1. Introduction
Rare earth
elements, or lanthanides, play a key role in modern day technological applications, such as computer memories and permanent magnets. Similarly,
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
-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
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
-electron lanthanides and
-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 promethium (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
shell. However, even with the availability of several EOS studies [
6], there is only one published study of the shock compression of Lu, by Al’tshuler et al. [
7]. Similarly, there is only one published study of the shock compression of both terbium (Tb) and thulium (Tm), by Carter 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
shock velocity
pressure, internal energy
temperature
density
and degree of compression (
) 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
between the two main variables,
and
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):
where the subscript 0 indicates the initial (unshocked) state. For the principal Hugoniot,
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
began in 1955 with the classic papers of Walsh and Christian (1955) [
9] and Bancroft et al. (1956) [
10]. Walsh and Christian 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. Bancroft et al. described the first polymorphic phase change discovered in a solid via shock waves, namely, iron. Two years later, Al’tshuler et al. (1958) [
11] reported the first data for iron to
P of several Mbar, essentially in excess of
P in the center of the Earth. Since that time, the EOSs of virtually hundreds or even thousands of condensed materials have been studied, including elements, compounds, alloys, rocks and minerals, polymers, fluids, and porous media. These studies have employed both conventional and nuclear explosive sources, as well as impactors launched with a range of guns to speeds of order of 10 km/s. Recently, with the establishment of the NIF Gigabar platform at the Lawrence Livermore National Laboratory [
12], pressures of order of 1 Gbar (
GPa, or 100 TPa), deep into the atomic pressure regime, have been routinely achieved in their experiments [
13] (in this regime,
P exceeds
, the so-called atomic pressure required to significantly distort core electron orbitals and estimated as
Mbar, where
is the Hartree energy and
is the Bohr radius).
The lanthanides series includes lanthanum (La) and the 14 following elements in which the
-electron shell gets progressively filled. This shell is half-filled for europium (Eu) and fully filled for ytterbium (Yb), which is responsible for both of them being divalent, in contrast to the remaining lanthanides, which are all trivalent, similar to their transition metal counterparts, scandium (Sc) and yttrium (Y). Because the
-electrons are localized deep in the lanthanide atoms (specifically, in their Xe-cores), they do not participate in metallic bonding [
14], which is dominated by the
,
, and
valence electrons. The typical electron structure of a lanthanide atom is [Xe]
or [Xe]
The dominance of the
-valence electrons in the bonding is responsible for the lanthanides’ phase diagrams to vary slowly across the series so that their relations to one another is clearly seen [
15]. The well established fact about the lanthanides is that, with increasing
virtually all of them undergo the same sequence of phase transformations from one hexagonal polytype to the next; it is characterized by the varying hexagonal layer stacking of the corresponding crystal structure: hexagonal close-packed (hcp, AB) →
-Sm (9R, ABCBCACAB) → double-hcp (dhcp, ABAC) → face-centered cubic (fcc, ABC) → distorted fcc (dfcc) →… Evans et al. [
16] showed that the dfcc phase of praseodymium (Pr) is
another hexagonal polytype with 24 atoms per unit cell, i.e., with a stacking of 24 ABC-layers.
along with two alternative crystal structures, namely, orthorhombic
, and
were identified as dfcc for other lanthanides. Recently, a possible extension of the above phase transition sequence to include post-dfcc phases was suggested in Ref. [
17].
The actinides form a similar series, from Ac to lawrencium (Lr), in which the
-electron shell gets progressively filled. In this respect, a comparison of the phase diagrams of the two series is useful and informative, but in contrast to the lanthanides, the structural systematics for the actinides, in terms of a sequence of phase transformations on increasing
P, is much less obvious for the light actinides, but becomes more obvious for the heavier actinies, starting with americium (Am) [
18]: dhcp → fcc →
-U →… Note that, again, in contrast to the lanthanides,
-U is not one of the hexagonal polytypes.
