Journal of Magnetism and Magnetic Materials 44 (1984) 313-328 313 North-Holland, Amsterdam

D E T E R M I N A T I O N OF T H E 4f S H E L L MAGNETIC M O M E N T IN CUBIC RARE-EARTH INTERMETALLIC C O M P O U N D S

E. BELORIZKY, Y. B E R T H I E R

Laboratoire de Spectromktrie Physique (associk au CNRS), Boite postale no. 87, 38402 Saint-Martin-d'Hbres Cedex, France

and

R.A.B. DEVINE

CN.E. 7", Boite Postale No. 45, 38240 Meylan, France

Received 4 June 1984

Different methods of determining the magnitude of the 4f shell electronic magnetic moment for cubic, ferromagnetic rare-earth intermetallics are critically reviewed. In particular, determination of the 4f ground state from magnetization measurements, polarized neutron diffraction, elastic and inelastic neutron spectroscopy are successively reviewed. The interest in performing nuclear magnetic resonance quadrupolar hyperfine splitting analysis is then emphasized and limitations of the method underlined. A comparative set of 4f moments for ferromagnetic rare-earth-A12 and rare-earth-Zn obtained by the various methods discussed is presented.

1. Introduction

In ferromagnetic rare-earth (RE) intermetallics, the magnetic properties may arise from the localized 4f electrons and more extended 5d and 6s conduction band electrons (for example in REA12). If there is an induced moment on the second element (e.g. RECo2) or it is itself magnetic (e.g. REFe2) other contributions must be included. A full understanding of the magnetic properties of rare-earth containing compounds clearly requires an accurate determination of the 4f shell magnetic moment but it is less clear which experimental observation will enable its deduction.

The 4f properties may by described [1,2] in a simple crystal field and molecular field model leading to wavefunctions expressible in the [ J M ) scheme. We shall focus attention on the REAl 2 and REZn ferromagnetic intermetallics for which extensive experimental data exists [3,4]. The interest in these systems comes both from the simplified cubic symmetry at the rare-earth site and the wide range of exchange interactions found within them. Furthermore, those compounds exhibiting ferromagnetic order, do so at readily accessible temperatures, so explaining the degree of experimental interest shown in them. There are no complications arising from induced or localized moments at the second element, A1 or Zn, site.

It is particularly difficult to envisage an experiment in which effects arising from 4f and conduction band electrons may be separated. For example, in measurement of the bulk magnetization as a function of temperature or applied field the addition of both terms is recorded. A clear example is found in the magnetization of GdAl 2 at low temperatures where the total is 7 .2#B/formula unit [5] whilst the saturation value for Gd 3+ is 7/~a. Since crystal field effects are absent for this S-state ion, the 0.2#a magnetization excess must be due to band electron polarisation i.e. it is of the order of 3% of the total magnetization. For rare-earth intermetallics which have a significant orbital moment in the 4f shell, the relative contribution of

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314 E. Belorizky et al. / 4f moment in cubic RE intermetallics

the conduction electrons to the total magnetization may be considerably more important as is anticipated for HoA12 [6]. In this case, detailed knowledge of the 4f ground state is essential if the conduction electron contribution is to be determined accurately. Similarly, the total hyperfine field, H N, at a rare-earth nucleus is of the order of 106 Oe, that arising from the 4f electrons is about 90% of this figure whilst the conduction electrons contribute of the order of 105 Oe. Since the latter contribution is usually obtained by subtraction of the 4f part from the total hyperfine field [7], it is clear that an uncertainty --- 5% in the 4f contribution leads to 50% uncertainty in deducing the conduction electron part.

Further examples of the importance of 4f ground state wavefunction determination are readily found. Evaluation of the higher tensorial order exchange interactions in the ferromagnetic state such as bi- quadratic [8] require precise knowledge of 4f wavefunctions. Similarly, a full understanding of the magnetic properties of ternary compounds such as [9] TbxDyl_xFe2 or Sml_xGd~Al 2 [10] require an accurate determination of the individual 4f moments and consequently, the wavefunctions.

In section 2, we shall briefly review the different methods used to determine 4f ground states and distinguish mainly between two groups: i) - Magnetization measurements on single crystals at high fields and low temperatures;

- Measurements of the magnetization anisotropy in single crystals as a function of temperature in the ordered state;

- Susceptibility measurements in the paramagnetic region together with inelastic neutron spectroscopy in powder samples.

The common features of these approaches result in simultaneous observations of the localized and diffuse electronic magnetization making it difficult to separate the two effects.

ii) - Polarized neutron diffraction on single crystals; - Quadrupolar effects in rare-earth hyperfine spectra by nuclear magnetic resonance, Mrssbauer or

other nuclear spectroscopic methods. In section 3, a detailed description is given of the nuclear magnetic resonance (NMR) method with a

critical analysis of the different possible sources of error. Finally, section 4 is devoted to a compilation of data for REAl2 and REZn systems available at the present time.

2. Different approaches to determination of the ground state 4f moment

2.1. Magnetization measurements on single crystals at low temperatures

Although, in principle, this is the simplest experimental method, one must not underestimate the problem posed by the necessity of obtaining RE intermetallic, single crystals. In the first instance, there may be strong magnetocrystalline anisotropy so that the easy axis of magnetization must be ascertained. Then, measurements over a wide range of magnetic fields are required in order to obtain accurately the function M ( H ) from whose extrapolation to zero external field, the saturation magnetization may be obtained, M s. The expectation value (J~) of the total angular momentum in the 4f groundstate along the easy axis of magnetization, z, is usually deduced from the saturation moment assuming M s = gj(Jz)l.ta. This is justified on the assumption that at low temperatures, only the ground state of the 4f electron is populated. Unfortunately, the measured magnetization, M s, includes, indistinguishably, both the localised 4f moment, Mar and the diffuse conduction band polarization, Mcond. Thus the relative accuracy of (J~) is, independent of experimental error, at least of the order of M¢ond/M4f.

