Neodymium

Section 4.3 Section 4.3 deals with measurements on anomalous lanthanide systems, where significant departures from the normal systems are encountered

4.2.2. Neodymium

Stirling et al. (1986) report seeing a peak at 240 meV which decreases smoothly with K. A more detailed investigation has confirmed that the transition at 237 meV is the 419/2--->4Ill/2 transition, since the K-variation of the intensity is in good agreement with a free-ion calculation (McEwen 1988). The corresponding value in Nd3+:LaF3 is about 234.4 meV (Carnall et al. 1989). The 419/2----* 4113/2 transi- tion is at about 479 meV. At that energy, a weak feature is seen in the neutron spectra with an integrated intensity consistent with the calculated estimate (fig.

3). However, it is on the limits of the observable.

Q) D O 3

O 0) E

161

10 -2

10 -3 _

4]:%

~ 419, 2--~ 4Ii1,2 "~

4 I9/2-~ 4113/2

/ I i

5 10

Wavevector K (~-1)

1(3-41

0 15

Fig. 3. Neutron inelastic structure factors for intermultiplet transitions in neodymium. The lines represent the calculated intensities of the 419/2--->419/2 (smooth line), 4Ial/2 (dashed line) and 4113/2

(chain line) transitions. The structure factors are normalised to the

419/z--->419/2

intensity at K = 0,

which has a cross-section of 634 mb sr -1. The filled circles are measured intensities of the 237 meV transition using incident energies of 515 and 828 meV. The open circle is the measured intensity of a

small peak observed at 479 meV.

24 R. OSBORN et al.

4.2.3. Samarium

The atomic form factor of 8m 3+ ions has a maximum at non-zero i¢ as a consequence of the antiparallel alignment of L and S. The orbital contribution to the form factor is more extended in K-space than the spin contribution, and the partial cancellation of the two components produces a minimum at K = 0. By contrast, the structure factor of the 6F5/2----> 6F7/2 spin-orbit transition is calcu- lated to fall monotonically with increasing K. This has been verified in measure- ments on polycrystalline samples of the trivalent samarium compound, SmPd3, and samarium metal, both performed on HET (Williams et al. 1987, Needham 1989). A major problem with neutron experiments on samarium systems are the resonant absorption peaks at 100 and 870meV in 1498m. Unless sufficient quantities of the expensive 1548m isotope are available, the only option is to make use of a transmission window between 400 and 700 meV, within which the absorption is less than 100 b. Using an incident energy of 611 meV, Williams et al.

(1987) successfully observed the spin-orbit peak at an energy transfer of 129 meV, about 2.5meV less than the value in Sm3+:LaF3 (Carnall et al. 1989). The K-variation of the measured intensity is once again in good agreement with the calculation, although this time there is no independent normalisation of the intensity. The results of measurements on samarium metal, made under identical experimental conditions, are similar, though the spin-orbit peak is much broader.

The full widths at half maximums (FWHMs) are 17 and 30 meV for SmPd 3 and Sm, respectively. The instrumental resolution is estimated to be about 12 meV.

This is clear evidence for intrinsic broadening of the transitions, probably because of the crystal-field splitting of the excited level.

4.2.4. Terbium

There are no unambiguous observations of the 7F 6-~ 7F 5 transition in terbium metallic systems. Stirling et al. (1986) measured a sample of terbium metal which was strongly contaminated with hydrogen, producing three strong vibrational peaks at 132, 256 and 370 meV. They interpreted an anomalous K-dependence of the central peak as evidence of magnetic scattering, which is consistent with the value of 254 meV for the transition energy in free Tb 3+ ions (Martin et al. 1978).

4.2.5. Thulium

Trivalent thulium (f12) is the "hole" analogue of the pr3+f 2 configuration. The splitting of the 3H term is therefore the inverse of praseodymium with 3H 6 as the ground state and 3H 5 as the first spin-orbit-split level at 1017 meV in Tm 3+:LaF 3 (Carnall et al. 1989). An indication of the strength of ~4t in the heavy rare-earths is that this is not the lowest transition energy, as spin-orbit-split levels from the 3H and 3F terms have crossed with the 3H6--~ 3F 4 transition occurring at only 690 meV. The only metallic system so far investigated is thulium metal. The spin-orbit transition is at too high an energy for a wide K-range to be measured, but the relative intensities of all the observed transitions (including the Coulomb transitions, see sect. 5) are in good agreement with calculation. Another con- sequence of the strength of ~4f is that the J = 4 levels are strongly mixed. This has

INTERMULTIPLET TRANSITIONS 25 the effect of increasing the intensity of the J---~ J - 2 (i.e., 3H 6--~ 3H4) transition, making it readily observable at 1560 meV.

