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176 J. Schweizer Fig. 8. Enaminoketon C 18 H 28 O 2 N 2 Cu(NO) 2 spin density projection along the axis [24]: (a) T =40 K, contour step 0.008µ B /Å 2 ,(b)T = 4 K, contour step 0.090µ B /Å 2 . striking fact is the quasidisappearance of the spin density on N 2 O 2 and N 6 O 6 . These two NO groups belong to the two different molecules: N 2 O 2 to molecule A and N 6 O 6 to mole- cule B. They are facing each other at a distance of 3.40 Å. It is obvious that the negative coupling between adjacent molecules, revealed by the drop of the magnetic susceptibility, corresponds actuallyto a negative coupling between these two spin carriers which provokes an almost complete dimerization of their spins. 4.2.3. Modeling the magnetic wave function. A convenient way to retrieve the spin den- sity distribution, while avoiding the problems due to Fourier inversion, is to model the spin density and to determine the parameters involved in the model by a refinement procedure from the experimental data. A first and natural model to represent a spin density is to consider it as the square of the wave function of the magnetic electrons. The structure factor can be expressed by F M =Ψ |e i Qr |Ψ = Ψ ∗ e i Qr Ψd 3 r. (78) |Ψ is in the general case a molecular wave function |Ψ = atoms α j ϕ j , Polarized neutrons and polarization analysis 177 where the sum runs over all the magnetic atoms. Two types of terms enter the expression of F M : • one center integrals ϕ ∗ j e i Qr ϕ j d 3 r (79) which represent the main contribution; • two center integrals ϕ ∗ j e i Qr ϕ j d 3 r (80) which are correcting terms corresponding to the overlap between the wave functions of neighboring atoms. In the following we will restrict ourselves to the one center integrals, considering the magnetic amplitude scattered by one atom only. We shall express this amplitude in term of magnetic form factor and we shall consider successively the case of p and d electrons with a magnetic moment which is mainly of spin origin, and the case of f electrons (rare earths and actinides) where spin and orbit couple together to give a total angular momentum. The general treatment can be found in [25,26] or in [27]. (i) Form factor of p and d electrons in the pure spin case. For the pure spin case, the magnetic form factor f( Q) can be defined by µf Q = ϕ ∗ e i Qr ϕd 3 r, (81) where µ is the magnetic moment. The one electron atomic wave function ϕ is expanded in a radial part R(r) and in an angular part ϕ(r)= l R l a (r) l m=−l a lm Y m l (θ, ϕ), (82) where θ, ϕ are the angular coordinates of r and where the Y m (θ, ϕ) are the usual spherical harmonics. In this sum = 0, 1, 2, for s,p,d, electrons. In most of the cases only one value of is concerned by magnetism. One expands the exponential e i Q·r , e i Q·r =4π ∞ L=0 i L j L (Qr) L M=−L Y M L (θ Q ,ϕ Q ) ∗ Y M L (θ, ϕ), (83) where, besides θ , ϕ, the angular coordinates of r , θ Q and ϕ Q are the angular coordinates of Q and where the j L (z) are the spherical Bessel functions (see the Appendix). 178 J. Schweizer Introducing this expansion in (63), one obtains f Q = ∞ L=0 j L (Q) L M=−L C LM Y M L (θ Q ,ϕ Q ) ∗ , (84) where the j L (Q) are the radial integrals (Bessel–Fourier transform) of the magnetic elec- trons, j L (Q) = ∞ 0 r 2 R l a (r) 2 j L (Qr) dr. (85) The radial integrals have been tabulated by Brown [28] for the 3d and 4d electrons of the transition metals and for the 4f and 5f electrons of the rare earth and actinides. A rather popular way to express the radial part of the wave function, in the molecular magnetism community, is the use of the atomic Slater functions R a (r) = (2α) (n a +1/2) √ (2n a )! r (n a −1) e −αr (86) an expression which depends on the first quantum number n a and where the Slater ex- ponent α, characteristic of the two quantum numbers n a and of the orbital, have been tabulated, for instance, by Clementi and Roetti [29]. In these conditions, the radial in- tegrals can be expressed as analytical functions of Q and α (see the Appendix for the expressions of j 0 and j 2 for the 2p electrons and expressions of j 0 , j 2 and j 4 for the 3d electrons). Such radial integrals are displayed in Figure 9 for the 2p electrons of oxygen. Fig. 9. The radial integrals and the form factor f( Q) calculated for oxygen. Polarized neutrons and polarization analysis 