4 Relativistic Density – Functional Study of Nuclear Fuels

Among nuclear materials used in both research and commercial power reactors, very highdensity dispersion fuels have been used for requirement of a large proportion of uranium per unit volume in order to compensate to the reduction of enrichment (Meyer et al., 2002, Kim et al., 1999, 2002). For examples, uranium (U) with 10 wt% molybdenum (Mo) dispersed in aluminum (Al) matrix (Meyer et al., 2002, Kim et al., 2002) and U3Si2/ Al dispersion fuel (Kim et al., 1999) have been hitherto examined. Furthermore, zirconium (Zr) based U/ plutonium (Pu) alloys have recently been focused as a promising fuel for advanced reactors (Chernock & Horton, 1994).


Introduction
Among nuclear materials used in both research and commercial power reactors, very highdensity dispersion fuels have been used for requirement of a large proportion of uranium per unit volume in order to compensate to the reduction of enrichment (Meyer et al., 2002, Kim et al., 1999, 2002).For examples, uranium (U) with 10 wt% molybdenum (Mo) dispersed in aluminum (Al) matrix (Meyer et al., 2002, Kim et al., 2002) and U 3 Si 2 / Al dispersion fuel (Kim et al., 1999) have been hitherto examined.Furthermore, zirconium (Zr) based U/ plutonium (Pu) alloys have recently been focused as a promising fuel for advanced reactors (Chernock & Horton, 1994).
U metals have α (orthorhombic)-, β (tetragonal)-, and γ (body centered cubic)-phases that depend on the ambient temperature (Chiotti et al., 1981).The αand β-U phases form solid solutions with other metal elements to a limited extent (Chiotti et al., 1981), whereas the γ-U phase forms solid solutions with other elements to any extent (Chiotti et al., 1981).In addition, the α-U phase forms many intermetallic compounds with other metal elements (Kaufman, 1961).In these reasons, some αand/ or γ-U/ transition metal (TM) alloys have been used in research reactors (Chiotti et al., 1981, Kaufman, 1961).To use advanced nuclear fuels practically, it is critical to understand the dissolution process of TM atoms into γ-U on the basis of their phase diagrams.
The alloying behavior of U compounds has been hitherto theoretically investigated by considering the relative stabilities of electronic configurations (Buzzard, 1955, Park & Buzzard, 1957) and the thermodynamics (Ogawa et al., 1995).Hume-Rothery has focused on the metallic radius of TM elements as a parameter for understanding their alloying behavior (Hume-Rothery & Raynor, 1954).According to their empirical findings, when the difference in the metallic radius between solute and solvent atoms is less than 15%, TM elements are very soluble to each other.Figure 1 shows the correlation between U/ TM radius and the maximum solid solubility (MSS) of TM into γ-U.Here, the dashed line denotes the metal radius smaller by 15% than that of γ-U (Zachariasen, 1973, Pauling, 1960).depends on compatibility factors considering ionic radius, electron negativity, and the number of d-electrons (Buzzard, 1955, Park & Buzzard, 1957).However, because many empirical parameters should be determined, this evaluation process of their alloying behaviors is not applicable to other actinide alloy systems.On the other hand, Ogawa et al. investigated the alloying behavior by using the ChemSage program, and concluded that the difference in the excess free energy (∆G E ) of the U/ Mo, U/ rhodium (Rh), and U/ palladium (Pd) alloying systems may be due to the contribution of the U6d-Pd4d orbital interactions (Ogawa et al., 1995).Although the interactions indeed play an important role of their alloying behaviors, it is necessary to clarify what interactions between γ-U and TMs contribute to the alloying behavior quantitatively in order to satisfactorily understand the alloying behavior of the 3d, 4d and 5d TMs into γ-U solid.
Fig. 1.The correlation between the metallic radius of TMs and the maximum solid solubility for γ-U alloyed with (a) 3d TMs, (b) 4d TMs, and (c) 5d TMs [Kurihara et al., (2008)] Recently, some groups have theoretically studied the thermodynamic properties of U/ TM alloys.Landa et al. have examined the phase equilibrium of U/ Zr alloys, using the scalarrelativistic Green's function technique based on Korrings-Kohn-Robstoker (KKR) method (i.e.without spin-orbit coupling) (Landa et al., 2009).They obtained a good agreement between theoretical and experimental results for the ground-state properties of γ-(bcc) and δ-(C32) phases of U/ Zr alloys.On the other hand, Li et al. have obtained the thermodynamic assessments for thorium (Th)/ U and Th/ Zr binary and Th/ U/ Zr ternary alloys (Li et al., 2009), using the CALPHAD (calculation of phase diagrams) method based on experimental data including the phase equilibria and thermodynamic properties of the alloys.They have also obtained a good agreement between the calculated phase equilibria and experimental data.Ab initio calculations based on density-functional theory have also been used to examine the thermodynamic properties of U/ Al systems (Alonso et al., 2009, Sedmidudbsky et al., 2010).To understand the dissolution of elements into solid U in addition to obtain the phase diagram or phase equilibria, it is indespensable to determine which electronic terms contribute to the thermodynamic properties of U alloys by comparing density-functional calculations with experimental results.
We have previously investigated the alloying behavior of 3d-, 4d-and 5d-TMs into γ-U, using the discrete-variational Dirac-Fock-Slater molecular orbital (MO) method (R-DFT) which takes into account fully relativistic effects including the spin-orbit coupling (Kurihara et al., 2004(Kurihara et al., , 2008(Kurihara et al., , 2011)).We have found that the maximum solid solubility (MSS) of TMs into γ-U is exponentially proportional to both the d-orbital energy of TM (Md) and the orbital overlap population (OOP) between TMd and U6d atomic orbitals.
In this chapter, we firstly introduce the relativistic discrete-variational Xα molecular orbital method (R-DFT) (Rosen et al., 1975, Adachi et al., 1977, Nakamatsu et al., 1991), secondary introduce our recent works on analysis of the valence photoelectron spectra of uranium carbides (UC) and α-U metal, and thirdly show the elucidation of the alloying behavior of TMs into γ-U metal using the R-DFT, and finally summarize the results obtained using R-DFT and describe the perspective for the quantum design of nuclear fuels for advanced reactors.

