Nottingham Trent University ILL/ESRF Summer School Programme Lattice - - PDF document

Nottingham Trent University

ILL/ESRF Summer School Programme

Lattice Dynamics studies of UO2 using IXS

Myron Huzan

Supervisor: Luigi Paolasini Summer School Coordinators: Paul Steffens Patrick Bruno Laurence Tellier 4th September 2017 - 29th September 2017

ILL/ESRF Summer School Programme

1

Contents

1 Introduction 3 1.1 Uranium Dioxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Inelastic X-Ray Scattering 4 2.1 Lattice Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 ID28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 Data Analysis 6 3.1 Transverse Acoustic Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Multi-Peak Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4 Conclusions 11 5 Acknowledgements 11

ILL/ESRF Summer School Programme

2

1 Introduction

1.1 Uranium Dioxide

The most common use for Uranium Dioxide (UO2) is as fuel in control rods, and since the 1960s extensive research of the lattice dynamics have been undertaken due to the observed unique elec- tronic and magnetic properties of the material [1]. There exist numerous competing and coinciding structural and magnetic phenomena, of which the main processes will be discussed in this introduction. A magnetic transition temperature, TN, was first discovered to exist in antiferromagnetic materials by Louis N´ eel in 1934 [2] and is now know as the N´ eel Temperature. This transition occurs when the thermal energy of the sample is great enough to overcome the macroscopic magnetic domain ordering. UO2 transition temperature exists at 30.7K below which the material orders from paramagnetic to anti-ferromagnetic, Figure 1, 2 [3]. Figure 1: Paramagnetic FCC Fluorite structure Figure 2: Anti-Ferromagnetic ordering above TN The simplest anti-ferromagnetic structuring exists as collinear 1-k, in which planes of magnetic mo- ments are parallel with respect to each other. This ordering can extend to 2 dimensions (e.g. x-y) producing a non-collinear 2-k ordering and even, as exists in UO2, a 3-dimensional anti-ferromagnetic

• rdering, Triple-K magnetic ordering [4]. Below TN this phenomenon adds additional complexi-

ties in the understanding of the material. Figure 3: Representation of multiple ordering within anti-ferromagnetic structures. In addition to the 3-k magnetic ordering there exists a 3-k structural ordering of the atoms below TN due to the Jahn-Teller Effect [5]. This spontaneous structural reordering of the atoms occurs to reduce the energy and symmetry of the material by removing any degeneracy in the molecule.

ILL/ESRF Summer School Programme

3

Symmetrically across the molecule two O2− atoms along the 111 plane move δ=0.014˚ A [3] closer to each U4+ atoms, however still keeping the same symmetry and volume as the initial unit cell. Uranium Dioxide possesses additional material properties which are responsible for its unique struc- tural and magnetic properties; however, of primary interest of this investigation is the proposed quadrupolar waves [6]. Quadrupole moments exist within the atoms of UO2 due to an asymmetri- cal distribution of charge within the nucleus resulting in nuclear spin, I> 1

• 2. The propagation of these

waves are theorised to travel due to distortion of the atoms, and it is this phenomenon Inelastic X-Ray Scattering (IXS) hopes to probe.

2 Inelastic X-Ray Scattering

IXS probes the motion of the electronic cloud of a nucleus, and the 92 electrons present in uranium greatly masks the scattering contribution of the 8 electrons from oxygen atoms. Therefore, IXS will predominantly provide structural information about the uranium atoms. Comparatively, Inelastic Neutron Scattering (INS) probes the nuclear motion of the atoms, and the nuclear scattering cross-section of uranium and oxygen is significantly more comparable and provides separation between the two atom sites. Additionally, INS polarisation analysis provides magnetic or- dering information within the material, which is unable to be probed with IXS. However, comparison

• f IXS and INS can distinctly prove if excited modes are magnetic or vibrational.

