PHYSICAL REVIEW B 84, 104409 (2011)

Multipolar, magnetic, and vibrational lattice dynamics in the low-temperaturephase of uranium dioxide

R. Caciuffo,1 P. Santini,2 S. Carretta,2 G. Amoretti,2 A. Hiess,3,* N. Magnani,4 L.-P. Regnault,5 and G. H. Lander1

1European Commission, Joint Research Centre, Institute for Transuranium Elements, Postfach 2340, DE-76125 Karlsruhe, Germany2Dipartimento di Fisica, Unita CNISM di Parma, Universita di Parma, via G.P. Usberti 7/a, IT-43100 Parma, Italy

3Institut Max von Laue-Paul Langevin, 6 rue Jules Horowitz, BP 156, FR-38042 Grenoble Cedex 9, France4Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California, 94720-8175, USA

5ISPSMS-MDN, UMR-E 9001, CEA/UJF-Grenoble 1, INAC, FR-38054 Grenoble, France(Received 17 June 2011; revised manuscript received 12 August 2011; published 6 September 2011)

We report the results of inelastic neutron scattering experiments performed with triple-axis spectrometers toinvestigate the low-temperature collective dynamics in the ordered phase of uranium dioxide. The results are invery good agreement with the predictions of mean-field random phase approximation calculations, emphasizingthe importance of multipolar superexchange interactions. By comparing neutron scattering intensities in differentpolarization channels and at equivalent points in different Brillouin zones, we show the mixed magnetovibrational-quadrupolar character of the observed excitations. The high-energy resolution afforded by the cold triple-axisspectrometer allowed us to study in detail the magnon-phonon interaction giving rise to avoided crossings alongthe [00ξ ] reciprocal space direction.

DOI: 10.1103/PhysRevB.84.104409 PACS number(s): 75.30.Ds, 71.70.Gm, 75.25.Dk, 75.50.Ee

I. INTRODUCTION

Systems with active orbital degrees of freedom displaypeculiar dynamical properties. These include well-knownsingle-ion phenomena such as the dynamical Jahn-Teller orthe quadrupolar Kondo effect, but also propagating excitationsmore complex than conventional magnetic spin waves orexcitons, and involving quadrupoles or higher-rank multipolesalone or in combination with dipoles. The identification andmodeling of such collective excitations may shed light ontothe scarcely known world of multipolar two-ion interactions,the nature and form of which usually remain elusive.1

Indeed, static properties (e.g., the character of multipolarorder or the response to macroscopic external perturbations)are typically affected by only a few of the many possiblemicroscopic couplings between active multipoles, and, more-over, only space-integrated information on these couplingscan be extracted (e.g., the value of the mean-field constant).When quadrupoles are order parameters in a phase transition,electronic and phononic contributions to two-ion quadrupolarinteractions can not usually be unambiguously separated. Asa consequence, in d or f systems with quadrupolar order, therelative importance of phonons (Jahn-Teller) and exchange(electronic) mechanisms in the phase transition has beendebated for years and can not be inferred from static properties.Conversely, in the dynamics, many active multipoles and thedetailed microscopic interactions among them play a role,and much more information can be extracted by modelingexperimental data on multipolar dynamics, in much the sameway as the detailed form of spin exchange couplings can beextracted by modeling spin-wave spectra.

Uranium dioxide is an ideal compound to study the effectsof multipolar interactions on the collective lattice dynamics.The first inelastic neutron scattering (INS) investigationsof antiferromagnetic (AF) UO2 were performed in the late1960s (Refs. 2–4) and gave convincing evidence of a strong

interaction between magnon and phonon excitations belowthe Neel temperature TN = 30.8 K. However, interpretationsof the results in terms of simple Heisenberg-type spin-wavetheories were unsatisfactory. Later INS studies performedwithout5 and with neutron polarization analysis6–9 provideda detailed picture of the low-energy quasiparticle dynamicsin UO2, revealed a large anisotropy of the dipolar exchangetwo-ion coupling, and confirmed the presence of features notcompatible with a description including only vibrational andmagnetic exchange interactions. Indeed, further experimentalevidence suggested that quadrupolar interactions play animportant role in stabilizing the ground state of UO2, as a staticJahn-Teller distortion of the oxygen cage10 and long-rangeorder of electric quadrupoles11 occur along with the first-ordermagnetic transition at TN . The fact that the magnetic order isof the 3-k type12 indicates that quadrupolar interactions haveboth magnetoelastic (ME) and purely electronic two-ion con-tributions, produced by the same mechanism responsible forthe anisotropy of the dipolar exchange, namely, superexchange(SE) interactions in the presence of orbital degeneracy.1 WithME coupling alone, in fact, the 1-k structure would be morestable than the 3-k one,13 the latter being stabilized by the SEcoupling.14 These quadrupolar interactions have a significantimpact also on the dynamics of UO2.

The magnetic low-temperature dispersion curves providedby previous investigations using various triple-axis spec-trometers at different neutron sources are shown in Fig. 1.The experimental data contain evidence of magnon-phononinteractions at several positions in reciprocal space, the mostevident one being the avoided crossing (AC) between themagnon and the transverse acoustic (TA) phonon branchespropagating in the [00ξ ] direction at ξ = 0.45 (ξ = aq/2π ,a being the lattice parameter of the cubic unit cell). Alongthe same [00ξ ] direction, two distinct optical branches andadditional excitations occur in the energy range between 9 and12 meV. These features are not compatible with a scenario

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R. CACIUFFO et al. PHYSICAL REVIEW B 84, 104409 (2011)

