# Unified Analytic Melt-Shear Model in the Limit of Quantum Melting
## 2 Methods
### Quantum Melting of OCP  
The melting of OCP was addressed in several studies using quantum Path-Integral Monte Carlo (PICM) simulations. Two distinct regimes were identified which convert to each other smoothly. First, classical melting regime: the melting of non-degenerate ions in low-density limit (\(k_{B}T/E_{F}\gg 1\)). In this regime the (classical) melting curve is increasing normally and is defined by the (classical) OCP coupling parameter  
\[\Gamma=\frac{(Ze)^{2}}{ak_{B}T}=2.275\cdot 10^{5}\frac{Z^{2}}{A^{1/3}}\frac{ \rho^{1/3}}{T} \tag{20}\]  
such that along the melting curve \(\Gamma=175\)[12] (and references therein). Hence, the equation for the classical OCP melting curve is  
\[T_{m}^{\mathrm{cl}}(\rho)=1300\ \frac{Z^{2}}{A^{1/3}}\rho^{1/3}\ \mathrm{K}. \tag{21}\]  
Second, cold quantum melting regime: the melting of highly degenerate ions in high-density limit (\(k_{B}T/E_{F}\ll 1\)). In this regime the melting curve reaches the turnaround point and goes down; the thermodynamics properties depend solely on \(R_{s}\), which is the quantum OCP coupling parameter. Figure 1 shows the phase diagram of OCP. The two regimes discussed above are clearly distinguishable. The turnaround point of the OCP melting curve corresponds to the maximum melting point \(T_{m\max}\). The OCP melting curves based on the results of [13; 14] have \(T_{m\max}\) different from each other by a factor of \(\sim\)2; nonetheless, they converge on the critical value of \(R_{s}\) (the termination point of the melting curve) of \(100/R_{s}^{*}=0.66\pm 0.03\), or \(R_{s}=152\pm 7\). It was independently established [15] that at the critical density for the quantum melting \(R_{s}\approx 160\pm 10\) for bosons and \(\approx 100\pm 20\) for fermions. The above OCP value of \(R_{s}=152\pm 7\) is consistent with boson's \(160\pm 10\). Ref. [13] does not offer a functional form for the best fit to the PIMC data. In the absence of a consistent theory of OCP that, if existed, would describe its entire melting curve analytically, we have to suggest the analytic form of the OCP melting curve based on the relevant physical arguments. Such arguments are: (i) the OCP melting curve terminates (hits zero) at a critical density \(\rho^{*}\) that corresponds to some critical \(R_{s}^{*}\) in Figure 1, (ii) it is a function of \(100/R_{s}\sim\rho^{1/3}\) which is (iii) singular at \(\rho=\rho^{*}\); the singularity is related to quantum melting. This point is discussed in more detail in Conclusions. Thus, the following (simplest analytic) functional form immediately suggests itself:  
\[T_{m}\left(\frac{100}{R_{s}}\right)=T_{m}^{\mathrm{cl}}\left(\frac{100}{R_{s} }\right)\left(1-\frac{100/R_{s}}{100/R_{s}^{*}}\right)^{\upsilon},\ \ \upsilon<1, \tag{22}\]  
where \(T_{m}^{\mathrm{cl}}(100/R_{s})\) is the classical OCP melting curve defined in Equation (21). The main features of the above functional form is that \(T_{m}(100/R_{s})\) does terminate at \(R_{s}=R_{s}^{*}\) and does exhibit a singularity as its derivative diverges. Now, the best fit to the data brings up the value of \(\
u\). Before we carry out our own fitting, let us rewrite \(T_{m}^{\rm cl}(100/R_{s})\) in a more suitable form. As follows from Equations (19) and (21),  
\[T_{m}^{\rm cl}\bigg{(}\frac{100}{R_{s}}\bigg{)}=3.313\cdot 10^{4}\,A\,Z^{4}\, \frac{100}{R_{s}}. \tag{23}\]  
Hence, we look for the best fit to the PIMC data of Ref. [13] of the functional form  
\[T_{m}\bigg{(}\frac{100}{R_{s}}\bigg{)}=3.313\cdot 10^{4}\,A\,Z^{4}\,\frac{100}{R _{s}}\bigg{(}1-\frac{100/R_{s}}{100/R_{s}^{*}}\bigg{)}^{\
u}. \tag{24}\]  
With \(\frac{100}{R_{s}^{*}}=0.66\pm 0.03\), the best fit of the above form leads to  
\[\
u=0.18\pm 0.03. \tag{25}\]  
The turnaround point corresponds to the solution of \(\frac{dT_{m}(100/R_{s})}{d(100/R_{s})}=0\), i.e.,  
\[\frac{100}{R_{s\,\rm max}}=\frac{1}{1+\
u}\,\frac{100}{R_{s}^{*}}\,=\,0.56\pm 0.03. \tag{26}\]  
Hence, \(R_{s\,\rm max}=179\pm 9\). Note that in Figure 1 the maximum melting temperature \(T_{m\,\rm max}\) corresponds to \(100/R_{s}\approx 0.55\).