## 2 Principal Hugoniot in a Wide Pressure Range  
We will construct the analytic models of the principal Hugoniots of the five substances, namely, Pm, Tb, Tm, Lu, and Ac, using the analytic framework established in our previous publication [22]. In this framework, a wide \(P\) range is divided into three regimes and the Hugoniot is constructed in each of these regimes and then interpolated smoothly between them. These regimes are: (i) the low-\(P\) regime in which the Hugoniot is described by \(U_{s}\,=\,C\,+\,B\,U_{p}\), where the values of \(C\) and \(B\) come from the experiment; (ii) the intermediate-\(P\) regime (discussed in more detail below) where the Hugoniot is described by the Thomas-Fermi-Kalitkin (TFK) model [23; 24]\(U_{s}\,=\,c\,+\,b\,U_{p}\,+\,a\,U_{p}^{2}\), with the values of \(c\), \(b\), and \(a\) determined virtually for all \(Z\)s (\(Z\) being the atomic number) [23; 24; 25]; and (iii) the high-\(P\) regime in which the Hugoniot is described by the Debye-Huckel-Johnson (DHJ) model [26; 27; 28]. The only assumption made was that the principal Hugoniot is goverened by some function \(U_{s}\,=\,U_{s}(U_{p})\) (which is linear at low \(P\) and quadratic at intermediate \(P\); its high-\(P\) form was established in [22]) and that this function is continuous and smooth (the first derivative \(dU_{s}/dU_{p}\) is continuous) at all \(U_{p}\). Then, it follows from the RH relations [22] that \(\eta\), \(P\), and \(E\) are all continuous and smooth as well. No other assumption, and no additional free parameter except the six mentioned above, \(C\), \(B\), \(c\), \(b\), \(a\), and \(Z_{\
u}\), are required for the construction of the analytic model of the principal Hugoniot. In order to match the next regime, the linear form of the low-\(P\) regime is modified into \(U_{s}\,=\,C\,+\,B\,U_{p}\,+\,A\,U_{p}^{2}\), where \(A\) is an additional 7 th parameter which introduces a very small non-linearity and is obtained using the formula \(A\,=\,a\,-\,(B\,-\,b)^{2}/(4\,(c\,-\,C))\)[22]. Since \(a=\mathcal{O}(10^{-5}\,\mathrm{s/km})\) and \(c-C=\mathcal{O}(1\,\mathrm{km/s})\) (see, e.g., Table 1 below), the value of \(A\) depends on \(B-b\). Typically, this difference is of the order of 0.1; hence, \(A\sim 10^{-2}\) s/km. However, if this difference is small, so is \(A\), but \(U_{p}^{*}\) in Equation (1), which determines the low-\(P\)-med-\(P\) transition point, is large, thus pushing the upper boundary of the low-\(P\) regime closer to the turnaround point. This is, e.g., the case of thulium considered in what follows for which \(U_{p}^{*}\approx 160.5\) km/s is \(\sim 2/3\) of \(U_{p}^{\mathrm{max}}\approx 233.4\) km/s at the turnaround point (see Table 1); typically, \(U_{p}^{*}\sim 0.1\,U_{p}^{\mathrm{max}}\).  
The names of the three \(P\) regimes of the principal Hugoniot may sound confusing and may not correspond to those adapted in high-pressure research in general and phase diagram and EOS studies in particular. Specifically, it is generally adopted that high-\(P\) corresponds to pressures in excess of \(\sim\)100 GPa. In our case, the low-\(P\)-med-\(P\) transition point corresponds to shock velocities of \(\sim\)10 km/s (see, e.g., Table 1); with ambient density of \(\sim\)1-10 g/cm\({}^{3}\), corresponding to a pressure of \(\sim\)100-1000 GPa. Thus, our med-\(P\) regime is analogous to the more familiar high-\(P\) range of EOS studies. Our med-\(P\)-high-\(P\) transition point corresponds to \(P\sim\)1-10 Gbar (10\({}^{5}\)-10\({}^{6}\) GPa).