FIG. 1. (Color online) The dispersion of the mixed magnetovibrational-quadrupolar modes in the low-temperature phase of UO2 is shownalong the principal crystallographic directions of the cubic lattice (Refs. 5 and 6). Neutron groups have been collected in the constant-Q mode,at the reciprocal lattice points (0,0,2 − ξ ), (1 − ξ,1 − ξ,1), and (1 + ξ,1 + ξ,1 + ξ ). Gray symbols indicate qualitatively smaller intensitythan the black symbols. The broken lines and open squares indicate acoustic phonon branches measured at 270 K (Ref. 6). Excitations around12 meV correspond to an optical quadrupolar branch. The experiments provide evidence of mixing between magnons and phonons, in theforms of large avoided crossings (ACs), for instance, at ξ ∼ 0.45 along the [00ξ ] direction. A portion of the UO2 (110) reciprocal latticeplane is shown in the upper panel. Filled and empty circles are points corresponding to crystal and magnetic structure Bragg reflections,respectively. Red (solid) arrows indicate neutron momentum transfer vectors; dark blue (dashed) arrows indicate quasi-momenta of collectiveexcitations.

assuming 3-k magnetic order and dipole-only interactions.Indeed, the 3-k magnetic structure of UO2 has Pa3 symmetryand the magnetic modes along the �-X direction shouldbelong to one acoustic branch and a single doubly degenerateoptical branch [see Fig. 2(a)]. The experimental picture isnot reproduced in a dipole-dipole scheme even if all the SEanisotropic coupling parameters allowed in the distorted Pa3structure are taken into account.1 Giannozzi and Erdos triedto solve the puzzle by attributing the optical branch at about12 meV to quadrupole excitations.13 However, assuming aspecial form of phonon-mediated quadrupole-quadrupole cou-pling, they obtained calculated intensities for the quadrupolarmodes smaller than the experimental values by two order ofmagnitudes.

A satisfactory model was eventually provided byCarretta et al. in Ref. 15, where they report the outcome ofextensive theoretical calculations that take into account SEmagnetic-dipole and electric-quadrupole coupling, togetherwith free phonons and ME interactions. These calculationsshow that quadrupolar interactions are an essential ingredient

to understand the dynamics of UO2 in its magnetoquadrupolarordered phase. The optical branch at ∼12 meV correspondsto propagating quadrupolar fluctuations carrying along amagnetic component. These excitations can be visualizedas wavelike oscillations of the two locally orthorhombicquadrupoles (Qxy , Qx2−y2 , where the z axis is along the localmoment direction) accompanied by the precession of a smallspin component.15 The model also predicts the existence ofa dispersive low-energy acoustical quadrupolar branch in the[00ξ ] direction, characterized by a very weak INS intensity.In addition, the calculations confirm the presence of theexcitations in the energy range between ∼8 and ∼12 meVreported in Ref. 5.

In this paper, we present the results of triple-axis INSexperiments performed with thermal neutrons and polarizationanalysis, and with unpolarized cold neutrons. The main goalsof these experiments were to find evidence of the weakintensity modes predicted in Ref. 15, including the acousticquadrupole-wave branch, and to better characterize the ACbetween magnons and TA phonons at (0,0,0.45). The results

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FIG. 2. (Color online) Dipolar magnetic INS cross section along[00ξ ], calculated with the model described in Ref. 15 and discussedin this paper. The color scale represents intensity in arbitrary units.(a) Intensity map showing the results of RPA spin-wave calculationsneglecting both superexchange and magnetoelastic quadrupole-quadrupole interactions. Bare acoustic phonon dispersion curves areshown by dashed white lines. (b) Intensity map obtained by takinginto account quadrupole-quadrupole couplings. Dashed white linesshow modes with sizable phonon component and cross section. Labelsspecify the character of the excitation branches: SA = spin acoustic,SO = spin optical, QA = quadrupolar acoustic, QO = quadrupolaroptical, TA = transverse acoustic phonon, LA = longitudinal acousticphonon. Vertical double arrows indicate avoided crossing positions.(c) Same calculation as in panel (b) with the ME coupling removed,i.e., by setting gA = gB = 0 in Eq. (2). (d) Same calculation asin panel (b) with high-energy phonons removed. This is done byartificially inserting a smooth cutoff in �ασ (q) in Eq. (3) for phononswith energy εσ (q) > 25 meV. Panels (c) and (d) are commented atthe end of Sec. V.

we obtained are in very good agreement with the theoreticalpredictions. By comparing neutron scattering intensities indifferent polarization channels and at equivalent points indifferent Brillouin zones, we show the mixed magnetovi-brational character of the observed excitations. The highenergy resolution afforded by the cold triple-axis spectrometerallowed us to clearly separate the two modes participating tothe AC at (0,0,0.45) and to follow the change from magnonliketo phononlike character of the two branches.

The remainder of this paper is structured as follows. InSec. II, we briefly summarize the theoretical model developedin Ref. 15; experimental details are given in Sec. III, and theresults obtained are reported and discussed in Secs. IV and V.A short summary and conclusions are found in Sec. VI.

II. THEORETICAL OUTLINE

In this paper, collective excitations in UO2 are calculated inthe mean-field random phase approximation15,16 (MF-RPA).The dynamics are driven by time-dependent effective fieldstransmitted by phonons and two-ion electronic exchange.Fields that are dipolar in character, such as those associ-ated with magnetic exchange, couple to the total angularmomentum J of the U ions and give rise to conventionalmagnon modes. Higher-rank multipolar interactions transmitfields that couple with higher-rank multipoles and may drivemultipolar waves. For quadrupoles or even-rank multipoles,these effective fields are physically equivalent to a time-dependent crystal field. Fields transmitted by exchange arenearly instantaneous on the time scale of the collectiveexcitations, whereas fields transmitted by phonons (necessarilyeven rank) may display important retardation effects. Theresponse of each ion to the fluctuating fields is encoded in theset of energy-dependent single-ion susceptibilities χAB(ω) (Aand B denote dipoles or multipoles), which are calculated fromself-consistent MF eigenstates. Poles in χAB(ω) are locatedat the MF energy gaps and describe single-ion excitations,which become dispersive excitation branches in the crystal.We label a branch as multipolar if the matrix element betweenthe corresponding MF levels is zero for dipole operators, andnonzero for higher-rank operators.

