From 3fa3c9ad160f7b428290c0c0b5e227647a3e6f5c Mon Sep 17 00:00:00 2001 From: bradramshaw undefined Date: Thu, 25 Jun 2020 01:04:17 +0000 Subject: Update on Overleaf. --- ref_response.tex | 54 +++++++++++++++++++++--------------------------------- 1 file changed, 21 insertions(+), 33 deletions(-) (limited to 'ref_response.tex') diff --git a/ref_response.tex b/ref_response.tex index 27b3468..69764dd 100644 --- a/ref_response.tex +++ b/ref_response.tex @@ -17,7 +17,7 @@ \pacs{} \maketitle -We thank the editor for organizing the review of our manuscript and are grateful to the referees for their valuable comments, which have strengthened the impact of our work. +We thank the editor for organizing the review of our manuscript and are grateful to the referees for their valuable comments, which have strengthened our work and made it more impactful. We respond in detail to the referees' comments below. \\[12pt] To summarize, we have: @@ -60,8 +60,7 @@ Jaron Kent-Dobias, Michael Matty, and Brad Ramshaw as T decreases, but appears immediately below TN, where eta is small. }\\ -The gross disagreement between the data and theory below the transition is -resolved by the addition of another interaction in the mean-field free energy +We agree, it's not at all obvious that it is a failure of the small $\eta$ and $\epsilon$ expansion that leads to the gross disagreement between the data and theory below the transition. To investigate this, we added the interaction in the mean-field free energy of the form $\epsilon^2 \eta^2$. In a new appendix, we have worked through the mean field modulus implied with this new interaction and a fit is now shown in Fig. 2 as a dashed black line. @@ -75,9 +74,9 @@ Higher order corrections to the mean-field free energy produce arbitrary analytic corrections to the low-temperature behavior, but do not influence predictions above the transition. Those can only be further fit by more complicated dependence of the present free-energy parameters on temperature and -pressure. +pressure. We have also left the original fit (without the extra term) since the model is more simply described in the text. -An inset with a zoom on the critical region has been added to the figure.\\ +As suggested, an inset with a zoom on the critical region has been added to the figure.\\ {\color{blue} Is it not clear how discriminatory is the agreement above TN in 2a, 2b @@ -86,7 +85,7 @@ An inset with a zoom on the critical region has been added to the figure.\\ the presence of a maximum at 120 K in 2b). }\\ -In order to check the agreement in our fit, we preformed the fit with a moving +In order to check the agreement in our fit, we performed the fit with a moving temperature window that cuts off at $T_\text{max}$. Our fits' parameters are $x_i$ for $i = 1, \ldots, 5$ for $C_0 = x_1 - x_2 (T / \mathrm K)$, $x_3 = b^2/a$, $x_4 = b^2/Dq_*^4$, and $x_5 = b \sqrt{-g/u}$. The variation of these @@ -110,7 +109,7 @@ down to cutoffs of $\sim90\,\mathrm K$. The fit functions that result from varyi More insight into the consistency of the fit comes from examining the linear combinations of parameters that form eigenvectors of the fit covariance matrix, -since—unlike the natural parameters of the mean field theory—these have +since---unlike the natural parameters of the mean field theory---these have uncorrelated uncertainties. For the fit including all temperatures (up to $275\,\mathrm K$), these are (in order of fit uncertainty): \begin{align*} @@ -135,7 +134,7 @@ window means that transformation is likewise stable.\\ \centering \includegraphics[width=0.7\textwidth]{referee_response_cutoff_curves.pdf} \caption{ - Fit function with color given by value of $T_\text{max}$ that produced it. Data shown as black line. + Fit to the data with the color given by value of $T_\text{max}$ (the maximum of temperature range of the data used in the fit) that produced the fit. Data shown as black line (mostly underneath the red curve). } \label{fig:parameter_curve} \end{figure} @@ -209,7 +208,7 @@ but we do not believe that every scientific advance has to be accompanied by new mathematical machinery for its own sake. Mean field theory happens to work quite well here, and allows us to make clear symmetry-based statements. In addition, the incorporation of gradient terms into the mean-field free energy -in the context of interpreting ultrasound data appears novel.\\ +in the context of interpreting ultrasound data does appear to be somewhat novel.\\ {\color{blue} 2) One can generically expect several sources for softening elastic @@ -230,15 +229,17 @@ $T_{HO}$. The data we show in figure 2b, obtained with resonant ultrasound, also shows a maximum at around $130\, \mathrm K$, and also softens to its room-temperature value by $T_{HO}$. The contamination in ref.\ 25 is likely an artifact of the pulse-echo ultrasound technique, which can mix between $c_{66}$ and -$(c_{11}-c_{12})/2$ when the crystal is not perfectly aligned. Perhaps more +$(c_{11}-c_{12})/2$ when the crystal is not perfectly aligned. + +Perhaps more importantly, the fit shown in figure 4 of ref 25 does not show very good agreement with the data at any temperature. The model used is one for -thermally-populated crystal field levels, and has nothing to do with the phase +thermally-populated crystal field levels, and does not directly relate to the phase transition at $T_{HO}$. This model does not produce the sharp change in slope of $(c_{11}-c_{12})/2$ at $T_{HO}$, which is an essential singularity in the -thermodynamic free energy and must appear in the elastic moduli, and it does +thermodynamic free energy and must appear in the elastic moduli at a second order phase transition, and it does not produce $1/(T-T_0)$ strain susceptibility above $T_{HO}$, which is a -signature of strain and order parameter coupling. \\ +signature of strain and order parameter coupling. To summarize, while ref. 25 does indeed propose a model to describe the softening seen in $(c_{11}-c_{12})/2$, it does not attribute the softening to the presence of an order parameter, does not capture the singularity at the phase transition, and does not provide a good match to the Curie-Weiss behaviour of the elastic constant. \\ {\color{blue} 3) The agreement of C[B1g] in the region $T