179 Coming back to (84), the coefficients C LM are given by C LM = i L (2 +1) 4π(2L +1) 1/2 × L 000 mm (−1) m a ∗ m a m L −mm M (87) using the 3j symbols abc def which are closely related to the Clebsch–Gordan coeffi- cients. Because of the triangular relations which exist for the 3j symbols L 2 −m +m +M =0 , one is restricted to: • for p electron ( =1) f Q = j 0 (Q) +A(θ Q ,ϕ Q ) j 2 (Q) , (88) • for d electron ( =2) f Q = j 0 (Q) +A(θ Q ,ϕ Q ) j 2 (Q) +B(θ Q ,ϕ Q ) j 4 (Q) . (89) From the knowledge of the magnetic wave function, the calculation of the form factor is straightforward. In particular, for a 2p orbital of oxygen, aligned along Oz (a 2p z orbital) the form factor is expressed as f Q = j 0 (Q) + 1 −3cos 2 θ Q j 2 (Q) . It is a very anisotropic form factor, depending on θ Q , the angle between Q and the Oz di- rection. It is limited by the two functions j 0 (Q)−2j 2 (Q) and j 0 (Q)+j 2 (Q) for θ Q =0 and θ Q =π/2, as represented by Figure 9. In a general way, the wave function is a very convenient model to describe a spin density distribution. The adjustable parameters such as the wave function coefficients a m ,orthe Slater exponents α which modify the expansion of the radial part R(r), can be refined. In particular, such a model is very suitable to detect the weak spin density which exists on certain atoms. An example of such a treatment is given in Figure 10. It concerns the spin density ob- tained by wave function modeling of the NitPy (C≡C−H) free radical [30], a compound where the shortest contact (2.14 Å) between molecules correspond to a weak C≡C−H···O hydrogen bond. The main part of the spin density is carried by the ONCNO fragment, with a negative sign on the central carbon, but weaker contributions, with alternated signs, also appear on the skeleton of the molecule. A significant spin density has been found on the hydrogen atom of the hydrogen bond, an indication of the active role played by this bond in the magnetic coupling of the molecules. 180 J. Schweizer (a) (b) Fig. 10. Projection onto the nitroxide mean plane of NitPy (C≡C−H) of the spin density as analyzed by wave function modeling: (a) high-level contours (step 0.04µ B /Å 2 ); (b) low-level contours (step 0.008µ B /Å 2 ). In the former example, the magnetic wave function was refined from the experimental magnetic structure factors F M (h,k,l). It is also possible to use programs which are adapted for acentric structures as they start directly from the flipping ratios R. (ii) Form factors of p and d electrons, with an orbital moment contribution. When the orbital contribution to the magnetic moment is not completely quenched, the magnetic moments includes not only a spin part but also an orbital part, µ =µ S +µ L , (90) f Q =a S f S Q +a L f L Q , (91) where a S and a L are the spin and orbit proportions of the magnetic moment. Within a spherical approximation the spin and the orbital form factors are expressed as f S Q = j 0 (Q) , (92) f L Q = j 0 (Q) + j 2 (Q) , (93) which gives for the total form factor f Q = j 0 (Q) +a L j 2 (Q) . (94) Let us note that as the orbital magnetization is being produced by orbital currents, it is more localized than the spin magnetization due to the same magnetic electrons and therefore its form factor falls down less rapidly in the reciprocal space. Polarized neutrons and polarization analysis 181 (iii) Form factors of f electrons: Rare earths and actinides. For rare earths and actinides the spin-orbit coupling is large. Spin and orbit couple to- gether to give a total angular momentum, J = L + S · J is a good quantum number (J =L −S for the first half and L +S for the second half ), and the magnetic moment can be expressed as µ =g J J · A complete formalism of the atomic form factor is exposed in [25–27]. Practical expres- sions can be found in [31]. The important point concerning rare earth and actinides is the presence of strong anisotropies in the magnetization distribution resulting from the strong orbital contribution. These anisotropies depend on the direction of the applied field as illustrated in Figure 11 for cerium [32]. A simplified expression for the form factor is given by the spherical approximation f Q = j 0 (Q) +c 2 j 2 (Q) (95) with c 2 = 2 g J −1 = J(J +1) +L(L +1) −S(S +1) 3J(J +1) −L(L +1) +S(S +1) . (96) Table 1 displays the values of c 2 for the different fillings of the f shell. One can note the particular case of five electrons where the spin part and the orbital part almost cancel, giving unusual shapes for the form factor. This is illustrated in Figure 12 for SmCo 5 , where the maximum at Q = 0 results from the different spatial extensions for the spin part and for the orbital part, the sign of both contributions being opposed [33]. Fig. 11. The form factor and the magnetization density of Ce 3+ calculated for the two cubic states Γ 7 and Γ 8 [32]. 