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High energy magnetic inelastic neutron scattering at ISIS
Physica B 159 (1989) 151-160 North-Holland, Amsterdam
HIGH ENERGY MAGNETIC INELASTIC NEUTRON SCATTERING AT ISIS R. OSBORN Neutron Science Division, Rutherford Appleton Laboratory,
Chilton, Didcot, Oxon OX11 OQX, UK
Received 20 January 1989
Two recent experiments on the HET spectrometer of the UK spallation neutron source ISIS are described, as examples of the potential of high energy magnetic neutron scattering. The measurements of intermultiplet transitions with energies up to 810 meV in praseodymium metal give direct evidence of the screening of the intra-4f electrostatic interaction for the first time in a metal. The results are discussed in the light of atomic orbital calculations of the transition intensities. Crystal field splittings in UO, establish both the symmetry of the ordered phase and the degree of J-mixing in the ground state wavefunctions
1. Introduction The magnetic dipole moment of the neutron makes it an extremely powerful probe of the dynamic magnetic susceptibility of condensed matter. The scattering cross-section is directly related to the wavevector and energy dependent magnetic response functions of theoretical interest, so that the information obtained from neutron scattering data is largely model-independent [l]. In recent years, the energy range explored by inelastic scattering measurements has been extended beyond the thermal energies (a few tens of meV) available on conventional reactor sources [2], to several hundreds of meV Whilst there have been some important experiments in this energy domain on reactor ‘hot’ source spectrometers, mainly in single crystal studies of itinerant magnetism, the development of high energy magnetic scattering is mainly due to the advent of a new generation of accelerator-based pulsed neutron sources, which are characterised by a substantial epithermal flux extending well into the eV range. In this paper, I will discuss some recent results obtained on the High Energy Transfer spectrometer HET of the UK spallation neutron source ISIS, which is now the most intense pulsed neutron source in the world. The experiments reported here are both on
isotropic scattering from localised f-electron systems. Although this only forms a part of the experimental programme of HET, the two examples illustrate the potential of high energy neutron scattering very well. On the one hand, the results on the intermultiplet splittings of praseodymium represent a dramatic increase in the energy scale so far accessible even on pulsed sources, with magnetic peaks measured above 800 meV [3]. The peak energies give information on the screening of the intra-4f electrostatic potential for the first time in a metal and open up new avenues of research, some of which are outlined in this paper. On the other hand, the measured crystal field splittings of UO, [4] show the necessity of good resolution even at these energies. The splittings of the cubic crystal field levels by the combined effects of lattice distortions and exchange interactions are only of the order of a few meV compared to an overall splitting of over 1.50meV. Calculations show that the splittings depend on whether the ordered phase is 2-k or 3-k, with the measurements indicating that the 3-k structure is correct [5]. In section 2, I will briefly describe HET and discuss some of the problems associated with high energy neutron scattering. There follows in sections 3 and 4, a report on the recent measurements on praseodymium and UO,, respec-
0921-4526/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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tively, with a discussion of what such experiments can reveal. There is a brief conclusion in section 5.
2. The high energy transfer spectrometer HET is a direct geometry chopper spectrometer i.e. the incident beam is monochromated and the scattered energy is determined by the measured time-of-flight [6]. Incident energies, in the range 30 to 2000 meV, are selected by phasing a fast Fermi chopper to the pulse of protons on the spallation target. High resolution, in neutron terms, is achieved by matching the burst time of the chopper, which spins at 400 to 600 Hz, to the intrinsic time resolution of the neutron pulse. The pulse width is a function of the moderator design; in the case of HET, it has been tightened at lower energies by the use of a poisoning layer, giving an effective moderator depth of 1.5 cm. The resolution is then in the range 1 to 2% of incident energy depending on the energy transfer and scattered flight path (fig. 1). The scattered neutrons are detected by two arrays of ‘He detectors, one lying at 4 m from the sample
hw/E, Fig. 1. Fractional lated for the 2 and of LO/E,, E, being necessary to obtain the intersections straight lines.
energy transfer resolution fi Am/E, calcu4 m detector banks of HET as a function the incident energy. The values of fiw/E, various percentages of Aoio are given by between the resolution curves and the
covering scattering angles from 3” to 7”, the other being at 2.5 m from the sample at scattering angles of 9” to 29”. A third 4 m detector array at 136” is used, in the context of this paper, to measure the shape of the phonon spectrum. In any discussion of high energy magnetic scattering, it is necessary to consider the severe kinematic constraints on the measurements. These arise from the need simultaneously to have a large scalar difference in the incident and scattered wavevectors (i.e. large &J = 2.07(k’ - k:), where ho is the energy transfer in meV and ki and k, are the incident and scattered neutron wavevectors in A-‘), whilst keeping the vector difference relatively small (i.e. small Q = ki - k,). The restriction to small values of Q is required by most magnetic scattering measurements, because their intensities are modulated by the atomic form factor, which falls approximately as exp(- (YQ’). For rare earth and actinide systems, (Yis in the range 0.04 to 0.07 with Q in A-‘. It is clear that scattering angles 4 must be kept small, which is why the main detector arrays of HET are concentrated at low angles. However, even as d, -+ 0, the limitations are severe. As discussed in more detail by Loewenhaupt [7], in order to obtain a particular value of Q, the necessary incident energy increases as the square of the maximum energy transfer. On HET for instance, keeping Q to 4 A-‘, an energy transfer of 100 meV requires an incident energy of only 135 meV Energy transfers of 400 meV require incident energies in excess of 2000 meV. As will become clear in the examples reported here, it is often possible to relax the constraint on the value of Q. The cost in form factor may be offset by the improved resolution, obtained by using a lower incident energy, and the reduced background. Noise levels on pulsed source spectrometers are intrinsically low, because the main neutron production occurs at one time. The background from fast neutrons, which are difficult to shield, only arises from the small delayed neutron fraction. The main source of background is then from the sample itself in the form of multiple scattering. In samples with a strong nuclear cross section, this can be a serious problem even well above the maximum single phonon
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energies, since multiple inelastic processes are appreciable. Fortunately, multiple scattering at high energies usually consists of a broad decaying tail so that sharp magnetic peaks from localised excitations can be readily identified. Broader magnetic responses, such as occur in spin fluctuation systems, require a more careful estimate of the scale of the multiple scattering, by for example using Monte Carlo techniques [8]. Nevertheless, experience has shown that, even when measuring sharp peaks, the best course is to minimise the amount of multiple scattering by selecting the lowest incident energy compatible with the estimated cross section.
