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 ðer 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)
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. Part of the motivation for this work was a suggestion by Marshall and Lovesey [I21 that the magnetic form factor for itinerant electrons at general Q positions might drop below the value given by a smooth interpolation between the Bragg positions. As shown in Fig. 4 such behavior is seen for Fe along [loo] and [llOl directions, but it was not seen in Ni along the [I001 direction. In an effort to understand these results Moon [13,11] reconsidered the formalism for calculating the magnetic form factor for hybridized d bands formed of linear combinations of atomic 3d orbitals and orthogonalized plane waves for the case when near-neighbor overlap of the atomic orbitals is not negligible. This calculation led to two interesting results. First, at Bragg positions the form factor is the sum of two terms, one of which is a scaled atomic form factor and the other is the Fourier transform of the total overlap spin density. This overlap density is spread widely in real space resulting in a Fourier transform which makes a very small contribution at the Bragg peak positions, as shown in Fig. 5. Numerical calculations by Cooke [I41 of the magnitude of the net overlap spin density showed that it was negative for Fe and Ni. Thus the atomic model used in analyzing the experimental results is fully justified within the framework of LCAO bands with the parameter a a measure of the total 3d overlap spin density. Second, for general Q positions, a third term appears which results in a reduced form factor between Bragg positions. Unfortu- nately, this term does not quantitatively reproduce the behavior shown in Fig. 4. The observed behavior may not be a static property of itinerant bands, but rather a dynamic distortion of the 3d electron distribution as near-neighbor atoms move toward and away from each other during a thermal vibration. To distinguish between these possibilities, it would be of great interest to have the Fourier transform of calculated spin densities at general Q vectors along the principal symmetry direc- tions in Fe and Ni. The physical significance of the parameter a has been severely questioned by van Laar et al. [I51 on two grounds. They have developed a sophisticated technique for Fourier analysis of the experimental data and find no evidence for a#O.
Fe0.96Si 0.04
ELASTIC 2.0 0 INELASTIC 1 Fig. 4 : M(Q) for Fe as measured in inelastic polarized beam, mea- surements on longitudinal acoustic phonons a1 ong different crystal lographic directions. The. solid lines are interpolations of the experimental elastic form factors for these directions. For data points without error bars the uncertainties are equal to or smaller than the size of the points. (From reference 11)
1 2 3 4 151 RECIPROCAL LATTICE UNITS 0.45 Fig. 5 : Fourier trans- form of the near-neighbor overlap spin density in .Fe for two different 0.40 atomic 3d wave functions. -'a The calculation is based ;+- in part on numerical sums a I over occupied band states 0.05 by J. F. Cooke. The 0- z* curves are normalized by I5 dividing by the total I unpaired spin. Note the 0 very small value at reciprocal lattice posi - tions. (See reference 13) 0 4 2 3 10) (LATTICE UNITS)
Secondly, they cite the lack of anything like a negative constant density in band calculations as an argument against trying to split the total density in two parts. They may we1 1 be correct in their call to abandon a, but it would be interesting to test their method of analysis for sensitivity to a. This could be done by using a fabricated form factor made up of an atomic part and a negative constant as input for their calculation. To the extent that the observed data are accurately repre- sented by the atomic model, they have already done this test and concluded that there is no evidence for a negative constant density. My conclusion is that their method of analysis is not sensitive to the negative constant. Secondly, a band cal- culation which results in the total spin density doesn't address the question of whether it makes any physical sense to try to split the total into a 3d-like part and a diffuse part. It would be of some interest to see the contributions to the total form factor from the "unpdired" spin, the "paired" spin, and the sp density. In the band calculation by Cooke et al. El61 for Fe they find a negative sp moment of -0.1 VB resulting from spd hybridization. A very useful review of spin distributions in transition metal elements and alloys has been given by Menzinger and Sacchetti 1171.