Relativistic density-functional method
The present R-DFT method has been shown to be a powerful tool for the study of the electronic structures of molecules containing heavy elements such as uranium (Onoe et al., 1992a, 1994b, Hirata et al., 1997a, Kurihara et al., 1999, 2000).The one-electron molecular Hamiltonian, H, in the R-DFT method is written as, (1) Here, c, P, m, α, β and V(r) respectively denote the velocity of light, the operator of momentum, the mass of electron, Dirac matrices and the sum of Coulomb and exchange potentials.The molecular orbitals (MOs) are obtained by taking a linear combination of atomic orbitals (AOs).Details of the R-DFT method have been described elsewhere (Rosen et al., 1975, Onoe et al., 1993).
Basis functions, which are numerical solutions of the atomic Dirac-Fock-Slater equations for an atomic-like potential, are obtained at the initial stage of individual iterations of self-consistent procedures (Adachi et al., 1977).The atomic-like potentials used to generate the basis functions are derived from the spherical average of the molecular charge density around nuclei.One-center (atomic) charges are estimated using Mulliken population analysis (Mulliken et al., 1955a(Mulliken et al., , 1955b(Mulliken et al., , 1955c(Mulliken et al., , 1995d) ) for application to the self-consistent charge (SCC) method (Rosen et al., 1976) that is used to approximate the self-consistent field.
Morinaga et al have first found that the d-electron energy (Md) plays an important role for the alloying behavior (Morinaga et al., 1984(Morinaga et al., , 1985a(Morinaga et al., , 1985b(Morinaga et al., , 1985c(Morinaga et al., , 1991(Morinaga et al., , 2005)).In a similar manner, we have evaluated the Md of TMs in γ-U/ TM alloys from a weighted average of each component (d 3/ 2 and d 5/ 2 ), along with the U6d orbital energy.To compare Md values obtained for all TMs and U with each other, Md was shifted with respect to the Fermi level (E F ) of γ-U used as a reference (Morinaga et al., 1984(Morinaga et al., , 1985a)).
Since the orbital overlap population (OOP) expresses the strength of covalent interactions between AOs, it is a powerful tool to clarify the contribution of individual AOs to covalent bonding (Onoe et al., 1997c).In order to obtain information of the covalent bonding, we employed the Mulliken population analysis (Mulliken et al., 1955a(Mulliken et al., , 1955b(Mulliken et al., , 1955c(Mulliken et al., , 1995d)).In the present analysis, the number of electrons (n i ) was partitioned into the gross of the ith AO, (,) , Here, l φ is the occupancy of the lth MO, ( ) CC is the coefficient of the linear combination of AOs, and ij S is the overlap integral between the ith and jth AOs.Two-center charges were estimated from the overlap populations (ith AO of γ-U and jth AO of TM), The effective charges for γ-U/ TM alloys were evaluated, because they are strongly related to the charge transfer (CT) between TM and γ-U.

