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Spin densities and form factors: Past, present, and future
Physica 137B (1986) 19-30 North-Holland, Amsterdam
19
SPIN DENSITIES A N D FORM FACTORS: PAST, PRESENT, AND FUTURE * R.M. MOON Solid State Dit,ision, Oak Rtdge National Laboratory, Oak Ridge. TN .¢7831. USA
The development of polarized neutron diffraction as a standard method for the study of magnetic moment densities was largely due to the efforts of C.G. Shull. The early experiments by Shull, his students, and co-workers are reviewed. A comparison of his early results on the magnetic form factor of Fe with very recent theoretical work is presented and excellent agreement is found. Some more recent experimental results are described to illustrate the many types of magnetic systems to which this technique has been applied. Brief speculations about future directions are offered.
1. Introduction
The applicati~.n of polarized neutrons to the study of spin densities and magnetic form factors has been an important part of the total scientific impact of neutron scattering. As in so many aspects of neutron scattering, Cliff Shuli was there at the beginning and guided much of the early polarized beam work. After leaving Oak Ridge and before he was able to begin experiments at the MIT reactor, Shull was a frequent visitor at the Brookhaven Graphite Research Reactor. During this period he guided the development of the first dedicated polarized beam diffractometer [1], and shortly thereafter built a similar machine as one of his early instruments at MIT. To appreciate the importance of the polarized beam technique, a little background in the fundamentals of magnetic scattering is necessary. The cross section for Bragg scattering from a ferromagnet can be written as
o - b 2 + 2bpP" q + pZq2,
(1)
where b is the nuclear scattering amplitude, P is the neutron polarization vector, and q is the magnetic interaction vector (the projection of a unit vector along the magnetization onto the plane * Research sponsored by the Division of Materials Sciences, U.S. Department of Energy under contract DE-AC05840R21400 with Martin Marietta Energy Systems. inc. 0378-4363/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
perpendicular to the scattering vector). The quantity of interest here is the magnetic scattering amplitude p, which is proportional to the Fourier transform of the magnetic moment density. In eq. (1) we assume an elemental material with atoms on equivalent sites: for a more complex material the nuclear and magnetic scattering amplitudes should be replaced by the nuclear and magnetic structure factors. The polarized beam experiment is so arranged that the magnetization is perpendicular to the scattering vector (q2 = 1) and that P can be either parallel or antiparallel to the magnetization ( P . q = +1). The ratio of Bragg intensities for these two directions of the neutron spin is then given by
(b+p/"
R=~b_p
} .
(2)
Fhis so-called flipping ratio is the measured quantity in polarized beam diffraction; from it one obtains the ratio p / b . The polarization dependent term in eq. (1), being linear in the magnetic amplitude, greatly increases the sensitivity to small values of the magnetic amplitude. With an unpolarized beam the polarization dependent term would be absent and the best one could do, by controlling the direction of magnetization (q2= 0 or 1), would be to measure the ratio
U= 1 + ( p / b ) 2.
(3)
]'he importance of the polarized beam technique is
R M. Moon / S p i n den~itu,s and [orm [actot:~
20
Table I .Hlustrating the great increase in sensitivity of the polarized b e a m technique over an unpolarized neutron m e a s u r e m e n t for Fe
hk l
p/ h
Polarized R
Unpolarized /5
110 220 330 440
(/.388 0.107 0.0083 - 0.0096
5.144 1.537 1.034 0.962
1.151 1.0114 1 .(XIXI7 I .(XI(X)9
clearly evident from table 1 which gives a comparison of the ratios R and U for a few reflections for the case of Fe. Note too that the sign of p relative to h is given by the polarized beam experiment. It is obvious that without a polarized beam diffractometer, meaningful measurements of the magnetic amplitude for Fe would be restricted to the first few Bragg peaks. Because the nuclear interaction has such a short range, the nuclear amplitude is independent of the scattering vector Q, so that the Q dependence of p/b accurately reflects the spatial variation of the magnetic moment density through a Fourier transform. The data may be converted to Bohr magneton units by
2b P(O) M ( Q ) - 3'to b
(4)
where y is the magnitude of the neutron moment in nuclear magnetons and r0 is the classical electron radius. The data are frequently reduced to
normalized form factor values given bv
f ( Q ) = M(Q)/M(O).
(5)
As a further indication of the sensitivity of the polarized-beam technique and as a commentary on one of its chief practioners, I list in table II some of the experiments which are discussed in this volume. In each case the largest (or smallest for p/h < 1) value of the measured flipping ratios is given with corresponding values for M(Q). In most cases this was the easiest measurement to obtain; usually the experiment required observation of several Bragg peaks with decreasing magnetic amplitude. I believe this table illustrates two important characteristics of Cliff Shull: the ability and determination t o get the most out of any experimental technique, and great patience. One does not measure intensity ratios to five significant figures without waiting to count a very large number of neutrons.
2. 3d ferromagnets After constructing the S-3 polarized beam diffractometer at MIT, Shull chose Fe as the first material to study with this new and very sensitive machine. A few Fe reflections had been measurcd earlier in the work at Brookhaven, but the MIT work was much more comprehensive in terms of number of reflections and much more precise. The polarized beam technique, involving a measurement of the intensity ratio as the neutron spin is
T a b l e II A chronological list of some of the C.G. Shull e x p e r i m e n t s giving an i n d i c a t i o n of the largest magnetic signal detected m each experiment Sample
Student or colleague
Publ. date
Extreme R
Maximum M( Q I ( p- ~))
Fe V Co I-e(jm~Pd0~7 Ni V~Si C u ( F e 1250 p p m ) C u ( F e 387 p p m ) Bi
Yamada Ferrier Moon Phillips M(x'Jk Wedgewood Stassis Dickens Collins
62 63 64 65 66 66 72 75 79
5.14 0.988 32.6 1.13 1.61 (}.989 1.00043 1.00101 I).99944
1.37 5.3 × 1.32 7.3 x 4.5 x 4.8 × 3.1 × 7.1 x -4.4 x
10 "~ 10 10 10 10 10 10
"? I "~ 4 4 -1
R.M. Moon / Spin densities and form factors
reversed, is clean compared to most diffraction measurements. Many factors which influence the intensity, such as absorption and the DebyeWaller factor, cancel in forming the intensity ratio. Extinction was minimized by distorting the crystals in a controlled manner and by working with thin samples. Nevertheless, in this "clean" experiment, Shull noticed very early that the results on some reflections were sample dependent and there was even a lack of reproducibility for the same sample following remounting in the diffractometer! In consultation with O.J. Guentert and F.P. Ricci over these disturbing results, it was concluded that
21
they were probably the result of simultaneous Bragg reflections. Subsequent experiments and calculations confirmed this conclusion [2]. A new goniometer which allowed continuous, motordriven rotation of the sample around the scattering vector was added to the polarized beam diffractometer. Henceforth, all flipping ratio measurements would be made as a function of this rotation angle and anomalous results discarded. In addition, samples were shaped to minimize the effects of simultaneous reflections (long in only one dimension). With the assistance of Y. Yamada, Shull proceeded to collect data, shown in fig. 1, on
~6
~5 0.4
~XPERIMENT
I0.5
3d64s2(calc) WATSON-FREEMAN (SPHERICAL SPINONLY)
~ ~dae~)
WATS(
o
QI
::..__
o
"°Jo
d,
Fig. I. Experimental magnetic form factor of Fe compared with spin-polarized Hartree-Fock atomic calculations for the spherically averaged spin density. (From ref. 4.)
22
R.M. Mo,m / 5"pro demme~ and.lbrm /,(t,,r.~
the first 26 reflections for Fe [3.4]. These data were analyzed in two ways: by Fourier inversion to give maps of the magnetization density, and by comparison with calculated atomic form factors. Pronounced departures from spherical symmetry were evident in the magnetization maps, as shown in fig. 2. The magnetization was concentrated along [100] directions with minima along [111} directions. A quantitative measure of this aspherical behavior was obtained by fitting an atomic model to the data following the work of Weiss and Freeman [5] and using spin-polarized atomic form factors calculated by Watson and Freeman [6]. It was found that 53g,~ of the spin density was in Eg orbitals, to be compared with an Eg population of 40% for spherical symmetry. To obtain a good fit with the experimental data, the 3d spin moment had to be increased beyond that expected for the total moment
per atom as determined by magnetization measurements. Within the atomic model, this implies the existence of a diffuse negative spin component which has an appreciable contribution to the form factor at only the origin. Negative regions of magnetic moment density at positions well removed from atomic sites were also revealed by direct Fourier transformation of the data {7]. In the atomic model, it was natural to ascribe this diffuse negative polarization to 4s electrons, but we will return to this point later. Shull and his students, R.M. Moon and H.A. Mook, continued this work on the 3d ferromagnets. obtaining detailed form factor measurements on hexagonal Co [8] and Ni [9]. The atomic model also gave remarkable fits to the data in these two cases, with Co being nearly spherically symmetric and Ni showing strong T_,~: 181~) behavior. To achieve good fits it was again necessary to scale up
L~ 4.~.% ~
DIRECTIONJM. ~TIC MOM(NT ~NS|TY
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F i g 2. Magnetic m o m e n t densit?, as a function of distance from the l-"e nucleus along the principle cr?,stallographic directions. (From ref. 4.)