3. Intermultiplet transitions in praseodymium One of the first uses of the epithermal spectrum of pulsed neutron sources has been to explore the multiplet structure of rare earth metals and compounds. This is a natural extension of the work already performed on the crystal field potential of these systems using reactor spectrometers over the last two decades [2, 91. Whilst the crystal field splittings are a measure of the relatively weak interactions of the 4f electrons with the surrounding lattice, the splittings of the different *‘+lL, levels result from the much stronger interactions within the 4fN configuration itself. As demonstrated by optical spectroscopy on ionic systems, they provide direct information on the dominant intra-4f electrostatic and spinorbit interactions, as well as on weaker higherorder terms [lo]. Indeed, the observation of a well defined multiplet structure is important in establishing the localised nature of the 4f shell in a metallic environment, and validates to a large degree the approximation of the 4f electron states by atomic wavefunctions. Given the undoubted success of the atomic model in explaining both bulk magnetic measurements and low energy spectroscopy in the lanthanides, it was not anticipated that measurements of intermultiplet transitions would require a radical change in the accepted picture. However, the results to date are beginning to set limits on how
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far the model can be taken without appreciable modification. There are now a number of reported observations of the J-, J + 1 transition in metallic systems, in Sm2+ (‘F, +‘F, at 36meV) [ll], Eu3+ (7F,,+ ‘F, at 46.5 meV) [12], Pr3+ (3H,+ 3H, at 261 meV) [3, 13, 141 and Sm3’ (6H5,2+ 6H7,2 at 124 meV) [15] ions, the first two on reactor spectrometers, the last two on pulsed source spectrometers at IPNS and ISIS. In all cases, the energies are very close to those determined by optical spectroscopy in both the free-ion and ionic compounds. That the energies of these transitions are similar in such a range of systems is not altogether surprising. The spin-orbit splitting is approximately given by &r(J + 1) /2S for J+ J + 1 transitions (5,,J/2S for J+ J - l), where the spin-orbit parameter &, is largely determined by the form of the 4f-electron wavefunctions in the vicinity of the ionic core. It is not therefore very sensitive to the local environment and so provides little new information. The main value of these neutron experiments has been to show that the measured transition intensities are in good agreement with the calculations of Balcar and Lovesey (BL) [16]. Having a reliable framework with which to predict the magnitude and Q-dependence of the neutron scattering is vital to the reliable identification of high energy magnetic peaks, which may occur at similar energies to hydrogen impurity vibrational modes, for example. The results of a recent experiment on the spin-orbit transitions in praseodymium metal at ISIS [3] are compared with the BL calculation in fig. 2. BL express the neutron cross section of intermultiplet transitions, summed over all states in both manifolds, as
x ~(LIJ + E, - E,) , where Ei and E, are the incident and scattered neutron energies, ri = 0.29 b, ho = Ei - E, and Q is the scattering vector. The structure factor,
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/
0
I
I
I
5
10
15
WAVEVECTOR
Q &‘I
Fig. 2. Neutron inelastic structure factors for transitions within the ‘H term. The lines represent calculated intensities of the ‘H,+ ‘H,, 3H, and 3H, transitions. The structure factors are normalised to the ‘H, -+ 3H, intensity at Q = 0, which has a cross section of 620 mb/sr. The points represent the intensity of the 260meV transition measured with incident energies of 370meV (filled square), 515 meV (open triangles), 830meV (open squares) and 13OOmeV (filled circles).
G( Q; EL,v), for transitions between “+lL, levels labelled by the quantum numbers p and v, may be expressed as a function of the Q-dependent n = 0,2,4,6, atomic radial integrals, (j,), which have been tabulated for all the rare-earths by Freeman and Desclaux [17]. G(Q; p, v) contains both dipolar and higher multipolar terms, only the former being finite at Q = 0. In the ‘dipole’ approximation (i.e. valid as Q-0) the intensity is proportional to (( j,) - ( j,))'. Because of the negative sign of the ( j,) ( j,) coefficient, the spin-orbit transition intensity falls much more sharply with increasing Q than the form factor characterising the Q-dependence of intra-multiplet scattering. In addition, it is an order of magnitude weaker at Q = 0. This can be seen by comparing the 3H,+ 3H, structure factor of Pr3+ with the calculated intramultiplet (3Hq+= ‘Hq) cross section (fig. 2). Fortunately, the strength of the quadrupolar terms in the cross-section produces a plateau region from about 7 to 10 A-’ but the intensity is still very
small. The ability to measure such weak scattering is mainly due to the extremely low noise levels achievable on pulsed source spectrometers, in which the neutron source is effectively ‘turned off’ at the time the neutrons are being counted. To achieve the lowest values of Q in the structure factor measurements shown in fig. 2, it was necessary to use an incident energy of 1300meV. According to the results of optical spectroscopy, the 3H,+ 3F, (n = 2,3,4) transitions are in the range 600 to 850meV [lo] and fig. 3 shows that three peaks are indeed observed at 578, 747 and 809 meV. The justification for assigning these peaks to the three 3H,+ 3F transitions, and for neglecting the 3H,+ 3H, transition, is given in more detail by Taylor et al. [3]. Since only J+ J * 1 transitions within a “+‘L term are dipole-allowed, these transitions are non-dipolar, i.e. their intensities fall to 0 as Q --, 0 and are predicted, by an extension of BL’s calculations, to reach a broad maximum at around 8 A-‘. The 3H,+ 3F, transition intensity was measured in the range Q = 6.5 to 12.5 A-’ with only a small drop in signal. Unusually for magnetic scattering, the peak intensities do not Q (A-‘)
0
200 ENERGY
400 TRANSFER
600
800
1000
ImeV)
Fig. 3. Neutron scattering cross section of praseodymium at 17 K, measured at an angle of 5” with an incident neutron energy of 1300meV. The change in the scattering vector across the spectrum is shown along the top axis. The data have been fitted by 4 Gaussians and tail of low energy scattering.
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fall sharply with Q; indeed, it is inadvisable to make such measurements at too low a scattering wavevector for two reasons. Firstly, the cross section falls rapidly below 3 A-’ and, secondly, the requisite high incident energy degrades the resolution and background, as discussed earlier. Whilst the J+ J + 1 transition is hardly shifted in energy from the measured free ion value, the three 3H+= 3F transitions are all lower by about 40 meV This observation can be readily explained in terms of the parametrised Hamiltonian used to model the effect of the intra-4f Coulomb interaction. In addition to J&, three extra parameters, F@), Fc4) and Fc6), are required [lo]. In principle, these are defined as radial (Slater) integrals over the 4f-electron wavefunctions. However, Hartree-Fock calculations substantially overestimate their magnitude by up to a factor 1.5, even in the free ion, so they are usually treated as empirical parameters. Providing the spin-orbit coupling is reasonably small, F(*), Fc4) and Fc6) determine the energies of the Russell-Saunders “+‘L terms whilst [,, determines the relative splittings the different J manifolds within each term. It is possible, to first order, to shift the energies of all three 3F levels rigidly up and down by adjusting the values of the Slater integrals. The transition energies are well reproduced by reducing the F(*) parameter by 10% from its free ion value, keeping all other parameters fixed. A reduction in the effective Slater integrals corresponds to a decrease in the electrostatic potential between the two 4f electrons of the Pr3+ ion. This is the first time that this screening effect has been observed in a rare earth metal. However, it has been observed in ionic systems by optical spectroscopy [lo]. For instance, a reduction of about 20 meV in the 3H* 3F transition energies has been observed in Pr3+ doped in LaCl, [18]. Slater parameter shifts are observed in the spectra of transition metal ions as well as 4f and 5f ions and it is possible to establish a correlation between the size of the shift and the type of ligand, producing the ‘nephalauxetic’ series (e.g. AF(Pr : LaCl,) > AF(Pr : LaF,), etc.) [19]. Moreover, the observed shift has been shown to be proportional to the polarisability of
