research papers
Neutron and hard Xray diffraction studies of the isothermal transformation kinetics in the research reactor fuel candidate U–8 wt%Mo
^{a}Heinz MaierLeibnitz Zentrum (MLZ), Technische Universität München, Lichtenbergstrasse 1, D85748 Garching, Germany, and ^{b}PhysikDepartment, Technische Universität München, JamesFranckStrasse 1, D85748 Garching, Germany
^{*}Correspondence email: steffen.saeubert@frm2.tum.de
Exposing uranium–molybdenum alloys (UMo) retained in the γ phase to elevated temperatures leads to transformation reactions during which the γUMo phase decomposes into the thermal equilibrium phases, i.e. U_{2}Mo and αU. Since αU is not suitable for a exposed to high burnup, it is necessary to retain the γUMo phase during the production process of the fuel elements for modern highperformance research reactors. The present work deals with the isothermal transformation kinetics in U–8 wt%Mo alloys for temperatures between 673 and 798 K and annealing durations of up to 48 h. Annealed samples were examined at room temperature using either Xray or neutron diffraction to determine the phase composition after thermal treatment, and in situ annealing studies disclosed the onset of phase decomposition. While for temperatures of 698 and 673 K the start of decomposition is delayed, for higher temperatures the first signs of transformation are already observable within 3 h of annealing. The typical Cshaped curves in a time–temperature–transformation (TTT) diagram for both the start and the end of phase decomposition could be determined in the observed temperature regime. Therefore, a revised TTT diagram for U–8 wt%Mo between 673 and 798 K and annealing durations of up to 48 h is proposed.
Keywords: uranium; nuclear fuels; uranium–molybdenum alloys; isothermal transformation kinetics; Xray diffraction; neutron diffraction.
1. Introduction
In order to reduce the amount of highly enriched uranium (HEU) fuel in the civilian _{3}Si_{2} and UAl_{x} do not provide the uranium density which is required to convert highperformance research reactors, a new fuel has to be developed. Pure metallic uranium, which would offer the highest uranium density possible, is known to show unfavourable behaviour during irradiation to high burnup (Frost, 1994; Hofman & Walters, 1994; Paine & Kittel, 1956; Rest et al., 1998). Only the γ phase of uranium has adequate properties to be used as a (Frost, 1994). Both UMo and UZrNb alloys retain the uranium γ phase in a metastable state at room temperature and have a sufficient uranium density.
cycle, efforts are being made to develop a fuel with a higher uranium density which would allow the conversion of research and test reactors from HEU to lower enriched uranium (LEU) while maintaining an equivalent neutron and quality. Since uranium compounds like UHowever, compared with UMo the ternary alloy UZrNb has a lower uranium density and γ stability and hence shows a poorer performance in terms of predictable swelling behaviour and high fission rate during both annealing and inpile irradiation experiments under research reactor conditions, i.e. high burnup and high temperature (Meyer et al., 2014; Snelgrove et al., 1996). Therefore the UMo alloy, well known since the 1950s, is currently the subject of renewed interest among the international research reactor fueldeveloping community. The addition of 7–10 wt% of Mo to the U is the best compromise between a high uranium density and a good γ stability of the UMo alloy (Hofman et al., 1998).
Nevertheless, the high temperatures to which the fuel element is exposed during the manufacturing process may lead to a decomposition of the UMo γ phase into its thermal equilibrium microstructures, i.e. αU and U_{2}Mo. Although it has been shown that the decomposition is reversed during inpile and heavyion irradiation (Bleiberg et al., 1956; Konobeevskii et al., 1967; Jungwirth, 2011), it is preferable to avoid it during fuel plate production. Therefore, the precise kinetics of the γUMo phase decomposition need to be understood. Since the available time–temperature–transformation (TTT) diagrams are based on data from the 1950s and 1960s, a new study applying more modern techniques seemed to be advisable.
Therefore, both neutron and Xray diffraction studies at room temperature were performed on annealed samples in order to obtain detailed crystallographic information on the state of decomposition as a function of time and temperature. Additionally, in situ annealing studies with neutron diffraction were used for the investigation of peakgrowth behaviour and hence the transformation kinetics of single phases.
2. Sample preparation
All samples analysed in this work originated from the same U–8 wt%Mo ingot provided by the AREVACERCA company (Romans, France). The samples were cut down from the ingot, melted in an electric arc furnace and cast into a cylindrical shape. After that, all the samples were homogenized at 1173 K for 48 h in vacuo to minimize oxidation and then water quenched to room temperature. This ensured that only the γUMo phase was captured in all the specimens before heat treatment. The presence of a single γUMo phase was verified by neutron on one sample which was prepared as described.