R.M. Moon / Spin densities and form factors
the moment attributed to 3d spin by a factor of (1 + a). The parameter a represented the fractional increase in 3d spin over that expected from magnetization measurements needed to give agreement with the neutron results. This increase was rather large: for Fe, a = 0.10, for Co, a = 0.18; and for Ni, a = 0.19. The experiments seemed to be telling us that the spatial variation of the magnetization was almost exactly that calculated in spin-polarized Hartree-Fock theory for a 3d"-24sZ configuration, but that there was more of this 3d-like moment than is revealed in a bulk magnetization measurement. In all three cases, direct Fourier inversion showed negative densities away from atomic sites. The success of the atomic model led some people to attach greater physical significance to the negative, nearly constant spin density component than was warranted. Van Laar and colleagues [10-12] have been critical of the concept of a constant negative component and have concluded, based on their projection operator method, that there is no experimental evidence for the existence of such a component. I must take this opportunity to retract a suggestion [13] that their method may not be sensitive to the presence of a constant component; they have demonstrated that the sensitivity is adequate [12]. One of the original publications [8] contained a specific warning on this point which also seems appropriate today. " O n e should be cautious in ascribing significance to the form-factor analysis beyond the fact that it is a convenient way to describe the observed spin distribution. In particular, we cannot claim that this analysis proves the existence of a constant negative polarization in the conduction band or that the spin density in the 3d band is just like that calculated for the free atom. All that can be said with certainty is that the sum of the polarization in the 3d and conduction bands has the spatial dependence described by the model." It is perhaps worth remarking that the decomposition of a periodic function into a superposition of "atomic" functions is not unique. That is, there are any number of ways to select #a(r) such that a given periodic function is formed by g(,) = E~G(ri
r,).
(6)
23
The Fourier transforms of different g~(r) which satisfy eq. (6) will be different for a general value of Q, but will be identical for the set of Q constituting the reciprocal lattice. A more significant discussion would center on why the spin density goes negative in regions of the unit cell which are most remote from atomic sites. This is a model independent fact given by Fourier inversion of the data for Fe, Co, and Ni. Probably the most important contribution to this negative density comes from paired electrons near the bottom of the 3d bands where the wave functions are significantly expanded relative to states near the top of the 3d bands. There is also a tendency for minority spin wave functions to be more expanded than corresponding majority spin wave functions, so that there should be a negative polarization at large distances from atomic sites from these paired 3d electrons. Band calculations do not support the idea of negative (sp) polarization from exchange splitting, but there is a negative (sp) polarization produced by (spd) hybridization. In the case of Fe, Cooke et al. [14] have calculated this to be -0.1gH. Finally, the atomic model used in fitting the data does have some theoretical credibility when the crystal wave functions are written in the tight-binding formalism [15]. The negative density is associated with overlap of the 3d atomic functions used in synthesizing the periodic wave functions. Theoreticians have made great progress in calculating energy bands since these experiments were performed. The self-consistent calculations of Callaway and Wang for Fe [16] and Ni [17] using the yon Barth-Hedin exchange-correlation potential gave relatively good agreement with the experimental results. Unfortunately the direct comparison of these calculated form factors with the experimental results is not quite proper because the calculation has no orbital moment and the experimental results do contain a contribution from the small orbital moment. Van Laar et al. [10] have extracted a "spin only" form factor from the experimental data to compare with the band calculation. They find a small disagreement between calculated and observed asphericities for Ni but good agreement with the spherically averaged form factor, while for Fe the calculated asphericity is
R.M. Moon / Spin densities and form lactor.~
24
correct but the spherically averaged calculated form factor is slightly high. The same conclusions can be reached by inspection of the difference between experimental and calculated form factors [13]. The first band calculations for Fe and Ni incorporating a spin-orbit interaction and hence an orbital moment have recently been completed by Cooke and Blackman [18]. Their initial calculations give rather good agreement with the experimental results, as shown in fig. 3 for Fe. It would seem that all the theoretical tools are finally in hand for a detailed confrontation between theory and experiment on the magnetic moment density in the 3d ferromagnets. There are at least two additional experimental results on 3d spin densities which merit some theoretical attention. Shull [4] and Maglic [19] measured the temperature dependence of the asphericity in Fe and found it to be negligibly small. This is not surprising since the major temperature effect would be a small repopulation of states near the Fermi level. However, Cable [20] has found for Ni a rather large temperature dependence of the fraction of spin associated with Eg states. This effect is too large to be explained by a thermal repopulation of states and suggests that the exchange splittings of Eg and Teg bands have
0.8
i
Fe EXP x
-SHULL
CALC-COOKE
& YAMADA & BLACKMAN
0.6 nO co 0.4
112
0 u_ 0.2
0.0 i
-0.1
0.0
0.2
0.4
I 0.6
0.8
1.0
1.2
sm 8/,~
l'ig. 3. C o m p a r i s o n of the e x p e r i m e n t a l Fe form factc, r with a b a n d c a l c u l a t i o n i n c l u d i n g the orbital contribution.
different temperature dependences. Another unexplained result is found in the work of Steinsvoll et al. [21] who used the polarized-beam technique to measure the magnetic amplitudes for phonon scattering in Fe and Ni. These measurements were not restricted to reciprocal lattice positions. Systematic deviations from a smooth interpolation of M ( Q ) between Bragg positions was found in Fe along [100] and [110] directions, but was not found in Ni along the [100] direction. It is not clear whether the deviations found in Fe are a static property of the band wave functions as suggested by Marshall and Lovesey [22] or result from a dynamic distortion of the 3d electron distribution during thermal vibrations. To distinguish between these possibilities, it would be of great interest to have the Fourier transform of calculated spin densities as a continuous function of Q along the principal symmetry directions in Fe and Ni.
3. Small moment systems The sensitivity of the polarized-beam technique has been used by Shull and co-workers in several pioneering experiments on ferromagnets with small moments, paramagnets, and even a diamagnet. Small amounts of Fe alloyed with Pd cause ferromagnetic behavior at low temperatures. While the total moment per Fe atom is around 10 P-u, the average moment per atom, for Fe concentrations of about 1%, is only 0.1p,~. The accurate measurement of the average spatial distribution of such small moments had not been possible before the development of the polarized-beam technique. Walter Phillips, one of Shull's early students at MIT, made such measurements on a crystal of Pd with 1.3% Fe [23]. The distribution of magnetic moment showed a strong departure from spherical symmetry favoring T2g behavior. After completing the Fe work, Shull was quick to fully exploit the capabilities of the polarizedneutron technique by beginning a study of vanadium [24], a paramagnetic metal with susceptibility nearly independent of temperature. With the available magnetic field, a moment of only' 1 m / ~ per atom could be induced, yet the sensitivity of the technique coupled with the unusually small
R.M. Moon / Spin densities and form factors
coherent nuclear amplitude of V allowed fairly accurate measurements to be obtained. These are discussed elsewhere in this volume [25]. It is worth noting that in the course of this work nuclear polarization (PN = 0.0004) was detected for the first time with this technique, and the neutron spin-neutron orbit interaction [26] was first observed. More than a decade would pass before similar work on paramagnetic metals could be taken up elsewhere with superconducting magnets and neutron sources of higher flux. The Kondo effect was the subject of extensive experimental and theoretical activity in the early '70s. An important issue was whether a compensating cloud of oppositely polarized conduction electrons formed around localized magnetic impurities in a nonmagnetic host. The classic Kondo system was dilute Fe in Cu, and Stassis and Shull [27] used the polarized beam technique to measure the 3d susceptibility in a Cu crystal containing 1250 p p m of Fe. They found that the 3d susceptibility was equal to the bulk susceptibility measured on the same sample, apparently leaving no room for a cloud of oppositely polarized conduction electrons. However, the issue was not clearly resolved because, for the Fe concentration used in this experiment, it was estimated that only 35% of the magnetization arises from isolated impurities and the remainder from pairs of Fe atoms ferromagnetically coupled. Subsequently, Dickens and Shull [28] measured the (111) scattering amplitude as a function of concentration (down to 387 ppm) and temperature, and used the concentration dependence to extrapolate to the isolated impurity case. At temperatures below 10 K they found an isolated 3d moment per impurity substantially larger than the total moment per impurity given by magnetization measurements. This does suggest the presence of a compensating polarization of conduction electrons that develops at low temperature, in apparent contradiction with conclusions based on Knight shift measurements. This situation is still unresolved. The first and only polarized beam study of an elemental diamagnet was by Collins [29] and Shuli. Bismuth has an anomalously high diamagnetic susceptibility attributed to five valence electrons outside a core of 78 electrons. The crystal structure
25 94
~
T
I
l'-Intercaptt-291
r
1
from susceptibility
BISMUTH, 2 9 0 K II BInQry (a) axis H'4.25 T
-80 X E -60 o "o
"~'\
~
"-..//
_o - 4 0
\ \
o c
7-
, 8 3 f r e e atom electrons
-20
N
no
o O
2 I
g~ 0_ 2
J l 0
0.1
o ~ _ _0 0 [ IJ[ 0.2 sine/),
0.3 (~-i~
_0 t 0.4
0.5
Fig. 4. Residual flipping ratios (R -1) 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 susceptibility measurements, is far off scale. (From ref. 29.)
is rhombohedral and the susceptibility is strongly anisotropic. Neutron data were obtained with the field along three different crystallographic directions. One of these sets of data is shown in fig. 4; the quantity plotted here is R - 1 . The contribution from the core electrons can be accurately calculated and is shown in the figure. Note that the outer electron contribution is evident only in the first Bragg peak, and the point at the origin, from bulk susceptibility measurements, is far off scale. No band calculation is available for comparison with these results, but fig. 4 illustrates an important characteristic of the Landau diamagnetism of outer electrons, namely that it is almost zero at Bragg peak positions but may be significant at Q = 0. This is important in the measurement and analysis of paramagnetic susceptibilities.