The principal Hugoniots of lanthanides have been experimentally investigated in several studies, e.g., refs. [
7,
19], but only a few substances at a time. The systematic study of the Hugoniots for the lanthanide series as a whole was carried out in Refs. [
8,
20]. Specifically, 13 lanthanides out of 15 were studied in [
8] (in addition to Sc and Y), and 9 in [
20] (in addition to Sc, Y, and hafnium (Hf)). Both sets of data are generally consistent with each other: the numerical values of the parameters of the least-square linear fits to the corresponding
-
data agree to within ∼15%. Here,
and
are particle and shock velocities, respectively. The unique feature of both sets of data is that for each substance, except cerium (Ce), the adequate description of the data in the whole
region of up to ∼3.6 km/s requires a two-segment fit; that is, the lower and upper portions of the data sets are described by two distinct linear segments such that the combined two-segment graph is continuous, but not smooth, as the slope of the second segment (∼1.2–1.6) is ∼1.5 times higher than that of the first one (∼0.8–1.0). In each case, the change of slope happens at a compression of ∼1.6; the corresponding transition
P increases from ∼20 GPa for the light lanthande members to ∼40 GPa for the heavy ones [
8]. Eu and Yb are two exceptions, for which the transition
P is ∼10 GPa. Additionally, the Eu data show the existence of a two-wave structure in the shock front, indicating a significant volume change at the transition, whereas in most of other cases, the data indicate that the volume change associated with this transition is either zero or very small [
8]. Such
-
behavior is typical of a solid–solid phase transformation or melting. In any event, the
-
slope being ∼50% higher indicates that the compressibility of the emerging phase is changing more rapidly (the value of the
P-derivative of the bulk modulus is higher). Direct evidence of phase transitions and melting along the principle Hugoniot in several lanthanide members using in-situ laser shock diffraction at the Dynamic Compression Sector was recently presented in [
21].
In this work, we derive the analytic forms of the principal Hugoniots of Ac and Pm, which have both never been measured or calculated before, as well as those of Tb, Tm, and Lu, the three least studied of the remaining lanthanides. Indeed, the existing Hugoniot data on each of these three substances can only be found in a single literature source, namely, Ref. [
8] for both Tb and Tm and Ref. [
7] for Lu.
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
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]
with the values of
b, and
a determined virtually for all
Zs (
Z being the atomic number) [
23,
24,
25]; and (iii) the high-
P regime in which the Hugoniot is described by the Debye-Hückel-Johnson (DHJ) model [
26,
27,
28]. The only assumption made was that the principal Hugoniot is goverened by some function
(which is linear at low
P and quadratic at intermediate
its high-
P form was established in [
22]) and that this function is continuous and smooth (the first derivative
is continuous) at all
Then, it follows from the RH relations [
22] that
P, and
E are all continuous and smooth as well. No other assumption, and no additional free parameter except the six mentioned above,
and
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
where
A is an additional 7th parameter which introduces a very small non-linearity and is obtained using the formula
[
22]. Since
and
(see, e.g.,
Table 1 below), the value of
A depends on
Typically, this difference is of the order of 0.1; hence,
s/km. However, if this difference is small, so is
but
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
km/s is
of
km/s at the turnaround point (see
Table 1); typically,
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 ∼100 GPa. In our case, the low-
P–med-
P transition point corresponds to shock velocities of ∼10 km/s (see, e.g.,
Table 1); with ambient density of ∼1–10 g/cm
, corresponding to a pressure of ∼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
1–10 Gbar (
–
GPa).
4. Results
The five principal Hugoniots of the substances discussed in this work are shown in
Figure 3,
Figure 4,
Figure 5,
Figure 6 and
Figure 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 Green’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 Lambert’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
points that map out the Hugoniot (presented in the
-
P coordinates upon conversion of the
points into the
ones) are found from the RH relation for internal energy:
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].