2.1. Heat treatment
Depending on the annealing duration and temperature, the decomposition of the γUMo phase is captured at different stages of transformation. Therefore, after homogenization, the samples were annealed at different temperatures between 673 and 773 K and with annealing times of 3, 6, 16, 24 or 48 h in order to obtain a reasonable grid of measurement points with various stages of γUMo phase decomposition. During the heat treatment the samples were again kept in vacuo in order to minimize oxidation.
3. Crystallographic phase analysis
Neutron diffraction experiments were performed at the ForschungsNeutronenquelle Heinz MaierLeibnitz (FRM II) (Garching, Germany). The phase composition in the preannealed samples was studied on the highresolution structure powder diffractometer SPODI at the FRM II (Hoelzel et al., 2012). For the experiment, a germanium monochromator Ge(551) was chosen, together with a takeoff angle of 155° and a 5 m distance to the sample. Measurement of the NIST Si640c standard along with a of the diffraction pattern determined the wavelength to be λ = 1.548 Å. In total, 12 samples were analysed, which were heat treated according to Table 1. A 2θ step width between 0.05 and 0.1° was chosen, along with scan times between 6 and 8 h. Data were collected in the angular range 1.0–151.8° in 2θ, i.e. 0.006–0.626 Å^{−1} in sin(θ)/λ.

Xray diffraction measurements were carried out at the Deutsches ElektronenSynchrotron (DESY) (Hamburg, Germany). The phase composition in the preannealed samples was studied on the highenergy materials science beamline P07 (HEMS) of the Positron–Electron Tandem Ring Accelerator III (PETRA III) at DESY (Schell et al., 2014). With an energy of E = 100 keV the wavelength is calculated to be λ = 0.124 Å, which was confirmed by measurement of the NIST LaB_{6}660a standard. In total, seven samples were analysed, which were heat treated before measurement according to Table 2. Data were collected in the angular range 0.0025–7.6400° in 2θ, i.e. 0.0004–0.537 Å^{−1} in sin(θ)/λ.

3.1. Data analysis
Diffraction data collected by either neutron or Xray diffraction were analysed using the ) and the FULLPROF software package (RodríguezCarvajal, 1993). Five phases were included in the process. Four of them describe the UMophases: γUMoa (space group ), γUMob (space group ), αU (space group Cmcm) and U_{2}Mo (space group I4/mmm). γUMoa represents the initial γ phase and γUMob a molybdenumenriched γ phase which precipitates during the phase decomposition reactions. Hence, the latter phase has smaller lattice parameters (Dwight, 1960). αU and U_{2}Mo are the final products of decomposition, and the α phase can also be in the distorted states α′U or α′′U, depending on the reaction temperature. α′U is a distorted α phase characterized by a contraction of the b parameter together with an expansion of the parameters a and c, whereas α′′U is a further distortion of the parameters a, b and c along with a phase change from an orthorhombic to a monoclinic structure. An enrichment in Mo leads to a distortion of the lattice and hence the formation of the phases α′U and α′′U (Lehmann & Hills, 1960; Orlov & Teplinskaya, 1999; Stewart & Williams, 1966). Owing to the inclusion of carbide and nitride during the production process of the material by the AREVACERCA company, one other phase was added. Since UC and UN have the same i.e. , and very similar lattice parameters, only one phase representing both of them was included and named UC. Fig. 1 shows the Bragg peak position of each included phase, along with the results of the and an example diffraction pattern.
method (Rietveld, 1969For the
pseudoVoigt functions were chosen to fit the Bragg peak shapes. The background was described by selected background points and a linear interpolation between these, rather than by mathematical functions. The analysis included the of scale factors, lattice parameters and peakshape parameters. Moreover, to improve the quality of the fit, the background points were refined as well.3.2. Crystallographic composition
Example Rietveld refined diffraction patterns for XRD and neutron diffraction are shown in Figs. 1 and 2, respectively. Both diffraction patterns were taken after annealing at 748 K for either 16 or 48 h. After 16 h of annealing at this temperature the decomposition is already in an advanced stage. Between 16 and 48 h of annealing, most of the remaining γUMo is decomposed into the equilibrium microstructures, i.e. αU and U_{2}Mo, as can be seen from the pattern taken after 48 h.