The U ions in UO2 are tetravalent, with a 5f 2 electronicconfiguration. The crystal-field (CF) ground state is a �5

triplet,17 carrying magnetic dipoles (Jx , Jy , Jz), �5 electricquadrupoles Qα transforming as xy, yz, and zx, and adoublet of �3 electric quadrupoles transforming as 3z2 − r2

and x2 − y2. In the ordered 3-k phase, the �5 triplet issplit into three singlets, labeled by the z components ofan effective S = 1 pseudospin operator (M = 0, ± 1). In areference frame with the z axis along the local magneticmoment direction, which is one of the equivalent 〈111〉directions, the operators Jz and Q3z2−r2 are diagonal and,therefore, static (see Fig. 3). The remaining six operators arenondiagonal and fluctuating: the first MF transition, betweenM = −1 and M = 0, is associated with fluctuations of Jx ,Jy , Qzx , and Qyz. This mixed magnetic-quadrupolar modeyields the conventional spin waves of Fig. 2(a) if quadrupolarinteractions are neglected. However, the two quadrupolaroperators provide coupling of this excitation to phonons andthis is the origin, for example, of the anticrossings around7 meV near ξ = 0.45 [TA-SA mixing, see Figs. 2(b) and5] and around 11 meV near ξ = 0.6 [TA-SO mixing, seeFig. 2(b)]. The second MF excitation from M = −1 to M = 1is purely quadrupolar at the MF level being associated withfluctuations of Qx2−y2 and Qxy . The dispersion of collectiveexcitations stemming from this MF mode is due entirely totwo-ion quadrupolar interactions. The dynamics are againfurther complicated by the coupling with phonons provided

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FIG. 3. (Color online) Arrangement of the ordered magneticmoments (cones) and quadrupoles (ellipsoids) in the 3-k phase.The four sublattices are labeled by different colors. The figure alsoillustrates the splitting of the ground �5 triplet below TN for twosublattices. The labels of nonvanishing off-diagonal matrix elementsare referred to the local z axis.

by the electric quadrupoles and manifested as ACs betweenelectronic and vibrational modes.

As the ordered lattice of UO2 contains four independentU sites,1 a total of eight branches originate from the twosingle-ion MF excitations. Four of them are acoustic branches,two spin (SA) and two quadrupolar (QA), and four are opticalbranches, again two spin (SO) and two quadrupolar (QO)branches. The number of acoustic phonon branches is twelve,when their folding into the smaller Brillouin zone of theordered phase is considered. The Hamiltonian to be consid-ered, H = HSS + HQQ + HP + HME, is the sum of termscorresponding to SE dipole-dipole (HSS) and quadrupole-quadrupole (HQQ) interactions, free phonons (HP ), and mag-netoelastic phonon-quadrupole couplings (HME). The largenumber of parameters in H cannot be fixed unambiguouslywith the available experimental information. Therefore, weadopt the simplest possible model capturing the key featuresof the dynamics: we limit SE coupling to nearest neighbors,and we include only the quadrupoles of �5 symmetry. In fact,these quadrupoles order below TN and must be included forreproducing static properties. For the pair of U ions at (0,0,0)and (1/2,1/2,0), we write

HSS = J {Sz(1)Sz(2) + δ[Sx(1)Sx(2) + Sy(1)Sy(2)]},HQQ = KSE

{QS

xy(1)QSxy(2)

+ δ[QS

yz(1)QSyz(2) + QS

xz(1)QSxz(2)

]}, (1)

where Sα are components of the effective spin operatorS = 1, and QS

α ∝ Qα are quadrupolar operator equivalents.For instance, QS

xy = SxSy + SySx . The Hamiltonian for the

remaining 11 bonds is obtained by rotating Eq. (1). In (1),we have kept only three of the six symmetry-allowed SEparameters: we have neglected off-diagonal couplings (theseplay no role along �X) and we have used a unique parameterδ for the anisotropies of both dipolar and quadrupolar SEinteractions. These anisotropies do not appear in the mean-fieldmodel derived from Eq. (1) and are, hence, irrelevant for staticproperties.15 Good descriptions of the observed dynamicsrequire dipolar and quadrupolar anisotropies to be both sizable.Since available information does not allow us to fix these twoparameters independently, we have assumed them to be equal.

The phonon term HP is calculated within the rigid-ionmodel of Ref. 2, and, in HME, only phonon-induced modulationof the oxygen cage is considered:

HME =∑

R

∑

α=xy,xz,yz

QSα(R)[gAγA,α(R) + gBγB,α(R)], (2)

where γA,α(R) and γB,α(R) (α = xy,xz,yz) are the α com-ponent of the two �5 normal modes (A, B) of the oxygencage around the U ion at R.18 These are linear combinationsof the oxygen displacements transforming as �5, which arethen expressed in terms of the normal modes of HP .19

The five free parameters (J , KSE, δ, gA, and gB) weredetermined by comparison with the main features of theprevious experimental results (see Table I below).

Dynamical susceptibilities are calculated in RPA by consid-ering fluctuations around the four-sublattice MF configuration,explicitly including spins, quadrupoles, and phonons.15 TheRPA system is solved numerically and the inelastic neutroncross section is then obtained from the absorptive part of thedynamical magnetic susceptibility.20 Within this approxima-tion, phonons act as a transmitting medium for effective two-ion quadrupolar couplings, which can be described in termsof a long-range retarded interaction K

ph

α,α′ (R,R′; ω) betweenthe quadrupoles QS

α(R) and QSα′(R′). The physical picture

is simple and recalls the mechanism by which conductionelectrons transmit the RKKY interaction: the quadrupoleQS

α(R) by Eq. (2) induces a distortion [γA/Bα(R)] of theoxygen cage centered at (R). This distortion corresponds toa superposition of lattice phonons and, hence, propagates withfinite speed through HP to a different site (R′). If the resultingdisplacement pattern of the oxygen cage at (R′) has nonzeroprojection onto the local normal modes [γA/Bα′(R′)], it inducesa polarization of QS

α′(R′). This is equivalent to an effectivequadrupolar retarded interaction: By writing R = l + h, wherel spans the Bravais lattice of the ordered phase and h spans thefour sublattices, and by Fourier-transforming on l, we find

Kph

α,α′ (q; h,h′; ω) = ei(q−G)·(h′−h)∑

G,σ

[gAFA

ασ (−q + G)

+ gBFBασ (−q + G)

][gAFA

α′σ (q − G)

+ gBFBα′σ (q − G)

]Dσ (q − G; ω), (3)

TABLE I. Value of the parameters in Eqs. (1) and (2).