Cluster model
Unlike band structure calculations of condensed matters obtained using their unit cells with a periodic boundary condition, the present R-DFT method employed a cluster model reflecting the crystal structure.

Uranium carbide
For a cluster model of uranium carbides (UC) with a NaCl-type structure, we made neutral UC 6 , CU 6 , and CU 6 C 18 cluster models with Oh symmetry and with a U-C bond length of 248.1 pm taken from the experimental results (Erode, P., 1983), as shown in Fig. 2.
In case of the CU 6 C 18 cluster, the CU 6 cluster was embedded with eighteen carbon atoms.
Since the spin function is included in the Dirac equation explicitly, the Oh symmetry reduces to the Oh* double group (Bethe, H., 1929).Symmetry orbitals for the irreducible representations of Oh* symmetry were constructed from AOs by the projection operator method (Meyer, J. et al., 1989).All the calculations were performed with the Slater exchange parameter α of 0.7.The DV sample points of 6000 were used for the UC 6 and CU 6 models , while 16,000 points for the CU 6 C 18 model.The basis functions up to the 7p orbital were used for U atom and those up to the 2p orbital were used for C atom.All the calculations were carried out self-consistently until the difference between the initial and final orbital populations in the iteration was less than 0.01.[Kurihara et al., (2000)].
The geometry of the present cluster has a C 2v symmetry with the U-U bond length of 5.219 au for (1)-( 2), 5.389 au for (1)-( 4), and 6.165 au for (1)-( 6) and ( 1)-( 8) on the basis of the experimental results of α-U metal (Holden, A. N., 1958).Here, " au" denotes the atomic unit (1 au = Bohr radius).Symmetry orbitals for the irreducible representations of C 2V * symmetry were constructed from AOs by the projection operator method.All the calculations were performed with the Slater exchange parameter α of 0.7.The DV sample points of 18,000 were used for the U 9 cluster model.The basis functions up to the 7p orbital were used for U atom.All the calculations were carried out under the same conditions as for UC.

γ-Uranium/transition metal alloy
Figure 4 shows schematic representation of a cluster model of γ-U alloyed with TMs.As shown in Fig. 4, the central U atom, U(1), was substituted with TM atoms.To compare the difference in the electronic structure among the γ-U/ TM alloys, the MO energy level structure and chemical bonding of the alloys were examined (Kurihara et al., 2004).In the present works, the lattice relaxation in association with TM element substitution to γ-U solid was ignored, because the Md and OOP may not be affected significantly by the lattice relaxation (Morinaga et al., 1984(Morinaga et al., , 1985a(Morinaga et al., , 1985b)).The lattice constant of γ-U crystal was taken from the experimental result of 6.659 au (Holden 1958).
Symmetry orbitals for the irreducible representations of D 4h * symmetry were constructed from the AOs by the projection operator method.The present R-DFT calculations were performed with the Slater exchange parameter α of 0.7 and with 18,000 DV sample points for the U 8 -TM cluster model.The basis functions were used to be the 1s-7p AOs for U atom, the 1s-4p AOs for 3d TMs, the 1s-5s AOs for 4d TMs, and the 1s-6s AOs for 5d TMs.All the calculations were carried out self-consistently under the same conditions as for UC and α-U.