For comparison of IXS and INS, an adiabatic approximation is considered, which assumes that the electrons of each atom travel with the nuclei. This approximation is particularly valid for hard X-Rays since scattering occurs between core electrons. X-Ray focussing is achieved by Beryllium compound refractive lenses to produce a minimum incident spot size of 20x10µm2 (H x V) on the ID28 beamline. This focusing is significantly smaller than anything feasible on INS beamlines, and as a consequence, the smaller sample environments enable diamond anvil cells to investigate pressures in excess of 100GPa and experiments undertaken with smaller, single crystal samples. Currently, 3rd generation synchrotrons provide a significantly increased Brilliance, (Eqn. 1) to any neutron spallation or reactor sources, and with the further upgrades of ESRF and 4th generation synchrotrons the flux is substantially higher with X-Rays. However, both IXS and INS provide very complimentary methods of investigating the lattice dynamics of materials and within this report results from both techniques will be compared and discussed for UO2. B = Photons/sec [mrad2] × [mm2] × [0.1%BW] (1)

2.1 Lattice Dynamics

Phonons are the propagation of vibrational energy within the crystal due to oscillations of atoms. There exist three modes of vibration, one longitudinal and two transverse. However, due to the exper- imental configuration, only 1 of the transverse phonon excitations should be excited since the reciprocal space being investigated was such that the reduced momentum transfer, q, is nearly perpendicular to the reciprocal lattice vector Ghkl. Additionally, since there exist two atoms within the unit cell, Uranium and Oxygen, there will exist an acoustic and optic phonon branch due to the in-phase and

ILL/ESRF Summer School Programme

4

• ut-of-phase vibrational motions respectively.

Magnons are quantised spin waves travelling through the crystal due to disturbances of the electron’s

• spin. X-Rays do not possess a magnetic moment, therefore cannot directly probe the magnetic struc-

ture in the form of spin waves. However, there do exist additional X-Ray techniques such as X-Ray Magnetic Circular Dichroism (XMCD) which provides magnetic ordering from X-Rays. Disruption to quadrupole moments within uranium will distort the oxygen cage in which surrounds

• it. This distortion is proposed to propagate through the sample in the form of a quadrupolar wave

and is hoped to be probed directly with IXS. Simulations calculated using a mean-field random phase approximation from Cacuiffo [7], present energy modes of Phonons, Magnons and Quadrupolar excitations neglecting all coupling interactions, Figure 4 a). However, complications of these modes are increased due to the couplings between the modes causing avoided crossings within the dispersion curves Figure 4 b). These branches of the simulation are what INS and IXS hope to validate or disprove to further develop the current theoretical model of UO2. Figure 4: Intensity map of the theory of dispersion curves for UO2 as modelled by Caciuffo a) Independent branches of non-interacting modes b) Interacting modes creating avoided crossings, yellow arrows.

2.2 ID28

X-Rays generated by guiding electrons from the synchrotron beam through undulators are directed towards the respective beamlines. Collimation and pre-monochromators reduce the bandwidth of the X-Ray spectrum before incidence onto the main backscattering monochromator. Silicon is chosen as the monochromator and selection of the high-order Bragg reflection enables control over the X-Ray flux, Energy Resolution and Energy. Bragg reflection (999) was selected for investigation of UO2 which provided an X-Ray energy of 17.794KeV and energy resolution of 3meV [8]. Selection of the

ILL/ESRF Summer School Programme

5

high-order Bragg reflection requires a compromise of the necessary resolution and flux, which varies dependent on the experiment undertaken. Figure 5: Schematic of the ID28 beamline specialising in Inelastic Scattering at ESRF Complete mapping of the available reciprocal space is possible if calibration of the sample is performed to two known Bragg reflections. One known reflection occurs at (600) for UO2 which satisfies the Laue condition and hence an elastic reflection. For investigation of the lattice dynamics, an inelastic scan was undertaken from G600 to G610. An inelastic scan is achieved by selecting the desired q value within the reciprocal space, fixing the analysers energy value (kf), performing an energy scan of incident X-Rays (ki) and measuring the intensity of reflections. Energy selection is achieved by thermally controlling at mK precision the silicon monochromator to control the lattice space, dhkl from which reflects different X-Ray energies

• nto the sample due to Bragg reflections, Eqn. 2.