4. Paramagnetic
metals
With the addition of superconducting magnets to polarized beam diffractometers it has been pos-
R.M. Moon / Spin denstties andJorm f~lctors
26
sible to continue the work started by Shull with vanadium by measuring 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 a few milli-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 arc usually negligible when dealing with ferromagnets. These arc nuclear polarization, the neutron spin-neutron orbit interaction [26], diamagnetic scattering, sample purity. and the effects of the applied magnetic field on the neutron trajectory and velocity [30,31 ]. References to most experimental results are given in summaries by Moon [13,31] and Stassis [32]. To these should be added recent results on V [331 and Rh [34]. From the flipping ratio measurements we obtain the zz component of the generalized susceptibility, evaluated at zero energy transfer and at reciprocal lattice positions. This can be written as the sum of several terms X : : ( Q ) = - x ~ .•,. . . .(.O )
+ x,,~(¢2),
Oh et al. [36]. "]'his lack of theoretical effort has made the interpretation of the experimental results difficult. Fortunately, Cooke. Liu and t,iu [37,38] have now developed a computational technique to calculate the orbital contribution for both cubic and hexagonal metals. A detailed comparison with all the experimental results has not yet been completed but preliminary results show reasonable agreement in most cases. This comparison will serve to validate band wave functions and will give a measure of the exchange enhancement of the spin susceptibilities. As expected the experimental results show marked deviations from free atom behavior for elements near the beginning of the 3d and 4d transition series. This is associated with spreading of the wave functions near the bottom of the d bands. A comparison of observed and calculated x ( Q ) is shown in fig. 5 for Zr. Their calculations give the orbital contribution and the uncnhanced spin contribution in absolute units. In fig. 5 the enhancement factor has been adjusted to give agreement with the first few Bragg peaks.
"~'""" xJ., ~t) + x,r(Q)
(7) 200
where the first 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 different contributions to the total induced moment. As already noted in the case of Bi, the Landau term may affect only the point at the origin and is always of unknown magnitude. Therefore, the practice of reporting normalized form factors can only lead to confusion and should be abandoned. The most useful quantities to report are absolute values of X(O)The core diamagnetic term is readily calculated [35] and is usually subtracted from the observed results. The spin contribution is related to ~2(r) for states at the Fermi surface and is easy for band theorists to calculate. The orbital contribution is much more difficult to calculate and, until quite recently, had only been done for the case of Cr by
'
I
180 160
1
I
I
i
x~o~, 74 : ?
X(o). H .
140
Zr
T = 100 K
~, -6 E
120 100
E
80
EXPERIMENT:
STASSIS
et 8 1
CALCULATION COOKE, LIU. AND LIU 7 SPIN ENHANCEMENT
60
~ 1 77
40
20 0 I
-20 0.0
[
0.1
I
0.2
I
I
0.3
0.4
sin O/h
Fig. 5. Comparison of observed and calculated susceptibilities for zirconium. The spin enhancement factor has been set at 1.77 to give agreement with the first three Bragg peaks. Note the large bulk susceptibility anisotropy, which is not observed at the first Bragg peak.
R.M. Moon / Spin densitiesand form factors
27
5. Rare-earth systems It was expected that the radial dependence of the 4f electrons in metals would be very little changed from their free atom distributions. Gd was the ideal test case; with a half-filled shell there would be no complicating orbital moment and the 4f spin density would be spherically symmetric. At the time of the polarized beam work [39] the best available atomic calculations were nonrelativistic Hartree-Fock. It was immediately apparent that these calculations were in poor agreement with the experimental results. Davis and Cooke [40] showed that much better agreement could be obtained with a relativistic H a r t r e e - F o c k - S l a t e r calculation, and shortly thereafter Freeman and Desclaux [41] obtained really excellent agreement with relativistic D i r a c - F o c k calculations. It was clear that something in addition to the 4f electrons is influencing the first few experimental points. The obvious conclusion was that the conduction electrons are responsible. It was possible to separate the conduction electron contribution by extrapolating to lower Q the pure 4f behavior seen at the higher Bragg positions. Knowing that the 4f radial functions are accurately given by the D i r a c - F o c k calculations [42], measurements of the magnetic scattering amplitudes can be used to determine the angular properties of the magnetization density and, therefore, the 4f ground state wave function. A summary of such work has been given by Boucherle, Givod and Schweizer [43]. It is well known that crystal-field effects can result in a reduction of the 4f moment relative to that of the free ion. In intermetallic compounds the moment determined by magnetization measurements is not a reliable measure of the 4f moment because of conduction electron polarization. Measurements of M ( Q ) at moderately high values of Q can be used to establish the 4f ground state and 4f moment. The contribution of the 4f electrons to M ( Q ) for the inner reflections can then be calculated and used to obtain M ( Q ) for the conduction electrons by subtraction from the experimental results. Maps of the conduction electron polarization have been obtained in this way for a number of compounds [43]; examples are
.AI '..:")" ~..-'.".'.:-:**~"."::-.... ---."."-t
~
:' i :".'~' "':. ': ':' "'" :'
o
", "'
I
!
0.0
o Fig. 6. Conduction spin polarization in NdAI2 and HoFe2. The local moment density has been subtracted. Note the change of sign in the conduction polarization around Nd and Ho. (From ref. 43.)
shown in fig. 6 for NdA12 and H o F 2. Note that the conduction density is negative (opposite to the 4f moment) for NdAI 2 and positive for HoA12. Except for Ce compounds, it is found that the conduction polarization is always parallel to the 4f spin. The spatial extent of the conduction polarization is very similar for the two cases of fig. 6, suggesting a common electronic shell which is believed to be the 5d shell.
6. Aetinides The 5f actinide compounds are an intermediate case between the rare-earth compounds, with well localized 4f wave functions which are little affected in their radial extent by the crystalline
R.M. Moon / Spin den,s'tttesandJorm fiwtors
28
environment, and the 3d transition metals, with extended wave functions which may be strongly affected. Understandably, most neutron experiments involving actinides have been performed on uranium compounds, but there have been a few notable exceptions. One of these is the recent work by Lander et al. [44] on the magnetization density of PuSb. The observed values of M ( Q ) are shown in fig. 7 together with calculated values based on D i r a c - F o c k atomic wave functions with fitting parameters which determine the ground state wave function. The shape of M ( Q ) strongly resembles that of Sm x÷ and shows convincingly that the Pu ion has a 5f 5 configuration. The ground state wave function is inconsistent with Russeil-Saunders coupling but is close to that expected for intermediate coupling. The difference between the extrapolated 4f M(0) and the magnetization indi-
0.9 0.8
\\
::L 6 0.e
\
"O -i .,,.,
\0
2 ~ 0.3
o 0.2-
"<, x
0.1 0
¢'~'0,, I OA
I
0.2
.~ 0.3
.~_ 0.4 sin
I
0.5
I .... _!_
0.6
0.7
. _.t ......
0.8
IO'~'O'-t
0.9
1.0
01~, A
F i g 7. The magnetic amplitude MCQ) for ferromagnetic PuSb at l0 K. The moment direction is [001]. Open points correspond to reflections (hkO), i.e. orthogonal to M. and the broken curve represents a calculated fit to these points. Reflections not orthogonal to M are represented by solid points, the calculated values for them by crosses. The arrow indicates the bulk magnetization. (From ref. 44.)
cates the presence of conduction polarization parallel to the 5f spin. Another impressive work has just been completed by Lander et al. on U('I 4 [45]. They found a small decrease in the 5f moment density near the U site and a small increase at about 1 ,~ from this site along U C1 bonds. This is thought to be caused by electron transfer from 5f to a 6d antibonding orbital formed as a result of a covalent interaction between 6d states of U and p states of CI. This is the first direct observation of covalencv in 5f, 6d systems.