Theoretical x-ray photoelectron spectra
XPS (X-ray photoelectron spectroscopy) intensity was estimated using the following equation (Gellius, U., 1974), Here, I l is the probability of photo-ionization from the lth MO level, σ i is the photoionization cross-section of the ith atomic orbital, and P il is the population of the ith atomic orbital in the lth MO.The photo-ionization cross-section of individual AOs was taken from the data estimated by ab initio calculations (Scofield, J. H., 1976).The cross-section of the U7p was ignored in the present calculations, because it was too small to contribute to the XPS intensity.For evaluating P il , the gross population of individual AOs for each MO was evaluated by Mulliken population analysis (Mulliken et al., 1955a(Mulliken et al., , 1955b(Mulliken et al., , 1955c(Mulliken et al., , 1995d)).Theoretical XPS spectra were obtained by replacing each stick peak with a Lorentzian curve with a full width at the half maximum (FWHM) of 1.46 eV.We then compared these with the experimental XPS spectra of UC solid (Ejima, T., et al., 1993).
In a similar manner to that for UC solid, theoretical XPS spectra of α-U solid were obtained by replacing each stick peak with that with a FWHM of 1.6 eV.Then, we compared these with the experimental XPS for α-U metal (Fuggle, J. C., et al., 1982).

Relativistic effects in molecules
In this section, we briefly explain relativistic effects on the electronic structure and chemical binding in molecules for understanding the results of uranium compounds introduced in Sections 4 and 5. Prior to describing the effects in molecules, we begin to explain the relativistic effects in atoms.The Bohr radius (a) and the energy (E) of hydrogen-like atoms are respectively given by the following equations, and Here, m is the mass of electron, n is the principal quantum number, Z is the atomic number, e is the elementary charge, ℏ is Planck's constant.In addition, the ratio of the speed of 1s electron (v 1s ) to that of light (c) can be written by Z/ 137.For U atom with Z=92, the v 1s can be estimated to be 0.67c, which indicates that the relativistic effects should be considered in quantum formula.Even for the U6s valence atomic orbital, the v 6s is calculated to be 0.1c, implying that the relativistic effects can appear in the valence region that is related to chemical bonding.According to the theory of relativity, the mass of electron is influenced by the relativistic effects, which can be given by the following equation, Here, m 0 and v are the rest mass and speed of electron, respectively.Accordingly, one can see from Eqs. ( 5) and ( 6) that when m increases with increasing Z, Bohr radius and energy become contracted and increased, respectively.A heavier element exhibits more remarkable change in both a and E by the relativistic effects.
The relativistic effects on AOs can be classified into two categories: direct and indirect effects.
The former effects result in the relativistic contraction of inner-shell orbitals (e.g.s and p orbitals).In addition, since the total angular momentum j (= l + s) is the quantum number in Dirac's equation, the energy splitting due to the spin-orbit interactions is simultaneously taken into account (e.g.6p splits to 6p 3/ 2 and 6p 1/ 2 ) in the equation.On the other hand, the indirect effects cause the expansion of outer AOs (e.g.d and f AOs) due to the screening of nuclear charges by the contraction of the inner-shell AOs.As described above, these relativistic effects become magnified with increasing Z, and remarkably affect both valence electronic structure and chemical bonding of condensed matters (molecules, complexes, and solids) containing heavy elements such as uranium.We next explain how the relativistic effects affect the valence electronic structure and chemical bonding of UF 6 as an example.

Relativistic effects on valence electronic structures
Figure 5 shows the comparison between non-relativistic and relativistic one-electron energies of valence MOs for the ground state of UF 6 that is often used as a material for uranium enrichment.Here, MOs were classified into two parts based on inversion symmetry: " gerade" and " ungerade" shown on the left and right sides, respectively.
Fig. 5. Non-relativistic and relativistic one-electron energies for the ground state of UF 6 [Onoe et al., (1993)] As shown in Fig. 5, two kinds of relativistic effects appear in the valence electronic structure: the energy splitting for several MOs and the upward or downward shift in the one-electron energy for individual MOs.In particular, since the energy splitting (∆E) between the U6p 1/ 2 and U6p 3/ 2 (∆E = 8.9 eV) is larger than that for both U5f (∆E = 0.8 eV) and U6d (∆E = 0.5 eV) AOs, the non-relativistic 7t 1u and 8t 1u MOs caused by the U6p-F2s interactions exhibit the noticeable splitting to form the four corresponding relativistic MOs.Indeed, we succeeded in assigning these four relativistic MOs in the valence photoelectron spectra of UF 6 (Onoe et al., 1994b).

