Electronically driven collapse of the bulk modulus in δ-plutonium

Significance A long-standing mystery in the material science of actinides concerns the question of why the bulk modulus of plutonium metal undergoes an anomalously large softening with increasing temperature compared to other metals. We show that a crucial step to understanding this phenomenon is taking into consideration the compressibility of thermally excited electronic configurations. We find this to lead to a previously unknown electronic softening contribution to the bulk modulus and a collapse of the bulk modulus when there exists a large partial pressure between different configurations.


Supporting Information Text
Prior models of the bulk modulus softening. In a prior model of the bulk modulus based on the invar model, the Debye temperature, and consequently the bulk modulus, was assumed to be the probability-weighted sum of the ground state bulk modulus and an excited invar configuration bulk modulus (1)(2)(3), which is equivalent to the first term of Eq. 4 of the main text. In the absence of any other terms contributing to the bulk modulus, Lawson et al were able to approximately account for the experimentally observed softening of the bulk modulus by setting " = 0 for the excited invar configuration. Such a value is expected to occur only at a very large positive volume strain of = &' &( + " , which we obtain by equating the second derivative of Eq. 6 in the Methods to zero. A volume strain this large ∼ 100% corresponds to an actual volume of order ∼ 50 Å, which is significantly larger than the value " ≈ 20 Å for the excited configuration of d-Pu obtained in fitting the invar model to the thermal expansion (1,2).
In another more recent model of the bulk modulus based on the disordered local moment model (4), the volume was assumed to remain approximately constant with the different configurations corresponding to a continuum of states with different degrees of orbital compensation of the local moment (5). In this model, the softening of the bulk modulus occurs in response to a reduction in the moment with increasing temperature. However, since the degree of softening in this model is predicted to be reduced for samples with a larger negative contribution to the thermal expansion, which generally occurs for samples with lower concentrations x of Ga (1), the predicted Ga-dependent trend is opposite to that found experimentally (plotted in Fig. 1B of the main text) (6).
Applicability of electronic structure methods. Advanced electronic structure methods utilizing Quantum Monte Carlo (QMC) or Dynamical Mean Field Theory (DMFT) inform us of how the 5felectron shells intermix with conduction electrons (7,8). These methods are essential for an accurate interpretation of spectroscopic measurements such as photoemission and quantum oscillations. However, they do not provide the most practical means for understanding thermodynamic quantities such as the bulk modulus.
A central point of this manuscript is that in order to understand the origin of the softening, the bulk modulus and its pressure derivative must be calculated from the second and third order derivatives, respectively, of the free energy with respect to volume -and over a broad range of temperatures. While QMC and DMFT can be used to calculate the energy-versus-volume curves at different temperatures, to do this energy over a sufficiently dense grid of volumes and temperatures to accurately determine the second and third order volume derivatives would constitute a formidable computational task. A single DMFT calculation performed at a single volume and temperature typically takes of order a week using state-of-the-art code and computer systems. Very approximate treatments have, instead, generally been advocated, which include the invar model (3) and atomistic model (9).
Questions regarding pressure-dependent data. There are uncertainties in how some of the experimental compositions in Fig. 4 of the main text can be compared with the model. For instance, the x = 3.5% Ga-stabilized sample, for which ' ~ + 4 is positive in contrast to the other d-Pu samples (10), has a substantially lower volume ( ∼ 24.2 Å) than that ( ∼ 24.6 Å) previously found for samples of nominally the same composition (interpolating between x = 2 and 4%) (1). Also, the lowest error bar ( ' = -4 ± 2) is obtained for a heavily Am-stabilized d-Pu sample (likely due to the wider range of accessible pressures), yet its thermal expansion and elastic properties remain largely unexplored (11).