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Experimental attempts to measure non-collinear local magnetisation
Journal of Physics and Chemistry of Solids 65 (2004) 1977–1983 www.elsevier.com/locate/jpcs
Experimental attempts to measure non-collinear local magnetisation P.J. Browna,b,* b
a Institut Laue Langevin, BP 156, 38042 Grenoble, France Physics Department, Loughborough University, Loughborough, UK
Abstract Representing atomic magnetic moments as simple vectors has limitations when applied to systems in which orbital moments, or significant spin–orbit coupling is present. These phenomena are associated with interactions leading to non-collinearity in the magnetisation distribution. Several experimental techniques are available to probe such non-collinearity. The most direct is to measure both the magnitude and direction of the magnetic interaction vectors, which give its Fourier components. Such measurements are now becoming possible for antiferromagnets with the advent of a new generation of neutron polarimeters, which will allow both greater geometric flexibility and higher precision. However, up to now non-collinear magnetisation distributions have been revealed by more indirect means. Polarised neutron flipping ratio measurements can give only a single component of the magnetic interaction vector directly. However, the special geometric properties of the interaction vector and the symmetry breaking properties of an applied field can be exploited to obtain evidence of non-collinearity in the magnetisation distribution even from such limited data. q 2004 Elsevier Ltd. All rights reserved. Keywords: C. Neutron scattering; D. Magnetic structure PACS: 75.25.Cz; 75.30.GW
1. Introduction The representation of atomic magnetic moments as simple vectors, shown as arrows in the pictorial representation of a magnetic structure, has proved a very useful one; but it should not be pushed too far. In a solid, the magnetisation is a continuous vector function, a property of the electron wave-functions, and there is no requirement that it should be collinear. Nevertheless, the assumption of a collinear magnetisation distribution is widely made and is often a very good first approximation. Careful thought should therefore be given to the conditions under which the approximation might not be valid and the circumstances in which the resulting non-collinearity would be observable. The validity of a model of magnetisation based on atomic magnetic moments requires that the intra-atomic * Address: Institut Laue Langevin, BP 156, 38042 Grenoble, France. Tel.: C33 4 76 20 70 40; fax: C33 4 76 48 39 06. E-mail address: [email protected]. 0022-3697/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2004.08.009
exchange energy: that which couples the spins of electrons within an atom, be very much greater than the interatomic exchange energies which lead to magnetic ordering. One may distinguish two extremes of non-collinearity: that inherent in the intra-atomic exchange and that resulting from the interaction of atomic moments with internal or external fields. The spin–orbit interaction is by far the most important cause of non-collinearity in magnetic systems being the principal means by which an atom’s spin direction is coupled to the orientation of its ligand environment. In all the examples in which non-collinear magnetisation distributions have been sought or found, such non-collinearity is due primarily to the presence of unquenched orbital magnetisation. The spin–orbit interaction can manifest itself in different ways. It can lead to a degree of intra-atomic noncollinearity, which may be present even in very simple systems. In antiferromagnets it may favour a non-collinear magnetic structure and in ferromagnetic and paramagnetic materials will give rise to magnetic anisotropy.
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2. The magnetisation around an atom or ion The magnetisation is given by the operator LC2S. Suppose for simplicity that the magnetisation can be approximated by the sum of magnetisations due to the constituent magnetic atom and that their wave function are a sum of single electron functions each of which can be expressed in terms of orbital quantum numbers l, m, and spin sZG(1/2). X ^ jZ uðrÞam;s Yml ðrÞ
transverse components of magnetisation with angular 2 4 and YG1 : distribution varying as combinations of YG1 Combined with the dominant z component this will give rise to a non-collinear spin density as illustrated in Fig. 1. The orbital contribution to the magnetisation for this simple model can be obtained from the equations given by Balcar [1] as " ML ðrÞ Z <
m 00 ;m 0 ;m;s
m 0 ;s 0 ;m;s
The magnetisation parallel to the quantisation axis due to spin moment is X MSz ðrÞ Z ðam0 ;1=2 am;1=2 K am0K1=2 am;K1=2 Þ m 0 ;m;
^ ml ðrÞ ^ !u2 ðrÞYml0 ðrÞY and the transverse spin magnetisation X ðam0 ;K1=2 am;1=2 C am 0 ;1=2 am;K1=2 Þ MSx ðrÞ Z m 0 ;m;
^ ml ðrÞ ^ !u2 ðrÞYml0 ðrÞY It will be zero unless there are some non-zero products am0 ;1=2 am;K1=2 which implies that the j are not eigenfunctions of Sz and that there are terms in the Hamiltonian which mix states with different values of s. Suppose lZ2 (3d electrons) and z is chosen parallel to the direction of an applied field. In the above representation the Hamiltonian is diagonal with respect to the Zeeman energy and leads to a collinear magnetisation parallel to z. Adding a spin-orbit coupling term to the Hamiltonian introduces mixing of states; j2;K1=2i with j1; 1=2i; j1;K1=2i with j0; 1=2i; jK 1; 1=2i with j0;K1=2i and jK 2; 1=2i with j1;K1=2i: These can give rise to
# l ^ ml 00 ðrÞ ^ am* 0 ;s am;s hm 00 jLjmig2 ðrÞYm* 0 ðrÞY
with g2 ðrÞ Z
m;s
The spin part of the magnetisation is then given by X ^ ml ðrÞ ^ am0 ;s0 am;s hs 0 jSjsiu2 ðrÞYml0 ðrÞY MS ðrÞ Z 2
X
1 2
ðr
u2 ðxÞdx
0
That parallel to z is MLz ðrÞ Z
X m 0 ;m;s
^ ml 0 ðrÞ ^ am0 ;s am;s mg2 ðrÞYml0 ðrÞY
and the transverse component parallel to x is " MLx ðrÞ Z <
X m 0 ;m;s
# l l ^ C LlmKYmK1 ^ am0 ;s am;s g2 ðrÞYml0 ðLlmCYmC1 ðrÞ ðrÞÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w hp e ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r e Ll mCZ 1=2 ffi ðlK mÞðlC mC 1Þ a n d Ll mKZ 1=2 ðlC mÞðlK mC 1Þ: There will be a z component of orbital moment if am,ssaKm,s for some value or values of m, and in that case a transverse orbital magnetisation varying as l l ðam;s am;s K aKm;s aKm;s ÞðLlmCYml YmC1 C LlmKYml YmK1 Þ
will co-exist with the z component. As for the transverse spin distribution the transverse orbital distribution MLx is an odd function of x which for lZ2 varies as combinations 2 4 of YG1 and YG1 : Several experimental techniques are available to probe such non-collinearity in the distribution of magnetisation. These methods and the degree of success obtained using them are described in the following sections.
