Calculated Phonon Spectra of Plutonium at High Temperatures

X. Dai
2003 Science  
We constructed computer-based simulations of the lattice dynamical properties of plutonium using an electronic structure method, which incorporates correlation effects among the f-shell electrons and calculates phonon spectra at arbitrary wavelengths. Our predicted spectrum for the face-centered cubic ␦ phase agrees well with experiments in the elastic limit and explains unusually large shear anisotropy of this material. The spectrum of the body-centered cubic phase shows an instability at zero
more » ... temperature over a broad region of the wave vectors, indicating that this phase is highly anharmonic and can be stabilized at high temperatures by its phonon entropy. Plutonium (Pu) is a material with very unusual solid-state properties. Despite its scientific and technological importance, many of its key properties, such as the spectrum of lattice vibrations, remain uninvestigated. It has not been possible to measure that spectrum experimentally because of Pu's extreme toxicity and radioactivity. It has not been possible to compute the spectrum theoretically, because Pu is strongly correlated, and the traditional electronic structure methods fail to describe it even qualitatively. These studies are, however, essential to be able to address the factors that govern the lattice stability of Pu, an issue that is important for Pu's storage and disposal over long time scales. Pu has six crystallographic allotropes with puzzling volume variations among them (1). Starting from the low-temperature ␣ structure with 16 atoms in the elementary cell, it undergoes a series of phase transitions ending in relatively simple face-centered cubic (fcc) (␦) and body-centered cubic (bcc) (ε) phases at temperatures greater than 500 K. The 25% volume increase during the transition from ␣ to ␦ is followed by a volume contraction upon further heating (2) through the ␦-ε transition, occurring by way of an intermediate body center tetragonal ␦Ј phase, which exists in a very narrow temperature interval. It is the unusual behavior of both the electronic and lattice degrees of freedom that determines the rich phase diagram of Pu. The experimental information about the lattice dynamical properties of this element is very limited. Pu has relatively soft elastic constants and a Debye temperature near 100 K (3). Using this information, phenomenological studies of the thermodynamics of Pu have been carried out (4). However, the role phonons play in the thermodynamics of Pu is unclear because of the lack of appropriate theoretical and experimental tools to study its lattice-dynamical properties. We address this issue based on a new approach that is capable of microscopically calculating phonons in strongly correlated systems. Our method is based on a recently developed (5) electronic structure algorithm, which allows us to include dynamical selfenergy effects in calculating total energies and spectra of materials with correlated electrons. Its foundation is provided by the dynamical mean field theory (DMFT) (6), which treats systems with competing localization and delocalization tendencies of the electrons, where such methods as the density functional theory (7) in its local density approximation (LDA) or generalized gradient approximations (GGAs) have limited applicability. For example, within the LDA, the theoretical volumes of ␦ Pu and ε Pu are 30% too small, the bulk modulus is one order of magnitude too large, and long-range magnetic order is predicted that is not experimentally observed (8-12). Many of these difficulties have been corrected by using DMFT-based calculations (5). The approach we used to compute the phonon spectrum of strongly correlated systems combines DMFT and linear response theory. It can be used to study the vibrations that involve arbitrary wave vectors of atomic displacements (13). A perturbative method with respect to small movements of atoms from their equilibrium positions is used to evaluate changes in the electronic charge densities, potentials, local Green functions, and self-energies caused by lattice vibrations. The dynamical matrix is evaluated as the second-order derivative of the expression for the total energy within DMFT (14). This provides important information about how electronic correlations affect the lattice vibrations. Why are lattice dynamics studies required for understanding the phases of Pu? Here, the f-shell electrons are close to the Mott transition (15). The compressibility at a Mott transition end point diverges (16), which suggests anomalous elastic properties in its vicinity. In isostructural phase transitions such as the ␣-␥ transition in cerium (17, 18) , the entropy changes associated with the lattice deformations can be safely neglected. In Pu, however, these entropy changes can be large. Because Pu does not show long-range magnetic order, magnetic moments on Pu atoms can exist only locally. Therefore, only dynamical self-energy effects can describe the notion of the disordered local moments. They can be treated using the DMFT on the level of the so-called Hubbard I approximation (19) for the Anderson impurity model (20) , which describes the Pu's paramagnetic state. We calculated total energies of fcc and bcc structures of Pu as a function of volume using a self-consistent dynamical mean field method. We solved the impurity model within various schemes ranging from simple Hartree-Fock-like magnetically ordered LDAϩU (where U is the total energy) (21, 22) to paramagnetic Hubbard-I and more sophisticated iterative perturbation theory-based interpolation methods (5, 6). We used U ϭ 4 eV in our simulations as approximately given by the atomic spectral data (23), constrained density functional studies (24), and our previous work (5, 21). All techniques gave us a good agreement in predicting the equilibrium volumes for ␦ Pu and ε Pu: V ␦ was obtained to be 15.9 cm 3 / mol, which is only 6% larger than the experiment in (2), and the ε-phase volume is predicted to be 3% smaller than V ␦ , in accordance with the experiment in (2). The emerging physical picture indicates an ε phase that has slightly more itinerant electrons than the ␦ phase. Why then does the more itinerant phase, with its smaller volume, become favorable at higher temperatures? The answer lies in the spectrum of vibrations of these two phases. We calculated the phonon spectra in ␦ and ε Pu by using the linear response technique (13, 25). Self-energy effects in the calculation of the dynamical matrix were included, using the Hubbard-I approximation (19). In the calculated frequencies, as a function of wave vector along high-symmetry directions in the Brillouin zone for the ␦ phase (Fig. 1) , we saw a considerable softening of the transverse
doi:10.1126/science.1083428 pmid:12738856 fatcat:z2ptw3sgrfeejbui2a4hdagvvi