J (meV) KSE (meV) δ gA (meV/A) gB (meV/A)

1.55 0.95 0.25 6.7 42.9

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where Fασ (q) = h�ασ (q)/√

8mOεσ (q). G spans zero and thethree basis vectors of the reciprocal lattice of the orderedphase and σ labels the nine phonon branches. Dσ (q; ω) isthe free phonon propagator and � is the projection of theα component of the local normal mode on the displacementpattern of phonon qσ . Finally, mO is the oxygen mass andεσ (q) are the bare phonon energies. Figure 2(a) shows thedipolar magnetic INS cross section calculated along [00ξ ]by taking into account the most general symmetry-allowedexpression of magnetic anisotropic exchange, but neglectingtwo-ion quadrupole-quadrupole coupling in the RPA system.The SA and the SO branches are degenerate at the � point,and the SO branch has the same energy gap at � and X.In this calculation, magnetic and vibrational modes are notcoupled and no ACs are observed. Note that, in the orderedphase, the zone boundary corresponds to ξ = 0.5, whereas theparamagnetic zone boundary ξ = 1 (X point) is a zone-centerpoint equivalent to ξ = 0 (� point). Hence, all excitationenergies are symmetric around ξ = 0.5 and a second, specularSA branch also exists. However, the INS cross section is notsymmetric around ξ = 0.5 because of the unit-cell structurefactor, which weights the four sublattices with different phases.

The dynamics change drastically if superexchange andmagnetoelastic quadrupolar interactions are included. This isshown in Fig. 2(b), where the intensity map corresponds to theset of parameters given in Table I.

Many more excitations emerge in the INS cross section.Actually, at low T , there are as many as 20 elementaryexcitations for each wave vector: 8 having a dominantelectronic component and 12 having a dominant phononcomponent. Yet, only a few of them are actually detected bymagnetic-dipole INS because of tiny dipolar matrix element(e.g., the QA branch) or because of tiny or vanishing INSstructure factor. The dynamics show a complex interplayof all these modes and the excitations usually display amixed character. In particular, branches marked by white linesin Fig. 2(b) have a sizable vibrational component. In thepresence of quadrupolar coupling, the longitudinal acoustic(LA) phonon branch remains purely vibrational along the [001]direction, whereas the two transverse acoustic (TA) branchesacquire mixed spin-phonon character. The mixing is maximalnear AC positions, approximately indicated by yellow arrows.The AC at ∼7.2 meV and ξ ∼ 0.45 involves one of the two TAand the SA branches. It is due to the Qyz and Qzx quadrupoles,coupling phonons with the M = 1 MF transition. The samequadrupoles are responsible for a second AC around 11 meVand ξ = 0.6, mixing TA and SO modes. Further, ACs occur atξ ∼ 0.55 around 9 meV, and at ξ ∼ 0.6 around 12 meV. Bothof them involve the M = 2 MF transition and are due to cou-pling promoted by Qxy and Qx2−y2 quadrupoles. The formermixes TA and QA modes, the latter TA and QO branches.

We stress that Fig. 2 shows the dipolar magnetic INS crosssection and, hence, the quadrupolar M = 2 modes emergeonly where they acquire a magnetic component due to themixing with M = 1 spin-wave branches. This mixing isproduced by two-ion quadrupolar interactions, which also setthe dispersion of quadrupolar waves. Hence, these waves are avery direct and accurate probe of the form of these otherwiseelusive interactions. The mechanism producing the mixingcan be visualized in the following way: If on a given ion (e.g.,

the one on the yellow sublattice at the corner of the cubebasis in Fig. 3) Qxy and Qx2−y2 fluctuate, for the presenceof two-ion quadrupolar interactions a different ion (e.g., theone on the purple sublattice at the center of the cube basisin Fig. 3) feels a time-dependent quadrupolar molecular field,equivalent to a time-dependent distortion. In the local (purple)z axis, tilted with respect to that of the yellow site, this time-dependent field will induce oscillations of the local Qxy andQx2−y2 quadrupoles but also (off resonantly) of the local Qxz

and Qyz quadrupoles. The latter are locked to Jx and Jy and,hence, a magnetic oscillation is also induced on the purplesite. The result is a certain amount of M = 1 component inthe M = 2 branches, which turns out to be large for the QObranches, and small or nearly vanishing for the QA branch. Ofcourse, quadrupolar modes can also be indirectly detected atlarge Q through the vibrational INS cross section where theydisplay sizable mixing with phonons (see below).

The spectrum of UO2 at low temperature is clearlycomplicated. We compare the observed excitations with amodel that has only five parameters, so a precise fit shouldnot be expected in every case. Nevertheless, as shown in thefollowing figures, the agreement is good.

III. EXPERIMENTAL DETAILS

The experiments were performed on the same single crystalused previously.6–9 The sample, of ∼9 cm3 in volume, wascut from a melt-growth crystal boule of depleted uraniumdioxide and aligned with the (1,1,0) vector perpendicularto the scattering plane. Rocking curve measurements showa homogeneous mosaic spread of about 0.4◦, with a smallportion of the sample misoriented by 7◦.

Measurements were carried out at the Institut Laue-Langevin (ILL) in Grenoble, France. The unpolarized INSdata were collected on the IN14 triple-axis cold-neutron spec-trometer, using vertically focusing pyrolytic graphite (PG-002)crystal as monochromator and analyzer. The instrument was inthe long-chair geometry, with open-60′-60′-open collimation.A beryllium filter was inserted between the sample positionand the analyzer’s collimator. Data were collected in theconstant-Q mode, with kf fixed at 1.5 A−1.

