Equation of State for Bismuth at High Energy Densities

Abstract

The purpose of this work is to describe the thermodynamic properties of bismuth in a broad scope of mechanical and thermal effects. A model of the equation of state in a closed form of the functional relationship between pressure, specific volume, and specific internal energy is developed. A new expression is proposed for the internal energy of a zero-temperature isotherm in a wide range of compression ratios, which has asymptotics to the Thomas–Fermi model with corrections. Based on the new model, an equation of state for bismuth in the region of body-centered cubic solid and liquid phases is constructed. The results of calculating the thermodynamic characteristics of these condensed phases with the new EOS are compared with the available experimental data for this metal in waves of shock compression and isentropic expansion. The parameters of shock waves in air obtained earlier by unloading shock-compressed bismuth samples are reconsidered. The newly developed equation of state can be used in modeling various processes in this material at high energy densities.

1. Introduction

Knowledge of the thermodynamic properties of materials in a wide range of high-energy states is required for the analysis of physical phenomena in various processes under intense pulse action on condensed matter [1,2,3]. These processes include high-velocity impact [4,5,6,7], interaction with the material of intense laser radiation [8,9,10,11] or particle flows of high power density [12,13,14], and the electrical explosion of conductors [15,16,17]. For numerical simulations of emerging hydrodynamic flows, it is necessary to know the relationships of the thermodynamic characteristics of the medium in the entire range of realizable states [18,19,20,21,22]. Interest in thermodynamic models is also associated, in particular, with the problems of studying the thermoelastic properties of various materials at high pressures [23,24,25,26].

Bismuth is used in nuclear power technology as a component in liquid metal coolants or solvents for fuel in nuclear reactors [27], as well as a target for charged particle beam irradiation in an accelerator-driven nuclear waste transmutation system [28,29,30,31]. In this regard, the equation of state (EOS) for bismuth in a wide range of changes in specific volume (V) and specific internal energy (E) is of interest for solving problems related to the numerical simulation of the dynamics of flows in the liquid phase and dense plasma.

In this paper, a model of the EOS for matter in the form of a function of pressure P = P (V, E) is developed. In contrast to the previously known EOSs (for bismuth) [32,33,34,35,36,37,38,39,40,41], a new expression is proposed for the internal energy of matter on the zero-temperature isotherm (cold curve) in a wide range of densities ($\rho =V^{-1}$), which has asymptotics to the Thomas–Fermi model with quantum and exchange corrections [42,43,44,45]. A new EOS is developed for a body-centered cubic (bcc) solid phase and a melt of bismuth in the high-pressure region. The results of the calculations of the shock adiabats for samples of different initial densities and release isentropes for shock-compressed samples using the new EOS are compared with the available experimental data for bismuth. In addition, the relationship between the parameters of plane shock waves in the metal and in the air adjacent to it when the front of the first wave reaches the sample–gas interface is reconsidered.

2. EOS Model

The general form of the dependence of pressure on specific volume and internal energy is chosen on the basis of the quasi-harmonic approximation:

$$P(V,E)=P_{\mathrm{c}}\left(V\right)+\frac{\mathsf{\Gamma }(V,E)}{V}[E-E_{\mathrm{c}}\left(V\right)],$$

(1)

where $E_{\mathrm{c}}$ is the specific cold energy, i.e., specific internal energy at zero temperature, T = 0; and $P_{\mathrm{c}}$ is the cold pressure, i.e., pressure at T = 0, that is, $P_{\mathrm{c}}=-\mathrm{d}{E}_{\mathrm{c}}/\mathrm{d}V$. The coefficient $\mathsf{\Gamma }$ is the ratio of thermal pressure to thermal energy density, $\mathsf{\Gamma }=(P-P_{\mathrm{c}})V/(E-E_{\mathrm{c}})$.