Relativistic effects on chemical bonding
To understand how the relativistic effects influence the chemical bonding of UF 6 , we have investigated the contribution of individual AOs to the chemical bonding by examining the non-relativistic and relativistic radial wave functions [Onoe et al., 1993].Figure 6 shows the radial wave functions of the uranium valence atomic orbitals (5f, 6s, 6p, 6d) obtained using non-relativistic and relativistic density-functional calculations.For the relativistic radial wave functions, the small components were omitted and only the large ones were shown in Fig. 6, because the former ones play a minor role of contributing to chemical bonding.Figure 6 suggests that the relativistic contraction and expansion of the valence AOs at the U-F bond length significantly affect the strength of chemical bonding of UF 6 .Since the orbital overlap population (OOP) is a good indicator for the strength of chemical bonding between AOs, we examined the OOP between individual U and F valence AOs, along with the bond overlap population (Bo) between U and F atoms, in order to clarify the contribution of the relativistic contracted and expanded AOs to the chemical bonding of UF 6 .
Table 1 summarizes the OOP and Bo obtained from Mulliken population analysis for the non-relativistic and relativistic calculations.Here, the positive and negative signs of OOP imply the bonding and anti-bonding interactions, respectively.For an example, the strength of anti-bonding U6s-F2s and U6s-F2p interactions becomes weakened from non-relativistic to relativistic calculations, because the relativistic contraction of the U6s radial wave function reduces the U6s-F2s and U6s-F2p OOPs in the U-F bond region, as shown in Fig. 6.This results in the strengthening of the U-F bond.On the other hand, the relativistic expansion of the U5f and U6d radial wave function in the U-F bond region strengthens the U5f-F2p and U6d-F2p bonding interactions.The relativistic changes in the U valence AOs cause a large difference in the U-F chemical bond, as indicated in Table 1.
Table 1.The overlap populations between U and F valence AOs for non-relativistic and relativistic calculations, along with the U-F bond overlap population [Onoe et al., (1993)]

Uranium carbides
Figure 7 shows the experimental valence XPS spectra of UC (Ejima, et al., 1993) (a) and theoretical spectra for the UC 6 (b), CU 6 (c), and CU 6 C 18 (d) cluster models shown in Fig. 2, along with the partial density-of-states (pDOS) of the U5f and U6d AOs (e) and of the C2s and C2p AOs (f).As shown in Fig. 7(a), UC has the intense peak "1" at the Fermi level (E F ), the shoulder peak "2" at around -2 eV, and the broad weak peak "3" at around -10 eV.We next compared theoretical spectra [Fig.7(b)-(d)] with the experimental result.As shown in Fig. 7(b), the UC 6 cluster model well reproduced both relative intensity and position (binding energy) for the peaks "1" and "3", but the shoulder peak "2" unfortunately seemed to be reproduced.On the other hand, the CU 6 cluster model showed a peak corresponding to the peak "2" besides the peaks "1" and "3", but it was completely separated (not a shoulder peak) from the peak "1".Accordingly, it is found that the CU 6 cluster model reproduces the experimental spectra better than the UC 6 cluster model.When the CU 6 cluster is surrounded with eighteen C atoms, a CU 6 C 18 embedded cluster model is expected to reproduce the experimental spectra better than the CU 6 cluster.Comparison between Figs. 7(a) and 7(d) indicates that the theoretical spectra well reproduce the whole experimental one.Fig. 7. Theoretical and experimental valence x-ray photoelectron spectra for UC: (a) experiment [Ejima. T., et al., (1993)] and theoretical spectra obtained using (b) UC 6 , (c) CU 6 , and (d) CU 6 C 18 cluster models, along with partial density-of-states (pDOS) for (e) the U5f (dashed line) and U6d (solid line) AOs and (f) the C2s (solid line) and C2p (dashed line) AOs [Kurihara et al.,(1999)] To assign the three peaks shown in Fig. 7(a), we next examined the pDOS based on the results of the CU 6 C 18 embedded cluster model.Figures 7(e) shows the pDOS of the U5f (dashed line) and U6d (solid line) AOs, whereas Figure 7(f) shows that of the C2s (solid line) and C2p (dashed line) AOs.Comparison between Figs. 7(a) and 7(e, f) indicates that the intense peak "1" is mainly attributed to the U5f AO which has a large photo-ionization cross-section and the shoulder peak "2" is attributed to the U6d-C2p AOs, whereas the broad weak peak "3" is mainly attributed to the C2s AO.
In previous reports by the other groups, Schalder et al reported that the intense peak "1" was due to the U5f-U6d bands, while the shoulder "2" due to the U6d-C2p band (Schadler, G. H., 1990).On the other hand, Ejima et al concluded that a small amount of the U5d component contributes to the shoulder "2".However, because the U5d 5/ 2 and U5d 3/ 2 AOs are respectively located at -91 eV and -99 eV, the U5d components can be considered to play a minor role for the shoulder "2".In fact, the present calculations show that the U5d components have no contribution to the valence electronic structure of UC.

