## 4 Results  
The five principal Hugoniots of the substances discussed in this work are shown in Figures 3-7, in which the five Hugoniots are compared to both independent theoretical calculations and the available experimental data. In addition to the available experimental data (for Tb, Tm, and Lu), for the purpose of comparison to the new analytic model, we have carried out theoretical Hugoniot calculations using: (i) the relativistic [PERSON]'s function quantum average atom code Tartarus [31; 32] for Pm and Lu; and (ii) the Thomas-Fermi model with corrections [33] for Tb, Tm, and Ac, for which [PERSON]'s orbital-free molecular dynamics (OFMD) code used, e.g., in a study on the transport properties of lithium hydride [34], was modified appropriately. These simulations are similar to those of Ref. [35] for platinum. The \((V,P)\) points that map out the Hugoniot (presented in the \(\eta\)-\(P\) coordinates upon conversion of the \((V,P)\) points into the \((\eta,P)\) ones) are found from the RH relation for internal energy: \(E-E_{0}=P\,(V_{0}-V)/2\). We also used three Hugoniots from the TEFIS database [36; 37] for Tb, Tm, and Ac for the comparison to the new model, as well as to the Thomas-Fermi model with corrections. We note that, although Lu is an example of the application of Tartarus to a real material considered in detail in Ref. [32], the Lu Hugoniot shown in Figure 4 is not presented in [32]. Figure 4: The principal Hugoniot of Tb: our new analytic model vs. the experimental data of Ref. [8] ([PERSON] et al. ), the Tb Hugoniot from TEFIS database (TEFIS), and our theoretical calculations using the Thomas–Fermi model with corrections (Thomas–Fermi Corr.). Figure 3: The principal Hugoniot of Pm: new analytic model vs. theoretical calculations using the relativistic Green’s function quantum average atom code Tartarus [31; 32]. Figure 5: The principal Hugoniot of Tm: our new analytic model vs. the experimental data of Ref. [8] ([PERSON] et al. ), the Tm Hugoniot from TEFIS database (TEFIS), and our theoretical calculations using the Thomas–Fermi model with corrections (Thomas–Fermi Corr.). Figure 6: The principal Hugoniot of Lu: our new analytic model vs. the experimental data of Ref. [7] ([PERSON] et al.) and our theoretical calculations using Tartarus. Similar to \(P=P(\eta)\), \(E\) along the Hugoniot can be calculated using the corresponding RH relation (1) for \(E\),  
\[E\,=\,E_{0}\,+\,\frac{P}{2\,\rho_{0}}\,\bigg{(}1\,-\,\frac{1}{\eta}\bigg{)}, \tag{7}\]  
and Equations (3) for \(P=P(\eta)\) in the corresponding \(U_{p}\) intervals. Figure 8 shows \(E=E(\eta)\) for Ac (we assumed \(E_{0}=0\), for simplicity). Figure 8: Internal energy along the principal Hugoniot for Ac: the new analytic model (curve) vs. our own theoretical calculations using the Thomas–Fermi model with corrections (symbols). Figure 7: The principal Hugoniot of Ac: our new analytic model vs. the Ac Hugoniot from TEFIS database (TEFIS) and our theoretical calculations using the Thomas–Fermi model with corrections (Thomas–Fermi Corr). It is clearly seen that the forms of \(P=P(\eta)\) and \(E=E(\eta)\) are very similar to each other. The form of \(T=T(\eta)\) is also expected to be similar to both \(P=P(\eta)\) and \(E=E(\eta)\). However, the accurate calculation of \(T=T(\eta)\) requires the knowledge of the specific heat along the Hugoniot; hence, this calculation goes well beyond the scope of this work. We can, however, estimate the value of \(T\), based on that of \(E\), and compare it to the corresponding value from the TEFIS tables. For an ideal gas (which the system well above the turnaround point represents), \(E\approx 3/2\,k_{B}T\), at the upper limit of \(E=E(\eta)\,k_{B}T\approx 2/3\,E\). The uppermost TEFIS point in Figure 7 corresponds to \(\eta\approx 5.04\). At this \(\eta\), the TEFIS table of \(T=T(\eta)\) gives \(T=10^{4}\) eV (a table of \(E=E(\eta)\) does not exist), and our value of \(E\) is \(E(\eta=5.04)\approx 3.8\times 10^{4}\) eV/mol; therefore, the above relation gives \(T\approx 2.5\times 10^{4}\) eV.