Pressure-induced bcc-rhombohedral phase transition in vanadium metal

Vanadium is reported to undergo a pressure-induced bcc-rhombohedral phase transition at 30–70 GPa, with a transition pressure that is sensitive to the hydrostaticity of the sample environment. However, the experimental evidence for the structure of the high-pressure phase being rhombohedral is surprisingly weak. We have restudied vanadium under pressure to 154 GPa using both polycrystalline and single-crystal samples, and a variety of different pressure transmitting media (PTM). We find that only when using single-crystal samples does one observe a rhombohedral high-pressure phase; the high-pressure diffraction profiles from the polycrystalline samples do not fit a rhombohedral lattice, irrespective of the PTM used. The single-crystal samples reveal two rhombohedral phases, with a continuous transition between them, and distortions from cubic symmetry are much smaller than previously calculated.

Figure 1

The body-centred unit cell of vanadium (solid lines) and the corresponding primitive rhombohedral cell (dashed lines). A cubic body-centred cell with lattice parameter $a_c$ can be considered a rhombohedral cell with $a_r=a_c\frac{1}{2}\sqrt{3}$ and $\alpha$=$109.47{}^{\circ }$ (rhombohedral axes) or a triple hexagonal cell with $a_h=\sqrt{2}{a}_c, c_h=a_c\frac{1}{2}\sqrt{3}$, and $c_h/a_h=\sqrt{3/8}$ (hexagonal axes).

Figure 2

Calculated diffraction patterns for two rhombohedral V structures with (a) $a_r=2.4284$ Å, $\alpha =109.{59}^{\circ }$ and (b) $a_r=2.4213$ Å, $\alpha =109.{35}^{\circ }$. Both structures have the same density. The peaks in the main part of the figure are indexed with their bcc Miller indices. The left-hand inset illustrates how the change in $\alpha$ from $109.{47}^{\circ }$ results in the (110)${}_c$ peak splitting evenly about its position in a bcc phase with the same density, shown by the vertical dashed line. The right-hand inset illustrates how the (211)${}_c$ peak splits into a triplet in the rhombohedral phase, the peaks of which are in different positions in the two different structures. Whether $\alpha$ is $>$ or $<109.{47}^{\circ }$ can therefore be determined from the positions and relative intensities of the peaks that form this triplet in the rhombohedral phase.

Figure 3

The $d$-spacing ratios of the ($1\overline{1}{0)}_r$/($1\overline{1}{1)}_r$ and (100)${}_r$/($1\overline{1}{1)}_r$ reflections (primitive rhombohedral indexing), as $calculated$ from the rhombohedral fits to the nonhydrostatic data reported by Ding et al. [6]. At the transition at 69 GPa, the ratios split equally about the value of $\sqrt{2}$, with the degree of splitting depending only on the rhombohedral angle. If the same ratios derived from the $observed$ peak positions do not exhibit a similar symmetric splitting, then the positions of the three peaks are inconsistent with a rhombohedral lattice.

Figure 4

(a) Diffraction profiles obtained from V compressed without a PTM up to 139 GPa. The incident x-ray wavelength was 0.4119 Å.The peaks in the ambient pressure (AP) profile are indexed using their bcc indices. (b) Enlarged view of the high-angled region of the profiles. The splitting of the bcc peaks above 30 GPa is clear, and is highlighted with dashed lines for the (220) peak.

Figure 5

The pressure dependence of $d$-spacing ratio of the ($1\overline{1}{0)}_r$ and (100)${}_r$ reflections relative to the ($1\overline{1}{1)}_r$ reflection for V compressed without a PTM. In the bcc phase the ratio is $\sqrt{2}$, while in the rhombohedral phase the two ratios change such that one is $>\sqrt{2}$ and the other is $<\sqrt{2}$, but their average remains $\sqrt{2}$, as shown by the dashed line. The deviation from this behavior means that a rhombohedral lattice cannot fit the ($1\overline{1}{0)}_r$, (100)${}_r$ and ($1\overline{1}{1)}_r$ reflections simultaneously. Also shown are the values calculated from the diffraction profile shown by Ding et al. at 155 GPa [6] and the diffraction profile of Takemura at 154 GPa [1]. The agreement of these two previous studies with the current data is excellent.

