# Phase Stability in U-6Nb Alloy Doped with Ti from the First Principles Theory

^{*}

## Abstract

**:**

^{0}(tetragonal) can form, depending on Nb concentration. Important mechanical properties depend on the crystal structure and, therefore, an understanding of the effect of impurities on phase stability is essential. Insights on this issue are obtained from quantum-mechanical DFT calculations. The DFT framework does not rely on any material-specific assumptions and is therefore ideal for an unbiased investigation of the U-Nb system.

## 1. Introduction

_{c}= 950 ± 20 °C, and a critical concentration, c

_{c}= 52.3 at.% Nb. At 647 °C an equilibrium monotectoid reaction is exhibited according to the formula γ

_{1}(BCC) → γ

_{2}(BCC) + α(orthorhombic). The γ

_{1}and γ

_{2}phases contain 13.3 and 70 ± 2 at.% Nb, respectively, whereas the α phase has less than 1 at.% Nb in solution. Upon slow cooling under equilibrium conditions, the γ

_{1}phase undergoes diffusional decomposition at 647 °C in order to form a two-phase structure consisting of unalloyed α-uranium and niobium-rich γ-phase (γ

_{2}). At this composition, microstructures tend to consist of discrete niobium-rich γ-phase (γ

_{2}) with a matrix of unalloyed α-uranium. This structure offers poor corrosion resistance due to the lack of niobium in solid solution, and also the two-phase structure gives rise to microscopic anodic and cathodic regions [3].

_{2}(see Figure 2a). DC corresponds to discontinuous coarsening.

_{2}. Due to the former LE, γ would decompose partially into α and metastable γ′, if its initial composition γ was greater than γ′. This explains the occurrence of DP. After a suitably long incubation time, the γ

_{2}phase will nucleate in the system and, since it corresponds to the stable phase in equilibrium with α, it would evolve spontaneously, resulting in DC. As the isothermal holding temperature increased, Djuric [25] observed that the composition of the γ′ precipitate gradually shifted towards Nb-lean compositions. Djuric schematically described energy profiles reproduced in Figure 2b to demonstrate their hypothesis [20,25]. In this Figure the common tangent to the α and γ′ (γ

_{1–2}on Figure 2b) phase curves is also given (dashed line). During the decomposition of the γ′ solid solution below the eutectoid temperature, the α + γ′ phase mixture, corresponding to this tangent, is formed. Of course, being metastable, it continues to transform to the equilibrium α + γ

_{2}(the common tangent solid line). The Gibbs energy schematic put forward by Djuric [25] illustrates the generation of competing common tangent equilibria, but in all cases the common tangent between α and γ′ is always metastable with respect to the α and γ

_{2}equilibrium. Djuric [25] mentioned that, upon Cahn’s theoretical argument [26], during eutectoid decomposition the equilibrium composition of the product phases could only be achieved in the ideal case when the reaction rate approaches zero.

^{0}) [5]. The reactions producing α″ and γ

^{0}are generally regarded as martensitic in nature, occurring by shear and/or displacive mechanisms requiring only cooperative, short-range motions of the atoms [5,7,27]. The highly supersaturated metastable α-phase solid solution exhibits far better corrosion resistance than unalloyed uranium and it is amenable to subsequent age hardening, permitting a wide range of mechanical properties through the selection of aging temperature and time [16,28].

^{0}(tetragonal) → α″(monoclinic) → α′ (orthorhombic)

^{0}has been suggested by Lehmann and Hills [33].

_{2}, depending on the alloy content, form as indicated in Figure 3 [16,29]. In this figure, corresponding paired lines represent the martensite start (M

_{S}) and martensite finish (M

_{F}) temperatures on cooling. Higher cooling rates favor martensitically-formed metastable phases by avoiding diffusional decomposition of γ solution during cooling to room temperature. Because of the sluggish solid-state diffusion in this system, the monotectoid reaction is usually bypassed at even moderate cooling rates (20 °C/s or greater), which are easily achieved during water quenching [3,29].

^{s}, in a 22.2 at.% Nb alloy that was quenched from the γ phase region and annealed for 300 h at 1200 °C. The structure was studied by a single-crystal XRD pattern and identified as a distorted cubic structure. The γ

^{s}phase might be interpreted as a doubled BCC cell in two directions due to periodic displacement of central atoms in the unit cells.

^{0}phase that is formed in U-Mo alloys at low temperature. Upon heating, the tetragonal structure transforms into the cubic γ

^{s}phase [34]. γ

^{s}phase transforms to γ phase with a quasi-body-centered cubic (q-BCC) lattice at elevated temperatures, according to Starikov et al. [35].

^{s}→ γ

^{0}→ α″ → α′.

^{s}has a quasi-body-centered cubic (q-BCC) lattice. The local positions of uranium atoms in γ correspond to γ

^{s}, but the orientations of the central atom displacements are disordered. At a low level of disorder (i.e., at low values of temperature or Mo concentration), the system exhibits long-range correlations in the “anisotropy direction”, and formation of γ

^{0}takes place. At a high level of disorder, long-range correlation in the “anisotropy direction” is disturbed, and the formation of γ

^{s}with q-BCC structure occurs. In this terminology [35], the γ

^{s}-phase is similar to the para-elastic state. It resembles the transformation between neptunium phases (β-Np → BCC-Np), which has been described in a similar manner [37].

^{0}, γ

^{s}and γ.

^{0}, and γ

^{s}, can be obtained by rapid quenching to room temperature, as has been discussed above [4,7,9,19,27,30,31,32,34]. Wu et al. [29] found that increasing a Nb equivalent (Nb

_{eq}) of the U-6 wt.% Nb alloy (adding some small amount of titanium at the expense of uranium) promotes the formation of the γ

^{0}-tetragonal phase at the expense of the α″-monoclinic phase, resulting in a doubling of the yield strength relative to the water quenched (WQ) α″ phase and a strain induced transformation to the α″ phase with superelastic strains to 4.5%.

