Determination of indium melting curve at high pressure by picosecond acoustics

Picosecond acoustics combined with a diamond anvil cell is used to study liquid indium and to determine with high accuracy both the sound velocity and the melting curve over an extended pressure and temperature range. The sound velocities, determined by phonon surface imaging, complement previous inelastic x-ray scattering determinations and are in good agreement with estimations according to a thermodynamic model. Based on exact thermodynamic relations, the equation of state of the liquid phase is obtained using the isothermal bulk modulus $B_{T,0}$ and its first pressure derivative $B_T^{\prime }$. These quantities are derived from the precise experimental determination of the variation of the sound velocity as a function of pressure. Melting is determined via the detection of abrupt changes in the elastic properties between solid and liquid phases and through the monitoring of the solid-liquid coexistence. The melting curve constrained up to 6 GPa and 673 K is shown to be well described by the Simon-Glatzel equation in the full $(p,T)$ range explored.

Figure 1

Top: Schematic view of the setup. PBS: Polarizing beam splitter, $\lambda /2$: Half-wave plate, $\lambda /4$: Quarter-wave plate, PD: Amplified photodetector, “A.O.M.” stands for acousto-optic modulator, and “ref. mirror” for reference mirror as a part of a Michelson interferometer. Blue ellipses represent optical objectives. Bottom: Interferometric signal as a function of delay time obtained in liquid indium at high pressure and temperature with collinear pump and probe beams. Numbers identify consecutive echoes corresponding to the $n\mathrm{th}$ wave arriving at the sample surface on the probe side. Inset: Magnification of the third echo.

Figure 2

Temperature estimated from the in situ optical sensors $T_{\mathrm{calib}}$ (see text and [58], Sec. 1) as a function of the temperature value $T_{\mathrm{th}}$ provided by the thermocouple glued to one of the diamonds of the DAC. The analysis of the ruby fluorescence line shift is performed according to two models, Datchi et al. [55] (red empty circles) and McCumber et al. [59] (blue crosses). The equality relation is indicated by the dashed line.

Figure 3

(a) Phonon imaging pattern of the acoustic wave front at the liquid-diamond interface in liquid indium at 3 GPa and 682 K with collinear pump and probe at circular pattern center for 11.9606 ns pump-probe delay. (b) Integrated profile. (c) Integrated profiles stacked versus time. For clarity, the final image between 0 and $T_{\mathrm{laser}}$ is duplicated 3 times. The colored ripples and the lines show the radius of the circles detected at the sample surface versus time. The nonlinear $R\left(t\right)$ curves are linked to the volume waves propagating inside the sample and appearing at the surface, whereas the linear $R\left(t\right)$ curves are due to the pure surface waves (possibly surface skimming bulk waves propagating in the diamond at the interface diamond-sample [60]). Only the first nonlinear curve is fitted by the function $R\left(t\right)$, and the calculation with the obtained results is shown as the blue dotted line. The red and green dotted lines are calculations corresponding to multiple reflections in the sample ($n=2$ and $n=3$, respectively) and demonstrate the quality of the fit procedure.

Figure 4

Sound velocity in liquid indium as a function of pressure and temperature, measured by the phonon imaging method. The data are compared with velocities derived from the thermal EOS of Komabayashi et al. [36] calculated along three reference isotherms. Inset: Sound velocity as a function of temperature at ambient pressure, measured by the phonon imaging method [average thickness $e_0=68.3\left(12\right)$ $\mu \mathrm{m}$, with imposed parameter $z_R=16.2\left(4\right)$ $\mu \mathrm{m}]$ above melting temperature ($T_M$). The data are compared with the average linear relation of Blairs [33] (blue dotted line) and its dispersion (gray area; see Fig. S3 in [58]), and with the sound velocity calculated from the thermal EOS.

Figure 5

Equation of state in liquid indium calculated from the Vinet EOS and thermodynamic parameters given in Table 1 (solid lines) compared to the EOS of Komabayashi (dashed lines). The experimental data are from Shen [73] and Takubo [74].

Figure 6

Transition between liquid and solid indium (gray region) as recorded by the large and sharp variation of the arrival time of the first acoustic echo at $T=510$ K. The transition pressure, measured both for increasing $p$ (top panel) and decreasing $p$ (bottom panel), is in this case 1.70(5) GPa.

Figure 7

Indium melting line measured by picosecond acoustics. As explained in the text, experimental $(p,T)$ points are obtained in two ways. Method 1: Large and sharp shift of the arrival time of the first acoustic echo upon pressure increase (blue solid triangles) and upon pressure decrease (blue empty triangles) along isothermal paths; method 2: Through monitoring the liquid-solid equilibrium (empty black squares). All the data are fitted by a Simon-Glatzel equation [dash-dotted red line; see text and Eq. (9)]. In addition, an extrapolation of the Clausius-Clapeyron relationship at ambient pressure is shown [Eq. (11); green line], as well as the polynomial function given by Richter extrapolated above 3.5 GPa (black dotted line; see Table 1 in Ref. [78]).

Figure 8

The melting curve obtained in this work by picosecond acoustics (red line) and extrapolated above 6 GPa (red dotted line) is compared with the literature data: Dudley (1960) [32], Dudley corrected in this work, McDaniel (1962) [81], Jayaraman et al. (1963) [79] and their correction of the Dudley data, Millet (1968) [82] (data points are taken from Cannon (1974) [83]), Richter (1980) [78], Höhne (1996) [84], Shen (2002) [85] (red filled squares: Solid, red empty squares: Liquid), and Errandonea (2010) [45]. The inset shows the low-pressure region.