High-pressure structure of praseodymium revisited: In search of a uniform structural phase sequence for the lanthanide elements

Angle-dispersive x-ray powder diffraction experiments have been performed on praseodymium metal to a pressure of 205 GPa. Between 20 and 165 GPa only the $oC4$ ($\alpha$-uranium) phase is observed, in agreement with previous studies. At 171(5) GPa we find a transition to a tetragonal $tI2$ phase which is isostructural with the high-pressure post-$oC4$ phase seen in the neighboring lanthanide cerium above 12 GPa, and with the high-pressure phase of the actinide thorium seen above 100 GPa. Electronic structure calculations determine the $oC4\to tI2$ transition to occur at 130 GPa at 0 K, but find another phase, with the $hP1$ (simple hexagonal) structure, to have a lower enthalpy than both the $oC4$ and $tI2$ structures above 20 GPa at 0 K.

Figure 1

The pressure distribution over the central 10 $\mu \mathrm{m}\times 10 \mu \mathrm{m}$ area of sample 1 at 174 GPa. Also shown for comparison is the 0.85 $\mu \mathrm{m}$ x-ray beam diameter (FWHM) used to obtain the diffraction data at PETRA III. Despite a pressure gradient of 6 GPa within the mapped region, the pressure variation within the sample volume probed by the 0.85 $\mu \mathrm{m}$ diameter beam is negligible.

Figure 2

The pressure distribution over the central 10 $\mu \mathrm{m}\times 10$ $\mu \mathrm{m}$ area of sample 2 at 193 GPa. Also shown for comparison is the 0.85 $\mu \mathrm{m}$ x-ray beam diameter used to obtain the diffraction data at PETRA III. Despite a pressure gradient of 11 GPa within the mapped region, the pressure variation within the sample volume probed by the 0.85 $\mu \mathrm{m}$ diameter beam is negligible.

Figure 3

Rietveld refinement of the $oC4$ structure to a background-subtracted diffraction profile from Pr at 165 GPa, showing the observed (crosses) and calculated (line) diffraction patterns, the calculated reflection positions, and the difference profile [$R_P=3.5\%, R_{wP}=5.8\%$, GoF = 1.19, $R\left(F^2\right)=16.6\%$, and preferred orientation in the (021) direction]. The first seven peaks of the $oC4$ phase are labeled with their Miller indices. The asterisks identify weak peaks from the W gasket, and the $+$ symbol identifies a peak from the Cu pressure calibrant.

Figure 4

Background-subtracted diffraction profiles collected from Pr on pressure increase from 165 GPa to 205 GPa. The data were collected from the two samples at PETRA III using $\lambda =0.4832$ Å. Tick marks beneath profile (a) show the best-fitting peak positions for the $oC4$ phase at this pressure (as obtained from Fig. 3), while the tick marks beneath profile (e) show the calculated peak positions for the reported $oP4$ structure at 205 GPa, assuming lattice parameters of $a=2.380$ Å, $b=4.673$ Å, and $c=4.499$ Å. The peak marked with an arrow in profile (b) is the most evident new peak from the post-$oC4$ phase, while the arrow in profile (e) identifies the (012) peak of the $oP4$ structure, which we do not observe. The peaks marked with asterisks in profiles (a)–(c) are from the W gasket, while peaks marked with $+$ are from the Cu pressure marker. Profiles (a)–(c) were obtained from sample 1, in which the diamonds failed at 191 GPa, and profiles (d) and (e) were obtained from sample 2.

Figure 5

Rietveld refinement of the $tI2$ structure to a background-subtracted diffraction profile from Pr at 205 GPa, showing the observed (crosses) and calculated (line) diffraction patterns, the calculated reflection positions, and the difference profile [$R_P=0.9\%, R_{wP}=1.7\%$, GoF = 0.5, $R\left(F^2\right)=5.2\%$, preferred orientation in the (120) direction]. All of the peaks from the $tI2$ phase are labeled with their Miller indices, while peaks marked with $+$ symbols are from the Cu pressure calibrant.

Figure 6

The compressibility of Pr to 205 GPa. The data from our previous measurements to 22 GPa [15] and the current study of $oC4$-Pr from 65 to 192 GPa are plotted with solid symbols, while the data for $oC4$-Pr from the study of Chesnut and Vohra [29] are shown as crosses. The current data on $tI2$-Pr are shown with unfilled triangles to distinguish them from $oC4$-Pr. Inset (i) highlights the volume change of $\sim 6$% at the $oI16\to oC4$ transition, while inset (ii) highlights the volume change of $\sim 0.5$% at the $oC4\to tI2$ transition. The dashed lines through the data points are the AP2 EOS fits.

Figure 7

Linearization of the compression of Pr shown in the form of an ${\eta }_{\mathrm{APL}}-\sigma$ plot, where $\sigma ={\sigma }_{0}x$. The data from the lower-pressure phases ($hP4, cF4, hR24$, and $oI16$) are plotted with filled triangles, while our data from $oC4$-Pr are plotted with filled squares and the $oC4$-Pr data from the study of Chesnut and Vohra [29] are shown as crosses. For clarity, the current data for $tI2$-Pr are plotted with unfilled triangles.

Figure 8

The enthalpies of the $tI2$-Pr, $P{2}_1{2}_1{2}_1$-Pr, and $Pnma$-Pr relative to that of $oC4$-Pr from 20 to 300 GPa. $tI2$-Pr becomes significantly more stable than $oC4$-Pr at pressures above 130 GPa. Also shown is the enthalpy curve for the second $Pnma$ structure, the true structure of which is simple hexagonal, $hP1$.

Figure 9

The enthalpy gain per atom of $oC4$-Tb (red line) and $tI2$-Tb (blue line) relative to that of $oF16$-Tb. The $oC4$ phase becomes more stable than $oF16$ at pressures above approximately 300 GPa. However, the enthalpy of $tI2$-Tb is always significantly above those of the other two phases. Also shown is the enthalpy gain per atom of $tI2$-Nd relative to $oC4$-Nd (green line) from 40 to 500 GPa. The $oC4$ phase remains the most stable of the two over the whole range examined. The kink at 80 GPa is associated with the loss of magnetization in $oC4$-Nd at this pressure.

Figure 10

The different phases reported in the lanthanide elements up to 310 GPa at ambient temperature, including the current results in Pr. With the exception of Ce, Eu, and Yb, the elements adopt a similar phase transition sequence, with the transition pressures increasing with increasing $Z$. The observation of the $tI2$ structure in Pr above 170 GPa strengthens the systematics between the different lanthanides, and suggests that this is the next common structure in the phase transition sequence. The phases are labeled using Pearson notations except for the two incommensurate phases of Eu which are denoted Incomm1 and Incomm2 [50, 51].