α-Uranium metal
Figure 8 shows (a) the experimental (Fuggle, J. C., et al., 1974) and theoretical x-ray photoelectron spectra obtained using the U 9 cluster model (Kurihara, M., et al., 2000) for α-U metal, (b) the pDOS of U valence AOs, and (c) the magnified pDOS in the vicinity of E F .As shown in Fig. 8(a), it is interesting to note that the theoretical spectra well reproduced the experimental one of α-U metal, in spite of using the minimum U 9 cluster model shown in Fig. 8.By comparison between Figs. 8(a) and 8(b), it is clearly found that the peaks " B" and " C" are only attributed to the U6p 3/ 2 and U6p 1/ 2 AOs, respectively.In a similar manner, the Fig. 8. (a) Experimental [Fuggle, et al., (1974)] and theoretical spectra, (b) partial density-of-states (pDOS) of U valence AOs and (c) magnified pDOS in the vicinity of E F [Kurihara et al., (2000)] intense peak " A" appearing at E F is mainly attributed to the U5f 7/ 2 and U5f 5/ 2 AOs.As shown in Fig. 8(c), since the other valence AOs such as the U6d, U7s, and U7p are located around E F , they contribute to the peak " A" to some extent.However, because their photoionization cross-sections are much smaller than that of the U5f AOs, they are considered to have minor contributions.
In previous band calculations (Yamagami. H., & Hasegawa. A., 1990), they compared the DOS structures with the experimental spectra, and discussed the assignment of each peak qualitatively.On the contrary, the present method can calculate XPS intensity of individual MOs for the cluster model used as the minimum unit of α-U metal.Thus we have quantitatively obtained the valence XPS spectra, together with pDOS of individual AOs contained in each MOs.This enables us to assign individual peaks satisfactorily.

γ-Uranium/transition metal alloys
As introduced in the previous section, the present R-DFT method well reproduced the experimental photoelectron spectra of UC and α-U solid and assigned the origins of individual peaks.In this section, we applied this method to understand what electronic factors play significant roles of the alloying behavior for γ-U/ TM metal alloys that are one of the candidates as nuclear fuels for advanced reactors.

Correlation between the Md (or effective charges) and the maximum solid solubility
The Md has been found to play an important role of alloying for Ni 3 Al (Morinaga et al., 1984) and bcc Fe ( (Morinaga et al., 1985a).In this section, we first present the correlation between the maximum solid solubility (MSS) and Md for γ-U/ TM alloys (Kurihara et al., 2004(Kurihara et al., , 2008)).
Figure 9 shows the plot of MSS as a function of Md, where the regression lines were obtained using a least square method.It is found that MSS is exponentially dependent on Md except for γ-U/ Ta and γ-U/ W alloys, though the reason of this exception is not still clearly understood.This exponential dependence will be discussed from a thermodynamic standpoint in Section 5.4.
Since Md is related to the charge transfer (CT) between the TM and γ-U, we next examined the correlation between MSS and CT (Kurihara et al., 2008).Figure 10 shows the plot of Md as a function of atomic number (Z) for individual TMs in γ-U/ TM alloys, along with the U6d energy of U(1) prior to TM substitution.The sign and amount of the difference between Md and U6d energies determine the direction and amount of CT between TM and γ-U, respectively.The CT takes place from TMs to γ-U when Md level is higher than the U6d one, whereas CT from γ-U to TMs takes place when Md level is lower than the U6d one.In addition, a larger difference between Md and U6d energies results in a larger amount of CT between them.From the results of Figs. 9 and 10, a smaller difference between the Md and U6d levels results in a larger MSS, as previously reported by Morinaga et al. (Morinaga et al., 1984(Morinaga et al., , 1985a,b),b).Namely, a smaller amount of CT between TM and γ-U provides a larger MSS for γ-U/ TM alloys.
Since the amount of CT is directly related to the effective charge, we next examined the correlation between MSS and the effective charges on TM and U atoms (Kurihara et al., 2008).Figure 11 shows that MSS exhibits a negatively exponential relationship with respect to the effective charge.This implies that MSS exhibits a negatively exponential dependence on the amount of CT between the TM and U atoms.Furthermore, because the effective charge indicates the degree of ionic bonding interaction between TMs and γ-U, it can be said that MSS shows a negatively exponential dependence on the strength of ionic bonding between TM and γ-U atoms.This indicates that a large ionic bonding between TM and γ-U atoms results in a smaller MSS for γ-U/ TM alloys.1) prior to TM substitution [Kurihara et al., (2008)] Fig. 11.The correlation between the maximum solid solubility (MSS) and the charge transfer (CT) between TM and U atoms for γ-U/ TM alloys [Kurihara et al., (2008)] Since the bonding between TMs and γ-U contain not only ionic but also covalent interactions, we next discuss the correlation between MSS and the covalent interactions between TMs and γ-U.