Figure 6

A two-phase Rietveld profile fit to the diffraction profile from V and the Pt pressure marker compressed without a PTM at 61 GPa, showing the observed (crosses) and calculated (line) diffraction profiles. The calculated reflection positions for both the V and Pt are shown beneath the profiles. The positions of all of the Pt peaks are predicted precisely by the lattice parameter of $a=3.729$(1) Å. In contrast, the peak positions of the V are clearly $not$ correctly predicted by the best-fitting rhombohedral lattice parameters of $a=2.420$(3) Å and $\alpha =109.68{\left(5\right)}^{\circ }$.

Figure 7

(a) Diffraction profiles obtained from V powder compressed with a He PTM up to 154 GPa. The incident x-ray wavelength was 0.4119 Å. The peaks in the AP profile are indexed using their bcc indices. (b) Enlarged view of the high-angled region of the profiles. The broadening of the bcc peaks on compression is clear.

Figure 8

The pressure dependence of $d$-spacing ratio of the ($1\overline{1}{0)}_r$ and (100)${}_r$ reflections relative to the ($1\overline{1}{1)}_r$ reflection for V powder compressed in a He PTM. In the bcc phase the ratio is $\sqrt{2}$, while in the rhombohedral phase the two ratios change such that one is $>\sqrt{2}$ and the other is $<\sqrt{2}$, but their average remains $\sqrt{2}$, as shown by the dashed line. The deviation from this behavior means that a rhombohedral lattice cannot fit the ($1\overline{1}{0)}_r$, (100)${}_r$, and ($1\overline{1}{1)}_r$ reflections simultaneously. Comparison with Fig. 5 reveals that the deviations from the expected behavior start at lower pressures than in the sample compressed without a PTM, but that the behavior of the $d{\left(1\overline{1}0\right)}_r/d{\left(1\overline{1}1\right)}_r$ ratio is very similar. While the magnitude of the deviation of the $d{\left(100\right)}_r/d{\left(1\overline{1}1\right)}_r$ ratio from $\sqrt{2}$ is similar in both samples, the displacements are in opposite directions.

Figure 9

Rotation ($\pm {25}^{\circ }$ in $\omega$) diffraction image from a V/Mo [001]-oriented single crystal at 8.7 GPa. The most intense reflections from the diamond anvils are identified with “D,” while the V/Mo reflections pairs (with the weaker Mo reflections being the lower-angled of the two) are labeled with their bcc Miller indices. To ensure consistency of indexing, the same indices were used to analyze both the cubic and rhombohedral phases, using the nonprimitive body-centred rhombohedral cell to analyze the latter. While the V reflections are elongated azimuthally even at this pressure, their individual $d$ spacings are still measurable precisely without interference from other reflections with the same or similar $d$ spacing. The thin vertical lines on the image are detector artifacts. The two faint Debye-Scherrer rings marked with “W” are from the tungsten gasket.

Figure 10

(a) Pseudo-powder-diffraction profiles obtained by integrating the 2D rotation images obtained from sample 2 of single-crystal V compressed in a mineral oil PTM up to 118 GPa. The peaks from the V are indexed with their cubic Miller indices, while the peaks arising from the Mo substrate are identified with asterisks. (b) Enlarged view of the high-angled region of the profiles.

