zero-field neutron polarimetry and resonant X-ray magnetic scattering for magnetic structure studies. Finally in Section 5 we quote recent diffraction data on a series of RERu2X2 (X=Ge, Si) compounds with long-range somewhat unusual antiferromagnetic order.
2. Techniques for determination of magnetic structures Relatively simple magnetic structures may be investigated by neutron diffraction techniques using conventional crystallographic methods like four-circle diffractometry or powder diffractometry. However, in order to unravel the details of the magnetic structures of compounds with more than one kind of magnetic atoms (or one symmetry site)
Fig. 2. Influence of a magnetic field on the magnetic helix structure that is observed in several heavy rare-earth metals. After Ref. [10].
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and determine the complex magnetic structures mentioned above (spin-slip structures of Ho [9], the multi-q structures of Nd [11,12] and CeAl2 [13], magnetic flux line lattices [14]), it is preferable to use single crystalline samples and untraditional crystallographic methods such as two- and threeaxis diffractometry and small angle neutron scattering. Therefore, although some magnetic structures have been determined exclusively by using powder neutron diffraction data, we limit our discussion to magnetic structures studied by single crystal methods. The reason for this is that knowledge of a magnetic structure implies the determination of the length and directions of magnetic propagation vectors, determination of the direction, length and mutual orientation of the magnetic moments of the different kinds of magnetic atoms and finally, determination of the coupling between the propagation vectors and the magnetic moments. Correct determination of the size of the magnetic moments further requires determination of higher harmonics; correct indexing of the magnetic Bragg peaks and correct estimation of the intensity in these peaks. All of this is best done using single crystals with welldefined axes orientations and preferably wellcharacterised magnetic domain distributions. A neutron diffraction study of a magnetic structure does not start from scratch. One usually base the study on available information from bulk magnetic measurements, which means that one at least knows the ordering temperature and the temperature dependence of the average magnetic susceptibility. For the complex magnetic structures a symmetry analysis, which predicts the possible incipient magnetic structures consistent with the crystallographic space group, is necessary but not sufficient. A useful, but unfortunately not userfriendly tool for this is the recently developed PCprogramme MODY [15]. An optimal way to study a magnetic structure would be to get an estimate of the dimension of the magnetic unit cell from powder diffraction data and switch to single crystal methods for more detailed studies. From neutron diffraction scans in reciprocal space along and parallel to the high symmetry directions through a number of nuclear reciprocal lattice points, either by step scanning using a single
detector or by use of area sensitive or multidetector techniques there is a rather high probability to collect the maximum information about the direction of the propagation vectors, the number of harmonics and the intensities of the magnetic peaks. Presently, step scan data using a single detector are the simplest to analyse in terms of intensities and peak positions and use for magnetic structure refinements. The only difference from conventional crystallography is that instead of letting the change of instrument settings correspond to equidistant steps in angle (degrees) between data points, the change of instrument settings correspond to equidistant steps in reci( 1). The implications for the procal lattice units (A width of the Bragg peaks and their integrated intensity are discussed in Ref. [16]. However, while data from area sensitive detectors or multidetector instruments yield a wealth of information about the magnetic structure, the crystallographic software for analysis of this type of data is presently sparse and needs to be developed in order to extract relevant data for a crystallographic analysis. Until rather recently most magnetic structures were solved by means of un-polarised neutron diffraction although polarised neutrons were used to elucidate special features [17,18], e.g. helix chiralities, separate magnetic from nuclear Brag peaks, etc. However, with the development of the CRYOPAD polarimeter [19] and the possibility to perform zero-field neutron polarimetry on a polarised neutron triple-axis spectrometer, it is now possible to measure all three components of the magnetic polarisation vectors resulting in a magnetic Bragg peak and perform a magnetic structure analysis. In principle this gives a complete experimental dataset where only the relative phases between the moment components are undetermined and must be selected by physical arguments, e.g. energy calculations. X-ray magnetic scattering is much weaker than neutron magnetic scattering, but resonant X-ray magnetic scattering is indeed powerful. Hereby one can study the magnetic order of specific elements by tuning the X-ray energy to an absorption edge of the particular element and hence supplement the result obtained from neutron diffraction [20].