λ = 2dhklsin(θ) (2)

3 Data Analysis

3.1 Transverse Acoustic Fitting

Raw data gathered from the ID28 beamline was initially processed within a custom-made macro within MatLab, addIXS, from which calibration of the incident energies (ki) are scaled to their normalised values. Impurities within a sample cause an elastic peak to exist at ∆E=0, thus providing a zero-energy

• calibration. However, the purity of the single crystal UO2 dictated that the elastic peak was indis-

tinguishable within the inherent background of the IXS data. Fortunately, scans at 300K produce symmetric inelastic peaks corresponding to ±Energy. +E indicates energy being gained by the sample from the X-Ray, whereas -E symbolises the sample was providing additional energy to the X-Ray. Due to the assumed symmetric nature of the peaks, a median was calculated and scaled to provide ∆E=0. Temperature scans at 40K and 5K, however, did not possess the -E peaks due to the sample not having enough thermal energy to provide to the X-Ray. Therefore the same calibration shifts for equivalent q scans were repeated for the lower temperature scans.

ILL/ESRF Summer School Programme

6

Figure 6: Raw Data (a), Gaussian fits to -E (b) and +E (c) peaks and scaled calibration (d) of 300K (6 0.5 0) IXS scan of UO2 performed in MatLab The analysis was achieved within an ID28 data analysis program, fit28, in which Damped Harmonic Oscillation (DHO) was fit to the +E peaks. Measured fitting parameters included Intensity, Peak Energy and FWHM. A 3meV resolution was limited by the monochromator and analyser, however deconvolving the data and calibration of the instruments provided a measure of the FWHM of the lattice modes. Intensity is dictated by the Bose-Einstein statistics, and the reflection intensity data is scaled by the Bose-Factor (Eqn. 3), and the corresponding Peak Energy represents the appropriate energy mode of the excitation. nq + 1 = 1 e

E kBT − 1

+ 1 (3)

ILL/ESRF Summer School Programme

7

Figure 7: Single peak DHO analysis of UO2 at 300K a), 40K b) and 5K c) at (6 0.5 0) 3.1.1 Results Due to the purity of the UO2 sample, a calibration shift calculated for 300K was repeated for the 40K and 5K measurements. This assumption was validated due to the intensities of each temperature experiment overlaying each other, due to the normalised Bose-Factor, Fig 8 a). As the transition temperature for UO2 exists at 30.7K, dispersion curves for 300K and 40K scans are expected only to show transverse acoustic (TA) phonon modes. Therefore, the anomaly at 5K q-0.6 shows an indicative peak shift likely due to additional dispersion modes. Further validated by the corresponding FWHM increase at 5K when compared to 300K. Intuitively, the broadening

• f the FWHM shows that there should exist additional peaks, and thus modes occurring at the

corresponding reciprocal lattice position. Additionally, there exists an increase of FWHM for 40K; this could be explained as a pre-cursor effect occurring close to the transition temperature; however, further investigation would be required.

ILL/ESRF Summer School Programme

8

Figure 8: a) Intensity relationship b) Dispersion curve c) FWHM of temperature measurements 300K (—) 40K (—) and 5K (—)

3.2 Multi-Peak Fitting

Preliminary analysis investigating multiple-peak fittings of 5K UO2 IXS data was undertaken by ex- tending the fit28 DHO fitting to include multiple peaks. Comparison of INS multi-peak spectra fitting [7] provided an outline of the number of peaks and estimations of energy peaks for the additional modes expected. Additionally, vertical slices of corresponding q values provided approximations of the peaks from simulated theory, Figure 4 b). Multi-peak fitting shows an improvement of the DHO analysis to provide additional lattice dynamic

• modes. There is a debate about the validity of these fittings due to the low statistics that are being fit

to, however, as a preliminary analysis of multi-peak fitting it provides an adequate baseline of where to extend the analysis and experimental method in future experiments.