A special case of particular interest, mentioned previously, is the half filled 4f shell i.e. Gd 3+ intermetallic compounds (and Eu 2+ compounds). Here the 4f moment of the RE ion in its ground state, arises only from spin and one can expect a saturation value of M4f = 7.0/~ B since there are no crystalline electric field effects. The measured saturation moment gives a direct indication of the extra conduction

E. Belorizky et aL / 4]" moment in cubic RE intermetallics 315

electron moment, Mcond. For example, in GdA12, M s = (7.2 + 0.05)# a [11,12] yielding a value of Mcond = (0.2 + 0.05)# a. Even in this apparently straightforward case, recent polarized neutron studies [13] on GdA12 have deduced a value of 6.6#a for M4r instead of 7/~ B and in consequence, M~ond = 0.6/~ B. This result is rather surprising. One possible explanation of the low value of M4f might be that GdAI/, like GdMg [14], is not a true ferromagnet but presents a slightly canted structure. Another explanation might be related to recent band structure calculations for Gd metal [15] in which it was shown that hybridization brings f conduction band character below E F leading to Mar = 6.67~a, significantly lower than the 7.0/t B expected.

2.2. Experimental approaches using crystalline electric field and molecular field parameters

All attempts concerning the interpretation of the magnetic properties of the RE intermetallic compounds with Laves phase structure were made within the framework of a simple crystal field and molecular field model for the rare earths. The single RE ion Hamiltonian is then

~= W [ O4x-~4-4- (1--lx])-~6]+gjttaJ.(Heff+H),06 (la)

where the first term is the usual cubic crystal field Hamiltonian with

Wx = B4F 4, W(1 - Ix l ) = B6F 6 ( lb)

and where the second term is the sum of the molecular field and external field interactions. Hef t is given by

Heft = hgj#a( J>, (2)

where ( J ) is the thermal average of the total 4f angular momentum and X is a coefficient measuring the magnitude of the exchange interactions, related to the paramagnetic Curie temperature, Op by

X = 3kOp/g~#2J(J + 1). (3)

The paramagnetic Curie temperatures, Op, are generally determined by extrapolating the reciprocal susceptibility, X-~(T), to zero. However, it is clear that such extrapolations should not be made on X-1 for temperatures of the order of the crystalline electric field splitting (A -- 100 K) since the form of X-1 is strongly influenced by crystal field effects. When crystal field parameters are known, they may be used to improve the accuracy in determining Op using the following simple approach [16]: denoting X¢¢(T) the Van Vleck crystal field susceptibility in the paramagnetic regime

M = Xcc(T) (n + XM) = x ( T ) H , (4)

from which we deduce

x- ' (r)=x:o'(r)-x . (5) At the ferromagnetic Curie temperature, T~, we have X-1(T¢)= O, so

2~ = Xc-c' (To). (6)

If T¢ is measured accurately, h is simply obtained using eq. (6) when the crystal field parameters, W and x, are known. Diagonalization of the Hamiltonian (la) is thus a self-consistent process since the average value of ( J ) is involved and the validity of this three parameter model (IV, x, ~) is checked by the convergence of the procedure. The most favourable case for determining the crystal field parameters arises when inelastic neutron scattering data are available. However, these measurements have to be performed at low temperatures and in a paramagnetic sample in order to avoid line broadening due to exchange interactions.

316 E. Belorizky et al. / 4f moment in cubic RE intermetallics

It is thus usually necessary to dilute the RE 3+ ions in a non-magnetic, isostructural matrix (e.g. YAI 2 or LaAI2) to simulate the REAl 2. This method, applied to Er diluted in YA12 yielded crystal field parameters [17] close to those deduced for ErA12 where a neutron spectrum was observable at low temperature (-- 6 K) in the ferromagnetic phase because the ferromagnetic ordering temperature is particularly low (14 K). In general, care must be taken since the nature of the diluting matrix and the paramagnetic/ferromagnetic transition may affect the crystal field parameters. A further difficulty arises from the non-uniqueness of the pairs of 14" and x obtained by fitting to the, in general, limited number of observed crystal field transitions between the populated electronic levels.

It is also possible to obtain information about crystal field parameters from careful analysis of magnetization behaviour for single crystals when the applied magnetic field is along the principal axes of the cube. The most accurate results are those obtained when it is possible to measure the difference [17] in magnetization, AM, between the three principal axes at low temperatures - two relations are obtained leading to a unique set for W and x. Unfortunately, this case is the exception rather than the rule since AM is usually only measurable between two directions. It is consequently necessary to try to obtain another independent relationship which might be provided by specific features of M ( H ) such as a jump in the magnetization for a given field along a particular direction (e.g. DyAI 2 [18]) or a change in the easy axis of magnetization for a given temperature (e.g. HoA12 [19]). Finally, starting from eqs. (la) and (6), a systematic fit of M ( H , T ) for the three principal directions in single crystals may provide approximate values of the crystal field parameters [20,21].

In all of the methods based upon magnetization measurements, it is not possible to separate the contributions arising from localized 4f electrons and conduction electrons and this is the main limitation of the approach. In fact, even measurements of A M can only be used to provide accurate information on the 4f ground state if the conduction electron magnetization is isotropic and this is not the general case, at least for the 5d electrons [22]. If the single ion Hamiltonian, (la) is incapable of explaining the magnetic properties of the system under investigation, some improvement of the model may be made by introducing two additional terms: (i) magneto-elastic coupling which is present when the symmetry of the RE 3÷ site is not exactly cubic;

(ii) quadrupole-quadrupole coupling [23,24] between 4f electrons on different sites in the presence of exchange and external magnetic fields which may be treated in the molecular field approximation [19]. The existence of these interactions is represented by adding to the Hamiltonian (la) terms proportional to T~(J~) ( - 2 ~ < / ~ < 2 ) where T2 ~ are the standard second order spherical tensor components constructed from the J~ angular momentum components of the system. These terms all depend upon two independent constant and have been used to fit the magnetic properties of DyA12 [18], HoA12 [19] and in REZn compounds [25] where magnetoelastic effects are important.