4f Metals.-The case of Gd is a splendid example of an experimental advance revealing inadequacies in theoretical calcul ations, followed rapidly by theoretical advances of escalating complexity in a generally successful attempt to understand the experimental results. It was expected that the radial dependence of the 4f electrons in metals should be very little changed from their free atom distribu- tions. For Gd, with a half-filled 4f shell, there would be no complicating orbital moment and the 4f spin density should be spherically symmetric. From magnetization measurements the magnitude and direction of the conduction electron polarization was known. At the time of the polarized-beam work El81 the best available atomic calcu- lations were nonrelativistic Hartree-Fock. It was immediately apparent that these calculations were in poor agreement with the experimental results. Davis and Cooke [I91 showed that much better agreement could be obtained with a relativistic Hartree-Fock-Sl ater calculation, and shortly thereafter Freeman and Descl aux [20] obtained really excel lent agreement with relativistic Di rac-Fock calculations. The comparison between the experimental results, the Hartree-Fock calculation and the Dirac-Fock calculation is shown in Fig. 6. It is clear that something in addition to the 4f electrons is influencing the first few Bragg peaks. The obvious con- clusion is that the conduction electron polarization is responsible for this dif- ference. Harmon and Freeman [21] performed APW band calculations in an attempt to understand the conduction polarization but quantitative agreement with the experi- mental results was not obtained. Freeman and Desclaux [20] actually published Dirac-Fock form factors for both ~d~+and GdO. In an attempt to determine which of these came closest to the C7-192 JOURNAL DE PIIYSIQUE
Fig. 6 : Comparison of Gd results with non-relativ- CALCULATION ~d~+HARTREE - FOCK istic Hartree-Fock and 1.o ,/b relativistic Dirac-Fock 4f form factors. The cal- culated curves are normal- ized to 6.42 u~,the 0.5 expected value for the 4f moment at 96 K. (See references 18, and 20)
0 0.2 0.4 0.6 0.8 4.0 1.2 sin @/A experimental results, a quantitative comparison of the data with various 4'f form factors was undertaken. To make this comparison as meaningful as possible the core polarization was considered. A reasonable approximation of the Gd core polarization was obtained from spi n-polarized core wavefunctions for EU~+obtained from Watson [22]. The core form factor obtained' by Fourier inversion of the net core polariza- tion is shown in Fig. 7. Least-square fits of the experimental (uf) values were performed using the relationship
where p is the only adjustable parameter and represents the 4f moment. The first three data points were excluded to avoid any conduction electron influence. The standard deviation, a, based on the weighted rms deviations from the mean of 11 is a measure of the goodness of fit. A second criterion is that u should be equal to 6.42 LIB at 96 K, assuming that the temperature dependence of the 4f moment is the same as that of the total moment observed in magnetization measurements. The results for a number of 4f form factors are shown in Table 1. The best fit is that which produces the smallest a and the closest approach to 6.42 ug. This is clearly the Dirac-Fock GdO calculation with the core polarization included. Among the relativistic Hartree-Fock-Slater family of form factors obtained by varying the
l\\\l~l~l[~ - - SPIN POLARIZED CORE FORM FACTOR - Fig. 7 : Calculated form factor for the core polari- - zation in EU~+based on spi n pol arized Hartree-Fock - wavefunctions from R. E. Watson.
- - IIIIIIIIIII coefficient a in the Slater P~/~exchange potential, probably the best compromise is reached with a=0.70.
Table 1. Summary of fits of Gd 4f form factors to neutron data
Core -Pol. -u e 6-42-11 Ha rtree-Fock ~d3+ Yes 5.763 0.021 0.657 Hartree-Fock-S1 ater Gdo, ~~~0.76 Yes 6.257 0.030 0.163 GdO, a=0.72 Yes 6.378 0.028 0.042 Gdo, az0.68 Yes 6.477 0.024 -0.057 Di rac-Fock ~d3+ Yes 6.350 0.014 0.070 Gd 0 Yes 6.399 0.014 0.021 GdO No 6.363 0.016 0.057
The final conclusion of the Gd story is that the relativistic Dirac-Fock wave- functions for the free atom can be used with confidence to describe the radial 4f electron densities in the rare-earth metal s. The conduction electron polarization is a much more difficult problem.