Polarization analysis measurements were carried out onthe thermal neutron IN22 triple-axis spectrometer, operatedin the fixed-kf mode (kf = 2.662 A−1), with Cu2MnAlHeusler crystals as monochromator and analyzer. A pyrolyticgraphite filter was used for second harmonic rejection. TheCRYOPAD device21,22 was used to control the polarization Pi

of the incident neutron beam and to analyze the longitudinalcomponents of Pf , the polarization of the scattered beam.A flipping ratio R = 10.5±0.5 was estimated from the peakintensity of the transverse acoustic phonon mode measuredat Q = (2,2,0.3). A standard liquid-He cryostat was used tomaintain the sample at 2 K on both spectrometers.

In this work, the x axis of the reference frame is definedto be parallel to the momentum transfer Q = ki − kf , the zaxis is perpendicular to the scattering plane, and y forms aright-handed system.

A general expression of the cross section for polarizedneutrons, describing the final polarization Pf of a scatteredneutron beam as a function of the incident beam polarization

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Pi , was derived by Blume23 and Maleyev.24 For Pi and Pf

parallel to the scattering vector Q (x-x geometry), the magneticsignal contributes only to the spin-flip (SF) channel, that is, theneutron spin is reversed upon scattering. This does not occurin the case of vibrational scattering, which appears only in thenon-spin-flip (NSF) channel.

Note that, in a general case, multipolar waves mightbe detected directly by nondipolar INS. Magnetic neutronscattering is produced by fluctuations of time-odd multipolemoments such as magnetic dipoles or octupoles, and themultipolar contributions to the neutron cross section mayenable purely nondipolar excitations (with vanishing dipole-moment matrix element between initial and final states) tobe directly probed.25,26 This cross section is rather smallbut may be detectable, especially its octupolar component.However, in the specific case of UO2, there are no fluc-tuations of magnetic multipoles associated with the M =2 transition. Indeed, magnetic multipoles with nonvanishingmatrix elements within the CF ground state have �4 symmetry.Since �5 × �5 contains �4 only once, all these multipolesare proportional to one another and map onto the threepseudospin operators Sx , Sy , and Sz. Hence, the same selectionrules valid for dipoles hold for high-rank �4 multipoles, and M = 2 transitions are forbidden. Thus, as already stressedabove, the dispersion of quadrupolar waves can be mappedby conventional dipolar or vibrational neutron scattering byexploiting the fact that, when these M = 2 excitationspropagate, they may acquire a magnetic or phonon componentfrom nearby M = 1 or phonon branches.

IV. EXPERIMENTAL RESULTS

The theoretical model described in Sec. II predicts thepresence of more than two branches propagating along[00ξ ], with appreciable magnetic INS intensity near thecrystallographic zone center �. The extra branches have amixed spin, quadrupole, and phonon character. Aside from thequadrupolar-optical branch around 12 meV already discussedin Ref. 15, there are three peculiar features deserving anexperimental investigation by polarized neutrons. These are(i) the nature of the calculated excitations in the energyrange between ∼8 and ∼12 meV, the energies of which areconsistent with the experimental results reported in Ref. 5;(ii) the large magnon-phonon AC around ξ = 0.45; (iii) thepredicted quadrupolar-acoustical branch.

As far as point (i) is concerned, we focus on a representativevalue ξ = 0.3, where the different excitations between 8 and12 meV are sufficiently split apart to be identified with theIN22 energy resolution. In this energy and wave-vector range,the model predicts excitations belonging to SA, SO, and QObranches with small phonon cross section and, hence, onlymeasurable in the SF channel. As shown in Fig. 4(a), reportingspectra at Q = (1,1,0.7), intensity is indeed observed only inthe SF channel, and the spectra can be fitted to the sum of threeintense and not fully resolved Gaussian lineshapes. Hints ofa weak higher-energy peak are also found. The lowest-energypeak corresponds to the SA mode, whereas the central peakhas an energy equal to the one reported in Ref. 5. In ourmodel, the latter is associated with SO excitations. The peakat about 12 meV corresponds to the magnetic component of

FIG. 4. (Color online) Constant-Q energy scan recorded withIN22 at (a) Q = (1,1,1) − (0,0,0.3), and (b) Q = (2,2,0) + (0,0,0.3)(P||Q; T = 2 K). Red (filled) circles represent neutron counts in thespin-flip (SF) channel, open circles are non-spin-flip (NSF) counts.The SF data are fitted to the sum of four Gaussian lineshapes shownby dashed (blue) lines. The peak at 5 meV in the NSF channel atQ = (2,2,0.3) is a TA phonon (black dashed line). The correspondingportion of the calculated intensity map is shown as a guide to theenergies of the excitations. The vertical dashed (white) line gives theq value at which the energy scan has been performed.

the QO branch, which would vanish in the lack of two-ionquadrupolar interactions. The weak highest-energy featuremight correspond to a mixed SO-TA mode, involving a foldedTA branch having vanishing phonon cross section. Theseresults are confirmed by the data shown in Fig. 4(b), where wereport equivalent spectra measured under the same conditionsaway from G = (2,2,0). The peak at 5 meV, correspondingto a TA phonon, is observed only in the NSF channel atQ = (2,2,0.3), its intensity being negligible at Q = (1,1,0.7)because of the unfavorable structure factor ratio of about 20compared to Q = (2,2,0.3).

One of the most striking features in the low-energycollective mode spectra of UO2 is the large avoided crossingbetween SA and TA modes at ξ = 0.45 along [00ξ ] [point(ii) above]. Evidence of magnon-phonon interactions at thisposition was clearly given in the earliest investigation,3 butthe energy resolution of previous studies was not sufficientto separate the interacting modes. Here, using the IN14cold-neutron triple-axis spectrometer, we have been able toresolve the two excitations involved all along the AC region.