Fig. 1. Examples showing the from of the non-collinear magnetisation distributions which can arise in the spin density from spin orbit coupling between d functions (lZ2) with (a) mZ0, sZ 1=2 and mZ1, sZK1=2 and (b) mZ2, sZK1=2 and mZ1, sZ 1=2: orbital density is shown in (c) when the orbital moment is due to unequal occupancy of d states with mZG2.
P.J. Brown / Journal of Physics and Chemistry of Solids 65 (2004) 1977–1983
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3. Direct observation from intensity measurement
4. Direct measurement using neutron polarimetry
A non-collinear magnetisation distribution must occur in a non-collinear antiferromagnetic structure. When the atomic moments themselves are not parallel it would be naı¨ve to suppose that there is a well-defined boundary between atoms at which the magnetisation changes abruptly from one general direction to another. An example illustrating the kind of structure in which such noncollinearity might be observed directly is CoSO4. The crystal structure of the a form is isomorphous with orthorhombic CrVO4: space group Cmcm. The Co2C ions occupy the 4(a) special positions (000), ð00 12 Þ; ð 12 12 0Þ and ð 12 12 12 Þ: In the antiferromagnetic phase the propagation vector is (0,1,0) and the moments lie in the b–c plane at angles G258 to [010] as shown in Fig. 2 [2]. The structure factors of the antiferromagnetic reflections have the form
To avoid the ambiguity in separating the different vector components contributing to a magnetic reflection the most direct solution is to determine the direction as well as the amplitude the of the magnetic interaction vectors. These are directly related to the Fourier components of the vector magnetisation. The directions of the interaction vectors of a purely magnetic reflection can be determined using zero field neutron polarimetry. This technique [3,4] allows the direction of the polarisation scattered by a reflection to be determined for any chosen direction of the incident polarisation. The incident polarisation direction is rotated in the plane perpendicular to the scattering vector until the incident and scattered polarisations are parallel; this direction is that of the interaction vector. Early attempts to measure non-collinearities with a fixed polarisation direction and rotating the crystal about the scattering vector, were unsuccessful both because of multiple scattering effects and incomplete mastery of antiferromagnetic domain formation. The magnitude of deviations from collinearity to be expected from unquenched orbital moments in transition metal salts such as FeCO3 is of the order of 18 [5]. Measurements with this degree of precision are only now becoming possible for antiferromagnets. The new generation of neutron polarimeters now being commissioned should have the required angular accuracy and allow greater geometric flexibility.
8 pl pl > > 4Sy fy ðhklÞcos2 C4Sz fz ðhklÞsin2 > > 2 2 > < 4Sy fy ðhklÞ for k even l odd MðhklÞZ > > 4Sz fz ðhklÞ for k odd l even > > > : 0 otherwise 4Sz fz ðhklÞ (1)
So that for this special structure the intensities of reflections with odd l give information about the distribution of the y component of moment and those with even l information about the z component. The vector distribution within the y–z plane might be constructed by combining intensity measurements of the two sets of reflections. Unfortunately the situation is not quite so straight-forward because Eq. (1) assumes that the form factors are spherically symmetric; if they are not the convenient separation between the y and z components is no longer complete. This makes it impracticable to determine the non-collinear magnetisation distribution, in this rather direct way, from intensity measurements alone.
Fig. 2. The magnetic structure of a-CoSO4.
5. Anisotropic site susceptibilities Intra-atomic non-collinear magnetisation distributions have been revealed by more indirect means. Polarised neutron flipping ratio measurements can give directly only a single component of the magnetic interaction vector. However, the special geometric properties of the interaction vector and the symmetry breaking properties of an applied field can be exploited to obtain evidence of non-collinearity in the magnetisation distribution even from such limited data. Recent polarised neutron studies of the magnetisation distributions in some ferromagnetic and paramagnetic materials have shown that the moments at crystallographically equivalent sites may be very different [6–8]. Such behaviour can arise when the local symmetry of the magnetic ion is lower than the overall symmetry of the crystal. The moment induced on each magnetic ion by the internal or external magnetic field depends on the orientation of the field with respect to the local symmetry of the magnetic site. The effect can be described by attributing to each magnetic atom a site susceptibility tensor cij which gives the magnetic response of the atom to an applied magnetic field. The symmetry of the tensor cij is the same as that of the tensor uij describing the thermal motion of atoms. The components of the uij tensor represent the mean square atomic displacement parameters (ADPs) and by analogy
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one can introduce atomic magnetisation parameters (AMPs). The response of an atom to a magnetic field can then be conveniently visualised as a magnetisation ellipsoid constructed from the six independent AMPs in much the same way as thermal ellipsoids are constructed from the ADPs [9]. In the absence of local anisotropy the magnetic ellipsoids reduce to spheres with their radii proportional to the induced magnetisation but in many cases anomalous (elongated or flattened) ellipsoids can appear. The presence of such anisotropic magnetic ellipsoids can account for the anomalous magnetic moments mentioned above. The overall response of a paramagnetic material to an applied magnetic field is described by the tensor equation Bi Z mij Hj Z m0 ðHi C cij Hj Þ and the bulk magnetisation M is Mi Z cij Hj The number of independent components of the susceptibility tensor cij is determined by the crystal class, being one for cubic groups, two for all uniaxial groups and, three, four and six for orthorhombic, monoclinic and triclinic, respectively. In a paramagnet containing localised moments the bulk magnetisation is the vector sum of the magnetisations induced on each of the constituent magnetic atoms and these will depend on the local site symmetry, rather than the overall symmetry. The power of the diffraction experiment is that it gives access to the wave-vector dependent magnetic response and hence to the individual site susceptibilities. Consider a single magnetic atom (a) with an anisotropic susceptibility arising from its local environment; the magnetisation induced on it by a field H is given by Ma Z ca H an equivalent atom (b) related to the first by the symmetry operator fR~ : tg would have magnetisation ~ a R~ H M Z cb H Z Rc K1
If the position of the first atom is ra and the distribution of moment about each atom (assumed spherical) is r(r) then the magnetisation distribution in the unit cell becomes X K1 MðrÞ Z R~ p ca R~ p Hrðr K R~ p ra C tp Þ p
where the sum is over all Ng operators fR~ p : tp g in the Ð space group. If f(k) is the magnetic form factor given by 0N 4pr 2 rðrÞdr 3 ; the corresponding magnetic structure factor is X 1 K1 R~ p ca R~ p H expðikðR~ p ra C tp ÞÞ f ðkÞ MðkÞ Z Na p Where Na is the number of operators q for which R~ q ra C tq Z ra so that Ng/Na is the multiplicity of the site a.