^{s}phase in the U

_{3}Nb compound that was discovered by Chebotarev and Utkina [34] and modeled by Starikov et al. [35].

## 2. Computational Methods

_{max}= 3. The EMTO orbitals, consecutively, consist of the spdf partial waves (solutions of the radial Schrödinger equation for the spherical OOMT potential wells) and the spdf screened spherical waves (solutions of the Helmholtz equation for the OOMT muffin-tin zero potential). The completeness of the muffin-tin basis was discussed in detail in [56], and it was demonstrated that for metals crystallizing in close-packed lattices l

_{max}= 3 (spdf orbitals) leads to well-converged charge density and total energy. The generalized gradient approximation (GGA) is selected for the electron exchange and correlation energy functional [43]. Integration over the Brillouin zone is implemented while using a 25 × 25 × 25 grid of k-points resolved according to the Monkhorst–Pack technique [58]. The moments of the density of states, needed for the kinetic energy and valence charge density, are computed by integrating the Green’s function over a complex energy contour (with 2.8–3.0 Ry diameter) while using a Gaussian integration technique with 40 points on a semi-circle encircling the occupied states.

_{U}= α

_{Nb}= 0.724 and β = 1.411 for U-Nb alloys; α

_{U}= 0.742, α

_{Nb}= 0.694, α

_{Ti}= 0.767, and β = 1.101 for U-Nb-Ti alloys. The EMTO-CPA technique has been previously successfully utilized in order to define thermodynamic properties of the uranium-based metallic nuclear fuels, including: U-Zr, U-Pu-Zr, U-Mo, U-Ti, and U-Nb [19,65,66,67,68,69].

## 3. Results

^{0}-tetragonal phase at the expense of the α″-monoclinic phase [29]. However, we demonstrate the applicability of the DFT approach more generally for unalloyed uranium metal before addressing this particular alloy. For elemental metals, the most exact results are obtained from the FPLMTO method because of its lack of any structure-geometrical approximations. In Figure 5, we show total energies for α, α″, γ

^{0}, and γ uranium, as functions of the atomic volume for both FPLMTO and EMTO methods. In these calculations, the axial ratios, and internal parameters for α, α″, and γ

^{0}have been optimized (relaxed) in order to produce the lowest energy state utilizing the FPLMTO approach. The same structure (no further relaxation) is adopted for the EMTO computations. The details of these structures can be found in Refs. [7,13,16,18,27,31] (see Appendix A). Importantly, we find that the ranking of the phases in terms of energy is the same for both methods with the ground-state α phase lowest and the BCC (γ) phase highest. The energy differences are also about the same, except for EMTO over-estimating γ

^{0}relative to FPLMTO. There are also slight shifts in the equilibrium volumes between these two methods, where EMTO predicts smaller values than both FPLMTO and experiments. For example, FPLMTO predicts an α-phase volume of 20.67 Å3, while EMTO concludes 19.94 Å3. The smaller volumes that were obtained from EMTO calculations originate from neglecting the spin-orbit coupling [49,67] (in the present study only scalar relativistic EMTO calculations are performed). The experimental zero-Kelvin volume is very close to the FPLMTO value, 20.55 Å

^{3}[70].

^{0}(tetragonal) phases at a constant atomic volume 20.3 Å

^{3}. Interestingly, there are no local maxima or minima, except at the locations of γ

^{0}and γ phases, on this two-dimensional energy surface. The optimized γ

^{0}structure (c/a = 0.94 and z = 0.13) is adopted for the EMTO calculations of this phase. Uranium metal also has a complicated 30-atom tetragonal β phase that only exists at high temperatures [1] and it is less important for the U-Nb system. Therefore, we have not included β-U in our current modeling.

^{0}, and γ structures, mentioned above for uranium, calculated for pure uranium and niobium metals within both FPLMTO and EMTO methods. The total energy of the α (monoclinic) structure is assumed to be equal to zero (as the reference point). As in the case of uranium, for niobium metal, the axial ratios and internal parameters for α, α″, and γ

^{0}have been optimized (relaxed) in order to produce the lowest energy state utilizing the FPLMTO approach. The same structure (no further relaxation) is adopted for the EMTO computations for niobium. The γ (BCC) structure is the ground state for niobium that lies 13 and 14 mRy/atom, for FPLMTO and EMTO, respectively, below the energy of the optimized α-U-type structure for niobium (the reference point), as can be seen from Figure 7. The energy of the γ

^{0}-type uranium structure for niobium converges to the energy of the ground state γ structure when the structure is relaxed (or optimized) according to both FPLMTO and EMTO calculations. The optimized α″-uranium structure for niobium has the energy of +7.9 mRy/atom and +12.7 mRy/atom with respect to the reference system (α-Nb), for FPLMTO and EMTO calculations, respectively, although the corresponding energies are very “close” in the case of uranium metal.

_{1-x}Nb

_{x}alloys require the optimization of the axial ratios and internal parameters for α, α″, and γ

^{0}structures at each value of the concentration “x”, the result for the γ (BCC) structure (no optimization is necessary) brings us to an interesting conclusion. According to FPLMTO and EMTO calculations, the γ structure becomes the stable structure in the U-Nb system when the amount of niobium exceeds 60 at.%, as can be seen from Figure 7. The U-Nb phase diagram, Refs. [15,19], at 647 °C an equilibrium monotectoid reaction occurs according to the formula γ

_{1}(BCC) → γ

_{2}(BCC) + α(orthorhombic), as is mentioned in the Introduction section [4]. The γ

_{1}and γ

_{2}phases contain 13.3 and 70 ± 2 at.% Nb, respectively, whereas, the α phase has less than 1 at.% Nb in solution [15,19]. From FPLMTO and EMTO calculations, the estimated range of the stable niobium rich γ-phase (>60 at.% of Nb) of the U-Nb solid solution corresponds to the upper Nb concentration limit (70 ± 2 at.% Nb) of the stable, γ

_{2}phase, the so-called “niobium-rich BCC solid solutions” formed as the result of the diffusional monotectoid reaction: γ

_{1}(BCC) → γ

_{2}(BCC) + α(orthorhombic).