Figure 5 shows a sequence of constant-Q energy scanscollected at T = 2 K. The spectra have been measured at

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FIG. 5. (Color online) Dispersion along the [00ξ ] direction of themixed magnon-phonon modes in UO2 at 2 K, near the anticrossingposition at q = (0,0,0.45). The diagrams show detector countsnormalized to monitor as a function of the energy transfer measuredwith IN14 at Q = (1,1, − 1 + ξ ). A sloping background has beensubtracted from the raw data. The inset shows calculated intensitymaps in the energy-momentum transfer space [zoom of Fig. 2(b)].For ξ = 0.6, a small peak is visible at 6.5 meV (see Fig. 8).

reciprocal space positions Q = G + q, with G = (1,1,−1)and q = (0,0,ξ ). Close to the magnetic zone center, thecrystallographic X point at ξ = 1, only the SA excitation isobserved, but the two interacting modes are clearly visible inproximity of the AC position. For ξ > 0.45, the phononlikeexcitation is on the high-energy side, whereas it occurs onthe low-energy side for ξ < 0.45. The two modes are clearlydistinguished by the difference in width, which is due to thefact that their dispersion curves have a slope of opposite sign,corresponding to focusing and defocusing configurations.

The results of a longitudinal scan at Q = (0,0,−2) +(0,0,0.45) are shown in Fig. 6. In this spectrometer config-uration, a purely vibrational TA mode would have zero crosssection because the phonon polarization is perpendicular to Q.The fact that two peaks with comparable intensity are observedindicates that both excitations have 50% magnonlike and 50%phononlike character. As stated above, the magnon-phononcoupling is here mediated byQyz andQzx quadrupoles, havingnonzero matrix elements between the M = −1 and M = 0 MFsingle-ion states.

The origin of the excitations away from the AC canbe probed by comparing spectra measured at equivalentpositions in different Brillouin zones, and making use ofpolarization analysis. Magnetic and vibrational excitationsare well separated, the first appearing in the SF channel

FIG. 6. (Color online) Longitudinal constant-Q energy scan atthe anticrossing position q = (0,0,0.45). The experimental data havebeen measured with the IN14 spectrometer at Q = (0,0, − 1.55),with the sample kept at T = 2 K. A sloping background has beensubtracted from the raw data. The two Gaussian curves (solid, redlines) have equal width.

and the latter in the NSF one. Moreover, phonon intensitiesincrease with the square of the momentum transfer Q, whereasmagnetic intensities decrease with increasing Q following thesquare of the magnetic form factor f (Q). However, near an ACwave vector, peaks are visible in both SF and NSF channelsdue to their mixed magnetovibrational character.

Figure 7 shows transverse constant-Q energy scans takenwith IN22 at equivalent positions, away from G = (1,1,1)and G = (2,2,0) reciprocal lattice points. Data have beenmeasured in the x-x (P||Q) polarization geometry. The spectrain Figs. 7(a)–7(d) correspond to the quasiparticle wave vectorq = (0,0,0.40), and those in Figs. 7(e)–7(h) to q = (0,0,0.50).

The spectra can be fitted to the sum of three Gaussians.The peak at about 12 meV corresponds to the QO branch,whereas the two peaks at lower energy correspond to the sameexcitations shown in Fig. 5. In the fit, their energy has beenfixed to the values observed with better resolution on IN14.

At ξ = 0.4, the peak at 6.4 meV is strong in intensityonly in the NSF channel at Q = (2,2,0.4), and is thereforeof mainly vibrational character. On the other hand, the peakat 8.2 meV has the largest intensity in the SF channel atQ = (1,1,0.6), and is therefore of dominant magnetic origin.Nevertheless, the mixed character of the excitations is revealedby the observation in both polarization channels of intensitieslarger than leakage corresponding to a flipping ratio R of about10. At ξ = 0.5, the situation is reversed. The peak at 6.2 meVis now strongest in the SF channel at Q = (1,1,0.5), and ismainly magnetic, whereas the peak at 8.2 meV has the largestintensity in the NSF channel at Q = (2,2,0.5), and is thereforemainly vibrational in origin.

The theoretical model described in Sec. II predicts also theexistence of a QA branch [point (iii) above] that should becomenearly pure for ξ > 0.7. This mode can only be observed inthe INS spectra by mixing with either phonon or magneticbranches. Figure 8 shows the results of IN22 measurements atQ = (2,2,0.6), carried out to verify this possibility. In the SFchannel, we observe three peaks, corresponding to the spin-acoustic, spin-optical, and quadrupolar-optical branches, inincreasing order of energy. The SO branch is here mixed withTA phonons due to the large AC marked by the arrow in Fig. 2.Also, the NSF spectrum shows three peaks and, as discussed in

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0

2

4

6

8

FS

-1 ,1 ,1( = Q ξ)(a

rb. u

nits

)

(a)

(c) (d) (g) (h)

(b) (e) (f)FS

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ξ 4.0 =

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6

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FSN

-1 ,1 ,1( = Q ξ)

Inte

nsity FSN

+0 ,2 ,2( = Q ξ)

4121018642refsnarT ygrenE

FS

-1 ,1 ,1( = Q ξ)

FS

+0 ,2 ,2( = Q ξ)

ξ 5.0 =

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6141210186424121018642

FSN

-1 ,1 ,1( = Q ξ)

)Vem(

FIG. 7. (Color online) Constant-Q energy scans recorded with IN22 at (a)–(d) Q = (1,1,0.6) and Q = (2,2,0.4); (e)–(h) Q = (1,1,0.5)and Q = (2,2,0.5), corresponding to the quasiparticle wave vector q = (0,0,0.4) and q = (0,0,0.5), respectively. Experimental data are fittedto the sum of three Gaussian lineshapes, shown by dashed (blue) lines, and a constant background.

the following section, the peak at about 6.6 meV correspondsto a mixed QA-TA branch. We emphasize that the intensityobserved at 6.6 meV in the NSF channel is a factor 10 largerthan the leakage from the SF channel expected for a flippingratio R of about 10.5. The weak feature appearing at about6.5 meV in the equivalent scan recorded with IN14 at Q =

FIG. 8. (Color online) Constant-Q energy scan recorded withIN22 at Q = (2,2,0) + (0,0,0.6) (P||Q; T = 2 K). (a) Neutron countsin the spin-flip (SF) channel; individual components of the spectrumare shown by dashed lines. (b) Non-spin-flip (NSF) neutron counts.The central peak at ∼9 meV (blue dashed line) is a transverse acousticphonon, whereas the weaker peaks at ∼6.6 meV and ∼11.4 meV(red full lines) correspond to mixed phonon-quadrupolar acousticmodes.