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3. Some special features existing in magnetic structures Magnetic structures may be classified as commensurate or incommensurate with the crystallographic unit cell. In most cases it is relatively easy to distinguish between the two types. However, if the repeat distance becomes very long the modulation type is more difficult to establish and for descriptive purposes these structures are most easily considered as incommensurate. In general, the symmetry of the magnetic structure is lower than the symmetry of the crystal structure and a magnetic unit cell, which is defined as integer multipla of the chemical cell, can describe a commensurate magnetic structure. Nevertheless, the more general way of describing the periodic magnetic structure is in terms of Fourier components with magnetic propagation vectors nj qj that describe the magnetic moment with amplitude S jl of the jth magnetic atom at position rjl in the lth magnetic unit cell. In most cases, the directions of the propagation vectors are parallel to or rather close to equivalent high symmetry directions and the direction and length of jqj j may vary with the environment, in particular as function of temperature. This dependence may be continuous for instance through incommensurate values that eventually can lock in at a commensurate value or change abruptly by jumps from one commensurate value to another. The former is observed in the light rare-earth element Nd [21] and the latter in the rare-earth monopnictide CeSb [22]. Even though a magnetic structure seems to be incommensurately modulated in one dimension it may in fact be commensurately modulated in two dimensions. Such behaviour has been suggested for Nd. As function of temperature the length of jqj varies while q turns away from the a reciprocal lattice direction and back again. Whenever q is along a ; jqj attains commensurate values 3 ð17; 18; 25 ?Þ and it has further been argued that the temperature dependence of q ¼ ðq8 ; q> Þ agrees with a sequence of commensurate values in the a b -plane [21]. In the language of group theory, a magnetic structure is a single-q structure if it is described by only one modulation vector. When more modulation vectors are needed, the magnetic
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structure is a multi-q structure. This means that for theorists a structure described by a fundamental first harmonic q and its higher harmonics nq and a structure described by non-collinear vectors q1 and q2 are both multi-q structures. This can lead to confusion because experimentalists call the former structure a squared-up single-q structure and the latter a two-q structure. The reason for this is that experimentally the fingerprints of these two structures are quite different. The squared-up single-q structure, which is usually formed well below the Ne! el temperature, will, for a single magnetic domain, give first order magnetic peaks at collinear distances 7jqj; 73jqj; y; 7njqj from nuclear Bragg reflections in the q-direction of reciprocal space with intensities given by the moment amplitude S q : The higher harmonic magnetic peaks will appear at collinear distances|, 73jqj; y; 7njqj from nuclear Bragg reflections in the q-direction of reciprocal space with intensities given by the moment amplitudes S 3q ; y; S nq : In contrast, the fingerprints for a single magnetic domain two-q structure are first order magnetic peaks at non-collinear positions 7q1 and 7q2 from the nuclear Bragg reflections with intensities given by S q1 and S q2 : The higher harmonic magnetic peaks appear at positions 7ðnq1 þ nq2 Þ and 7ðnq1 nq2 Þ which are mutually non-collinear and also non-collinear with q1 and q2 : A good example of a squared up structure is NdCu2 [23] but many other compounds show a similar behaviour. Elemental Nd is perhaps the prime example of a multi-q structure. From group theoretical considerations and experimental evidence it was originally suggested that the magnetic structure is single-q just below TN ; changes to a three-q structure B0.5 K below TN followed by further changes to other multi-q structures below 10 K [11]. At the lowest temperatures the diffraction pattern becomes quite complex with 36 first order magnetic satellites around each Bragg point in the double hexagonal closed packed structure (P63/mmc) plus an immense number of higher order satellites [12]. Based on a series of neutron diffraction studies in applied magnetic field or under pressure [12] it has been shown that the magnetic structure of Nd is indeed single-q just below TN ; but it changes to a two-q structure
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B0.5 K below TN : This order that is predominantly on the 2d-sites (so-called hexagonal) of the crystal structure, persists to B10 K where a three-q magnetic structure is formed. This structure combines moments on both the 2d- and the 2asites (so-called nearly cubic) to a magnetic structure, which transforms to a four-q structure at the lowest temperatures. The structural sequence is very well described in Ref. [12] but a complete data set with a formal magnetic structure analysis has not yet been attempted despite the fact that compound is just a single element!
4. Mixed 2d- and 4f- or 5f-compounds with more than one magnetic atom The magnetic properties of inter-metallic compounds with the ThMn12 tetragonal type crystal structure and composition MFe12xXx where M is a 4f- or 5f-element and X a p-element (Al, Si) have attracted much interest. The reason for this is that some of the compounds were seen as possible high-Tc materials. Others, predominantly the compounds with higher Fe concentration like SmFe10Si2 were considered to be potential magnet-material because they are ferromagnetic with high Curie temperatures and relatively large magnetic anisotropies. The stoichiometric compounds of composition MFe4Al8 exhibit interesting magnetic properties whose interpretation has been controversial for many years. Early studies of the magnetic order in polycrystalline material and/ or non-stoichiometric samples had led to a series of conflicting proposals including spin-glass behaviours at low temperatures. These conflicts are now essentially resolved through a combination of theoretical considerations and advanced experimental techniques. To illustrate this, two examples are chosen—the 5f-compound UFe4Al8 [24] and the 4f-compounds DyFe4Al8 and HoFe4Al8 [25]. UFe4Al8 has two magnetic sublattices, which contribute to the magnetic scattering. The U atoms contribute to the magnetic intensity in all space-group allowed reflections. No extra peaks were observed in the neutron diffraction data indicating that any magnetic order of the U atoms would be ferromagnetic. By symmetry, the Fe
atoms only add magnetic intensity to some reflections and hence the observed modification of certain peaks in the diffraction pattern below a particular temperature is a strong indication of the magnetic order on the Fe sites. Un-polarised neutron diffraction studies of a stoichiometric single crystal showed that the principle interaction is a commensurate so-called G-type antiferromagnetic ordering of the Fe sublattice having moments in the ab-plane. However, symmetry arguments suggested that the Fe sublattice had a week ferromagnetic component also in the basal plane. Polarised neutron diffraction experiments in a field established a small basal plane ferromagnetic moment on the U atoms. Furthermore, a magnetic field in the b-direction showed that the Fe sublattice anti-ferromagnetism is aligned perpendicular to the field, i.e. in the basal plane a-direction. Hence, UFe4Al8 is an example of a commensurate magnetic structure where the magnetic cell is identical to the chemical unit cell. The weak ferromagnetic basal plane component of the Fe moments corresponds to a canting of 161. Relatively small magnetic fields are sufficient to move the moments further towards the applied fields and 4.6 T along the b-direction results in a canting of 251. The magnetic structure is compatible with symmetry and shows beyond doubt that UFe4Al8 is not a spin-glass. There is little anisotropy in the basal plane and the canting of the Fe moments must be a result of the exchange field established by the ferromagnetic order on the U atoms. In the sister compounds DyFe4Al8 and HoFe4Al8 the Fe sublattice moments order at 175 K and form circular cycloids with the propagation 2 vectors along [1 1 0] (jqjB15 r.l.u. and jqjB17 r.l.u., respectively) and with moments rotating in the abplane. The Fe sublattice structure is a small modification of the G-type mkmk antiferromagnetic structure propagating along (1 1 0) with a long-wavelength modulation superimposed. At B50 K, respectively B80 K, the rare-earth moments starts to order and follow the modulation of the Fe sublattice but with a phase difference (Df) between the cycloids of the two sublattices. The observed temperature dependencies are shown in Fig. 3. The top panel (DyFe4Al8) and the centre panel (HoFe4Al8) shows the intensity variation for
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Fig. 3. Top and centre panels: Temperature dependencies of the magnetic intensity of the first order satellites around the reflections (1 2 1)+ and (1 1 0)+ of DyFe4Al8 (top; third and fifth orders are shown in the inset) and HoFe4Al8 (centre). The dashed curves are the extrapolated Fe-sublattice contributions to the (1 1 0)+ satellite. The solid curves below B50 K are the calculated temperature dependencies of the (1 1 0)+ using the procedure sketched in the text and described in detail in Ref. [24]. Bottom panel: Temperature dependencies of the magnetic modulation vector q for DyFe4Al8 and HoFe4Al8.
