James T.
Pegg
*ab,
Ashley E.
Shields
c,
Mark T.
Storr
b,
Andrew S.
Wills
a,
David O.
Scanlon
ade and
Nora H.
de Leeuw
af
aDepartment of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK. E-mail: pegg.james.t@gmail.com
bAtomic Weapons Establishment (AWE) Plc, Aldermaston, Reading RG7 4PR, UK
cOak Ridge National Laboratory, One Bethel Valley Road, Oak Ridge, Tennessee 37831, USA
dDiamond Light Source Ltd., Diamond House, Harwell Science and Innovation Campus, Didcot, Oxfordshire OX11 0DE, UK
eThomas Young Centre, University College London, Gower Street, London WC1E 6BT, UK
fCardiff University, School of Chemistry, Main Building, Park Place, Cardiff, CF1D 3AT, UK
First published on 26th July 2018
A thorough understanding of the chemistry of PuO2 is critical to the design of next-generation nuclear fuels and the long-term storage of nuclear materials. Despite over 75 years of study, the ground-state magnetic structure of PuO2 remains a matter of much debate. Experimental studies loosely indicate a diamagnetic (DM) ground-state, whereas theoretical methods have proposed either a collinear ferromagnetic (FM) or anti-ferromagnetic (AFM) ground-state, both of which would be expected to cause a distortion from the reported Fmm symmetry. In this work, we have used accurate calculations based on the density functional theory (DFT) to systematically investigate the magnetic structure of PuO2 to resolve this controversy. We have explicitly considered electron-correlation, spin–orbit interaction and noncollinear magnetic contributions to identify a hereto unknown longitudinal 3k AFM ground-state that retains Fm
m crystal symmetry. Given the broad interest in plutonium materials and the inherent experimental difficulties of handling this compound, the results presented in this paper have considerable implications for future computational studies relating to PuO2 and related actinide structures. As the crystal structure is coupled by spin–orbit interactions to the magnetic state, it is imperative to consider relativity when creating computational models.
The low-temperature characterisation of the PuO2 magnetic ground-state is challenging, due to: nucleonic radioactive decay,25 the toxicity of the metals,26 and the inhomogeneity of samples.7 A number of inconsistencies in the literature have been recorded.27 For example, a singlet Γ1 DM ground-state (in accordance with the one-electron crystal-field model) has been inferred by: magnetic susceptibility (T = 4 K),4 inelastic neutron scattering (INS) (T > 30 K),5,6 and nuclear magnetic resonance (NMR) (T > 4 K)7,8 measurements. Although a number of inconsistencies in the literature have been recorded.27 A singlet Γ1 ground-state has been inferred from the temperature-independent magnetic susceptibility of PuO2 (5.4 × 10−4 cm3 mol−1).1 In contrast, the temperature-dependent magnetic susceptibility of Th1−xPuxO2 (∼1.1 × 10−3 cm3 mol−1) indicates a magnetic response that is inconsistent with earlier results.28 INS measurements of PuO2 indicate a singlet Γ1 to triplet Γ4 transition of 123 meV, which corresponds to a temperature-dependent magnetic susceptibility of ∼1 × 10−3 cm3 mol−1 when T > 400 K. The magnetic coupling of proximal cations by intermediary DM anions is described by magnetic superexchange.29 The full width at half maximum (FWHM) INS peak results of 25 meV is indicative of AFM superexchange.1,30 In addition, magnetic susceptibility measurements of U1−xPuxO2 indicate AFM superexchange with high concentrations of plutonium.31 Finally, the low-temperature 17O NMR measurements of PuO2 result in an apparent DM state at 6 K and it is conceivable that even lower temperatures may result in a different magnetic ground-state. For instance, low temperature measurements of U1−xPuxO2 indicate a Néel temperature under 5 K when x = 0.5.31 To compensate for experimental issues (thermal energy released by the decay of actinide nuclei), theoretical methods offer an alternative and complementary means of investigation.