2.3. Polarized neutron diffraction experiments on single crystal

Direct observation of the spatial distribution of the magnetization density is of particular interest. This microscopic measurement may provide a direct picture of the state of the 4f ion which results from competition between the crystalline electric field and magnetic interactions. Magnetization density mea- surement is also, in principle, a means of visualizing and analysing the contribution due to conduction electrons. Magnetic neutron scattering allows measurement of the magnetic structure factors, F M ( H ), which are the Fourier components of the magnetization density. When polarized neutrons are used, measurement of the flipping ratio, R ( t t ) = I ÷ / I _ , between the scattered intensities for the two possible neutron polarizations of the beam, yields directly the value of the ratio 2t = F M ( H ) / F N ( H ) where FN ( H ) is the nuclear form factor.

In rare-earth alloys, due to the strong localization of the 4f magnetization density, the magnetic form factor is significantly expanded in reciprocal space. In consequence, in order to obtain a reliable estimate of

E. Belorizky et al. / 4f moment in cubic RE intermetallics 317

the local density around the nucleus, short wavelength neutrons must be used. Wavelengths down to X = 0.42 A currently available (ILL, Grenoble) allow measurements of Bragg reflections with values of sin 0/X up to 1.5 A-1 [26,27]. The main difficulties encountered in measuring F M ( H ) accurately arise from extinction phenomena and from the lack of exact knowledge of FN(H ) [28].

Contributions to the magnetic form factor arising from 4f and conduction electrons may. be separated as follows: since conduction electrons are not strongly localized (as for example, 4f electrons), they contribute to the magnetic form factor only for small values of the scattering vector, H. For large H, only the 4f electrons contribute. For the case of NdA12, for example [29], 23 reflections are observed corresponding to sin 0/X > 0.39 A-~. This significantly large number suggests that an accurate fit to the 4f contribution can be expected.

In the crystal field and molecular field model described in section 2.2, the 4f ground state [~o), for a RE ion with a ground multiplet, J, may be written as

where R(r) is the radial part of the 4f wavefunction and a M are numerical coefficients normalised such that

E laMI 2 = 1. M

In cubic symmetry, the allowed values of M are defined with a periodicity 4, 3 or 2 depending upon the easy axis of magnetization, fourfold, threefold or twofold. Obviously

(JA = E laMI 2M (8) M

and the 4f magnetic moment at low temperatures, when only the ground state, Itko), is appreciably populated, is

(M) , f = ga(J~)t%. (9)

The relationship between the values of the spherical components of the magnetic amplitudes, FM~ (with q -- 1, 0, - 1), determined from experiment and the unknown coefficients, aM, is given [28] by

F M , ( H ) = Ye2 4v/4-~ E Y ~ , " ( H ) E E a•aM' meC2 K", Q'" K', Q" M, M'

× [A(K"K') + B(K"K')] ( g ' o ' J g ' l J M ) ( g " o " g ' O ' l l q ) , (10)

~, is the neutron gyromagnetic ratio, Y~:' are spherical harmonics depending on the direction of t h e scattering vector H. A(K"K ' ) and B(K "K') represent, respectively, the orbital and spin contributions to the magnetic scattering; their form has been given by Marshall and Lovesey [30]. They involve numerical coefficients which have been tabulated for rare-earth ions by Lander and Brun [31] and radial integrals ( j x (H) ) defined by

(JK(H)> = fo°°r2R2(r)jK(Hr) dr, (11)

wherejx(Hr ) is a spherical Bessel function. The integrals given in eq. (11) may be evaluated using 4f radial wavefunctions, R(r), obtained from

atomic Hartree-Fock calculations [32]. More accurate R(r) 's may be obtained by including relativistic effects in Dirac-Fock calculations [33]. When the radial integrals (11) are known, this method permits

318 E. Belorizky et al. / 4.[ moment in cubic RE intermetallics

determination of the coefficients a M of the wavefunction given by eq. (7). Comparison between theory and experiment is made on the basis of the form factors and in practice, when the scattering amplitudes are strongly anisotropic, precise determination of a M can be expected. On the other hand, if the scattering amplitudes vary rather monotonically, the experimental information is usually insufficient to yield accurate results for a M which are strongly correlated. Using these techniques, 4f ground state wavefunctions have been obtained for NdAI 2 [29], CeA12 [32], HoAI 2 [35], SmA12 [36] and SmZn [37]. We note in passing that for Sm based intermetallics, significant ground multiplet ( J = 5/2) and first excited multiplet ( J = 7/2) mixing occurs and must be taken into account.

We must also mention a method of determination of (Jz) developed by Furrer and Purwins [38] for NdAI 2 based on the analysis of magnetic excitations spectra at 4.2 K by neutron inelastic scattering on single crystals. The crystal field and exchange interaction parameters were fitted to the observed excitation energies using exchange interactions of the Heisenberg type exhibiting a RKKY like qualitative behaviour.

2.4. Quadrupolar effects in rare-earth hyperfine interactions

"A priori", a very simple method enabling determination of the 4f electron ground state magnetic moment would be via direct measurements at low temperature of the magnetic hyperfine field, H t, at the rare-earth site. This is typically of the order of 102 T and this method has, indeed, been frequently employed by MiSssbauer spectroscopists to ascertain the 4f moment. Unfortunately, a refined analysis shows that the measured field, H t, contains not only the field produced by the localised 4f electrons, H4f, but also a field, Hsp, of the order of 10 T, created by local polarization of the 5d/6s conduction band electrons through 4f -5d /6s exchange interactions. Finally, there is a transferred hyperfine field created through rare-earth a n d / o r other neighbours [7]. This field may be more or less important than Hsp depending upon the nature of the neighbours (e.g. magnetic/non-magnetic) and the strength of the transfer mechanism. A potentially more appropriate means of probing the 4f moment is provided by careful analysis of quadrupolar effects in the hyperfine spectrum. 6s electrons do not contribute to this interaction and, as will be demonstrated, corrections due to 5d electrons may be realistically estimated.