4. In situ annealing studies
Diffraction studies of γUMo samples during in situ annealing were performed on the materials science diffractometer STRESSSPEC at the FRM II (Hofmann et al., 2006). For the experiment, a germanium monochromator Ge(311) and a 1.065 m distance to the sample were chosen. Measurement of the NIST Si640c standard showed a wavelength of λ = 1.914 Å. Data were collected in the angular range of 37.8–56.7° in 2θ, i.e. 0.169–0.248 Å−1 in sin(θ)/λ. This angular range was chosen because the most distinctive peaks of each individual phase are located in this range. Therefore, it is theoretically possible to observe the following peaks (underlined peaks overlap with other peaks; bold peaks are unaffected by other peaks; roman peaks are prohibited):
Diffraction patterns were collected every 5 min while annealing the samples in a hightemperature furnace evacuated to a high vacuum (∼10^{−6} mbar; 1 bar = 100 kPa). The temperature was monitored using a Ctype thermocouple pressed on top of the sample, separated only by a thin vanadium foil. Besides a second Ctype thermocouple next to the first one as a reference, the reliability of the temperature control was verified by observing the structural in a lead(II) titanate (PbTiO_{3}) standard. The samples were heated at a rate of 10 K min^{−1}, which allowed the sample to adapt to the set temperature in a controlled way. Because of this high heating rate, the samples spent only a short time in the temperature regime where decomposition starts early, and hence there are only negligible effects on the transformation process. Aluminium windows around the sample position allowed the neutrons to penetrate the experimental setup easily. In total, five specimens were investigated as shown in Table 3.

4.1. Data analysis
Consecutive collected diffraction patterns (an example is shown in Fig. 3) were analysed using StressTextureCalculator (Randau et al., 2011). This software processes all diffraction patterns sequentially. The growth of a single peak was determined by observing the sum of intensities over an angular range as a function of time. Thus, the angular range is the same for all diffraction patterns of one sequential measurement. This method was preferred over fitting each individual peak, since the fit was not good for very small peaks, i.e. the beginning of peak growth.
The obtained curves of peak growth with time were then analysed with a modified , 1940, 1941):
(Avrami, 1939where the parameters n and k describe the process [for a detailed description of the kinetics, see Christian (2002) and Avrami (1939, 1940, 1941)], A is a scale factor for the intensity, B describes the background intensity, and t_{0}, with a lower limit of t_{0} = 0, is introduced in order to describe peakgrowth curves which do not start at t = 0. The peak intensity as a function of annealing time could only be described quantitatively for the αU phase since, because of overlapping peaks, the quantitative extraction of the intensities of the other phases, i.e. the growth of U_{2}Mo and the decrease of γUMo, was not feasible.
Peakgrowth curves were obtained for αU at the different temperatures shown in Table 3. On the basis of the information obtained from these measurements, Avrami curves were calculated in order to determine the beginning and end of αU phase growth as a function of annealing temperature. An example of this peak growth is shown in Fig. 4.
With the obtained fit parameters, definitions of the times τ and τ_{e} were found by taking the intersections of the tangent to the inflection point with the background and saturation, respectively. Moreover, with the any fraction of the transformed phase can be calculated at any time. Similar calculations were made for each individual measurement. The most significant durations for transformed αU as a function of annealing temperature are shown in Table 4.

The diffraction patterns collected in the angular range of 0.169–0.248 Å^{−1} in sin(θ)/λ showed that, because of the overlap of the peaks 110γUMo, 110U_{2}Mo and 103U_{2}Mo, it was not possible to observe the correct peak intensity increase and decrease of these peaks. It should be noted that a second γUMo phase was not included in the evaluation since it is not possible to distinguish a second γUMo phase at about the same position as the first one. Also, the peaks 021αU, 002αU and 004U_{2}Mo overlap each other. Since all of them increase with time, the start of phase decomposition could be observed. Owing to the abovementioned shortterm measurements and the small angular range, the exact content of αU could not be calculated via Therefore, it was not possible to determine the intensities of the single phases 021αU and 002αU and hence an estimated intensity for 004U_{2}Mo.
4.2. αU phasegrowth kinetics
The beginning of αU phase growth is also the beginning of phase decomposition, since αU is, together with U_{2}Mo, the first product of the intermediate and hightemperature reactions (Repas et al., 1964; Van Thyne & McPherson, 1957; Blake & Hehemann, 1976). Information on the beginning and end of peak growth is obtained by observing the peak intensities as a function of time. Thus, the most promising peaks are 110αU and 111αU, because these are the only peaks which do not overlap with any other. Also observable are the peaks 021 and 002αU, which are seen as one peak since it is not possible to distinguish between them. Both peaks grow equally and hence the result for the start of phase decomposition, obtained by observing the peak growths together, is correct.
5. Results
While the in situ annealing measurements performed here determined the onset of phase decomposition, as well as giving detailed information on αU phase growth, diffraction patterns at different points in time and temperature were used to derive the crystallographic phase composition at these points. Hence, in situ measurements gave information on the time a phase started to decompose, but not on the exact crystallographic composition as a function of time and temperature, because of the limited coverage in Crystallographic phase analysis, on the other hand, delivered detailed information on the phase composition in the samples at a certain time and temperature, but none about how this state was reached. Therefore, a complementary consideration of both methods is required in order to obtain a significant isothermal transformation diagram for U–8 wt%Mo.
5.1. Phase decomposition measurements during in situ annealing
The data show the strong temperature dependence of the onset of phase decomposition and the transformation itself. As expected, the transformation starts earlier for higher temperatures, while it is delayed and much slower for lower temperatures. Hence, the transformation is divided into hightemperature reactions (above 748 K) and intermediatetemperature reactions (698–748 K).