7. Spin densities and chemistn, Most of the magnetic moment density studies to date have been on rather simple materials of interest to solid-state physicists. There is now a growing use of polarized neutrons by chemists, particularly at the lnstitut Laue-Langevin. These experiments tend to involve rather complex molecules with large unit cells so that a very large number of reflections must be measured to produce accurate density maps. The use of polarized neutrons to measure the delocalization of electron density in transition metal complexes is probably the most direct experimental technique for studying covalencv. "[he motivation behind such measurements and a review of some of the results has been given by Day [46]. An interesting example is the work of Figgis, Reynolds and Mason [47] on N i ( N H ) ) 4 ( N O e ) > They fitted a model using atomic orbitals to the data on 303 Bragg peaks and concluded that 27¢~: of the total spin resides on ligand groups, with more on the nitro than the ammine. Spin density' maps of the fitted model in two planes are shown in fig. 8. Free radicals are another area of chemical interest for which spin density measurements are valuable. Examples can be found in the work of Brown et al. [48] on the NO radical where they found the unpaired electron to be shared almost equally between the two atoms, and in the work of Boucherle et al. [49] on DPPH (diphenylpicrylhydrazyl). ]'he latter study was especially challenging because the structure is non-centrosymmetric
R.M. Moon / S p i n densities and form factors
~
,.....'"'". ¢1 !
1
I
~,'--.. ~.
'
-
_
'~
~ \ ~~ \ ""-. "'...
E '/I/
~
Fig. 8. Spin density of Ni(NH.~)4(NO2)2 in ac plane (A) and bc* plane (B). Solid lines are positive contours; dashed lines are negative contours; dotted lines are the zero contours. Note the positive spin on the NO 2 and NH 3 groups and negative spin density in the overlap regions. (From ref. 47).
and the structure factor is therefore complex. Flipping ratio measurements cannot give both real and imaginary parts of the magnetic structure factor, so that a model was constructed to give an analytic description of the spin density with adjustable parameters to fit to the data.
8. Future directions
Speculations on the future of science are usually valuable only as a source of humor for scientists of the future. Nevertheless, here are some speculations. Most of these experiments involve weak magnetic signals a n d / o r the observation of many Bragg peaks, so that they would become more feasible with neutron sources of higher flux. A realistic comparison of induced magnetization densities in the 3d and 4d paramagnetic metals is now just beginning. More accurate measure-
29
ments of some of these materials may become desirable in order to make judgements on the accuracy of different calculational techniques. ~Similar measurements on the 3d ferromagnets at temperatures well above T~ may reveal some interesting information on the variation of the radial wave functions with energy. In part, this speculation is based on preliminary calculations by Cooke and Blackman [50]. Using the same procedure which gives good agreement with the experimental results for ferromagnetic Fe, as shown in fig. 3, but basing the spin density on wave functions at the Fermi level, the calculated form factor increases by about 20% for the first four reflections for Fe. As more chemists grow to appreciate the enhanced sensitivity of spin density measurements, relative to charge density measurements, to the behavior of the outer electrons, the general area of chemical applications will grow. The work on transition metal complexes and on free radicals will be continued. An area not covered in this survey, but in which interesting results have been obtained recently, is that of surface magnetism. The use of polarized neutrons to study the modification of bulk magnetic properties at surfaces and interfaces will grow in the future. For complex paramagnets involving magnetic atoms on different crystallographic sites, the bulk susceptibility gives only the average magnetic response. The magnitude and temperature dependence of the susceptibility for different sites can be quite different. Polarized beam measurements can yield separately the susceptibilities for each site, thus giving a more complete description of the magnetic response of the system. Finally, it seems clear that as new magnetic materials are developed, the need for accurate magnetic moment density measurements to fully understand these materials will remain strong.
References [1] R. Nathans, C.G. Shull, G. Shirane and A. Andresen, J. Phys. Chem. Solids 10 (1959) 138. [21 R.M. Moon and C.G. Shull, Acta. Cryst. 17 (1964) 805.
30
R.M. Moon ,/Spin densittes and fi)rm.factor~"
[3] C.CL Shull and Y. Yamada. J. Phys. Soc. Japan 17 Suppl. B-Ill (1962) 1. [4] C.G. Shull, in: Structure and Alloy ChemistD' of the Transition Elements. P.A. Beck, cd, (Interscicnce, Ne~ York, 1963). [5] R.J. Weiss and A.J. Freeman, J. Phys. Chem. Solids 10 (1959) 147. [6] R.E. Watson and A.J. Freeman, Phys. Rev. 123 (1961) 2027; and private communication. [7] C.G. Shull and FI.A. Mook, Phys. Rev. lett. 16 (1966) 184. [8] R.M. Moon, Phys. Rev. 136 (1964) A195. [9] H.A. Mook, Phys. Rev. 148 (1966) 495. [10] B. van Laar, F. Maniawski, S. Kaprzyk and L. Dobrzynski. J. Magn. Magn. Mat. 14 (1979) 94. [111 S. Kaprzyk. B. van I,aar and F. Maniawski. J. Magn. Magn. Mat. 23 (1981) 105. [12] B. van Laar, F. Maniawski and S. Kaprzyk. J. dc Physique 43 (1982) ('7-113. 113] R. Moon, J. de Physique 43 (1982) ('7-187. [14] J.F. Cooke. J.W. Lynn, and H.L. Davis. Phys. Rev. B21 (1980) 4118. [15] R.M. Moon, AIP Conf. Proc. 24 (1975) 425. [16] J. Callaway and C.S. Wang, Phys. Rev. B16 (1977) 2095. [17] C.S. Wang and J. Callaway, Phys. Rev. B15 (1977) 298. [18] J.F. Cc×3k and J.A. Blackman. to be published. [19] R.C. Maglic. AlP Conf. P r ~ . 5 (1972) 1420. [20] J.W. Cable. Phys. Rev. B23 (1981) 6168. 121] O. Steinsvoll, R.M. Moon, W.C. Koehler and ('.G. Windsor, Phys. Rev. B24 (1981) 4031. [22] W. Marshall and S.W. Lovesey, in: Theory of Thermal Neutron Scattering (Oxford University. Oxford, 1971) p. 124.
[23] W.C. Phillips, Phys. Rev. 138 (1965) A1649. [24] C.G. Shull and R.P. Ferrier, Phys. Rev. l,ett. 10 (1963) 295. [251 C. Stassis, Physica 137B (1986) 61 (these proceedings). [26] C.G. Shull. Phys. Rev. Left. 10 (1963) 297. [271 C. Stassis and C.G. Shull. Phys. Rev. B5 (1972) 1040. [281 M.H. Dickens, C.G. Shull, W.C. Koehler, and R.M. M~x~n, Phys. Rev. Lett. 35 (19751 595.
[29] S.R. Collins, Ph.l). Thesis, Massachusetts Institute of Technology (1979). [30] R.M. Moon. W.C. Koehler and ('.G. Shull, Nucl. Im, tr. Meth. 29 (1975) 515. [31] R.M. Moon, W.C. Koehler and J.W. Cable, Proc. ('onf. on Neutron ,~at tering, Gatlinburg (1976). (CON F-'~60601 -P2) p. 577. [32] 4". Stassis. Nuckleonika 24 (19791 765. [33] P.J. Brown and J.B. Forsyth. J. Phys. F. 114 (1984) 1715. [341 P. Radakrishna and P.J. Brown, Physica 120B (1983~ 216. [35] C. Stassis. Phys. Re*.. Lett. 24 (1970) 1415. 136] K.H. Oh, B. Harmon, S.H Liu and S.K. Sinha, Phys. Rev. B14 (1976) 1283. [37] J.F. Ctw~ke, S.tt. l,iu and A.J. Liu. J. Appl. Phys. 57 (1985) 3027. [38] S.H. Liu, A.J. Liu and J.[:. Cooke. to he published. [39] R.M. Moon, W.C. Koehlcr. J.W. Cable and H R . Child, Phys. Re','. B5 (1972) 997. [40] tt. Davis and J.F. ('ooke. AlP Conf. Proc. 5 (1972) 1441. 141] A.J. Freeman and J.P. I)esclaux. Int. Jour. Mag. 3 (1972~ 311. [42] A.J. Freeman and J.P. l)esclaux. J. Magn. Magn. Mat. 12 (1979) 11. [43] J.X. Bouchcrle, 1). (iivord and J. Schweizer. J. de Phystquc 43 (1982) C7-199. [44] G.H. Lander, A. Delapalmc. P.J. Brown. J.(. Spirlet, J. Rebizant and O. Vogt, J. Appl. Phys. 57 (1985) 3748. 145] G . H Lander, P.J. Brown, M.R. Spirlet, J. Rebizant and P,. Kanellakopulos, J. Chem. Phys.. in pre.ss. [46] P. Day. J. de Physique 43 (1982) 4"7-341. [47] B.N. Figgis, P.A. Reynolds, and R. Mason, J. Am. ('hem. Soc. 105 (1983) 440. [481 P.J. Brown, A. ('apiomont. B. Gillon and 3. ,~hweizcr, .1. Magn. Mam. Mat. 14 (19791 289. [49] J.X. Boucherle. B. Gillon. J. Maruani and J. Schweizcr. J. de Physique 43. (1982) C7-227. [5(I] J.F. Cooke, private communieation.