^{0}-tetragonal phase (recently established experimentally in Ref. [29]). Figure 8a shows the calculated (DFT-EMTO) energies for two metastable α″ and γ

^{0}phases of the U

_{86.23}Nb

_{13.77}alloy as functions of the atomic volume. The amount of Nb (13.77 at.%) in the U-Nb alloy corresponds to the value that is presented in Table 1 of Ref. [29] for the alloy named the “Powder 1” in the first row: “As-built = 137,000 ± 400” ppm. The metastable α″ phase of the U

_{86.23}Nb

_{13.77}alloy exhibits a lower energy than the metastable γ

^{0}phase indicating the stability of the α″ phase over the γ

^{0}phase observed in this alloy, as can be seen from Figure 8a [29].

^{0}phases of the U

_{83.56}Nb

_{13.84}Ti

_{2.60}alloy as function of the atomic volume. The amount of Nb (13.84 at.%) in the U-Nb-Ti alloy corresponds to the value that is presented in Table 1 of Ref. [29] for the alloy named the “Powder 2” in the first row: “As-built = 138,400 ± 300 ppm”. The metastable γ

^{0}phase of the U

_{83.56}Nb

_{13.84}Ti

_{2.60}alloy exhibits a lower energy than another metastable α″ phase indicating the stability of the γ

^{0}phase over α″ phase observed in this alloy, as can be seen from Figure 8b [29]. Hence, the effect of Ti is to favor γ

^{0}over α″.

_{eq}(at.%) = Nb (at.%) + Σ

_{impurities}(at.%)) increases from 13.77 at.% (U-Nb alloy) to 13.84 + 2.60 = 16.44 at.%. As has been mentioned in the Introduction, depending on the Nb concentration, the U-Nb alloys exhibit the following room-temperature structures on quenching from the γ phase: α′ orthorhombic (0 to 9 at.%); α″ monoclinic (9 to 15 at.%); and, γ

^{0}tetragonal (16 to 20 at.%). By keeping the amount of niobium metal almost the same as was in the original U-Nb alloy and adding 2.60 at.% of titanium, the authors [29] increased the Nb equivalent, Nb

_{eq}, from 13.77 at.% (U-Nb) to 16.44 at.% (U-Nb-Ti). Nb

_{eq}(at.%) = 13.77 at.% corresponds to the stability region of the metastable α″-monoclinic structure, but Nb

_{eq}(at.%) = 16.44 at.% corresponds to the stability region of the γ

^{0}-tetragonal structure, as has been mentioned above. These arguments explain why the addition of some small amount of titanium metal to the U-6Nb alloy promotes the stabilization of the γ

^{0}-tetragonal phase at the expense of the α″-monoclinic phase, resulting in better mechanical properties of U-6Nb alloys [28].

^{0}→ α″ → α′.

^{s}, could be quenched in the U-22.2 at.% Nb alloy, according to the experiments of Chebotarev and Utkina [34] and recent modeling of Starikov et al. [35]. Starikov et al. [35] suggested that the diffusion-less transitions in the U-Nb system can follow the scheme (see the expression (3)): γ → γ

^{s}→ γ

^{0}→ α″ → α′.

_{1-x}Nb

_{x}alloys, the metastable phase variants of the stable α and γ

_{2}phases, α′, α″, and γ

^{0}, form depending on the niobium content. According to Figure 3, increasing Nb content leads to a sequence of the metastable phases: α′ → α″ → γ

^{0}[16,29]. This progression is consistent with the results shown in Figure 5 where the stable α phase of uranium metal has the lowest energy followed by the metastable α″ and γ

^{0}phases, and finally the stable γ uranium phase. As was experimentally established in Ref. [34], another metastable phase, γ

^{s}, in a U-22.2 at.% Nb alloy could be quenched from the γ phase region. Of course, it makes sense that a relatively large amount of the BCC metal Nb (>20 at.%) can “stabilize” this metastable γ phase. Actually, a simple approximate model where one is substituting uranium with niobium atoms (without allowing for any disorder or structural relaxations) explains this stabilization, see Figure 9. In this model, it is clear that 25 at.% Nb favors γ over α uranium. Intuitively, one associates the “stabilization” of the metastable γ phase as being due to alloying with a metal of the same structure (Nb is also BCC). However, this is not the driving force of the stabilization, but rather more generally the presence of 4d electrons that belongs to the transition metal. The α phase originates from the bonding 5f states of the uranium atom [37,38], while the BCC phase is a result of a balance between 5f and 4d states, where the latter provides the preference for the BCC phase [39]. To show this, we illustrate in Figure 9 that replacing the Nb atom with an atom from a metal that is face-centered cubic (FCC), namely Rh, actually has a greater effect on stabilizing the γ phase than Nb itself (dashed lines in the middle panel). This might seem counterintuitive, but the reason is that Rh provides more 4d electrons than Nb and these electrons help in stabilizing the γ phase. However, for practical purposes, Rh is far too rare and expensive to be considered a viable stabilizer.