Correlation between orbital overlap populations and the maximum solid solubility
We examined the correlation between TMd-U6d OOP and MSS for γ-U/ TM alloys, because OOP is a good indicator for the strength of the covalent bonding between TM and U atoms.
Figure 12 shows the plot of MSS as a function of OOP between TMd and U6d AOs TMs [Kurihara et al., (2008)] (Kurihara et al., 2008), and demonstrates that MSS exhibits an exponential dependence on the strength of the U6d-TMd covalent bonding.Namely, a stronger covalent bonding between TM and U atoms provides a larger MSS for TMs alloyed into γ-U.This is opposite to the correlation between MSS and the TM-U ionic bonding.
We will discuss the reason why MSS is exponentially proportional to OOP as well as Md in Section 5.4.

Role of U6d-TMd orbital interactions in alloying of γ-U/TMs
Figure 13 shows the plot of MSS as a function of the energy difference (α U -α TM ) between the U6d and TMd AOs (Kurihara et al., 2011).The energy difference (α U -α TM ), which is often used to discuss the CT between U and TM atoms, shows that MSS exhibits an inversely exponential dependence on (α U -α TM ), thus a smaller value of (α U -α TM ) provides a larger MSS.Since the energy difference (α U -α TM ) is related to the magnitude of the CT between TM and γ-U, Figure 13 implies that a smaller CT between them gives rise to a larger MSS for γ-U/ TM alloys.
Fig. 13.Plot of the maximum solid solubility (MSS) as a function of the energy difference between the U6d (α U ) and TMd (α TM ) AOs [Kurihara et al., (2011)] Figure 14 shows the correlation between the TMd-U6d OOP and MSS for γ-U/ TM alloys (Kurihara et al., 2011), and demonstrates the exponential dependence of MSS on the OOP for all γ-U/ TM alloys except for Ta and W elements.Thus, an increase in the OOP results in an increase in MSS of γ-U/ TM alloys.
From the results of Figs. 13 and 14, one can see that the interactions between TMd and U6d AOs play an important role of determining the magnitude of MSS for γ-U/ TM alloys.In other words, the TMd-U6d orbital interactions become a key parameter for estimating MSS of γ-U/ TM alloys.We next discuss the physical meaning of the TMd-U6d orbital interactions in the magnitude of MSS.
Figure 15 schematically illustrates the stabilization energy (∆E) caused by the U6d-TMd orbital interactions from a viewpoint of molecular orbital theory (top) and the correlation between MSS and ∆E or the U6d-TMd energy difference (bottom).In the framework of the simple Hückel approximation (Hückel, E, 1931), ∆E is obtained as, ( ) ( ) Here, α U and α TM respectively denote the U6d and TMd energies, and α U is equal to or greater than α TM (α U ≥ α TM ).β denotes the resonance integral between the TMd and U6d AOs, which can be written as, ( ) Where, K is a constant, and S is the overlap integral between the TMd and U6d AOs.
Namely, β is proportional to S, thus proportional to the TMd-U6d OOP.
To  8).Then, ∆E can be rewritten as, By considering the range of the two quantities on the right-hand side of Eq. ( 7) (i.e., t ≧ 0 and 0 < F(t) ≦ 1), we obtained the range of the stabilization energy, 0 < ∆E ≦|β|.
Accordingly, ∆E is the maximum (=|β|) when F(t) is unity at t = 0, that is, α U = α TM .On the contrary, ∆E becomes the minimum (near equal to zero) when (α U -α TM ) is much larger than unity (≫1).Consequently, a larger MSS for γ-U/ TM alloys is obtained at a lower value of (α U -α TM ) (corresponding to a larger ∆E), whereas a small MSS is obtained at a larger value of (α U -α TM ) (corresponding to a smaller ∆E).Consequently, it can be concluded that the magnitude of ΔE caused by the TMd-U6d orbital interactions plays a key role of determining the magnitude of MSS.