19
SPIN DENSITIES A N D FORM FACTORS: PAST, PRESENT, AND FUTURE * R.M. MOON Solid State Dit,ision, Oak Rtdge National Laboratory, Oak Ridge. TN .¢7831. USA
The development of polarized neutron diffraction as a standard method for the study of magnetic moment densities was largely due to the efforts of C.G. Shull. The early experiments by Shull, his students, and co-workers are reviewed. A comparison of his early results on the magnetic form factor of Fe with very recent theoretical work is presented and excellent agreement is found. Some more recent experimental results are described to illustrate the many types of magnetic systems to which this technique has been applied. Brief speculations about future directions are offered.
1. Introduction
The applicati~.n of polarized neutrons to the study of spin densities and magnetic form factors has been an important part of the total scientific impact of neutron scattering. As in so many aspects of neutron scattering, Cliff Shuli was there at the beginning and guided much of the early polarized beam work. After leaving Oak Ridge and before he was able to begin experiments at the MIT reactor, Shull was a frequent visitor at the Brookhaven Graphite Research Reactor. During this period he guided the development of the first dedicated polarized beam diffractometer [1], and shortly thereafter built a similar machine as one of his early instruments at MIT. To appreciate the importance of the polarized beam technique, a little background in the fundamentals of magnetic scattering is necessary. The cross section for Bragg scattering from a ferromagnet can be written as
o - b 2 + 2bpP" q + pZq2,
(1)
where b is the nuclear scattering amplitude, P is the neutron polarization vector, and q is the magnetic interaction vector (the projection of a unit vector along the magnetization onto the plane * Research sponsored by the Division of Materials Sciences, U.S. Department of Energy under contract DE-AC05840R21400 with Martin Marietta Energy Systems. inc. 0378-4363/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
perpendicular to the scattering vector). The quantity of interest here is the magnetic scattering amplitude p, which is proportional to the Fourier transform of the magnetic moment density. In eq. (1) we assume an elemental material with atoms on equivalent sites: for a more complex material the nuclear and magnetic scattering amplitudes should be replaced by the nuclear and magnetic structure factors. The polarized beam experiment is so arranged that the magnetization is perpendicular to the scattering vector (q2 = 1) and that P can be either parallel or antiparallel to the magnetization ( P . q = +1). The ratio of Bragg intensities for these two directions of the neutron spin is then given by
(b+p/"
R=~b_p
} .
(2)
Fhis so-called flipping ratio is the measured quantity in polarized beam diffraction; from it one obtains the ratio p / b . The polarization dependent term in eq. (1), being linear in the magnetic amplitude, greatly increases the sensitivity to small values of the magnetic amplitude. With an unpolarized beam the polarization dependent term would be absent and the best one could do, by controlling the direction of magnetization (q2= 0 or 1), would be to measure the ratio
U= 1 + ( p / b ) 2.
(3)
]'he importance of the polarized beam technique is
R M. Moon / S p i n den~itu,s and [orm [actot:~
20
Table I .Hlustrating the great increase in sensitivity of the polarized b e a m technique over an unpolarized neutron m e a s u r e m e n t for Fe
hk l
p/ h
Polarized R
Unpolarized /5
110 220 330 440
(/.388 0.107 0.0083 - 0.0096
5.144 1.537 1.034 0.962
1.151 1.0114 1 .(XIXI7 I .(XI(X)9
clearly evident from table 1 which gives a comparison of the ratios R and U for a few reflections for the case of Fe. Note too that the sign of p relative to h is given by the polarized beam experiment. It is obvious that without a polarized beam diffractometer, meaningful measurements of the magnetic amplitude for Fe would be restricted to the first few Bragg peaks. Because the nuclear interaction has such a short range, the nuclear amplitude is independent of the scattering vector Q, so that the Q dependence of p/b accurately reflects the spatial variation of the magnetic moment density through a Fourier transform. The data may be converted to Bohr magneton units by
2b P(O) M ( Q ) - 3'to b
(4)
where y is the magnitude of the neutron moment in nuclear magnetons and r0 is the classical electron radius. The data are frequently reduced to
normalized form factor values given bv
f ( Q ) = M(Q)/M(O).
(5)
As a further indication of the sensitivity of the polarized-beam technique and as a commentary on one of its chief practioners, I list in table II some of the experiments which are discussed in this volume. In each case the largest (or smallest for p/h < 1) value of the measured flipping ratios is given with corresponding values for M(Q). In most cases this was the easiest measurement to obtain; usually the experiment required observation of several Bragg peaks with decreasing magnetic amplitude. I believe this table illustrates two important characteristics of Cliff Shull: the ability and determination t o get the most out of any experimental technique, and great patience. One does not measure intensity ratios to five significant figures without waiting to count a very large number of neutrons.
2. 3d ferromagnets After constructing the S-3 polarized beam diffractometer at MIT, Shull chose Fe as the first material to study with this new and very sensitive machine. A few Fe reflections had been measurcd earlier in the work at Brookhaven, but the MIT work was much more comprehensive in terms of number of reflections and much more precise. The polarized beam technique, involving a measurement of the intensity ratio as the neutron spin is
T a b l e II A chronological list of some of the C.G. Shull e x p e r i m e n t s giving an i n d i c a t i o n of the largest magnetic signal detected m each experiment Sample
Student or colleague
Publ. date
Extreme R
Maximum M( Q I ( p- ~))
Fe V Co I-e(jm~Pd0~7 Ni V~Si C u ( F e 1250 p p m ) C u ( F e 387 p p m ) Bi
Yamada Ferrier Moon Phillips M(x'Jk Wedgewood Stassis Dickens Collins
62 63 64 65 66 66 72 75 79
5.14 0.988 32.6 1.13 1.61 (}.989 1.00043 1.00101 I).99944
1.37 5.3 × 1.32 7.3 x 4.5 x 4.8 × 3.1 × 7.1 x -4.4 x
10 "~ 10 10 10 10 10 10
"? I "~ 4 4 -1
R.M. Moon / Spin densities and form factors
reversed, is clean compared to most diffraction measurements. Many factors which influence the intensity, such as absorption and the DebyeWaller factor, cancel in forming the intensity ratio. Extinction was minimized by distorting the crystals in a controlled manner and by working with thin samples. Nevertheless, in this "clean" experiment, Shull noticed very early that the results on some reflections were sample dependent and there was even a lack of reproducibility for the same sample following remounting in the diffractometer! In consultation with O.J. Guentert and F.P. Ricci over these disturbing results, it was concluded that
21
they were probably the result of simultaneous Bragg reflections. Subsequent experiments and calculations confirmed this conclusion [2]. A new goniometer which allowed continuous, motordriven rotation of the sample around the scattering vector was added to the polarized beam diffractometer. Henceforth, all flipping ratio measurements would be made as a function of this rotation angle and anomalous results discarded. In addition, samples were shaped to minimize the effects of simultaneous reflections (long in only one dimension). With the assistance of Y. Yamada, Shull proceeded to collect data, shown in fig. 1, on
~6
~5 0.4
~XPERIMENT
I0.5
3d64s2(calc) WATSON-FREEMAN (SPHERICAL SPINONLY)
~ ~dae~)
WATS(
o
QI
::..__
o
"°Jo
d,
Fig. I. Experimental magnetic form factor of Fe compared with spin-polarized Hartree-Fock atomic calculations for the spherically averaged spin density. (From ref. 4.)
22
R.M. Mo,m / 5"pro demme~ and.lbrm /,(t,,r.~
the first 26 reflections for Fe [3.4]. These data were analyzed in two ways: by Fourier inversion to give maps of the magnetization density, and by comparison with calculated atomic form factors. Pronounced departures from spherical symmetry were evident in the magnetization maps, as shown in fig. 2. The magnetization was concentrated along [100] directions with minima along [111} directions. A quantitative measure of this aspherical behavior was obtained by fitting an atomic model to the data following the work of Weiss and Freeman [5] and using spin-polarized atomic form factors calculated by Watson and Freeman [6]. It was found that 53g,~ of the spin density was in Eg orbitals, to be compared with an Eg population of 40% for spherical symmetry. To obtain a good fit with the experimental data, the 3d spin moment had to be increased beyond that expected for the total moment
per atom as determined by magnetization measurements. Within the atomic model, this implies the existence of a diffuse negative spin component which has an appreciable contribution to the form factor at only the origin. Negative regions of magnetic moment density at positions well removed from atomic sites were also revealed by direct Fourier transformation of the data {7]. In the atomic model, it was natural to ascribe this diffuse negative polarization to 4s electrons, but we will return to this point later. Shull and his students, R.M. Moon and H.A. Mook, continued this work on the 3d ferromagnets. obtaining detailed form factor measurements on hexagonal Co [8] and Ni [9]. The atomic model also gave remarkable fits to the data in these two cases, with Co being nearly spherically symmetric and Ni showing strong T_,~: 181~) behavior. To achieve good fits it was again necessary to scale up
L~ 4.~.% ~
DIRECTIONJM. ~TIC MOM(NT ~NS|TY
~..X
I_ \
-,w~
i
'
._J~-
|
F~OM F O ~ F . R
(x)o) ~RECT~
I
;Fu,,~no,, \'.,"\
\'. \
\
..,.~"- \ ( ' ) x'- \
\ \',
I t
\ "~
Ii 0
| 02
./°
I 0.4
" ~.
