Ab Initio Phase Diagram of Chromium to 2.5 TPa

Abstract

Chromium possesses remarkable physical properties such as hardness and corrosion resistance. Chromium is also a very important geophysical material as it is assumed that lighter Cr isotopes were dissolved in the Earth’s molten core during the planet’s formation, which makes Cr one of the main constituents of the Earth’s core. Unfortunately, Cr has remained one of the least studied $3d$ transition metals. In a very recent combined experimental and theoretical study (Anzellini et al., Scientific Reports, 2022), the equation of state and melting curve of chromium were studied to 150 GPa, and it was determined that the ambient body-centered cubic (bcc) phase of crystalline Cr remains stable in the whole pressure range considered. However, the importance of the knowledge of the physical properties of Cr, specifically its phase diagram, necessitates further study of Cr to higher pressure. In this work, using a suite of ab initio quantum molecular dynamics (QMD) simulations based on the Z methodology which combines both direct Z method for the simulation of melting curves and inverse Z method for the calculation of solid–solid phase transition boundaries, we obtain the theoretical phase diagram of Cr to 2.5 TPa. We calculate the melting curves of the two solid phases that are present on its phase diagram, namely, the lower-pressure bcc and the higher-pressure hexagonal close-packed (hcp) ones, and obtain the equation for the bcc-hcp solid–solid phase transition boundary. We also obtain the thermal equations of state of both bcc-Cr and hcp-Cr, which are in excellent agreement with both experimental data and QMD simulations. We argue that 2180 K as the value of the ambient melting point of Cr which is offered by several public web resources (“Wikipedia,” “WebElements,” “It’s Elemental,” etc.) is most likely incorrect and should be replaced with 2135 K, found in most experimental studies as well as in the present theoretical work.

1. Introduction

Chromium is the fourth $3d$ transition metal found in the periodic table. It is the first element known to violate one of the aufbau (building-up) principles, specifically the Madelung energy ordering rule, as its electron configuration is [Ar]3d${}^5$4s${}^1$ rather than [Ar]3d${}^4$4s${}^2$ predicted by this rule. Chromium is extremely hard, being the third hardest element behind carbon (diamond) and boron. Chromium is also valued for its high corrosion resistance which finds its application in stainless steel production. Chromium-based catalysts are widely used for industrial production of polyethylene and 1-hexene.

Chromium is also an important geophysical material. It is the Earth’s 10th most abundant element, behind iron, oxygen, silicon, magnesium, sulfur, etc. Comparison of the concentrations of elements found in the meteorites that are of the same age as the Earth (4.5 billion years) to those found in the Earth’s crust sheds light on how the Earth was formed. It appears that these meteorites contain more chromium than the Earth [1], which constitutes the so-called missing chromium problem. The resolution of this problem was found in [1]. Using Cr isotope measurements from both the meteorites and the Earth’s crust, it was shown [1] that lighter Cr isotopes sank into the Earth’s core during the planet’s formation. In addition, first-principles-based computer simulations revealed [1] that this process occurred in the early stages of the Earth’s formation when smaller bodies were coalescing to form the contemporary Earth.

Thus, the knowledge of the physical properties of chromium is very important for materials science and condensed matter physics as well as for geophysics, since being one of the main constituents of the Earth’s core, its physical properties are directly related to the Earth’s thermal history. Unfortunately, chromium has remained one of the least studied $3d$ transition metals. Most of the information on its physical properties is available at low pressure $\left(P\right)$ only. Its crystal structure at ambient conditions is body-centered cubic (bcc). With decreasing temperature $\left(T\right)$, bcc is modified by two first-order phase transitions: to antiferromagnetic (AF) phase with orthorhombically distorted bcc at the Néel T ($T_{\mathrm{N}}$) of 311 K, and to AF phase with tetragonally distorted bcc at the spin-flip T of 123 K [2]. In both phases, the bcc distortions are too small to be detected by X-ray diffraction. The physical reasons for the ground state of Cr being AF have been discussed in several papers; see, e.g., Ref. [3], which provides arguments for bcc-Fe being ferromagnetic (FM) but both bcc-Mn and bcc-Cr being AF. Above 311 K, Cr is paramagnetic (PM).