4.3. Anomalous lanthanides

4.3.1. Samarium and europium intermediate valence compounds

In sect. 4.2.3, SmPd 3 and samarium metal were treated as normal lanthanides since their magnetic properties are consistent with those of stable trivalent ions.

However, samarium ions are never far from a valence instability, and compounds such as SmB 6 and Sma_xYxS are amongst the most extensively studied of intermediate valence systems (Lawrence et al. 1981). Nevertheless, there have been only a few neutron scattering studies, mainly because of the need for 154Sm-enriched samples. To date, neutron scattering investigations have been confined to SmS (Shapiro et al. 1975, McWhan et al. 1978), Sml_xYxS (Mook et al. 1978, Holland-Moritz et al. 1988, Weber et al. 1989) and a rather inconclusive investigation on StuB 6 (Holland-Moritz and Kasaya 1986). Unfortunately, a similar absorption problem has also restricted neutron investigations of europium intermediate valence systems to a single study of EuPdzSi 2 (Holland-Moritz et al.

1987). This is a pity because in principle, neutron studies of spin-orbit transitions can provide fundamental information on the hybridisation between the two nearly degenerate configurations since they are usually at a high enough energy not to be washed out by the mixing interaction but not so high that the lifetime broadening from the spontaneous charge fluctuations cannot be resolved.

The spin-orbit splitting of the Sm 2+ ion is the smallest of all the lanthanide ground state configurations. The 7F 0----> 7F 1 transition of the 4f 6 configuration is at only 36 meV and so gives rise to a significant Van Vleck contribution to the magnetic susceptibility at room temperature. Shapiro et al. (1975) measured the dispersion of the 7F 1 level in a single crystal of SmS and showed that the system was an ideal example of the paramagnetic singlet-triplet model. The observed peaks are not intrinsically broadened and, using a mean-field random-phase approximation (RPA) model, the derived energy of the single-ion spin-orbit splitting is identical to the free-ion value, within the experimental error (Martin et al. 1978). The temperature dependence of the cross-section is also consistent with the RPA model (Mook et al. 1978). There is therefore nothing to suggest the proximity of a valence instability, even though SInS undergoes a first-order phase transition from a divalent semiconductor to a mixed valent metal at a pressure of only 6.5 kbar.

Hirst (1970, 1975) has discussed the conditions under which the intra-ionic excitations remain sharp even though the compound is close to a configuration instability, or crossover, the condition being that

El, + A < Efn-1 -k ~c, (38)

where Efn is the energy of the lowest level, i.e., the Hund's rule ground state in the fn configuration, A are the energies of the excited intra-configuration excita- tions for the fn configuration and Ec is the energy of the bottom of the empty conduction band. In SmS, n = 6 and A corresponds to the energies of the

26 R. OSBORN et al.

7F 0 ~ 7F~, 7F 2 etc. transitions. Equation (38) simply states that a particular ionic level of the f6 configuration will only show broadening provided that the decay into a level of the f5 configuration, with the emission of a conduction electron, is energetically allowed. Configuration crossover occurs when the inequality is no longer satisifed even for the ground state of the

f6

configuration, ie., with za = 0, or rather when the difference in the energies of the two configurations is less than

akf = ~ [ V , [ ~ , (39)

where

Iv.I

is the hybridisation matrix element between the f electrons and the conduction band, and p is the density of states of the conduction band.

In SmS, configuration crossover is induced by applying pressure which has the effect of lowering the conduction band, the transition being first order because of coupling to the lattice. In a second neutron scattering investigation under pressures of up to 10 kbar, the spin-orbit peak disappeared in the collapsed high-pressure phase, with no evidence of additional magnetic scattering, such as crystal-field splittings, from the 4f 5 configuration (McWhan et al. 1978). It cannot be ruled out that the spin-orit transition is still present but is considerably broadened. There was no evidence of the approach of the instability, the spin-orbit peak remaining resolution limited right up to the transition. It seems therefore that the 7F 0---> 7F 1 transition is at too low an energy to ever satisfy eq.