I 0.1 DISTANCE
I O.G
.
i LO"
i 1.2
I I.C;
I1
FROM NUCLEUS ( ~ )
F i g 2. Magnetic m o m e n t densit?, as a function of distance from the l-"e nucleus along the principle cr?,stallographic directions. (From ref. 4.)
R.M. Moon / Spin densities and form factors
the moment attributed to 3d spin by a factor of (1 + a). The parameter a represented the fractional increase in 3d spin over that expected from magnetization measurements needed to give agreement with the neutron results. This increase was rather large: for Fe, a = 0.10, for Co, a = 0.18; and for Ni, a = 0.19. The experiments seemed to be telling us that the spatial variation of the magnetization was almost exactly that calculated in spin-polarized Hartree-Fock theory for a 3d"-24sZ configuration, but that there was more of this 3d-like moment than is revealed in a bulk magnetization measurement. In all three cases, direct Fourier inversion showed negative densities away from atomic sites. The success of the atomic model led some people to attach greater physical significance to the negative, nearly constant spin density component than was warranted. Van Laar and colleagues [10-12] have been critical of the concept of a constant negative component and have concluded, based on their projection operator method, that there is no experimental evidence for the existence of such a component. I must take this opportunity to retract a suggestion [13] that their method may not be sensitive to the presence of a constant component; they have demonstrated that the sensitivity is adequate [12]. One of the original publications [8] contained a specific warning on this point which also seems appropriate today. " O n e should be cautious in ascribing significance to the form-factor analysis beyond the fact that it is a convenient way to describe the observed spin distribution. In particular, we cannot claim that this analysis proves the existence of a constant negative polarization in the conduction band or that the spin density in the 3d band is just like that calculated for the free atom. All that can be said with certainty is that the sum of the polarization in the 3d and conduction bands has the spatial dependence described by the model." It is perhaps worth remarking that the decomposition of a periodic function into a superposition of "atomic" functions is not unique. That is, there are any number of ways to select #a(r) such that a given periodic function is formed by g(,) = E~G(ri
r,).
(6)
23
The Fourier transforms of different g~(r) which satisfy eq. (6) will be different for a general value of Q, but will be identical for the set of Q constituting the reciprocal lattice. A more significant discussion would center on why the spin density goes negative in regions of the unit cell which are most remote from atomic sites. This is a model independent fact given by Fourier inversion of the data for Fe, Co, and Ni. Probably the most important contribution to this negative density comes from paired electrons near the bottom of the 3d bands where the wave functions are significantly expanded relative to states near the top of the 3d bands. There is also a tendency for minority spin wave functions to be more expanded than corresponding majority spin wave functions, so that there should be a negative polarization at large distances from atomic sites from these paired 3d electrons. Band calculations do not support the idea of negative (sp) polarization from exchange splitting, but there is a negative (sp) polarization produced by (spd) hybridization. In the case of Fe, Cooke et al. [14] have calculated this to be -0.1gH. Finally, the atomic model used in fitting the data does have some theoretical credibility when the crystal wave functions are written in the tight-binding formalism [15]. The negative density is associated with overlap of the 3d atomic functions used in synthesizing the periodic wave functions. Theoreticians have made great progress in calculating energy bands since these experiments were performed. The self-consistent calculations of Callaway and Wang for Fe [16] and Ni [17] using the yon Barth-Hedin exchange-correlation potential gave relatively good agreement with the experimental results. Unfortunately the direct comparison of these calculated form factors with the experimental results is not quite proper because the calculation has no orbital moment and the experimental results do contain a contribution from the small orbital moment. Van Laar et al. [10] have extracted a "spin only" form factor from the experimental data to compare with the band calculation. They find a small disagreement between calculated and observed asphericities for Ni but good agreement with the spherically averaged form factor, while for Fe the calculated asphericity is
R.M. Moon / Spin densities and form lactor.~
24
correct but the spherically averaged calculated form factor is slightly high. The same conclusions can be reached by inspection of the difference between experimental and calculated form factors [13]. The first band calculations for Fe and Ni incorporating a spin-orbit interaction and hence an orbital moment have recently been completed by Cooke and Blackman [18]. Their initial calculations give rather good agreement with the experimental results, as shown in fig. 3 for Fe. It would seem that all the theoretical tools are finally in hand for a detailed confrontation between theory and experiment on the magnetic moment density in the 3d ferromagnets. There are at least two additional experimental results on 3d spin densities which merit some theoretical attention. Shull [4] and Maglic [19] measured the temperature dependence of the asphericity in Fe and found it to be negligibly small. This is not surprising since the major temperature effect would be a small repopulation of states near the Fermi level. However, Cable [20] has found for Ni a rather large temperature dependence of the fraction of spin associated with Eg states. This effect is too large to be explained by a thermal repopulation of states and suggests that the exchange splittings of Eg and Teg bands have
0.8
i
Fe EXP x
-SHULL
CALC-COOKE
& YAMADA & BLACKMAN
0.6 nO co 0.4
112
0 u_ 0.2
0.0 i
-0.1
0.0
0.2
0.4
I 0.6
0.8
1.0
1.2
sm 8/,~
l'ig. 3. C o m p a r i s o n of the e x p e r i m e n t a l Fe form factc, r with a b a n d c a l c u l a t i o n i n c l u d i n g the orbital contribution.
different temperature dependences. Another unexplained result is found in the work of Steinsvoll et al. [21] who used the polarized-beam technique to measure the magnetic amplitudes for phonon scattering in Fe and Ni. These measurements were not restricted to reciprocal lattice positions. Systematic deviations from a smooth interpolation of M ( Q ) between Bragg positions was found in Fe along [100] and [110] directions, but was not found in Ni along the [100] direction. It is not clear whether the deviations found in Fe are a static property of the band wave functions as suggested by Marshall and Lovesey [22] or result from a dynamic distortion of the 3d electron distribution during thermal vibrations. To distinguish between these possibilities, it would be of great interest to have the Fourier transform of calculated spin densities as a continuous function of Q along the principal symmetry directions in Fe and Ni.
3. Small moment systems The sensitivity of the polarized-beam technique has been used by Shull and co-workers in several pioneering experiments on ferromagnets with small moments, paramagnets, and even a diamagnet. Small amounts of Fe alloyed with Pd cause ferromagnetic behavior at low temperatures. While the total moment per Fe atom is around 10 P-u, the average moment per atom, for Fe concentrations of about 1%, is only 0.1p,~. The accurate measurement of the average spatial distribution of such small moments had not been possible before the development of the polarized-beam technique. Walter Phillips, one of Shull's early students at MIT, made such measurements on a crystal of Pd with 1.3% Fe [23]. The distribution of magnetic moment showed a strong departure from spherical symmetry favoring T2g behavior. After completing the Fe work, Shull was quick to fully exploit the capabilities of the polarizedneutron technique by beginning a study of vanadium [24], a paramagnetic metal with susceptibility nearly independent of temperature. With the available magnetic field, a moment of only' 1 m / ~ per atom could be induced, yet the sensitivity of the technique coupled with the unusually small
R.M. Moon / Spin densities and form factors
coherent nuclear amplitude of V allowed fairly accurate measurements to be obtained. These are discussed elsewhere in this volume [25]. It is worth noting that in the course of this work nuclear polarization (PN = 0.0004) was detected for the first time with this technique, and the neutron spin-neutron orbit interaction [26] was first observed. More than a decade would pass before similar work on paramagnetic metals could be taken up elsewhere with superconducting magnets and neutron sources of higher flux. The Kondo effect was the subject of extensive experimental and theoretical activity in the early '70s. An important issue was whether a compensating cloud of oppositely polarized conduction electrons formed around localized magnetic impurities in a nonmagnetic host. The classic Kondo system was dilute Fe in Cu, and Stassis and Shull [27] used the polarized beam technique to measure the 3d susceptibility in a Cu crystal containing 1250 p p m of Fe. They found that the 3d susceptibility was equal to the bulk susceptibility measured on the same sample, apparently leaving no room for a cloud of oppositely polarized conduction electrons. However, the issue was not clearly resolved because, for the Fe concentration used in this experiment, it was estimated that only 35% of the magnetization arises from isolated impurities and the remainder from pairs of Fe atoms ferromagnetically coupled. Subsequently, Dickens and Shull [28] measured the (111) scattering amplitude as a function of concentration (down to 387 ppm) and temperature, and used the concentration dependence to extrapolate to the isolated impurity case. At temperatures below 10 K they found an isolated 3d moment per impurity substantially larger than the total moment per impurity given by magnetization measurements. This does suggest the presence of a compensating polarization of conduction electrons that develops at low temperature, in apparent contradiction with conclusions based on Knight shift measurements. This situation is still unresolved. The first and only polarized beam study of an elemental diamagnet was by Collins [29] and Shuli. Bismuth has an anomalously high diamagnetic susceptibility attributed to five valence electrons outside a core of 78 electrons. The crystal structure
25 94
~
T
I
l'-Intercaptt-291
r
1
from susceptibility
BISMUTH, 2 9 0 K II BInQry (a) axis H'4.25 T
-80 X E -60 o "o
"~'\
~
"-..//
_o - 4 0
\ \
o c
7-
, 8 3 f r e e atom electrons
-20
N
no
o O
2 I
g~ 0_ 2
J l 0
0.1
o ~ _ _0 0 [ IJ[ 0.2 sine/),
0.3 (~-i~
_0 t 0.4
0.5
Fig. 4. Residual flipping ratios (R -1) 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 susceptibility measurements, is far off scale. (From ref. 29.)
is rhombohedral and the susceptibility is strongly anisotropic. Neutron data were obtained with the field along three different crystallographic directions. One of these sets of data is shown in fig. 4; the quantity plotted here is R - 1 . The contribution from the core electrons can be accurately calculated and is shown in the figure. Note that the outer electron contribution is evident only in the first Bragg peak, and the point at the origin, from bulk susceptibility measurements, is far off scale. No band calculation is available for comparison with these results, but fig. 4 illustrates an important characteristic of the Landau diamagnetism of outer electrons, namely that it is almost zero at Bragg peak positions but may be significant at Q = 0. This is important in the measurement and analysis of paramagnetic susceptibilities.