With increasing P, chromium is expected to exhibit a polymorphism, similar to both molybdenum and tungsten, its partners in Period 6B of the periodic table (PT), the high-P crystal structure of which is double-hexagonal-close-packed (double-hcp, or dhcp) [4]. Indeed, a solid–solid (s-s) phase transition into hcp is predicted theoretically for Cr, but the values of the transition P that can be found in the literature span an interval of 7–12.5 Mbar [5]. The melting curve of chromium was measured experimemtally in [6], along with that of vanadium and other transition metals. It appeared to be the so-called “low” melting curve which virtually flattens out at P∼1 Mbar and for which the value of T at the melting point $\left(T_m\right)$ on the shock Hugoniot is much lower than that extracted from the shock compression data. Moreover, it appeared to virtually coincide with the melting curve of V. The latter was very thoroughly re-measured recently [7] and shown to be actually the so-called “high” melting curve for which the value of the shock $T_m$ is in agreement with the shock data. Thus, the melting curve of Cr of Ref. [6] is most likely erroneous, just like that of V, and therefore needs to be corrected via being carefully re-measured experimentally and/or calculated theoretically.

Even the value of the ambient $T_m$ of Cr is ambiguous. $T_m$ = 2136 K can be found in Young’s book [8]. Very accurate experimental measurements of Refs. [9,10] give, respectively, $T_m=2136$ and $2131\pm 5$ K. According to [11], the experimental values of the ambient $T_m$ of Cr that can be found in the literature span an interval of $2085\left(10\right)$- $2180\left(20\right),$ i.e., ∼$2130\pm 50.$ The value $T_m$ = 2180 K which is offered by several public web resources (“Wikipedia,” “WebElements,” “It’s Elemental,” etc.) comes from Ref. [12] and is most likely incorrect. Specifically, in [12] this value was assessed, along with that of the melting enthalpy $\Delta H_m$ (latent heat of melting), based on some literature data. The latter turns out to be $\Delta H_m=21$ kJ/mol [12]. The value of $\Delta H_m$ from the experiment of Ref. [9] is 29,674(574) J/mol, and that calculated in [13] based on the accurate literature data is ≈28,000 J/mol. Thus, $\Delta H_m$∼$(29\pm 1)$ kJ/mol, so that the assessment of Ref. [12] leading to 21 kJ/mol is very likely incorrect, and so is their value of $T_m$ of 2180 K. In the experiment of Ref. [11], Cr melts at T between 2120 and 2150 K, i.e., at $2135\pm 15$ K. Our own theoretical results on the melting of Cr discussed below lead to $T_m$ = 2135 K.

To address all of the above points of concern related to the phase diagram of chromium, in this work we carry out its extensive theoretical study to a pressure of 2.5 TPa (25 Mbar). This work continues a previous combined experimental and theoretical study of Cr in which its equation of state and melting curve were determined to a pressure of 150 GPa [14]. It is necessitated by the importance of the knowledge of the physical properties of Cr, specifically its phase diagram, to higher $P.$

2. Methods

Our theoretical calculations of the equation of state and melting curve of Cr are based on density-functional theory (DFT) with the projector-augmented-wave (PAW) [15] implementation and the generalized gradient approximation (GGA) for exchange-correlation energy, in the form known as Perdew-Burke-Ernzerhof (PBE) [16]. For these calculations, Vienna Ab initio Simulation Package (VASP) [17] was used. The reason for choosing GGA instead of another implementation of DFT, namely, local density approximation (LDA), is that it has been known that LDA does not accurately describe the properties of 3d transition metals, specifically chromium, for which its predictions are in drastic disagreement with experiment [18,19], and iron, for which it produces a wrong ground state: AF or nonmagnetic (NM) bcc [20,21] or NM-hcp [22,23,24] instead of FM-bcc. LDA also fails in assessing the strength of the magnetovolume effect. On the other hand, GGA correctly predicts the structural and magnetic phase diagrams [25].