the (1 2 1)+ and (1 1 0)+ satellites and the bottom panel the corresponding dependence of the magnetic modulation vector q: Similarly to UFe4Al8,
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the rare-earth atoms contribute to the magnetic intensity in all space-group allowed reflections, whereas the Fe atoms only contribute to the intensity in some reflections (e.g. (1 1 0)). Because it is the temperature dependent scattering amplitudes of each sublattice that adds coherently, the magnetic cross section contains a Df-dependent interference term between the two magnetic sublattices. Hence, the temperature dependence of the magnetic intensity depends on whether the interference between the sublattices is constructive (01oDfo901) or destructive (901oDfo1801). The results of model calculations with this in mind (see Ref. [25] for details) are included in Fig. 3 (top) and (centre). The model calculation reproduces reasonably well the observed temperature dependencies below 50 K for the (1 1 0) magnetic satellites in DyFe4Al8 (Df ¼ 431) and HoFe4Al8 (Df ¼ 1451). At the lowest temperatures there is some disagreement between model and experiment for DyFe4Al8, presumably because the model calculation neglects the relatively intense higherorder harmonics of the modulation that develop below B15 K and change the magnetic structure of the rare-earth ion to a bunched elliptical cycloid. Similarly to the circular cycloid, the elliptical cycloid follows the modulation of the Fe sublattice with a relative phase between the cycloids of the two sublattices. Another interesting feature is that although the antiferromagnetic coupling of the rare-earth magnetic moments has long-range order giving sharp magnetic satellites in the diffraction patterns, a non-negligible fraction of the 4f-moment does not contribute to these peaks but appears as diffuse scattering beneath the Bragg peaks—an indication of short-range ferromagnetic correlations between neighbouring rare-earth moments. The results sketched above represent a tour-de-force [20,25] effort. It involved magnetisation measurements, unpolarised neutron diffraction, magnetic X-ray scattering and zero-field neutron polarimetry on the same well-characterised, stoichiometric single crystals. It clearly illustrates the importance of access to and the ability of mastering many different microscopic techniques when attempting to understand complex magnetic behaviours.
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5. Magnetic structures of RERu2 X 2 (RE=Tb, Ho, Dy; X=Si, Ge)
1.40 1.20 1.00
MILLER INDEX k
0.80 0.60 0.40 0.20 0.00 -0.20 -0.40 0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.20
1.40
MILLER INDEX h 1.40 1.20 1.00 0.80
MILLER INDEX k
The ternary rare-earth compounds, RERu2X2 (RE=Tb, Ho, Dy; X=Si, Ge) with the tetragonal ThCr2Si2-type structure (space group: I4/mmm), show a variety of interesting magnetic properties. Amongst them are successive metamagnetic transitions caused by a frustration of the RKKY-type exchange interactions and the large uniaxial anisotropy [26]. The magnetic order arise from the rare-earth atom only, which form a body centred tetragonal lattice. Bulk magnetic measurements have shown that huge crystalline anisotropy confine the rare-earth moments to the c-axis. A series of recent neutron diffraction studies of several of these compounds [27] have revealed complex diffraction patterns which may be interpreted as commensurately modulated in two dimensions at the lowest temperatures. Just below TN the fundamental magnetic modulation vec3 tors are long ranged with q ¼ Bð13 0 0Þ; 4 2 6 Bð17 0 0Þ; Bð9 0 0Þ and Bð29 0 0Þ for TbRu2Si2, TbRu2Ge2, DyRu2Si2 and HoRu2Si2, respectively. Although the ordering temperatures differ, the diffraction patterns are similar but complex compared to the diffraction patterns observed for most other magnetic compounds. The magnetic satellites are not only observed along reciprocal lattice lines related by symmetry and parallel to high symmetry directions through nuclear Bragg peaks ðh 0 0Þ: They appear also along several low symmetry lines of the form ðh7n=13 0 0Þ: For simplicity we limit the further discussion to TbRu2Si2 and describe models for the magnetic structures of this compound. These models agree qualitatively with our experimental findings but structure factor calculations are needed to confirm the models. TbRu2Si2 orders antiferromagnetically at TN B56 K into a commensurately modulated 3 structure with q0 Bð13 0 0Þ which persists to Tt B7:5 K. The diffraction pattern is relatively simple with satellites which may be indexed by q0 and its odd harmonics. The high temperature magnetic structure has a 13a a c magnetic unit cell and involves 26 (1 0 0) Tb-atom planes perpendicular to the a-axis and separated by a=2: The spin configuration in a single domain may be
0.60 0.40 0.20 0.00 -0.20 -0.40 0.00
0.20
0.40
0.60
0.80
1.00
MILLER INDEX h
Fig. 4. Equal intensity contour maps (logarithmic scale) of the Bragg reflections observed in the a*b*-reciprocal lattice plane of TbRu2Si2 at 4.5 K (top panel) and 1.8 K (bottom panel). The (1 1 0) peak is nuclear, while all other peaks are magnetic satellites. The ring of scattering in the upper right-hand corners is caused by the cryostat. The numbers marked on the fat contour lines correspond to log10 ðIÞ:
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described by d4 4% 4d4% 4 4% : Here ‘4’ and ‘4% ’ denote four consecutive spin-up ferromagnetic planes and spin-down ferromagnetic planes, respectively. ‘d’ a frustrated or effectively paramagnetic plane. In the intermediate phase (7:5 K > T > Tt0 B4:0 K), satellites appear on low symmetry lines as described above and illustrated by the equal intensity contour maps in Fig. 4. We suggest that the magnetic modulation vector change to 3 3 q0 Bð13 0Þ with a the magnetic unit cell equal to 13 13a 13a c: The spin configuration in a single domain still involves 26 (1 0 0) Tb-atom planes perpendicular to the a-axis and separated by a=2: It is described by 3 4 4% 4 3 4% 4 4% ; i.e. a sequence similar to the high temperature magnetic phase with the very important difference that ‘3’ denotes antiferromagnetic planes shifted by p every 13th layers containing frustrated or effectively paramagnetic Tb-atoms [27]. These antiferromagnetic planes are in a sense discommensurations, defect planes or spin-slip planes similar to those found in Ho [9]. The low temperature phase which appears below B4.0 K is more complex because the satellites splits and moves. The magnetic structure is presumably similar to that of the intermediate phase but a model cannot yet be proposed.