182 J. Schweizer Table 1 Coefficients C 2 for the different fillings of an f shell f shell LS J c 2 =2/g J −1 f 1 (Ce 3+ )31/25/2 1.333 f 2 (Pr 3+ ) 5 1 4 1.500 f 3 (Nd 3+ )63/29/2 1.750 f 4 (Pm 3+ ) 6 2 4 2.333 f 5 (Sm 3+ )55/25/2 6.000 f 6 (Eu 3+ ) 3 3 0 No moment f 7 (Gd 3+ )07/27/2 0.000 f 8 (Tb 3+ ) 3 3 6 0.333 f 9 (Dy 3+ )55/215/2 0.500 f 10 (Ho 3+ ) 6 2 8 0.600 f 11 (Er 3+ )63/215/2 0.667 f 12 (Tm 3+ ) 5 1 6 0.714 f 13 (Yb 3+ )31/27/2 0.750 Fig. 12. The form factor of Sm measured in SmCo 5 at 300 K [33]. 4.2.4. Modeling the spin density: The multipolar expansion. It is possible to obtain a more versatile model of the magnetization (spin) distribution by parametrizing the magne- tization density directly rather than parametrizing the wave function. A well-adapted model results from a multipolar expansion of the density around the nuclei at rest [34]. It consists of a superposition of aspherical atomic densities, each one described by a series expansion in real spherical harmonic functions y m (ˆr), m(r) = ∞ =0 R d (r) m=− P m y m (θ, ϕ), (97) Polarized neutrons and polarization analysis 183 where the P m are the population coefficients and R d (r) are radial functions of the spin density of the Slater type. As the density is the square of the wave function, we can take the radial functions R d (r) of the density as the square of the radial functions R a (r) of the amplitude and write them in the following way: R d (r) = ζ (n d +3) (n d +2)! r n d e −ζr , (98) with ζ = 2α and n d = 2(n a − 1). This would give for 1s orbitals (n a = 1)n d = 0, for 2s and 2p orbitals (n a =2)n d =2, for 3s,3p and 3d orbitals (n a =3)n d =4, The magnetic structure factors become F M Q = atoms ∞ =0 Φ (Q) m=− P m y m Q e i Qr e −W , (99) where Φ (Q) are the radial integrals defined by Φ (Q) =i ∞ 0 R d (r)j (Qr)r 2 dr (100) with the spherical Bessel functions j (z). The thermal motion of the atoms is taken into account through the term e −W . The real spherical harmonics y m (ˆr)are linear combinations of the usual spherical harmonics Y m (ˆr), y m+ = 1 2 Y m +Y −m , (101) y m− = 1 2i Y m −Y −m . (102) A set of parameters ζ , P m for each atom characterizes the magnetization distribution. These parameters are fitted by a least-square refinement of the data, in the general case, for centric structures, the experimental structure factors. This determines the spin density. An example of application of this method is given for the spin distribution of the tanol suberate (C 13 H 23 O 2 NO) 2 . This molecule is a binitroxide free radical where the unpaired electrons are localized on the NO groups located at the two ends of the chain molecules. The flipping ratios of reflections (0kl),uptosinθ/λ =0.45 Å −1 , were investigated [35]. Actually, only reflections with a large nuclear amplitude were measured, giving a partial set of 69 magnetic structure factors F Mobs . Figure 13 compares two spin density recon- structions obtained by Fourier inversion and by multiple expansion. The last map clearly shows less noise, an enhanced resolution and also values of the density closer to reality than the partial Fourier summation. Another example of interest is the reconstruction of the spin density of the radical-based cyano-acceptor tetracyanoethylene (TCNE) •+ [36]. The main part of the density is local- ized on the central sp 2 carbon atoms but, due to both the spin delocalization and the spin 184 J. Schweizer (a) (b) Fig. 13. Tanol suberate: the comparison of the spin distribution projected along the a direction (a) by Fourier inversion, (b) by multipole expansion [35]. polarization, the spin density extends also on the other atoms: 33%, −5% and 13% of the total spin on the sp 2 carbons, sp carbons and the nitrogen atoms, respectively. However, when reconstructing the spin density from a refinement of the magnetic molecular wave function, the results were not satisfactory