The point group Q formed by the rotational parts of the set of operators q gives the symmetry of the site a. The quantity measured in a classical polarised neutron experiment is the flipping ratio, RZIC/IK, between the intensity of neutrons scattered when polarised parallel (IC) and anti-parallel (IK) to the external magnetic field. The scattered intensities are given by 0 0 IG Z N 2 C 2PGðN 0 Mt C N 00 Mt Þ C M2t ;
where N is the nuclear structure factor with real and 0 00 imaginary parts N 0 and N 00 , Mt and Mt are the real and imaginary parts of the magnetic interaction vector Mt defined by Mt(k)Zk!M(k)!k and PG are the two neutron polarisation vectors. R can easily be expressed in terms of the susceptibility parameters using the equations given above and hence the flipping ratios can be used to determine these parameters using least squares refinement techniques. The presence of a high degree of anisotropy in the site susceptibilities implies considerable non-collinearity in the magnetisation distribution. The polarised neutron study of Nd0.88S4 which has the cubic Th3P4 structure provides an and the example [9]. The space group of Nd0.88S4 is I 43d Nd atoms occupy 12(a) sites with 4 site symmetry; they can be divided into three groups of four with their local four-fold axes along x, y and z, respectively. When the crystal was magnetised parallel to one of the tetrads the Nd atoms whose local tetrad was parallel to the field was found to carry less than half the moment found on the other eight. Sets of polarised neutron flipping ratios measured with different directions of magnetisation could be refined together using the site susceptibility concept. For 4 symmetry there are only two independent components c11Zc22 and c33. The least squares refinement gave c11Z0.207(7), c33Z0.078(8) mBTK1. When the applied field is not parallel to one of the principal axes of the magnetisation ellipsoid, the direction of magnetisation will not be parallel to the field. Fig. 3 shows how the two are related. The ellipse (1) with axes of length c33 and c22 is a section through the magnetic ellipsoid, the radius vectors of which give the magnitude of the magnetisation which can be induced in that direction by unit field. The ellipse (3) with pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi axes 1= c33 and 1= c22 is a section through the representation surface and the dashed circle (2) is a section through a sphere of unit radius representing the applied field. A unit field applied parallel to OP induces magnetisation in the direction of n perpendicular to thepffiffiffiffiffiffiffiffi surface (3) at P. Its magnitude is given by OMZ 1= OQ where OM is parallel to n. Fig. 4 illustrates schematically the magnetisation induced in Nd0.88S4 by a magnetic field applied parallel to [110]. Only on the Nd atoms whose local tetrad is parallel to [001] is the local magnetisation parallel to the field. On the others the relevant sections of the magnetisation ellipsoids are foreshortened in the direction of the local
P.J. Brown / Journal of Physics and Chemistry of Solids 65 (2004) 1977–1983
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Fig. 3. A principal section through (1) the magnetisation ellipsoid; (2) A unit sphere and (3) the representation quadric for susceptibility. The vector OM gives the magnetisation induced by unit field parallel to OP.
tetrad and the local magnetisation is rotated away from [110] towards the long axis so the final magnetisation distribution is non-collinear.
6. Orbital anisotropies In Section 5 it was seen how classical polarised neutron diffraction can reveal non-collinearity in the magnetisation aligned on different atoms by an external field. The method of analysis, however, assumes collinear alignment of the moment within each atom. However, the interactions which lead to magnetic anisotropy, are just those, which can give rise to non-collinearity of the magnetisation distribution within a single ion. The methods of analysis have not yet been sufficiently elaborated to deal with simultaneous inter and intra-atomic non-collinearity. However, it was seen in Section 2 that when there is a finite orbital moment, transverse components of magnetisation may be present giving intra atomic non-collinearity even in very simple systems and without invoking spin–orbit interaction. The paramagnetic form factors of the 3d, 4d and 5d transition metals have been extensively studied using polarised neutron scattering [10] and the results have been interpreted using difference between the angular dependencies of the spin and orbital form-factors to estimate the ratio between the orbital and spin moments. Particularly large proportions of orbital magnetisation in the induced moments have been found in studies of chromium and vanadium [11–14]. In chromium the results suggest that 60% of the total induced moment is of orbital origin. It might seem unpromising to search for transverse components of moment using the classical polarised neutron technique since, when the moment is induced by an applied field the method measures only the component of the interaction vector parallel to the field direction z. However, it may be recalled that the interaction vector is proportional
to the projection of the magnetisation on the plane perpendicular to the scattering vector. When the scattering vector lies in the plane perpendicular to z the transverse components do not contribute, but if the scattering vector is inclined to z at an angle qk the transverse component contributes MT cos qk sin qk, and the longitudinal component Mz sin2qk to the polarisation dependent term in the cross-section (Fig. 5) [15]. This property has been used to measure the transverse components of orbital magnetisation induced by a magnetic field in chromium and vanadium [16]. The orbital magnetisation in chromium and vanadium is due to a small imbalance induced by the field, in the energies of 3d orbitals with opposite values of the projection m. Theq perturbed functions of lowest energy have the form ffiffiffi jGm Z 12 ðð1C xm ÞjmiGð1K xm ÞjK miÞ; with mean orbital
Fig. 4. The [110] projection of the unit cell of Nd0.88S4 showing the direction of the moments induced in the Nd atoms and the active sections of their magnetisation ellipsoids when a magnetic field is applied parallel to [110].