The cold energy under compression ($\varsigma =V_{0\mathrm{c}}/V⩾1$; $V_{0\mathrm{c}}$ is the specific volume at T = 0 and P = 0) is proposed in the following form:

$$E_{\mathrm{c}}\left(V\right)=\frac{9}{2}{V}_{0\mathrm{c}}{B}_{0\mathrm{c}}{\varsigma }^{2/3}{\zeta }^{2}exp\left[K\left(\zeta \right)\right].$$

(2)

Here, $B_{0\mathrm{c}}$ is the cold bulk modulus $B_{\mathrm{c}}=-V\mathrm{d}{P}_{\mathrm{c}}/\mathrm{d}V$ at $\varsigma =1$; $\zeta =1-{\varsigma }^{-1/3}$;

$$K\left(\zeta \right)=\sum _{k=1}^{N_{\mathrm{k}}}{a}_k{\zeta }^k.$$

(3)

The form of Equation (2) ensures that the following conditions are met:

$$E_{\mathrm{c}}\left(V_{0\mathrm{c}}\right)=0,$$

(4)

$$P_{\mathrm{c}}\left(V_{0\mathrm{c}}\right)=0,$$

(5)

$$B_{\mathrm{c}}\left(V_{0\mathrm{c}}\right)=B_{0\mathrm{c}}.$$

(6)

The fulfillment of the condition for the value of the cold bulk modulus with respect to cold pressure $B_{\mathrm{c}}^{\prime }=\mathrm{d}{B}_{\mathrm{c}}/\mathrm{d}{P}_{\mathrm{c}}$ at $\varsigma =1$ is required:

$$B_{\mathrm{c}}^{\prime }\left(V_{0\mathrm{c}}\right)=B_{0\mathrm{c}\prime }^{\prime }$$

(7)

which gives $a_1=B_{0\mathrm{c}}^{\prime }-3$. In the high-density limit $\varsigma \to \infty$, $\zeta \to 0$, an asymptotic pressure behavior

$$P_{\mathrm{c}}\left(V\right)\approx b_2{\varsigma }^{5/3}+b_1{\varsigma }^{4/3}$$

(8)

is needed, where the coefficients $b_2$ and $b_1$ are determined from consideration of the Thomas–Fermi model with quantum and exchange corrections [42,43,44,45,46]:

$$b_2=\frac{3^{2/3}{\pi }^{4/3}{Z}^{5/3}}{5}\frac{a_{\mathrm{B}}^2{E}_{\mathrm{H}}}{{\left[A{m}_{\mathrm{u}}{V}_{0\mathrm{c}}\right]}^{5/3}},$$

(9)

$$b_1=-\left(\frac{3^{2/3}{\pi }^{1/3}{Z}^2}{2^{1/3}5}+\frac{11Z^{4/3}}{2^2{3}^{5/3}{\pi }^{1/3}}\right)\frac{a_{\mathrm{B}}{E}_{\mathrm{H}}}{{\left[A{m}_{\mathrm{u}}{V}_{0\mathrm{c}}\right]}^{4/3}},$$

(10)

where $a_{\mathrm{B}}$ is the Bohr radius; $E_{\mathrm{H}}$ is the Hartree energy; $m_{\mathrm{u}}$ is the unified atomic mass unit (u); A it the atomic mass in u; and Z is the atomic number. Equation (8) gives two conditions for the coefficients $a_k$ in Equation (3):

$$\sum _{k=1}^{N_{\mathrm{k}}}{a}_k=ln\frac{b_2}{3B_{0\mathrm{c}}},$$

(11)

$$\sum _{k=1}^{N_{\mathrm{k}}}k{a}_k=-\frac{2b_1}{b_2}-2.$$

(12)

The cold energy of matter in the expansion region ($\varsigma <1$) is taken as the sum of the power functions:

$$E_{\mathrm{c}}\left(V\right)=V_{0\mathrm{c}}{B}_{0\mathrm{c}}\left(\frac{a_{\mathrm{m}}}{m}{\varsigma }^m+\frac{a_{\mathrm{n}}}{n}{\varsigma }^n-\frac{a_{\mathrm{m}}+a_{\mathrm{n}}}{l}{\varsigma }^l\right)+E_{\mathrm{s}}’$$

(13)

which provides the value of the sublimation energy $E_{\mathrm{s}}$ in the limit of zero density $\varsigma \to 0$ as well as the condition of Equation (5). Satisfying Equations (4), (6), and (7) leaves only two free parameters m and n in Equation (13).