In general terms, for a rare-earth ion whose total 4f angular momentum may be characterized by J, the total hyperfine interaction may be written as [39]

,~h f= A l * J + B { [3J~2 -J ( J + 1 ) ] ( I 2 - - ½ 1 2)

1[ j 2 1 2 + j 2 1 2 ) } . (12a) +½(JzJ+ + J + J z ) ( I j - + I - I z ) + c . c . + ~ + - _

The first term is the magnetic hyperfine interaction and the second, the quadrupole interaction with

- 3 e 2 Q B = 4~-2-I --- ] ) ( rq3) (J I ]aHJ) ' (lZb)

where Q is the nuclear quadrupolar moment of the RE, (rq 3) the expectation value of r - 3 for the 4f electron and (J l la l l J ) a reduced matrix element. At low temperatures, where only the ground electronic level is populated and treating.,affhf as a first order perturbation, we obtain after averaging over the ground state electronic wavefunction, I~P0), a purely nuclear Hamiltonian-"fiN-

When the easy axis of magnetization is a fourfold or threefold symmetry axis we obtain

~g'N = aI~ + P,, [12 - ½ I ( I + 1)], (13a)

with

a = A ( J z ) , (13b)

PI ' = B ~ [ 3 J g 2 - - J(J -~ 1)] ). (13c)

E. Belorizky et al. / 4f moment m cubic RE intermetallics 319

Introducing, as does MacCausland [40], a 0 and P0 corresponding to the values of a and/~, for the fully saturated free ion moment (IJ l = s ) , we have

a°(J~) and P,=Po (3J~2-J(J+ 1)) (14) a = - - - f - - J ( 2 J - 1)

The transition energy between two adjacent nuclear levels, m and m - 1, is, from (13a):

hvm.,._ , =a+ P, ,(2m- 1). (15)

When the easy axis of magnetization is a twofold symmetry axis, an additional term is present in ovt°N which may be written as

1 B ( / j 2 \ 1 2 j 2 12 ~\ +/ _ + ( _ ) +). (16)

This term, however, may be taken as a perturbation when compared with the magnetic contribution to the hyperfine energy levels and produces only second order effects thus having negligible importance for the hyperfine spectrum [41].

Experimentally, for a nuclear spin I, one observes 2 I resonance lines corresponding to the transition energies hvm, m_ l(Iml < I) which may consequently be analysed using the following expression [41]:

hvm,,,_ , = a t + Pt(2m - 1) + W(3m 2 - 3m + 1), (17)

in which at, the main term, represents the energy of the central transition ( I half integer), Pt refers to the magnitude of the quadrupolar splitting and W, usually small, is characteristic of the asymmetry of the overall spectrum. W originates from the second order effects present in the magnetic hyperfine interaction. As has been shown elsewhere [7], a t includes not only a defined by eq. (13b) but also contributions from 5d and 6s electrons which are difficult to estimate with any accuracy. A pseudo-dipole contribution arising from second order perturbation effects of the magnetic hyperfine interaction of the order of a2/AE (AE = splitting between ground and excited electronic levels) is also included in a t. It is consequently very difficult to obtain the expectation value of (Jz) directly from measurements of a t.

We are thus essentially concerned here with the quadrupole term, Pt which may be written in the general form

Pt = P, +p<2) + Pco,d + Plat (18)

(in cubic symmetry, the lattice term, Plat, is zero), where P,, is the main term, as defined by eq. (14). p(2) is the pseudo-quadrupole term [41] arising essentially from second order effects of the magnetic hyperfine interaction, again of the order of a2/AE. Strictly speaking, there are also second order effects arising from cross terms of the quadrupolar and hyperfine interactions as well as third order terms of the magnetic hyperfine interactions - their contribution to p(2), however, is negligible. An explicit expression for p(2) has been given by Waind et al. [41]. When excited electronic levels are present close to the ground state or the magnetic hyperfine constant a is large, corrections p(2) may be important. This is typically the case for 165Ho in HoAI 2 where pt2)/p, is estimated to be = 10%. Pcond in eq. (18) arises through quadrupolar polarisation of the 5d band electrons due to 4f -5d exchange interactions. This polarization induces an additional quadrupolar splitting which may not be "a priori" negligible. This term is similar in nature to //so in the magnetic interaction except that here, s electrons do not participate - we will evaluate this using linear response theory in the next section.

Assuming that p(2) + Pcond may be correctly estimated (and fortunately they are often rather weak) it is possible to deduce from the experimental value of Pt a rather accurate value of P , and thus a good estimate

320 E. Belorizky et al. / 4f moment m cubic RE intermetallics

of the average value (3J, 2 - J ( J + 1)) and hence, ( j 2 ) . Details of the determination, subsequently, of (J~) are presented in the next section.

3. Ground state 4f moment determination from quadrupolar hyperfine experiments

3.1. Experimental techniques

Nuclear magnetic resonance, M6ssbauer effect and perturbed angular correlation are the three main available experimental techniques. In the case of NMR, a wide range of variable frequencies is required in order to cover the various RE nuclei. Two pulsed NMR spectrometers working in zero external field have been built for this purpose, one in Grenoble [42] over the range 30 to 4000 MHz, the other in Manchester [40] for the range 4000 to 7000 MHz. The essential advantages of NMR result from the ability to measure with high accuracy the resonance line positions and to separate directly the magnetic and quadrupolar hyperfine components of all the RE nuclei except Cerium. On the other hand, the M6ssbauer effect gives significant information for a more limited range of RE nuclei - the resolution is generally less than in the NMR case but hyperfine parameters may be measured from low temperatures up to T~. In certain cases, e.g. Tm compounds, the nuclear ground state may be spin I = 1 /2 (169Tm) in which case no quadrupolar splitting is observed. Valuable information may be obtained, however, on the quadrupolar coupling, through nuclear transitions between the ground and first excited nuclear level ( I = 3 /2) [43].

3. 2. Determination of (J~) f rom ( J r )

Let us assume that ( j 2 ) has been obtained from quadrupolar hyperfine data analysis and proceed to derive approximate values for (J~). In the absence of crystal field, with pure exchange interactions, the 4f ground-state wavefunction in the basis I J, M ) = IM), is I~k0) = I J ) and the excited states I~M) = I M ) with M varying from J - 1 to - J . In the presence of crystal field, the ground-state wavefunction will have the form

- J

[~b0) = ~ aMIM ), (19) M = J

with M = J modulo N and N = 4, 3, 2 for a fourfold, threefold and twofold symmetry axis of the easy direction of magnetization. We wish to demonstrate that to a good approximation, the first two terms in the expansion of eq. (19) only need be considered i.e. a j I J ) + a j_ N I J - N ) . Normalizing, the approximate ground state, Ilk0), it is trivial to deduce (Jz) from ( j 2 ) . We propose to justify this method and study its limitations.