Cr was modeled using the core-valence representation [${}^{12}$Mg]3p${}^6$3d${}^5$4s${}^1,$ i.e., we assign the 12 outermost electrons to the valence. The valence electrons are represented with a plane-wave basis set with a cutoff energy of 350 eV, while the core electrons are represented by projector augmented-wave (PAW) pseudopotential (PP). The core radius (RCUT) for this PP is 2.3 a.u., or 1.22 Å. Since numerical errors in the VASP calculations remain almost negligible until the nearest neighbor distance reaches 2 × RCUT/$(1.25\pm 0.05),$ with this PP one can study systems with densities up to ∼$22.5$ g/cm${}^3,$ at least at low $T.$ With our EOS of Cr, this corresponds to P∼$2.5$ TPa.

Cold ($T=0$) EOSs and enthalpies were calcualted using unit cells with very dense k-point meshes for high accuracy. Specifically, for bcc-Cr and hcp-Cr, k-point meshes of $50\times 50\times 50$ and $50\times 50\times 30$ were used. In all the non-cubic cases, we first relaxed the structure to determine its unit cell parameters at each volume. The AF bcc-Cr was studied in the framework of fully unconstrained noncollinear magnetism developed in Ref. [25]. In all the calculations including the simulations of the bcc-Cr and hcp-Cr melting curves and the inverse-Z solidification simulations discussed below, periodic boundary conditions were used.

The theoretical melting curve of chromium (as the combination of those of both bcc-Cr and hcp-Cr) was calculated via ab initio quantum molecular dynamics (QMD) simulations using the Z method implemented with VASP, which is described in detail in Refs. [26,27,28]. We used a 432-atom ($6\times 6\times 6$) bcc-Cr supercell and a 392-atom ($7\times 7\times 4$) hcp-Cr supercell with a single $\mathsf{\Gamma }$-point in each case. This was sufficient to ensure full energy convergence (to $\stackrel{<}{\sim }$1 meV/atom) which was checked for each simulation. We obtained a total of 12 melting points (six for bcc-Cr and six for hcp-Cr), and for each of them, 10 $NVE$ (fixed total number of atoms N, system volume V, and total energy E; V corresponds to one of the six densities from Table 1) runs of 10,000–20,000 time steps of 1.0 fs each. For bcc-Cr, increments of the initial T were 125 K for the 1st, 250 K for the 2nd, 375 K for the 3rd and 4th, and 500 K for the 5th and 6th $T_m.$ For hcp-Cr, those increments were also 125 K for the 1st, 250 K for the 2nd, 375 K for the 3rd, 500 K for the 4th, and 625 K for the 5th and 6th $T_m$. The corresponding error in $T_m$ is half of the increment of the initial T [28] so that it does not exceed ∼4% for each of the six bcc-Cr $T_m$s and ∼5% for each of the six hcp-Cr $T_m$s. The errors in P are negligibly small: $\stackrel{<}{\sim }$$0.5$ GPa for the first point, and 1–2 GPa for the remaining five in either case. Hence, our melting results on both bcc-Cr and hcp-Cr are very accurate.

Examples of our Z method melting simulations can be found in Ref. [14] (Figures 3 and 4 of [14]).

For inverse-Z solidification simulations described in detail in Ref. [27], we used a computational cell of 512 atoms prepared by melting a $8\times 8\times 8$ solid simple cubic (sc) supercell, which would eliminate any bias towards solidification into bcc or any other solid structure (fcc, hcp, dhcp, etc.). We used sc unit cells of 1.810, 1.770, 1.735 and 1.700 Å; the corresponding densities are (in g/cm${}^3$), respectively, 14.561, 15.570, 16.532 and 17.574, which corresponds to P of ∼650, 800, 1000 and 1200 GPa. We carried out $NVT$ simulations using the Nosé–Hoover thermostat with a timestep of 1 fs, with the initial T increment of 2500 K. Complete solidification typically required from 15 to 25 ps, or 15,000–25,000 timesteps.