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the ligand involved, and therefore presumably to its ability to screen any electrostatic potential within the rare earth ion. It would be interesting to see if a similar sensitivity to the surrounding lattice could be observed in metallic systems where the screening mechanism is more effective. Another striking feature of the experiment is that the measured intensities of the three 3H+ 3F. transitions are nearly a factor two lower than the calculations of BL, when normalised to the 3H,-+ 3H4 scattering. Moreover, we also failed to observe the 3H,-+ ‘G, transition at about 1170 meV for which the calculated cross section is comparable to the 3H4+ 3F2 transition, and sufficiently large to have been measurable. This discrepancy is in marked contrast to the excellent agreement in the case of the 3H4+ 3H, transition. There are a number of possible reasons that need to be explored further. For instance, Balcar [20] has pointed to the need to perform the calculations fully relativistitally. The 3H and 3F states will have different radial wavefunctions because of the differing amounts of i = 5 /2 and j = 7 /2 character whereas the BL method assumes them to be the same. It is difficult to estimate the magnitude of this effect because it involves the numerical calculations of several radial integrals which are factorised out in the semi-relativistic treatment. However, it would seem surprising if it could have such dramatic consequences. Another possibility is that the reduction in intensity is related to the cause of the Slater parameter shift. I have already mentioned that the absolute value of the Slater parameters are severely overestimated by Hartree-Fock calculations. This is known to be due to the neglect of configuration interactions, i.e. perturbations by electron configurations other than 4fN, which give rise to terms in the Hamiltonian which have the same symmetry as the bare intra4f Coulomb potential [21]. In other words, most of the effect is absorbed into the empirical Slater integral parameters. There are also other parameters introduced into the Hamiltonian, but given the limited number of levels observed in the neutron scattering, we are not in a position to comment
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on them. As a consequence of configuration interactions, then, the 4f wavefunctions are hybridised with other higher-lying orbitals. The electrons partially occupy more delocalised levels and so experience a reduced mutual electrostatic repulsion. This same hybridisation may be responsible for the reduction in the 3H-+ 3F transition intensities. If this were so there should be a direct correlation between the Slater parameter shifts and the reductions in the neutron cross sections. Further neutron experiments on praseodymium in ionic systems, where the shifts are smaller than in praseodymium metal, are therefore planned. In conclusion, it seems that the main effect of a metallic environment on the 4f shell in Pr is an enhanced screening of the electrostatic interaction between the two f-electrons. The multiplet structure is preserved, at least in the energy range so far accessible. The transition peaks are broader than the instrumental resolution, though without knowing the extent of the crystal field splittings within each multiplet it is difficult to draw any conclusions from this. It would be useful to explore the other rare-earth metals to look for systematic trends in the Slater parameter shifts. Unfortunately, there are only two trivalent rare earths in which levels derived from different *“IL terms are below 1 eV, Sm3’ and Tm3+ [lo], both of which present problems of neutron absorption. However, these ions show intermediate valent behaviour in a number of systems, so any information on the degree of hybridisation of the f-shell is of special interest. Observations of the multiplet structure would also be of value in actinide intermetallic compounds, where the aim would be to establish the valence of the actinide ion. Work has already begun on USb and UTe [22], but has concentrated on the validity of proposed crystal field models in these compounds. The measured magnetic response has so far been extremely broad, stretching to 80 meV with no well-defined localised excitations about the spin wave bands, but the experiments need to be extended to much higher energy if intermultiplet transitions are to be observed. I should mention two further avenues of pos-
neutron
scattering
sible research in this field. Firstly, it should be of value to study the spin-orbit transition in cerium compounds, in particu(*Fs,* -+ 2F 7,2) lar to explore the development of the peak as a function of hybridisation. Whilst it may be anticipated that heavy fermion systems, for which the appropriate energy scales are a few Kelvin, will show a well defined spin-orbit peak, the more intermediate valent compounds should show a marked broadening of the level and possibly a renormalisation of the energy. Such questions as whether the degree of hybridisation of the spinorbit level is the same as in the ground state could then be addressed. Secondly, it may yet be possible to directly measure interband transitions. This has long been an aim of high energy neutron scattering, but the predicted cross sections have been discouragingly small. Also, since what is measured is a joint density of states, the signal will be spread over an energy range comparable to the bandwidth, in most systems a few eV. However, in some actinide compounds, such as PuTe [23], band structure calculations suggest that the fbands consist of two spin-orbit split bands (i.e. of predominant j = 5 /2 and j = 7/2 character). The cross section will be considerably enhanced by the spin-orbit coupling and spread over a limited energy range, perhaps a few hundred meV. It will probably still be a few years before such experiments are attempted, experience with more localised actinide systems is required first, but it is an interesting prospect.
4. Crystal fields in UO, Another field m which pulsed neutrons have begun to play a prominent role is the crystal field spectroscopy of actinide oxides. Crystal field splittings in these systems are frequently greater than 100 meV for a number of reasons. Firstly, oxygen ligands give rise to larger crystal field potentials than for example chlorine and bromine. The relation of this spectrochemical series with the nephalauxetic series referred to earlier is discussed by Newman [19]. Even in rare earth oxides, such as Pro, and BaPrO,,
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both of which have been studied at IPNS [24], the excitation energies are well above 100 meV Secondly, it is well known that the greater spatial extent of the Sf-electron wavefunctions of actinide ions, compared to those of lanthanide 4f electrons, leads to a factor two or more increase in the crystal field potential. Finally, within the actinide series, the tetravalent ions experience stronger potentials than their trivalent counterparts [25]. It may be thought that insulating, or in the case of UO, semiconducting, compounds would be much better studied by classical optical spectroscopy, which offers a wider energy range and generally superior resolution. However, there is a real complementarity here between the two techniques. The assignments of optical transitions in the trivalent rare earths and actinides often rely on the spectra being split into bands of crystal field excitations each derived from a different 2s+1L, level. It is then possible to fit the free ion parameters, for instance the spin-orbit parameter and the Slater integrals, first before determining the crystal field parameters. In the tetravalent oxides, such a clean separation is not possible because the crystal field potential is comparable in magnitude to the spin-orbit coupling. All the transitions are of mixed character and do not generally form identifiable bands. The only good quantum numbers are the irreducible representations of the local point group symmetry. A further problem is that the spectra are often contaminated by excitations of vibronic origin. Calculations of optical transition probabilities are particularly difficult since intra5f transitions are not allowed by electric dipole selection rules. To be measurable, they depend on small admixtures of other electronic configurations. Reliable assignments are therefore a problem. This point is exemplified by the case of UO,, whose tetravalent uranium ions have a localised 5f2 configuration [26] (cf. Pr3+ has a 4f2 configuration). Optical absorption studies had already been made of this compound [26]. Structure in the energy range 150 to 180 meV was attributed to two-phonon processes because of its anomalous temperature dependence. Peaks above
500 meV were also observed but were too numerous and evenly distributed in energy to provide a reliable comparison with the only available predictions of Rahman and Runciman [27]. It was not until the first neutron measurements at IPNS [24] that it was finally established, by the Q-dependence of their intensity, that the peaks below 200 meV were in fact magnetic. Following the more recent ISIS results [4], which were performed with better resolution, we now know that their unusual temperature dependence is due to the presence of a strong JahnTeller coupling responsible for the combined quadrupolar and magnetic phase transition at 30.8 K. As seen in fig. 4, this gives rise to a strong splitting of the cubic crystal field levels in the ordered phase. Optical spectra have since been reanalysed in the light of the neutron results [28]. Figure 5 shows how sensitive the crystal field splittings of the ground state multiplet are to both the type of potential (the ratio of V, to V,) and the presence of other multiplets. The fact that the crystal field splittings are not directly proportional to V, is a consequence of J-mixing effects. The stronger the crystal field potential, the greater the admixture of, for instance, 3F2 character into the wavefunctions of the lowest multiplet. 3F2 is the predominant component of the first excited multiplet in the U4+ free ion,
hw (mev) Fig. 4. Neutron scattering cross section of UO, at 6.5 K, measured at an angle of 5” with an incident neutron energy of 290meV. The data have been fitted by 4 Gaussians and a sloping background. The peak at lowest energy corresponds to the r, + r, transition. The remaining peaks correspond to the r,+r, transition split by the quadrupolar and exchange field of the ordered phase.