^{s}phase of the U

_{3}Nb compound in Figure 10. In this figure, the parameter x distorts the BCC γ phase (x = 0.25) and defines the γ

^{s}phase. As opposed to the calculation in Figure 9, the structures are all fully relaxed and for the U

_{3}Nb compound, the α phase remains slightly below the BCC (γ) phase. The metastable γ

^{s}phase is observed for the specific U

_{77.5}Nb

_{22.2}alloy with the specific value of the parameter, x = 0.241 ± 0.001, according to Chebotarev and Utkina [34]. In Figure 10, we notice that energetics of the metastable γ

^{s}phase of the U

_{3}Nb compound, within the interval 0.242 ≤ x ≤ 0.248, is only slightly higher either then that of the γ or α forms of this compound.

## 4. Summary and Conclusions

^{0}(tetragonal) in high Nb regions of the microstructure [16,29]. Wu et al. [29] investigated the microstructure-property behaviors of two U-6Nb alloys with different impurity levels that were fabricated while using laser powder bed-based additive manufacturing (AM). Two powder alloys have been investigated, which differ in composition, such that one alloy (referred to as the high impurity alloy, HI) contains a significantly higher amount of Ti than the other (referred to as the low impurity alloy, LI) [29]. Both HI and LI alloys contain 13.6–13.9 at.% (5.9–6.0 wt.%) Nb; however, the HI alloy contains 2.37 at.% (0.49 wt.%) of Ti and LI alloy contains almost no Ti − 0.0022 at.% (0.00044 wt.%). Neutron diffraction measurements of the LI alloy indicated a mixed phase in the as-built condition, resembling 42% γ

^{0}and 58% α″ [29]. After homogenization and water quenching from high to room temperatures (WQ), the LI alloy had approximately 100% α″ structure and this structure was not affected by aging at 200 °C for 2 h [29]. The HI alloy possessed approximately 50% γ

^{0}and 50% α″ in the as-built condition; however, after homogenization and WQ, the HI alloy had 100% γ

_{0}structure and aging also did not affect this structure [29]. The mechanical properties of these two alloys were affected by their different microstructures: after homogenization/WQ/aging, the HI alloy with 100% γ

^{0}-structure showed a yield strength increase of 550 MPa, over 2× that of the LI alloy, which contained 100% α″-structure. This increase in yield strength was accompanied by super-elastic strains up to 4.5% [29].

_{86.23}Nb

_{13.77}alloy with 7.80 wt.% Nb in order to simplify our modeling conditions. We also assumed that the HI alloy had no impurities, except 2.60 at.% of Ti, and it is represented as the U

_{83.56}Nb

_{13.84}Ti

_{2.60}alloy with 6.04 wt.% Nb.

^{0}-tetragonal phase at the expense of the α″ monoclinic phase, which is in perfect accord with the experimental results [29]. We explain this phenomenon by comparing the Nb equivalent for the U

_{86.23}Nb

_{13.77}alloy (Nb

_{eq}(at.%) = 13.77 at.%) with one for the U

_{83.56}Nb

_{13.84}Ti

_{2.60}alloy (Nb

_{eq}(at.%) = 16.44 at.%). At intermediate Nb content (9–15 at.% Nb), the metastable α″-monoclinic structure is formed and at the higher Nb content (16–20 at.% Nb) the metastable γ

^{0}-tetragonal structure is formed. We argue that the U

_{86.23}Nb

_{13.77}alloy has a Nb

_{eq}(at.%) = 13.77 at.%, which corresponds to the stability region of the α″-monoclinic structure, but the U

_{83.56}Nb

_{13.84}Ti

_{2.60}alloy has a Nb

_{eq}(at.%) = 16.44 at.%, which corresponds to the stability region of the γ

^{0}-tetragonal structure.

^{0}-tetragonal structure over another metastable α″-monoclinic structure observed [29] in the WQ U-6Nb-Ti alloy.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. The α-uranium Structure

- Primitive Vectors
**A**= ½ a_{1}**X**− ½ b**Y**+ 0**Z****A**= ½ a_{2}**X**+ ½ b**Y**+ 0**Z****A**= 0_{3}**X**+ 0**Y**+ c**Z**

- Basis Vectors
**B**= 0_{1}**X**+ y b**Y**+ ¼ c**Z****B**= 0_{2}**X**− y b**Y**− ¼ c**Z**

- Uranium: c/a = 1.736; b/a = 2.057; y = 0.1025
- Niobium: c/a = 1.860; b/a = 2.080; y = 0.1340

#### Appendix A.2. The α″-uranium Structure

- Primitive Vectors
**A**= a_{1}**X**+ 0**Y**+ 0**Z****A**= b cosγ_{2}**X**+ b sinγ**Y**+ 0**Z****A**= 0_{3}**X**+ 0**Y**+ c**Z**

- Basis Vectors
**B**= 0_{1}**X**+ 0**Y**+ 0**Z****B**= 0_{2}**X**+ y_{1}b**Y**+ ½ c**Z****B**= ½ a_{3}**X**+ y_{2}b**Y**+ ½ c**Z****B**= ½ a_{4}**X**+ ½ b**Y**+ 0**Z**

- Uranium: c/a = 1.753; b/a = 2.022; α = 90°; β = 90°; γ = 91.30°; y
_{1}= 0.833333; y_{2}= 0.333333 - Niobium: c/a = 1.820; b/a = 1.940; α = 90°; β = 90°; γ = 91.30°; y
_{1}= 0.833333; y_{2}= 0.333333

#### Appendix A.3. The γ^{0}-uranium Structure

- Primitive Vectors
**A**= a_{1}**X**+ 0**Y**+ 0**Z****A**= 0_{2}**X**+ b**Y**+ 0**Z****A**= 0_{3}**X**+ 0**Y**+ c**Z**