3. Results

3.1. Cold ($T=0$) Enthalpies of Different Solid Structures of Cr

Our calculated cold enthalpies of three other solid structures of Cr in relation to that of bcc-Cr assumed to be identically zero are shown in Figure 1. We note that the enthalpy of bcc-Cr combines its AF and NM counterparts, below and above ∼20 GPa, respectively. It is seen that, with inceasing P, three other solid structures become thermodynamically more stable than bcc, but it is hcp that does it first, at a pressure of 709.7 GPa. Hence, in contrast to both Mo and W (each having dhcp as a high-P solid structure), in the case of Cr, the s-s phase transition boundary is the bcc-hcp one, and its starting point is $(P,T)\approx (710,0).$ Pressure as high as this is nowadays achievable in a diamond anvil cell [29]; thus, our prediction can be tested experimentally.

3.2. Equations of State of bcc-Cr and hcp-Cr

The $(P=0,T=0)$ ground state corresponds to a bcc-Cr density of 7.229 g/cm${}^3$ (a lattice constant of 2.88 Å or an atomic volume of 11.944 Å${}^3$), or a molar volume of 7.193 cm${}^3$, in excellent agreement with 7.2 cm${}^3$ from experiment [30]. At this density, the NM bcc-Cr is higher in energy by ∼20 meV/atom, i.e., the ground state of bcc-Cr is correctly predicted to be AF. However, the calculated magnetic moment per atom (MMA), ∼$1.1$ ${\mu }_{\mathrm{B}}$, is somewhat larger than the experimental 0.62 ${\mu }_{\mathrm{B}}$ [31], which is typical as regards a discrepancy between theory and experiment for this type of theoretical calculation [31,32]. MMA depends almost linearly on the bcc-Cr lattice constant, $a:$$\mu \left(a\right)\approx 1.1+14(a-2.88),$ and turns zero at a density of 7.87 g/cm${}^3$ (or $a\approx 2.8$ Å); the corresponding P is $\sim 22$ GPa. This P is higher than ≈10 GPa (or $a\approx 2.84$ Å) where the experimental Néel T turns zero as a function of P (or a) [33] but is consistent with another theoretical work [32] which shows that the calculated MMA is suppressed considerably at a pressure of $\sim 20$ GPa. At $P\stackrel{>}{\sim }20$ GPa, the EOS of AF bcc-Cr merges into that of NM bcc-Cr, so that the two EOSs become virtually identical. Both AF bcc-Cr below 20 GPa and NM bcc-Cr above 20 GPa combined are described by the third-order Birch–Murnaghan EOS (here $\rho$ stands for density, in g/cm${}^3,$ and B and $B^{\prime }$ for bulk modulus, in GPa, and its pressure derivative, and the subscript 0 implies $(P=0,T=0))$.

$$P\left(\rho \right)=\frac{3}{2}{B}_0\left({\eta }^{7/3}-{\eta }^{5/3}\right)\left[1+\frac{3}{4}(B_0^{\prime }-4)\left({\eta }^{2/3}-1\right)\right],$$

(1)

where $\eta =\rho /{\rho }_0$ and ${\rho }_0=7.229,B_0=215.2,B_0^{\prime }=4.1.$ This EOS is expected to be reliable to ∼$2.5$ TPa as discussed above. In Figure 2, we compare our theoretical results on the cold EOS of bcc-Cr to the most recent experimental data by Anzellini et al. [14] and to the EOS obtained from the shock Hugoniot data by McQueen et al. [34]. It is clearly seen that agreement between theory and the most recent experiment is excellent.

