MAGNETIC FORM FACTORS R. Moon To cite this version: R. Moon. MAGNETIC FORM FACTORS. Journal de Physique Colloques, 1982, 43 (C7), pp.C7-187- C7-197. 10.1051/jphyscol:1982727. jpa-00222334 HAL Id: jpa-00222334 https://hal.archives-ouvertes.fr/jpa-00222334 Submitted on 1 Jan 1982 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. JOURNAL DE PHYSIQUE Colloque C7, supplément au n°12, Tome 43, décembre 1982 page C7-187 MAGNETIC FORM FACTORS R.M. Moon Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, U.S.A. Résumé.—La diffraction des neutrons polarisés est une méthode très puissante pour étudier, en mesurant le facteur de forme magnétique, la distribution des électrons qui se trouvent dans les couches extérieures des atomes de corps magnétiques. Nous donnons ici les résultats typiques pour quelques exemplaires choisis de la série de métaux ferromagnétiques 3d, des métaux paramagnétiques, des terres-rares, et finalement, des corps diamagnétiques. Pour chacun, nous établissons une comparaison entre l'expérience et la théorie. Nous pouvons en conclure qu'il nous manque une compréhension théorique profonde du moment magnétique orbital des métaux paramagnétiques. Abstract.—Polarized neutron diffraction has been an extremely valuable tech­ nique for studying outer electron distributions in magnetic materials through measurements of magnetic form factors. Representative results are presented on the following classes of elemental materials: 3d ferromagnetic metals, paramagnetic metals, rare-earth metals, and diamagnetic materials. In each case a comparison with the best available theoretical result is given. It is concluded that more theoretical effort is needed on the orbital moment in transition metals, particularly for the paramagnetic metals, in order to have a meaningful confrontation between theory and experiment. Introduction.—The principal motivation in undertaking the measurement of mag­ netic form factors with polarized neutrons has always been to gain a better under­ standing of solid-state wavefunctions. How are the outer electron densities of free atoms changed when the atoms are brought together in condensed matter? Because the atomic magnetic moments are carried by the outer electrons, the magnetic interaction of the neutron can be used to gain unique information on the spatial dependence of these electrons. Experimentalists hoped that their measurements could serve as a check point for theorists as they developed more sophisticated computing methods. In many cases these hopes have been fulfilled. Our purpose in this paper is to take a brief look backward, focussing on those areas where significant confrontations between experiment and theory have occurred, or should occur. To limit the scope of the review to manageable proportions, we will restrict it to elemental metals. 3d Ferromagnets.—The 3d ferromagnets were the first class of materials to be studied with the polarized-beam technique. These measurements were completed in the early 1960's under the supervision of Shull at MIT [1-3]. Prior to this time band calculations had revealed that the 3d wavefunctions were atomic-like near the top of the band, but were greatly expanded near the bottom of the band. The general expectation, therefore, was that an atomic form factor might give reasonable agree­ ment with the experimental results for Ni but that the degree of agreement would deteriorate in moving to the left in the periodic table. The most remarkable feature of these experimental results was that they could be reproduced with a very high degree of accuracy using free-atom form factors, in particular the unrestricted, or spin-polarized, Hartree-Fock calculations of Watson and Freeman [4]. This is demonstrated in Fig. 1 for the case of Co. To achieve such a fit it was necessary to scale up the moment attributed to 3d spin by a factor Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982727 C7-168 JOURNAL DE PHYSIQUE P 0.7 $ CALCULATED: $1.48 f,:d + -2 fs Gs '3d + 0.5 Fig. 1 : Comparison of the spherical 2 part of the observed form factor in 0.4 hcp cobalt with the atomic model. 