In Equation (1), the coefficient $\mathsf{\Gamma }$ depends on the volume and internal energy [46]:

$$\mathsf{\Gamma }(V,E)={\gamma }_{\mathrm{i}}+\frac{{\gamma }_{\mathrm{c}}\left(V\right)-{\gamma }_{\mathrm{i}}}{1+{\sigma }^{-2/3}[E-E_{\mathrm{c}}\left(V\right)]/E_{\mathrm{a}}},$$

(14)

where $\sigma =V_0/V$; $V_0$ is the specific volume under normal conditions, $E=E_0$ and $P=P_0$; ${\gamma }_{\mathrm{c}}$ is the Grüneisen coefficient $\gamma =V{(\partial P/\partial E)}_V$ at $T=0$; ${\gamma }_{\mathrm{i}}$ is the value of the Grüneisen coefficient in the case of high thermal energies, $E-E_{\mathrm{c}}\gg E_{\mathrm{a}}{\sigma }^{2/3}$; and $E_{\mathrm{a}}$ is a parameter. The coefficient ${\gamma }_{\mathrm{c}}$ is represented by the volume function [47,48]

$${\gamma }_{\mathrm{c}}\left(V\right)=2/3+({\gamma }_{\mathrm{c}0}-2/3)\frac{{\delta }_{\mathrm{n}}+{ln}^2{\sigma }_{\mathrm{m}}}{{\delta }_{\mathrm{n}}+{ln}^2(\sigma /{\sigma }_{\mathrm{m}})},$$

(15)

where the value of ${\gamma }_{\mathrm{c}0}$ corresponds to the normal volume $V_0$; and ${\sigma }_{\mathrm{m}}$ and ${\delta }_{\mathrm{n}}$ are parameters. The chosen forms of Equations (1), (14), and (15) provide the following relation:

$${\gamma }_{\mathrm{c}0}={\gamma }_{\mathrm{i}}+\left({\gamma }_0-{\gamma }_{\mathrm{i}}\right){\left[1+\frac{E_0-E_{\mathrm{c}}\left(V_0\right)}{E_{\mathrm{a}}}\right]}^2,$$

(16)

where ${\gamma }_0$ is the Grüneisen coefficient under normal conditions, $\gamma (V_0,E_0)={\gamma }_0$.

3. Thermodynamic Properties of Bismuth

At atmospheric pressure, crystalline bismuth has a rhombohedral structure and melts at 544.5 K [49]. Under compression at room temperature, the rhombohedral structure transforms into a monoclinic structure at 2.55 GPa [50], which, in turn, transforms into an incommensurate host–guest structure of two nested body-centered tetragonal cells at 2.7 GPa [51,52]. A further increase in pressure leads to a transition to a phase with a bcc structure at $7.67\pm 0.18$ GPa [50,51,52], which is observed up to the maximum pressure of 222 GPa reached under static conditions [53,54].