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3F~,_,_~~:_-__________-__---_‘-‘==-_-_---_~_-_~_
4oc
-vl
CmeV)
Fig. 5. Calculated crystal field splittings of UO, as a function of V, when (a) V,lV, = -0.06 and (b) V,lV, = -0.21 using the Slater integral parameters Fz = 23.73 meV, F, = 4.19 meV, F, = 0.49 meV and the spin-orbit parameter &, = 222.7 meV. The curves are labelled by the predominant 2s”L, component of wavefunctions when V, = 0. The arrows mark the magnetic dipole-allowed transitions at the value of V, estimated by (a) Rahman and Runciman and (b) Osborn et al.
predicted to be at 450 meV. It splits into levels of Is and I?, symmetry in a cubic crystal field, as shown in fig. 5. These levels, along with those derived from the free-ion 3H, state at 700 meV, effectively suppress the lowest crystal field levels derived from the 3H, multiplet. Long before there were any spectroscopic measurements available, Rahman and Runciman [27] predicted a Is ground state with the I3 crystal field level at 169 meV and the I, level at 624 meV (The Is -+ I’, transition is not neutron dipole-allowed and so will be much weaker than the other two). In arriving at this estimate, they assumed a ratio of V, to V, of -0.06 and chose V4 to reproduce the observed ordered moment of 1.8~~. The reduction from the 3H, Is moment of 2~~ is, in their treatment, a consequence of
J-mixing. It can be seen from fig. 5 that a different ratio of -0.21 produces a radically different spectrum with the two levels close to each other for all values of V,. The latest measurements at ISIS have now shown that the latter ratio is correct, with V, = - 123 meV [4], much lower than the Rahman and Runciman estimate, but in line with more recent ab initio estimates [28]. With this choice of potential, the ground state moment is not reduced by J-mixing. In fact, it is marginally increased, but this is consistent with Allen’s suggestion that the moment reduction is due to the quadrupolar coupling in the ordered phase [29]. It is clear from fig. 4 that the neutron measurements on HET show that high resolution is required in order to understand the nature of the quadrupolar coupling. The I?, triplet is split by about 10 meV by the lattice distortion of the ordered phase, whereas the I, doublet is only split by about 2 meV. This is just resolved in another scan using an incident energy of 229 meV. There are two possible models for this distortion which are compatible with neutron diffraction results [30]. In one, the moments are aligned along [llO] directions in a 2-k structure, with a monoclinic distortion of the unit cell. In the other, the magnetic structure is 3-k, with the moments aligned along [ll l] directions in a trigonally distorted crystal field. These two models give quite different predictions for the splittings of the I3 doublet and I, triplet, according to the calculations of Amoretti et al. [5]. In the former, the I3 doublet has a much larger quadrupole moment than the I4 triplet and so is more strongly split. In the latter, this situation is reversed. The neutron scattering therefore allows us to distinguish the two cases and leads to the conclusion that the 3-k model is correct. These results throw light, therefore, on two different aspects of the magnetic properties of UO,, the degree of J-mixing in the ground-state wavefunctions and the symmetry of the ordered state. On the one hand, the overall scale of the crystal field splittings shows that the Is ground state is given fairly accurately by the intermediate coupling approximation with relatively small admixtures of other J states. On the other
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hand, fine splittings of the crystal field levels are only consistent with a 3-k magnetic structure. Both are discussed in more detail in recent publications [4, 51.
5. Conclusion Given the context of this paper, it is perhaps appropriate that a recurring theme has been the complementarity of neutrons and photons, although the comparisons are with classical optical spectroscopy. In both measurements, we are exploring the nature of the interactions within the f-electron configuration and how they interact with other configurations or the crystalline environment. Neutron spectroscopy cannot hope to match the energy range of optical techniques, in spite of the increasing overlap between the two, and so the detailed consideration of high order (e.g. orbit-orbit and spin-other orbit) interactins will not be possible. However, there are two reasons why it is still important. Firstly, it is possible to make reliable calculations of cross sections and their Q-dependence on the basis of atomic calculations. This is valuable both for the purposes of peak assignments and, potentially, in determining deviations from simple atomic wavefunctions. Secondly, neutrons can be used to probe metallic systems and perhaps reveal the degree of f-electron hybridisation with the conduction electrons. This will be particularly interesting in both actinides, where the extent of intra-5f correlations is still unclear, and in intermediate valent materials. These projects are still at an early stage, but the results have so far been very promising.
Acknowledgements This work was done in collaboration with A.D. Taylor, G. Amoretti, M.T. Hutchings, K.A. McEwen and several others. I would also like to acknowledge stimulating discussions with S.W. Lovesey, E. Balcar and G.L. Goodman.
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References 111 S.W. Lovesey, Theory of Neutron Scattering from Condensed Matter, vol. 2 (Oxford Univ. Press, Oxford, 1984). PI W.G. Stirling and K.A. McEwen, in: Methods of Experimental Physics, vol. 23, K. Skiild and D.L. Price, eds. (Academic Press, New York, 1987) chap. 20, p. 159. 131 A.D. Taylor, R. Osborn, K.A. McEwen, W.G. Stirling, Z.A. Bowden, W.G. Williams, E. Balcar and S.W. Lovesey, Phys. Rev. Lett. 61 (1988) 1309. [41 R. Osborn, A.D. Taylor, Z.A. Bowden, M.A. Hackett, W. Hayes, M.T. Hutchings, G. Amoretti, R. Caciuffo, A. Blaise and J.M. Fournier, J. Phys. C 21 (1988) L931. [51 G. Amoretti, A. Blaise, R. Caciuffo, J.M. Fournier, M.T. Hutchings, R. Osborn and A.D. Taylor, Rutherford Appleton Laboratory Report, RAL-88-096 (1988); and Phys. Rev. B, in press. 161 A.D. Taylor, B.C. Boland, Z.A. Bowden and T.J.L. Jones, Rutherford Appleton Laboratory Report, RAL87-012 (1987). 171 M. Loewenhaupt, in: Proc. Conf. on Neutron Scattering in the Nineties (IAEA, Vienna, 1985) p. 435. PI M.W. Johnson, Harwell Report No. AERE-R7682 (1974). (91 M. Loewenhaupt and P. Fulde, Adv. Phys. 34 (1985) 589. WI Z.B. Goldschmidt, in: Handbook of the Physics and Chemistry of the Rare Earths, vol. 1, K.A. Gchneidner and L. Eyring, eds. (North-Holland, Amsterdam, 1978) p. 1. C.A. Morrison and R.P. Leavitt, ibid., vol. 5 (1982) p. 461. [Ill D.B. McWhan, S.M. Shapiro, J. Eckert, H.A. Mook and R.J. Birgeneau, Phys. Rev. B 18 (1978) 3623. WI E. Holland-Moritz, E. Braun, B. Roden, B. Perscheid, E.V. Sampathkumaran and W. Lange], Phys. Rev. B 35 (1987) 3122. P31 M. Loewenhaupt, Physica B 130 (1985) 347. [I41 W.G. Stirling, K.A. McEwen and C.-K. Loong, Physica B 136 (1986) 420. [I51 W.G. Williams, B.C. Boland, Z.A. Bowden, A.D. Taylor, S. Culverhouse and B.D. Rainford, J. Phys. F 17 (1987) L151. Ml E. Balcar and S.W. Lovesey, J. Phys. C 19 (1986) 4605. u71 A.J. Freeman and J.P. Desclaux, J. Magn. Magn. Mat. 12 (1979) 11. 1181 G.H. Dieke, Spectra and Energy Levels of Rare Earth Ions in Crystals (Interscience, New York, 1968). [I91 D.J. Newman, Austr. J. Phys. 30 (1977) 315. PO1 E. Balcar, private communication. PI E. Rajnak and B.C. Wybourne, Phys. Rev. 132 (1963) 280. WI R. Osbom, M. Hagen, D.L. Jones, W.G. Stirling, G.H. Lander, K. Mattenberger and 0. Vogt, J. Magn. Magn. Mat. 76 & 77 (1988) 429.