as the agreement between observed and calcu- lated structure factors was by far too poor. To know the reason for this mismatch, more flexibility was given to the model, and a multipolar expansion of the spin density was per- formed and refined up to the octupoles. The result is presented in Figure 14, which shows clearly what happens. The π ∗ molecular orbital is antibonding and, on the sp 2 carbons, the projected spin density is pushed away from the center of the C–C bond. Such an effect was impossible to see in the molecular orbital refinement, the model being not flexible enough. This method of modeling the spin density can be extended to acentric structures (see [37]) for the DPPH (diphenyl picryl hydrazil) spin density. In such cases, the least- square refinement compares directly the experimental and the calculated flipping ratios. 4.3. Investigation of noncollinear magnetic structures The flipping ratio method can also be used to investigate complex distributions, either when the magnetic structures are naturally canted or when the noncollinearity results from the applied magnetic field. In such situations, expressions (61) and (62) of the flipping ratio are no more right for two reasons: (i) the scheme of Figure 3 being not legitimate, relations (59) and (60) are not fulfilled and the sin 2 α simplification cannot be applied; (ii) for noncollinear structures, F M⊥ is not supposed to be parallel to F ∗ M⊥ , and the Polarized neutrons and polarization analysis 185 Fig. 14. Projection of the spin density of the tetracyanoethylene plane, projection reconstructed by multipolar expansion [36]. chiral term ( F M⊥ ∧ F ∗ M⊥ ) z cannot be ignored. This term, being of the order two in magnetic amplitude, could be neglected only if F M is small compared to F N . Therefore, the exact expression (57) for the flipping ratio has to be applied, and the flipping ratio method is still very accurate and may be very useful. This has been done to understand the action of a magnetic field on the very anisotropic compound Ce 3 Sn 7 . This compound is unusual. As already seen (Figure 5), it is an interme- diate valence compound: there are two sites for the cerium atoms, but only atoms of the site Ce II carry a moment; atoms Ce I do not carry any moment. Below 5.1 K, the moments of Ce II order antiferromagnetically, splitting the four positions of this site in two sublattices: m 1 and m 3 in one direction and m 2 and m 4 in the opposite direction. The unusual feature is that axis c is the spontaneous axis, the axis along which the antiferromagnetic moments align themselves, but with a very small moment of 0.36µ B only. When a field is applied along the a axis, the antiferromagnetic structure is broken and an average magnetization of 3.5µ B /Ce 3 Sn 7 unit is already reached at 1.5 T, while when the same field is applied along the spontaneous axis c, the induced magnetization is lower by almost two orders of magnitude (see Figure 15(a)). Furthermore, at higher fields, two transitions occur. Flipping ratios were measured below the first transition under a field of 4.6 T applied along the spontaneous axis c [38]. The result is surprising and displayed in Figure 15(b): though the field is applied along c, both sublattices lean over axis a, which is not the spontaneous axis but seems to be “easy” in the sense that moments, when they are aligned along this axis, are larger than when they are aligned, even spontaneously, along any other axis. Paramagnetic Sm 3 Te 4 is another case of very anisotropic compound. The properties of this mixed valence compound were described by the superposition of magnetic Sm 3+ and nonmagnetic Sm 2+ . In the acentric cubic symmetry I ¯ 43d, the Sm atoms occupy a single site (12a) with a ¯ 4-local symmetry and with an easy axis which is along x, y or z. A valence order between Sm 3+ and Sm 2+ had been proposed, with an atom ratio Sm 3+ :Sm 2+ equal to 2 : 1 [39]. For a field applied along [110], it has been observed that the Sm atoms are [...]