The ground state equilibrium structure was evaluated using QE code. We assumed electronic configurations of 5f³, 6d¹, 7s² for U, 5f⁰, 6d², 7s² for Th, 2s², 2p⁴ for O, 2s², 2p³ for N and 2s², 2p² for C atoms. The conjugate gradient diagonalization was used, and the following parameters were applied: Methfessel-Paxton (MP) smearing; a mixing beta parameter of 0.15 in Broyden charge density mixing; a convergence threshold of 10 - 12 Ry. We adopted the kinetic cutoff energies determined previously for the norm-conserved pseudopotentials of 250 Ry (3401 eV) for thorium compounds [8] [9] [19] and 150 Ry (2041 eV) for U compounds [5] [9] . ThN, ThC, UN and UO₂ have cubic structures ( $Fm\bar{3}m$) with cations (U, Th) forming stable fcc type frames. U atoms in UO₂ are surrounded by eight O atoms in tetrahedral positions while octahedral interstitials are not occupied. In contrast, in the metallic fuels discussed here, U and Th have four nearest neighbours (O or N) in octahedral positions while tetrahedral interstitials are not occupied.

In Table 1 , for each compound listed in column one, the U/Th atomic density is shown in column two as evaluated using equilibrium lattice constants calculated with implementation in the QE GGA-PBEsol functional and listed in column three. The previously calculated lattice constants [8] [9] using GGA-PBE are shown in column four while experimental results are listed in the last column. We find a very good agreement between the calculated and measured values.

The results in Table 1 , column two, confirm that UN has 42% more U atoms per unit volume than UO₂. Therefore, UN is more economical since it will require lower enrichment, which is expensive. ThN and ThC also show a higher density of Th atoms than the calculated U atom density in UO₂.

We demonstrated in Section 3.1 that the metallic ceramic fuels investigated here have higher U/Th atom density than traditionally used Urania. Therefore, it is important to evaluate if this leads to enhanced fuel swelling.

We investigated here the effect of incorporating He and Xe in one interstitial in (2 × 2 × 2) supercells as illustrated in Figures 1 and Figures 2 respectively for UO₂ and UN. Both He and Xe atoms were incorporated in each compound separately. All supercells have the same number (32) of U/Th atoms. The supercell for UO₂-He is shown in Figure 1 : It contains 32 U atoms and 64 O atoms and one He placed in an octahedral interstitial site.

We present in Figure 2 an exemplary picture of the supercells used for all metallic ceramic compounds with 64 atoms total and one Xe atom incorporated in a tetrahedral site (0.25, 0.25, 0.25).

In Table 2 , column two, the calculated lattice constants of the respective supercells listed in column one are shown. The respective formation energies ( $\Delta E_{\mathrm{int}}^F$) as evaluated from Eq. 1 for incorporated He and Xe atoms in interstitials are shown in the last column. The formation energy decreases linearly (R² = 0.999 for Xe and 0.954 for He) with increasing lattice constants and it is up to three times

larger for incorporation of Xe than for He atoms. Additionally, the evaluated distances between incorporated He/Xe atoms in interstitial sites (0.25, 0.25, 0.25). to their nearest neighbour (O/N) and the nearest U/Th atoms are shown in columns three and four, respectively.

