Neutron Scattering from Magnetic Materials part 8 pptx

206 J. Schweizer magnetic scattering, shows that a large fraction of the magnetic moments are not aligned along the applied magnetic field but is canted away from the field direction. The randomly directed strong crystalline fields in the amorphous material has the effect that the applied field can orient only the component of the magnetic moment that is parallel to it. Moreover some correlations exist between the directions of crystalline fields in adjacent cells,enough to produce a ferromagnetic short-range order of the transverse components of the magnetic moments. A more subtle arrangement of the transverse components is met in magnetically ordered amorphous materials which contain rare-earth atoms associated with a transition metal. In these systems with two types of magnetic atoms, the huge anisotropy of the rare earth plays a fundamental role. For instance, in the Er–Co amorphous alloys, the cobalt moments are ferromagnetically coupled and are parallel to one another, aligned by the external field, and the Er moments are on the average antiparallel to the Co moments, but strongly cor- related to the local crystal field axes, in spite of the fact that these axes are distributed in all directions. Such a structure with rare-earth moments distributed in space, but still more or less antiparallel to the cobalt moments, has been called “sperimagnetic” [52]. The rare- earth moments are also aligned when a magnetic field is applied, but only partly, because of their magnetic interaction with the cobalt moments; the magnetic anisotropy, very large at low temperatures, prevents the alignment from being complete. Concerning their trans- verse components, are they randomly distributed or are they correlated? The question has been answered by a uniaxial polarization analysis experiment [53]. In this experiment, the polarization was parallel to the scattering vector, with a horizontal field of 0.2 T, enough to align the magnetization of the sample. The non-spin-flip and the spin-flip scattering at low temperature are reported in Figures 30(a) and 30(b)–(c). In this configuration, if we take apart the nuclear spin incoherent scattering, the non-spin-flip scattering is only nuclear and the spin-flip scattering is only magnetic and due to the transverse components of the mo- ments (formulae (119  ) and (120  )). There are two striking results concerning the magnetic scattering. First, a noticeable amount of these transverse components are correlated, about 25% of the transverse magnetic scattering. Second, there are “ferromagnetic” rings, at the same position as those which exist for the nuclear scattering, but there are also, in between the ferromagnetic rings, other rings that have been called “antiferromagnetic”. These two types of rings are been explained by the following exchange scheme: • large and positive exchange J Co–Co is responsible for the long range order of the Co moments, • lower and negative exchange J Co–Er forces most of the Er moments to be opposed to the Co ones. However, because of their very strong anisotropy, the Er moments lie very close to their easy magnetization axes, • the last exchange J Er–Er, being smaller and negative, when the angle θ between the Co moments and the Er local axes is small, the Co–Er interactions dominate the Er–Er interactions and the Er moments order ferromagnetically at short distances as shown in Figure 31(a), but on the contrary, for large values of θ , the Co–Er interactions are weakened as in Figure 31(b), and the negative Er–Er interactions cause the Er moments to order antiferromagnetically. Polarized neutrons and polarization analysis 207 Fig. 30. (a) Non-spin-flip; (b) spin-flip; (c) spin-flip corrected for the erbium form factor. Fig. 31. Ferromagnetic (a) and antiferromagnetic (b) short range distance arrangements of the Er moments. 