4. Paramagnetic
metals
With the addition of superconducting magnets to polarized beam diffractometers it has been pos-
R.M. Moon / Spin denstties andJorm f~lctors
26
sible to continue the work started by Shull with vanadium by measuring 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 a few milli-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 arc usually negligible when dealing with ferromagnets. These arc nuclear polarization, the neutron spin-neutron orbit interaction [26], diamagnetic scattering, sample purity. and the effects of the applied magnetic field on the neutron trajectory and velocity [30,31 ]. References to most experimental results are given in summaries by Moon [13,31] and Stassis [32]. To these should be added recent results on V [331 and Rh [34]. From the flipping ratio measurements we obtain the zz component of the generalized susceptibility, evaluated at zero energy transfer and at reciprocal lattice positions. This can be written as the sum of several terms X : : ( Q ) = - x ~ .•,. . . .(.O )
+ x,,~(¢2),
Oh et al. [36]. "]'his lack of theoretical effort has made the interpretation of the experimental results difficult. Fortunately, Cooke. Liu and t,iu [37,38] have now developed a computational technique to calculate the orbital contribution for both cubic and hexagonal metals. A detailed comparison with all the experimental results has not yet been completed but preliminary results show reasonable agreement in most cases. This comparison will serve to validate band wave functions and will give a measure of the exchange enhancement of the spin susceptibilities. As expected the experimental results show marked deviations from free atom behavior for elements near the beginning of the 3d and 4d transition series. This is associated with spreading of the wave functions near the bottom of the d bands. A comparison of observed and calculated x ( Q ) is shown in fig. 5 for Zr. Their calculations give the orbital contribution and the uncnhanced spin contribution in absolute units. In fig. 5 the enhancement factor has been adjusted to give agreement with the first few Bragg peaks.
"~'""" xJ., ~t) + x,r(Q)
(7) 200
where the first 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 different contributions to the total induced moment. As already noted in the case of Bi, the Landau term may affect only the point at the origin and is always of unknown magnitude. Therefore, the practice of reporting normalized form factors can only lead to confusion and should be abandoned. The most useful quantities to report are absolute values of X(O)The core diamagnetic term is readily calculated [35] and is usually subtracted from the observed results. The spin contribution is related to ~2(r) for states at the Fermi surface and is easy for band theorists to calculate. The orbital contribution is much more difficult to calculate and, until quite recently, had only been done for the case of Cr by
'
I
180 160
1
I
I
i
x~o~, 74 : ?
X(o). H .
140
Zr
T = 100 K
~, -6 E
120 100
E
80
EXPERIMENT:
STASSIS
et 8 1
CALCULATION COOKE, LIU. AND LIU 7 SPIN ENHANCEMENT
60
~ 1 77
40
20 0 I
-20 0.0
[
0.1
I
0.2
I
I
0.3
0.4
sin O/h
Fig. 5. Comparison of observed and calculated susceptibilities for zirconium. The spin enhancement factor has been set at 1.77 to give agreement with the first three Bragg peaks. Note the large bulk susceptibility anisotropy, which is not observed at the first Bragg peak.
R.M. Moon / Spin densitiesand form factors
27
5. Rare-earth systems It was expected that the radial dependence of the 4f electrons in metals would be very little changed from their free atom distributions. Gd was the ideal test case; with a half-filled shell there would be no complicating orbital moment and the 4f spin density would be spherically symmetric. At the time of the polarized beam work [39] the best available atomic calculations were nonrelativistic Hartree-Fock. It was immediately apparent that these calculations were in poor agreement with the experimental results. Davis and Cooke [40] showed that much better agreement could be obtained with a relativistic H a r t r e e - F o c k - S l a t e r calculation, and shortly thereafter Freeman and Desclaux [41] obtained really excellent agreement with relativistic D i r a c - F o c k calculations. It was clear that something in addition to the 4f electrons is influencing the first few experimental points. The obvious conclusion was that the conduction electrons are responsible. It was possible to separate the conduction electron contribution by extrapolating to lower Q the pure 4f behavior seen at the higher Bragg positions. Knowing that the 4f radial functions are accurately given by the D i r a c - F o c k calculations [42], measurements of the magnetic scattering amplitudes can be used to determine the angular properties of the magnetization density and, therefore, the 4f ground state wave function. A summary of such work has been given by Boucherle, Givod and Schweizer [43]. It is well known that crystal-field effects can result in a reduction of the 4f moment relative to that of the free ion. In intermetallic compounds the moment determined by magnetization measurements is not a reliable measure of the 4f moment because of conduction electron polarization. Measurements of M ( Q ) at moderately high values of Q can be used to establish the 4f ground state and 4f moment. The contribution of the 4f electrons to M ( Q ) for the inner reflections can then be calculated and used to obtain M ( Q ) for the conduction electrons by subtraction from the experimental results. Maps of the conduction electron polarization have been obtained in this way for a number of compounds [43]; examples are
.AI '..:")" ~..-'.".'.:-:**~"."::-.... ---."."-t
~
:' i :".'~' "':. ': ':' "'" :'
o
", "'
I
!
0.0
o Fig. 6. Conduction spin polarization in NdAI2 and HoFe2. The local moment density has been subtracted. Note the change of sign in the conduction polarization around Nd and Ho. (From ref. 43.)
shown in fig. 6 for NdA12 and H o F 2. Note that the conduction density is negative (opposite to the 4f moment) for NdAI 2 and positive for HoA12. Except for Ce compounds, it is found that the conduction polarization is always parallel to the 4f spin. The spatial extent of the conduction polarization is very similar for the two cases of fig. 6, suggesting a common electronic shell which is believed to be the 5d shell.
6. Aetinides The 5f actinide compounds are an intermediate case between the rare-earth compounds, with well localized 4f wave functions which are little affected in their radial extent by the crystalline
R.M. Moon / Spin den,s'tttesandJorm fiwtors
28
environment, and the 3d transition metals, with extended wave functions which may be strongly affected. Understandably, most neutron experiments involving actinides have been performed on uranium compounds, but there have been a few notable exceptions. One of these is the recent work by Lander et al. [44] on the magnetization density of PuSb. The observed values of M ( Q ) are shown in fig. 7 together with calculated values based on D i r a c - F o c k atomic wave functions with fitting parameters which determine the ground state wave function. The shape of M ( Q ) strongly resembles that of Sm x÷ and shows convincingly that the Pu ion has a 5f 5 configuration. The ground state wave function is inconsistent with Russeil-Saunders coupling but is close to that expected for intermediate coupling. The difference between the extrapolated 4f M(0) and the magnetization indi-
0.9 0.8
\\
::L 6 0.e
\
"O -i .,,.,
\0
2 ~ 0.3
o 0.2-
"<, x
0.1 0
¢'~'0,, I OA
I
0.2
.~ 0.3
.~_ 0.4 sin
I
0.5
I .... _!_
0.6
0.7
. _.t ......
0.8
IO'~'O'-t
0.9
1.0
01~, A
F i g 7. The magnetic amplitude MCQ) for ferromagnetic PuSb at l0 K. The moment direction is [001]. Open points correspond to reflections (hkO), i.e. orthogonal to M. and the broken curve represents a calculated fit to these points. Reflections not orthogonal to M are represented by solid points, the calculated values for them by crosses. The arrow indicates the bulk magnetization. (From ref. 44.)
cates the presence of conduction polarization parallel to the 5f spin. Another impressive work has just been completed by Lander et al. on U('I 4 [45]. They found a small decrease in the 5f moment density near the U site and a small increase at about 1 ,~ from this site along U C1 bonds. This is thought to be caused by electron transfer from 5f to a 6d antibonding orbital formed as a result of a covalent interaction between 6d states of U and p states of CI. This is the first direct observation of covalencv in 5f, 6d systems.