- Basis Vectors
**B**= 0_{1}**X**+ 0**Y**+ 0**Z****B**= ½ a_{2}**X**+ ½ b**Y**+ 0**Z****B**= 0_{3}**X**+ ½ b**Y**+ (½ c + z)**Z****B**= ½ a_{4}**X**+ 0**Y**+ (½ c − z)**Z**

- Uranium: b/a = c/a = 0.94; z = 0.13

#### Appendix A.4. The Body Centered Cubic structure

- Primitive Vectors
**A**= − ½ a_{1}**X**+ ½ a**Y**+ ½ a**Z****A**= ½ a_{2}**X**− ½ a**Y**+ ½ a**Z****A**= ½ a_{3}**X**+ ½ a**Y**− ½ a**Z**

- Basis Vectors
**B**= 0_{1}**X**+ 0**Y**+ 0**Z**

## References

- Donohue, J. The Structures of the Elements; John Wiley & Sons: New York, NY, USA, 1974. [Google Scholar] [CrossRef]
- Söderlind, P.; Grabowski, B.; Yang, L.; Landa, A.; Björkman, T.; Souvatzis, P.; Eriksson, O. High-temperature phonon stabilization of γ-uranium from relativistic first-principles theory. Phys. Rev. B
**2012**, 85, 060301. [Google Scholar] [CrossRef] [Green Version] - Eckelmeyer, K.H.; Romig, A.D.; Weirick, L.J. The effect of quench rate on the microstructure, mechanical properties, and corrosion behavior of U-6 wt pct Nb. Metall. Trans. A
**1984**, 15, 1319–1330. [Google Scholar] [CrossRef] - Vandermeer, R. Phase transformations in a uranium + 14 at.% niobium alloy. Acta Metall.
**1980**, 28, 383–393. [Google Scholar] [CrossRef] - Vandermeer, R.A.; Ogle, J.C.; Northcutt, W.G. A phenomenological study of the shape memory effect in polycrystalline uranium-niobium alloys. Metall. Trans. A
**1981**, 12, 733–741. [Google Scholar] [CrossRef] - Brown, D.W.; Bourke, M.A.M.; Stout, M.G.; Dunn, P.S.; Field, R.D.; Thoma, D.J. Uniaxial tensile deformation of uranium 6 wt pct niobium: A neutron diffraction study of deformation twinning. Metall. Mater. Trans. A
**2001**, 32, 2219–2228. [Google Scholar] [CrossRef] - Tangri, K.; Williams, G. Metastable phases in the uranium molybdenum system and their origin. J. Nucl. Mater.
**1961**, 4, 226–233. [Google Scholar] [CrossRef] - Howlett, B. A study of the shear transformations from the gamma-phase in uranium-molybdenum alloys containing 6.0–12.5 at % molybdenum. J. Nucl. Mater.
**1970**, 35, 278–292. [Google Scholar] [CrossRef] - Anagnostidis, M.; Colombie, M.; Monti, H. Phases metastables dans les alliages uranium-niobium. J. Nucl. Mater.
**1964**, 11, 67–76. [Google Scholar] [CrossRef] - Takahashi, Y.; Yamawaki, M.; Yamamoto, K. Thermophysical properties of uranium-zirconium alloys. J. Nucl. Mater.
**1988**, 154, 141–144. [Google Scholar] [CrossRef] - Kahana, E.; Talianker, M.; Landau, A. Formation of the monoclinic α “phase in quenched U-3.6 at.% Ti-4.7 at.% W alloy. J. Nucl. Mater.
**1997**, 246, 144–149. [Google Scholar] [CrossRef] - Speer, J.; Edmonds, D. An investigation of the γ → α martensitic transformation in uranium alloys. Acta Metall.
**1988**, 36, 1015–1033. [Google Scholar] [CrossRef] - Zhang, C.; Xie, L.; Fan, Z.; Wang, H.; Chen, X.; Li, J.; Sun, G. Straightforward understanding of the structures of metastable α″ and possible ordered phases in uranium–niobium alloys from crystallographic simulation. J. Alloys Compd.
**2015**, 648, 389–396. [Google Scholar] [CrossRef] - Lillard, J.A.; Hanrahan, R.J., Jr. Corrosion of uranium and uranium alloys. In Corrosion: Materials; Cramer, S.D., Covino, B.S., Eds.; ASM International: Materials Park, OH, USA, 2005; Volume 13B, pp. 370–384. [Google Scholar] [CrossRef]
- Koike, J.; Kassner, M.; Tate, R.E.; Rosen, R.S. The Nb-U (Niobium-Uranium) system. J. Phase Equilibria
**1998**, 19, 253–260. [Google Scholar] [CrossRef] - Hackenberg, R.E.; Brown, D.W.; Clarke, A.J.; Dauelsberg, L.B.; Field, R.D.; Hults, W.L.; Kelly, A.M.; Lopez, M.F.; Teter, D.F.; Thoma, D.J.; et al. U–Nb Aging Final Report; Report No. LA-14327; Los Alamos National Laboratory: Los Alamos, NM, USA, 2007. [Google Scholar]
- Liu, X.; Li, Z.; Wang, J.; Wang, C. Thermodynamic modeling of the U–Mn and U–Nb systems. J. Nucl. Mater.
**2008**, 380, 99–104. [Google Scholar] [CrossRef] - Field, R.; Thoma, D.J.; Dunn, P.S.; Brown, D.W.; Cady, C.M. Martensitic structures and deformation twinning in the U–Nb shape-memory alloys. Philos. Mag. A
**2001**, 81, 1691–1724. [Google Scholar] [CrossRef] - Duong, T.C.; Hackenberg, R.; Landa, A.; Honarmandi, P.; Talapatra, A.; Volz, H.M.; Llobet, A.; Smith, A.I.; King, G.; Bajaj, S.; et al. Revisiting thermodynamics and kinetic diffusivities of uranium–niobium with Bayesian uncertainty analysis. Calphad
**2016**, 55, 219–230. [Google Scholar] [CrossRef] [Green Version] - Duong, T.C.; Hackenberg, R.E.; Attari, V.; Landa, A.; Turchi, P.E.; Arroyave, R. Investigation of the discontinuous precipitation of U-Nb alloys via thermodynamic analysis and phase-field modeling. Comput. Mater. Sci.
**2020**, 175, 109573. [Google Scholar] [CrossRef] - Volz, H.; Hackenberg, R.; Kelly, A.; Hults, W.; Lawson, A.; Field, R.; Teter, D.; Thoma, D. X-ray diffraction analyses of aged U–Nb alloys. J. Alloys Compd.
