# Unified Analytic Melt-Shear Model in the Limit of Quantum Melting
## 3 Results
### Quantum Melting Added to the Unified Analytic Melt-Shear Model  
In view of the results of the previous section on the entire melting curve of OCP with its both classical and quantum segments combined, and taking into account the fact that in the \(\rho\to\infty\) limit the melting curve of the unified analytic melt-shear model, Equation (14), smoothly converts to the classical OCP melting curve (21) (having the correct prefactor of \(\rho^{1/3}\) in the formula for \(T_{m}(\rho)\) in the \(\rho\to\infty\) limit is one of the conditions on the parameters \(\gamma_{3}\) and \(q_{3}\) (with \(\gamma_{1}=\frac{7}{40}\,Z^{2/3}\) fixed), the entire melting curve having both classical and quantum segments combined is  
\[T_{m}(\rho)=T_{m}^{\rm cl}(\rho)\left[1-\left(\frac{\rho}{\rho^{*}}\right)^{1 /3}\right]^{\
u}\!\!\!, \tag{30}\]  
with \(\
u\) from (25) and \(\rho^{*}\) from (27), where \(T_{m}^{\rm st}(\rho)\) is the (classical) model melting curve (14). Indeed, since \(\rho\) scales with \(1/R_{s}^{3}\), in view of (19), the analogs of formulas (22)-(24) in terms of \(\rho\) are those in which \(R_{s}\) is replaced with \(\rho^{-1/3}\). Thus,  
\[T_{m}(\rho)=T_{m}(\rho_{m})\left(\frac{\rho}{\rho_{m}}\right)^{1/3}\exp\! \left\{6\gamma_{1}\left(\frac{1}{\rho_{m}^{1/3}}-\frac{1}{\rho^{1/3}}\right)+ \frac{2\gamma_{3}}{q_{3}}\,\left(\frac{1}{\rho_{m}^{q_{3}}}-\frac{1}{\rho^{q_{ 3}}}\right)\right\}\left[1-\left(\frac{\rho}{\rho^{*}}\right)^{1/3}\right]^{\
u}\!\!\!. \tag{31}\]  
Equation (31) is the entire melting curve of a solid substance including the region of its quantum melting. Let us derive the corresponding Gruneisen parameter. As follows from (9),  
\[\gamma_{T_{m}}(\rho)\,=\,\frac{1}{2}\,\frac{d\ln T_{m}(\rho)}{d\ln\rho}\,+\, \frac{1}{3}\,=\,\frac{1}{2}\,+\,\frac{\gamma_{1}}{\rho^{1/3}}\,+\,\frac{\gamma _{3}}{\rho^{q_{3}}}\,+\,\frac{\
u}{6}\,\,\frac{1}{1-(\rho^{*}/\rho)^{1/3}}\]  
\[\equiv\gamma_{T_{m}}^{\rm cl}(\rho)\,+\,\frac{\
u}{6}\,\frac{1}{1-(\rho^{*}/ \rho)^{1/3}}\!\!, \tag{32}\]  
where \(\gamma_{T_{m}}^{\rm cl}(\rho)\) is the (classical) Gruneisen parameter (11). The maximum melting \(T\) corresponds to the solution of \(dT_{m}(\rho)/d\rho=0\), or \(\gamma_{T_{m}}(\rho)=1/3\). Hence, at \(\rho_{T_{m},\max}\) that corresponds to \(T_{m\max}\), \(\gamma_{T_{m}}(\rho_{T_{m},\max})=1/3\). At densities \(\rho\sim\rho_{\max}\sim\rho^{*}\), the second and third terms of \(\gamma_{T_{m}}^{\rm cl}(\rho)\) are negligibly small (that is, \(\gamma_{T_{m}}^{\rm cl}\approx 1/2\)). Hence, in view of (32), \(\gamma_{T_{m}}(\rho_{\max})=1/3\) reduces to  
\[\frac{1}{2}+\frac{\
u}{6}\,\frac{1}{1-(\rho^{*}/\rho_{T_{m},\max})^{1/3}}=\frac {1}{3}\!\!\!, \tag{33}\]  
or  
\[\rho_{T_{m},\max}=\frac{\rho^{*}}{(1+\
u)^{3}}\!\!\!. \tag{34}\]  
With \(\
u\) from (25) and \(\rho^{*}\) from (27), Equation (34) gives \(\rho_{T_{m}\max}=(2.9\pm 0.4)\,10^{3}\;A^{4}\,Z^{6}\), in agreement with (28) obtained based on the corresponding values of \(100/R_{s}^{*}\) and \(100/R_{s\max}\). We can now repeat the above procedure for the cold shear modulus and suggest its modification because of quantum melting. First, we have to choose the corresponding critical density at which the cold shear modulus terminates. The most obvious choice is that the critical density for cold \(G\) is the same as that for \(T_{m}\), i.e., \(\rho^{*}\). Indeed, if it were larger than \(\rho^{*}\) then, since \(T_{m}\) terminates at \(\rho^{*}\) and only liquid exists at higher density, there would have been a density region in which liquid has a non-zero shear modulus, apparently quite an unphysical situation. If it were smaller than \(\rho^{*}\), there would have been a density region in which a solid has zero shear modulus, as unphysical situation as the previous one. Thus, the critial density for both \(G\) and \(T_{m}\) must be the same. Second, we have to choose the exponents in the formula analogous to Equation (30).