(1,1,−0.4) (see box in Fig. 5) provides further confirmationof this finding. On the other hand, a scan with high countsstatistics performed at the magnetic zone center Q = (0,0,1)only shows the SA branch at about 2.4 meV, with no evidenceof further excitations between 3 and 7 meV (Fig. 9).

V. DISCUSSION

As Fig. 2(b) well illustrates, the dynamics of UO2 can notbe rationalized in terms of a simple paradigm, being the resultof an intricate interplay of several sorts of motions. While allterms in Eqs. (1) and (2) contribute in a complex nonlinearway in determining the actual composition and dispersion ofelementary excitations, a few general qualitative remarks canbe made. The first one is that ACs involving phonons are adirect manifestation of the magnetoelastic coupling Eq. (2), asthese would clearly not exist in the lack of such coupling. Forinstance, Fig. 2(c) shows how the INS cross section in Fig. 2(b)

FIG. 9. (Color online) Constant-Q energy scan recorded withIN14 at Q = (0,0,1), corresponding to the X point of the crystallo-graphic cell. The experimental data are fitted to the sum of a Gaussianlineshape and a sloping background. The experimental error on theintensity is smaller than the size of the symbols.

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changes when the ME coupling is removed by keeping allremaining parameters fixed. Phonon and electronic modes areseparated, and in the latter the bare SA, SO, QA, and QObranches are clearly recognizable. Even in the lack of MEcoupling, however, the quadrupolar branches get a magneticcomponent by the two-ion quadrupolar superexchange. Thealternative calculation with quadrupolar SE removed whilekeeping the ME coupling is more problematic. In fact, SE hasthe leading role in stabilizing the ordered phase as it accountsalmost entirely for the quadrupolar MF constant.15 Hence,removing quadrupolar SE in the dynamics by leaving all otherparameters unchanged seriously spoils the self-consistency ofthe calculation.

The theoretical model predicts several features that werenot clearly observed in previous experiments and that areconfirmed by the results presented here. In particular, asidefrom confirming the QO character of the highest-energy branchpropagating in the [00ξ ] direction, the experiments provideevidence for the presence of a QA branch. Its magneticcross section is, in general, very weak, apart from ξ ∼0.3–0.4, where it joins the QO excitations and is unfortunatelyindistinguishable from the latter in view of the finite INSenergy resolution. However, in between ξ ∼ 0.3 and ξ ∼0.7, the QA branch experiences a very large AC with TAphonons to reemerge as a nearly pure quadrupolar mode forξ ∼ 0.7. The expected magnetic intensity is extremely weak,and in fact this part of the branch is hardly visible in themap shown in Fig. 2 and it is not observed at Q = (0,0,1)(Fig. 9). On the other hand, more favorable conditions aremet inside the AC range, where the QA mode is expected tomix with one of the TA phonon branches. In particular, ξ ∼0.6 is a good possible choice as the mode is fairly isolatedin energy. Indeed, in Fig. 8(b), the central peak at ∼9 meV(blue dashed line) is due to the TA-phonon component of themixed TA-SO mode, the peak at ∼11.4 meV corresponds tothe phonon component of the three mixed branches markedby white dashed lines in Fig. 2, and the weaker peak at∼6.6 meV corresponds to the mixed QA-TA branch. At themeasured position, q = (0,0,0.6), this branch has a sizablevibrational component, and can therefore be detected in theNSF polarization channel. The peak at 6.6 meV provides,therefore, indirect evidence for the QA mode, as its intensitywould be zero if the QA branch did not exist. The peak isalso detected at ξ = 0.7 at an energy of about 5.8 meV. Themeasured energy of the mode is slightly smaller than predicted(∼ 7.7 meV for ξ = 0.6 and ∼ 7.2 meV for ξ = 0.7).

As Fig. 10 shows, the excitations so far identified by INSalong �X fit well the structure of the calculated branches: themodes near 12 meV correspond to the QO branch, whereasthe modes in the energy range between ∼8 and ∼12 meV,first reported in Ref. 5, correspond to SO branches and,possibly, to the first part of the QA branch. With increasingξ , the latter experiences a large AC with phonons, whichenables its detection around ξ = 0.6 and becomes nearlypurely quadrupolar as ξ → 1. Of course, given the simplifiedparametrization, the model can not be expected to provide aprecise quantitative fit. The calculated SA branch is slightlytoo low at X, and the comparison with the data from Ref. 5points to an underestimation of the SO dispersion. Indeed, inmoving away from � and X, the energy of these branches

FIG. 10. (Color online) Left panel: excitations in the low-temperature phase of UO2 along �X. Circles are the same as inthe left panel of Fig. 1, whereas blue diamonds are the excitationsdiscussed in this paper. Right panel: dipolar magnetic INS crosssection along (1,1,1 − ξ ), calculated by the present model. Dashedwhite lines show modes with sizable phonon component and crosssection. Note that, with (1,1,1 − ξ ) being equivalent to (0,0,ξ ), thispanel describes the same excitations as panel (b) in Fig. 2. However,the INS cross section is slightly different and both SO modes arevisible.

appears to vary more quickly than calculated. For the QAbranch also, the dispersion appears to be underestimated. Suchdiscrepancies are entirely reasonable in view of the assump-tions we made and of the complex way dispersion curvesemerge.

An interesting and somewhat counterintuitive result of oursimulations is the key role played by high-E optical phononsin the low-E dynamics. One might expect only minor effects inview of the large energy mismatch between these phonons andthe bare spin and quadrupolar branches. However, the largemodulation of the oxygen cage provided by the displacementpattern of some optical vibrations [i.e., a large �ασ (q) inEq. (3)] enables important second-order mixing effects inspite of the large energy differences. For instance, we showin Fig. 2(d) a calculation where high-E phonons have beenartificially removed by leaving all parameters unchanged.The spectrum is very different from that of Fig. 2(b) and isdominated by first-order mixing between SA, QA, and SObranches on one side, and TA phonon branches on the otherside, leading to three clearly recognizable ACs. If opticalphonons are included, these ACs evolve into the three largestACs marked in Fig. 2(b). Note how the second-order mixingwith these phonons pushes down in energy all branches, andparticularly the QO. The result is that the mode intersects theTA phonons and produces the fourth smallest AC in Fig. 2(b).Both the center and the width of the SA-TA AC at ξ ∼ 0.45are affected by the second-order mixing.