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geometry by tilting the detectorpout ffiffiffi of the horizontal plane by an angle nZGsinK1 ðlðhK lÞ= 2aÞ: The hhl values giving cos2 qkO0.03 accessible within the geometric constraints imposed by the magnet are just 110, 002 and 112. The magnetic structure factors M were obtained from the flipping ratios measured in the experiment using the known values of the nuclear structure factor N and the equation RðhklÞ Z
NðhklÞ2 C2pCNðhklÞMðhklÞsin2 qk CMðhklÞ2 sin2 qk NðhklÞ2 C2PKNðhklÞMðhklÞsin2 qk CMðhklÞ2 sin2 qk ð3Þ
By expressing the differences in the values of M between hhl and lhh reflections in terms of an asymmetry factor Fig. 5. A vertical section including the scattering vector for a reflection inclined at angle qk to the direction of the applied field. If qksp/2 there is a non-zero contribution from the transverse magnetisation Mx to Mtz.
angular momentum hLziZx1C2x2. With these functions the z-component of the magnetic interaction vector can be written as Mt ðkÞz orbital
1 3 ð2x2 C x1 Þðhj0 i C hj2 iÞ C ðx1 K 2x1 Þ 2 28
!ð5 cos2 qk K 1Þðhj2 i C hj4 iÞ ð2Þ
Z sin2 qk
The first term is just the usual dipole approximation for the orbital scattering, but part of the second term which depends on cos2 qk is due to the transverse orbital moment, and can be detected in a polarised neutron experiment. In the experiment single crystals of pure chromium and axis parallel to the vanadium were mounted with a ½110 field direction of a 10 T superconducting magnet. With this crystal orientation, reflections of the form hhl have scattering vectors at qkZ908 to the field axis, whereas for the cubically hlh and lhh reflections qk Z pequivalent ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosK1 ðhK lÞ= 2ð2h2 C l2 Þ : The latter type of reflection can be measured in normal beam diffraction
SðhhlÞ ZðFM ðhhlÞKFM ðlhhÞÞ=ðFM ðhhlÞCFM ðlhhÞÞ
(4)
the effects of extinction and other systematic errors may be largely eliminated. The magnetic structure factors and asymmetry factors measured for the three reflections 110, 002, and 112 are listed in Table 1. The asymmetry factors measured for chromium and vanadium follow the same trend as one another being small and negative for 110, larger but still negative for 002 and not significantly different from zero for 112. The values for chromium are roughly twice those for vanadium implying that the orbital moment in chromium is twice that in vanadium in good agreement with the results of the band structure calculations [17]. If the induced moment due to spin is d it contributes dsin2qkhj0i to Mtz. Incorporating the spin scattering into Eq. 2 and recalling that the orbital moment is 2x2Cx1 the asymmetry factor can be written Sðk; qk Þ Z Gðk; xÞy cos2 qk ZK
15ðhj2 i C hj4 iÞ !y cos2 qk 28ðhj0 ið1 C xÞ C hj2 iÞ
(5)
Where xZd/(2x2Cx1) is the ratio of spin to orbital moment and y is the ratio (x2K2x1)/(2x2Cx1). The factors G(k,x) calculated from Eq. (5) with the values xZ0.22 and 1.11 obtained in band structure calculations for chromium and
Table 1 Magnetic structure factors FM and the corresponding asymmetry factors between hhl and lhh reflections of Cr and V measured in 9.5 T. The relevant formfactors integrals are also listed hkl
Chromium FM
a
Vanadium S(hhl)
b
hj0i 11 10 00 20 11 21 a b
0 1 2 0 2 1
312(9) 342(10) 215(12) 268(23) 178(16) 175(19)
FMa
Form factors hj2i
hj4i
K0.046(20) 0.457
0.166
0.031
K0.110(54) 0.249
0.187
0.058
0.008(71) 0.146
0.178
0.074
The magnetic structure factors are in units of 10K5 mB/cell. The asymmetry factor S(hhl)Z(FM (hhl)KFM (lhh))/(FM (hhl)CFM (lhh)).
S(hhl)b
Form factors hj0i
361(6) 375(12) 223(9) 225(10) 178(8) 180(6)
hj2i
hj4i
K0.019(19) 0.451
0.169
0.031
K0.047(29) 0.240
0.190
0.059
K0.007(27) 0.136
0.180
0.076
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vanadium, respectively [17], have been used to determine the values of y needed to obtain the observed asymmetry. The values found for y are consistent with oneanother within experimental error giving mean values yZ0.65(24) (x 1/x 2ZK0.11(17)) for Chromium and yZ0.46(1) (x 1/x 2 Z0.03(11)) for vanadium. Since a negative value for x1/x2 is unphysical, these results suggest that in both chromium and vanadium xz0, and effectively the whole of the orbital moment is due to orbitals with jmjZ2.
7. Conclusion The usual assumption of collinearity of the magnetisation distribution may be questioned in almost any case in which there is a significant orbital moment, or in which the spin– orbit interaction couples the spin moment to the crystal lattice. For crystals in which the single ion anisotropy is large the non-collinearity may be large, and readily detectable using the classical polarised neutron method. In most other cases the effects are very subtle and special methods in conjunction with well developed models must be used to detect them.