(38). However, higher energy transitions, J = 0--~ J = 2, 3, 4, 5, 6, may show precursor effects below 6.5 kbar (Hirst 1975). The latter are all non-dipolar, with the 7F0--~ 7F 2 transition being particularly weak, but the 7F0---~ 7F 3 transition at 185meV is reasonably strong at intermediate values of K (table 4) so this experiment may be practicable.

Whilst no magnetic scattering has been identified in the mixed valence phase of SmS, it has been seen in Sm0.75Y0.258 by Mook et al. (1978) who observed a very broad peak with a F W H M of 15 meV at 30 meV. The addition of yttrium has a similar effect to that of pressure in contracting the lattice. At this composition, the valence changes with temperature from 2.3 to 2.45 at about 200 K (Weber et al. 1989). This means that the compound is at a configuration crossover with the f level effectively pinned at the Fermi level ( E l , - E~n-~ = Ev) such that the ground state consists of a hybridised mixture of the two configurations. The valence is then determined by thermodynamic considerations in which the ionic excitation

TABLE 4

Coefficients of radial integrals in the neutron scattering cross-sections of Sm2+, J = O - - - - ~ J ' =

1, 2 , . . . , 6 transitions.

J' (Jo) 2 (Jo)(J2) (J;) 2 (J2)(J4) (J4) 2 (J4)(J6) (J6) 2

1 2.00000 -3.33333 1.38889 0.00000 0.00000 0.00000 0.00000

2 0.00000 0.00000 0.05556 0.00000 0.00000 0.00000 0.00000

3 0.00000 0.00000 0.79365 0.47619 0.07143 0.00000 0.00000

4 0.00000 0.00000 0.00000 0.00000 0.22727 0.00000 0.00000

5 0.00000 0.00000 0.00000 0.00000 0.33058 0.05510 0.00230

6 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.88384

INTERMULTIPLET TRANSITIONS 27 structure of each configuration plays an important role (Wohlleben 1984). For instance, at high temperatures (kBT ~ A~f) the valence adjusts to maximise the entropy from the two configurations, which are determined by the excitation energies and their degeneracies. The intensity of the peak seen by Mook et al.

falls faster with temperature than the RPA model used for SmS predicts, in particular at the valence transition. This is mostly accounted for by the reduction is Sm 2÷, though not entirely. Nevertheless, this seems to be the first neutron scattering evidence for the persistence of intra-ionic correlations in a strongly mixed valent phase.

The observation of large widths of intraconfigurational peaks is not surprising since they are expected to be broadened by some function of A~f. The widths also depend on the precise valence of the compound, the degeneracy of the levels (Wohlleben and Wittershagen 1985) and the effects of projecting the interconfigu- rational mixing matrix elements on to the ionic ]L, S, J) basis (Hirst 1975). This can lead to different widths for different ionic transitions. More surprising is that the peak is shifted to lower energies than in SmS and the free ion, even after allowing for dispersion. From table 3 it can be seen that the intermediate valence compounds Sm0.75Y0.25S and EuPdzSi 2 are the only systems to show a significant renormalisation of the spin-orbit splitting. Indeed this may be a signature of intermediate valence, because the 4f wavefunction close to the ionic core is only affected by substantial mixing of two different electronic configurations (see sect.

2.1.2). It is interesting to note that the spin-orbit coupling in Sm 3÷ ions is lower than in Sm 2÷ : ~4f = 191 meV and 216 meV, respectively. Using the lower value, the J = 0--~ J = 1 transition would occur at 32 meV which is where it is centred in more recent measurements (Holland-Moritz et al. 1988). This agreement is, of course, fortuitous but it does give some insight into the nature of the hybridised ground state wavefunction.

There appears to be a strong coupling to optic phonons in Sm0.75Y0.25S (HoUand-Moritz et al. 1988) which could also lead to an effective energy shift but single-crystal experiments are required to clarify the issue. Incidentally, Weber et al. (1989) plot the K-dependence of the transition intensity and compare it to the Eu 2+ atomic form factor. The correct comparison is with the 7F0----~ 7F 1 inelastic structure factor and gives an equally good agreement. However, the K-range is not sufficiently extended to be sensitive to any differences between the two.

