# Unified Analytic Melt-Shear Model in the Limit of Quantum Melting
## 5 Conclusions  
We have presented the modification of the unified analytic melt-shear model of a multi-phase material by the inclusion of quantum melting, and derived new analytic forms for the melting temperature, cold shear modulus, and the corresponding Gruneisen parameters as functions of density including the density region of quantum melting. We have demonstrated how the new analytic form for the cold \(G\) combines with EOS in the unified analytic cold EOS-shear modulus model, on an example of a known EOS, namely, [PERSON]'s one. The modification of the unified analytic melt-shear model by the inclusion of quantum melting implies that the modified model becomes applicable in the entire density range of the existence of the solid state, up to the critical density of quantum melting above which the solid state does not exist. Our approach can be generalized to model melting curves and cold shear moduli of different solid phases of a multi-phase material over the corresponding ranges of mechanical stability; for instance, hcp-Be in Figure 3 is mechanically unstable (\(G<0\)) at densities larger than \(\sim 7.3\) g/cm\({}^{3}\). This will be the subject of further studies. In addition to formulas for \(\rho^{*}\), \(\rho_{T_{m}\max}\) and \(T_{m\max}\) already discussed in the literature before, we have derived the formulas for \(\rho_{G\max}\) and \(G_{\max}\), Equations (40) and (42). We have also obtained the analytic form of the melting curve of OCP, Equations (21) and (30), which will be useful for numerous applications associated with the thermodynamics of stellar objects. The new unified analytic melt-shear model with quantum melting included can be used to analyze physical phenomena in white dwarfs and other dense stellar objects. As noted by [PERSON] and [PERSON] [21], deviations from the classical melting curve of OCP towards the region of quantum melting become significant for densities \(\rho\stackrel{{>}}{{\sim}}400\,Z^{6}\,A^{4}\,\)g/cm\({}^{3}\). This corresponds to \(\rho\stackrel{{>}}{{\sim}}10^{7}\) g/cm\({}^{3}\) for a He white dwarf and \(\rho\stackrel{{>}}{{\sim}}4\cdot 10^{11}\) g/cm\({}^{3}\) for a C white dwarf. Since these white dwarfs can exits in nature, the phenomenon of quantum melting may occur in their cores. Another phenomenon, the opposite of quantum melting but ultimately related to it, namely, quantum crystallization, may play an important role in the evolution of dense stellar objects such as white dwarfs since the cooling rate for a white dwarf strongly depends on the phase of the matter in the star: whether free thermal motion (in a liquid) of lattice vibrations (in a solid) are the central cooling mechanism [22]. We will now dwell on one of the features of the new modified analytic melt-shear model. As follows from Equations (32) and (37) which describe Gruneisen function \(\gamma(\rho)\approx\frac{1}{2}+\frac{\
u}{6}\,\frac{1}{1-(\rho^{*}/\rho)^{1/3}}\) common to both \(G\) and \(T_{m}\), as \(\rho\) approaches \(\rho^{*}\), \(\gamma(\rho)\to-\infty\), i.e., this Gruneisen function diverges to minus infinity. In fact, \(\gamma(\rho)\) starts going down from its classical \(\rho\to\infty\) limiting value of \(1/2\) at densities much lower than \(\rho^{*}\); at \(\rho_{T_{m}\max}\), \(\gamma(\rho)=1/3\), at a slightly higher \(\rho_{G\max}\), \(\gamma(\rho)=-1/6\), etc. The divergence of a Gruneisen parameter at a cold quantum phase transition is a well-known phenomenon; see, e.g., Ref. [23] for a review, or a more recent Ref. [24] for the case of a symmetry-enhanced 1 st-order quantum phase transition. Let us consider this phenomenon in some more detail. The Mie-Gruneisen-Debye equation of state of a solid reads \((P-P_{c})=\gamma\,\rho\,(E-E_{c})\), where \(E\) is the total energy, and the Gruneisen \(\gamma\) is identical to that in Equation (8) or Equation (46).