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References [1] E. Balcar, J. Phys. C: Solid State Phys. 8 (1975) 1581. [2] B.C. Frazer, P.J. Brown, Phys. Rev. 125 (1962) 1283. [3] F. Tasset, P.J. Brown, E. Lelie`vre-Berna, T. Roberts, S. Pujol, J. Alibon, E. Bourgeat-Lami, Physica B 267–268 (1999) 69. [4] P.J. Brown, J.B. Forsyth, F. Tasset, Proc. R. Soc. Lond. A 442 (1993) 147. [5] P.J. Brown, J.B. Forsyth, J. Phys. C: Solid State 10 (1977) 3157. [6] A. Gukasov, P. Wisniewski, Z. Henkie, J. Phys. Condens. Matter 8 (1996) 10589. [7] J.-X. Boucherle, F. Givord, J. Schweizer, A. Gukasov, J.-M. Mignot, E. Lelie`vre-Berna, H. Aoki, A. Ochiai, Physica B 267–268 (1999) 37. [8] J.-X. Boucherle, F. Givord, J. Schweizer,, A. Gukasov, J.M. Mignot, E. Lelie`vre-Berna, E.H. Aoki, A. Ochiai, Physica B 281–282 (2000) 139. [9] A. Gukasov, P.J. Brown, J. Phys. Condens. Matter 14 (2002) 8831. [10] R.M. Moon, W.C. Koehler, J.W. Cable, Proceedings of the ICNS, ORNL Publication, CONF-760601, 1976, p. 577. [11] C. Stassis, G.R. Kline, S.K. Sinha, Phys. Rev. B 11 (1975) 2171. [12] J. Strempfer, T. Bruckel, G.J. McIntyre, F. Tasset, T. Zeiske, K. Burger, W. Prandl, Physica B 267–268 (1999) 56. [13] C.G. Shull, Phys. Rev. Lett. (1963) 297. [14] P.J. Brown, J.B. Forsyth, J. Phys. F: Metal Phys. 14 (1984) 1715. [15] P.J. Brown, J.B. Forsyth, R. Masonhil, Phil. Trans. R. Soc. Lond. (1980) B290481. [16] P.J. Brown, E. Lelie`vre-Berna, K.R.A. Ziebeck, J. Phys. Condens. Matter 15 (2003) 1747. [17] J.F. Cooke, S.H. Liu, A.J. Liu, Phys. Rev. B 37 (1988) 289.
Experimental attempts to measure non-collinear local magnetisation P.J. Browna,b,* b
a Institut Laue Langevin, BP 156, 38042 Grenoble, France Physics Department, Loughborough University, Loughborough, UK
Abstract Representing atomic magnetic moments as simple vectors has limitations when applied to systems in which orbital moments, or significant spin–orbit coupling is present. These phenomena are associated with interactions leading to non-collinearity in the magnetisation distribution. Several experimental techniques are available to probe such non-collinearity. The most direct is to measure both the magnitude and direction of the magnetic interaction vectors, which give its Fourier components. Such measurements are now becoming possible for antiferromagnets with the advent of a new generation of neutron polarimeters, which will allow both greater geometric flexibility and higher precision. However, up to now non-collinear magnetisation distributions have been revealed by more indirect means. Polarised neutron flipping ratio measurements can give only a single component of the magnetic interaction vector directly. However, the special geometric properties of the interaction vector and the symmetry breaking properties of an applied field can be exploited to obtain evidence of non-collinearity in the magnetisation distribution even from such limited data. q 2004 Elsevier Ltd. All rights reserved. Keywords: C. Neutron scattering; D. Magnetic structure PACS: 75.25.Cz; 75.30.GW
1. Introduction The representation of atomic magnetic moments as simple vectors, shown as arrows in the pictorial representation of a magnetic structure, has proved a very useful one; but it should not be pushed too far. In a solid, the magnetisation is a continuous vector function, a property of the electron wave-functions, and there is no requirement that it should be collinear. Nevertheless, the assumption of a collinear magnetisation distribution is widely made and is often a very good first approximation. Careful thought should therefore be given to the conditions under which the approximation might not be valid and the circumstances in which the resulting non-collinearity would be observable. The validity of a model of magnetisation based on atomic magnetic moments requires that the intra-atomic * Address: Institut Laue Langevin, BP 156, 38042 Grenoble, France. Tel.: C33 4 76 20 70 40; fax: C33 4 76 48 39 06. E-mail address: [email protected]. 0022-3697/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2004.08.009
exchange energy: that which couples the spins of electrons within an atom, be very much greater than the interatomic exchange energies which lead to magnetic ordering. One may distinguish two extremes of non-collinearity: that inherent in the intra-atomic exchange and that resulting from the interaction of atomic moments with internal or external fields. The spin–orbit interaction is by far the most important cause of non-collinearity in magnetic systems being the principal means by which an atom’s spin direction is coupled to the orientation of its ligand environment. In all the examples in which non-collinear magnetisation distributions have been sought or found, such non-collinearity is due primarily to the presence of unquenched orbital magnetisation. The spin–orbit interaction can manifest itself in different ways. It can lead to a degree of intra-atomic noncollinearity, which may be present even in very simple systems. In antiferromagnets it may favour a non-collinear magnetic structure and in ferromagnetic and paramagnetic materials will give rise to magnetic anisotropy.
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2. The magnetisation around an atom or ion The magnetisation is given by the operator LC2S. Suppose for simplicity that the magnetisation can be approximated by the sum of magnetisations due to the constituent magnetic atom and that their wave function are a sum of single electron functions each of which can be expressed in terms of orbital quantum numbers l, m, and spin sZG(1/2). X ^ jZ uðrÞam;s Yml ðrÞ
transverse components of magnetisation with angular 2 4 and YG1 : distribution varying as combinations of YG1 Combined with the dominant z component this will give rise to a non-collinear spin density as illustrated in Fig. 1. The orbital contribution to the magnetisation for this simple model can be obtained from the equations given by Balcar [1] as " ML ðrÞ Z <
m 00 ;m 0 ;m;s
m 0 ;s 0 ;m;s
The magnetisation parallel to the quantisation axis due to spin moment is X MSz ðrÞ Z ðam0 ;1=2 am;1=2 K am0K1=2 am;K1=2 Þ m 0 ;m;
^ ml ðrÞ ^ !u2 ðrÞYml0 ðrÞY and the transverse spin magnetisation X ðam0 ;K1=2 am;1=2 C am 0 ;1=2 am;K1=2 Þ MSx ðrÞ Z m 0 ;m;
^ ml ðrÞ ^ !u2 ðrÞYml0 ðrÞY It will be zero unless there are some non-zero products am0 ;1=2 am;K1=2 which implies that the j are not eigenfunctions of Sz and that there are terms in the Hamiltonian which mix states with different values of s. Suppose lZ2 (3d electrons) and z is chosen parallel to the direction of an applied field. In the above representation the Hamiltonian is diagonal with respect to the Zeeman energy and leads to a collinear magnetisation parallel to z. Adding a spin-orbit coupling term to the Hamiltonian introduces mixing of states; j2;K1=2i with j1; 1=2i; j1;K1=2i with j0; 1=2i; jK 1; 1=2i with j0;K1=2i and jK 2; 1=2i with j1;K1=2i: These can give rise to
# l ^ ml 00 ðrÞ ^ am* 0 ;s am;s hm 00 jLjmig2 ðrÞYm* 0 ðrÞY
with g2 ðrÞ Z
m;s
The spin part of the magnetisation is then given by X ^ ml ðrÞ ^ am0 ;s0 am;s hs 0 jSjsiu2 ðrÞYml0 ðrÞY MS ðrÞ Z 2
X
1 2
ðr
u2 ðxÞdx
0
That parallel to z is MLz ðrÞ Z
X m 0 ;m;s
^ ml 0 ðrÞ ^ am0 ;s am;s mg2 ðrÞYml0 ðrÞY
and the transverse component parallel to x is " MLx ðrÞ Z <
X m 0 ;m;s
# l l ^ C LlmKYmK1 ^ am0 ;s am;s g2 ðrÞYml0 ðLlmCYmC1 ðrÞ ðrÞÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w hp e ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r e Ll mCZ 1=2 ffi ðlK mÞðlC mC 1Þ a n d Ll mKZ 1=2 ðlC mÞðlK mC 1Þ: There will be a z component of orbital moment if am,ssaKm,s for some value or values of m, and in that case a transverse orbital magnetisation varying as l l ðam;s am;s K aKm;s aKm;s ÞðLlmCYml YmC1 C LlmKYml YmK1 Þ
will co-exist with the z component. As for the transverse spin distribution the transverse orbital distribution MLx is an odd function of x which for lZ2 varies as combinations 2 4 of YG1 and YG1 : Several experimental techniques are available to probe such non-collinearity in the distribution of magnetisation. These methods and the degree of success obtained using them are described in the following sections.
