# Unified Analytic Melt-Shear Model in the Limit of Quantum Melting
## 3 Results
### Quantum Melting Added to the Unified Analytic Melt-Shear Model
In the absence of the corresponding theory that would guide us as regards this choice we make the assumption that the corresponding exponents are exactly the same as for \(T_{m}\), i.e.,  
\[G(\rho)=G^{\rm cl}(\rho)\left[1-\left(\frac{\rho}{\rho^{*}}\right)^{1/3}\right] ^{\
u}, \tag{35}\]  
where \(G^{\rm cl}(\rho)\) is the (classical) model shear modulus (13). Formula (35) implies that the relation (15) is exact, i.e., \(G(\rho,0)/(\rho T_{m}(\rho))=\rm const\). The constancy of this combination of \(\rho\), \(G\) and \(T_{m}\) derives in the dislocation-mediated melting model and is the main signature of this model [16] based on which the original unified analytic melt-shear model was developed [4]. The assumption that the behavior of \(G\) is exactly the same as that of \(T_{m}\) implies the identical assumption that dislocation melting mechanism operates in OCP. In fact, if was demonstrated in Ref. [17] that dislocation-mediated melting model describes the melting of alkali metals and reproduces its main features such as the \(T_{m}\) values, dislocation densities at melt, etc. With respect to many of its physical properties, an alkali metal can be regarded as a body-centered cubic (bcc) lattice of point positive ions in a uniform background of free electrons, i.e., a one-component plasma. Indeed, it is well known that the deviations of alkali-metal Fermi surfaces from perfect spheres are of order 1% or less, clear evidence that the valence electrons are very nearly free. In addition, the ratio of ionic radius to half the interatomic distance increases from 0.4 in Li to only 0.7 in Cs [17], hence the overlap between alkali ions is small, and so to a good approximation the ions are effectively point charges. Hence, alkali metals do represent a good approximation of OCP. Therefore, if dislocation-mediated melting mechanism operates in alkali metals, it may well operate in the actual OCP, which fully justifies our choice of the corresponding exponents in Equation (35). Then, as in the case of \(T_{m}\) considered above,  
\[G(\rho,0)=G(\rho_{0},0)\left(\frac{\rho}{\rho_{0}}\right)^{4/3}\exp\left\{6 \gamma_{1}\left(\frac{1}{\rho_{0}^{1/3}}-\frac{1}{\rho^{1/3}}\right)+\frac{2 \gamma_{2}}{d_{2}}\,\left(\frac{1}{\rho_{0}^{q_{2}}}-\frac{1}{\rho^{q_{2}}} \right)\right\}\left[1-\left(\frac{\rho}{\rho^{*}}\right)^{1/3}\right]^{\
u}, \tag{36}\]  
which is the entire cold shear modulus of a solid substance including the region of its quantum melting. The corresponding Gruneisen parameter comes from Equation (8):  
\[\gamma_{G}(\rho)=\frac{1}{2}\,\frac{d\ln G(\rho)}{d\ln\rho}-\frac{1}{6}=\frac {1}{2}+\frac{\gamma_{1}}{\rho^{1/3}}+\frac{\gamma_{2}}{\rho^{q_{2}}}+\frac{ \
u}{6}\,\,\frac{1}{1-(\rho^{*}/\rho)^{1/3}}\equiv\gamma_{G}^{\rm cl}(\rho)+ \frac{\
u}{6}\,\,\frac{1}{1-(\rho^{*}/\rho)^{1/3}}, \tag{37}\]  
where \(\gamma_{G}^{\rm cl}(\rho)\) is the (classical) Gruneisen parameter (10). The maximum value of \(G\) corresponds to the solution of \(dG(\rho,0)/d\rho=0\), or \(\gamma_{G}(\rho_{G\max})=-1/6\). Again, as in the case of \(T_{m}\), at densities \(\rho\sim\rho_{\max}\sim\rho^{*}\), the second and third terms of \(\gamma_{G}^{\rm cl}(\rho)\) are negligibly small (that is, \(\gamma_{G}^{\rm cl}\approx 1/2\)).