A computational investigation of the PuO2 systems is highly complicated. This is due to exchange–correlation effects, relativistic influences and magnetic behaviour. It is now well recognised that in highly correlated f-electron systems, conventional DFT32,33 methods fail to reproduce the correct electronic structure. For instance, identifying finding well-known AnO2 insulators as metallic by underestimating the band-gap.17,21 To calculate highly-correlated materials, a number of methods offer a correction to the self-interaction error. These include the self-interaction correction (SIC) method,34 modified density functional theory (DFT+U),32,33,35–37 dynamic mean field theory (DMFT),38,39 and hybrid density functionals.12,40,41 In condensed matter physics, the DFT+U method offers a computationally tractable means of study, where the on-site Coulomb repulsion of the f-electrons is treated by tuneable U and J modifiers. In contrast, although the use of hybrid functionals are computationally expensive, they represent one of the more accurate methods available.
DFT studies have consistently reported either a collinear ferromagnetic (FM) or a collinear antiferromagnetic (AFM) ground-state.3 The ordered magnetic states have been attributed to approximations inherent to DFT methods.42 In response to the highly-controversial ordered magnetic structures, a number of investigations have focused on the DM solution.42,43 As a combination of the exact diagonalization of the Anderson impurity model and LDA+U (U = 6.5 eV), correlated band theory yielded a DM solution with an f-shell occupation of ∼4.5 at the Pu atoms. The fully spherical symmetric Coulomb interaction, crystal field, and SOI were included in the multiorbital impurity model.43 The authors found an exited triplet state at 126 meV; however, a complete investigation of ordered magnetic states is needed.
In cases where an ordered magnetic state has been incorporated within the DFT methodology, the investigators have limited the discussion to FM(001) and collinear 1k AFM states.2,3,9–21 As the actinide elements are heavy-fermion systems, SOI is imperative for the computational treatment of nuclear materials.44 A limited number of studies have included SOI (due to the high computational cost) in their calculations.18–21,42,43 The inclusion of noncollinear magnetic behaviour PuO2 has been ignored.
In addition to general complications associated with highly-correlated systems, the magnetic ground-state of the AnO2 is influenced by superexchange, multipolar and noncollinear interactions.1,27 Other methods have been used to model PuO2 systems. The complete active space (CAS) wavefunctions to correct the exchange–correlation errors common to DFT can be used to achieve a DM ground-state.45 A limitation of the embedded cluster method, however, is that the quantum cluster may not be sufficiently large enough to capture the superexchange interactions in PuO2, and therefore the findings are incomplete. In addition, the ground-state electronic structure and crystal-field levels of PuO2 have been calculated by DFT in terms of the total energy differences.30 The total energy of the crystal-field states are calculated by the Clebsch–Gordon method from a standard model of rare-earth elements and as a function of their single-electron components. The calculated ground Γ1 state-excited Γ4 state transition of 99 meV compares with the measured INS result of 123 meV.
In the current study, the magnetic structure of PuO2 has been investigated by noncollinear relativistic calculations. We have implemented three levels of theory. These include density functional theory (PBEsol),32,33 DFT with the Hubbard correction (PBEsol+U),32,33,35–37 and hybrid (HSE06) functionals.12,40,41,46 To date, noncollinear contributions to the magnetic ground-state have remained unexplored, and we have therefore investigated the noncollinear interactions illustrated in Fig. 2. To highlight the importance of SOI, we have compared with non-SOI HSE06 calculations (see additional information, ESI†).
As a computationally intensive method, hybrid functionals incorporate Hartree–Fock (HF) exchange energy into the DFT formulism. In this study, the hybrid Heyd–Scuseria–Ernzerhof (HSE06) functional is employed.12,40,41,46 The functional offers improved: thermochemical data, band-gaps (for low-medium semi-conductors) and lattice constants.
EHSEXC = (a)EHF,SRX(μ) + (1 − a)EPBE,SRX(μ) + EPBE,LRX(μ) + EPBEC. | (1) |
As a computationally more tractable means of investigation, the DFT+U32,33,35–37 method addresses the on-site Columbic repulsion of the Pu f-electrons. The correction of the electron self-interaction error (SIE) implicit in DFT calculations, is treated by the rotationally invariant Liechtenstein et al.36 formulism. In the rotationally invariant Liechtenstein et al.36 formulism, the on-site Coulombic (U) and exchange (J) modifiers are treated as independent variables.
