## 1 Introduction  
Rare earth \(4f\) elements, or lanthanides, play a key role in modern day technological applications, such as computer memories and permanent magnets. Similarly, \(5f\) element actinides, being the backbone of nuclear fission technologies for the production of energy, also find applications in many non-power strategic fields, from space exploration to medical diagnostic and treatments. Specifically, the first member of the \(5f\)-series, actinium (Ac), is a potential therapeutic agent for cancer and infectious diseases [1, 2]. Additionally, rare-earth superhydrides have recently been discovered to have near room temperature \(T_{c}\) for superconductors above megabar pressures (1 Mbar = 100 GPa) [3, 4, 5]. As such, understanding and predicting the material behavior of lanthanides is crucial. This can be done using equation of state (EOS) information; however, both \(4f\)-electron lanthanides and \(5f\)-electron actinides present a unique challenge in performing first-principles based ab initio calculations due to their complex electronic and lattice structures. Therefore, experimental EOS measurements are indispensable in providing benchmark data to tightly constrain the ab initio calculations. Research efforts on Ac, as well as some lanthanides, have been hampered by the lack of supply and the high costs of current production methods, especially for Ac, and the radioactivity of proomethium (Pm). In fact, Pm is the only radioactive element among the lanthanides, and the second radioactive element after technetium (Tc) that has stable neighboring elements. Thus, the EOS data on both Pm and Ac are scarce and virtually non-existent. Lutetium (Lu), in contrast, has been the subject of intense studies, mainly because it is the last element in the lanthanide series and has a completely filled \(4f\) shell. However, even with the availability of several EOS studies [6], there is only one published study of the shock compression of Lu, by [PERSON] et al. [7]. Similarly, there is only one published study of the shock compression of both terbium (Tb) and thulium (Tm), by [PERSON] et al. [8]. The shock Hugoniot, both the principal and second-shock one, is a continuous curve which describes the locus of all possible thermodynamic states a substance can exist in behind a shock propagating through it. It is usually given as a projection onto a two-dimensional plane and is described in terms of two coordinates which can be chosen among particle velocity behind the shock front (\(U_{p}\)), shock velocity (\(U_{s}\)), pressure, internal energy (\(E\)), temperature (\(T\)), density (\(\rho\)), and degree of compression (\(\eta\)) called \"compression\" for simplicity. Up to \(P\) of a few hundred GPa, the Hugoniot is known from experiments, and in the vast majority of cases is described by a linear relation \(U_{s}\,=\,C\,+\,B\,U_{p}\), between the two main variables, \(U_{s}\) and \(U_{p}\), in terms of which all other relevant quantities can be expressed via the Rankine-Hugoniot (RH) relations (which represent the conservation of mass, momentum, and energy, respectively):  
\[\eta\equiv\frac{\rho}{\rho_{0}}=\frac{U_{s}}{U_{s}-U_{p}},\quad P=P_{0}+\rho_{ 0}\,U_{s}\,U_{p},\quad E-E_{0}=\frac{1}{2}(P+P_{0})\bigg{(}\frac{1}{\rho_{0}}- \frac{1}{\rho}\bigg{)}, \tag{1}\]  
where the subscript \(0\) indicates the initial (unshocked) state. For the principal Hugoniot, \(P_{0}=0\). Over decades of research, shock Hugoniot measurements have been one of the most important sources of EOS data. The use of planar shock waves to determine the EOS of condensed materials to very high pressure (\(P\)) began in 1955 with the classic papers of [PERSON] and [PERSON] (1955) [9] and [PERSON] et al. (1956) [10]. [PERSON] and [PERSON] described the use of in-contact explosives to determine dynamic pressure-volume relations for metals and compared these to the then available static compression data. [PERSON] et al. described the first polymorphic phase change discovered in a solid via shock waves, namely, iron. Two years later, [PERSON] et al.