Paramagnetic Metal s.--Wi th the addition of superconducting magnets to polarized beam diffractometers it has been possible to measure the induced moment form factor of a number of paramagnetic metals. These are difficult experiments, involving long counting times because the induced moment may be only several mi 11i-Bohr magnetons. In addition to the purely statistical problem of accumulating sufficient counts, there are a distressingly large number of side effects which can influence the experimental results and which are usually negligible when dealing with ferro- magnets. These are nuclear polarization [30], the neut ron-spi n-neutron-orbit inter- action C231, diamagnetic scattering, sample purity, and the effects of the applied magnetic field on the neutron trajectory and velocity [24,25]. In addition, there is an urge to use larger than normal crystals in order to reduce counting times, resulting in extinction problems. In my opinion, the field effects set a lower limit on the ultimate accuracy which can be achieved in flipping ratio measurements of about AR = 5 x This estimate was made for an applied field of 60 kOe and would increase for higher fields. This estimate can be used to survey the transition metal paramagnets in order to identify promi sing candidates for this experimental technique. The residual flipping ratio for an elemental paramagnet is given by
where X is the susceptibility in emu/mole, H is the applied field in Oe and b is the nuclear scattering amplitude in 10-12 cm. A reasonable goal is to measure the form factor f(Q) to an accuracy of +0.025, which means we must measure r to an accuracy of
In Fig. 8 we show the desired accuracies for the paramagnetic metals of the 3d, 4d, and 5d transition series for an applied field of 60 kOe. The dashed line is my estimate of the non-statistical errors due to the field effects. Elements shown above the line can be measured to the desired accuracy by accumulati ng a sufficient number of counts, shown on the right. For elements below the line, the non- statistical errors are larger than the desired accuracy. Note that the majority of favorable cases have already been investigated. A higher magnetic field would move each of the points upward but would a;so move the dashed line upward, with the C7-194 JOUPSJAL DE PHYSIQUE
Fig. 8 : A1 lowable error in the flipping ratio to - determine the normalized form factor to t0.025 for Mn - (2 X~O')~ - b- transition metal para- z magnets. Closed circles - P.d 3 indicate elements for g which data have been - - to8 TJ c: w obtained. Dashed line m -S." pi gives estimated level of ------.- O-RI---:~Jhy.aon~-= do9 3 non-statistical errors no - 10'0 = due to field effects. 3d I 4d I 5d
result that data could be obtained faster but new materials would not become more accessible to experimentation. From the flipping ratio measurements we can obtain the zz component of the generalized susceptibility, evaluated at zero energy transfer and at reciprocal lat- tice positions. This can be written as the sum of several terms Lan -+ xzZ(:,o) = -x:::~(?) - xdia(:) + xSp(~)+ xOrb(i),
where the first term is the diamagnetic contribution of the core electrons, the second is the Landau diamagnetism of the outer electrons, the third is the Pauli spin paramagnetism and the fourth is the Van Vleck orbital paramagnetism. Each of these terms will have characteristic form factors associated with the shape of the various contributions to the induced magnetization. The diamagnetic core contri bu- tion can be calculated using a formula derived by Stassis [26] and it is common practice to correct the observed results by subtracting this term, leaving a net susceptibility due to the outer electrons. The calculation of the remaining terms has been discussed by Oh et al. [27]. The spin contribution has a form factor related to the average of $(r) for states at the Fermi surface and is relatively easy for band theorists to calculate. The orbital contribution involves a summation of off-diagonal matrix elements over the entire band structure and is a much more difficult thing to calculate. There are now a number of APW band calculations for the spin contribution, only one calculation (by Oh et al. [27] for Cr) of the orbital contribution, and no calculations of the Landau diamagnetic term for realistic band structures. The lack of theoretical activity on the orbital form factor, coupled with a frequent lack of experimental evidence on the relative size of the spin and orbital susceptibilities, has made it difficult to properly inter- pret the available magnetic form factor data. Summaries of the experimental results on a variety of paramagnetic metals have been published by Moon et al. [25] and by Stassis [281. Measurements now exist for Sc [29], Ti [251, V [301, Cr [311, Y [25], Zr [32], Nb [25], Pd [33], Pt [34], Lu r351, and a-U [36]. Certain interesting generalities have emerged from this work. In the 3d series it is surprising that for Cr and V the observed form factor is very much like the free atom with an appropriate mixture of spin and orbital contribu- tions. Ti shows definite evidence of the expected wavefunction expansion as we move to the left in the periodic table, and Sc is definitely expanded. The Sc results are shown in Fig. 9 together with the APW calculated spin form factor of Gupta and Freeman [37]. Notice the rapid fall of the first three Bragg peaks and a shift upward on the fourth. This behavior has now been observed in Sc, Y, and Lu and seems to be a signature for three electrons in ds bands in hcp materials. It is also evident in APW calculations for these materials. In the Lu case the agreement between the experimental results and APW calculations of the spin form factor is even better than displayed in Fig. 9 for Sc. The. agreement is so good that it turns into an embarrassment because it leaves no room for an orbital contribution. This is inconsistent with the observed anisotropy in static susceptibility measurements [38] and with calculations by Das [39] showing a substantial Van Vleck suscepti- bility. We thus have a peculiar situation where good agreement between theory and experiment calls into question the accuracy of the theoretical calculations. The Fig. 9 : The paramagnetic form factor of metallic scandium. The closed circles are spin form factors from an APW calculation by Gupta and Freeman [37] and the solid line is a free-ion Hartree-Fock calculation. (From ref- erence 29) t