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[23] M.S.S. Brooks, Phys. Scripta 35 (1987) 742.
[24] S. Kern, C.-K. Loong and G.H. Lander, Phys. Rev. B 32 (1985) 3051. [25] W.T. Carnal], G.L. Goodman, K. Rajnak and R.S. Rana, Argonne National Laboratory Report, ANL-88-8 (1988). [26] J. Schoenes, Phys. Rep. 63 (1980) 301.
[27] H.U. Rahman and W.A. Runciman, .I. Phys. Chem. Solids 27 (1966) 1833. [28] Z. Gajek, M.P. Lahalle, J.C. Krupa and J. Mulak, J. Less-Common Metals 139 (1988) 351. [29] S.J. Allen, Phys. Rev. 166 (1968) 530, 167 (1968) 492. [30] J. Faber and G.H. Lander, Phys. Rev. B 14 (1976) 1151.
HIGH ENERGY MAGNETIC INELASTIC NEUTRON SCATTERING AT ISIS R. OSBORN Neutron Science Division, Rutherford Appleton Laboratory,
Chilton, Didcot, Oxon OX11 OQX, UK
Received 20 January 1989
Two recent experiments on the HET spectrometer of the UK spallation neutron source ISIS are described, as examples of the potential of high energy magnetic neutron scattering. The measurements of intermultiplet transitions with energies up to 810 meV in praseodymium metal give direct evidence of the screening of the intra-4f electrostatic interaction for the first time in a metal. The results are discussed in the light of atomic orbital calculations of the transition intensities. Crystal field splittings in UO, establish both the symmetry of the ordered phase and the degree of J-mixing in the ground state wavefunctions
1. Introduction The magnetic dipole moment of the neutron makes it an extremely powerful probe of the dynamic magnetic susceptibility of condensed matter. The scattering cross-section is directly related to the wavevector and energy dependent magnetic response functions of theoretical interest, so that the information obtained from neutron scattering data is largely model-independent [l]. In recent years, the energy range explored by inelastic scattering measurements has been extended beyond the thermal energies (a few tens of meV) available on conventional reactor sources [2], to several hundreds of meV Whilst there have been some important experiments in this energy domain on reactor ‘hot’ source spectrometers, mainly in single crystal studies of itinerant magnetism, the development of high energy magnetic scattering is mainly due to the advent of a new generation of accelerator-based pulsed neutron sources, which are characterised by a substantial epithermal flux extending well into the eV range. In this paper, I will discuss some recent results obtained on the High Energy Transfer spectrometer HET of the UK spallation neutron source ISIS, which is now the most intense pulsed neutron source in the world. The experiments reported here are both on
isotropic scattering from localised f-electron systems. Although this only forms a part of the experimental programme of HET, the two examples illustrate the potential of high energy neutron scattering very well. On the one hand, the results on the intermultiplet splittings of praseodymium represent a dramatic increase in the energy scale so far accessible even on pulsed sources, with magnetic peaks measured above 800 meV [3]. The peak energies give information on the screening of the intra-4f electrostatic potential for the first time in a metal and open up new avenues of research, some of which are outlined in this paper. On the other hand, the measured crystal field splittings of UO, [4] show the necessity of good resolution even at these energies. The splittings of the cubic crystal field levels by the combined effects of lattice distortions and exchange interactions are only of the order of a few meV compared to an overall splitting of over 1.50meV. Calculations show that the splittings depend on whether the ordered phase is 2-k or 3-k, with the measurements indicating that the 3-k structure is correct [5]. In section 2, I will briefly describe HET and discuss some of the problems associated with high energy neutron scattering. There follows in sections 3 and 4, a report on the recent measurements on praseodymium and UO,, respec-
0921-4526/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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tively, with a discussion of what such experiments can reveal. There is a brief conclusion in section 5.
2. The high energy transfer spectrometer HET is a direct geometry chopper spectrometer i.e. the incident beam is monochromated and the scattered energy is determined by the measured time-of-flight [6]. Incident energies, in the range 30 to 2000 meV, are selected by phasing a fast Fermi chopper to the pulse of protons on the spallation target. High resolution, in neutron terms, is achieved by matching the burst time of the chopper, which spins at 400 to 600 Hz, to the intrinsic time resolution of the neutron pulse. The pulse width is a function of the moderator design; in the case of HET, it has been tightened at lower energies by the use of a poisoning layer, giving an effective moderator depth of 1.5 cm. The resolution is then in the range 1 to 2% of incident energy depending on the energy transfer and scattered flight path (fig. 1). The scattered neutrons are detected by two arrays of ‘He detectors, one lying at 4 m from the sample
hw/E, Fig. 1. Fractional lated for the 2 and of LO/E,, E, being necessary to obtain the intersections straight lines.
energy transfer resolution fi Am/E, calcu4 m detector banks of HET as a function the incident energy. The values of fiw/E, various percentages of Aoio are given by between the resolution curves and the
covering scattering angles from 3” to 7”, the other being at 2.5 m from the sample at scattering angles of 9” to 29”. A third 4 m detector array at 136” is used, in the context of this paper, to measure the shape of the phonon spectrum. In any discussion of high energy magnetic scattering, it is necessary to consider the severe kinematic constraints on the measurements. These arise from the need simultaneously to have a large scalar difference in the incident and scattered wavevectors (i.e. large &J = 2.07(k’ - k:), where ho is the energy transfer in meV and ki and k, are the incident and scattered neutron wavevectors in A-‘), whilst keeping the vector difference relatively small (i.e. small Q = ki - k,). The restriction to small values of Q is required by most magnetic scattering measurements, because their intensities are modulated by the atomic form factor, which falls approximately as exp(- (YQ’). For rare earth and actinide systems, (Yis in the range 0.04 to 0.07 with Q in A-‘. It is clear that scattering angles 4 must be kept small, which is why the main detector arrays of HET are concentrated at low angles. However, even as d, -+ 0, the limitations are severe. As discussed in more detail by Loewenhaupt [7], in order to obtain a particular value of Q, the necessary incident energy increases as the square of the maximum energy transfer. On HET for instance, keeping Q to 4 A-‘, an energy transfer of 100 meV requires an incident energy of only 135 meV Energy transfers of 400 meV require incident energies in excess of 2000 meV. As will become clear in the examples reported here, it is often possible to relax the constraint on the value of Q. The cost in form factor may be offset by the improved resolution, obtained by using a lower incident energy, and the reduced background. Noise levels on pulsed source spectrometers are intrinsically low, because the main neutron production occurs at one time. The background from fast neutrons, which are difficult to shield, only arises from the small delayed neutron fraction. The main source of background is then from the sample itself in the form of multiple scattering. In samples with a strong nuclear cross section, this can be a serious problem even well above the maximum single phonon
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energies, since multiple inelastic processes are appreciable. Fortunately, multiple scattering at high energies usually consists of a broad decaying tail so that sharp magnetic peaks from localised excitations can be readily identified. Broader magnetic responses, such as occur in spin fluctuation systems, require a more careful estimate of the scale of the multiple scattering, by for example using Monte Carlo techniques [8]. Nevertheless, experience has shown that, even when measuring sharp peaks, the best course is to minimise the amount of multiple scattering by selecting the lowest incident energy compatible with the estimated cross section.