2 The 3d spin form factor is based on U spi n-pol arized Hartree-Fock cal cu- $ 0.3 lations by Freeman and Watson for a ::I 3d74s2 configuration. The term with 0.2 fc is a small core polarization con- t ribution. (From reference 2) 0.1 0 0.2 0.4 0.6 0.8 4.0 -.In e A (l+a). The experiment seemed to be telling us that the spatial variation of the magnetization is almost exactly that expected for 3d electrons, but that there is more of this 3d moment than is revealed in a bulk magnetization measurement. It follows that there is then a spatially diffuse contribution of negative polarity that has a form factor which makes a negligible contribution at all reciprocal lat- tice points except the origin. This atomic model was highly successful for Fe (a=0.10), Co (az0.18) and Ni (a=0.19). An interesting question, which I believe is still unresolved, is &ether to attach any physical significance to the parameter a. Is it a measure of (sp) polarization, or of 3d overlap, or of the contribution of "paired" 3d electrons, or is it just an arbitrary fitting parameter with no physical significance? The other striking feature of the experimental results was the departure from spherical symmetry revealed by scatter in the high-Q values of the form factor. The data showed a preference for e symmetry in the case of Fe, t2 symmetry for Ni, and nearly spherical symmetry for fo. These symmetry properties 09 the data could also be accounted for on the basis of the atomic model. A number of band calculations have been reported for Fe and Ni which include a comparison of calculated spin density form factors with the experimental results. The most recent and most sophisticated of these are the self-consistent calculations of Cal laway and Wang [5-61 using the spin-polarized exchange-correlation potential of von Barth and Hedin. The differences between these calculations and the observed form factors are shown in Fig. 2. In his original work Mook used 0.606 ug as the low-temperature saturation moment of Ni, whereas the current best value is 0.616 pb 7 In preparing Fig. 2 we have renormal ized the Ni data to be consistent with the revised value for the total moment. The calculated form factors do not include a contribution from the orbital moment, so we should not expect exact agreement with the experimental results. As an estimate of the expected difference we have plotted (1 - 2/g)<j2>, with <j2> taken from the free-atom calculations of Watson and Freeman and with g taken from the work of Meyer and Asch [8]. First of all, note that the deviations are small, only a few percent of the forward scattering amplitude, so that the band calculation gives a reasonably good representation of the observed spin density (although not as good as the free atom model with two adjustable param- eters). The scatter in the Ni plot shows that the band calculation underestimates the asymmetry in this case, but it seems to do a good job in the Fe case in correctly predicting the asymmetry. On the other hand, the radial dependence seems about right for Ni but there are definite systematic differences in the Fe case. l~l~l[l~l~l - Ni fexp MOOK (RENORMALIZED1 - fcOlcWANG AND CALLAWAY - Fig. 2 : The difference between the observed form factors for Ni [3] and Fe C11 and the band calcula- - tions of Callaway and Wang [5,6]. The Ni experimen- - tal results have been re- IIIlIl11111. normalized slightly to - 1~1(1~1~1~1- agree with the current Fe f,,, SHULL AND YAMADA best value of the satu- rated moment. The solid - f,,,, CALLAWAY AND WANG - 1ines are estimates of the - - expected difference due to Q orbital contributions which were not considered in the band calculations. - P - - I I IIIIII- 0 0.2 0.4 0.6 0.8 4.0 4.2 sin B/X Our estimate of the orbital contribution is certainly crude and it would be desir- able to include a proper orbital contribution in band calculations for a more pre- cise comparison wZth the experimental results. There are two recently published experimental results which merit some theo- retical attention. One is the report by Cable [9] of the temperature dependence of the asymmetry in Ni, as shown in Fig. 3. There should be a thermal effect on the asymmetry due to smearing of the Fermi distribution function but calculations for a realistic band structure show that this effect is too small to explain the results of Fig. 3. It would appear that the spin splitting of the eg and tzg subbands have Fig. 3 : Temperature de- pendence of the eg popul a- tion in Ni. The solid point is from Mook [31. (From reference 9) C7-190 JOUP,NAL DE PHY S IQUE different temperature dependences. A similar experiment for Fe by Maglic [lo] showed no temperature dependence of the symmetry properties. In another recent pub1 ication, Steinsvoll et a1 . 1111 report on polari zed-beam measurements on phonons in Fe and Ni which yield magnetic form factor results at general positions in reciprocal space, that is, not restricted to reciprocal lattice positions.