We note that the above EOS does not explicitly contain PM-Cr. However, we did calculate the EOS of PM bcc-Cr using the GGA + U scheme with spin-orbit coupling developed for the calculation of the properties of paramagnetic materials [35]. Specifically, we used the Dudarev approach [36] in which only the difference $U-J$ is meaningful rather than the values of the two parameters separately. We adopted $U-J=4.5$ eV from [37]. We found that the EOSs of both PM bcc-Cr and NM bcc-Cr virtually coincide, because the difference in the two values of P at the same density decreases with increasing density, and eventually becomes negligibly small. Furthermore, we calculated two melting points of PM bcc-Cr (the calculation of the melting curve is discussed above). Although the corresponding values of P and T are slightly different from those of NM bcc-Cr having the same densities, these two melting points lie on the melting curve of NM bcc-Cr, Equation (3) below. Thus, the melting curves of both PM-Cr and NM-Cr are virtually identical. Since their EOSs are virtually identical, too, for the purpose of the present study there is no difference in the choice of the structure of bcc-Cr as either paramagnetic or nonmagnetic. For the calculation of the melting curve of Cr presented in what follows, the nonmagnetic structure was chosen.

Cold EOS of hcp-Cr is described by the BM3 form, Equation (1), with

$${\rho }_0=7.244{\mathrm{g}/\mathrm{cm}}^3,B_0=198.7\mathrm{GPa},B_0^{\prime }=4.1.$$

(2)

The P dependence of the $c/a$ ratio for the lattice constants of the hcp-Cr unit cell is shown in Figure 3. Above 1000 GPa, the hcp-Cr structure is virtually ideal ($c/a=2\sqrt{2/3}\approx 1.633$). At the transition P of 710 GPa, the two density values predicted by these EOSs are, respectively (in g/cm${}^3)$, 14.967 for bcc-Cr and and 15.348 for hcp-Cr; hence the bcc-hcp transition corresponds to a small volume change of ∼2.5%.

3.3. Ab Initio Melting Curve of bcc-Cr

The six melting points of bcc-Cr that we obtained are listed in Table 1.

Table 1. The six ab initio melting points of bcc-Cr, $(P_m,T_m\pm \Delta T_m),$ obtained from the Z method implemented with VASP.

Lattice Constant (Å)	Density (g/cm${}^3$)	${\mathit{P}}_{\mathit{m}}$ (GPa)	${\mathit{T}}_{\mathit{m}}$ (K)	$\mathbf{\Delta }{\mathit{T}}_{\mathit{m}}$ (K)
3.0	6.3957	−5.2	1950	62.5
2.8	7.8665	47.9	3330	125.0
2.65	9.2793	124	4460	187.5
2.5	11.052	266	5840	187.5
2.35	13.306	526	7720	250.0
2.25	15.160	815	9310	250.0

Table 1. The six ab initio melting points of bcc-Cr, $(P_m,T_m\pm \Delta T_m),$ obtained from the Z method implemented with VASP.

Lattice Constant (Å)	Density (g/cm${}^3$)	${\mathit{P}}_{\mathit{m}}$ (GPa)	${\mathit{T}}_{\mathit{m}}$ (K)	$\mathbf{\Delta }{\mathit{T}}_{\mathit{m}}$ (K)
3.0	6.3957	−5.2	1950	62.5
2.8	7.8665	47.9	3330	125.0
2.65	9.2793	124	4460	187.5
2.5	11.052	266	5840	187.5
2.35	13.306	526	7720	250.0
2.25	15.160	815	9310	250.0

Interpolating between the corresponding six points in Table 1 leads to $T_m=T_m\left(\rho \right)$, such that at the experimental ambient melting density of solid Cr of ${\rho }_S=6.575$ g/cm${}^3$ [10] $T_m=2134.9$ K. Hence, in our simulations the ambient melting point is ≈2135 K. Fitting the six melting points in Table 1 by the Simon–Glatzel functional form with $T_m\left(0\right)=2135$ K fixed gives

$$T_m\left(P\right)=2135{\left(1+\frac{P}{25.7}\right)}^{0.42},$$

(3)

