forgot to add bib file, and some spot changes

2 files changed, 23 insertions, 5 deletions

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.
 
diff --git a/hidden_order.bib b/hidden_order.bib
new file mode 100644
index 0000000..9822fa0
--- /dev/null
+++ b/hidden_order.bib
@@ -0,0 +1,18 @@
+
+@article{fisher_specific_1990,
+	title = {Specific heat of {URu}$_{2}${Si}$_{2}$: {Effect} of pressure and magnetic field on the magnetic and superconducting transitions},
+	volume = {163},
+	issn = {0921-4526},
+	shorttitle = {Specific heat of {URu}2Si2},
+	url = {http://www.sciencedirect.com/science/article/pii/092145269090229N},
+	doi = {10/ck6dj9},
+	abstract = {Specific heats were measured in the range 0.3 ?T?30 K for 0?H?7T and P=0, and for H=0 and 0?P?6.3 kbar. For H=0 and P=0, the measurements were extended to 0.15K. Above the superconducting transition the H=0 and 7T data can be superimposed. For the magnetic transition near T0 = 18K, T0 increased with increasing P accompanied by a broadening and attenuation of the specific heat anomally. The superconducting transition near Tc = 1.5 K was broadened, attenuated and shifted to lower temperatures for both increasing P and H. The superconducting transition is similar to that of UPt3, and both the temperature dependence of the superconducting state specific heat and the derived parameters are consistent with an unconventional polar-type pairing.},
+	number = {1},
+	urldate = {2019-06-17},
+	journal = {Physica B: Condensed Matter},
+	author = {Fisher, R. A. and Kim, S. and Wu, Y. and Phillips, N. E. and McElfresh, M. W. and Torikachvili, M. S. and Maple, M. B.},
+	month = apr,
+	year = {1990},
+	pages = {419--423},
+	file = {Fisher et al. - 1990 - Specific heat of URu2Si2 Effect of pressure and m.pdf:/home/pants/.zotero/data/storage/HHVDKMSP/Fisher et al. - 1990 - Specific heat of URu2Si2 Effect of pressure and m.pdf:application/pdf}
+}
\ No newline at end of file