Similar to
E along the Hugoniot can be calculated using the corresponding RH relation (1) for
and Equations (3) for
in the corresponding
intervals.
Figure 8 shows
for Ac (we assumed
for simplicity).
It is clearly seen that the forms of
and
are very similar to each other. The form of
is also expected to be similar to both
and
However, the accurate calculation of
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
based on that of
and compare it to the corresponding value from the TEFIS tables. For an ideal gas (which the system well above the turnaround point represents),
at the upper limit of
The uppermost TEFIS point in
Figure 7 corresponds to
At this
the TEFIS table of
gives
eV (a table of
does not exist), and our value of
E is
eV/mol; therefore, the above relation gives
eV. Thus, our value of
T is about 2.5 times larger than TEFIS’ one, which directly corresponds with the fact that our
P is larger than TEFIS’ one by roughly the same amount, as can be clearly seen in
Figure 7.
5. Discussion
As
Figure 3,
Figure 4,
Figure 5,
Figure 6 and
Figure 7 clearly demonstrate, in each of the five cases, the agreement between the new analytic model and independent data is very good except for the three Hugoniots from the TEFIS database. As a matter of fact, the three TEFIS Hugoniots appear to violate Johnson’s constraint
, which is based on rigorous theoretical grounds. (We have determined that this is the case for the vast majority of TEFIS Hugoniots for other substances.)
It is worthwhile to dwell on Johnson’s constraint in some more detail. In Ref. [
38], Johnson derives the formula for the maximum compression on the princial Hugoniot:
where
As can be clearly seen in
Figure 4 of [
22], this
and the one given by the new analytic model are in very good agreement with each other, which lends further support to both formulations. Johnson’s constraint now follows directly from (8):
and:
We believe, the reason for this behavior of TEFIS Hugoniots is the way these Hugoniots are constructed; specifically, the scheme used in Refs. [
39,
40] for the interpolation between the portion of the Hugoniot below and across the turnaround point described by the TFK model, and that above the turnaround point, which incorporates electron shell effects, such shell effects are clearly seen in
Figure 4,
Figure 5, and
Figure 7 for Tb, Tm, and Ac, respectively. In other words, the violation of Johnson’s theoretical constraint by the TEFIS Hugoniots may be the artifact of the interpolation scheme used for their construction. Otherwise, for
P below and above the turnaround point,
and
agreement between the TEFIS Hugoniots and both the new analytic model and the Thomas–Fermi model with corrections is generally good.
Let us note that in a more realistic case of a shock compression of a substance beyond the corresponding turnaround point, the Hugoniot must be modeled by taking into account the well-known effects of the contribution of both the equilibrium radiation of hot plasma and relativistic effects [
41,
42,
43,
44]. Indeed, at a turnaround point,
–250 km/s (see
Table 1), which constitutes
% of the speed of light, and at the med-
P–high-
P transition point,
km/s, about 0.5% of the speed of light. Hence, as the system enters the high-
P regime, relativistic effects are expected to start manifesting themselves and eventually to dominate the evolution of the system at even higher
Our model can, in principle, be modified to incorporate these effects. For instance, taking into account the radiation-dominated (or the so-called strong shock) regime can be done by replacing equations for
and
describing the high-
P regime with their counterparts stemming from the physics of a photon gas. This goes beyond the scope of the present work, but will be undertaken in one of our subsequent studies.
The new model discussed in this work does not incorporate potential electronic shell effects in the high-
P regime. If present, they manifest themselves in terms of some “irregularities”, that is, changes of the sign of
of the continuous line
over some (small) regions of
such “irregularities” are seen, e.g., in the Tartarus Hugoniots of both Pm and Lu around
–
GPa in
Figure 3 and
Figure 6, respectively, since Tartarus takes electronic shell effects into account explicitly. Similar irregularities are present in the TEFIS Hugoniots, but the corresponding changes of the sign of
are very abrupt. This is because these Hugoniots are constructed the way that causes the violation of Johnson’s theoretical constraint, as discussed above. In contrast to Tartarus, the orbital-free procedure of the OFMD code treats all electrons on an equal footing, albeit approximately, with no distinction between bound and ionized electrons. This is why any shell-ionization related effects are absent in the corresponding Hugoniots, as seen in
Figure 4,
Figure 5, and
Figure 7. Such effects cannot be predicted by the new model, but, if firmly established, they can be added to the model by considering additional region(s) of
P described by the corresponding
functional forms. We plan to undertake such an addition of electronic shell effects to the new model in one of our subsequent studies on this subject.