5.1.1. Hightemperature reactions
In order to investigate the early beginning of phase decomposition, in situ annealing studies were performed at 798 K. We expected to observe the phase transformation in less than 2 h and hence the total annealing time was set to 4.5 h. Diffraction patterns were collected every 5 min. Even though the total annealing time of 4.5 h was too short to observe the end of peak growth for αU, a full Avrami fit was possible and allowed us to derive not only τ but also τ_{e}: τ ≃ 100 min and τ_{e} ≃ 275 min. Another specimen was annealed for 7 h at 773 K and diffraction patterns were collected every 5 min. The Avrami fit delivers τ ≃ 135 min and τ_{e} ≃ 335 min.
By regarding the peakgrowth curves (an example curve of αU peak growth is shown in Fig. 4), it becomes apparent that the nucleation period τ does not describe the very beginning of phase decomposition: it is obvious that there is already some transformed material before the nucleation period is reached. This is a result of the definition of τ, using the intersection of the linear part with the background. A comparison of the data obtained during in situ annealing and the data obtained by crystallographic phase analysis will be conducted in §5.3, and the definition of the start of phase decomposition will be discussed there in more detail.
Sample A773K16h, which has already been measured with XRD, was annealed in situ for another 20 h at 773 K and diffraction patterns were collected every 5 min. Fig. 5 shows the 110αU peak intensities for the measurements from 16 to 36 h, together with the measurements from 0 to 6 h. It can easily be seen that the saturation described by the Avrami curve does not exactly define the end of peak growth. After the fast growth described by the Avrami curve, the peak keeps on growing slowly with a linearlike increase. According to this, the Avrami theory does not describe peak growth in UMo exactly, but it does describe the fast growth, up to the point where the linearlike growth starts. For further discussions the Avrami curve always describes this fast growth at the beginning of phase decomposition rather than the total peak growth. This will be of great importance for the discussion and comparison of data obtained from diffraction patterns and peakgrowth curves.
5.1.2. Intermediatetemperature reactions
One sample was annealed for 10 h at 723 K and diffraction patterns were again collected every 5 min. Fitting the τ ≃ 230 min and τ_{e} ≃ 510 min. As a final investigation of the intermediatetemperature regime, a specimen was annealed for 10 h at 698 K. After 10 h of annealing only the first signs of decomposition are visible. Both the αU peaks and the U_{2}Mo peak start to grow, while the γUMo peak decreases. The collected data were not sufficient for an Avrami fit and therefore the beginning and end of peak growth could not be determined. Hence, the data given for 698 K only give an estimate for the beginning of phase decomposition.
givesTable 4 summarizes the results derived from the Avrami fit for all investigated temperatures in the high and intermediatetemperature regimes. It is noteworthy that, for all Avrami fits, n ≃ 3 within the margin of error, whereas for a threedimensional process n ≥ 3. For a zero nucleation rate it is n = 3 and for a decreasing nucleation rate it is 3 < n < 4. Therefore, the results suggest either a zero or a decreasing nucleation rate (Christian, 2002).
5.2. Crystallographic phase analysis via Rietveld refinement
5.2.1. Hightemperature reactions (748 and 773 K)
After annealing for 3 h at 748 K, already a large part of the initial γUMo is transformed. Besides αU, another product at the beginning of decomposition during hightemperature reactions is a molybdenumenriched phase γUMob. As expected for hightemperature reactions, the decomposition started with forming αU and enriched γUMob. This can be seen by the relatively high content of these two phases. Furthermore, some U_{2}Mo has already formed.
The two γUMo peaks are very close together, and therefore distinguishing between these two phases is prone to error. Hence, there is the possibility that some calculated content of γUMoa belongs to γUMob, or vice versa.
It is significant that the amount of γUMoa drops fast along with the fast growth of αU. The αU growth already approaches its saturation before 24 h. The amount of γUMob increases very fast but then decreases fast shortly after reaching its maximum. This can be explained by the fact that a gradually increasing enrichment of the γUMob phase in Mo leads to the formation of U_{2}Mo at the cost of γUMob.
Table 5 summarizes the calculated lattice constants for the two γUMo, U_{2}Mo and αU phases for each measurement. The behaviour of γUMob shows the molybdenum enrichment with time. The longer the specimen was annealed, the smaller are the lattice parameters, and hence the higher is the molybdenum content in γUMob.

The calculated lattice parameters for αU show expanded parameters a and c, together with a contracted parameter b, compared with a pure αU phase. This indicates that the present uranium phase is the distorted phase α′U. The lattice parameters for γUMo, αU and the suggested αU phase are in good agreement with results previously obtained by Palancher et al. (2012) for U–8 wt%Mo samples. It should be noted that Palancher and coworkers investigated UMo/Al(Si) plates rather than pure UMo samples.