**2007**, 444–445, 217–225. [Google Scholar] [CrossRef] - Clarke, A.; Field, R.; Hackenberg, R.; Thoma, D.; Brown, D.; Teter, D.; Miller, M.; Russell, K.; Edmonds, D.; Beverini, G. Low temperature age hardening in U–13 at.% Nb: An assessment of chemical redistribution mechanisms. J. Nucl. Mater.
**2009**, 393, 282–291. [Google Scholar] [CrossRef] - Hackenberg, R.; Volz, H.M.; Papin, P.A.; Kelly, A.M.; Forsyth, R.T.; Tucker, T.J.; Clarke, K. Kinetics of Lamellar Decomposition Reactions in U-Nb Alloys. Solid State Phenom.
**2011**, 172–174, 555–560. [Google Scholar] [CrossRef] - Zhang, J.; Brown, D.W.; Clausen, B.; Vogel, S.C.; Hackenberg, R.E. In Situ Time-Resolved Phase Evolution and Phase Transformations in U-6 Wt Pct Nb. Metall. Mater. Trans. A
**2019**, 50, 2619–2628. [Google Scholar] [CrossRef] - Djurič, B. Decomposition of gamma phase in a uranium-9.5 wt % niobium alloy. J. Nucl. Mater.
**1972**, 44, 207–214. [Google Scholar] [CrossRef] - Cahn, J. The kinetics of cellular segregation reactions. Acta Metall.
**1959**, 7, 18–28. [Google Scholar] [CrossRef] - Tangri, K.; Chaudhuri, D.K. Metastable phases in uranium alloys with high solubility in the BCC gamma phase. Part—The system U-Nb. J. Nucl. Mater.
**1965**, 4, 278–287. [Google Scholar] [CrossRef] - Brown, D.W.; Bourke, M.; Clarke, A.; Field, R.; Hackenberg, R.; Hults, W.; Thoma, D. The effect of low-temperature aging on the microstructure and deformation of uranium-6 wt% niobium: An in-situ neutron diffraction study. J. Nucl. Mater.
**2016**, 481, 164–175. [Google Scholar] [CrossRef] [Green Version] - Wu, A.S.; Brown, D.W.; Clausen, B.; Elmer, J.W. The influence of impurities on the crystal structure and mechanical properties of additive manufactured U–14 at.% Nb. Scr. Mater.
**2017**, 130, 59–63. [Google Scholar] [CrossRef] [Green Version] - D’Amato, C.; Saraceno, F.; Wilson, T. Phase transformations and equilibrium structures in uranium-rich niobium alloys. J. Nucl. Mater.
**1964**, 12, 291–304. [Google Scholar] [CrossRef] - Yakel, H.L. Crystal structures of transition phases formed in U/16.60 at% Nb/5.64 at% Zr alloys. J. Nucl. Mater.
**1969**, 33, 286–295. [Google Scholar] [CrossRef] - Jackson, R.J. Metallographic study of segregation in uranium-base niobium alloys. Metallography
**1973**, 6, 347–359. [Google Scholar] [CrossRef] - Lehmann, J.; Hills, R.F. Proposed nomenclature for phases in uranium alloys. J. Nucl. Mater.
**1960**, 2, 261–268. [Google Scholar] [CrossRef] - Chebotarev, N.T.; Utkina, O.N. Crystal structure of the γ
^{s}phase in uranium- molybdenum, uranium-rhenium, and uranium-niobium alloys. Atomnaya Energiya**1980**, 48, 76–80. [Google Scholar] - Starikov, S.; Kolotova, L.; Kuksin, A.Y.; Smirnova, D.; Tseplyaev, V. Atomistic simulation of cubic and tetragonal phases of U-Mo alloy: Structure and thermodynamic properties. J. Nucl. Mater.
**2018**, 499, 451–463. [Google Scholar] [CrossRef] - Ivanov, O.S.; Badaeva, T.A.; Sofronova, R.M.; Kishenevskiy, V.B.; Kushnir, N.P. Phase Diagrams and Phase Transitions in Uranium Alloys; Nauka: Moscow, Russia, 1972. [Google Scholar]
- Söderlind, P. Theory of the crystal structures of cerium and the light actinides. Adv. Phys.
**1998**, 47, 959–998. [Google Scholar] [CrossRef] - Söderlind, P. First-principles phase stability, bonding, and electronic structure of actinide metals. J. Electron Spectrosc. Relat. Phenom.
**2014**, 194, 2–7. [Google Scholar] [CrossRef] [Green Version] - Skriver, H.L. Crystal structure from one-electron theory. Phys. Rev. B
**1985**, 31, 1909–1923. [Google Scholar] [CrossRef] - Hohenberg, P.; Kohn, W. Inhomogeneous electron gas. Phys. Rev.
**1964**, 136, B864–B871. [Google Scholar] [CrossRef] [Green Version] - Kohn, W.; Sham, L. Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev.
**1965**, 140, A1133–A1138. [Google Scholar] [CrossRef] [Green Version] - Perdew, J.P. Electronic Structure of Solids; Ziesche, P., Eschrig, H., Eds.; Springer: Berlin/Heidelberg, Germany, 1991; pp. 11–20. [Google Scholar]
- Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett.
**1996**, 77, 3865–3868. [Google Scholar] [CrossRef] [Green Version] - Hedin, L.; Lundqvist, B.I. Explicit local exchange-correlation potentials. J. Phys. C Solid State Phys.
**1971**, 4, 2064–2083. [Google Scholar] [CrossRef] - Perdew, J.P.; Ruzsinszky, A.; Csonka, G.I.; Vydrov, O.A.; Scuseria, G.E.; Constantin, L.A.; Zhou, X.; Burke, K. Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces. Phys. Rev. Lett.
**2008**, 100, 136406. [Google Scholar] [CrossRef] [Green Version] - Xie, W.; Xiong, W.; Marianetti, C.A.; Morgan, D. Correlation and relativistic effects in U metal and U-Zr alloy: Validation of ab initio approaches. Phys. Rev. B
**2013**, 88, 235128. [Google Scholar] [CrossRef] - Söderlind, P.; Landa, A.; Turchi, P.E.A. Comment on “Correlation and relativistic effects in U metal and U-Zr alloy: Validation of ab initio approaches”. Phys. Rev. B
**2014**, 90, 157101. [Google Scholar] [CrossRef] [Green Version] - Söderlind, P.; Sadigh, B.; Lordi, V.; Landa, A.; Turchi, P. Electron correlation and relativity of the 5f electrons in the U–Zr alloy system. J. Nucl. Mater.