6. Conclusions
Here is a brief summary of the findings of this work. We have presented the principal Hugoniots of actinium and the lanthanide promethium, which have both never been measured or calculated before, as well as those of terbium, thulium, and lutetium, the three least studied of the remaining lanthanides. We used the analytic framework established in our previous publication [
22]. All five sets of the relevant parameters are summarized in
Table 1. The five principal Hugoniots are compared to the available experimental data (for Tb and Tm only) and independent theoretical calculations in
Figure 3,
Figure 4,
Figure 5,
Figure 6 and
Figure 7. As can be clearly seen in these figures, in each of the five cases, the agreement between the new analytic model and independent data is very good except for the three Hugoniots from the TEFIS database, which all violate Johnson’s theoretical constraint on the maximum compression. Their behavior is analyzed in some detail in our paper. We anticipate that new experimental measurements could shed more light on a potential systematics of the lanthanide Hugoniots in terms of the values of the parameters
C and
the maximum compression
etc. We expect our results to serve as the initial guidance for such experiments.
Our new analytic model of the principal Hugoniot [
22] can be used for the validation of the
P-
V-
T EOS by comparing the Hugoniot produced by the EOS to that given by the model. Additionally, the new model itself can be used as a basis for EOS construction. Indeed, if the Grüneisen parameter along the Hugoniot is available, e.g., using the approach discussed in this work, then it can be used in the Mie-Grüneisen-type EOS
[
45], where the subscript “H” implies that the corresponding variable is in shock-compression conditions. This EOS can then be brought in direct correspondence to the more familiar Mie-Grüneisen (M-G) EOS,
where the subscript “c” implies the cold
conditions, since there exists a direct algebraic connection between
(of this work) and
of the M-G EOS [
45].
The analytic model developed in our previous study [
22] and applied to five substances in this work can be used to calculate of the Hugoniots of other substances, specifically, other lanthanides and/or actinides. In this respect, the analytic knowledge of the regimes of the Hugoniot past the turnaround point is very important. In a very recent publication [
46], a team of astronomers report a detailed study of a pair of shock waves produced by a collision of two clusters of galaxies that occurred roughly a billion years ago. The shocks that are associated with cluster mergers are known as radio relics and they can be used to probe the properties of the intergalactic space within the cluster, known as the intracluster medium, as well as intracluster dynamics. The study focused on a particular cluster called Abell 3667, which at least 550 galaxies are associated with and which is still coming together. It was concluded that the shock waves are propagating through it at velocities of ∼1500 km/s, which are 5–6 times larger than the velocities corresponding to turnaround points [
22],
250–300 km/s (see
Table 1). Moreover, at such shock velocities, many elements, especially the low-
Z ones, will be under the conditions that are beyond the validity of Kalitkin’s parabolic representation (5), i.e., in the high-
P regime considered in this work. Hence, to predict the properties of the intracluster medium and to describe intracluster dynamics, an analytic model of the principal Hugoniot in the high-
P regime is a must. Once the analytic formulas describing the high-
P regime are available (the last lines of both systems of Equations (2) and (3)), the basic mechanical and thermodynamic properties of a material under intergalactic shock, such as the bulk modulus, the Grüneisen parameter, energy, temperature, etc., along the principal Hugoniot can be derived from the RH relations and Equation (16) using the new model and some additional assumptions on the Grüneisen