208 J. Schweizer 5.6. Investigation of antiferromagnetic structures In their paper, Moon et al. [3] illustrated the separation between the nuclear and the mag- netic scattering of an antiferromagnet in showing the nuclear and the magnetic patterns of a powder of α-Fe 2 O 3 by applying the polarization along the scattering vector (Figure 32). Considering that equations (114)–(117) separate the components of  M ⊥ which are par- allel or perpendicular to the uniaxial polarization  P , we can expect specific informations to analyze complex magnetic structures. A first example of information brought by the polarization analysis concerns the collinearity of the antiferromagnetic structure in dilute semiconductors (Cd, Mn)Te. The crystal structure of CdTe is of zincblende type with the Cd atoms occupying one FCC lattice and the Te atoms the second FCC lattice. Mn can replace the Cd atoms up to Fig. 32. α-Fe 2 O 3 nuclear and magnetic powder patterns separation obtained with the polarization along the scattering vector [3]. Polarized neutrons and polarization analysis 209 about 70%. In the concentration range between 17% and 60%, a spin-glass behavior is found, attributed to the lattice frustration mechanism. Between 60% and 70% Mn, addi- tional peaks are observed at low temperatures, indicating the onset of antiferromagnetic ordering, but a short-range ordering only, as the line width of the magnetic peaks are much larger than the experimental resolution width. The propagation vector is of the type (1/2, 0, 0), permitting the existence of 3 magnetic domains. Among other experiments, Steigenberger and Lindley [54] investigated the polarization of the magnetic reflections of a single crystal of composition Cd 0.35 Mn 0.65 Te with a non- polarized incident neutron beam. They found that, for a number reflections with the scat- tering vector along the direction of the polarization analysis, the scattered intensities had a nonzero polarization, but on the contrary, for those reflections with the scattering vector perpendicular to this direction, the scattered reflections were not polarized. As explained before, the only term able to produce a polarization is the cross-term i(  M ⊥ ∧  M ∗ ⊥ ), a po- larization that is parallel to the scattering vector. If a polarization is detected, such a term exists, and if such a term exists, the magnetic structure is chiral, not necessarily helicoidal, but at least not collinear. This is the case for Cd 0.35 Mn 0.65 Te. A second example concerns MnP for which Moon [55] took advantage of uniaxial polar- ization analysis to answer the questions which remained open about its helicoidal magnetic structure. In this orthorhombic crystal, an incommensurate helix propagates at low temper- atures along a, the hard axis, and the moments rotate in the (  b, c) plane. The questions that arose were: (i) does the anisotropy between the  b and the c axes of the crystal modify the helix? (ii) if yes, would it result in an elliptical helix with different  b and c components, or in a circular but distorted helix with equal components of the moments but with a bunching of the moments along the easy direction c as proposed by Hiyamizu and Nagamiya [56]? In this last case, third-order satellites should exist, but they had not been seen. Before Moon’s experiment, the best neutron data collection [57] had given, in the frame of the elliptical model, 1.29 ± 0.10µ B along c and 1.20 ± 0.05µ B along  b,givingno clear answer to the open questions. To remove this uncertainty, Moon compared the spin- flip and the non-spin-flip intensities of the satellites (2 ± δ, 0, 0) of a single crystal. For this comparison, the  b axis of the crystal was put vertical and the uniaxial polarization as well (Figure 33). With this arrangement (formulae (104)–(107)), and still in the frame of the elliptical model, the spin-flip cross-section measures the component of the moments along c while the non-spin-flip cross-section measures the component along  b.Forthe two satellites, the spin-flip intensity associated with c was found higher than the non-spin- flip intensity associated with  b, and their ratio was measured with an excellent accuracy (R =1.091 ±007). But the observed difference could also result of the bunching of the moments. If this were the case, the bunching parameter, which is related to the exchange and anisotropy constants, can be determined from the former measurement. Its value would imply that the third-order satellites are 10 −3 or 10 −4 the first-order satellites. With unpolarized neutrons, Moon looked for these very weak reflections and he then concluded that the bunching model of Hiyamizu and Nagamiya was correct. 210 J. Schweizer Fig. 33. Neutron polarization analysis of the satellites (2 ±δ,0, 0) of MnP [55]. 