... (119)–(120)), all the non-spin-flip scattering is of nuclear origin and all the spinflip scattering is magnetic The nuclear scattering shown in Figure 27( a) exhibits broad diffuse peaks at (1, 0, 1/2) and at other symmetry related positions This nuclear scattering is strong, favored by the high nuclear contrast between Mn and Cu The magnetic scattering shown in Figure 27( b) has intensity distributed around... amorphous materials and glasses Before the emergence of neutron polarization analysis, amorphous ferromagnets were assumed to be collinear arrangements of atomic magnetic moments The occurrence of a tool that allows one not only to separate magnetic from nuclear scattering, but also the transverse component magnetic scattering from the longitudinal component scattering, reveals a reality which is more complex... patterns obtained for MnF2 [3] Fig 24 The paramagnetic scattering of Pd2 MnSn [44] 201 202 J Schweizer Fig 25 The paramagnetic scattering observed along the direction [h, h, 0] of a Fe 5 at.% Si single crystal [45] Fig 26 The paramagnetic scattering of γ oxygen [48] temperatures by the addition of 5% of silicon [45] As seen in Figure 25, the magnetic pattern is far from being constant: it is enhanced in the... nuclear and magnetic short range order The separation between nuclear and magnetic scattering is done by the same methods as for paramagnetic scattering A very good example of the impact of polarization analysis is given by the Mn–Cu alloys, an FCC system where the antiferromagnetism of manganese is perturbated by nonmagnetic copper substitutions and exhibits a spin-glass behavior Earlier neutron measurements... and magnetic contributions Finally, Cable et al [50] obtained the most informative results from polarization analysis experiments on a single crystal Figure 27 represents the separation between nuclear and magnetic scattering for a crystal containing 25% Mn atoms at T = 10 K This separation was obtained by aligning the polarization along the scattering vector This way, if we neglect the incoherent scattering. .. temperature the scattering is entirely quasielastic with an energy range less than 5 meV All the paramagnetic scattering had then been measured and the value of the effective moment had a real meaning A totally different example of paramagnetic scattering is given by a single crystal of iron at 1 273 K (1.25TC ), a crystal where the α phase of iron has been stabilized at high Polarized neutrons and polarization... to ferromagnetic order at concentrations greater than 10% The magnetic patterns of YMn1.9 Fe0.1 and YCo1.9 Fe0.1 , measured at low temperature on spectrometer D7 by the XY Z difference method [50], are displayed in Figure 28 The correlations found here look very similar Polarized neutrons and polarization analysis 205 to those found for the paramagnetic materials described above: antiferromagnetic... technique, they have shown the scattering patterns of MnF2 , patterns reproduced in Figure 23: with the polarization aligned along the scattering vector all the non-flip-spin scattering is nuclear while the spin-flip pattern is mainly paramagnetic, also containing some spin incoherent scattering In practice, most of the authors use a differential method to measure the paramagnetic scattering: the XZ method... nuclear spin incoherent (equation (122 )) is displayed in the lower part of the figure The magnitude of the spin-flip scattering, about 30% of the Fig 29 Upper part: non-spin-flip and spin-flip intensities scattered from amorphous Fe0.83 B0. 17 with a vertical field, lower part: spin-flip cross-section after removal of the nuclear spin incoherent scattering [51] ... examples of antiferromagnetic correlations Figure 26 represents the magnetic pattern obtained, also on D5, by an XZ difference on a polycrystalline sample of Polarized neutrons and polarization analysis 203 γ oxygen [48] The solid line on the figure represents the expected scattering for uncorrelated oxygen moments (S = 1, g = 2) The dip at small scattering vectors clearly shows the antiferromagnetic correlations . = Ψ ∗ e i Qr Ψd 3 r. (78 ) |Ψ is in the general case a molecular wave function |Ψ = atoms α j ϕ j , Polarized neutrons and polarization analysis 177 where the sum runs over all the magnetic atoms. Two. depend Polarized neutrons and polarization analysis 1 87 Fig. 16. Magnetization in Sm 3 Te 4 : the magnetic field is along [110] (perpendicular to the page). Fig. 17. Magnetization in Sm 3 Te 4 : the magnetic. complete separation between the nuclear and the magnetic scattering: the non-spin-flip scattering is only nuclear and the spin-flip scattering is only magnetic. On the other hand, when the incident
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