For cubic RE intermetallics and in particular the REAl2 and REZn we shall consider that the ground state [~k0) has always its largest projection on the saturated state, I J ) . This can be verified by considering approximate wavefunctions obtained by other methods such as magnetic measurements combined with inelastic neutron diffraction experiments [17] or polarized neutron scattering as described in sections 2.2, and 2.3, respectively. This is readily understandable if one considers that in all these compounds the overall crystal field splitting is smaller than or of the same order of magnitude as the overall spliting due to pure molecular field except for the case of Praseodimium which requires special treatment. It can also be demonstrated that, for all the RE, addition to the molecular field Hamiltonian, ~MF, of the diagonal part of the crystal field Hamiltonian, ~CF, (terms in O ° and O ° ) never changes the ground state [J) , although

E. Belorizky et al. / 4f moment in cubic RE intermetallics 321

permutations in excited states may be predicted. Considering the off-diagonal part of o~cv (terms in 044 and 0 4 for tetragonal symmetry) as a perturbing term as compared to the sum of molecular field "~MF and diagonal crystal field "~CF parts, we see that the first order correction to the [J) state is the addition of a component a j _ a l J - 4 ) and that the state a s _ s l J - 8 ) arises only from second order corrections of the off-diagonal components. Consequently, it is a satisfactory approximation to write the wavefunction for a fourfold symmetry axis of magnetization as

{~bo) = as lJ ) + aj_4lJ - 4). (20)

This state being normalized, in terms of the experimentally determined quantity (Jz z) we have

(Jz) = [ ( J~) + J( J - 4 ) ] / ( 2 J - 4). (21)

For the case of DyAI2 as an example, the uncertainty arising in (J~) using this method is estimated to be less than 0.2%. The method cannot be applied to the case of PrAI 2 because the overall crystal field splitting is greater than the total molecular field and the ground state has the form

I•0) = a4~4) + a010) + a - a [ - 4).

Now a 4 > a 0 > a_ 4 but since a o does not contribute to ( j 2 ) or (Jz), the term in a_ 4 plays a non-negligible role.

In trigonal symmetry the off-diagonal elements of the crystal field, 0 3 and 063 contribute in first order to a component as_31J - 3) whilst the term in 066 gives rise to a j _ 6 1 J - 6). The latter term is, however, always very small and the approximation of maintaining only the first two terms leads to

( Jz) = [ ( j 2 ) + j ( j _ 3 ) ] / ( 2 J - 3). (22)

The relative uncertainty arising from this method of deducing (J~) is estimated to be about 10-4 for TbAI 2 and 6 × 10-3 for ErAI 2, two typical trigonal easy axis examples. Finally, in the case of twofold symmetry, this method is less applicable because the off-diagonal terms O4 z and 063 on one hand and 0 4 and 0 4 on the other, contribute similarly to the components aj_zlJ - 2 ) and aj_4lJ - 4 ) . In consequence we have three components which cannot be deduced straightforwardly from quadrupolar studies. However, even in that case, rather accurate values of (J~) may usually be obtained from ( j z) (e.g. HoAI2).

3.3. Accuracy in determination of ( J~)

As indicated in section 2.4, experiments provide values of Pt defined by eqs. (17) and (18). The major contribution to Pt is given by P, defined by eqs. (13c) and (14). The values of P0 for the trivalent rare-earth ions were deduced by Bleaney [39] from atomic beam magnetic resonance experiments assuming

Po (ion) 3+ { ( r q 3 ) 3 + ( J l l a l l J ) J ( 2 J - 1)}io"

P0 (atom) - { (% 3)atom(Jl[allJ)J(2J -- 1) } atom" (23)

Very accurate values of Po are available for the most of the rare-earths. Values of (rq3)3+ and (rq3)atom were taken from the calculations of Lindgren [44]. Bleaney suggested an upper limit to the inaccuracy of deduction of P0(ion) of 5% based upon the most unfavourable case, that of Tb, where difficulties were observed in deducing P0(atom) [45]. For most of the other rare-earths, given the precision of determination of P0(atom)_< 1%, the accuracy in P0(ion) depends upon the validity of the assumption that the ratio (rq3)3+//(rq-3)ato m calculated using Lindgren's results, is correct. Questions have, in fact, been raised about the accuracy of Lindgren's calculations [46]. Recently, accurate relativistic calculations of the 2 ÷ and 3 + states of the rare-earth ions have been carried out by Freeman and Desclaux [47] and values o f ( r -3) tabulated. Given that the 4f electrons are well inside the closed 5s and 5p shells, and thus screened from the

322 E. Belorizky et aL / 4.[ moment m cubic RE intermetallics

effects of 6s outer electrons, we can take the calculated values of ( r -3)2 + to be representative of the atomic case. In table 1 we list the r a t i o ( r - 3 ) 3 + / ( r - 3 ) 2 + obtained for various rare-earth ions together with the results of Lindgren. In all cases the difference between the two ratios does not exceed 1.5%. In consequence, with the exception of Tb mentionned previously, we believe the value of P0(ion) estimated from P0(atom) [39] to in fact be accurate to better than 2.5%.

3.4. Evaluation of different sources of error

We wish to evaluate different possible sources of error involved in the determination of (Jz 2) from the experimental values of Pt. As suggested in eq. (18), they arise essentially from the pseudo-quadrupolar interaction described by the term pt2) and from quadrupolar polarization of the 5d conduction band and electrons represented by the term Pcond" We first focus our attention on pt2) which arises through second order perturbation effects of the magnetic hyperfine interaction and the quadrupolar interaction; we write it in the form

p~2) = a~a + aoPofl, (24)

where a and fl are factors inversely proportional to the energy splitting of the electronic levels. Analytic expressions for a and fl are given by Waind et al. [41]. Strictly speaking, determination of a and fl requires precise knowledge of the wavefunctions of the ground J multiplet levels. However, since pt2) is usually small with respect to P , , a rough estimate is adequate. The latter may be obtained using approximate wavefunctions deduced by other methods. For example, diagonalization of the Hamiltonian (la) in zero external field, using W and x values deduced from inelastic neutron scattering in the paramagnetic regime; alternatively one may use magnetic measurements. In both cases, h is deduced from the susceptibility at T~ and a self-consistent value of (Jz) obtained.