Fig. 1. Examples showing the from of the non-collinear magnetisation distributions which can arise in the spin density from spin orbit coupling between d functions (lZ2) with (a) mZ0, sZ 1=2 and mZ1, sZK1=2 and (b) mZ2, sZK1=2 and mZ1, sZ 1=2: orbital density is shown in (c) when the orbital moment is due to unequal occupancy of d states with mZG2.
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3. Direct observation from intensity measurement
4. Direct measurement using neutron polarimetry
A non-collinear magnetisation distribution must occur in a non-collinear antiferromagnetic structure. When the atomic moments themselves are not parallel it would be naı¨ve to suppose that there is a well-defined boundary between atoms at which the magnetisation changes abruptly from one general direction to another. An example illustrating the kind of structure in which such noncollinearity might be observed directly is CoSO4. The crystal structure of the a form is isomorphous with orthorhombic CrVO4: space group Cmcm. The Co2C ions occupy the 4(a) special positions (000), ð00 12 Þ; ð 12 12 0Þ and ð 12 12 12 Þ: In the antiferromagnetic phase the propagation vector is (0,1,0) and the moments lie in the b–c plane at angles G258 to [010] as shown in Fig. 2 [2]. The structure factors of the antiferromagnetic reflections have the form
To avoid the ambiguity in separating the different vector components contributing to a magnetic reflection the most direct solution is to determine the direction as well as the amplitude the of the magnetic interaction vectors. These are directly related to the Fourier components of the vector magnetisation. The directions of the interaction vectors of a purely magnetic reflection can be determined using zero field neutron polarimetry. This technique [3,4] allows the direction of the polarisation scattered by a reflection to be determined for any chosen direction of the incident polarisation. The incident polarisation direction is rotated in the plane perpendicular to the scattering vector until the incident and scattered polarisations are parallel; this direction is that of the interaction vector. Early attempts to measure non-collinearities with a fixed polarisation direction and rotating the crystal about the scattering vector, were unsuccessful both because of multiple scattering effects and incomplete mastery of antiferromagnetic domain formation. The magnitude of deviations from collinearity to be expected from unquenched orbital moments in transition metal salts such as FeCO3 is of the order of 18 [5]. Measurements with this degree of precision are only now becoming possible for antiferromagnets. The new generation of neutron polarimeters now being commissioned should have the required angular accuracy and allow greater geometric flexibility.
8 pl pl > > 4Sy fy ðhklÞcos2 C4Sz fz ðhklÞsin2 > > 2 2 > < 4Sy fy ðhklÞ for k even l odd MðhklÞZ > > 4Sz fz ðhklÞ for k odd l even > > > : 0 otherwise 4Sz fz ðhklÞ (1)
So that for this special structure the intensities of reflections with odd l give information about the distribution of the y component of moment and those with even l information about the z component. The vector distribution within the y–z plane might be constructed by combining intensity measurements of the two sets of reflections. Unfortunately the situation is not quite so straight-forward because Eq. (1) assumes that the form factors are spherically symmetric; if they are not the convenient separation between the y and z components is no longer complete. This makes it impracticable to determine the non-collinear magnetisation distribution, in this rather direct way, from intensity measurements alone.
Fig. 2. The magnetic structure of a-CoSO4.