which is our theoretical melting curve of bcc-Cr. It is supported by the most recent experimental melting data [14]. Its initial slope, $d{T}_m{\left(P\right)/dP|}_{P=0},$ is 34.9 K/GPa. Let us compare this value to that given by the Clausius–Clapeyron (CC) formula $d{T}_m{\left(P\right)/dP|}_{P=0}=T_m\left(0\right)·\Delta V_m/\Delta H_m$ using the corresponding experimental input. According to [10], the ambient melting density of liquid Cr is ${\rho }_L=6.170$ g/cm${}^3.$ This, along with ${\rho }_S=6.575$ g/cm${}^3$, leads to volume change at melt $\Delta V_m=0.519$ cm${}^3$/mol. According to another set of thermal expansion data on Cr [38], ${\rho }_L=6.156$ g/cm${}^3$ and ${\rho }_S=6.509$ g/cm${}^3,$ so $\Delta V_m=0.458$ cm${}^3$/mol. To accommodate both data sets, we take $\Delta V_m=0.49\pm 0.03$ cm${}^3$/mol. With the latent heat of melting of $\Delta H_m=29\pm 1$ kJ/mol (see above), the CC formula gives $d{T}_m{\left(P\right)/dP|}_{P=0}=36.1\pm 2.5,$ which our theoretical value is in good agreement with. Another theoretical value of the initial slope, 33.5 K/GPa [39], is also consistent with the CC fomula. We note that the use of $\Delta H_m=21$ kJ/mol from [12] would result in the value of $d{T}_m{\left(P\right)/dP|}_{P=0}$ being about 50% higher and would contradict the experimental data. This strongly suggests that the assessment of Ref. [12] is incorrect, and so is the value of $T_m\left(0\right)$ of 2180 K proposed in [12].

3.4. Ab Initio Melting Curve of hcp-Cr

The six melting points of hcp-Cr that we obtained are listed in

Table 2. The six ab initio melting points of hcp-Cr, $(P_m,T_m\pm \Delta T_m),$ obtained from the Z method implemented with VASP.

a (Å)	c (Å)	Density (g/cm${}^3$)	${\mathit{P}}_{\mathit{m}}$ (GPa)	${\mathit{T}}_{\mathit{m}}$ (K)	$\mathbf{\Delta }{\mathit{T}}_{\mathit{m}}$ (K)
2.6053	4.7355	6.2036	1.7	1270	62.5
2.3593	4.0556	8.8329	129	3710	125.0
2.2125	3.5521	11.468	347	5860	187.5
2.0737	3.2715	14.174	692	8170	250.0
1.9630	3.1422	16.468	1093	10190	312.5
1.8565	2.9999	19.285	1744	12890	312.5

.

The best fit to these six hcp-Cr points gives the melting curve of hcp-Cr in the Simon–Glatzel form ($T_m$ in K, P in GPa):

$$T_m\left(P\right)=1205{\left(1+\frac{P}{15.4}\right)}^{0.50}.$$

(4)

Figure 4 and Figure 5 demonstrate the time evolution of T and $P,$ respectively, in the Z-method runs of the hcp-Cr melting point (P in GPa, T in K) (1093, 10,190), the fifth of the six $T_m$s in Table 2. They are similar to Figures 3 and 4 of Ref. [14] for the bcc-Cr melting point (124, 4460), the third of the six $T_m$s in Table 1.

The other two solid structures that we considered, namely, fcc-Cr and dhcp-Cr, melt below bcc at lower P and below hcp at higher $P.$

The melting curves of bcc-Cr and hcp-Cr cross each other at (P in GPa, T in K) $(P,T)=(1370.9,11433)$ which is the bcc-hcp-liquid triple point. The choice of $T\left(P\right)=a{(710-P)}^b,0<b<1$ as a functional form for the bcc-hcp solid–solid phase transition boundary leads to:

$$T\left(P\right)=33.1{(P-710)}^{0.90},$$

(5)

which crosses the triple point and lies within the bounds imposed by the solidification simulations using the inverse Z method. We discuss these inverse Z simulations in the following section. Note that the value $b=0.9$ for Cr is exactly the same as for both of its PT partners Mo and W [4].