Acknowledgements The examples of magnetic structures included in this paper have been selected mainly from collaborative work, which has involved scientists from many different institutions and countries. It is my hope that the list of references is at least representative enough to give them some of the credit they rightly deserve.
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[20] S. Langridge, J.A. Paix*ao, N. Berhoeft, C. Vettier, G.H. Lander, D. Gibbs, S.Aa. S rensen, A. Stunault, D. Wermeille, E. Talik, Phys. Rev. Lett. 82 (1999) 2187. [21] B. Lebech, J. Wolny, R.M. Moon, J. Phys.: Condens. Matter 6 (1994) 5201. [22] P. Fischer, B. Lebech, G. Meier, B.D. Rainford, O. Vogt, J. Phys. C 11 (1978) 345; J. Wolny, B. Lebech, A. Wne-k, Hae Seop Shim, J. Phys. Soc. Japan 70 (2001) (Suppl. A) 115. [23] M. Loewenhaupt, Th. Reif, P. Svoboda, S. Wagner, . M. Wafferschmidt, H.v. Lohneysen, E. Gratz, M. Rotter, B. Lebech, Th. Hauss, Z. Phys. B 101 (1996) 499. [24] J.A. Paix*ao, B. Lebech, A.P. Gonc-alves, P.J. Bown, G.H. Lander, P. Burlet, A. Delapalme, J.C. Spirlet, Phys. Rev. B 55 (1997) 14370 and references therein.
[25] J.A. Paix*ao, M. Ramos Silva, S.Aa. S rensen, B. Lebech, G.H. Lander, P.J. Bown, S. Langridge, E. Talik, A.P. Gonc-alves, Phys. Rev. B 61 (2000) 6176 and references therein. [26] D. Gignoux, D. Schmitt, in: K.H.J. Bushow (Ed.), Handbook of Magnetic Materials, Vol. 10, Elsevier Science B.V, Amsterdam, 1997, p. 239. [27] S. Kawano, Y. Kosugi, M. Takahashi, T. Shigeoka, N. Iwata, J. Phys. Soc. Japan 70 (2001) 209; S. Kawano, B. Lebech, M. Takashi, T. Shigeoka, N. Iwata, T. Kosugi, these proceedings, Physica B (2002); S. Kawano, B. Lebech, T. Shigeoka, N. Iwata, Proceedings of ICNS2001, Munich September 9–13, 2001 and references therein.
Bragg diffraction from magnetic materials Bente Lebech Department of Materials Research, Ris National Laboratory, DK-4000 Roskilde, Denmark
Abstract Neutrons form a penetrating neutral probe, which makes it possible to use neutrons scattering techniques to study bulk materials, localise both light and heavy atoms and to distinguish between isotopes (e.g. hydrogen and deuterium). These properties make neutron scattering complementary to X-ray scattering when studying crystalline properties of matter. In addition, neutrons possess magnetic moments of the same order of magnitude as the atomic magnetic moments in elements and this makes neutrons highly suited for studies of the order and interactions between the magnetic moments in magnetic materials. This will be illustrated by reviewing the ordered magnetic structures found in some simple elements and in some chemically more complex systems containing several magnetic elements. The different scattering techniques (two- and three-axis neutron scattering, small angle neutron scattering, polarised neutron scattering (polarisation analysis) and magnetic X-ray scattering) and their relevance for the particular study will be elucidated. r 2002 Elsevier Science B.V. All rights reserved. PACS: 75.10.b; 75.25.+z; 75.30.m Keywords: Neutron scattering; Magnetic structures; Rare earths; Transition metals; Actinides
1. Introduction Neutrons form a penetrating neutral probe, which makes it possible to use neutron scattering techniques to study bulk material, localise both light and heavy atoms and to distinguish between isotopes (hydrogen and deuterium). This feature is caused by the fundamental properties of the neutron: It has a mass similar to hydrogen, energies from room temperature and below and wavelengths comparable with distances in solid matter. These are properties which make neutron scattering complementary to X-ray scattering when studying crystalline properties of matter and for many years neutron scattering has been E-mail address: [email protected] (B. Lebech).