5. Anisotropic site susceptibilities Intra-atomic non-collinear magnetisation distributions have been revealed by more indirect means. Polarised neutron flipping ratio measurements can give directly only a single component of the magnetic interaction vector. However, the special geometric properties of the interaction vector and the symmetry breaking properties of an applied field can be exploited to obtain evidence of non-collinearity in the magnetisation distribution even from such limited data. Recent polarised neutron studies of the magnetisation distributions in some ferromagnetic and paramagnetic materials have shown that the moments at crystallographically equivalent sites may be very different [6–8]. Such behaviour can arise when the local symmetry of the magnetic ion is lower than the overall symmetry of the crystal. The moment induced on each magnetic ion by the internal or external magnetic field depends on the orientation of the field with respect to the local symmetry of the magnetic site. The effect can be described by attributing to each magnetic atom a site susceptibility tensor cij which gives the magnetic response of the atom to an applied magnetic field. The symmetry of the tensor cij is the same as that of the tensor uij describing the thermal motion of atoms. The components of the uij tensor represent the mean square atomic displacement parameters (ADPs) and by analogy
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one can introduce atomic magnetisation parameters (AMPs). The response of an atom to a magnetic field can then be conveniently visualised as a magnetisation ellipsoid constructed from the six independent AMPs in much the same way as thermal ellipsoids are constructed from the ADPs [9]. In the absence of local anisotropy the magnetic ellipsoids reduce to spheres with their radii proportional to the induced magnetisation but in many cases anomalous (elongated or flattened) ellipsoids can appear. The presence of such anisotropic magnetic ellipsoids can account for the anomalous magnetic moments mentioned above. The overall response of a paramagnetic material to an applied magnetic field is described by the tensor equation Bi Z mij Hj Z m0 ðHi C cij Hj Þ and the bulk magnetisation M is Mi Z cij Hj The number of independent components of the susceptibility tensor cij is determined by the crystal class, being one for cubic groups, two for all uniaxial groups and, three, four and six for orthorhombic, monoclinic and triclinic, respectively. In a paramagnet containing localised moments the bulk magnetisation is the vector sum of the magnetisations induced on each of the constituent magnetic atoms and these will depend on the local site symmetry, rather than the overall symmetry. The power of the diffraction experiment is that it gives access to the wave-vector dependent magnetic response and hence to the individual site susceptibilities. Consider a single magnetic atom (a) with an anisotropic susceptibility arising from its local environment; the magnetisation induced on it by a field H is given by Ma Z ca H an equivalent atom (b) related to the first by the symmetry operator fR~ : tg would have magnetisation ~ a R~ H M Z cb H Z Rc K1
If the position of the first atom is ra and the distribution of moment about each atom (assumed spherical) is r(r) then the magnetisation distribution in the unit cell becomes X K1 MðrÞ Z R~ p ca R~ p Hrðr K R~ p ra C tp Þ p
where the sum is over all Ng operators fR~ p : tp g in the Ð space group. If f(k) is the magnetic form factor given by 0N 4pr 2 rðrÞdr 3 ; the corresponding magnetic structure factor is X 1 K1 R~ p ca R~ p H expðikðR~ p ra C tp ÞÞ f ðkÞ MðkÞ Z Na p Where Na is the number of operators q for which R~ q ra C tq Z ra so that Ng/Na is the multiplicity of the site a.
The point group Q formed by the rotational parts of the set of operators q gives the symmetry of the site a. The quantity measured in a classical polarised neutron experiment is the flipping ratio, RZIC/IK, between the intensity of neutrons scattered when polarised parallel (IC) and anti-parallel (IK) to the external magnetic field. The scattered intensities are given by 0 0 IG Z N 2 C 2PGðN 0 Mt C N 00 Mt Þ C M2t ;
where N is the nuclear structure factor with real and 0 00 imaginary parts N 0 and N 00 , Mt and Mt are the real and imaginary parts of the magnetic interaction vector Mt defined by Mt(k)Zk!M(k)!k and PG are the two neutron polarisation vectors. R can easily be expressed in terms of the susceptibility parameters using the equations given above and hence the flipping ratios can be used to determine these parameters using least squares refinement techniques. The presence of a high degree of anisotropy in the site susceptibilities implies considerable non-collinearity in the magnetisation distribution. The polarised neutron study of Nd0.88S4 which has the cubic Th3P4 structure provides an and the example [9]. The space group of Nd0.88S4 is I 43d Nd atoms occupy 12(a) sites with 4 site symmetry; they can be divided into three groups of four with their local four-fold axes along x, y and z, respectively. When the crystal was magnetised parallel to one of the tetrads the Nd atoms whose local tetrad was parallel to the field was found to carry less than half the moment found on the other eight. Sets of polarised neutron flipping ratios measured with different directions of magnetisation could be refined together using the site susceptibility concept. For 4 symmetry there are only two independent components c11Zc22 and c33. The least squares refinement gave c11Z0.207(7), c33Z0.078(8) mBTK1. When the applied field is not parallel to one of the principal axes of the magnetisation ellipsoid, the direction of magnetisation will not be parallel to the field. Fig. 3 shows how the two are related. The ellipse (1) with axes of length c33 and c22 is a section through the magnetic ellipsoid, the radius vectors of which give the magnitude of the magnetisation which can be induced in that direction by unit field. The ellipse (3) with pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi axes 1= c33 and 1= c22 is a section through the representation surface and the dashed circle (2) is a section through a sphere of unit radius representing the applied field. A unit field applied parallel to OP induces magnetisation in the direction of n perpendicular to thepffiffiffiffiffiffiffiffi surface (3) at P. Its magnitude is given by OMZ 1= OQ where OM is parallel to n. Fig. 4 illustrates schematically the magnetisation induced in Nd0.88S4 by a magnetic field applied parallel to [110]. Only on the Nd atoms whose local tetrad is parallel to [001] is the local magnetisation parallel to the field. On the others the relevant sections of the magnetisation ellipsoids are foreshortened in the direction of the local
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Fig. 3. A principal section through (1) the magnetisation ellipsoid; (2) A unit sphere and (3) the representation quadric for susceptibility. The vector OM gives the magnetisation induced by unit field parallel to OP.
tetrad and the local magnetisation is rotated away from [110] towards the long axis so the final magnetisation distribution is non-collinear.
6. Orbital anisotropies In Section 5 it was seen how classical polarised neutron diffraction can reveal non-collinearity in the magnetisation aligned on different atoms by an external field. The method of analysis, however, assumes collinear alignment of the moment within each atom. However, the interactions which lead to magnetic anisotropy, are just those, which can give rise to non-collinearity of the magnetisation distribution within a single ion. The methods of analysis have not yet been sufficiently elaborated to deal with simultaneous inter and intra-atomic non-collinearity. However, it was seen in Section 2 that when there is a finite orbital moment, transverse components of magnetisation may be present giving intra atomic non-collinearity even in very simple systems and without invoking spin–orbit interaction. The paramagnetic form factors of the 3d, 4d and 5d transition metals have been extensively studied using polarised neutron scattering [10] and the results have been interpreted using difference between the angular dependencies of the spin and orbital form-factors to estimate the ratio between the orbital and spin moments. Particularly large proportions of orbital magnetisation in the induced moments have been found in studies of chromium and vanadium [11–14]. In chromium the results suggest that 60% of the total induced moment is of orbital origin. It might seem unpromising to search for transverse components of moment using the classical polarised neutron technique since, when the moment is induced by an applied field the method measures only the component of the interaction vector parallel to the field direction z. However, it may be recalled that the interaction vector is proportional
to the projection of the magnetisation on the plane perpendicular to the scattering vector. When the scattering vector lies in the plane perpendicular to z the transverse components do not contribute, but if the scattering vector is inclined to z at an angle qk the transverse component contributes MT cos qk sin qk, and the longitudinal component Mz sin2qk to the polarisation dependent term in the cross-section (Fig. 5) [15]. This property has been used to measure the transverse components of orbital magnetisation induced by a magnetic field in chromium and vanadium [16]. The orbital magnetisation in chromium and vanadium is due to a small imbalance induced by the field, in the energies of 3d orbitals with opposite values of the projection m. Theq perturbed functions of lowest energy have the form ffiffiffi jGm Z 12 ðð1C xm ÞjmiGð1K xm ÞjK miÞ; with mean orbital
Fig. 4. The [110] projection of the unit cell of Nd0.88S4 showing the direction of the moments induced in the Nd atoms and the active sections of their magnetisation ellipsoids when a magnetic field is applied parallel to [110].