At 773 K, similar behaviour to the reactions at 748 K is expected because both temperatures belong to the hightemperature reaction regime. However, a noticeable difference can be seen between samples A748K3h and A773K3h. Owing to the very sharp peak of the γUMo Bragg reflection, small changes to the peakshape parameters have a huge effect on the crystallographic composition without influencing the quality of the fit in a significant manner. Moreover, it was not possible to refine two different γUMo phases inside these diffractograms. Therefore, no quantitative information is given for sample A773K3h apart from the knowledge that decomposition has already started.
The calculated lattice parameters summarized in Table 5 show no distinctive features except for the missing second γUMo phase in sample A773K3h. Again, the distortion of the pure αU phases suggests the presence of α′U rather than αU.
5.2.2. Intermediatetemperature reactions (673, 698 and 723 K)
The decomposition takes place differently in the intermediatetemperature regime compared with the hightemperature regime. While at 723 K a noticeable decomposition already took place after 3 h of annealing, at the lower temperatures the beginning of phase decomposition is delayed. And not only is the onset of phase decomposition delayed, but also the transformation itself takes place at a much slower rate than at higher temperatures.
At 723 K, the scan of A723K3h shows that the decomposition has already started. Rietveld analysis of the diffraction data for this sample was difficult, owing to an uneven background and UC inclusions at the position of the beam spot in the experiment. Since Xray measurements were only performed on small areas (0.2 × 0.2 mm) on the samples, any inclusions or grain structure present strongly influenced the results for these measurements. Therefore, the results for the amount of αU and U_{2}Mo in the A723K3h sample differ from measurements where the grain size is negligible.
Since the intermediatetemperature reactions contain both the low and hightemperature reactions, and because 723 K is at the top of the intermediatetemperature regime, a comparable result to the hightemperature regime was expected for decomposition. The irregularity in the decrease of the γUMo peaks and the increase of U_{2}Mo are again due to overlapping peaks of these phases.
Small lattice parameters are apparent for γUMob after 16 h of heat treatment (Table 5), where the other lattice parameters for this measurement show no anomalies. Moreover, the lattice constants for the other measurements show the expected behaviour. The αU phase is still α′U.
The nucleation process is very sluggish for temperatures of 698 K or less and hence the decomposition starts much later than for higher temperatures. After 3 h of annealing there is still no evidence for decomposition. The composition of sample A698K3h shown in Table 6 is similar to that of sample Ainitial. Therefore, after 3 h of heat treatment, the sample still consists of one highly homogeneous γUMo phase.

These data suggest that the nucleation period is somewhere between 3 and 6 h. A698K6h shows a slight decomposition of the γUMo phase compared with A698K3h. The results for A698K24h reveal differences between the progress of decomposition in the high and intermediatetemperature regimes. And not only the nucleation but also the decomposition itself is more sluggish: after 24 h, a large amount of γUMo is still left. A striking aspect is that the amount of U_{2}Mo is much lower compared with the results obtained after annealing at 723–773 K, while the amount of αU is only slightly lower. A retarded decomposition explains the difference in the amount of αU but not the difference in the U_{2}Mo content. Although the reactions in the intermediatetemperature regime start with a cellular reaction as in the hightemperature regime, they take a different course because of the lowtemperature reactions and the formation of a Widmannstätten α structure (Repas et al., 1964). These differences explain the rather high amount of αU together with the rather low amount of U_{2}Mo. The amount of UC is higher than for most of the samples and can again be explained by the measurement conditions with Xrays, where the lack of a homogeneous distribution of the carbides causes disproportionally high amounts of UC. The lattice constants listed in Table 5 show no distinctive features and the suggested αU phase is still α′U.
Owing to the sluggish nucleation and hence the retarded decomposition at lower temperatures, only two samples were annealed at 673 K, for 24 h (A673K24h) and 48 h (A673K48h).
The specimen A673K24h shows only a slight decomposition and therefore it was not possible to fit the neutron diffraction data with a second γUMo phase. The most distinctive features compared with the higher temperatures can be seen in the crystallographic compositions shown in Table 6. The increase in the rising phases is not only retarded but also starts with U_{2}Mo rather than with αU. This is due to the lowtemperature reactions, where the decomposition is initiated by the formation of U_{2}Mo without the presence of αU, whereas the latter is subsequently precipitated. Since 673 K is still in the intermediatetemperature regime, the hightemperature reactions occur as well and ensure the formation of αU at the very beginning of decomposition. After 48 h, only half of the initial γUMo has decomposed into equal parts of αU and U_{2}Mo, as well as some enriched γUMo.