5.7. Conclusions on the uniaxial polarization analysis Uniaxial polarization analysis opened huge possibilities of investigations in magnetic scat- tering. The main fields of application are the separation between nuclear and magnetic scattering and the study of all the magnetic contributions for which the separation from the nuclear scattering is not straightforward: paramagnetic scattering, magnetic short range order, transverse components. It is also very useful to investigate complex magnetic struc- tures, in spite of the fact that today spherical polarization analysis offers more possibilities as will be explained in the next chapter [10]. Uniaxial polarization is rather simple to adapt on a spectrometer. Considering that po- larizing monochromators and analyzers are less efficient in term of luminosity than the nonpolarizing ones, the cost in neutron intensity is rather high, particularly at short wave- lengths. However, with the development of polarizing filters, particularly the 3 He polariz- ing filters, this inconvenience is not as strong as it was and uniaxial polarization analysis is a very powerful tool, able to solve many problems in magnetism. Appendix: Atomic Slater functions and radial integrals As seen in the text, the one electron atomic wave function can be expanded in a radial part and an angular part: ϕ(r) =   R  a (r)   m=− a m Y m  (θ, ϕ), (82) Polarized neutrons and polarization analysis 211 where θ, ϕ are the angular coordinates of r and where the Y m  are the usual spherical harmonics. Very commonly, R  a (r) is expressed as an atomic Slater function, R  a (r) = (2α) (n a +1/2) √ (2n a )! r (n a −1) e −αr ; (86) for n a =1 (1s) R  a (r) =2α 3/2 e −αr , for n a =2 (2s,2p) R  a (r) = 2α 5/2 √ 3 re −αr , for n a =3 (3s,3p, 3d) R  a (r) = 2 3/2 α 7/2 3 √ 5 r 2 e −αr . The atomic Slater exponent α is characteristic of the two quantum numbers n a and . They have been tabulated for all the electronic shells of all the elements by Clementi and Roetti [29] (attention: values given in atomic units and not in Å −1 ). Radial integrals are the Bessel–Fourier transform of these radial functions,  j L (Q)  =  ∞ 0 r 2  R  a (r)  2 j L (Qr) dr, (85) where the j L (z) are the spherical Bessel function, j 0 (z) = 1 z sinz, j 1 (z) = 1 z 2 sinz − 1 z cosz, j 2 (z) =  3 z 3 − 1 z  sinz − 3 z 2 cosz, j 3 (z) =  15 z 4 − 6 z 2  sinz +  −15 z 3 + 1 z  cosz, j 4 (z) =  105 z 5 − 45 z 3 + 1 z  sinz −  −105 z 4 + 10 z 2  cosz. Introducing these functions in (85), one obtains • for n a =1 (1s)  j 0 (x)  = 1 (1 +x 2 ) 2 , • for n a =2 (2s,2p)  j 0 (x)  = 1 −x 2 (1 +x 2 ) 4 ,  j 2 (x)  = 2x 2 (1 +x 2 ) 4 , 212 J. Schweizer • for n a =3 (3s,3p, 3d)  j 0 (x)  = 1 −(10/3)x 2 +x 4 (1 +x 2 ) 6 ,  j 2 (x)  = (56/15)x 2 −(8/5)x 4 (1 +x 2 ) 6 ,  j 4 (x)  = (16/3)x 4 (1 +x 2 ) 6 with x =Q/2α(Å −1 ). Considering that α(Å −1 ) =α(atomic units)/a 0 , with a 0 =0.52918 Å, x = Q 2α(Å −1 ) = Qa 0 2α(atomic units) = 2πa 0 α(atomic units) sinθ λ . References [1] C.G. Shull and J.S. Smart, Phys. Rev. 76 1256 (1949). [2] R. Nathans, C.G. Shull, G. Shirane and A. Andresen, J. Phys. Chem. Solids 10 138 (1959). [3] R.M. Moon, T. Riste and W.C. Koehler, Phys. Rev. 181 920 (1969). [4] S.V. Maleyev, V.G. Baryakhtar and R.A. Suris, Fiz. Tverd. Tela 4 3461 (1962), English translation: Sov. Phys. Solid State 4 2533 (1963). [5] M. Blume, Phys. Rev. 130 1670 (1963). [6] F. 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Brown Institut Laue–Langevin, B.P. 156X, 38042 Grenoble cedex, France and Physics Department, Loughborough University, Loughborough, UK E-mail: brown@ill.fr Contents Notation 217 1.Neutronpolarimetry 217 1.1.Neutronprecessioninanexternalfield 218 1.2.Classicalpolarisationanalysis 219 1.3. Multidirectional polarisation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 1.4.Sphericalneutronpolarimetry 220 2.Polarisedneutronscattering 222 2.1.Magneticscattering 222 2.2.Nuclearscattering 223 2.3.Nuclear–magneticinterference 224 2.4.TheBlume–Maleevequations 225 2.5.Tensorrepresentationofthescatteredpolarisation 225 3.Magneticdomains 226 3.1.Configurationdomains 227 3.2. 180 ◦ domains 227 3.3.Orientationdomains 229 3.4. Chirality domains . . . . . 230 4.MagneticstructuredeterminationusingSNP 232 4.1.Experimentalstrategy 232 4.2.Commensuratestructureswithnonzeropropagationvectors 233 4.3.Incommensuratestructures 234 4.4.Magneticstructureswithzeropropagationvector 237 5.Determinationofantiferromagneticformfactors 241 References 244 NEUTRON SCATTERING FROM MAGNETIC MATERIALS Edited by Tapan Chatterji © 2006 Elsevier B.V. All rights reserved 215 [...]