Although the rigorous derivation of the thermal equations of state of bcc-Cr and hcp-Cr goes beyond the scope of this work, we note that the finite-T counterparts of the above two EOSs can be written approximately as $P(\rho ,T)=P\left(\rho \right)+\alpha T,$ where ${\alpha }_{\mathrm{bcc}}=7.9$ · ${10}^{-3}$ and ${\alpha }_{\mathrm{hcp}}=1.9$ · ${10}^{-2}.$ The resulting “approximate” thermal EOSs turn out to be quite accurate. For example, for the six bcc-Cr melting points in Table 1, the corresponding thermal EOS gives pressures of −5.11, 47.9, 125.1, 265.6, 525.7 and 815.5, which are basically identical to those in the third column of Table 1. For the six hcp-Cr melting points in Table 2, the six P values are 1.693, 129.7, 347.2, 691.9, 1094.2 and 1745.4, in excellent agreement with those in the fourth column of Table 2.

3.5. Inverse-Z Solidification Simulations of Liquid Cr

To constrain the location of the bcc-hcp solid–solid phase boundary on the P-T plane between the points $(P=710,T=0)$ and $(P=1371,T=11430),$ we carried out two sets of independent inverse Z runs to solidify liquid Cr and to confirm that liquid Cr solidifies into bcc on one side of this boundary and into hcp (or any other solid structure) on the other side, such that the location of this phase boundary may be constrained.

The inverse Z runs indicate that liquid Cr only solidifies into bcc at ∼650 GPa in the whole temperature range from 0 to essentially the corresponding $T_m$. However, at ∼800–1200 GPa it solidifies into bcc above the transition boundary in Figure 6, while below this boundary it solidifies into another solid structure. The radial distribution functions (RDFs) of the final solid states are typical of hexagonal polytypes (red bullets in Figure 6 in the 1200–1400 GPa range), but because they are noisy it is difficult to associate the structure of the final solid state with any of the polytypes (hcp, dhcp, 9R, etc.). To make these RDFs more discriminating, we fast-quench them to low $T.$ Then, by comparing them to the RDFs of fcc, hcp, dhcp, 9R, etc., we conclude that hcp is the closest strucure to those that liquid Cr solidies into below the transition boundary.

The RDFs of the solidified states at ∼1000 GPa above the transition boundary are shown in Figure 7, and of those solidified below the transition boundary in Figure 8. The 6000 K state virtually lies on the boundary. We tentatively assign it to bcc, because it definitely has features of bcc (RDF peaks at $R\sim 50$, 60 and 80, etc.). At the same time, it certainly has some features that are both uncharacteristic of bcc (e.g., the disappearance of the bcc peaks at $R\sim 115$ and 160, etc.) and characteristic of hcp (peaks at $R\sim 120,$ 135, 150, etc.). Most likely, this 6000 K state is bcc with some admixture of hcp.

4. Discussion

The theoretical results described in the previous sections can now be combined into the theoretical phase diagram of chromium. This phase diagram, to 2.5 TPa, is shown in Figure 6. It includes the two, bcc and hcp, melting curves, the bcc-hcp solid–solid phase transition boundary, and the results of the solidification of liquid Cr into final states of either solid bcc or solid hcp, using the inverse Z method.

We note that this is the phase diagram of paramagnetic (nonmagnetic) Cr. The magnetic part takes its very tiny portion in the lower left corner and represents a triangle bounded by $P=0,$ $T=0$ and $T_{\mathrm{N}}\left(P\right)$, which starts at $(P=0,T=311$ K) and terminates at $(P\sim 20$ GPa, T=0).

Our theoretical study of the phase diagram of chromium now concludes with a brief discussion of melting on its shock Hugoniot. According to calculations by McQueen et al. ([34], p. 541), the theoretical shock Hugoniot of Cr in the P-T coordinates is given by

$$T_{\mathrm{H}}\left(P\right)=293+0.167P^{1.89}.$$

(6)

It crosses the theoretical melting curve of bcc-Cr, Equation (3), at $(P,T)=(244$ GPa, 5730 K), which is the Hugoniot melting point of Cr. It is consistent with the Hugoniot melting point of V, the nearest neighbor of Cr on the left in PT, namely, (225 GPa, 5690 K) [7], as well as with that of Fe, one of the neighbors of Cr on the right in PT, namely, $(242.5\pm 19.5$ GPa, $5600\pm 850$ K) [40]. The Hugoniot melting point of Mn, the nearest neighbor of Cr on the right in PT, is yet to be determined.