Whilst Weber et al. have succeeded in identifying quasielastic scattering from the Sm 3÷ configuration with a linewidth of about 7 meV, there has been no study of the Sm 3+ spin-orbit peak at 130meV. Since the transition is well above any phonon energies, its observation would be useful in determining to what extent the resonant phonon coupling is responsible for the renormalisation of the Sm 2÷

spin-orbit level.

EuPdzSi 2 is an intermediate valence compound in which Eu undergoes a shift in valence from 2.33 at 170K to 2.76 at 130K (Kemly et al. 1985). Eu 2+ is isoelectronic to Gd 3÷ which has an 8S7/2 ground state, i.e., L = 0 . There are therefore no spin-orbit transitions, and the first excited level is at nearly 3.5 eV (Martin et al. 1978). Eu 3+ has a 7F 0 ground state separated by only 46 meV from the 7F 1 level in Eu:LaC13 (Martin et al. 1978). At low temperature, Holland-

28 R. OSBORN et al.

Moritz et al. (1987) observe an excitation at 38meV in EuPd2Si2, but this disappears at the valence transition consistent with the reduction in Eu 3+ charac- ter in the high-temperature region. This drop in intensity with temperature is correlated with a large increase in quasielastic scattering arising from fluctuations within the Eu 2+, 8S7/2 ground state. In this instance, it is unlikely that the reduction in the peak position from the stable trivalent value is due to phonon coupling since the peak is quite sharp and shows no extra structure. The scale of the shift is similar to Sm0.75Y0.25S and could therefore also be due to a renormalisation of ff4f, but is unusual since its value (8 meV) is much larger than the intrinsic linewidth (2 meV). It does not appear reasonable to ascribe the shift to an interconfigurational energy difference since this would require the neutron spectra to be dominated by transitions between the

f7

and f6 configurations rather than within them.

4.3.2. Cerium heavy-fermion compounds

The underlying physics of heavy-fermion and intermediate valence phenomena is believed to be similar, both involving hybridisation between the localised 4f electron wavefunctions and the conduction band whilst preserving the strong intra-atomic correlations of the 4f electrons. In spite of this, charge fluctuations are not significant in the cerium heavy-fermion compounds, all of which are close to being integral valent (R6hler 1987). Instead, the hybridisation produces an effective antiferromagnetic exchange coupling between the localised and itinerant electrons (Schrieffer and Wolf 1966) which can either suppress the magnetism altogether, through spin fluctuations, as in the Kondo effect, producing a Fermi liquid of extremely heavy quasiparticles, or lead to magnetic order through an RKKY-like mechanism. Since the characteristic energy scale of the spin fluctua- tions, usually called the Kondo temperature, TK, is only of the order of 10 K, the spin-orbit transition at 279 meV is not of thermodynamic significance. In that case, the hybridisation is confined within the sixfold-degenerate 2F5/2 ground state, or even a subset of that manifold if the crystal-field splittings are larger than T K. On the other hand, this does not apply to cerium intermediate valence systems, such as a-Ce and CeRu2, in which the spin fluctuation temperature may be several thousand Kelvins and the number of f levels available for hybridisation is closer to 14.

Whilst there have been several theoretical investigations of the effect of hybridisation on the crystal-field excitations within the ground multiplet (Maekawa et al. 1985, Lopes and Coqblin 1986), there have been relatively few in which the spin-orbit level is explicitly included. Cox et al. (1986) have shown, in the context of the Anderson impurity model, that when T K is comparable to the spin-orbit splitting, the inelastic peak is broadened and shifted to lower energies.

Given that the cross-section is weak, at about half the intensity of the praseodymium spin-orbit cross-section, they concluded that the transition was unlikely to be seen except in heavy-fermion compounds with low values of T~.

This appears to be confirmed by the failure to observe such a transition in CePd 3 in recent measurements on H E T (Osborn, unpublished). On the other hand, the

INTERMULTIPLET TRANSITIONS 29 spin-orbit transitions in low-T K materials should not be dramatically shifted from their flee-ion values.