0 , 0 0.f 0.2 0.3 0.4 sin @/A (A;') calculated form factors were based on non-self-consistent APW calculations which may not yield adequate wavefunctions. Direct measurements of the gyromagnetic ratios in Sc and Lu would be highly desirable, as would self-consistent band calculations of both the spin and orbital form factors. The need for orbital calculations is not so great in cases such as Pd and Pt where the observation and the APW spin form factor are not greatly different from the free atom results. In the one case, Cr, where the orbital form factor has been calculated [27] for band electrons, the result was very close to the free atom calculation. The authors caution that this will probably not be the case near the beginning of the 3d series. Finally, there is the Landau diamagnetic susceptibility, of which we are almost comp7etely ignorant. It undoubtedly falls very rapidly as a function of Q. It certainly makes a significant contribution at the origin and may also affect the first few Bragg points. Oh et al. [27] also point out that there are certain sur- face integrals over the unit cell boundary which appear in the theory and which probably affect only XZZ(O,O) and which have been universally ignored. In the analysis of the Cr results by Oh et al. there is a suggestion that these neglected terms may be significant and positive. All this is an introduction to a plea to experimentalists and theorists, but especially experimentalists, to stop reporting normalized form factors. Whenever we are dealing with different electron groups, or with effects which have substantially different Q dependences, the point at the ori- gin may not be proper normalization for the effect we are measuring at finite Q. I mean not proper in the sense that it can lead to great confusion. We a1 1 have a tendency to glance at experimental form factors compared to a calculated curve and pronounce that the density is contracted or expanded depending on whether the points are above or below the curve. In fact, the distributions may be identical but one is improperly normalized. It would be far better to report M(Q) for ferromagnets and X(Q) for paramagnets. Diama nets.-The first polarized beam study of a diamagnetic material, Bi , was by CO&J and Shull. Bi has an anomalously high diamagnetic susceptibility attributed to five valence electrons outside a core of 78 electrons. The crystal structure is rhombohedra1 and the susceptibility is strongly anisotropic. Neutron data were obtained with the field a1 ong three different crystal lographic directions. One of these sets of data is shown in Fig. 10. Note that the outer electron contribution is evident only for the first Bragg peak and the point at the origin is far off scale. No band calcuiation has been performed but these data should serve as a valuable bench mark for theorists as they work toward a thorough understanding of metal 1ic suscept+hilities. C7-196 JOURNAL DE PHYSIQUE
tlntercapt=-291 from ru~ceptibility I BISMUTH, 290 K H II Binary (a) axis I-H 14.25 T i Fig. 10 : Residual flipping ratios for bismuth. The solid line 'is the calculated contribution for the 78 core electrons, the dashed line is the free atom calculation for all 83 electrons. Note that the point at the origin, based on bulk electrons -- susceptibility measurements, is far off scale. (From reference 40)
Complex Paramagnets.-+ inal ly, I would like to mention a slightly different application of the polarized beam technique. Many interesting magnetic materials have more than one type of magnetic atom, or have the same atom on several dirferent crystal lographic sites. Bul k susceptibility measurements average over the different atoms or different sites, but by observing several Bragg peaks in neutron diffrac- tion experiments we can determi ne each site susceptibility separately. We must know the form factors involved and make use of the structure factor variations to deter- mine induced moments for each site. An example is the case of Nd where half the atoms are on cubic sites and half on hexagonal sites. By making flipping ratio measurements on (102) reflections the desired separation was obtained. For L = 1,3, FM % phex and for 2 = 0,4, FM.% (2pcub - vhex). The separate susceptibilities are shown in Fig. 11. This techn~quegives a much more complete description of the mag- netic susceptibility of complex paramagnets than does the bulk susceptibility.
I I I I
Nd
0 CUBIC SITE HEXAGONAL SITE Fig. 11 : The cubic and hexagonal suscepti bili- > TRANSITION TEMPERATURE ties in Nd metal deter- -5 0.2 FOR H-0 243 T ALONG [T20] mined by pol arized m t- neutron diffraction. P W (From Moon, tebech, and 0m - Thompson, unpubl ishM) 2 O.!
0 I I I I 0 0 10 20 30 40 50 TEMPERATURE (K) This research was sponsored by the Division of Materi a1 Sciences, U .S. Depart- ment of Energy under Contract No. W-7405-eng-26 with the Union Carbide Corporation.
