diff options
Diffstat (limited to 'main.tex')
| -rw-r--r-- | main.tex | 45 |
1 files changed, 4 insertions, 41 deletions
@@ -479,47 +479,10 @@ This is typically done by computing the Ginzburg temperature \cite{ginzburg_remarks_1961}, which gives the proximity to the critical point $t=(T-T_c)/T_c$ at which mean field theory is expected to break down by comparing the magnitude of fluctuations in a correlation-length sized box to -the magnitude of the field. In the modulated phase the spatially averaged -magnitude is zero, and so we will instead compare fluctuations in the -\emph{amplitude} at $q_*$ to the magnitude of that amplitude. Defining the -field $\alpha$ by $\eta(x)=\alpha(x)e^{-iq_*x_3}$, it follows that in the -modulated phase $\langle\alpha(x)\rangle=\alpha_0$ for $\alpha_0^2=|\delta -\tilde r|/4u$. In the modulated phase, the $q$-dependant fluctuations in -$\alpha$ are given by -\begin{equation} - G_\alpha(q)=k_BT\chi_\alpha(q)=\frac1{c_\parallel q_\parallel^2+D_\perp(4q_*^2q_\perp^2+q_\perp^4)+2|\delta r|}, -\end{equation} -An estimate of the Ginzburg criterion is then given by the temperature at which -$V_\xi^{-1}\int_{V_\xi}G_\alpha(0,x)\,dx=\langle\delta\alpha^2\rangle_{V_\xi}\simeq\langle\alpha\rangle^2=\alpha_0^2$, -where $V_\xi=\xi_\perp\xi_\parallel^2$ is a correlation volume. The parameter $u$ -can be replaced in favor of the jump in the specific heat at the transition -using -\begin{equation} - c_V=-T\frac{\partial^2f}{\partial T^2} - =\begin{cases}0&T>T_c\\Ta^2/12 u&T<T_c.\end{cases} -\end{equation} -The integral over the correlation function $G_\alpha$ can be preformed up to -one integral analytically using a Gaussian-bounded correlation volume, yielding -$\langle\Delta\tilde r\rangle^2\simeq k_BTV_\xi^{-1}|\Delta \tilde r|^{-1}\mathcal -I(\xi_\perp q_*)$ for -\begin{equation} - \begin{aligned} - \mathcal I(x)=-2^{3/2}\pi^{5/2}\int& dy\,e^{[2+(4x^2-1)y^2+y^4]/2} \\ - &\times\mathop{\mathrm{Ei}}\big(-(1+2x^2y^2+\tfrac12y^4)\big). - \end{aligned} -\end{equation} -This gives a transcendental equation -\begin{equation} - \mathcal I(\xi_{\perp0}q_*t_\G)\simeq3k_B^{-1}\Delta c_V\xi_{\parallel0}^2\xi_{\perp0}t_\G^{3/4}, -\end{equation} -with $\xi=\xi_0|t|^{-\nu}$ defining the bare correlation lengths. -Experiments give $\Delta c_V\simeq1\times10^5\,\J\,\m^{-3}\,\K^{-1}$ -\cite{fisher_specific_1990}. Our fit above gives $\xi_{\perp0}q_*=(D_\perp -q_*^4/aT_c)^{1/4}\simeq2$. Further supposing that -$\xi_{\parallel0}\simeq\xi_{\perp0}$, we find $t_\G\sim0.4$, though this -estimate is sensitive to uncertainty in $\xi_{\parallel0}$: varying our -estimate for $\xi_{\parallel0}$ over one order of magnitude yields changes in -$t_\G$ over nearly four orders of magnitude. The estimate here predicts +the magnitude of the field. + +Our fit above gives $\xi_{\perp0}q_*=(D_\perp +q_*^4/aT_c)^{1/4}\simeq2$. The estimate here predicts that an experiment may begin to see deviations from mean field behavior within around several degrees Kelvin of the critical point. A \rus\ experiment with more precise temperature resolution near the critical point may be able to resolve a |