Role of the Lattice in the $\gamma \to \alpha$ Phase Transition of Ce: A High-Pressure Neutron and X-Ray Diffraction Study

The temperature and pressure dependence of the thermal displacements and lattice parameters were obtained across the $\gamma \to \alpha$ phase transition of Ce using high-pressure, high-resolution neutron and synchrotron x-ray powder diffraction. The estimated vibrational entropy change per atom in the $\gamma \to \alpha$ phase transition, $\Delta S_{\mathrm{vib}}^{\gamma -\alpha }\approx (0.75\pm 0.15)k_{\mathrm{B}}$, is about half of the total entropy change. The bulk modulus follows a power-law pressure dependence that is well described using the framework of electron-phonon coupling. These results clearly demonstrate the importance of lattice vibrations, in addition to the spin and charge degrees of freedom, for a complete description of the $\gamma \to \alpha$ phase transition in elemental Ce.

Figure 1

Neutron powder diffraction patterns of Ce at $P=0.527\mathrm{GPa}$ for (a) $\gamma$ phase ($T=295\mathrm{K}$) and (b) $\alpha$ phase ($T=149\mathrm{K}$) with corresponding Rietveld fits (solid line). Tick marks indicate positions of the Bragg reflections. Also shown are difference curves.

Figure 2

(a) Isotropic thermal displacements of Ce (circles) vs pressure at 300 K. The square is calculated from Born-von Kármán force model at ambient conditions [13, 26]. (b) Thermal displacements vs temperature. Data at $T=260\mathrm{K}$ have mixed $\gamma$ and $\alpha$ phases. Solid lines are fits to Eq. (1) with Debye temperatures ${\Theta }_{\mathrm{D}}^{\gamma }=104(3)\mathrm{K}$ and ${\Theta }_{\mathrm{D}}^{\alpha }=133(3)\mathrm{K}$.

Figure 3

(a) $P$-$V$ room temperature isotherm. The solid line is a cubic polynomial fit to the $\gamma$ phase of Ce. (b) Bulk modulus vs pressure. The symbols are calculated from finite differences. The solid line is obtained from the polynomial fit to the $\gamma$ phase $P$-$V$ data in (a). The dashed line in the $\alpha$ phase is from Olsen et al. [35]. Inset: The dashed line is a fit to the solid line near the transition ($0.7\le P\le 0.83\mathrm{GPa}$), where $B_T^2\sim |P-P_c{|}^{2\alpha }$ with $P_c\approx 0.83\mathrm{GPa}$ and $\alpha \approx 0.46$.