3. Intermultiplet transitions in praseodymium One of the first uses of the epithermal spectrum of pulsed neutron sources has been to explore the multiplet structure of rare earth metals and compounds. This is a natural extension of the work already performed on the crystal field potential of these systems using reactor spectrometers over the last two decades [2, 91. Whilst the crystal field splittings are a measure of the relatively weak interactions of the 4f electrons with the surrounding lattice, the splittings of the different *‘+lL, levels result from the much stronger interactions within the 4fN configuration itself. As demonstrated by optical spectroscopy on ionic systems, they provide direct information on the dominant intra-4f electrostatic and spinorbit interactions, as well as on weaker higherorder terms [lo]. Indeed, the observation of a well defined multiplet structure is important in establishing the localised nature of the 4f shell in a metallic environment, and validates to a large degree the approximation of the 4f electron states by atomic wavefunctions. Given the undoubted success of the atomic model in explaining both bulk magnetic measurements and low energy spectroscopy in the lanthanides, it was not anticipated that measurements of intermultiplet transitions would require a radical change in the accepted picture. However, the results to date are beginning to set limits on how
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far the model can be taken without appreciable modification. There are now a number of reported observations of the J-, J + 1 transition in metallic systems, in Sm2+ (‘F, +‘F, at 36meV) [ll], Eu3+ (7F,,+ ‘F, at 46.5 meV) [12], Pr3+ (3H,+ 3H, at 261 meV) [3, 13, 141 and Sm3’ (6H5,2+ 6H7,2 at 124 meV) [15] ions, the first two on reactor spectrometers, the last two on pulsed source spectrometers at IPNS and ISIS. In all cases, the energies are very close to those determined by optical spectroscopy in both the free-ion and ionic compounds. That the energies of these transitions are similar in such a range of systems is not altogether surprising. The spin-orbit splitting is approximately given by &r(J + 1) /2S for J+ J + 1 transitions (5,,J/2S for J+ J - l), where the spin-orbit parameter &, is largely determined by the form of the 4f-electron wavefunctions in the vicinity of the ionic core. It is not therefore very sensitive to the local environment and so provides little new information. The main value of these neutron experiments has been to show that the measured transition intensities are in good agreement with the calculations of Balcar and Lovesey (BL) [16]. Having a reliable framework with which to predict the magnitude and Q-dependence of the neutron scattering is vital to the reliable identification of high energy magnetic peaks, which may occur at similar energies to hydrogen impurity vibrational modes, for example. The results of a recent experiment on the spin-orbit transitions in praseodymium metal at ISIS [3] are compared with the BL calculation in fig. 2. BL express the neutron cross section of intermultiplet transitions, summed over all states in both manifolds, as
x ~(LIJ + E, - E,) , where Ei and E, are the incident and scattered neutron energies, ri = 0.29 b, ho = Ei - E, and Q is the scattering vector. The structure factor,
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_0 !z
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/
0
I
I
I
5
10
15
WAVEVECTOR
Q &‘I
Fig. 2. Neutron inelastic structure factors for transitions within the ‘H term. The lines represent calculated intensities of the ‘H,+ ‘H,, 3H, and 3H, transitions. The structure factors are normalised to the ‘H, -+ 3H, intensity at Q = 0, which has a cross section of 620 mb/sr. The points represent the intensity of the 260meV transition measured with incident energies of 370meV (filled square), 515 meV (open triangles), 830meV (open squares) and 13OOmeV (filled circles).
G( Q; EL,v), for transitions between “+lL, levels labelled by the quantum numbers p and v, may be expressed as a function of the Q-dependent n = 0,2,4,6, atomic radial integrals, (j,), which have been tabulated for all the rare-earths by Freeman and Desclaux [17]. G(Q; p, v) contains both dipolar and higher multipolar terms, only the former being finite at Q = 0. In the ‘dipole’ approximation (i.e. valid as Q-0) the intensity is proportional to (( j,) - ( j,))'. Because of the negative sign of the ( j,) ( j,) coefficient, the spin-orbit transition intensity falls much more sharply with increasing Q than the form factor characterising the Q-dependence of intra-multiplet scattering. In addition, it is an order of magnitude weaker at Q = 0. This can be seen by comparing the 3H,+ 3H, structure factor of Pr3+ with the calculated intramultiplet (3Hq+= ‘Hq) cross section (fig. 2). Fortunately, the strength of the quadrupolar terms in the cross-section produces a plateau region from about 7 to 10 A-’ but the intensity is still very
small. The ability to measure such weak scattering is mainly due to the extremely low noise levels achievable on pulsed source spectrometers, in which the neutron source is effectively ‘turned off’ at the time the neutrons are being counted. To achieve the lowest values of Q in the structure factor measurements shown in fig. 2, it was necessary to use an incident energy of 1300meV. According to the results of optical spectroscopy, the 3H,+ 3F, (n = 2,3,4) transitions are in the range 600 to 850meV [lo] and fig. 3 shows that three peaks are indeed observed at 578, 747 and 809 meV. The justification for assigning these peaks to the three 3H,+ 3F transitions, and for neglecting the 3H,+ 3H, transition, is given in more detail by Taylor et al. [3]. Since only J+ J * 1 transitions within a “+‘L term are dipole-allowed, these transitions are non-dipolar, i.e. their intensities fall to 0 as Q --, 0 and are predicted, by an extension of BL’s calculations, to reach a broad maximum at around 8 A-‘. The 3H,+ 3F, transition intensity was measured in the range Q = 6.5 to 12.5 A-’ with only a small drop in signal. Unusually for magnetic scattering, the peak intensities do not Q (A-‘)
0
200 ENERGY
400 TRANSFER
600
800
1000
ImeV)
Fig. 3. Neutron scattering cross section of praseodymium at 17 K, measured at an angle of 5” with an incident neutron energy of 1300meV. The change in the scattering vector across the spectrum is shown along the top axis. The data have been fitted by 4 Gaussians and tail of low energy scattering.