The lattice parameters for the 673 K measurements in Table 5 are in good agreement with the results for the higher temperatures and show no distinctive features. Therefore, the suggested phase for αU is still α′U. Previous results by Palancher and coworkers found α′′U in samples with about 7 wt% molybdenum after annealing at temperatures of 698 K or less, and measurements on U–8 wt%Mo samples delivered an α′U (Palancher et al., 2012, 2013). This might be due to the formation of phases including U and Al in the dispersed UMo/Al samples examined by Palancher and coworkers. Hence, relatively more Mo remains inside the UMo kernels, which might lead to the formation of molybdenumenriched α′′ instead of α′. The relationship between the lattice parameters and the crystallographic structure was discussed in their reports. The same behaviour of αU for temperatures below 698 K was observed in this work. Moreover, refining the of αU, which is the same for the distortion α′U but not for α′′U, was successful and caused no problems. This leads us to conclude that the specimens annealed at 698 and 673 K still contain α′U rather than α′′U.
The final agreement factors of the Rietvield refinements, which define the quality of the fit, are shown in Table 7. A striking aspect is that the refinements of the Xray patterns have larger χ^{2} values, which is explained by one major factor. Although R_{exp} decreases as the number of counts collected for the diffraction pattern is increased, the difference between R_{exp} and R_{wp} becomes larger. Hence, χ^{2} is worse even though the model fitted to the diffraction pattern is improved. The reason for this is that, for diffraction patterns with a very large number of counts, even minor imperfections in the peak shape or peak position and unmodelled features of the background can make it impossible to obtain small values for R_{wp} and hence for χ^{2}. Such imperfections can be seen in the diffraction patterns collected by XRD, where the scattering at the aperture of the experiment induced small peaks in the diffraction pattern at 0.24 Å^{−1} which are not included in the fit.

Besides the agreement factors, the difference between the data and the calculated pattern, Y_{obs} − Y_{calc}, can be taken as an indicator of the quality of the fit. This shows that the models obtained for the Xray patterns are not worse than those obtained for the neutron patterns, as suggested by the final agreement factors. The discrepancy is well explained by the fact of a much higher number of counts for the measurements with Xrays.
5.3. Isothermal transformation diagram for U–8 wt%Mo
Comparing the growth curves with the data on the crystallographic compositions of the annealed samples shows that, after the growth of the αU peak has stopped as described by the Sshaped Avrami curve, the peak intensity keeps growing with a linearlike behaviour and much more slowly compared with the Sshaped growth. The linearlike growth slowly approaches the saturation of this phase.
Fig. 6 shows the αU phase growth according to Table 4 and is described by the Since αU is the first product of the transformation, Fig. 6 describes the beginning of phase decomposition along with detailed information on αU growth.
In Fig. 7 the crystallographic phase compositions at different measurement points are displayed. The data suggest that the Sshaped growths of U_{2}Mo and αU start together but increase differently. The growth of U_{2}Mo takes much longer. Comparing the data obtained for measurements on samples annealed for 24 and 48 h, respectively, for temperatures between 723 and 748 K suggests that U_{2}Mo keeps growing with a linearlike increase after the Sshaped growth. This is the same behaviour as observed for αU. Therefore, the blue area drawn in Fig. 7 indicates the beginning of the linearlike increase of U_{2}Mo and the approach to the end of the Avramilike phase decomposition.
Different definitions for the start and end of phase decomposition define the positioning of the Cshaped curves in a TTT diagram. While earlier work used the first signs of phase decomposition to define transition curves, a definition via growth curves can be found in the current literature. The Avrami fit allows us to extract the time for any fraction of transformed phases from the data obtained during in situ annealing.
Comparing the growth curves with data on the crystallographic composition of the annealed samples shows that, after the fast growth of a peak has stopped at τ_{e}, the peaks keep growing with a linearlike behaviour and much more slowly than the fast growth. The latter statement is supported by the peakgrowth behaviour observed at 773 K discussed in §5.1, where not only fast growth was observed but also the growth for t > τ_{e}. This linearlike growth of a peak slowly approaches the saturation of this phase. The time when the fast growth reaches the linearlike growth could not be measured exactly, since the measurements were not performed over such a long period for 773 K, so it was calculated using the for each temperature.
Hence, the following information for the TTT diagram is obtained by in situ annealing studies:
(i) t_{0}: the first signs of phase decomposition, i.e. 1% of the αU peak fast growth;
(ii) τ: the start of phase decomposition, 10% of fast peak growth according to Avrami;
(iii) t_{50%}: 50% of the fast growth is reached at the point of inflection;
(iv) τ_{e}: the end of phase decomposition, 90% of fast peak growth according to Avrami;
(v) t_{e}: the start of the linearlike increase after fast growth is finished, i.e. 99% of the αU peak fast growth.
The contribution of in situ annealing studies to the isothermal transformation diagram is depicted in Fig. 6. Plotted are the abovedescribed times describing the fast growth of αU and hence the beginning of phase decomposition, along with detailed information on αU growth.