**2014**, 444, 356–358. [Google Scholar] [CrossRef] [Green Version] - Sadigh, B.; Kutepov, A.; Landa, A.; Söderlind, P. Assessing Relativistic Effects and Electron Correlation in the Actinide Metals Th to Pu. Appl. Sci.
**2019**, 9, 5020. [Google Scholar] [CrossRef] [Green Version] - Lejaeghere, K.; Bihlmayer, G.; Björkman, T.; Blaha, P.; Blügel, S.; Blum, V.; Caliste, D.; Castelli, I.E.; Clark, S.J.; Corso, A.D.; et al. Reproducibility in density functional theory calculations of solids. Science
**2016**, 351, aad3000. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wills, J.M.; Eriksson, O.; Andersson, P.; Delin, A.; Grechnyev, O.; Alouani, M. Full-Potential Electronic Structure Method; Springer Series in Solid-State Science; Springer: Berlin/Heidelberg, Germany, 2010; Volume 167. [Google Scholar]
- Wills, J.M.; Eriksson, O. Crystal-structure stabilities and electronic structure for the light actinides Th, Pa, and U. Phys. Rev. B
**1992**, 45, 13879–13890. [Google Scholar] [CrossRef] - Fast, L.; Eriksson, O.; Johansson, B.; Wills, J.M.; Straub, G.; Roeder, H.; Nordström, L. Theoretical Aspects of the Charge Density Wave in Uranium. Phys. Rev. Lett.
**1998**, 81, 2978–2981. [Google Scholar] [CrossRef] - Söderlind, P.; Landa, A.; Sadigh, B. Density-functional theory for plutonium. Adv. Phys.
**2019**, 68, 1–47. [Google Scholar] [CrossRef] - Vitos, L. Total-energy method based on the exact muffin-tin orbitals theory. Phys. Rev. B
**2001**, 64, 014107. [Google Scholar] [CrossRef] - Vitos, L. Computational Quantum Mechanics for Materials Engineers: The EMTO Method and Application; Springer: London, UK, 2007. [Google Scholar]
- Kollár, J.; Vitos, L.; Skriver, H. From ASA Towards the Full Potential. In Lecture Notes in Physics; Dreyssé, H., Ed.; Springer: Berlin/Heidelberg, Germany, 2007; pp. 85–113. [Google Scholar]
- Monkhorst, H.J.; Pack, J.D. Special points for Brillouin-zone integrations. Phys. Rev. B
**1976**, 13, 5188–5192. [Google Scholar] [CrossRef] - Faulkner, J. The modern theory of alloys. Prog. Mater. Sci.
**1982**, 27, 1–187. [Google Scholar] [CrossRef] - Vitos, L.; Abrikosov, I.; Johansson, B. Anisotropic Lattice Distortions in Random Alloys from First-Principles Theory. Phys. Rev. Lett.
**2001**, 87, 156401. [Google Scholar] [CrossRef] [PubMed] - Ruban, A.; Skriver, H. Screened Coulomb interactions in metallic alloys. I. Universal screening in the atomic-sphere approximation. Phys. Rev. B
**2002**, 66, 024201. [Google Scholar] [CrossRef] [Green Version] - Ruban, A.; Simak, S.I.; Korzhavyi, P.A.; Skriver, H. Screened Coulomb interactions in metallic alloys. II. Screening beyond the single-site and atomic-sphere approximations. Phys. Rev. B
**2002**, 66, 024202. [Google Scholar] [CrossRef] [Green Version] - Ruban, A.; Simak, S.I.; Shallcross, S.; Skriver, H. Local lattice relaxations in random metallic alloys: Effective tetrahedron model and supercell approach. Phys. Rev. B
**2003**, 67, 214302. [Google Scholar] [CrossRef] [Green Version] - Abrikosov, I.; Simak, S.; Johansson, B.; Ruban, A.V.; Skriver, H. Locally self-consistent Green’s function approach to the electronic structure problem. Phys. Rev. B
**1997**, 56, 9319–9334. [Google Scholar] [CrossRef] [Green Version] - Landa, A.; Söderlind, P.; Turchi, P.E. Density-functional study of the U–Zr system. J. Alloys Compd.
**2009**, 478, 103–110. [Google Scholar] [CrossRef] - Landa, A.; Söderlind, P.; Turchi, P.E.A.; Vitos, L.; Ruban, A. Density functional study of Zr-based actinide alloys. J. Nucl. Mater.
**2009**, 385, 68–71. [Google Scholar] [CrossRef] - Landa, A.; Söderlind, P.; Turchi, P.E.A.; Vitos, L.; Ruban, A. Density-functional study of Zr-based actinide alloys: 2. U-Pu-Zr system. J. Nucl. Mater.
**2009**, 393, 141–145. [Google Scholar] [CrossRef] - Landa, A.; Söderlind, P.; Turchi, P. Density-functional study of U–Mo and U–Zr alloys. J. Nucl. Mater.
**2011**, 414, 132–137. [Google Scholar] [CrossRef] [Green Version] - Bajaj, S.; Landa, A.; Söderlind, P.; Turchi, P.E.; Arroyave, R. The U–Ti system: Strengths and weaknesses of the CALPHAD method. J. Nucl. Mater.
**2011**, 419, 177–185. [Google Scholar] [CrossRef] - Söderlind, P.; Young, D.A. Assessing Density-Functional Theory for Equation-Of-State. Computation
**2018**, 6, 13. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**The equilibrium U-Nb phase diagram [15,16]. The γ phase is the body-centered-cubic solid solution phase, whereas α and β are the orthorhombic and tetragonal uranium allotropes that have very limited solubility of Nb. Below the 647 °C monotectoid isotherm, the equilibrium state of U-6Nb consists of ~80 mole % α (~0 at.% Nb) + ~20 mole % γ

