set $T = 1\mu$K and $n_{0} = 10^{14}$cm$^{-3}$.](pmicro.eps "fig:")]{} (-70, 35)

**$(\omega/c_{0}n_{0})^{2}$**

(90, -6)[**$k \Lambda_{T}$**]{}

The spin and quadrupole modes in the $m_{F} = 0$ state are degenerate. For $c_{2} > 0$, or for large enough $q$, one obtains a gapped quadratic mode. For $c_{2} <0$ however, the dispersion goes from quadratic to linear at $q=4|c_{2}|n_{0}$. Upon lowering $q$ further, the $m_{F} = 0$ state undergoes a [*dynamic*]{} instability where the excitation frequency is complex. At $q=2|c_{2}|n_{0}$, the leading order term in the small $k$ expansion vanishes.

The structure of the modes should be contrasted with those in the $m_{F} = 1$ state. Even though either case may not be a thermodynamically stable starting point for anti-/ferromagnetic interactions (respectively), only in the antiferromagnetic case do small fluctuations grow exponentially.

Taking the $\tau_{D} \rightarrow \infty$ limit and solving (\[eq:5\]), the dispersion relation is determined by the equation:

$$\begin{aligned}
\label{eq:8}
\Big((\omega - \Omega_{s})(\omega + \Omega_{d}) - \frac{v^{2}_{T}k^{2}}{2}\Big)\Big((\omega + \Omega_{s})(\omega  - \Omega_{d}) - \frac{v^{2}_{T}k^{2}}{2}\Big) = -4(c_{2}n_{0})^{2}(\omega - \Omega_{d})(\omega + \Omega_{d})\end{aligned}$$

where $\Omega_{s} = q + 2c_{2}n_{0}$. Expanding (\[eq:8\]) to lowest order in $k$ returns Eq.(\[eq:7\]).

The roots of (\[eq:8\]) for negative $c_{2}$ are plotted in Fig. \[fig:-1\](a) for various values of the quadratic 