used to locate hydrogen in organic compounds. In addition, neutrons possess magnetic moments of the same order of magnitude as the atomic magnetic moment in compounds containing magnetic elements. This makes neutrons superbly suited for studies of the order and interactions between the magnetic moments caused by the unpaired electrons in magnetic elements. With a quotation from the Nobel poster [1] in connection with award of the 1994 Nobel prize in physics to Clifford G. Shull and Bertram B. Brockhouse ‘Neutrons show where atoms are and what atoms do.’ Since Shull [2] and collaborators solved the first magnetic structure in 1949 a kaleidoscopic array of exotic magnetic structures has been revealed. Magnetic structures such as helices, cycloids, cones, spin-slips, multi-q and magnetic
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 0 7 8 4 - 6
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important to note that for neutrons the coherent nuclear scattering amplitudes vary irregularly with atom number and that the coherent magnetic scattering amplitudes is of the same order of magnitude as the nuclear scattering amplitudes. The neutron coherent nuclear scattering amplitudes arise from the nucleus only and are independent of sin y=l: This contrasts the neutron coherent magnetic scattering amplitudes which arise because of unpaired electrons in the electron cloud and therefore depend on sin y=l in a similar way as for X-ray or electron scattering. Most of the magnetic structures are periodic with magnetic unit cells that repeat themselves over distances ranging from a few times the extension of the chemical unit cell to more than hundred times the extension of the chemical unit cell. The very long period structures are rare and exist mainly in compounds having crystal structures, which lack a centre of inversion. In these symmetries a special term in the free energy, the so-called Dzyaloshinskii-Moriya term may cause a
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NEUTRONS
Magnetic scattering amplitudes 5f - experiments on mono pnictides and chalcogenides 4f - elements, calculated 3d - elements, calculated Nuclear scattering amplitudes Naturally occuring elements Isotopes
125
100
Ag
Au
X-RAYS
sin θ /λ = 0
75
sinθ /λ= 0.5
ELECTRONS
sin θ /λ = 0
Cu
sinθ /λ= 0.5
Al
50 58
Ni
25
0
-13
SCATTERING AMPLITUDE -9 cm for neutrons and x-rays) (10 cm for electrons)
flux line lattices exist in magnetic materials ranging from the the chemically simplest to very complex compounds. Many compounds that are vital for technological applications like high Tc super-conducting materials and permanent magnets order magnetically at low temperatures and the understanding of the magnetic sub-lattice interactions in these compounds on a microscopic level is essential. For this, neutron diffraction is the best-suited experimental technique but X-ray diffraction plays an important complementary role when it comes to elucidating certain aspects or predictions based on the neutron scattering results. In order to illustrate the complementarity between X-ray and neutron diffraction, Fig. 1 summarises the coherent neutron scattering amplitude for all atoms and isotopes [3] and indicates the scattering amplitudes of X-rays and electrons [4]. The magnetic scattering amplitudes calculated as described in Ref. [3] for some compounds containing 3d (8), 4f (X) and 5f (M) elements are also included in Fig. 1. It is
(10
62
Ni
0
20
40
60
80
100
ATOM NUMBER Fig. 1. Neutron coherent scattering amplitudes (nuclear and magnetic) for all the elements. Also shown are some corresponding values for X-rays and electrons. Note that the ordinate for electrons differ from the ordinate used for neutrons and X-rays.
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slowly varying instability of the ferromagnetic order. The most well known examples are cubic MnSi, FeGe and their alloys [5] with periods of ( B600 A ( up to B1800 A. ( A more recent B180 A, example is CsCuCl3 [6]. Many magnetic structures may conveniently be described by a sum of cosine functions including one fundamental harmonic and its higher harmonics. These are called modulated magnetic structures. They include helices, cones, helifans, cycloids and single-q or multi-q structures, some of which will be described below. Additional details may be found for instance in Ref. [7]. These structures can be changed, sometimes dramatically, with changed external conditions such as temperature, applied magnetic fields and applied pressure, which affect the magnetic interaction parameters. Fig. 2 shows an artists impression of how the magnetic order of the rare-earth element Ho changes when an applied magnetic field is increased. The crystal structure of Ho is hexagonal close packed. All the magnetic moments in planes normal to the hexagonal c-axis are parallel, but the direction of the moments in the different planes rotate. In Fig. 1, each layer is represented by a step in a staircase and the arrow and the step points in the direction of the magnetic moments in that layer. The zero field magnetic structure is called a helix structure (left) and is observed in heavy rare-earth elements close to the Ne! el temperature [8]. At lower temperatures, regular faults or missing steps occur and the helix changes to the so-called spinslip structure [9]. The fan (centre) is the precursor of the complete ferromagnetic alignment that
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occurs when very high fields are applied to a compound with the helix structure at zero fields. The helifan (right), composed of helix- and fanlike sections, may occur at intermediate fields. The latter structure was suggested in Ref. [10] and used to describe the magnetic structure in Ho at 50 K in an applied magnetic field of 1.2 T. In Sections 2 and 3 we describe briefly some experimental techniques for determination of magnetic structures and remark on some of the features mentioned above. In Section 4 follows a summary of the result of a recent magnetic structure study which resolves long-term controversial interpretations of the magnetic structure of REFe4Al8 compounds and illustrates the complementarity of the use of conventional neutron diffraction, zero-field neutron polarimetry and resonant X-ray magnetic scattering for magnetic structure studies. Finally in Section 5 we quote recent diffraction data on a series of RERu2X2 (X=Ge, Si) compounds with long-range somewhat unusual antiferromagnetic order.
2. Techniques for determination of magnetic structures Relatively simple magnetic structures may be investigated by neutron diffraction techniques using conventional crystallographic methods like four-circle diffractometry or powder diffractometry. However, in order to unravel the details of the magnetic structures of compounds with more than one kind of magnetic atoms (or one symmetry site)
Fig. 2. Influence of a magnetic field on the magnetic helix structure that is observed in several heavy rare-earth metals. After Ref. [10].
