Here $a_{2} (= c_{0} +c_{2}) < 0$, and the gas is dynamically unstable towards collapse. IV: When $c_{2} \ge c_{0}/3$, the polar state becomes metastable. In the presence of finite populations of $\pm1$, phase separation is expected to occur. In this region, the ferromagnetic state is thermodynamically unstable. Finally, the open and filled circles are typical interaction scales in $^{23}$Na and $^{87}$Rb for $n \sim 10^{14}$cm$^{-3}$.](phaseplot "fig:")]{} (30,7) (55, 8) (-80, 47) (-81, 70)

1.  For $c_{0} > 0$ and $c_{2}< 0$, the ground state is a ferromagnet.

2.  For $c_{0} > 0$ and $c_{2} > 0$, but $c_{2} \le c_{0}/3$, the ground state is polar.

3.  In region III the scattering length in the spin-$2$ channel ($a_{2}$) is negative, and the gas becomes mechanically unstable.

4.  In region IV, the polar state is metastable, in the presence of finite populations of $\pm1$ components, the system will phase separate [@mueller2]. The ferromagnet is thermodynamically unstable here.

5.  The regime of current day experiments on spin-$1$ gases is that of $|c_{2}| \ll c_{0}$, indicated by the open and filled circles in the figure. For $^{87}$Rb, at $n \sim 10^{14}$cm$^{-3}$, $c_{0}n \sim 50$nK and $|c_{2}|n \sim 0.25$nK (filled). For $^{23}$Na, at the 