The information obtained from crystallographic phase analysis contributes less significantly to the C curves for the beginning of phase decomposition in the TTT diagram. This has its reasons in the wide distribution of the measuring points in time. Although measurements at 3 and 6 h at a certain temperature were not sufficient to observe the fast growth of peaks, measuring points between t_{0} and t_{e} can still be used to complement the fastgrowth curves and support the results obtained from the in situ annealing studies.
Analysing severely decomposed material, on the other hand, gives insight into the stage of phase decomposition. The data suggest that the fast growths for U_{2}Mo and αU start together but increase differently. The fast growth of U_{2}Mo takes much longer. Comparing the data obtained for measurements on samples annealed for 24 and 48 h, respectively, at temperatures between 723 and 748 K suggests that U_{2}Mo keeps growing with a linearlike increase after the fast growth. Therefore, the data indicate the final stage of the U_{2}Mo fast growth and thereby the points after which the content of the transformation products increases only slightly with time. An exact determination of this transformation curve is not possible because measurements were only performed for 16 and 24 h, and not somewhere in between. Thus, a Cshaped area describing the final stage of U_{2}Mo fast growth and the approach of the end of phase decomposition can be estimated. Fig. 7 shows the isothermal transformation curve describing the final stage of U_{2}Mo fast growth, along with the measuring points and their determined crystallographic compositions. The dashed part of the curve thus describes an estimation, since no sample was prepared with annealing at 698 K for 48 h or for times greater than 48 h at 673 K.
The final isothermal transformation diagram in Fig. 8 comprises the curves from Figs. 6 and 7. The diagram shows the start of the cellular reaction where γUMo starts to transform into αU and U_{2}Mo, the end of the αU phase fast growth, the final stage of the U_{2}Mo fast growth, and the region where the remaining γUMo slowly vanishes as the αU and U_{2}Mo phases increase with a linearlike behaviour.
The results of this work can be compared with the previous results found in the literature. Fig. 9 shows the isothermal transformation curves obtained in this work and the TTT diagram for U–8 wt%Mo proposed by Repas et al. (1964). The latter work obtained the curves by preparing a grid of samples examined via metallurgical methods and dilatometric, microhardness and XRD data. Thus, an exact definition of the start of phase decomposition was not obvious. Moreover, the number of measurement points, and therefore the density of the measurement grid, was not given.
6. Conclusion
This work complements previous similar experiments considering neutron diffraction of UMo/Al systems exposed to elevated temperatures between 673 and 748 K and with annealing times between 2 and 52 h (Palancher et al., 2013). In the present work, a wider range of annealing temperatures and a more precise stepping in annealing time were used in order to provide a more detailed investigation of the growth kinetics.
The isothermal transformation curves obtained in this work contain detailed information on the fast growth of αU. The data show that fast growth for U_{2}Mo is much slower than that for αU. Despite suggestions found in the literature where first the αU precipitates and then U_{2}Mo starts to grow later, the diffraction data presented here clearly reveal that both transformation products start to grow simultaneously. The Cshaped area of the diagram at long annealing times describes the end of the U_{2}Mo fast growth and the approach of the end of phase decomposition. A 100% transformation of metastable γUMo could not be seen within 48 h of annealing at any temperature.
Acknowledgements
Parts of this research were carried out at the light source PETRA III at DESY, a member of the Helmholtz Association (HGF); the authors thank Uta Rütt and Olof Gutowski for assistance in using beamline P07. We also thank Herve Palancher from CEA Cadarache for valuable discussions and support during data treatment. This study was supported by a combined grant (No. FRM0911) from the Bundesministerium für Bildung und Forschung (BMBF) and the Bayerisches Staatsministerium für Bildung und Kultus, Wissenschaft und Kunst (StMBW).