In the present work, the EOS was constructed for the region of the bcc-solid and liquid phases of bismuth. The coefficients of Equations (1)–(16), which optimally generalize the information available at high pressures and high thermal internal energies, are obtained as follows: $V_0=0.087$ cm${}^3$/g, $V_{0\mathrm{c}}=0.086139$ cm${}^3$/g, $B_{0\mathrm{c}}=55.277026$ GPa, $a_1=2.48$, $a_2=0.826667$, $a_3=-3.051687$, $a_4=4.092720$, $a_{\mathrm{m}}=-0.662946$, $a_{\mathrm{n}}=-0.000316$, m = 1, n = 12, l = 2.512935, $E_{\mathrm{s}}=1.9$ kJ/g, ${\gamma }_{\mathrm{c}0}=2.3$, ${\sigma }_{\mathrm{m}}=1.3$, ${\delta }_{\mathrm{n}}=4$, ${\gamma }_{\mathrm{i}}=0.4$, and $E_{\mathrm{a}}=2.3$ kJ/g.

The calculated cold compression curve is presented in Figure 1. As can be seen in Figure 1, the calculated zero-temperature isotherm is in good agreement with the available experimental data [53,54] on the compression of the bcc phase of bismuth at room temperature at pressures up to 222 GPa. In addition, the obtained dependence of pressure at $T=0$ on the density at compression ratios above 20 agrees well with the results of calculations by Kalitkin and Kuzmina [45] using the Thomas–Fermi model with corrections [44]. The previously known cold-curve approximation [34,39] deviates noticeably from the results of pressure calculations using the Thomas–Fermi model with corrections at compression ratios above 500. The cold compression curves of the bcc phase of bismuth from the EOSs [38,40] are in good agreement with the experimental data [53,54], but they deviate greatly from the results of calculations using the Thomas–Fermi model with corrections in the range of applicability of the latter (at compression ratios above 50).

The shock compressibility of bismuth has been studied using traditional planar explosive systems [55,56,57,58,59,60,61,62,63,64,65,66,67,68] up to a pressure of 177 GPa. The use of hemispherical explosive systems [69,70] made it possible to obtain a pressure behind the shock wave front in this metal up to 385 GPa [56,61]. Higher shock-compression pressures in bismuth up to 675 GPa have been achieved using layered cumulative systems and cone-converging shock-wave generators [61]. A high-power laser [67] and a light-gas gun [68] were also used to study phase transformations of bismuth in shock waves.

The parameters of the shock adiabat of samples with an initial density ${\rho }_{00}$ at a given pressure P are calculated by solving a system consisting of the EOS P = P (V, E) (1)–(16) and the Rankine–Hugoniot relation [1] expressing the energy conservation law:

$$E=E_0+\frac{1}{2}(P+P_0)(V_{00}-V),$$

(17)

where $P_0$, $V_{00}$, and $E_0$ are the pressure, specific volume, and internal energy of the matter ahead of the shock-wave front; $V_{00}={\rho }_{00}^{-1}$; and P, V, and E are the pressure, specific volume, and internal energy of the shock-compressed matter behind the shock-wave front. The velocities of the front of the shock wave and particles behind the front are found from the Rankine–Hugoniot relations [1], which express the laws of conservation of mass and momentum:

$$U_{\mathrm{s}}=V_{00}\sqrt{\frac{P-P_0}{V_{00}-V}},$$

(18)

$$U_{\mathrm{p}}=\sqrt{(P-P_0)(V_{00}-V)}.$$

(19)

The results of the calculation of the shock adiabats for bismuth samples of different initial densities are shown in Figure 2, Figure 3 and Figure 4 in comparison with the available shock-wave data [55,56,57,58,59,60,61,62,64,65,66,67]. An analysis of the figures allows one to conclude that the calculated shock adiabats are in good agreement with the experimental data in the entire studied region of states of the bcc and liquid phases of bismuth.

In experiments [61], in addition to the parameters of the shock wave in the sample, the velocity of the shock wave in the barrier (anvil) located directly behind the sample was also measured. From the known shock adiabat of the barrier material, the particle velocity and the pressure behind the front of this shock wave in the barrier can be determined. The particle velocity and pressure also correspond to the state of the sample after expanding isentropically into the barrier (if the barrier material has a dynamic impedance less than that of the sample material). Such experiments on isentropic expansion of shock-compressed samples make it possible to study a wide range of states from highly-compressed and heated condensed matter to rarefied gas and plasma.