...Spherical neutron polarimetry 217 Notation a, b, c θB g γN κ l mN M M(r) M(Q) M⊥ (Q) M⊥ M∗ (Q) ⊥ M∗ ⊥ M⊥i N (Q) N N∗ ΩL P Pi P Pi P Pi P Pij P Pij Q Q r τ ˆ u unit cell vectors Bragg angle a reciprocal lattice vector gyromagnetic ratio of the neutron crystallographic scattering vector Q = −Q a real space lattice vector neutron mass magnetic moment vector magnetisation at r magnetic structure factor at Q magnetic. .. perpendicular to Q give rise to magnetic scattering The effect of magnetic scattering on the scattered polarisation is illustrated graphically in Figure 3 Figure 3(a) corresponds to the case where M⊥ and M∗ are parallel; OA is parallel ⊥ Spherical neutron polarimetry 223 Fig 3 Relationship between the incident and scattered polarisation directions P and P for pure magnetic scattering (a) When M⊥ is parallel... rise is fundamental in interpreting polarimetric data Magnetic domains can occur whenever the symmetry of the ordered magnetic structure is less than that of the paramagnetic phase In general if the order of the paramagnetic space group is p and that of the magnetic space group m, the number of different Spherical neutron polarimetry 227 domains is p/m Magnetic domains may usefully be classified into the... translation is necessary to make the two domains identical The Bragg reflections from magnetic structures with zero propagation vector can contain both nuclear and magnetic scattering They give the interference terms in the polarised neutron scattering which were introduced in Section 2.3 Figures 6(b) and 6(c) show two simple antiferromagnetic structures with zero τ In Figure 6(b) the domains can only be brought... the different magnetic species, it can not give the absolute magnitude of the magnetic moments unless nuclear and magnetic scattering are present in the same reflections Spherical neutron polarimetry 233 4.2 Commensurate structures with nonzero propagation vectors For structures with τ = 0 the magnetic and nuclear reflections are independent so SNP cannot determine the absolute magnitude of magnetic moments... the magnetic fields are parallel to a single direction usually vertical, and the neutrons are polarised and analysed with respect to this direction In the figure, a neutron Fig 1 Triple-axis spectrometer for polarisation analysis in a vertical magnetic field 220 P.J Brown beam from a reactor enters the instrument from the left Neutrons of the chosen wavelength, polarised parallel to the polarising field,... can the individual cross-sections because they are obtained from ratios of intensities measured without having to move the sample 1.1 Neutron precession in an external field The behaviour of a neutron with spin S and gyromagnetic ratio γN inclined at an angle θ to a uniform magnetic field B can be represented classically as the precession of a magnetic dipole (γN S) about the field direction The precession... Spherical neutron polarimetry 231 magnetic moments on atoms at positions related by the centre are neither parallel or antiparallel to one another For example, suppose two atoms at vector distance ±r from a centre of symmetry have magnetic moments represented by M1 , M2 The magnetic structure factor M(Q) = (M1 + M2 ) cos(Q · r) + i(M1 − M2 ) sin(Q · r) and M⊥ (Q) = M⊥u cos(Q · r) + iM⊥v sin(Q · r), ( 18) ... the scattering process 3 Magnetic domains The squared modulus of the scattered polarisation P obtained from (13)–(16) is always greater than or equal to |P|2 which means that the amplitude of the polarisation is either increased or unchanged by scattering from any pure target state Real depolarisation of the scattered beam is an indication that a mixed state consisting of more than one type of magnetic. .. can be created in the scattering process 2.2 Nuclear scattering If any nuclear spin is assumed to be randomly oriented the interaction between an atomic nucleus and a neutron can be represented by the Fermi pseudopotential, a scalar field which is zero except very close to the nuclei The interaction with a scalar field cannot depend on the orientation of the neutron spin The nuclear scattering cross-section . Theory of Neutron Scattering from Condensed Matter, Clarendon, Oxford (1 983 ). [27] E. Balcar and S.W. Lovesey, in: Theory of Magnetic Neutron and Photon Scattering, Clarendon, Oxford (1 989 ). [ 28] P.J Rev. B 28 452 (1 983 ). [47] F. Dunstetter, V.P. Plakhty and J. Schweizer, J. Magn. Magn. Mater. 72 2 58 (1 988 ). [ 48] F. Dunstetter, V.P. Plakhty and J. Schweizer, J. Magn. Magn. Mater. 96 282 (1991). [49]. Magn. Mater. 50 1 78 (1 985 ). [50] R. Cywinski, S.H. Kilcoyne and J.R. Stewart, Physica B 267–2 68 106 (1999). [51] R.A. Cowley, N. Cowlam and L.D. Cussens, J. Phys. C 8 1 285 (1 988 ). [52] J.M. Coey,