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and determine the complex magnetic structures mentioned above (spin-slip structures of Ho [9], the multi-q structures of Nd [11,12] and CeAl2 [13], magnetic flux line lattices [14]), it is preferable to use single crystalline samples and untraditional crystallographic methods such as two- and threeaxis diffractometry and small angle neutron scattering. Therefore, although some magnetic structures have been determined exclusively by using powder neutron diffraction data, we limit our discussion to magnetic structures studied by single crystal methods. The reason for this is that knowledge of a magnetic structure implies the determination of the length and directions of magnetic propagation vectors, determination of the direction, length and mutual orientation of the magnetic moments of the different kinds of magnetic atoms and finally, determination of the coupling between the propagation vectors and the magnetic moments. Correct determination of the size of the magnetic moments further requires determination of higher harmonics; correct indexing of the magnetic Bragg peaks and correct estimation of the intensity in these peaks. All of this is best done using single crystals with welldefined axes orientations and preferably wellcharacterised magnetic domain distributions. A neutron diffraction study of a magnetic structure does not start from scratch. One usually base the study on available information from bulk magnetic measurements, which means that one at least knows the ordering temperature and the temperature dependence of the average magnetic susceptibility. For the complex magnetic structures a symmetry analysis, which predicts the possible incipient magnetic structures consistent with the crystallographic space group, is necessary but not sufficient. A useful, but unfortunately not userfriendly tool for this is the recently developed PCprogramme MODY [15]. An optimal way to study a magnetic structure would be to get an estimate of the dimension of the magnetic unit cell from powder diffraction data and switch to single crystal methods for more detailed studies. From neutron diffraction scans in reciprocal space along and parallel to the high symmetry directions through a number of nuclear reciprocal lattice points, either by step scanning using a single
detector or by use of area sensitive or multidetector techniques there is a rather high probability to collect the maximum information about the direction of the propagation vectors, the number of harmonics and the intensities of the magnetic peaks. Presently, step scan data using a single detector are the simplest to analyse in terms of intensities and peak positions and use for magnetic structure refinements. The only difference from conventional crystallography is that instead of letting the change of instrument settings correspond to equidistant steps in angle (degrees) between data points, the change of instrument settings correspond to equidistant steps in reci( 1). The implications for the procal lattice units (A width of the Bragg peaks and their integrated intensity are discussed in Ref. [16]. However, while data from area sensitive detectors or multidetector instruments yield a wealth of information about the magnetic structure, the crystallographic software for analysis of this type of data is presently sparse and needs to be developed in order to extract relevant data for a crystallographic analysis. Until rather recently most magnetic structures were solved by means of un-polarised neutron diffraction although polarised neutrons were used to elucidate special features [17,18], e.g. helix chiralities, separate magnetic from nuclear Brag peaks, etc. However, with the development of the CRYOPAD polarimeter [19] and the possibility to perform zero-field neutron polarimetry on a polarised neutron triple-axis spectrometer, it is now possible to measure all three components of the magnetic polarisation vectors resulting in a magnetic Bragg peak and perform a magnetic structure analysis. In principle this gives a complete experimental dataset where only the relative phases between the moment components are undetermined and must be selected by physical arguments, e.g. energy calculations. X-ray magnetic scattering is much weaker than neutron magnetic scattering, but resonant X-ray magnetic scattering is indeed powerful. Hereby one can study the magnetic order of specific elements by tuning the X-ray energy to an absorption edge of the particular element and hence supplement the result obtained from neutron diffraction [20].
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3. Some special features existing in magnetic structures Magnetic structures may be classified as commensurate or incommensurate with the crystallographic unit cell. In most cases it is relatively easy to distinguish between the two types. However, if the repeat distance becomes very long the modulation type is more difficult to establish and for descriptive purposes these structures are most easily considered as incommensurate. In general, the symmetry of the magnetic structure is lower than the symmetry of the crystal structure and a magnetic unit cell, which is defined as integer multipla of the chemical cell, can describe a commensurate magnetic structure. Nevertheless, the more general way of describing the periodic magnetic structure is in terms of Fourier components with magnetic propagation vectors nj qj that describe the magnetic moment with amplitude S jl of the jth magnetic atom at position rjl in the lth magnetic unit cell. In most cases, the directions of the propagation vectors are parallel to or rather close to equivalent high symmetry directions and the direction and length of jqj j may vary with the environment, in particular as function of temperature. This dependence may be continuous for instance through incommensurate values that eventually can lock in at a commensurate value or change abruptly by jumps from one commensurate value to another. The former is observed in the light rare-earth element Nd [21] and the latter in the rare-earth monopnictide CeSb [22]. Even though a magnetic structure seems to be incommensurately modulated in one dimension it may in fact be commensurately modulated in two dimensions. Such behaviour has been suggested for Nd. As function of temperature the length of jqj varies while q turns away from the a reciprocal lattice direction and back again. Whenever q is along a ; jqj attains commensurate values 3 ð17; 18; 25 ?Þ and it has further been argued that the temperature dependence of q ¼ ðq8 ; q> Þ agrees with a sequence of commensurate values in the a b -plane [21]. In the language of group theory, a magnetic structure is a single-q structure if it is described by only one modulation vector. When more modulation vectors are needed, the magnetic
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structure is a multi-q structure. This means that for theorists a structure described by a fundamental first harmonic q and its higher harmonics nq and a structure described by non-collinear vectors q1 and q2 are both multi-q structures. This can lead to confusion because experimentalists call the former structure a squared-up single-q structure and the latter a two-q structure. The reason for this is that experimentally the fingerprints of these two structures are quite different. The squared-up single-q structure, which is usually formed well below the Ne! el temperature, will, for a single magnetic domain, give first order magnetic peaks at collinear distances 7jqj; 73jqj; y; 7njqj from nuclear Bragg reflections in the q-direction of reciprocal space with intensities given by the moment amplitude S q : The higher harmonic magnetic peaks will appear at collinear distances|, 73jqj; y; 7njqj from nuclear Bragg reflections in the q-direction of reciprocal space with intensities given by the moment amplitudes S 3q ; y; S nq : In contrast, the fingerprints for a single magnetic domain two-q structure are first order magnetic peaks at non-collinear positions 7q1 and 7q2 from the nuclear Bragg reflections with intensities given by S q1 and S q2 : The higher harmonic magnetic peaks appear at positions 7ðnq1 þ nq2 Þ and 7ðnq1 nq2 Þ which are mutually non-collinear and also non-collinear with q1 and q2 : A good example of a squared up structure is NdCu2 [23] but many other compounds show a similar behaviour. Elemental Nd is perhaps the prime example of a multi-q structure. From group theoretical considerations and experimental evidence it was originally suggested that the magnetic structure is single-q just below TN ; changes to a three-q structure B0.5 K below TN followed by further changes to other multi-q structures below 10 K [11]. At the lowest temperatures the diffraction pattern becomes quite complex with 36 first order magnetic satellites around each Bragg point in the double hexagonal closed packed structure (P63/mmc) plus an immense number of higher order satellites [12]. Based on a series of neutron diffraction studies in applied magnetic field or under pressure [12] it has been shown that the magnetic structure of Nd is indeed single-q just below TN ; but it changes to a two-q structure
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B0.5 K below TN : This order that is predominantly on the 2d-sites (so-called hexagonal) of the crystal structure, persists to B10 K where a three-q magnetic structure is formed. This structure combines moments on both the 2d- and the 2asites (so-called nearly cubic) to a magnetic structure, which transforms to a four-q structure at the lowest temperatures. The structural sequence is very well described in Ref. [12] but a complete data set with a formal magnetic structure analysis has not yet been attempted despite the fact that compound is just a single element!