![]() | (2) |
In the past, the influence of J on noncollinear magnetic materials has been investigated.44,54 The introduction of J increased the anisotropic nature of the f-states.42 As such, in the current study, the J modifier is kept at a constant value of 0.00 eV.44 By comparison, when J = 0.00 eV, the Dudarev et al.35 formulism and Liechtenstein et al.36 formulism are equivalent.54 The integration of reciprocal space was completed with a Γ-centred k-point (5·5·5) mesh52 by Blöchl tetrahedron method.53 The iteration threshold for electronic and ionic convergence is 1 × 10−8 eV and 1 × 10−3 eV Å−1, respectively. The DFT32,33 and DFT+U35–37 calculations are performed with PBE revised for solids (PBEsol)55 exchange–correlation functional. In the calculation of actinide materials, the functional has shown improved results.44 For the optical absorbance calculation, the k-point mesh is 15 × 15 × 15. For the band structure calculations, the Fmm k-point pathway of high symmetry points in the Brillion zone is defined as Γ → L → W → X → Γ. The density of states and band structure have been illustrated using the SUMO code, a command-line plotting tool for ab initio calculations.56
In the Russell–Saunders coupling scheme, under the cubic crystal-field, the degeneracy of the 5I4 multiplet is split into ascending Γ1 (1), Γ4 (3), Γ3 (2), and Γ5 (3) crystal-field levels (the number in the parentheses denotes the degeneracy).30,42,43 The calculated singlet Γ1 ground-state to triplet Γ4 excited state transition is 123 meV as measured by neutron inelastic scattering.6 The crystal-field model is technically only valid for the one-electron case; nevertheless, the DM state inferred by experiment is often justified by crystal-field theory.
The majority of studies on the AnO2 are often limited by the computational expense of the systems and restricted to discussions of collinear 1k magnetism.2,3,9–21,57 In general, the actinide dioxides (AnO2) exhibit low-temperature noncollinear 3k AFM or PM behaviour (Table S1, ESI†).7,24,58–64 The low-temperature magnetic state can result in a distortion of the crystal structure.1,27,44,65 For instance, in uranium dioxide (UO2) transverse 3k AFM order results in Pa symmetry; whereas, in neptunium dioxide (NpO2) longitudinal 3k AFM order preserves Fm
m symmetry. As the crystal structure is coupled by spin–orbit interactions (SOI) to the magnetic state, it is imperative to incorporate relativistic influences when creating computational models.
Magnetic configuration | Relative energy (eV per F.U.) | Band-gap (eV) | Magnetic moment (μB/Pu ion) | Lattice volume (Å3) | Space group (number) | |
---|---|---|---|---|---|---|
Diamagnetic | 1.069 | 2.92 | 0.00 | 155.37 |
Fm![]() |
|
Ferromagnetic | (001) | 0.074 | 1.91 | 3.66 | 155.88 | I4/mmm (no. 139) |
(011) | 0.037 | 1.79 | 3.68 | 156.03 | Immm (no. 71) | |
(111) | 0.027 | 1.68 | 3.68 | 155.99 |
R![]() |
|
Antiferromagnetic | ||||||
Longitudinal | 1k | 0.058 | 2.83 | 3.63 | 155.69 | I4/mmm (no. 139) |
2k | 0.011 | 3.05 | 3.65 | 155.65 | I4/mmm (no. 139) | |
3k | 0.000 | 3.04 | 3.65 | 155.65 |
Fm![]() |
|
Transverse | 1k | 0.062 | 2.50 | 3.64 | 155.64 | Fmmm (no. 69) |
2k | 0.020 | 2.93 | 3.66 | 155.73 | Pbca (no. 61) | |
3k | 0.009 | 2.68 | 3.66 | 155.72 |
Pa![]() |
|
Experimental | — | 2.8066 | 0.004–8 | 157.2523 |
Fm![]() |
