## 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 \(T_{\rm N}(P)\), 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 [PERSON] et al. ([34], p. 541), the theoretical shock Hugoniot of Cr in the \(P\)-\(T\) coordinates is given by  
\[T_{\rm H}(P)=293\ +\ 0.167\ P^{1.89}. \tag{6}\]  
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  
Figure 8: RDFs of the final states of the solidification of liquid Cr at \(\sim\)1000 GPa at lower temperatures. 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. [PERSON] and [PERSON] [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.