The calculated isentropes of the unloading of shock-compressed bismuth samples of different initial densities are shown in Figure 5 along with the experimental data [61]. As can be seen, the proposed EOS is in good agreement with these data in the entire region of the achieved kinematic and dynamic parameters of the unloading waves, except for the region of the relatively low expansion velocities and pressures. According to the calculations made using another EOS model [33,34,39], which is thermodynamically complete and takes into account phase transitions, in this region of the thermodynamic parameters, bismuth melts and evaporates in isentropic expansion waves. Accounting for these effects is beyond the scope of the presented simple thermodynamically incomplete EOS model.

In experiments [61] upon unloading shock-compressed bismuth samples into air with an initial normal pressure, the measured parameter was the velocity of the shock wave in air, which was pushed by the free surface of the shock-compressed sample. In experiments [55,59], the velocity of the free surface of the sample (i.e., the interface with air) was taken as the measured parameter of the supersonic flow in air. The shock velocity (for the experiments in [55,59]) or particle velocity (for the experiments in [61]) and pressure in shock-compressed air can be determined from the known shock adiabat of this gas [71]. The results of such a calculation are presented in Figure 5, Figure 6 and Figure 7 and

Table 1. Data on isentropic release of shock-compressed bismuth samples into air: $U_{\mathrm{s}}$, $U_{\mathrm{p}}$, and P are the shock and particle velocities and the pressure behind the front of the first shock wave in the bismuth samples; $U_{\mathrm{s}\mathrm{air}}$, $U_{\mathrm{p}\mathrm{air}}$, and $P_{\mathrm{air}}$ are the shock and particle velocities and the pressure behind the front of the shock wave in air; ${\mathsf{\Delta }}_2=U_{\mathrm{p}\mathrm{air}}-2U_{\mathrm{p}}$.