It is noted that the experimental optical band-gap (2.80 eV) has been measured by epitaxial thin film absorbance.66 As the band-gap is influenced by the magnetic state, the experimental PM (high-temperature, T ∼ 298 K) result and computational longitudinal 3k AFM result (low-temperature, T ∼ 0 K) cannot be directly compared. In addition, the magnetic structure is coupled by SOI to the crystal symmetry. In our calculations, the DM and longitudinal 3k AFM states retain Fmm crystal symmetry, only. The collinear 1k AFM states (used in past studies) calculate an incorrect crystal structure. A tetragonal I4/mmm (no. 139) (longitudinal 1k AFM) or orthorhombic Fmmm (no. 69) (transverse 1k AFM) crystallographic distortion results.2,3,9–21 This inconsistent with experimental results. In addition, transverse 3k AFM order results in Pa
crystal symmetry (as observed in UO2 and AmO2).44
As a more computationally tractable method of theory (compared with the hybrid functionals), the results obtained from PBEsol+U are reported for different values of U (Fig. 3). In an identical manner to hybrid (HSE06) calculations, the magnetic state influences the electronic and crystallographic structure. PBEsol (U = 0 eV) calculates a degenerate metallic FM (111), (011), (001) ground-state; however, the metallic ground-state is inconsistent with the known insulating nature. The result exemplifies the failure of conventional DFT methods with highly-correlated systems. The band-gap is only replicated by the AFM and DM states when U = 5–7 eV.66 In particular, the band-gap of the experimental DM state and theoretical longitudinal 3k AFM ground-state is reproduced when U = 6.35 eV and U = 6.00 eV, respectively. In contrast, when calculating the band-gap for the FM states, at every value of U, the band-gap is underestimated relative to experimental values.
In experimental studies, DM order has been inferred from the absence of an effective Pu4+ magnetic moment.4–8 In this study, the effective magnetic Pu4+ moment of the ordered FM and AFM states is reported. In the FM states, the effective magnetic moment decreases from 3.85 to 3.75 μB/Pu ion in the U = 0–2 eV range; however, increases from 3.75 μB/Pu ion to 3.83 μB/Pu ion when U = 3–7 eV. In contrast, the effective magnetic moment of the AFM states continually increases from 3.55 μB/Pu ion to 3.80 μB/Pu ion over the range of U = 0–7 eV. However, when U = 6 eV, the effective magnetic moment converges at 3.80 μB/Pu ion for all magnetic states.
DFT+U | Band-gap (eV) | Lattice constant (Å) | Bulk modulus (GPa) | Magnetic moment (μB/Pu ion) | Crystal symmetry | Magnetic configuration | |
---|---|---|---|---|---|---|---|
Fundamental | Optical | ||||||
6.35 | 2.81 | 2.82 | 5.411 | 217 | 0.00 |
Fm![]() |
DM44 |
6.00 | 2.80 | 2.97 | 5.415 | 215 | 3.80 |
Fm![]() |
Longitudinal 3k AFM |
— | — | 2.8066 | 5.395–5.39823 | 178–379 | 0.00 |
Fm![]() |
Exp. |
A charge-transfer insulator with an indirect bandgap of 2.80 eV has been calculated; this contrasts against the Mott-insulator calculated by the fully relativistic linear combinations of Gaussian-type orbitals-fitting function (LCGTO-FF) method.19 The respective X–Γ valence band maximum (VBM) to L conduction band minimum (CBM) is comprised predominately of O p- and Pu f-states. The degeneracy of the bands in the longitudinal 3k AFM ground-state is perturbed by the presence of magnetic order. This key finding was made possible only through the inclusion of noncollinear and spin–orbit interactions.