4. Mixed 2d- and 4f- or 5f-compounds with more than one magnetic atom The magnetic properties of inter-metallic compounds with the ThMn12 tetragonal type crystal structure and composition MFe12xXx where M is a 4f- or 5f-element and X a p-element (Al, Si) have attracted much interest. The reason for this is that some of the compounds were seen as possible high-Tc materials. Others, predominantly the compounds with higher Fe concentration like SmFe10Si2 were considered to be potential magnet-material because they are ferromagnetic with high Curie temperatures and relatively large magnetic anisotropies. The stoichiometric compounds of composition MFe4Al8 exhibit interesting magnetic properties whose interpretation has been controversial for many years. Early studies of the magnetic order in polycrystalline material and/ or non-stoichiometric samples had led to a series of conflicting proposals including spin-glass behaviours at low temperatures. These conflicts are now essentially resolved through a combination of theoretical considerations and advanced experimental techniques. To illustrate this, two examples are chosen—the 5f-compound UFe4Al8 [24] and the 4f-compounds DyFe4Al8 and HoFe4Al8 [25]. UFe4Al8 has two magnetic sublattices, which contribute to the magnetic scattering. The U atoms contribute to the magnetic intensity in all space-group allowed reflections. No extra peaks were observed in the neutron diffraction data indicating that any magnetic order of the U atoms would be ferromagnetic. By symmetry, the Fe
atoms only add magnetic intensity to some reflections and hence the observed modification of certain peaks in the diffraction pattern below a particular temperature is a strong indication of the magnetic order on the Fe sites. Un-polarised neutron diffraction studies of a stoichiometric single crystal showed that the principle interaction is a commensurate so-called G-type antiferromagnetic ordering of the Fe sublattice having moments in the ab-plane. However, symmetry arguments suggested that the Fe sublattice had a week ferromagnetic component also in the basal plane. Polarised neutron diffraction experiments in a field established a small basal plane ferromagnetic moment on the U atoms. Furthermore, a magnetic field in the b-direction showed that the Fe sublattice anti-ferromagnetism is aligned perpendicular to the field, i.e. in the basal plane a-direction. Hence, UFe4Al8 is an example of a commensurate magnetic structure where the magnetic cell is identical to the chemical unit cell. The weak ferromagnetic basal plane component of the Fe moments corresponds to a canting of 161. Relatively small magnetic fields are sufficient to move the moments further towards the applied fields and 4.6 T along the b-direction results in a canting of 251. The magnetic structure is compatible with symmetry and shows beyond doubt that UFe4Al8 is not a spin-glass. There is little anisotropy in the basal plane and the canting of the Fe moments must be a result of the exchange field established by the ferromagnetic order on the U atoms. In the sister compounds DyFe4Al8 and HoFe4Al8 the Fe sublattice moments order at 175 K and form circular cycloids with the propagation 2 vectors along [1 1 0] (jqjB15 r.l.u. and jqjB17 r.l.u., respectively) and with moments rotating in the abplane. The Fe sublattice structure is a small modification of the G-type mkmk antiferromagnetic structure propagating along (1 1 0) with a long-wavelength modulation superimposed. At B50 K, respectively B80 K, the rare-earth moments starts to order and follow the modulation of the Fe sublattice but with a phase difference (Df) between the cycloids of the two sublattices. The observed temperature dependencies are shown in Fig. 3. The top panel (DyFe4Al8) and the centre panel (HoFe4Al8) shows the intensity variation for
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Fig. 3. Top and centre panels: Temperature dependencies of the magnetic intensity of the first order satellites around the reflections (1 2 1)+ and (1 1 0)+ of DyFe4Al8 (top; third and fifth orders are shown in the inset) and HoFe4Al8 (centre). The dashed curves are the extrapolated Fe-sublattice contributions to the (1 1 0)+ satellite. The solid curves below B50 K are the calculated temperature dependencies of the (1 1 0)+ using the procedure sketched in the text and described in detail in Ref. [24]. Bottom panel: Temperature dependencies of the magnetic modulation vector q for DyFe4Al8 and HoFe4Al8.