VI. CONCLUSIONS

In this paper, we report the results of extensive investi-gations on the collective quasiparticle dynamics in antiferro-magnetic UO2. Inelastic neutron scattering (INS) measure-ments have been performed with unpolarized-cold-neutronand polarized-thermal-neutron triple-axis spectrometers at the

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ILL. By exploiting the high-energy resolution offered by coldneutrons, we have characterized magnon-phonon interactionsin much greater detail than in previous investigations. Themagnon-phonon coupling is particularly strong along the [00ξ ]direction at ξ = 0.45. The avoided crossing involves the spinacoustic and one of the transverse acoustic phonon branches.Polarization analysis allowed us to shed light on the mixedcharacter of the observed excitations. The experimental resultshave been discussed against the predictions of a mean-fieldrandom phase approximation theory that provides convincingexplanations for the several anomalous features in the UO2

low-temperature dynamics that have resisted interpretation foralmost half a century.

Our results demonstrate the existence of an optical branchcorresponding to propagating quadrupolar waves, and provideevidence on a mixed vibrational-quadrupolar-acoustic branch.We also confirm the importance of superexchange multipolar

interactions in determining the dynamics of systems withunquenched orbital degrees of freedom. Quadrupolar wavesare not easily detected and have so far remained an elusiveentity. In our experiments, the uranium atomic spins arecaught in the act of surfing on quadrupolar waves thatcan therefore be detected by INS through the perturbationinduced in the spin system. In f -electron materials, orbitaldegrees of freedom play a crucial role in a number of exoticphenomena, ranging from unconventional superconductivityto heavy-fermion behavior. The observation of multipolarwaves is a key finding for the science and technology of thisimportant class of materials.

ACKNOWLEDGEMENT

We thank the ILL for the award of beamtime on the IN14,IN22, and IN3 spectrometers.

*Current address: European Spallation Source ESS AB, P.O. Box 176,SE-22100 Lund, Sweden.1P. Santini, S. Carretta, G. Amoretti, R. Caciuffo, N. Magnani, andG. H. Lander, Rev. Mod. Phys. 81, 807 (2009).

2G. Dolling, R. A. Cowley, and A. D. B. Woods, Can. J. Phys. 43,1397 (1965).

3G. Dolling and R. A. Cowley, Phys. Rev. Lett. 16, 683 (1966).4R. A. Cowley and G. Dolling, Phys. Rev. 167, 464 (1968).5W. J. L. Buyers and T. M. Holden, in Handbook on the Physicsand Chemistry of the Actinides, Vol. 2, edited by A. J. Freeman andG. H. Lander (Elsevier, Amsterdam, 1985), p. 239.

6R. Caciuffo, G. Amoretti, P. Santini, G. H. Lander, J. Kulda, andP. de V. Du Plessis, Phys. Rev. B 59, 13892 (1999).

7G. Amoretti, R. Caciuffo, P. Santini, G. H. Lander, J. Kulda, andP. de V. Du Plessis, J. Appl. Phys. 85, 4524 (1999).

8E. Blackburn, R. Caciuffo, N. Magnani, P. Santini, P. J. Brown,M. Enderle, and G. H. Lander, Phys. Rev. B 72, 184411 (2005).

9R. Caciuffo, N. Magnani, P. Santini, S. Carretta, G. Amoretti,E. Blackburn, M. Enderle, P. J. Brown, and G. H. Lander, J. Magn.Magn. Mater. 310, 1698 (2007).

10J. Faber, G. H. Lander, and B. R. Cooper, Phys. Rev. Lett. 35, 1770(1975).

11S. B. Wilkins, R. Caciuffo, C. Detlefs, J. Rebizant, E. Colineau,F. Wastin, and G. H. Lander, Phys. Rev. B 73, 060406 (2006).

12P. Burlet, J. Rossat-Mignod, S. Quezel, O. Vogt, J. C. Spirlet, andJ. Rebizant, J. Less-Common Met. 121, 121 (1986).

13P. Giannozzi and P. Erdos, J. Magn. Magn. Mater. 67, 75 (1987).14V. S. Mironov, L. F. Chibotaru, and A. Ceulemans, Adv. Quantum

Chem. 44, 599 (2003).15S. Carretta, P. Santini, R. Caciuffo, and G. Amoretti, Phys. Rev.

Lett. 105, 167201 (2010).16J. Jensen and A. R. Macintosh, Rare Earth Magnetism (Oxford

University Press, New York, 1991).17N. Magnani, P. Santini, G. Amoretti, and R. Caciuffo, Phys. Rev. B

71, 054405 (2005).18C. Y. Huang and M. Inoue, J. Phys. Chem. Solids 25, 889 (1964).19G. A. Gehring and K. A. Gehring, Rep. Prog. Phys. 38, 1 (1975).20P. Santini, S. Carretta, N. Magnani, G. Amoretti, and R. Caciuffo,

Phys. Rev. Lett. 97, 207203 (2006).21P. J. Brown, J. Forsyth, and F. Tasset, J. Phys. Condens. Matter 10,

663 (1998).22F. Tasset, Phys. B (Amsterdam) 297, 1 (2001).23M. Blume, Phys. Rev. 130, 1670 (1963).24S. V. Maleyev, V. G. Bar’yakhtar, and R. A. Suris, Sov. Phys. Solid

State 4, 2533 (1963) [Fiz. Tverd. Tela (Leningrad) 4, 3461 (1962)].25P. M. Levy and G. T. Trammell, J. Phys. C: Solid State Phys. 10,

1303 (1977).26M. J. Sablik, Phys. Rev. B 32, 5880 (1985).

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