The fundamental band-gap (the difference between the ionisation potential and the electron affinity, the valence band maximum (VBM) to conduction band minimum (CBM) transition) and optical band-gap (transitions are restricted by orbital symmetry) are influenced by the magnetic state. In our calculations, the longitudinal 3k AFM electronic structure is in excellent agreement with optical absorbance measurements.66 The direct and optical band-gaps differ by 0.17 eV, although both are in excellent agreement with the experimental band-gap of 2.80 eV (Table 3).66
In the absence of an ordered moment and of noticeable lattice anomalies, a phase-transition could be driven by an ordered parameter no longer invariant under time-reversal symmetry.1 The interaction leading to a phase-transition would have to arise from a purely electronic mechanism, as phonons may only carry interactions between time-reversal-invariant multipoles. A magnetic multipole (for which Kramers' theorem would no longer apply) by AFM superexchange is a candidate for an ordered mechanism.1,27,67 As an indication of superexchange, the magnetic susceptibility measurements for PuO2 and Th1−xPuxO2 are inconsistent.4,28,31 The interaction of the Pu4+ ions in Th1−xPuxO2 is reduced by dilution; this suppresses the superexchange mechanism and increases the magnetic susceptibility.1 In addition, anomalies in the INS measurements of PuO2 have been attributed to Jahn–Teller distortions and superexchange interactions.5,6 As a comparison, a superexchange interaction is theorised to occur in NpO2 where a magnetic octupole has been established.68
In another interpretation, PuO2 may be a small-moment system. This is observed in other actinide materials where the AFM response of URu2Si2 (TN = 17.5 K), UPt3 (TN = 5 K) and UPd3 (TN = 4.5 K) is 0.02–0.03 μB/U ion.1 Indeed, a small-moment magnetic state has been theorised for NpO2 where muon spin rotation measurements of the magnetic moment is 0.06–0.15 μB/Np ion.1,69 The nature of small-moment systems in unclear.1 The moment of a system may be reduced by competing magnetic interactions. In the localised PuO2 f-electron system, the interaction may be Rudermann, Kittel, Kasuya, Yoshida (RKKY) or Kondo in nature. If a small-moment in the PuO2 system exists, establishing the magnetic ground-state is a considerable challenge. In our results, the magnetic moment of the longitudinal 3k AFM ground-state is ∼3.65 μB/Pu ion. In this regard, first-principle DFT methods are known to overestimate small-moment actinide materials.1 It is also conceivable that PuO2 undergoes a second low-temperature phase transition, which is known to occur in other actinide compounds. For instance, the specific heat capacity of UPt3 has two distinct anomalies at TC1 = 0.49 K and TC2 = 0.44 K indicative of changing magnetic order. A detailed low-temperature experimental study of magnetic order in PuO2 is needed.
In our calculations, the metastable DM state (in contrast with the longitudinal 3k AFM ground-state) is inconsistent with current experimental information. In this paper, we have established a longitudinal 3k AFM ground-state with Fmm crystal symmetry. The fact that this magnetic ground-state has not been experimentally observed may be explained in two possible ways. First, PuO2 may be a small-moment material for which current experimental resolution is insufficient to identify an ordered magnetic state. If this is the case, dynamic mean field-theory (DMFT) may be needed to resolve the magnetic moment. Second, an AFM-DM phase transition may occur outside of the temperature range typically studied in the available literature. In theory a superexchange mechanism might result in a hidden magnetic phase transition. This paper calls for further experimental studies to resolve outstanding issues, in particular low-temperature specific heat capacity and muon spin relaxation measurements are needed.
Furthermore, we have shown how the crystal structure is influenced by the magnetic state. For instance, transverse 3k AFM behaviour results in Pa crystal symmetry,22,44 synonymous with calculations of UO2 and AmO2 when SOI are considered. Finally, the electronic structure of PuO2 is highly influenced by the magnetic state employed. These results have profound implications for investigations of bonding, reactivity, and surface structure. The re-orientation of magnetic vectors in surface calculations is often neglected (the direction of magnetism is usually defined by the z-axis), resulting in incomparable magnetic configurations between low-index surfaces. As the magnetic structure has a clear impact of the energetics, it is imperative that the re-orientation of magnetic vector is accounted for in computational surface investigations.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8cp03583k |
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