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fall sharply with Q; indeed, it is inadvisable to make such measurements at too low a scattering wavevector for two reasons. Firstly, the cross section falls rapidly below 3 A-’ and, secondly, the requisite high incident energy degrades the resolution and background, as discussed earlier. Whilst the J+ J + 1 transition is hardly shifted in energy from the measured free ion value, the three 3H+= 3F transitions are all lower by about 40 meV This observation can be readily explained in terms of the parametrised Hamiltonian used to model the effect of the intra-4f Coulomb interaction. In addition to J&, three extra parameters, F@), Fc4) and Fc6), are required [lo]. In principle, these are defined as radial (Slater) integrals over the 4f-electron wavefunctions. However, Hartree-Fock calculations substantially overestimate their magnitude by up to a factor 1.5, even in the free ion, so they are usually treated as empirical parameters. Providing the spin-orbit coupling is reasonably small, F(*), Fc4) and Fc6) determine the energies of the Russell-Saunders “+‘L terms whilst [,, determines the relative splittings the different J manifolds within each term. It is possible, to first order, to shift the energies of all three 3F levels rigidly up and down by adjusting the values of the Slater integrals. The transition energies are well reproduced by reducing the F(*) parameter by 10% from its free ion value, keeping all other parameters fixed. A reduction in the effective Slater integrals corresponds to a decrease in the electrostatic potential between the two 4f electrons of the Pr3+ ion. This is the first time that this screening effect has been observed in a rare earth metal. However, it has been observed in ionic systems by optical spectroscopy [lo]. For instance, a reduction of about 20 meV in the 3H* 3F transition energies has been observed in Pr3+ doped in LaCl, [18]. Slater parameter shifts are observed in the spectra of transition metal ions as well as 4f and 5f ions and it is possible to establish a correlation between the size of the shift and the type of ligand, producing the ‘nephalauxetic’ series (e.g. AF(Pr : LaCl,) > AF(Pr : LaF,), etc.) [19]. Moreover, the observed shift has been shown to be proportional to the polarisability of
15.5
the ligand involved, and therefore presumably to its ability to screen any electrostatic potential within the rare earth ion. It would be interesting to see if a similar sensitivity to the surrounding lattice could be observed in metallic systems where the screening mechanism is more effective. Another striking feature of the experiment is that the measured intensities of the three 3H+ 3F. transitions are nearly a factor two lower than the calculations of BL, when normalised to the 3H,-+ 3H4 scattering. Moreover, we also failed to observe the 3H,-+ ‘G, transition at about 1170 meV for which the calculated cross section is comparable to the 3H4+ 3F2 transition, and sufficiently large to have been measurable. This discrepancy is in marked contrast to the excellent agreement in the case of the 3H4+ 3H, transition. There are a number of possible reasons that need to be explored further. For instance, Balcar [20] has pointed to the need to perform the calculations fully relativistitally. The 3H and 3F states will have different radial wavefunctions because of the differing amounts of i = 5 /2 and j = 7 /2 character whereas the BL method assumes them to be the same. It is difficult to estimate the magnitude of this effect because it involves the numerical calculations of several radial integrals which are factorised out in the semi-relativistic treatment. However, it would seem surprising if it could have such dramatic consequences. Another possibility is that the reduction in intensity is related to the cause of the Slater parameter shift. I have already mentioned that the absolute value of the Slater parameters are severely overestimated by Hartree-Fock calculations. This is known to be due to the neglect of configuration interactions, i.e. perturbations by electron configurations other than 4fN, which give rise to terms in the Hamiltonian which have the same symmetry as the bare intra4f Coulomb potential [21]. In other words, most of the effect is absorbed into the empirical Slater integral parameters. There are also other parameters introduced into the Hamiltonian, but given the limited number of levels observed in the neutron scattering, we are not in a position to comment
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on them. As a consequence of configuration interactions, then, the 4f wavefunctions are hybridised with other higher-lying orbitals. The electrons partially occupy more delocalised levels and so experience a reduced mutual electrostatic repulsion. This same hybridisation may be responsible for the reduction in the 3H-+ 3F transition intensities. If this were so there should be a direct correlation between the Slater parameter shifts and the reductions in the neutron cross sections. Further neutron experiments on praseodymium in ionic systems, where the shifts are smaller than in praseodymium metal, are therefore planned. In conclusion, it seems that the main effect of a metallic environment on the 4f shell in Pr is an enhanced screening of the electrostatic interaction between the two f-electrons. The multiplet structure is preserved, at least in the energy range so far accessible. The transition peaks are broader than the instrumental resolution, though without knowing the extent of the crystal field splittings within each multiplet it is difficult to draw any conclusions from this. It would be useful to explore the other rare-earth metals to look for systematic trends in the Slater parameter shifts. Unfortunately, there are only two trivalent rare earths in which levels derived from different *“IL terms are below 1 eV, Sm3’ and Tm3+ [lo], both of which present problems of neutron absorption. However, these ions show intermediate valent behaviour in a number of systems, so any information on the degree of hybridisation of the f-shell is of special interest. Observations of the multiplet structure would also be of value in actinide intermetallic compounds, where the aim would be to establish the valence of the actinide ion. Work has already begun on USb and UTe [22], but has concentrated on the validity of proposed crystal field models in these compounds. The measured magnetic response has so far been extremely broad, stretching to 80 meV with no well-defined localised excitations about the spin wave bands, but the experiments need to be extended to much higher energy if intermultiplet transitions are to be observed. I should mention two further avenues of pos-
neutron
scattering
sible research in this field. Firstly, it should be of value to study the spin-orbit transition in cerium compounds, in particu(*Fs,* -+ 2F 7,2) lar to explore the development of the peak as a function of hybridisation. Whilst it may be anticipated that heavy fermion systems, for which the appropriate energy scales are a few Kelvin, will show a well defined spin-orbit peak, the more intermediate valent compounds should show a marked broadening of the level and possibly a renormalisation of the energy. Such questions as whether the degree of hybridisation of the spinorbit level is the same as in the ground state could then be addressed. Secondly, it may yet be possible to directly measure interband transitions. This has long been an aim of high energy neutron scattering, but the predicted cross sections have been discouragingly small. Also, since what is measured is a joint density of states, the signal will be spread over an energy range comparable to the bandwidth, in most systems a few eV. However, in some actinide compounds, such as PuTe [23], band structure calculations suggest that the fbands consist of two spin-orbit split bands (i.e. of predominant j = 5 /2 and j = 7/2 character). The cross section will be considerably enhanced by the spin-orbit coupling and spread over a limited energy range, perhaps a few hundred meV. It will probably still be a few years before such experiments are attempted, experience with more localised actinide systems is required first, but it is an interesting prospect.