_{2}(70–75 at.% Nb), indicated by the purple tie line. Decomposition of alloys with 4-8 wt.% Nb (9.65-18.22 at.%) in the 350 °C–650 °C temperature range takes place by cellular precipitation, initially to α + metastable γ′ (blue dotted tie line) before further evolution to α + equilibrium γ

_{2}. The decomposition paths and mechanisms of these same alloys below 350 °C are less clear, but will involve the formation of the metastable supersaturated γ

^{0}or α″ martensitic phase (see Figure 3) and its subsequent decomposition (tempering) as it evolves toward the α + γ

_{2}final equilibrium state [15].

**Figure 2.**(

**a**) Schematic representation of discontinuous monotectoid decomposition in the uranium–niobium system; and, (

**b**) schematic energies describing Djuric’s hypothesis, taken from Ref. [20].

**Figure 3.**The metastable U-Nb binary alloy phase diagram [29], developed by Thoma et al. and reproduced with permission from [16], with martensite start, M

_{S}, and martensite finish, M

_{F}, temperatures indicated for the metastable phases [29]. Composition points are plotted (squares) at room temperature. Alloys with 0.75–3 wt.% Nb (1.9 at.%–7.34 at.% Nb) show a single γ ↔ α′ transition, whereas alloys with 4–10 wt.% Nb (9.65 at.%–22.16 at.% Nb) show two transitions: γ ↔ γ

^{0}. and γ

^{0}↔ α″.

**Figure 4.**The T-x U-Mo diagram showing areas of the phase stability calculated for different structures: the γ

^{0}-phase and γ-phase [35]. The phase stability area for the γ

^{0}-phase is bounded by solid blue line. The blue shaded area shows estimations for stability of the γ

^{s}-phase. The black dashed curve shows the evaluated line of the phase transition between γ

^{0}and γ phases in the U-Nb alloy [16].

**Figure 5.**Density-functional-theory (DFT) total energies; full-potential linear muffin-tin orbital (FPLMTO) (solid), and EMTO (dashed) total energies for α, α″, γ

^{0}, and γ uranium.

**Figure 6.**DFT-FPLMTO energy landscape for uranium at a constant atomic volume of 20.3 Å

^{3}. The “+” indicates the lowest (optimized or relaxed) energy for γ

^{0}(shifted to zero energy) and the top right corner, “BCC”, that of the γ energy. For an explanation of the atomic displacement z, see Refs. [7,16,18,23,27,31].

**Figure 7.**FPLMTO and EMTO total energies for α″ (monoclinic), γ

^{0}(tetragonal), and γ (BCC) structures calculated for pure U and Nb. The total energy of α (monoclinic) structure is assumed to be equal to zero (as a reference).

**Figure 8.**DFT-EMTO energies for the α″ and γ

^{0}phases of the (

**a**) U

_{86.23}Nb

_{13.77}and (

**b**) U

_{83.56}Nb

_{13.84}Ti

_{2.60}alloys as functions of the atomic volume.

**Figure 9.**DFT-FPLMTO energies for α and γ (BCC) U-Nb alloys as functions of the atomic volume. At 25 at.% Nb the γ phase is stable over the α phase. The dashed lines in the middle panel refers to calculations where Nb is replaced by the FCC metal Rh.

**Figure 10.**DFT-FPLMTO energies for the U

_{3}Nb compound in the γ (BCC) and γ

^{s}structures relative to the orthorhombic α phase.

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Landa, A.; Söderlind, P.; Wu, A.
Phase Stability in U-6Nb Alloy Doped with Ti from the First Principles Theory. *Appl. Sci.* **2020**, *10*, 3417.
https://doi.org/10.3390/app10103417

**AMA Style**

Landa A, Söderlind P, Wu A.
Phase Stability in U-6Nb Alloy Doped with Ti from the First Principles Theory. *Applied Sciences*. 2020; 10(10):3417.
https://doi.org/10.3390/app10103417

**Chicago/Turabian Style**

Landa, Alexander, Per Söderlind, and Amanda Wu.
2020. "Phase Stability in U-6Nb Alloy Doped with Ti from the First Principles Theory" *Applied Sciences* 10, no. 10: 3417.
https://doi.org/10.3390/app10103417