the (1 2 1)+ and (1 1 0)+ satellites and the bottom panel the corresponding dependence of the magnetic modulation vector q: Similarly to UFe4Al8,
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the rare-earth atoms contribute to the magnetic intensity in all space-group allowed reflections, whereas the Fe atoms only contribute to the intensity in some reflections (e.g. (1 1 0)). Because it is the temperature dependent scattering amplitudes of each sublattice that adds coherently, the magnetic cross section contains a Df-dependent interference term between the two magnetic sublattices. Hence, the temperature dependence of the magnetic intensity depends on whether the interference between the sublattices is constructive (01oDfo901) or destructive (901oDfo1801). The results of model calculations with this in mind (see Ref. [25] for details) are included in Fig. 3 (top) and (centre). The model calculation reproduces reasonably well the observed temperature dependencies below 50 K for the (1 1 0) magnetic satellites in DyFe4Al8 (Df ¼ 431) and HoFe4Al8 (Df ¼ 1451). At the lowest temperatures there is some disagreement between model and experiment for DyFe4Al8, presumably because the model calculation neglects the relatively intense higherorder harmonics of the modulation that develop below B15 K and change the magnetic structure of the rare-earth ion to a bunched elliptical cycloid. Similarly to the circular cycloid, the elliptical cycloid follows the modulation of the Fe sublattice with a relative phase between the cycloids of the two sublattices. Another interesting feature is that although the antiferromagnetic coupling of the rare-earth magnetic moments has long-range order giving sharp magnetic satellites in the diffraction patterns, a non-negligible fraction of the 4f-moment does not contribute to these peaks but appears as diffuse scattering beneath the Bragg peaks—an indication of short-range ferromagnetic correlations between neighbouring rare-earth moments. The results sketched above represent a tour-de-force [20,25] effort. It involved magnetisation measurements, unpolarised neutron diffraction, magnetic X-ray scattering and zero-field neutron polarimetry on the same well-characterised, stoichiometric single crystals. It clearly illustrates the importance of access to and the ability of mastering many different microscopic techniques when attempting to understand complex magnetic behaviours.
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5. Magnetic structures of RERu2 X 2 (RE=Tb, Ho, Dy; X=Si, Ge)
1.40 1.20 1.00
MILLER INDEX k
0.80 0.60 0.40 0.20 0.00 -0.20 -0.40 0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.20
1.40
MILLER INDEX h 1.40 1.20 1.00 0.80
MILLER INDEX k
The ternary rare-earth compounds, RERu2X2 (RE=Tb, Ho, Dy; X=Si, Ge) with the tetragonal ThCr2Si2-type structure (space group: I4/mmm), show a variety of interesting magnetic properties. Amongst them are successive metamagnetic transitions caused by a frustration of the RKKY-type exchange interactions and the large uniaxial anisotropy [26]. The magnetic order arise from the rare-earth atom only, which form a body centred tetragonal lattice. Bulk magnetic measurements have shown that huge crystalline anisotropy confine the rare-earth moments to the c-axis. A series of recent neutron diffraction studies of several of these compounds [27] have revealed complex diffraction patterns which may be interpreted as commensurately modulated in two dimensions at the lowest temperatures. Just below TN the fundamental magnetic modulation vec3 tors are long ranged with q ¼ Bð13 0 0Þ; 4 2 6 Bð17 0 0Þ; Bð9 0 0Þ and Bð29 0 0Þ for TbRu2Si2, TbRu2Ge2, DyRu2Si2 and HoRu2Si2, respectively. Although the ordering temperatures differ, the diffraction patterns are similar but complex compared to the diffraction patterns observed for most other magnetic compounds. The magnetic satellites are not only observed along reciprocal lattice lines related by symmetry and parallel to high symmetry directions through nuclear Bragg peaks ðh 0 0Þ: They appear also along several low symmetry lines of the form ðh7n=13 0 0Þ: For simplicity we limit the further discussion to TbRu2Si2 and describe models for the magnetic structures of this compound. These models agree qualitatively with our experimental findings but structure factor calculations are needed to confirm the models. TbRu2Si2 orders antiferromagnetically at TN B56 K into a commensurately modulated 3 structure with q0 Bð13 0 0Þ which persists to Tt B7:5 K. The diffraction pattern is relatively simple with satellites which may be indexed by q0 and its odd harmonics. The high temperature magnetic structure has a 13a a c magnetic unit cell and involves 26 (1 0 0) Tb-atom planes perpendicular to the a-axis and separated by a=2: The spin configuration in a single domain may be
0.60 0.40 0.20 0.00 -0.20 -0.40 0.00
0.20
0.40
0.60
0.80
1.00
MILLER INDEX h
Fig. 4. Equal intensity contour maps (logarithmic scale) of the Bragg reflections observed in the a*b*-reciprocal lattice plane of TbRu2Si2 at 4.5 K (top panel) and 1.8 K (bottom panel). The (1 1 0) peak is nuclear, while all other peaks are magnetic satellites. The ring of scattering in the upper right-hand corners is caused by the cryostat. The numbers marked on the fat contour lines correspond to log10 ðIÞ:
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described by d4 4% 4d4% 4 4% : Here ‘4’ and ‘4% ’ denote four consecutive spin-up ferromagnetic planes and spin-down ferromagnetic planes, respectively. ‘d’ a frustrated or effectively paramagnetic plane. In the intermediate phase (7:5 K > T > Tt0 B4:0 K), satellites appear on low symmetry lines as described above and illustrated by the equal intensity contour maps in Fig. 4. We suggest that the magnetic modulation vector change to 3 3 q0 Bð13 0Þ with a the magnetic unit cell equal to 13 13a 13a c: The spin configuration in a single domain still involves 26 (1 0 0) Tb-atom planes perpendicular to the a-axis and separated by a=2: It is described by 3 4 4% 4 3 4% 4 4% ; i.e. a sequence similar to the high temperature magnetic phase with the very important difference that ‘3’ denotes antiferromagnetic planes shifted by p every 13th layers containing frustrated or effectively paramagnetic Tb-atoms [27]. These antiferromagnetic planes are in a sense discommensurations, defect planes or spin-slip planes similar to those found in Ho [9]. The low temperature phase which appears below B4.0 K is more complex because the satellites splits and moves. The magnetic structure is presumably similar to that of the intermediate phase but a model cannot yet be proposed.
Acknowledgements The examples of magnetic structures included in this paper have been selected mainly from collaborative work, which has involved scientists from many different institutions and countries. It is my hope that the list of references is at least representative enough to give them some of the credit they rightly deserve.
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