References
SHULL C. G. and YAMADA Y., J. Phys. Soc. Japan 11 Suppl . B-I11 (1962) 1. MOON R. M., Phys. Rev. 136 (1964) A195. MOOK H. A., Phys. Rev. 148 (1966) 495. WATSON R. E. and FREEMAm. J., Acta Cryst. 14, (1961) 27. CALLAWAY J. and WANG C. S., Phys. Rev. B 16 v977) 2095. WANG C. S. and CALLAWAY J., Phys. Rev. B (1977) 298. DANAN H., HERR A., and MEYER A.J.P., J. AEl . Phys. 2 (1968) 669. MEYER A.J.P. and ASCH G., J. Appl. Phys. 2 (1961) 330s. CABLE J. W., Phys. Rev. B 23 (1981) 6168. MAGLIC R. C., AIP Conf. Prz. No. 5 (1972) 1420. STEINSVOLL O., MOON R. M., KOEHLER W. C., and WINDSOR C. G., Phys. Rev. 84 (1981) 4031. MARSHALL W. and LOVESEY S. W., in Theory of Thermal Neutron Scattering (Oxford University, Oxford, 1971), p. 124. MOON R. M., AIP Conf. Proc. No. 24 (1975) 425. COOKE J. F., private communication. VAN LAAR B., MANIAWSKI F., KAPRZYK S., and DOBRZYNSKI L., J. Magn. Magn. Mater. -14 (1979) 94; KAPRZYK S., VAN LAAR B., and MANIAWSKI F., J. Magn. Magn. Mater. -23 (1981) 105. See also B. van Laar, F. Maniawski, and S. Kaprzyk in these proceedings. COOKE J. F., LYNN J. W., and DAVIS, H. L., Phys. Rev. B 21 (1980) 4118. MENZINGER F. and SACCHETTI F., Nukleoni ka 4 (1979) 737 .- MOON, R. M., KOEHLER W. C., CABLE J. W., and CHILD H. R., Phys. Rev. B 5 (1972) 997. DAV~SH. and COOKE J. F., AIP Conf. Proc. No. 5 (1972) 1441. FREEMAN A. J. and DESCLAUX J. P., Int. J. Magn. 3 (1972) 311. HARMON B. N. and FREEMAN A. J., Phys. Rev. B 10 T1974) 1979. WATSON R. E.. private communication. SHULL C. ~.,-~h~s.Rev. Lett. 10 (1963) 297. MOON R. M., KOEHLER W. C., and SHULL C. G., Nucl. Inst. Methods 129 (1975) 515. MOON R. M., KOEHLER W. C., and CABLE J. W., Proc. Conf. on Neutron Scattering, Gatlinburg, 1976 (CONF-760601-P2) p. 577. STASSIS C., Phys. Rev. Lett. (1970) 1415. OH K. H., HARMON B. N., LIU S. H., and SINHA S. K., Phys. Rev. B 14 (1976) 1283. STASSIS C., Nukleoni ka 3 (1979) 765. KOEHLER W. C. and MOON R. M., Phys. Rev. Lett. 2 (1976) 616. SHULL C. G. and FERRIER R. P., PHYS. Rev. Lett. 10 (1963) 295. STASSIS C., KLINE G. R., and SINHA S. K., Phys. Rev. B 11 (1975) 2171. STASSIS C., KLINE G., HARMON B. N., MOON R. M.. and KOE~ERW. C., J. Magn. Magn. ~ater.14 (1979) 303. CABLE J. W., WOLLAN E. O., FELCHER G. P., BRUN T. O., and HORNFELDT S. P., Phys. Rev. Lett. 34 (1975) 278. MAGLIC R., BRUN TFO., FELCHER G. P., and CHANG Y. K., J. Magn. Magn. Mater. 9 (1978) 318. STASSIS C., KLINE G. R., LOONG C.-K., and HARMON B. R., Solid State Commun. 23 (1977) 159. MAGLIC R. C., LANDER G. H., MUELLER M. H., and KLEB R., Phys. Rev. B 11 (1978) 308. GUPTA R. P. and FREEMAN A. J., Phys. Rev. Lett. 36 (1976) 613. SPEDDING F. H. and CROAT J. J., J. Chern. Phys. -59 (1973) 2451. DAS S. G., Phys. Rev. B 13 (1976) 3978. COLLINS S. R., Ph.D ThesE, Massachusetts Institute of Technology, 1979.