In Figure 3(a) , the percentage volume increases due to incorporation of He (grey circles) or Xe (dark blue squares) are shown as a function of the lattice constants of the respective supercells (2a) with the indicated name of the pure compound. Incorporation of FP in UO₂ has been extensively studied in magnetic and nongnetic states using various functionals as reviewed in [24] . Hybrid functionals were used for comparison to study incorporation of FP in interstitial [24] . However due to usage of small conventional unit cells for these computationally demanding functional the respective percentage volume increases were three times larger than evaluated here for (2 × 2 × 2) supercells when incorporating He (0.43% versus 1.2%.) and Xe (2.12% versus 6.90%). GGA+U was used to evaluate FP incorporation in antiferromagnetic (AFM) UN in ref. [25] therefore unphysical tetragonal distortion was observed. The studied here non magnetic (NM) cases and previous research on UO₂ with noncolinear magnetism [26] have found stable cubic structures in agreement with experiment. Alternatively UN with ferromagnetic (FM) ordering is also cubic and since it is a weak magnetism the evaluated formation energies for Xe in interstitials in UN AFM and FM only differed by 0.3% [27] . The volume expansion was not presented for UN in ref. [25] but elastic correction for (2 × 2 × 2) supercells for He and Xe incorporation in interstitials was found to be small (1% for He and 5.5% for Xe). The difference between energy incorporation for He and Xe in UN was close to the found here difference between the formation energies (11.66 eV [25] versus 12.27 eV). We confirm here that UN had 42% more U atoms per unit volume than UO₂ and a 55% higher volume increase when accommodating one Xe atom in one interstitial site of a (2 × 2 × 2) supercell. However, for He, the volume increase was 27% lower in UN, which can be explained by having lower number of nearest neighbor (NN) atoms (four N versus eight O). The previous evaluation [28] confirmed that going beyond GGA/LDA approximation or using spin polarized calculations was not necessary to evaluate structural properties of UC with defects and the found lattice constant change using GGA (PBE) of UC with incorporation of FP are very close to evaluated here for He (small −0.1% versus 0.1 %) and Xe (1.2% versus 1.1%) in ThC. The differences between formation energies of Xe and He calculated previously [28] and evaluated here differences for the respective formation energies are comparable (9.09 eV versus 6.86 eV). Interestingly, even though the Th atoms’ density in ThN and ThC was lower than that of U atoms in the UN compound, a similar trend was found in volume increase when incorporating Xe atoms ( Figure 3(a) ). In contrast, a lower percentage volume increase was observed in UO₂ (12%). We conclude, therefore, that for swelling, the local structural symmetry (tetrahedral versus octahedral sites) is more important than the density of atoms.

The percentage of energy increase (E pure compound) when incorporating He or Xe atoms in interstitials is presented in Figure 3(b) . The energy change for small He atoms is very small, and it is almost the same for tetrahedral interstitials of the considered metallic ceramic fuels. It is only slightly lower for incorporation into octahedral interstitial in UO₂. In contrast, the % the total absolute value of energy change of a larger gaseous atom, such as Xe, varies a lot and it is also much higher. ThC has the lowest binding energy and the percentage energy increases to a very high value ( Figure 3(b) ) where it is about four times higher than the percentage energy increase for positioning Xe in an octahedral site of UO₂. The 37 % higher total absolute value of energy increase for ThC versus ThN cannot be explained by crystal geometry since the ThC-Xe supercell has a higher lattice constant ( Table 2 , second column) than the ThN-Xe supercell. This can only be explained by investigating electronic structure, as discussed in Section 3.3.

In previous work, we studied the electronic structure of UN in detail [5] and we also compared the electronic structure of ThN with ThC [8] . In this work, we compare the effect on electronic structure of neighbouring atoms: small He atoms compared with much larger Xe atoms whose ground state gaseous neutral electronic configuration is: [Kr] 4d¹⁰, 5s², 5p⁶. In current calculations we use 1s² electrons for He and 4d¹⁰, 5s², 5p⁶ electrons for Xe. They are incorporated in one interstitial site in supercells as shown in Figures 1 and Figures 2 . In an octahedral interstitial of UO₂ they are surrounded by eight nearest neighbours (NN) of O atoms ( Figure 1 ) while in tetrahedral position they have four nearest neighbours: N atoms in UN ( Figure 2 ) and ThN, and C atoms in ThC. The electrons included in the calculations from NN atoms are 2s² and 2p with occupations varying from 4 to 2 for O, N and C, respectively.

In Figure 4 we show the respective electron densities of states of He and Xe together with their nearest neighbours’ (NN) electron densities of states of 2s and 2p electrons. To analyse the effect of interaction, we also plot electron densities of

states of O, N and C atoms, which are positioned away from He and Xe atoms (not NN) in the supercells. Figures 4(a) and Figures 4(c) illustrate that there is no visible interaction between 1s electrons of He (solid black line) with 2p electrons (dashed dot green line) of either O atoms in UO₂ or N atoms in UN. Only 2s electrons are slightly affected. This explains why incorporation of He into interstitials requires low energy as shown (grey diamonds) in Figure 3(b) .

In contrast, incorporation of Xe atoms demonstrates ( Figures 4(b) , Figures 4(d) , Figures 4(e) , Figures 4(f) ) strong interaction between 5p electrons (dashed dot dot dark blue line) of Xe and 2p electrons (dashed dot green line) of O, N and C.

