From 7550ebfb5245c71eeef8f7543c33e98d2c238c69 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 20 Jun 2019 10:44:32 -0400
Subject: forgot to add bib file, and some spot changes

---
 hidden-order.tex | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

(limited to 'hidden-order.tex')

diff --git a/hidden-order.tex b/hidden-order.tex
index 5031d69..4bf9bde 100644
--- a/hidden-order.tex
+++ b/hidden-order.tex
@@ -159,7 +159,7 @@ for
 \end{align*}
 and the susceptibility near the disordered--modulated transition is
 \[
-  \chi(q)=\frac12\big[c_\parallel q_\parallel^2+D(q_0^2-q_\perp^2)^2+|\Delta r|\big]^{-1}=\frac1{2D}\frac{\xi_\perp^4}{1+\xi_\parallel^2q_\parallel^2+\xi_\perp^{4}(q_0^2-q^2)^2}
+  \chi(q)=\frac12\big[c_\parallel q_\parallel^2+D(q_\parallel^4+2q_\parallel^2q_\perp^2)+D(q_*^2-q_\perp^2)^2+|\Delta r|\big]^{-1}=\frac1{2D}\frac{\xi_\perp^4}{1+\xi_\parallel^2q_\parallel^2+\xi_\perp^4(q_\parallel^4+2q_\parallel^2q_\perp^2)+\xi_\perp^{4}(q_*^2-q_\perp^2)^2}
 \]
 where $\xi_\perp=(|\Delta r|/D)^{-1/4}$ and $\xi_\parallel=(|\Delta r|/c_\parallel)^{-1/2}$. We're also interested in the elastic response, defined by
 \[
@@ -178,7 +178,7 @@ and the effective susceptibility for $\Aog$ has a response like
 \]
 At the disordered--modulated transition the responses are of the form
 \[
-  \frac{\lambda}{\tilde\lambda(q)}=1+\frac{b^2}{4\lambda}\big[c_\parallel q_\parallel^2+D(q_0^2-q_\perp^2)^2+|\Delta r|\big]^{-1}=1+\frac{b^2}{2\lambda}\chi(q)
+  \frac{\lambda}{\tilde\lambda(q)}=1+\frac{b^2}{4\lambda}\big[c_\parallel q_\parallel^2+D(q_\parallel^4+q_\parallel^2q_\perp^2)+D(q_*^2-q_\perp^2)^2+|\Delta r|\big]^{-1}=1+\frac{b^2}{2\lambda}\chi(q)
 \]
 and
 \[
@@ -192,11 +192,11 @@ The Ginzburg criterion gives the proximity $t_G$ of the critical point at which
 Experiments give $\Delta c_V\sim1\times10^5\,\mathrm J\,\mathrm m^{-3}\,\mathrm K^{-1}$ \cite{fisher_specific_1990} and $T_c\sim17.5\,\mathrm K$. A fit of $\tilde\lambda$ to experimental data (Fig.~\ref{fig:B1g.fit}) yields
 \begin{align*}
   \lambda=71\,\mathrm{GPa}-(0.10\,\mathrm{GPa}\,\mathrm{K}^{-1})T &&
-  \frac{b^2}{4\lambda Dq_0^4}=0.084&&\frac a{Dq_0^4}=0.0038\,\mathrm K^{-1}
+  \frac{b^2}{4\lambda Dq_*^4}=0.084&&\frac a{Dq_*^4}=0.0038\,\mathrm K^{-1}
 \end{align*}
-with $|r-r_c|=a|T-T_c|$. We suspect that the modulation at the transition at very low pressure is on the order of the lattice spacing, which would give $q_0^{-1}\sim9.568\,\text{\r A}$. Combined with our fit, this gives
+with $|r-r_c|=a|T-T_c|$. We suspect that the modulation at the transition at very low pressure is on the order of the lattice spacing, which would give $q_*^{-1}\sim9.568\,\text{\r A}$. Combined with our fit, this gives
 \[
-  \xi_0=(aT_c/D)^{-1/4}=\big[T_c(a/Dq_0^4)q_0^4\big]^{-1/4}\sim19\,\text{\r A}
+  \xi_0=(aT_c/D)^{-1/4}=\big[T_c(a/Dq_*^4)q_*^4\big]^{-1/4}\sim19\,\text{\r A}
 \]
 and therefore $t_G\sim1.2\times10^{-6}$, which means one would need to get within about $\Delta T=T_ct_G\sim20\,\mu\mathrm K$ of the critical point to see mean field theory break down.
 
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