Phase transitions and equations of state of alkaline earth fluorides ${\text{CaF}}_2$, ${\text{SrF}}_2$, and ${\text{BaF}}_2$ to Mbar pressures

Phase transitions and equations of state of the alkaline earth fluorides ${\text{CaF}}_2$, ${\text{SrF}}_2$, and ${\text{BaF}}_2$ were examined by static compression to pressures as high as 146 GPa. Angle-dispersive x-ray diffraction experiments were performed on polycrystalline samples in the laser-heated diamond-anvil cell. We confirmed that at pressures less than 10 GPa all three materials undergo a phase transition from the cubic $\left(Fm\overline{3}m\right)$ fluorite structure to the orthorhombic $\left(Pnam\right)$ cotunnite-type structure. This work has characterized an additional phase transition in ${\text{CaF}}_2$ and ${\text{SrF}}_2$: these materials were observed to transform to a hexagonal $\left(P{6}_3/mmc\right)$ ${\text{Ni}}_2\text{In}$-type structure between 63–79 GPa and 28–29 GPa, respectively, upon laser heating. For ${\text{SrF}}_2$, the ${\text{Ni}}_2\text{In}$-type phase was confirmed by Rietveld refinement. Volumes were determined as a function of pressure for all high-pressure phases and fit to the third-order Birch-Murnaghan equation of state. For ${\text{CaF}}_2$ and ${\text{SrF}}_2$, the fluorite-cotunnite transition results in a volume decrease of $8–10\mathrm{\%}$, while the bulk modulus of the cotunnite-type phase is the same or less than that of the fluorite phase within uncertainty. For all three fluorides, the volume reduction associated with the further transition to the ${\text{Ni}}_2\text{In}$-type phase is $\sim 5\mathrm{\%}$. The percentage increase in the bulk modulus $\left(\Delta K\right)$ across the transition is greater when the cation is smaller. While for ${\text{BaF}}_2$, $\Delta K$ is $10–30\mathrm{\%}$, $\Delta K$ values for ${\text{SrF}}_2$ and ${\text{CaF}}_2$ are $45–65\mathrm{\%}$ and $20–40\mathrm{\%}$. Although shock data for ${\text{CaF}}_2$ have been interpreted to show a transition to a highly incompressible phase above 100 GPa, this is not consistent with our static equation of state data.

Figure 1

Representative x-ray diffraction patterns for ${\text{CaF}}_2$. Peaks labeled with asterisks in fluorite and ${\text{Ni}}_2\text{In}$-type patterns are from minor amounts of the cotunnite-type phase.

Figure 2

Representative x-ray diffraction patterns for ${\text{SrF}}_2$.

Figure 3

Representative x-ray diffraction patterns for ${\text{BaF}}_2$.

Figure 4

(Color online) Full profile refinement of ${\text{SrF}}_2$ ${\text{Ni}}_2\text{In}$-type phase with $P{6}_3/mmc$ symmetry and atomic positions Sr: (1/3, 2/3, 1/4), F: (0, 0, 0), and (1/3, 2/3, 3/4). Refined fit curve is superimposed on measured intensities (dots), with the residual curve below. Vertical ticks mark peak positions for ${\text{SrF}}_2$ ${\text{Ni}}_2\text{In}$-type, NaCl B2, and Pt phases.

Figure 5

(Color online) Compression curves and Birch-Murnaghan equation of state fits for ${\text{CaF}}_2$, ${\text{SrF}}_2$, and ${\text{BaF}}_2$. Open symbols are from previous experimental work (Refs. 4, 12, 57). All data are plotted relative to $V_0$ of the fluorite phase.

Figure 6

Lattice parameters of the cotunnite-type phase of alkaline earth fluorides. Open triangles are from previous work in He medium (Ref. 12). Linear fits are shown as a guide to the eye. Near the upper stability limit, the structure distorts continuously toward the hexagonal phase: the $a$ parameter becomes more compressible while $c$ increases in magnitude.

Figure 7

Interplanar spacing variation across the cotunnite-${\text{Ni}}_2\text{In}$ transition. Open symbols represent cotunnite-type phase diffraction peaks. Closed symbols are ${\text{Ni}}_2\text{In}$-type phase peaks. ${\text{CaF}}_2$ and ${\text{SrF}}_2$ data are from this study. ${\text{BaF}}_2$ data are from molecular-dynamics simulation (Ref. 26).

Figure 8

(Color online) Bulk modulus vs pressure for ${\text{CaF}}_2$, ${\text{SrF}}_2$, and ${\text{BaF}}_2$ computed from Birch-Murnaghan equation of state fits to volume data. Gaps are left across phase transitions due to uncertainty in the compressibility in these regions.

Figure 9

Lattice parameters (upper panel) and $c/a$ ratio (lower panel) for the ${\text{Ni}}_2\text{In}$-type phase of the alkaline earth fluorides. Filled symbols are for data under quasihydrostatic conditions while open symbols represent nonhydrostatic conditions. Open right-pointing triangles are from a nonhydrostatic experiment with a silicone grease medium (Ref. 16). Filled down-pointing triangles are from a hydrostatic experiment with a He medium (Ref. 12). In upper panel, least-squares fits are shown as a guide to the eye.

Figure 10

For ${\text{BaF}}_2$, the $c/a$ ratio of the ${\text{Ni}}_2\text{In}$-type phase (open triangles), the ratio of differential stress to pressure in Pt (filled circles), and the full width at half maximum for the Pt 111 peak normalized to the $2\vartheta$ peak position (filled squares). The high degree of correlation $\left(\rho =0.69\right)$ between the $c/a$ ratio and these indicators of differential stress explains the systematic difference between measurements of this phase in different pressure media. Under nonhydrostatic conditions, the $c/a$ ratio of this phase increases while under quasihydrostatic conditions (after laser annealing at pressures indicated by arrows) it decreases.

Figure 11

(Color online) Shock compression data for ${\text{CaF}}_2$ (Refs. 37, 63) (open diamonds) and ${\text{BaF}}_2$ (Ref. 64) (open up-pointing triangles) displayed with static data for ${\text{CaF}}_2$ [filled squares, this study and Angel (Ref. 4)] and ${\text{BaF}}_2$ [filled down-pointing triangles, this study and Smith et al. (Ref. 12)].

Figure 12

(Color online) Phase stability range for fluoride compounds. Filled circles: experiment, diamonds: theory. Fluoride structures exhibit a systematic dependence on cation radius and pressure. ${\text{CaF}}_2$, ${\text{SrF}}_2$, and ${\text{BaF}}_2$ are constrained by this study (filled circles with black outlines). Other experimental data (filled circles without outlines) are from ${\text{PdF}}_2$ (Ref. 68), ${\text{MnF}}_2$ (Ref. 66), ${\text{BaF}}_2$ (Ref. 16), ${\text{PbF}}_2$ (Ref. 71), ${\text{MgF}}_2$ (Ref. 69), and ${\text{ZnF}}_2$ (Ref. 65). Transition pressures from theoretical calculations (diamonds) are shown for ${\text{PbF}}_2$ (Ref. 70), ${\text{CaF}}_2$ (Ref. 5), ${\text{CdF}}_2$ (Ref. 67), and ${\text{ZnF}}_2$ (Ref. 67).