Heat capacity and the good old days of uranium

Interpreting the charge-density wave in low-temperature alpha-uranium

Over the past forty years, neutron, X-ray, electron diffraction, and heat-capacity experiments have indicated that a charge-density wave occurs at low temperatures in alpha-uranium (α-U). This charge-density wave is considered unusual because the wave vector has components in all three dimensions. Despite numerous investigations, there has been a lack of unanimity on how these results should be interpreted. In particular, there has been discussion of the importance of surface effects in transmission electron microscopy measurements and the effects of spatial averaging in X-ray and neutron diffraction. Confusion about the interpretation of heat-capacity data has continued to the present day.

In collaboration with Los Alamos colleagues Edward Rosten, Ross McDonald, and John Singleton, I have shown that the different results can be understood in the context of a single model if peaks in the heat-capacity data are interpreted to arise from “dirty Peierls transitions”: transitions to a charge-density wave state, or changes in the periodicity of that state, in a disordered system. When analyzed in this way, it is also possible to use heat-capacity data as a noninvasive method for bulk microscopy of disorder in the sample. This method is potentially an important step forward in assessing uranium sample quality. Our analysis of historical data indicates that sample quality was highest in the 1950s, showing that those who complain that actinide crystals were better in the “good old days” may actually be correct.

Heat capacity of alpha-uranium (α-U) and the modeled background for various samples. Legend: (a) pseudo single crystal U PSC 91 (Crangle and Temporal, 1973), (b) polycrystalline sample (Hall and Mortimer, 1977), (c) single crystal (Mihaila et al., 2006), (d) polycrystalline sample (Mihaila et al., 2006), (e) polycrystalline sample U10 (Crangle and Temporal, 1973), and (f) polycrystalline sample U MM (Crangle and Temporal, 1973). The data are offset from each other by 3 Jmol-1K-1. When looking at low-temperature α-U heat-capacity data, it is easy to see that there is a smooth background above which three peaks are visible in the single-crystal data and one peak in the polycrystalline data.

When looking at low-temperature α-U heat-capacity data in the figure above, it is easy to see that there is a smooth background above which three peaks are visible in the single-crystal data and one peak is visible in the polycrystalline data. Such peaks are usually associated with phase transitions, and because it is known that a charge-density wave forms in α-U, one would expect each of the peaks to be attributable to either the appearance of a charge-density wave or a change in the wave vector (periodicity) of the charge-density wave.

The highest-temperature peak is close to 42 kelvin (K), a temperature at which neutron and X-ray experiments show all three components of the charge-density wave appearing with incommensurate wave vectors. The second peak, around 38 K, is close to the point at which the x component of the charge-density wave locks into a value of q_x = 0.5a*. At 22 K the y and z components of the charge-density wave lock into q_y = 1/6b* and q_z = 2/11c* (from X-ray measurements) or q_z = 5/27 c* (from neutron measurements). Different combinations of these wave-vector components give rise to two different charge-density wave vectors, (q_x, +q_y, +q_z) and (q_x, +q_y, −q_z).

Transmission electron microscopy studies have discovered that in any area of the sample only one of the wave vectors exists (i.e., the two wave vectors never coexist). However, the results also challenged the simple picture from X-ray and neutron diffraction, which found that the q_x component of the wave-vector never became fully commensurate.

Neutron, X-ray, and electron diffraction all measure the wave vector of the charge-density wave accurately, although neutron and X-ray techniques average over a large volume and transmission electron microscopy only gives information over very short length scales. However, as can be seen from the description of the measured values of the wave vectors, all three techniques produced conflicting results.

It has long been known that the properties of different uranium samples vary, and it was thought that this probably led to the differing conclusions. But there was no noninvasive method for measuring properties such as the presence of impurities or grain boundaries, which might influence the results. Transmission electron microscopy can be used to measure the properties of a sample, but it damages the sample and observations can be dominated by surface effects. In addition, because of the radioactive nature of uranium, a dedicated microscope is needed, of which there are only a few worldwide.

There have been a number of studies of the heat capacity of α-U; the data that we consider here come from studies by Hall and Mortimer (1977), Crangle and Temporal (1973), and Mihaila and others (2006). To make the transitions more visible, a smooth background was removed from the data. The calculation of the high-temperature background requires the Debye temperature, which was obtained by fitting the low-temperature (1.8-10 K) heat-capacity data to a polynomial equation. The high-temperature data were fitted with a Debye model and an Einstein model, where the Debye temperature obtained from the low-temperature fitting was used.