Here, TM denotes a TM atom in the TM bulk phase, while U 9 / U 8 TM and U respectively denote a cluster and a U atom in the γ-U bulk phase.As the equilibrium constant (K) in Eq. ( 11) increases, the concentration of U 8 TM increases, thus increasing MSS.On the other hand, as K decreases, the U 8 TM concentration correspondingly decreases, thus reducing MSS.
Accordingly, MSS is proportional to K. Then the relation between MSS and K can be written as, Where, A is a constant, ∆G denotes the difference in Gibbs free energy before and after the substitution, R is the gas constant, and T is the absolute temperature.Since the lattice relaxation associated with substitution of TM into γ-U solid was negligible in the present calculations, the entropy remained constant (∆S = 0) before and after the TM substitution.Thus, ∆G is equal to the enthalpy difference (∆H) before and after the substitution.Accordingly, Eq. ( 12) can be rewritten as, Since ∆H includes ∆E caused by the TMd-U6d orbital interactions, the following relationship is obtained, Since ∆E is proportional to both (α U -α TM ) and S, it can be understood that MSS depends exponentially on both Md and OOP, as shown in Figs. 9 and 12.
In the above discussion, Md and OOP can be regarded as a good parameter to determine the magnitude of MSS.Then we next made the OOP-Md map for γ-U/ TM alloys (Kurihara et al, 2011), as shown in Fig. 16.Interestingly, γ-U/ TM alloys with a smaller MSS appear in a Fig. 16.OOP-Md plot of γ−U/ TM alloys [Kurihara et al., (2011)] lower left hand on the map, whereas those with a higher MSS in an upper right hand.Consequently, the OOP-Md map could be useful for designing γ-U/ TM alloys.We believe that the present findings can be used for not only the γ-U/ TM alloy system but also other actinide alloys used as nuclear fuels for advanced reactors.

Summary and perspective
We have performed R-DFT calculations for understanding and designing γ-U/ TM alloys as a typical example of nuclear fuels.Md and OOP have a good correlation with MSS of γ-U/ TM alloys, and become a more suitable parameter to determine the magnitude of MSS than metallic radius as previously proposed by Hume-Rothery.The present parameters, Md and OOP, demonstrated that a stronger covalent bonding between TMd and U atoms gives rise to a larger MSS of γ-U/ TM alloys, whereas a stronger ionic bonding between them gives rise to a smaller MSS.
In summary, the magnitude of MSS was successfully explained in terms of the stabilization energy (∆E) caused by the U6d-TMd orbital interactions for γ-U/ TM alloys.In addition, the exponential dependence of MSS on Md and OOP was also interpreted by considering the equilibrium constant based on thermodynamics of the substitution model for γ-U/ TM alloys.The OOP-Md map was useful to determine the magnitude of MSS for γ-U/ TM alloys.Finally, the present approach could be applied to understand and design the other alloy systems used as nuclear fuels for advanced reactors.

Fig. 3 .
Fig. 3. Schematic illustration of α-U metal: (a) the orthorhombic structure of α-U metal and (b) a U 9 cluster model used as the minimum unit of α-U crystal structure[Kurihara et al., (2000)].

Fig. 12 .
Fig. 12.The correlation between the maximum solid solubility (MSS) and the orbital overlap population (OOP) between TMd and U6d atomic orbitals: (a) 3d TMs, (b) 4d TMs and (c) 5dTMs[Kurihara et al., (2008)] discuss the correlation between ∆E a n d M d/O O P m or e c l e a r l y , we i n t r odu c e t