In the commonly observed case where the molecular field splitting is significantly larger than the overall crystal field splitting, i.e., when the ground state I~0) and first excited states are almost pure I J ) , IJ - 1) . . . . . I J - M ) states, we have to a good approximation

a = h / 2 J A E ; # = 3 h / J A E , (25)

where AE is the ground state-first excited state splitting. In table 2 we tabulate the values of a 0, P0, a, fl and pt2) for various rare-earth ions in the REAl 2 and REZn series. It can be seen that the contribution of

Table 1 Values of the rat ios ( r - 3 ) 3 + / / ( r - 3 ) a t o m calcula ted by Lindgren [44] compared relat ivis t ic ca lcula t ions of F reeman and Desc laux [47] for rare ear th ions

to the rat ios ( r - 3 ) 3 + / ( r - 3 ) 2 + ob ta ined from the

R E i o n ( r - 3 ) 3 + / / ( r - 3 ) . . . . ( r - 3 ) + 3 / / ( r - 3)2 +

(Lindgren) (F reeman and Desclaux)

N d 3+ 1.133 1.116

Sm 3 + 1.094 1.094 Eu 3 + 1.082 1.085 Gd 3 + 1.071 1.079 Tb 3 + 1.065 1.073 Dy 3 ÷ 1.057 1.068 Ho 3 + 1.052 1.064 Er 3 + 1.049 1.061 Tm 3 + 1.044 1.057

Yb 3 + not ca lcu la ted 1.056

E. Belorizky et al. / 4f moment in cubic RE intermetallics 323

p ( 2 ) to Pt is small in all but two cases: NdAI 2 where P ( 2 ) / / P t --~ 2% [48] and H o A 1 2 where P ( 2 ) / / P t m 10%. We now consider Pco,d" The 5d and 6s electrons are polarized through 4f-5d and 4f-6s exchange

interaction. This effect has been extensively studied [22,27]. However, even though both polarizations contribute to the total hyperfine field at the rare-earth nucleus through Hsp, due to the orbital character of the quadrupolar interaction, only the quadrupolar polarization of the 5d electrons contributes to Pco,d- This contribution can be approximately estimated using linear response theory. This method was originally used by Caroli, Caroli and Fredkin [49], by Dworin and Narath [50] and then by Huang Liu, Ling and Orbach [51]. For our application we consider the polarization of the 5d electrons (which we will take to be in a virtual bound state [51]) by their interaction with the 4f electrons already assumed polarized in the ordered state. The 4f-5d exchange interaction averaged over the ground state of the single rare-earth ion Hamiltonian (crystal field plus molecular field) is written as [22]

4 p . 1 *

~ 4 f ) S d = - E E a~Y~(I)[2b.(Y~.(L)S)'s+~c.(Y~p(L))], ( 2 6 ) t t=0 v= - p

where a t are exchange integrals [51] and b~ and c~ are coupling coefficients [22]. L, S refer to the 4f electrons and l, s to he 5d band electrons. Y~(l) and Y~p(L ) are spherical tensor operators. The response of a conduction electron operator, (B) , is given by

(B) = E E (¢rmrlBlCjmj)(¢jmjlX~<4f>Sdl'krmr)S( ¢r, q'j), ij m,mj

(27)

where, in cubic symmetry, 0r (i = 1 . . . . . 5) are the t2g or e~ orbital 5d states considered as bound states and m r refers to the spin state of the 5d electron. S(¢ r, Oj) is a generalized susceptibility. For d electrons there are three independent susceptibilities, S(t, t), S(e, e) and S(t, e). In the simplest approach, the susceptibih- ties depend upon the 5d band width and relative positions of the t2g and % bands with respect to the Fermi level. Their expressions are given in the appendix. Note that, as in ref. [51], we neglect the effects of electron-electron interactions in the 5d band. We furthermore consider the susceptibilities to be indepen- dent of the rare-earth considered. For GdAI2, magnetization measurements provide a saturation moment of 7.2/~B/RE atom, i.e. a conduction electron polarization of 0.2/tB/RE. Since there are both 5d and 6s spin polarization contributions, obviously (sz) d < 0.1. From eq. (A.3) we obtain for GdA12

(Sz) d = 0.59a0 [3S(t, t) + 2S(e, e)]. (28)

Table 2 Values of a 0, Po (taken from ref. [39]), a, fl defined by eq. (24) and calculated (i) by diagonalization of the Hamil tonian ( la ) or (ii) f rom eq. (25), and corresponding values of pt2). Also included are the experimental values of Pt and the derived values of P** = Pt - p(2)

Compound Isotope I a 0 Po a /] p{2) Pt P .

(MHz) (MHz) (10 -12 s) (10 -12 s) (MHz) (MHz) (MHz)

NdAI 2 143 7 / 2 - 991.4 - 5.3 (i) 0.065 0.304 0.066 - 3.25 4- 0.03 - 3.32 TbAI 2 159 3 / 2 + 3180 + 386 (ii) 0.067 0.414 1.186 356.75 + 0.15 355.56 DyA12 163 5 / 2 +1143 +228 (ii) 0.054 0.322 0.154 221.8 -I-0.1 221.65 HoAl 2 165 7 / 2 +6497 +61.2 (i) 0.135 0.222 6.30 57.2 +0 .6 (54) 50.9 ErAl 2 167 7 / 2 - 940 - 67.6 (ii) 0.149 0.896 0.188 - 58.85 + 0.15 - 59.04 TbZn 159 3 / 2 - 3180 - 386 (ii) 0.0235 0.141 0.411 362.5 4- 0.5 362.1 DyZn 163 5 / 2 + 1143 - 228 205.25 4- 0.25 ErZn 167 7 / 2 - 940 - 67.6 (i) 0.335 0.936 0.355 - 31.25 4-1.25 - 31.6

324 IS. Belorizky et aL / 4f moment in cubic RE intermetallics

For DyAI 2 for which the easy direction of magnetization is [100], we have from eq. (A.4) and using the approximate 4f wavefunction obtained in a self-consistent way as in the determination of p(2),