${\mathit{U}}_{\mathbf{s}}$, km/s	${\mathit{U}}_{\mathbf{p}}$, km/s	P, GPa	${\mathit{U}}_{\mathbf{s}\mathbf{air}}$, km/s	${\mathit{U}}_{\mathbf{p}\mathbf{air}}$, km/s	${\mathit{P}}_{\mathbf{air}}$, MPa	${{\Delta }}_2$, km/s
$2.696\pm 0.019$1	$0.718\pm 0.005$1	$18.95\pm 0.19$	$1.708\pm 0.011$	$1.401\pm 0.010$1	$2.98\pm 0.04$	$-0.035\pm 0.014$
$2.585\pm 0.018$1	$0.676\pm 0.005$1	$17.11\pm 0.17$	$1.618\pm 0.010$	$1.318\pm 0.009$1	$2.67\pm 0.03$	$-0.034\pm 0.013$
$3.075\pm 0.022$1	$0.914\pm 0.006$1	$27.52\pm 0.27$	$2.134\pm 0.014$	$1.793\pm 0.013$1	$4.71\pm 0.06$	$-0.035\pm 0.018$
$3.084\pm 0.022$1	$0.922\pm 0.006$1	$27.84\pm 0.28$	$2.141\pm 0.014$	$1.800\pm 0.013$1	$4.74\pm 0.06$	$-0.044\pm 0.018$
$3.682\pm 0.026$1	$1.212\pm 0.008$1	$43.69\pm 0.43$	$2.864\pm 0.018$	$2.476\pm 0.017$1	$8.64\pm 0.11$	$\phantom{-}0.052\pm 0.024$
$3.659\pm 0.026$1	$1.222\pm 0.009$1	$43.77\pm 0.43$	$2.956\pm 0.019$	$2.564\pm 0.018$1	$9.23\pm 0.12$	$\phantom{-}0.120\pm 0.025$
$\phantom{0}3.69\pm 0.037$2	$\phantom{0}1.19\pm 0.012$2	$42.91\pm 0.61$	$2.889\pm 0.026$	$\phantom{0}2.50\pm 0.025$2	$8.80\pm 0.17$	$\phantom{0-}0.12\pm 0.035$
$\phantom{0}2.56\pm 0.026$2	$\phantom{0}0.69\pm 0.007$2	$17.27\pm 0.24$	$1.588\pm 0.014$	$\phantom{0}1.29\pm 0.013$2	$2.57\pm 0.05$	$\phantom{0}$$-0.09\pm 0.019$
$\phantom{0}2.54\pm 0.025$2	$\phantom{0}0.69\pm 0.007$2	$17.14\pm 0.24$	$1.642\pm 0.015$	$\phantom{0}1.34\pm 0.013$2	$2.75\pm 0.05$	$\phantom{0}$$-0.04\pm 0.019$
$\phantom{0}4.60\pm 0.046$2	$\phantom{0}1.90\pm 0.019$2	$85.6\pm 1.2$	$4.559\pm 0.044$	$\phantom{0}4.07\pm 0.041$2	$22.4\pm 0.4\phantom{0}$	$\phantom{0-}0.27\pm 0.056$
$\phantom{0}2.08\pm 0.021$2	$\phantom{0}0.46\pm 0.005$2	$\phantom{0}9.35\pm 0.13$	$1.157\pm 0.009$	$\phantom{0}0.89\pm 0.009$2	$1.34\pm 0.02$	$\phantom{0}$$-0.03\pm 0.013$
$\phantom{0}1.82\pm 0.018$2	$\phantom{0}0.32\pm 0.003$2	$\phantom{0}5.69\pm 0.11$	$0.978\pm 0.008$	$\phantom{0}0.72\pm 0.007$2	$0.95\pm 0.02$	$\phantom{0-}0.08\pm 0.010$
$4.81\pm 0.05$3	$2.03\pm 0.02$3	$95.7\pm 1.4$	$\phantom{{}^0}5.13\pm 0.13$3	$4.60\pm 0.12\phantom{{}^0}$	$28.5\pm 1.5\phantom{0}$	$\phantom{-}0.54\pm 0.13$
$6.80\pm 0.20$3	$3.60\pm 0.11$3	$240\pm 10$	$\phantom{{}^0}9.83\pm 0.29$3	$9.02\pm 0.27\phantom{{}^0}$	$107\pm 6\phantom{00}$	$\phantom{-}1.82\pm 0.34$
$7.25\pm 0.22$3	$3.93\pm 0.12$3	$279\pm 12$	$\phantom{{}^0}11.3\pm 0.3$3$\phantom{0}$	$10.31\pm 0.30\phantom{0^0}$	$140\pm 8\phantom{00}$	$\phantom{-}2.45\pm 0.38$
$\phantom{0}7.6\pm 0.23$3	$\phantom{0}4.2\pm 0.13$3	$313\pm 13$	$\phantom{{}^0}12.0\pm 0.4$3$\phantom{0}$	$10.9\pm 0.3\phantom{0^0}$	$158\pm 9\phantom{00}$	$\phantom{-}2.5\pm 0.4$
$\phantom{0}7.8\pm 0.23$3	$\phantom{0}4.4\pm 0.13$3	$336\pm 14$	$\phantom{{}^0}11.9\pm 0.4$3$\phantom{0}$	$10.8\pm 0.3\phantom{0^0}$	$155\pm 9\phantom{00}$	$\phantom{-}2.0\pm 0.4$
$\phantom{0}8.4\pm 0.25$3	$\phantom{0}4.9\pm 0.15$3	$403\pm 17$	$\phantom{{}^0}13.9\pm 0.4$3$\phantom{0}$	$12.6\pm 0.4\phantom{0^0}$	$211\pm 13\phantom{0}$	$\phantom{-}2.8\pm 0.5$
$\phantom{0}8.7\pm 0.26$3	$\phantom{0}5.0\pm 0.15$3	$426\pm 18$	$\phantom{{}^0}14.1\pm 0.4$3$\phantom{0}$	$12.8\pm 0.4\phantom{0^0}$	$218\pm 13\phantom{0}$	$\phantom{-}2.8\pm 0.5$
$\phantom{0}9.6\pm 0.29$3	$\phantom{0}5.8\pm 0.17$3	$546\pm 23$	$\phantom{{}^0}16.2\pm 0.5$3$\phantom{0}$	$14.7\pm 0.4\phantom{0^0}$	$286\pm 17\phantom{0}$	$\phantom{-}3.1\pm 0.6$
$10.0\pm 0.3$3$\phantom{0}$	$\phantom{0}6.2\pm 0.19$3	$608\pm 26$	$\phantom{{}^0}18.1\pm 0.5$3$\phantom{0}$	$16.3\pm 0.5\phantom{0^0}$	$356\pm 21\phantom{0}$	$\phantom{-}3.9\pm 0.6$
$10.6\pm 0.3$3$\phantom{0}$	$\phantom{0}6.5\pm 0.20$3	$675\pm 29$	$\phantom{{}^0}18.2\pm 0.5$3$\phantom{0}$	$16.4\pm 0.5\phantom{0^0}$	$360\pm 21\phantom{0}$	$\phantom{-}3.4\pm 0.6$