Figures 4(b) illustrates that 5p electrons with a sharp single peak are almost unperturbed by the Xe atom positioned in an octahedral interstitial of UO₂. However, the lower energy peak of the 2p electron density of states of a NN O atom is depressed and higher energy peaks are enhanced, which leads to the net energy increase. Furthermore, the interaction between 2p electrons of NN atoms and 5p electrons of Xe incorporated in a tetrahedral interstitial in UN ( Figures 4(d) ) becomes more significant with the broadening of the 5p electron density of states. The density of states of 2p electrons of NN O (dashed dot green line) shows a three peaks structure with the largest value at higher energies than the single peak shape of 2p electron density of states of not NN O (dashed brown line) electrons. N and C have fewer 2p electrons (three and two versus four); therefore 5p electrons shown in Figures 4(e) and Figures 4(f) occupy a wide range of energies and 2p electrons are modified having an additional peak at energies where the peak of the 5p electron density of states is. These states are at higher energies, and it explains why % increase of absolute value of the total energy due to incorporation of Xe atom increases in ThN and especially in ThC versus UO₂.

The presented results demonstrate of importance of DFT studies in evaluating the effect of FP on swelling and energy increase. The comparison of behaviour of various ceramics using the same methodology (DFT Quantum Espresso Code) is particularly useful. Our work complements existing research which usually focus on one compound as discussed above.

Our analysis demonstrate that behaviour of FP is complex and DFT calculations are needed. Furthermore we need to note that swelling in nuclear fuel is a complex phenomena and requires inclusion of interaction of various defects and diffusion. Molecular dynamics studies with a large number of atoms and a very accurate potentials based on DFT are very promising as presented in recent work [26] [29] .

ThN and ThC have similar crystal structures, as discussed above, but similarly to the energetics evaluated above, the electronic transport is significantly affected by the lower number (2 versus 3) of 2p electrons in C than in N atoms. The electronic properties for ThN, ThC and UN, using a slightly different approach, were previously evaluated [6] [8] [9] as discussed in Section 2, and we compare the results here. We used QE [12] and EPW [16] codes and found significant differences in electronic transport due to differences in the electronic structure of these compounds.

We compare the total electron density of states at Fermi energy (n(ε_F)), the electronic heat capacity coefficient (γ), and the electron-phonon coupling strength (λ) in Table 3 . The shown averages were calculated over the temperature range 300 - 1000 K for the number of mobility electrons (N_e (av)), and the evaluated electronic heat conductivity (κ_e) for UN, ThN and ThC at a typical fuel surface temperature of 700 K. N_e(av) values were obtained from experimental resistivity (exp) and from the solution of BTE using EPW code, as indicated, respectively. We summarized in Section 2 the methods used and the details of the calculations in our recent works [8] [9] .

We found that most of the calculated parameters are similar for both ThN and ThC except for the number of mobility electrons, which we found to be much higher in ThN (2.76 versus 1.20 as calculated from experimental resistivity or 3.04 versus 1.42 obtained using a solution from BTE). We also determined that electron-phonon coupling is about twice as high (0.89 versus 0.43) for ThC as for ThN. This results in ThN’s high electronic thermal conductivity of 40.3 Wm⁻¹K⁻¹ as evaluated using Eq. 2 or 38.2 Wm⁻¹K⁻¹ as calculated from BTE and Eq. 3, respectively. Therefore, ThN is currently considered for application in nuclear reactors as its high thermal conductivity allows for fast heat dissipation and that makes reactors safer and more economical in operation.

We need to point out that the previously calculated values [8] for both the electron density of states n(ε_F) and the electronic heat capacity coefficient (γ), as shown in Table 3 , are very similar and rather smaller: n(ε_F) in ThN is 1.225 versus 1.020 in ThC and (γ) in ThN is 0.00289 versus 0.00240 for ThC. However, it is incorrect to conclude that therefore electronic heat conductivity is low in both compounds as stated previously [11] . ThN has about four times higher electronic thermal conductivity than ThC, as shown in Table 3 . Electronic heat conductivity is a transport phenomenon that is dependent on the number of mobile electronic carriers (N_e (av)) and e-ph scattering rate (λ) as discussed previously [6] - [9] .