Osborn et al. (1990) have observed the 2F5/2---~ 2F7/2 spin-orbit transition in CeAI 3 (fig. 4), one of the canonical heavy-fermion materials characterised by an especially large electronic specific heat coefficient at 1620 mJ mo1-1 K -2 (Andres et al. 1975). T K is estimated to be about 5 K (Murani et al. 1980) so a well defined spin-orbit peak was anticipated, especially since the compound has a low nuclear scattering cross-section and therefore low multiple scattering backgrounds. The spin-orbit peak is split into two peaks at low K at 260 and 291 meV with identical intensities and F W H M s of 49 meV, compared to an estimated resolution width of 12 meV. This is a surprising result because there are no other instances of a splitting of a spin-orbit peak in neutron spectra so far. In general, the instrumen- tal resolution is not good enough to reveal crystal-field splittings of excited multiplets, particularly in metals where the crystal-field potential is well screened and hence rather weak. Overall splittings of the ground state multiplet are commonly in the range 10-20 meV with similar values expected for the spin-orbit levels. In CeA13, the ground state multiplet is split by just over 7 meV and, although the crystal-field potential has not been solved (in particular, the sixth- degree crystal-field parameters cannot be estimated from the ground multiplet splitting) it is unlikely to produce such a large splitting of the 2F7/2 multiplet.

The splitting of the spin-orbit level may be even greater than 30 meV since there is some evidence of inelastic scattering at higher energies (up to 360 meV) emerging at higher K. Whereas the peaks at 260 and 291 meV are of dipolar character, the higher energy scattering is clearly non-dipolar. This could arise

I I I

5 3 x5

L.A

-g 1-

I I I

200 300 4 0 0

~hw [meV]

Fig. 4. Neutron inelastic scattering from CeA13 at 20 K, measured at an angle of 5 ° with an incident neutron energy of 600 meV on HET. The data have been fitted by two Gaussians and a tail of low-energy scattering.

30 R. OSBORN et al.

from scattering from hydrogen impurities, but there does not appear to be any evidence of other vibrational overtones. Non-dipolar transitions are possible between the two multiplets, even though the total cross-section is dipolar. They arise from transitions between crystal-field levels in the two multiplets in which AM > 1, e.g., 2F5/2, M = 1 ~ 2F7/2 ' M = -~.

Goremychkin and Osborn conclude that the crystal-field potential acting on the spin-orbit level is larger than the potential acting on the ground state level. Since extensive optical studies of the crystal-field splittings in ionic systems have shown that a unique potential can be used to analyse the splittings in a large number of different multiplets with only minor discrepancies (Carnall et al. 1989), this observation suggests that a radically different mechanism for the crystal field is in operation, one that can vary from multiplet to multiplet. One possibility is that the crystal-field potential is due to the same hybridisation mechanism that produces the low-temperature Fermi liquid behaviour. It is already known that the hybridisation-mediated exchange between the f electrons and the conduction electrons, which will reflect the point group symmetry of the cerium ion, can give rise to a large contribution to the crystal field (Wills and Cooper 1987, Levy and Zhang 1989). The magnitude of the exchange coupling is given by (Schrieffer and Wolf 1966)

u (40)

3-kf = 2]Vkf]2 Efn(Ef, + U) '

where Vkf is the hybridisation matrix element, El, is the energy of the f level and U is the intra-4f Coulomb repulsion. The energies are with respect to the Fermi energy, so with El, negative and El, + U positive, 3-kf represents an anti- ferromagnetic exchange coupling. Equation (40) helps to explains why hybridisa- tion exchange is so important in cerium systems, Firstly, IVkt] is large because of the relatively expanded character of the cerium 4f wavefunction, and secondly El, is small, though not so small that the Schrieffer-Wolf transformation is invali- dated.

The other interesting feature of eq. (40) is that the absolute value of El, will be lower for t h e 2F7/2 multiplet than for t h e 2F5/2 level so that 3-kf is enhanced. The widths of the excited level will also be increased by any increase in ~-kf as is observed. One corollary of this suggestion is that 3-kf is probably the dominant component in the crystal-field splitting of the ground state manifold as well, in agreement with the suggestion of Levy and Zhang (1989).