( l 2 - 2)0 = 0.30a2 [as( t , t) + 4S(e, e)] - 0.41a4 [S(t , t) - S(e, e)]. (29)

The ratios a2(Dy)/ao(Gd ) and a4(Dy)/ao(Gd ) may be evaluated from Huang Liu et al. [51] and are 0.47 and 0.34, respectively. Thus

(1~ - 2)d(OyA1/) 0.48S(t, t) + 1.19S(e, e) (sz)a(GdA12) = 3S(t, t) + 2S(e, e) (30)

Using the various possible limits for the relative importance of S(t, t) and S(e, e) we find that the right-hand side of eq. (30) must lie in the limits 0.16 to 0.6 so that ( l ~ - 2)d(DyAl2)< 0.06. The ratio Pcond/P,, is given by

P¢o.d 2 1 ( r -3)sd (1~-- 2)5 d (31) P, 21 (J{l~llJ) (r-3)4f (Jz 2 - l j2 )4f"

Values for (r-3)5d and (r-3)4f are typically 3 and 9 au, respectively [52], so we obtain

Pcond (DyAI2) < 9 X 10 -3, P ,

i,e. P¢o,a < 2 MHz. We have performed similar calculations for TbAI 2 and ErAI 2 where the easy axis of magnetization is [111] and for which eq. (A.5) for ( 1 ~ - 2)d must be used. In these cases we obtain (/2 - 2)a < 0.03 (and 15 × 10 -3) and Pcond/P, < 4 × 10 -3 (and 6 × 10 -3) for TbA12 (and ERA12) so that Pcond < 0.9 MHz (and < 0.3 MHz). We conclude that Pco,d/P,, is always < 1% and this is probably much smaller since we have used extreme values. The conduction electron quadrupolar polarization effect can thus be neglected.

Finally, we will comment on errors which might arise due to magnetoelastic effects. Any departure from cubic symmetry at the RE site due to magnetostrictive effects will lead to a lattice contribution, Plat which must be subtracted from Pt to deduce P , . For pure RE metals, which are hexagonal, Bleaney [39] has estimated values of Plat of the order of - 0 . 6 MHz for a value of c / a - 1.633 = 0.05. (Note that c /a = 1.633 corresponds to the ideal case, Plat = 0.) For magnetostrictive distortions in cubic symmetry, Plat will be proportional to the ratio (c - a)/a. In the REAl 2 series, this ratio varies [53] from 10 -3 for NdAI 2 to 2.5 × 10 -3 for DyA12. By simple comparison with the pure rare-earth metal results one expects ]Plat] < 3 X 10 -3 MHz for NdAI 2 and < 1 MHz for DyA12. In these two cases, P l a t / P t < 10 -3 and 5 x 10 -3, respectively. For some of the REZn compounds, magnetostrictive effects are larger [25]. For ErZn (c - a ) /a --- 10 -3 whilst it reaches 8 x 10 -3 for DyZn, in c o n s e q u e n c e [Platl < 0.2 MHz for ErZn but may be of the order of 2.7 MHz for DyZn.

Based on the estimates presented in this section we believe the results for P,, given in the last column of table 2 to be precise to within 1% except for HoA12 where the precision may be 2% [54].

4. Results and discussion

All of the elements are now present permitting an accurate deduction of (Jz) for the RE's from N MR quadrupolar hyperfine spectra. From the values of P0 and P, given in table 2, we deduce immediately the values of (3J f - J ( J + 1)) with eq. (14). Using the method described in section 3.2 we deduce the value of (Jz) appropriate to the NMR data. The corresponding results are given in table 3 (column 7) and are

E. Belorizky et al. / 4f moment in cubic RE intermetallics 325

compared with values of ~J~) obtained by other techniques i.e. self-consistent diagonalization of the Hamiltonian (la) when IV, x are determined from magnetic measurements (column 8) or from inelastic neutron scattering spectroscopy (column 9), polarized neutron diffraction (column 10) and saturation magnetization measurements (column 11). Some care must now be taken about the accuracy of ~J~) determined from the NMR results. P, is known to better than 1% and P0 better than 2.5% so that we can expect the error in (3J 2 - J ( J + 1)) to be less than 3%, except for Tb compounds where accuracy --6% because P0 is less well known. The absolute uncertainties in ~3J~ 2 - J ( J + 1)) are given in brackets in column 4 of table 3. The uncertainty in ~J~) obtained via NMR is written in the form

A<L) = A1<L) + A2<L), (32)

where A 1 arises from the uncertainty in {3J~ 2 - J ( J + 1)) and may be evaluated simply from eqs. (21) and (22):

A l ( Jz) = A~ j 2 ) / ( 2 J -- N ), (33)

N being the order of the symmetry of the easy axis of magnetization, and A2(Jz) arises from the approximate method itself. As underlined in section 3.2, A 2 varies from system to system b u t quite generally one can state that the closer (Jz) tends to J, the smaller is A 2 because the a j _ N terms in eq. (19) become smaller. For an almost saturated system the method is applicable for cases having [110] easy axes such as the case of TbZn. Values of A 2 are always smaller than A 1 and then can be estimated by applying the method to the approximate ground state bk0) obtained by self-consistent diagonalization of the Hamiltonian (la). Values for A1(J~) and AE(Jz) a r e given in columns 5 and 6 of table 3. The total uncertainty, A(Jz), is given in brackets in column 7. The relative accuracy, A ~ J ~ ) / ( J ~ ) is always better than 2.5%.