Heat capacity above background and disordered Peierls transitions fits to the data. Legend: (a i) pseudo single crystal U PSC 91 (Crangle and Temporal, 1973), (a ii) polycrystalline sample (Hall and Mortimer, 1977), (b i) single crystal (Mihaila et al., 2006), (b ii) polycrystalline sample (Mihaila et al., 2006), (b iii) polycrystalline sample U10 (Crangle and Temporal, 1973), and (b iv) polycrystalline sample U MM (Crangle and Temporal, 1973). The data are offset from each other by 1 Jmol-1K-1. Three transitions were found in the single crystals as expected (below left), and in the polycrystalline samples an excess heat capacity was evident in the temperature range where the upper two transitions were observed in the single crystal, with the third transition being absent (below right).

The data were fitted with iterative reweighted least squares using the Levenberg–Marquardt method. This method has the advantage over the more-commonly used least-squares technique that low weights are automatically given to areas that have a poor fit to the model, eliminating the need for sections to be cut out of the data by hand before they are fitted. The heat capacity above background is shown in the figures below. Three transitions were found in the single crystals as expected, and in the polycrystalline samples an excess heat capacity was evident in the temperature range where the upper two transitions were observed in the single crystal, with the third transition being absent.

The heat capacity above background was fitted to a model of a Peierls transition, which is the type of second-order transition that occurs when a charge-density wave is formed. The model for a Peierls transition occurring in a disordered material (a dirty Peierls transition) was developed by Chandra (1981) and gives the form of the transition as a function of the level of disorder. The only fit parameters are the disorder length scale and a scale factor. It can be seen that this model fits well in the single-crystal materials. In the polycrystalline materials, the heat-capacity peak can be well modeled as two Peierls transitions with greater disorder than in the single crystals. The third transition observed in the single crystal is entirely absent in the polycrystalline samples. The fits to the transitions give a disorder length scale for each transition and are displayed in the table at right.

The disorder length scale gives an idea of the average distance that separates impurities or physical dislocations in the sample. Therefore, when the sample is better quality and so more ordered, this length scale will be longer; when the sample is poorer quality and so more disordered, this length scale will be shorter. Because all three transitions in one sample are occurring in the same disorder environment, one might expect the disorder parameters to be the same for all three transitions.

Disorder length-scale parameters, in angstroms (Å), for different alpha-uranium (α-U) samples. Names and dates refer to the heat-capacity measurements from which the parameters were extracted.

However, the disorder parameter indicates the length scale at which the disorder is affecting the transition; the three different transitions will each react differently to the various impurities and grain-boundary dislocations present. Because the same transitions are occurring in the different samples, it is possible to compare how the disorder length scale of a given transition varies from sample to sample. Some caution must be used with the results from polycrystalline samples because the problem is somewhat ill constrained. It is safest to look at the total of the two transitions rather than placing too much weight on the individual results for each length scale.

Because the length scale over which the charge-density wave orders at the third transition is much larger than that for the upper two transitions, it should be the transition most affected by disorder, explaining its disappearance in the polycrystalline samples. This result is consistent with the transmission electron microscopy observation that the charge-density wave is incommensurate at low temperatures (although the study was performed on a thin-film sample, and a considerable amount of disorder is usually introduced during the sample preparation stage).

This work is important because it provides a way of assessing the disorder level in a crystal using a nondestructive technique. It is particularly useful because transmission microscopy is problematic for radioactive materials and can give misleading results. By contrast, it is relatively easy to make probes for heat-capacity measurements that can contain radioactive material to an acceptable standard.

Our analysis method allows different uranium samples to be compared and gives a good idea of the relative quality of different single-crystal samples. From the characteristic disorder length scales given in the table, we can see that the single crystal with the longest disorder length scale, i.e., the least disorder, was the sample U PSC 91, which was grown by E.S. Fisher in 1957. The fact that the highest-quality sample was probably the oldest indicates the difficulty of uranium metallurgy and the little attention that this problem has received in recent decades.

The charge-density wave transitions in both single-crystal and polycrystalline α-U can be modeled as Peierls transitions in systems containing different levels of disorder. The disordered Peierls model links the previously contradictory pictures that arose from transmission electron microscopy studies and X-ray or neutron studies. We have recently found similar effects in another compound, the lanthanum-calcium-manganese-oxygen compound La_0.5Ca_0.5MnO₃, in which a charge-density wave coexists with a high level of disorder.

In the future, we hope to look at the low-temperature behavior of the heat capacity of α-U because it may be possible to observe the collective mode behavior at low temperatures. In addition, we will further explore the more-general characteristics of charge-density waves in disordered materials in other related systems.

Next: Plutonium Futures conference set for July in Dijon, France