Figure 11

The pressure dependence of $d$-spacing ratio of the ($1\overline{1}{0)}_r$ and (100)${}_r$ reflections relative to the ($1\overline{1}{1)}_r$ reflection (all primitive rhombohedral indexing) for the two samples of single-crystal V compressed in a mineral oil PTM. Note (i) the reduced $y$-axis scale (0.02 vs 0.06) compared to Figs 5. and 8; (ii) the symmetry of the displacements of the filled/unfilled data points about the ratio of $\sqrt{2}$; and (iii) the fact that the filled/unfilled points cross the $\sqrt{2}$ value at $\sim 100$ GPa.

Figure 12

The rhombohedral angle $\alpha$ in both V and Mo, as determined from the two single-crystal samples. The same indexing was used for both materials. The data from Mo show some scatter about the cubic value of $\alpha =109.{47}^{\circ }$, perhaps arising from the nonhydrostatic pressure environment, but shows no trend away from cubic symmetry. In V, the sample is initially cubic ($\alpha =109.{47}^{\circ }$) until $\sim 40$ GPa, above which the angle becomes systematically smaller than the ideal cubic value, reaching a minimum of 109.35(2)${}^{\circ }$ at 66 GPa. On further compression $\alpha$ then increases quasilinearly, passing back through the cubic value at 100 GPa, and increasing to $109.55{\left(2\right)}^{\circ }$ at 118 GP, the highest pressure reached with sample 2. The change in $\alpha$ reported by Ding et al. [6], and the value determined by Errandonea et al. [15] from a Rietveld refinement at 64 GPa, are also shown for comparison.

Figure 13

The rhombohedral angle $\alpha$ as determined from the two single-crystal samples in the current study, compared to that predicted by Wang et al. [12] in their two rhombohedral phases, RH1 and RH2, up to 320 GPa. The rhombohedral angles predicted by other computational studies are similar [8, 10, 11]. The calculated rhombohedral angles (at 0 K) are clearly very much larger than those observed (at 300 K), and the sense of those distortions, that is, whether $\alpha$ is $>$ or $<109.{47}^{\circ }$, is the reverse of that observed.

Figure 14

The compressibility of V to 120 GPa, as determined from the single-crystal data. A fit using an AP2 equation of state [26] is shown, from which the zero pressure bulk modulus $K_0$ and its pressure derivative $K^{\prime }$ were determined to be 158.9(13) GPa and 3.58(6), respectively. Also shown are the compressibility curves reported by Ding et al. [6] from their nonhydrostatic and quasihydrostatic powder data.

Figure 15

The $d$-spacing ratio of the ($1\overline{1}{0)}_r$/(100)${}_r$ doublet (as determined from its midpoint) relative to that of the ($1\overline{1}{1)}_r$ peak, as determine from the radial diffraction data reported by Xiong et al. at $\psi =54.7^{\circ }$ ($\circ$). The ratio should remain fixed at $\sqrt{2}$ in both the bcc and rhombohedral phases. Also shown for comparison are the same ratios obtained from our powdered sample compressed without a PTM ($\blacksquare$), the powdered sample compressed in a He PTM ($•$), and from the two single-crystal samples ($▴$). For the single-crystal data, the error bars are smaller than the symbols used to plot the points and have been omitted.

Figure 16

The calculated diffraction profiles from rhombohedral vanadium with $a=2.420$ Å, $\alpha =109.{68}^{\circ }$ experiencing differential strains of (a) $-5$% to (e) $+5$%. Profile (c) shows the profile unaffected by strain, with the calculated peak positions marked by tickmarks beneath the profile. The effects of the differential strain are to move the positions of the Bragg peaks relative to each other, but the peak $splittings$ are unaffected. The inset shows enlarged views of the (220)${}_c$ and (310)${}_c$ reflections [which split into the ($20\overline{2}{)}_r$/(200)${}_r$ and ($21\overline{2}{)}_r/(21\overline{1}{)}_r$ doublets], and also the observed diffraction data at 61 GPa (the same data shown in Fig. 6). It is clear that the larger peak splittings observed in the data cannot be explained by a differential strain model.