miscible-immiscible phase boundary: $$\label{eq:critical_trap}
    \begin{cases} 
    \text{core shell MIM} & \mbox{:} \
    \eta \gg \eta_c \\
    \text{core shell IMI} & \mbox{:} \
    \eta \ll \eta_c  \\
    \text{CF or SB} & \mbox{:} \
    \eta \approx \eta_c \\
    \end{cases}$$ For particle number ratios discussed in this section, the critical region lies at $a_I/a_M \approx 0.8$ (Fig. \[fig:MF\_few\_body.eps\]a) and at $a_I/a_M \approx 0.9$ (Fig. \[fig:MF\_few\_body.eps\]b).

Next, we want to find an estimate for the miscible-immiscible phase boundary $g^c_{_{MI}}$. To this end we combine the energetical separation criterion in with approximate effective potentials from and . Specifically, for a given particle number ratio $N_I/N_M$ we determine the critical trap ratio $\eta_c$. Then depending on the choice of $\eta$ we solve numerically for the ground state energy of a single particle inside the effective potential or . Finally, we compare this energy to the potential height at the trap center:

$$\label{eq:critical_g}
    \begin{cases} 
    a_I/a_M < \sqrt[7]{N_I/N_M} & \mbox{:} 
    \ H_M^{eff}= -\frac{1}{2} \frac{\partial^2}{\partial x^2} + V_{M}^{eff}(x)
        \begin{cases}
        E^{eff}_{_{0,M}} > g_{_{MI}} N_I \sqrt{\frac{\eta}{\pi}}  
        & \Rightarrow \ \text{M} \\
        E^{eff}_{_{0,M}} < g_{_{MI}} N_I \sqrt{\frac{\eta}{\pi}}  
        & \Rightarrow \ \text{MIM} \\
        \end{cases} \\
    \\
    a_I/a_M > \sqrt[7]{N_I/N_M} & \mbox{:} 
    \ H_I^{eff}=-\frac{1}{2} \frac{\partial^2}{\partial y^2} + V_{I}^{eff}(y)
        \begin{cases}
        E^{eff}_{_{0,I}} > g_{_{MI}} N_B \sqrt{\frac{1}{\pi}}  
        & 