The calculations for non-magnetic UN predict a much higher electron density of states at Fermi Energy and heat capacity coefficient than for thorium compounds, as shown in Table 3 and noted previously [6] [9] . There is also a lower number of electronic carriers in UN (0.34) as evaluated from experimental resistivity measurements, but lower electron-phonon coupling (0.25) still leads to a higher calculated electronic thermal conductivity for UN than for ThC as shown in Table 3 . It is again important to note ( Table 3 ) that while the electronic heat capacity coefficient is linearly dependent on the electron density of states at Fermi energy, this simple relation cannot be used to determine the electronic heat conductivity. However since UN is not NM but paramagnetic with non zero local moment we calculated the scaling factor equal to the ratio of the electron density of states at Fermi energy for NM and FM state (s = DOS_NM(E_F)/DOS_FM(E_F)/ = 2.67). We included in the bracket in Table 3 a scaled conductivity by this factor at 700 K (10.5 Wm⁻¹K⁻¹).

In the current version 5.4 of EPW code, only electrons within 3/2 k_BT region around Fermi energy are assumed to be involved in electronic transport. Therefore, as evaluated from BTE, non-magnetic UN has high electronic thermal conductivity [9] as indicated by the red dashed-dot-dot line (28.0 Wm⁻¹K⁻¹ at 700 K), as shown in Figure 5 . This does not agree with the experiment (red dashed-dot line [30] ). The generalization of EPW calculations for UN with any magnetic ordering (ferromagnetic or antiferromagnetic) and a paramagnetic state, in contrast to a non-magnetic state, would lead to a much lower electron density of states of U atoms around Fermi Energy as shown previously [4] , which would rectify these results. Note that in this work, for evaluation of the effect of incorporated FP atoms we analysed DOS close to non-U atoms, so therefore our evaluation is not affected by this discrepancy.

The electronic thermal conductivity (κ_e) of ThN and ThC as calculated from the experimental values for electrical resistivity via Wiedemann-Franz law [18] (Eq. 3) are shown in Figure 5 as green circles [31] , and black diamonds [32] , dotted black line [33] , respectively. They are in good agreement with the electronic thermal conductivities, calculated using Eq. 3, of non-magnetic ThN and ThC and evaluated by EPW (BTE) electrical conductivity [9] as indicated by a medium-dashed green line, and short-dashed black line, respectively. The agreement is particularly good for ThN. According to this prediction, UN should have lower electronic thermal conductivity than ThN, and the values obtained (red dashed-dot-dot line) are closer to the experimental data for ThN than for UN (red dashed dot line). ThN is non-magnetic and the assumption regarding electrons in UN that was incorrect as described above, was used in the calculations for UN. Therefore we also show in Figure 5 as grey short-long dashed line the scaled by the factor s = 2.67 the evaluated by EPW (BTE) thermal electrical conductivity of UN.

Additionally, the values of κ_e calculated using Eqs. 2 and 3 with the averaged over temperature the number of electronic carriers evaluated from the experimental resistivity as calculated previously [8] are shown in Figure 5 for the temperature range from 300 K - 1000 K (note not varying much with temperature) by solid lines, respectively for ThN, thin green, UN, medium red, ThC, thick black. They are in qualitative agreement with the experiment. Furthermore, the results from BTE calculations used together with Eq. 3 correctly predict that the κ_e of ThN is higher than for ThC as calculated from the first principles. It is important to note that the difference originates from the electronic shell structures of ThN and ThC. This results in a different affinity of 2p electrons in the half-filled shell of N (2p³) versus the less than half-filled respective shell of C (2p²) as discussed before [7] [8] and which leads to the different number of mobility electrons, as shown in Table 3 . Furthermore, more experimental attempts should be made to separate the electronic and thermal conductivities in these metallic ceramic fuels. The preliminary evaluation of the electronic thermal conductivity of ThN using “alloy separation” and “curve fitting” techniques gives κ_e~40 Wm⁻¹K⁻¹ at 300 K (Fig. 41 of [34] ) and this result supports our predictions of high electronic thermal conductivity in ThN, which remains almost insensitive to the temperature above 300 K ( Figure 5 ). It has also been confirmed previously [19] that thermal conductivity remains relatively flat at the temperature interval 130 - 300 K.

It is now well established that although magnetic ordering in uranium compounds disappears at low temperature the local moment does not. The accurate calculations for uranium compound must include the effect of local magnetic moments on DOS and noncolinear magnetism like for example in ref. [26] for UO₂ to prevent artificial distortion like observed in [25] .