Table 3 Values of <3J ) - J(J + 1))(column 4) and of <Jz) (column 7) determined by N M R with the estimated error between brackets for the series REAl 2 and REZn. The values of ( Jz ) are compared with the available data obtained by other techniques (columns 8 to 11)

Com- Easy J N M R ( J , ) (self-consistent methods)

pound direct- ( 3 j 2 - j ( J + l ) ) Al(J~) A2(Jz) (Jz) magnetic inelastic polarized saturation ion of measure- neutron scat- neutron dif- magnetiza- magne- ments tering fraction tion tization

1 2 3 4 5 6 7 8 9 10 11

NdA12 [100] 9 / 2 25.5 (0.7) 0.04 0.01 3.60(0.05) d) 3.56 a) 3.63 a) 3.58 c) 3.37 a) 3.57 S)

[111] 6 60.8 (3.6) 0.13 < 10 -4 5.81(0.13) e) 5.975 a) 5.96 a) _ 5.93 a) [100] 15 /2 102.1 (3.0) 0.09 2 x 10- 3 7.41(0.09) e) 7.22 a) 7.45 a) - 7.40 a) [110] 8 99.9 (1.8) 0.04 0.03 7.51(0.07) f) 7.39 ~) 7.40 6.90 c) 7.34 ~) [111] 15 /2 91.7 (2.7) 0.08 0.04 7.13(0.12) e) 6.99 ~) 7.05 a) 6.50 a) [110] 6 61.9 (3.7) 0.12 0.013 5.86(0.13) c) _ 5.79 b) 5.90 b) [100] 15 /2 94.5 (2.8) 0.09 0.03 7.18(0.12) ¢) -- - 6.79 b) [100] 15 /2 49.1 (3.2) 0.10 0.11 5.81(0.21) c) 5.6 b) 5.55 b) 5.67 b)

TbAI 2 DyAi 2 HoAI 2 ErA12 TbZn DyZn ErZn

a) M. Rossignol thesis, ref. [17]. b) p. Morin, ref. [25]. c) X. Boucherle, ref. [26]. d) y . Bcrthier and E. Belorizky, ref. [48]. e) This work. 00.M. Prakash, ref. [54]. g) A. Furrer and H.G. Purwins, ref. [38].

326 E. Belorizky et al. / 4 f moment in cubic R E intermetallics

The following general conclusions may be drawn from the results of this study: i) The magnetic moment of the RE 4f shell in cubic intermetallics may be obtained at low temperature

using NMR to relatively high accuracy. The experiments are relatively simple and may be performed on powdered samples, eliminating many experimental difficulties.

ii) The method of determination of (Jz) from (3Jr 2 - - J( J + 1)) deduced experimentally, is accurate when the easy axis of magnetization is fourfold or threefold. It is not generally applicable for the twofold case except when the moment is very nearly saturated.

iii) Experimentally, we have obtained accurate information for the REAl 2 series for which resonance lines are narrow - for REZn compounds, metallurgical problems give rise to broad lines and significant magnetostrictive effects are present, particularly in DyZn.

iv) Our results for (Jz) converge well with those obtained by diagonalization of Hamiltonian (la) in a self-consistent way when W and x were obtained through inelastic neutron scattering and )~ determined from the susceptibility at T c. For the cases considered (and diluting matrices used) this means that the crystal field parameters for the R E 3+ ions did not vary significantly between the concentrated and dilute magnetic host matrices. However NMR measurements made in the ferromagnetic phase, are simpler, more rapidly performed and require no approximations induced by magnetic dilution.

v) The NMR technique may be used in the case where very strong exchange interactions are present such as REFe 2. Because of the very high T~ inelastic neutron scattering experiments are not possible directly and the exchange interactions due to the Fe ions mean magnetic dilution replacing the RE with say Y, serves no purpose. Results concerning the REFe 2 series will be presented elsewhere. A preliminary study was given by Devine and Berthier [38].

Acknowledgements

We gratefully acknowledge helpfull discussions with Dr. J.J. Niez and Prof. P.M. Levy on the subject of 4f -5d exchange.

Appendix. Calculation of the polarization of the 5d electrons

We begin with eqs. (26) and (27) where the spherical tensors Y~(I) are normalized in such a way that

T r [ Y . * ( I ) Y ~ . ( I ) ] = 1.

The susceptibilities S(epi, ¢pj) entering eq. (27) are given by [51]

At " S(e, e ) = Ae (A.1) S(t, t) = ~(E2 + A2t) , .rr(E2 + A2e) ,

where A t and A~ are the t2g and eg bandwidths and E t and E e are the energies of the centre of these bands with respect to the Fermi level. S(t, e) is given by

S ( t , e ) = 1 ( e t - E e ) ( O t - O e ) + ( A t - A e ) l o g r t / G , (A.2) "~ ( E _ E ¢ ) 2 + ( A _A~) 2

with

tgO t = A t / E t and r t = A ~ t + E t 2

with similar definitions for O e and r e.

E. Belorizky et al. / 4f moment in cubic RE intermetallics 327

It is readi ly d e m o n s t r a t e d that for a n y phys ica l s i t ua t i on S(t , e) is i n t e r m e d i a t e be tween S(t , t) a n d S(e, e). In the l imi t ing case where A t = A e a n d E t = E e we o b t a i n S(t , t) = S(e, e) = S(t , e) = S. Express ions for the sp in po la r iza t ion , <S~)d a n d orb i t a l po la r i za t ion , ~l~)d were g iven in ref. [22] for [100] a n d [111] easy axes of magne t i za t i on . W e recall that , for the case of G d 3+

~s~) d = ~aoboN~l~Yto(J))[3S( t , t ) + 2 S ( e , e ) ] . (A .3)

S imi la r express ions were der ived for the q u a d r u p o l a r po l a r i za t ion of the c o n d u c t i o n electrons. We o b t a i n e d

[100] : ~l 2 - 2)d = ~va2c2d2 [3S( t , t) + a S ( e , e)] ( Y 2 0 ( J ) )

+v~a4c4d4[S( t , t ) - S ( e , e ) ] { ~ Y 4 o ( J ) - ~ o ~ [Y44(J)+ Y 4 _ 4 ( J ) ] > } , (A .4)

[111]: <12 - 2)d = v~a2c2d2 [3S( t , t) + a s ( t , e)] ~ Y 2 0 ( J ) )

+ ~(i~a,cad4[ S ( t , t) - S ( t , e)] ( ~ - Y40(J ) + ½ ~---70 [ Y43(J ) - Y4-3( J ) ] > }-

(A.5)

where in (A .3 ) - (A .5 ) d 2, d4, N01l are r ecoup l ing coeff ic ients de f ined in ref. [22] a n d ~ Y ~ ( J ) ) are average va lues of spher ical t ensor opera to r s o n the R E 4f g r o u n d state.

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