References
Avrami, M. (1939). J. Chem. Phys. 7, 1103–1112. CrossRef CAS Google Scholar
Avrami, M. (1940). J. Chem. Phys. 8, 212–223. CrossRef CAS Google Scholar
Avrami, M. (1941). J. Chem. Phys. 9, 177–182. CrossRef CAS Google Scholar
Blake, D. & Hehemann, R. F. (1976). Transformation in Uranium Base Alloys. Physical Metallurgy of Uranium Alloys, edited by J. J. Burke, D. A. Colling, A. E. Gorum & J. Greenspan, Vol. 9, pp. 189–218. Chestnut Hill: Brook Hill Publishing Co. Google Scholar
Bleiberg, M., Jones, L. J. & Lustman, B. (1956). J. Appl. Phys. 27, 1270–1283. CrossRef CAS Google Scholar
Christian, J. W. (2002). The Theory of Transformations in Metals and Alloys. London: Pergamon. Google Scholar
Dwight, A. E. (1960). J. Nucl. Mater. 2, 81–87. CrossRef CAS Web of Science Google Scholar
Frost, B. (1994). Editor. Nuclear Materials. Materials Science and Technology – A Comprehensive Treatment, series edited by R. W. Cahn, P. Huusen & E. J. Kramer, Vol. 10A, 1st ed. Weinheim: VCH. Google Scholar
Hoelzel, M., Senyshyn, A., Juenke, N., Boysen, H., Schmahl, W. & Fuess, H. (2012). Nucl. Instrum. Methods Phys. Res. A, 667, 32–37. CrossRef CAS Google Scholar
Hofman, G. L., Meyer, M. K. & Ray, A. E. (1998). 21st International Meeting on Reduced Enrichment for Research and Test Reactors (RERTR), 18–23 October 1998, Sao Paulo, Brazil, https://www.rertr.anl.gov/PAPERS98.html. Google Scholar
Hofman, G. L. & Walters, L. C. (1994). Metallic Fast Reactor Fuels. In Nuclear Materials, edited by B. Frost. Materials Science and Technology – A Comprehensive Treatment, series edited by R. W. Cahn, P. Huusen & E. J. Kramer, Vol. 10A, 1st ed. Weinheim: VCH. Google Scholar
Hofmann, M., Schneider, R., Seidl, G. A., RebeloKornmeier, J., Wimpory, R. C., Garbe, U. & Brokmeier, H.G. (2006). Phys. B Condens. Matter, 385–386, 1035–1037. CrossRef CAS Google Scholar
Jungwirth, R. (2011). PhD thesis, Technische Universität München, Germany. Google Scholar
Konobeevskii, S., Sokurskii, Yu. N., Bobkov, Ya. V., Dubrobin, K. P. & Protsenko, L. N. (1967). At. Energy, 22, 565–573. CrossRef Google Scholar
Lehmann, J. & Hills, R. F. (1960). J. Nucl. Mater. 2, 261–268. CrossRef CAS Google Scholar
Meyer, M. K., Gan, J., Jue, J. F., Keiser, D. D., Perez, E., Robinson, A., Wachs, D. M., Woolstenhulme, N., Hofman, G. & Kim, Y. S. (2014). Nucl. Eng. Technol. 46, 169–182. CrossRef CAS Google Scholar
Orlov, V. K. & Teplinskaya, V. M. (1999). At. Energy, 86, 118–125. CrossRef CAS Google Scholar
Paine, S. H. & Kittel, J. H. (1956). United Nations International Conference on the Peaceful Uses of Atomic Energy, Geneva, Switzerland, 8–20 August 1955, United Nations, New York, USA. Google Scholar
Palancher, H., Bonnin, A., Colin, C. V., Nassif, V., Honkimäki, V., Jungwirth, R., Ritter, C., Champion, G. & Calzavara, Y. (2013). Powder Diffr. 28, S371–S393. CrossRef CAS Google Scholar
Palancher, H., Bonnin, A., Honkimäki, V., Buslaps, T., Grasse, M., Stepnik, B. & Zweifel, T. (2012). J. Alloys Compd. 527, 53–65. CrossRef CAS Google Scholar
Randau, C., Garbe, U. & Brokmeier, H.G. (2011). J. Appl. Cryst. 44, 641–646. Web of Science CrossRef CAS IUCr Journals Google Scholar
Repas, P. E., Goodenow, R. H. & Hehemann, R. F. (1964). Trans. Am. Soc. Met. 57, 150–163. CAS Google Scholar
Rest, J., Hofman, G. L., Konovalov, I. & Maslov, A. (1998). 21st International Meeting on Reduced Enrichment for Research and Test Reactors (RERTR), 18–23 October 1998, Sao Paulo, Brazil, https://www.rertr.anl.gov/PAPERS98.html. Google Scholar
Rietveld, H. M. (1969). J. Appl. Cryst. 2, 65–71. CrossRef CAS IUCr Journals Web of Science Google Scholar
RodríguezCarvajal, J. (1993). Phys. B Condens. Matter, 192, 55–69. Google Scholar
Schell, N., King, A., Beckmann, F., Fischer, T., Müller, M. & Schreyer, A. (2014). Mater. Sci. Forum, 772, 57–61. CrossRef Google Scholar
Snelgrove, J. L., Hofman, G. L., Trybus, C. L. & Wiencek, T. C. (1996). 1996 International Meeting on Reduced Enrichment for Research and Test Reactors (RERTR), 6–11 October 1996, Seoul, Korea, https://www.rertr.anl.gov/PAPERS96.html. Google Scholar
Stewart, D. & Williams, G. I. (1966). J. Nucl. Mater. 20, 262–268. CrossRef CAS Google Scholar
Van Thyne, R. J. & McPherson, D. J. (1957). Trans. Am. Soc. Met. 49, 588–619. Google Scholar
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