4. Crystal fields in UO, Another field m which pulsed neutrons have begun to play a prominent role is the crystal field spectroscopy of actinide oxides. Crystal field splittings in these systems are frequently greater than 100 meV for a number of reasons. Firstly, oxygen ligands give rise to larger crystal field potentials than for example chlorine and bromine. The relation of this spectrochemical series with the nephalauxetic series referred to earlier is discussed by Newman [19]. Even in rare earth oxides, such as Pro, and BaPrO,,
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both of which have been studied at IPNS [24], the excitation energies are well above 100 meV Secondly, it is well known that the greater spatial extent of the Sf-electron wavefunctions of actinide ions, compared to those of lanthanide 4f electrons, leads to a factor two or more increase in the crystal field potential. Finally, within the actinide series, the tetravalent ions experience stronger potentials than their trivalent counterparts [25]. It may be thought that insulating, or in the case of UO, semiconducting, compounds would be much better studied by classical optical spectroscopy, which offers a wider energy range and generally superior resolution. However, there is a real complementarity here between the two techniques. The assignments of optical transitions in the trivalent rare earths and actinides often rely on the spectra being split into bands of crystal field excitations each derived from a different 2s+1L, level. It is then possible to fit the free ion parameters, for instance the spin-orbit parameter and the Slater integrals, first before determining the crystal field parameters. In the tetravalent oxides, such a clean separation is not possible because the crystal field potential is comparable in magnitude to the spin-orbit coupling. All the transitions are of mixed character and do not generally form identifiable bands. The only good quantum numbers are the irreducible representations of the local point group symmetry. A further problem is that the spectra are often contaminated by excitations of vibronic origin. Calculations of optical transition probabilities are particularly difficult since intra5f transitions are not allowed by electric dipole selection rules. To be measurable, they depend on small admixtures of other electronic configurations. Reliable assignments are therefore a problem. This point is exemplified by the case of UO,, whose tetravalent uranium ions have a localised 5f2 configuration [26] (cf. Pr3+ has a 4f2 configuration). Optical absorption studies had already been made of this compound [26]. Structure in the energy range 150 to 180 meV was attributed to two-phonon processes because of its anomalous temperature dependence. Peaks above
500 meV were also observed but were too numerous and evenly distributed in energy to provide a reliable comparison with the only available predictions of Rahman and Runciman [27]. It was not until the first neutron measurements at IPNS [24] that it was finally established, by the Q-dependence of their intensity, that the peaks below 200 meV were in fact magnetic. Following the more recent ISIS results [4], which were performed with better resolution, we now know that their unusual temperature dependence is due to the presence of a strong JahnTeller coupling responsible for the combined quadrupolar and magnetic phase transition at 30.8 K. As seen in fig. 4, this gives rise to a strong splitting of the cubic crystal field levels in the ordered phase. Optical spectra have since been reanalysed in the light of the neutron results [28]. Figure 5 shows how sensitive the crystal field splittings of the ground state multiplet are to both the type of potential (the ratio of V, to V,) and the presence of other multiplets. The fact that the crystal field splittings are not directly proportional to V, is a consequence of J-mixing effects. The stronger the crystal field potential, the greater the admixture of, for instance, 3F2 character into the wavefunctions of the lowest multiplet. 3F2 is the predominant component of the first excited multiplet in the U4+ free ion,
hw (mev) Fig. 4. Neutron scattering cross section of UO, at 6.5 K, measured at an angle of 5” with an incident neutron energy of 290meV. The data have been fitted by 4 Gaussians and a sloping background. The peak at lowest energy corresponds to the r, + r, transition. The remaining peaks correspond to the r,+r, transition split by the quadrupolar and exchange field of the ordered phase.
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3F~,_,_~~:_-__________-__---_‘-‘==-_-_---_~_-_~_
4oc
-vl
CmeV)
Fig. 5. Calculated crystal field splittings of UO, as a function of V, when (a) V,lV, = -0.06 and (b) V,lV, = -0.21 using the Slater integral parameters Fz = 23.73 meV, F, = 4.19 meV, F, = 0.49 meV and the spin-orbit parameter &, = 222.7 meV. The curves are labelled by the predominant 2s”L, component of wavefunctions when V, = 0. The arrows mark the magnetic dipole-allowed transitions at the value of V, estimated by (a) Rahman and Runciman and (b) Osborn et al.
predicted to be at 450 meV. It splits into levels of Is and I?, symmetry in a cubic crystal field, as shown in fig. 5. These levels, along with those derived from the free-ion 3H, state at 700 meV, effectively suppress the lowest crystal field levels derived from the 3H, multiplet. Long before there were any spectroscopic measurements available, Rahman and Runciman [27] predicted a Is ground state with the I3 crystal field level at 169 meV and the I, level at 624 meV (The Is -+ I’, transition is not neutron dipole-allowed and so will be much weaker than the other two). In arriving at this estimate, they assumed a ratio of V, to V, of -0.06 and chose V4 to reproduce the observed ordered moment of 1.8~~. The reduction from the 3H, Is moment of 2~~ is, in their treatment, a consequence of
J-mixing. It can be seen from fig. 5 that a different ratio of -0.21 produces a radically different spectrum with the two levels close to each other for all values of V,. The latest measurements at ISIS have now shown that the latter ratio is correct, with V, = - 123 meV [4], much lower than the Rahman and Runciman estimate, but in line with more recent ab initio estimates [28]. With this choice of potential, the ground state moment is not reduced by J-mixing. In fact, it is marginally increased, but this is consistent with Allen’s suggestion that the moment reduction is due to the quadrupolar coupling in the ordered phase [29]. It is clear from fig. 4 that the neutron measurements on HET show that high resolution is required in order to understand the nature of the quadrupolar coupling. The I?, triplet is split by about 10 meV by the lattice distortion of the ordered phase, whereas the I, doublet is only split by about 2 meV. This is just resolved in another scan using an incident energy of 229 meV. There are two possible models for this distortion which are compatible with neutron diffraction results [30]. In one, the moments are aligned along [llO] directions in a 2-k structure, with a monoclinic distortion of the unit cell. In the other, the magnetic structure is 3-k, with the moments aligned along [ll l] directions in a trigonally distorted crystal field. These two models give quite different predictions for the splittings of the I3 doublet and I, triplet, according to the calculations of Amoretti et al. [5]. In the former, the I3 doublet has a much larger quadrupole moment than the I4 triplet and so is more strongly split. In the latter, this situation is reversed. The neutron scattering therefore allows us to distinguish the two cases and leads to the conclusion that the 3-k model is correct. These results throw light, therefore, on two different aspects of the magnetic properties of UO,, the degree of J-mixing in the ground-state wavefunctions and the symmetry of the ordered state. On the one hand, the overall scale of the crystal field splittings shows that the Is ground state is given fairly accurately by the intermediate coupling approximation with relatively small admixtures of other J states. On the other
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hand, fine splittings of the crystal field levels are only consistent with a 3-k magnetic structure. Both are discussed in more detail in recent publications [4, 51.
5. Conclusion Given the context of this paper, it is perhaps appropriate that a recurring theme has been the complementarity of neutrons and photons, although the comparisons are with classical optical spectroscopy. In both measurements, we are exploring the nature of the interactions within the f-electron configuration and how they interact with other configurations or the crystalline environment. Neutron spectroscopy cannot hope to match the energy range of optical techniques, in spite of the increasing overlap between the two, and so the detailed consideration of high order (e.g. orbit-orbit and spin-other orbit) interactins will not be possible. However, there are two reasons why it is still important. Firstly, it is possible to make reliable calculations of cross sections and their Q-dependence on the basis of atomic calculations. This is valuable both for the purposes of peak assignments and, potentially, in determining deviations from simple atomic wavefunctions. Secondly, neutrons can be used to probe metallic systems and perhaps reveal the degree of f-electron hybridisation with the conduction electrons. This will be particularly interesting in both actinides, where the extent of intra-5f correlations is still unclear, and in intermediate valent materials. These projects are still at an early stage, but the results have so far been very promising.
Acknowledgements This work was done in collaboration with A.D. Taylor, G. Amoretti, M.T. Hutchings, K.A. McEwen and several others. I would also like to acknowledge stimulating discussions with S.W. Lovesey, E. Balcar and G.L. Goodman.
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