The calculated phase diagram of Cr will be useful for geophysics, specifically for modeling the thermal history of the Earth. At pressures corresponding to the Earth’s core ($P>130$ GPa), the melting curve of Cr is above the geotherm [41] determined from the PREM model [42]. Thus, at the conditions of the Earth’s core, Cr is expected to be solid.

Theoretical ab initio calculations of Ref. [4] and the present work demonstrate that the phase digrams of the three substaces that form Group 6B, i.e., Cr, Mo and W, are topologically similar (look-alike) and have one of the hexagonal polytypes, namely, hcp for Cr and dhcp for both Mo and W, as their high-$PT$ equilibrium solid structure. Additional support for the existence of a hexagonal polytype as a high-$PT$ structure of Group 6B substances comes from Ref. [43]. With decreasing volume (increasing P), the electron wave functions of Cr, Mo and W become more spherically symmetric, which favors a higher atomic coordination number than 8 (of bcc) and therefore a more symmetric structure than bcc. Since the highest possible coordination number is 12 (which comes from the solution of the famous 3D-sphere-packing problem), corresponding to all of the hexagonal polytypes, namely, hcp (layer stacking sequence AB), fcc (ABC), dhcp (ABAC), triple-hcp (ABCACB), etc., at high P, each of the three Group 6B substances is expected to have one of the hexagonal polytypes as equilibrium solid structure. In contrast to Cr, Mo and W, the study of Ref. [44] shows that the electron wave functions of Group 5B substances V, Nb and Ta behave in exactly the opposite way: with decreasing volume, they become less spherically symmetric, which favors a lower atomic coordination number than bcc’s 8, and therefore a less symmetric structure than bcc, e.g., an orthorhombic one. This appears to be in agreement with the fact that, in both Nb [45] and Ta [27], the high-P equilibrium solid structure seems to be an orthorhombic Pnma.

It is interesting to compare the (metastable) ambient melting points of the high-$PT$ hexagonal polytypes of Cr, Mo and W found in ab initio studies to those determined in other theoretical approach. Kaufman and Bernstein [46] estimated the metastable melting points of the hcp and fcc polytypes of Cr, Mo and W by extrapolating enthalpies in their binary phase diagrams to pure hcp and pure fcc. This way, they obtained $T_m$(hcp-Cr) = 1175 K, $T_m$(hcp-Mo) = 1900 K and $T_m$(hcp-W) = 2650 K. The first of the three values is in good agreement with our $T_m$(hcp-Cr) = 1205 K. Since the values of the metastable $T_m$ of dhcp-Mo and dhcp-W are not estimated in [46], we approximate them by the $T_m$ values for hcp polytypes, taking into account that the corresponding enthalpies are close to each other; see Figure 1 of this work for Cr or Figure 6 of Ref. [4] for W as two examples. Thus, $T_m$(dhcp-Mo) $\approx 1900$ K and $T_m$(dhcp-W) $\approx 2650$ K, which is in excellent agreement with $T_m$(dhcp-Mo) = 1880 K and $T_m$(dhcp-W) = 2640 K from ab initio calculations [4].

5. Conclusions

We have constructed the theoretical phase diagram of chromium to 2.5 TPa, using a suite of ab initio QMD simulations based on the Z methodology which combines both direct Z method for the simulation of melting curves and inverse Z method for the calculation of solid–solid phase boundaries. We determined that hcp-Cr becomes the thermodynamically stable solid structure of chromium at high-$PT$ and finds itself on the phase diagram of chromium, along with bcc-Cr, its solid structure at ambient conditions. We have calculated the melting curves of both bcc-Cr and hcp-Cr, determined the location of the bcc-hcp-liquid triple point, and obtained an equation for the bcc-hcp solid–solid phase transition boundary. Last but not least, we have proposed thermal equations of state for both bcc-Cr and hcp-Cr, which appear to be in good agreement with both experimental data and QMD simulations.