a^2}{\frac{\Omega}{f_S(h)} - J_0}}.$$ At low temperatures, $f_S(\Omega) \rightarrow 1$. In this case, the conventional TIM ($J_1 = J_0/Z$) gives $\xi\approx4.3$ Å, independent of $S$. For $S=1$, the range of correlation lengths from the modified TIM, where the $J_1$ values are taken from , is 2.9-6.1 nm, which is an order of magnitude larger. The pseudospin anisotropy $J_\mathrm{an}$ is therefore an essential part of the TIM.

Fitting $\Omega$, $J_0$, and $\mu$ for SrTiO$_3$ {#sec:fit}
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Most of the TIM parameters can be fit to existing susceptibility data. We do this for STO, as it will form the basis of our discussion in .

Inserting equation  into equation , we obtain the susceptibility $$\label{X}
\chi(T,0) = \frac{\mu^{2} \eta}{\epsilon_{0}} \frac{1}{L(h,T) - J_{0}/S} \Bigg\vert_{E=0},$$ where $h = |{\mathbf{h}}|$, ${\mathbf{h}}$ is given by equation , and $$\begin{aligned}
\fl L(h,T) = \Bigg[ S \left( \frac{1}{h} - \frac{\left( h^{(3)} \right)^{2}}{h^{3}} \right) f_S(h) + S \frac{\left( h^{(3)} \right)^{2}}{h^{2}} \frac{\partial f_S(h)}{\partial h} \Bigg]^{-1}.\end{aligned}$$

At high temperatures, this expression simplifies. Taking $L(h,T)|_{T \rightarrow \infty}=[\beta S(S+1)/3]^{-1}$, equation  obtains a Curie-Weiss form, $$\label{X_0}
\chi(T,0) = \frac{\mu^{2} \eta S(S+1) }{3 \epsilon_{0} k_B} \frac{1}{T - T_\mathrm{CW}},$$ where $T_\mathrm{CW} = (S+1) J_{0}/3k_\mathrm{B} \approx 30$ K [@sakudo71] is the transition temperature implied by the high-temperature susceptibility. 