7. Spin densities and chemistn, Most of the magnetic moment density studies to date have been on rather simple materials of interest to solid-state physicists. There is now a growing use of polarized neutrons by chemists, particularly at the lnstitut Laue-Langevin. These experiments tend to involve rather complex molecules with large unit cells so that a very large number of reflections must be measured to produce accurate density maps. The use of polarized neutrons to measure the delocalization of electron density in transition metal complexes is probably the most direct experimental technique for studying covalencv. "[he motivation behind such measurements and a review of some of the results has been given by Day [46]. An interesting example is the work of Figgis, Reynolds and Mason [47] on N i ( N H ) ) 4 ( N O e ) > They fitted a model using atomic orbitals to the data on 303 Bragg peaks and concluded that 27¢~: of the total spin resides on ligand groups, with more on the nitro than the ammine. Spin density' maps of the fitted model in two planes are shown in fig. 8. Free radicals are another area of chemical interest for which spin density measurements are valuable. Examples can be found in the work of Brown et al. [48] on the NO radical where they found the unpaired electron to be shared almost equally between the two atoms, and in the work of Boucherle et al. [49] on DPPH (diphenylpicrylhydrazyl). ]'he latter study was especially challenging because the structure is non-centrosymmetric
R.M. Moon / S p i n densities and form factors
~
,.....'"'". ¢1 !
1
I
~,'--.. ~.
'
-
_
'~
~ \ ~~ \ ""-. "'...
E '/I/
~
Fig. 8. Spin density of Ni(NH.~)4(NO2)2 in ac plane (A) and bc* plane (B). Solid lines are positive contours; dashed lines are negative contours; dotted lines are the zero contours. Note the positive spin on the NO 2 and NH 3 groups and negative spin density in the overlap regions. (From ref. 47).
and the structure factor is therefore complex. Flipping ratio measurements cannot give both real and imaginary parts of the magnetic structure factor, so that a model was constructed to give an analytic description of the spin density with adjustable parameters to fit to the data.
8. Future directions
Speculations on the future of science are usually valuable only as a source of humor for scientists of the future. Nevertheless, here are some speculations. Most of these experiments involve weak magnetic signals a n d / o r the observation of many Bragg peaks, so that they would become more feasible with neutron sources of higher flux. A realistic comparison of induced magnetization densities in the 3d and 4d paramagnetic metals is now just beginning. More accurate measure-
29
ments of some of these materials may become desirable in order to make judgements on the accuracy of different calculational techniques. ~Similar measurements on the 3d ferromagnets at temperatures well above T~ may reveal some interesting information on the variation of the radial wave functions with energy. In part, this speculation is based on preliminary calculations by Cooke and Blackman [50]. Using the same procedure which gives good agreement with the experimental results for ferromagnetic Fe, as shown in fig. 3, but basing the spin density on wave functions at the Fermi level, the calculated form factor increases by about 20% for the first four reflections for Fe. As more chemists grow to appreciate the enhanced sensitivity of spin density measurements, relative to charge density measurements, to the behavior of the outer electrons, the general area of chemical applications will grow. The work on transition metal complexes and on free radicals will be continued. An area not covered in this survey, but in which interesting results have been obtained recently, is that of surface magnetism. The use of polarized neutrons to study the modification of bulk magnetic properties at surfaces and interfaces will grow in the future. For complex paramagnets involving magnetic atoms on different crystallographic sites, the bulk susceptibility gives only the average magnetic response. The magnitude and temperature dependence of the susceptibility for different sites can be quite different. Polarized beam measurements can yield separately the susceptibilities for each site, thus giving a more complete description of the magnetic response of the system. Finally, it seems clear that as new magnetic materials are developed, the need for accurate magnetic moment density measurements to fully understand these materials will remain strong.
References [1] R. Nathans, C.G. Shull, G. Shirane and A. Andresen, J. Phys. Chem. Solids 10 (1959) 138. [21 R.M. Moon and C.G. Shull, Acta. Cryst. 17 (1964) 805.
30
R.M. Moon ,/Spin densittes and fi)rm.factor~"
[3] C.CL Shull and Y. Yamada. J. Phys. Soc. Japan 17 Suppl. B-Ill (1962) 1. [4] C.G. Shull, in: Structure and Alloy ChemistD' of the Transition Elements. P.A. Beck, cd, (Interscicnce, Ne~ York, 1963). [5] R.J. Weiss and A.J. Freeman, J. Phys. Chem. Solids 10 (1959) 147. [6] R.E. Watson and A.J. Freeman, Phys. Rev. 123 (1961) 2027; and private communication. [7] C.G. Shull and FI.A. Mook, Phys. Rev. lett. 16 (1966) 184. [8] R.M. Moon, Phys. Rev. 136 (1964) A195. [9] H.A. Mook, Phys. Rev. 148 (1966) 495. [10] B. van Laar, F. Maniawski, S. Kaprzyk and L. Dobrzynski. J. Magn. Magn. Mat. 14 (1979) 94. [111 S. Kaprzyk. B. van I,aar and F. Maniawski. J. Magn. Magn. Mat. 23 (1981) 105. [12] B. van Laar, F. Maniawski and S. Kaprzyk. J. dc Physique 43 (1982) ('7-113. 113] R. Moon, J. de Physique 43 (1982) ('7-187. [14] J.F. Cooke. J.W. Lynn, and H.L. Davis. Phys. Rev. B21 (1980) 4118. [15] R.M. Moon, AIP Conf. Proc. 24 (1975) 425. [16] J. Callaway and C.S. Wang, Phys. Rev. B16 (1977) 2095. [17] C.S. Wang and J. Callaway, Phys. Rev. B15 (1977) 298. [18] J.F. Cc×3k and J.A. Blackman. to be published. [19] R.C. Maglic. AlP Conf. P r ~ . 5 (1972) 1420. [20] J.W. Cable. Phys. Rev. B23 (1981) 6168. 121] O. Steinsvoll, R.M. Moon, W.C. Koehler and ('.G. Windsor, Phys. Rev. B24 (1981) 4031. [22] W. Marshall and S.W. Lovesey, in: Theory of Thermal Neutron Scattering (Oxford University. Oxford, 1971) p. 124.
[23] W.C. Phillips, Phys. Rev. 138 (1965) A1649. [24] C.G. Shull and R.P. Ferrier, Phys. Rev. l,ett. 10 (1963) 295. [251 C. Stassis, Physica 137B (1986) 61 (these proceedings). [26] C.G. Shull. Phys. Rev. Left. 10 (1963) 297. [271 C. Stassis and C.G. Shull. Phys. Rev. B5 (1972) 1040. [281 M.H. Dickens, C.G. Shull, W.C. Koehler, and R.M. M~x~n, Phys. Rev. Lett. 35 (19751 595.
[29] S.R. Collins, Ph.l). Thesis, Massachusetts Institute of Technology (1979). [30] R.M. Moon. W.C. Koehler and ('.G. Shull, Nucl. Im, tr. Meth. 29 (1975) 515. [31] R.M. Moon, W.C. Koehler and J.W. Cable, Proc. ('onf. on Neutron ,~at tering, Gatlinburg (1976). (CON F-'~60601 -P2) p. 577. [32] 4". Stassis. Nuckleonika 24 (19791 765. [33] P.J. Brown and J.B. Forsyth. J. Phys. F. 114 (1984) 1715. [341 P. Radakrishna and P.J. Brown, Physica 120B (1983~ 216. [35] C. Stassis. Phys. Re*.. Lett. 24 (1970) 1415. 136] K.H. Oh, B. Harmon, S.H Liu and S.K. Sinha, Phys. Rev. B14 (1976) 1283. [37] J.F. Ctw~ke, S.tt. l,iu and A.J. Liu. J. Appl. Phys. 57 (1985) 3027. [38] S.H. Liu, A.J. Liu and J.[:. Cooke. to he published. [39] R.M. Moon, W.C. Koehlcr. J.W. Cable and H R . Child, Phys. Re','. B5 (1972) 997. [40] tt. Davis and J.F. ('ooke. AlP Conf. Proc. 5 (1972) 1441. 141] A.J. Freeman and J.P. I)esclaux. Int. Jour. Mag. 3 (1972~ 311. [42] A.J. Freeman and J.P. l)esclaux. J. Magn. Magn. Mat. 12 (1979) 11. [43] J.X. Bouchcrle, 1). (iivord and J. Schweizer. J. de Phystquc 43 (1982) C7-199. [44] G.H. Lander, A. Delapalmc. P.J. Brown. J.(. Spirlet, J. Rebizant and O. Vogt, J. Appl. Phys. 57 (1985) 3748. 145] G . H Lander, P.J. Brown, M.R. Spirlet, J. Rebizant and P,. Kanellakopulos, J. Chem. Phys.. in pre.ss. [46] P. Day. J. de Physique 43 (1982) 4"7-341. [47] B.N. Figgis, P.A. Reynolds, and R. Mason, J. Am. ('hem. Soc. 105 (1983) 440. [481 P.J. Brown, A. ('apiomont. B. Gillon and 3. ,~hweizcr, .1. Magn. Mam. Mat. 14 (19791 289. [49] J.X. Boucherle. B. Gillon. J. Maruani and J. Schweizcr. J. de Physique 43. (1982) C7-227. [5(I] J.F. Cooke, private communieation.