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P.J. Brown / Journal of Physics and Chemistry of Solids 65 (2004) 1977–1983
geometry by tilting the detectorpout ffiffiffi of the horizontal plane by an angle nZGsinK1 ðlðhK lÞ= 2aÞ: The hhl values giving cos2 qkO0.03 accessible within the geometric constraints imposed by the magnet are just 110, 002 and 112. The magnetic structure factors M were obtained from the flipping ratios measured in the experiment using the known values of the nuclear structure factor N and the equation RðhklÞ Z
NðhklÞ2 C2pCNðhklÞMðhklÞsin2 qk CMðhklÞ2 sin2 qk NðhklÞ2 C2PKNðhklÞMðhklÞsin2 qk CMðhklÞ2 sin2 qk ð3Þ
By expressing the differences in the values of M between hhl and lhh reflections in terms of an asymmetry factor Fig. 5. A vertical section including the scattering vector for a reflection inclined at angle qk to the direction of the applied field. If qksp/2 there is a non-zero contribution from the transverse magnetisation Mx to Mtz.
angular momentum hLziZx1C2x2. With these functions the z-component of the magnetic interaction vector can be written as Mt ðkÞz orbital
1 3 ð2x2 C x1 Þðhj0 i C hj2 iÞ C ðx1 K 2x1 Þ 2 28
!ð5 cos2 qk K 1Þðhj2 i C hj4 iÞ ð2Þ
Z sin2 qk
The first term is just the usual dipole approximation for the orbital scattering, but part of the second term which depends on cos2 qk is due to the transverse orbital moment, and can be detected in a polarised neutron experiment. In the experiment single crystals of pure chromium and axis parallel to the vanadium were mounted with a ½110 field direction of a 10 T superconducting magnet. With this crystal orientation, reflections of the form hhl have scattering vectors at qkZ908 to the field axis, whereas for the cubically hlh and lhh reflections qk Z pequivalent ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosK1 ðhK lÞ= 2ð2h2 C l2 Þ : The latter type of reflection can be measured in normal beam diffraction
SðhhlÞ ZðFM ðhhlÞKFM ðlhhÞÞ=ðFM ðhhlÞCFM ðlhhÞÞ
(4)
the effects of extinction and other systematic errors may be largely eliminated. The magnetic structure factors and asymmetry factors measured for the three reflections 110, 002, and 112 are listed in Table 1. The asymmetry factors measured for chromium and vanadium follow the same trend as one another being small and negative for 110, larger but still negative for 002 and not significantly different from zero for 112. The values for chromium are roughly twice those for vanadium implying that the orbital moment in chromium is twice that in vanadium in good agreement with the results of the band structure calculations [17]. If the induced moment due to spin is d it contributes dsin2qkhj0i to Mtz. Incorporating the spin scattering into Eq. 2 and recalling that the orbital moment is 2x2Cx1 the asymmetry factor can be written Sðk; qk Þ Z Gðk; xÞy cos2 qk ZK
15ðhj2 i C hj4 iÞ !y cos2 qk 28ðhj0 ið1 C xÞ C hj2 iÞ
(5)
Where xZd/(2x2Cx1) is the ratio of spin to orbital moment and y is the ratio (x2K2x1)/(2x2Cx1). The factors G(k,x) calculated from Eq. (5) with the values xZ0.22 and 1.11 obtained in band structure calculations for chromium and
Table 1 Magnetic structure factors FM and the corresponding asymmetry factors between hhl and lhh reflections of Cr and V measured in 9.5 T. The relevant formfactors integrals are also listed hkl
Chromium FM
a
Vanadium S(hhl)
b
hj0i 11 10 00 20 11 21 a b
0 1 2 0 2 1
312(9) 342(10) 215(12) 268(23) 178(16) 175(19)
FMa
Form factors hj2i
hj4i
K0.046(20) 0.457
0.166
0.031
K0.110(54) 0.249
0.187
0.058
0.008(71) 0.146
0.178
0.074
The magnetic structure factors are in units of 10K5 mB/cell. The asymmetry factor S(hhl)Z(FM (hhl)KFM (lhh))/(FM (hhl)CFM (lhh)).
S(hhl)b
Form factors hj0i
361(6) 375(12) 223(9) 225(10) 178(8) 180(6)
hj2i
hj4i
K0.019(19) 0.451
0.169
0.031
K0.047(29) 0.240
0.190
0.059
K0.007(27) 0.136
0.180
0.076
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vanadium, respectively [17], have been used to determine the values of y needed to obtain the observed asymmetry. The values found for y are consistent with oneanother within experimental error giving mean values yZ0.65(24) (x 1/x 2ZK0.11(17)) for Chromium and yZ0.46(1) (x 1/x 2 Z0.03(11)) for vanadium. Since a negative value for x1/x2 is unphysical, these results suggest that in both chromium and vanadium xz0, and effectively the whole of the orbital moment is due to orbitals with jmjZ2.
7. Conclusion The usual assumption of collinearity of the magnetisation distribution may be questioned in almost any case in which there is a significant orbital moment, or in which the spin– orbit interaction couples the spin moment to the crystal lattice. For crystals in which the single ion anisotropy is large the non-collinearity may be large, and readily detectable using the classical polarised neutron method. In most other cases the effects are very subtle and special methods in conjunction with well developed models must be used to detect them.
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