Effects of substitutional Ga on the phonons of $\delta$-phase Pu based on density functional theory calculations

The delta phase of Pu is stabilized by Ga doping, but the mechanism of this stabilization remains an open question. Density functional theory calculations focused on how Ga doping affects the phonons sheds some light on the phonons' contribution to the stabilization. The calculated phonon modes of Ga-doped delta phase Pu fall into two distinct types: localized, high frequency Ga-dominated phonon modes, and Pu-dominated modes at lower frequencies. Increasing the Ga concentration has an effect on the Pu-dominated phonon modes opposite to that of compression: higher-frequency modes soften, and lower-frequency modes stiffen. The latter provides an indication that the stabilization mechanism is not due to a thermodynamic contribution from the phonons. Furthermore, the stiffened phonon modes include candidate modes that describe possible pathways into low-temperature phases, suggesting that doping with Ga could impede such pathways.

Figure 1

Comparison of the $\delta$-Pu phonon dispersion with experiment and other calculations. The dispersion in this paper is calculated with the AFM structure in a computational cell with 96 atoms at $a_0=4.64\AA$. Unlike experiments (Wong et al. [12]) and other calculations (Söderlind et al. [36], Dai et al. [43]), the AFM structure breaks the cubic symmetry by singling out the $\left[00x\right]$ direction for the wave vector of the magnetic structure: consequently, the results are shown along more high symmetry directions than required for the fcc structure. The experimental data is from measurements on a $\delta$-Pu sample with 2 at. % Ga.

Figure 2

Comparison of phonon densities of states (DOS) with experiment. The DOS calculated here employs the AFM structure in computational cells with 72 atoms; only phonons with commensurate wave vectors are shown, with the calculated frequencies convoluted with a Gaussian of width 0.05 THz. (a) The experimental data of Wong et al. [12] is calculated based on measurements on a $\delta$-Pu sample with 2 at. % Ga at room temperature. (b) The experimental data of McQueeney et al. [44] is measured on a $\delta$-Pu sample with 5 at. % Al at two temperatures; the sample at room temperature has a reported lattice constant of 4.58 Å.

Figure 3

Phonon densities of states calculated for $\delta$-Pu with and without Ga substitutions in a computational cell with 72 atoms at $a_0=4.64\AA$. The 213 ($3\times 72-3$) phonon frequencies are convoluted with a Gaussian of width 0.05 THz.

Figure 4

Frequencies of localized phonon modes calculated for $\delta$-Pu with a single substitution in a computational cell with 64 atoms at $a_0=4.64\AA$, plotted versus the weight of the phonon modes on the substitutional atom. The data denoted as mass-only substitution are calculated by changing the mass of one atom (to values 110 a.u. and below, in steps of 5 a.u.) and using the force constants from the DFT calculation of the 64-atom $\delta$-Pu supercell. The data denoted as Ga actual substitution are calculated from the force constants of the 64-atom $\delta$-Pu supercell with one Ga substitution and relaxed atomic positions.

Figure 5

Phonon mode frequencies calculated for $\delta$-Pu with a single Ga substitution in a computational cell with 72 atoms at $a_0=4.64\AA$, plotted versus the weight of the phonon mode on the Ga atom. Inset: Ga-dominated modes in computational cells with 16, 36, 64, and 72 atoms. Phonon modes with zero weight are completely described by a linear combination of the amplitudes of Pu atoms.

Figure 6

Phonon mode tracing for $\delta$-Pu with one ${\mathrm{Ga}}_{\text{Pu}}$ substitution in computational cell with 72 atoms at $a_0=4.64\AA$. Orange circles represent $\delta$-Pu modes with black circles highlighting the $T\left[111\right]$ modes; green diamonds represent Ga modes. The areas of the symbols are proportional to the weights as defined in Eq. (2); only weights larger than 0.01 are shown.

Figure 7

Average frequency of Ga-dominated phonon modes (a), phonon moments for Pu-dominated phonon modes (b), and phonon moments for all phonon modes (c), calculated for Ga-doped $\delta$-Pu in computational cells with 72 atoms at $a_0=4.64\AA$. Data include one calculation at 1.4 at. % Ga, ten at 2.8 at. % Ga, two at 4.2 at. % Ga, and one at 5.6 at. % Ga.

Figure 8

Low-frequency excerpt of phonon frequency tracing for $\delta$-Pu with (a) one, (b) two, (c) three, and (d) four substitution Ga atoms in computational cell with 72 atoms at $a_0=4.64\AA$. Orange circles represent the $\delta$-Pu modes with black circles highlighting the eight transverse [111] modes at $f=0.98$ THz (degenerate in undoped $\delta$-Pu). The areas of the symbols are proportional to the weights as defined in Eq. (2); only data with weight larger than 0.01 are shown.

Figure 9

Phonon frequency tracing from $\delta$-Pu at 4.64 Å (a) to $\delta$-Pu at $a_0=4.52\AA$ and (b) to Pu-4.2 at. % Ga $\delta$-Pu at $a_0=4.60\AA$ (the corresponding experimental lattice constant) in computational cells with 72 atoms. Orange circles represent the $\delta$-Pu modes with black circles highlighting the eight transverse [111] modes at $f=0.98$ THz (degenerate in undoped $\delta$-Pu). The areas of the symbols are proportional to the weights as defined in Eq. (2); only data with weights larger than 0.01 are shown.