¹ Values taken from [55]. ² Values taken from [59]. ³ Values taken from [61].

. The initial densities of the bismuth samples are taken as ${\rho }_{00}=9.79$ g/cm${}^3$ for the case in [55], 9.776 g/cm${}^3$ for [59], and 9.8 g/cm${}^3$ for [61]; the initial density of air is taken as ${\rho }_{00\mathrm{air}}=1.2040$ mg/cm${}^3$ [71] for all three cases; $P_0=0.1$ MPa. It should be noted that the values of pressure and particle velocity determined with the use of experimental data [61] in this work differ somewhat from those given in [61]. Apparently, this can be explained by the difference in the parameters of the shock adiabat and the initial density of air taken in [61] and in the present work.

The calculated relationship between the shock velocities in the bismuth samples and in air is shown in Figure 6 in comparison with the experimental points [55,59,61] from Table 1. The agreement of the calculated curve with the data above the velocity $U_{\mathrm{s}}=7.8$ km/s in bismuth within the experimental error can be seen.

It seems interesting to evaluate the deviation from the rule of doubling the velocity of matter during the isentropic expansion of a shock-compressed sample into air [72,73] for bismuth. The calculated dependence of the particle velocity of shock-compressed air ($U_{\mathrm{p}\mathrm{air}}$) minus the double mass velocity of shock-compressed bismuth ($U_{\mathrm{p}}$) on the latter is shown in Figure 7 in comparison with the experimental points [55,59,61] from Table 1. As can be seen in Figure 7, this calculated dependence agrees with the data within the experimental error at particle velocities in bismuth above $U_{\mathrm{p}}=4.4$ km/s.

4. Conclusions

Thus, the proposed EOS model as a function of pressure on the specific volume and specific internal energy is applicable for generalizing the available experimental and theoretical information about the properties of a substance in a wide range of thermodynamic parameters. The newly constructed EOS for the bcc and liquid phases of bismuth is in good agreement with the available experimental data, with the exception of the region of relatively low pressures achieved at low expansion velocities of the shock-compressed material into barriers with low dynamic impedance, where phase transitions (melting, evaporation) occur. The refined parameters of shock waves in air under the experimental conditions on the isentropic expansion of shock-compressed bismuth samples provide additional information that can be applied to the study of phase transitions in this metal. The developed EOS can be effectively used in numerical simulations of various processes in bismuth at high energy densities.