ILL/ESRF Summer School Programme

9

Figure 9: Multi-Peak DHO analysis at 5K of (6 0.4 0) a), (6 0.5 0) b) and (6 0.6 0) c). 3.2.1 Results Direct comparison of IXS and INS measurements presented comparable results of the expected disper- sion branches for UO2, Figure 10. TA phonons are expected to correspond to the dashed line with a branching off at 14meV, validated by the IXS results. However, spin waves cannot be directly investi- gated by IXS unless coupled to phonons or quadrupoles. The phonon-magnon branch is evident with an avoided crossing at 7meV, and there is an argument that a quadrupolar branch is observed (q-0.6, E=6meV). However, as stated previously the accuracy of this analysis is insufficient to conclude any clear results.

ILL/ESRF Summer School Programme

10

Figure 10: Dispersion curves with IXS data (•) overlaying INS data published by Caciuffo

4 Conclusions

INS and IXS provide complimentary techniques of investigating the dynamics of UO2. Single DHO fitting provides evidence of additional modes below the transition temperature due to the increase

• f FWHM which is indicative of additional peaks while intensity remains constant. These additional

modes are theorised to be the presence of quadrupolar waves within the sample in addition to the phonons and magnons present in UO2. Preliminary multi-peak DHO fitting was undertaken and provided comparisons of the IXS and INS dispersion curves; this will enable future IXS experiments a more refined region of interest to investigate with higher resolution and statistics to further refine the results and develop the theory to evolve the model currently accepted for UO2.

5 Acknowledgements

I would like to extend my gratitude to Luigi Paolasini who supervised me throughout my time at the ID28 beamline at ESRF; Paul Steffens, Patrick Bruno and Laurence Tellier for organising the summer school and the sixteen other students who attended and made Grenoble a thoroughly enjoyable experience.

ILL/ESRF Summer School Programme

11

References

[1] R. Cowley and G. Dolling. Magnetic Excitations in Uranium Dioxide. Physical Review, 1968. [2] L. Neel. Magnetism and Local Molecular Field. Science, 174(4013):985–992, 1971. [3] V.S Mironov, L.F Chibotaru, and A Ceulemans. First-order Phase Transition in UO2: The Inter- play of the 5f2–5f2 Superexchange Interaction and Jahn–Teller Effect. In Advances in Quantum Chemistry, vol. 44, pp. 599-616, volume 44, pages 599–616. 2003. [4] P. Burlet, J. Rossat-Mignod, S. Vuevel, O. Vogt, J. C. Spirlet, and J. Rebivant. Neutron diffraction

• n actinides. Journal of The Less-Common Metals, 1986.

[5] H. A. Jahn and E. Teller. Stability of Polyatomic Molecules in Degenerate Electronic States. I. Orbital Degeneracy. Proceedings of the Royal Society A: Mathematical and Physical Sciences, 1937. [6] S. Carretta, P. Santini, R. Caciuffo, and G. Amoretti. Quadrupolar waves in uranium dioxide. Physical Review Letters, 2010. [7] R. Caciuffo, P. Santini, S. Carretta, G. Amoretti, A. Hiess, N. Magnani, L. P. Regnault, and G. H.

• Lander. Multipolar, magnetic, and vibrational lattice dynamics in the low-temperature phase of

uranium dioxide. Physical Review B - Condensed Matter and Materials Physics, 2011. [8] Alfred Q R Baron. Introduction to High-Resolution Inelastic X-Ray Scattering.

ILL/ESRF Summer School Programme

12