diff options
| -rw-r--r-- | fig-stiffnesses.gplot | 22 | ||||
| -rw-r--r-- | fig-stiffnesses.pdf | bin | 331750 -> 332053 bytes | |||
| -rw-r--r-- | main.tex | 8 | ||||
| -rw-r--r-- | referee_comments.txt | 91 | ||||
| -rw-r--r-- | referee_response_cutoff_curves.pdf | bin | 81698 -> 80978 bytes | |||
| -rw-r--r-- | referee_response_cutoff_eigenvectors.pdf | bin | 9171 -> 9073 bytes | |||
| -rw-r--r-- | referee_response_cutoff_parameters.pdf | bin | 9444 -> 9194 bytes |
7 files changed, 85 insertions, 36 deletions
diff --git a/fig-stiffnesses.gplot b/fig-stiffnesses.gplot index 4c08763..52762dd 100644 --- a/fig-stiffnesses.gplot +++ b/fig-stiffnesses.gplot @@ -9,11 +9,11 @@ cc5 = "#000000" Tc = 17.26 -a1 = 71.13597475161484 -b1 = 0.010425748328804992 -c1 = 1665.064389018724 -d1 = 6.283065722894796 -e1 = 14.579664883988315 +a1 = 72.9479505025133 +b1 = 0.01227234014115862 +c1 = 2675.3172792640935 +d1 = 8.418493827841177 +e1 = 19.456906054458745 C10(T) = a1 - b1 * T C1(T) = C10(T) / (1 + 1 / (1 / d1 + abs(T - Tc) / c1) / C10(T)) @@ -88,11 +88,11 @@ set title '(c)' offset 12,-2.7 set xlabel '$T / \mathrm K$\\~' offset 0,0.5 set xtics 50, 50, 250 offset 0,0.5 mirror unset y2tics -set ytics 63,1,72 +set ytics 63,1,74 unset y2label set ylabel '$\textcolor{mathc3}{C_{\mathrm{B_{1\mathrm g}}}},\textcolor{mathc4}{C_{\mathrm{B_{1\mathrm g}}}^0} / \mathrm{GPa}$' offset 3 -set yrange [64.5:71.5] +set yrange [64.5:73.5] plot "data/c11mc12.dat" using 1:(100 * $2) with lines lw 6 lc rgb cc3, \ C10(x) dt 3 lw 4 lc rgb cc4 @@ -102,8 +102,8 @@ set y2label '$\Big[C^0_{\mathrm{B_{1\mathrm g}}}(C^0_{\mathrm{B_{1\mathrm g}}}/C set title '(d)' offset 12,-2.7 set format y2 '\scriptsize$%0.2f$' set format y '\scriptsize$%0.2f$' -set yrange [0.14:0.38] -set y2tics 0.15,0.05,0.39 offset -0.7 mirror +set yrange [0.11:0.25] +set y2tics 0.05,0.025,0.25 offset -0.7 mirror plot "data/c11mc12.dat" using 1:(1 / (C10($1)*(C10($1) / (100 * $2) - 1))) with lines lw 6 lc rgb cc3, \ x > Tc ? 1/(C10(x) * (C10(x) / C12(x) - 1)) : 1 / 0 lw 3 lc cc5, \ x <= Tc ? 1/(C10(x) * (C10(x) / C12(x) - 1)) : 1 / 0 dt (2,1) lw 3 lc cc5, \ @@ -127,8 +127,8 @@ plot "data/c11mc12.dat" using 1:(100 * $2) with points pt 2 ps 0.15 lc rgb cc3, unset margins set origin 0.5,0.26 -set yrange [0.154:0.166] -set y2tics 0.155,0.005,0.167 +set yrange [0.1155:0.123] +set y2tics 0.105,0.0025,0.131 set format y2 '\scriptsize$%0.3f$' plot "data/c11mc12.dat" using 1:(1 / (C10($1)*(C10($1) / (100 * $2) - 1))) with points pt 2 ps 0.15 lc rgb cc3, \ diff --git a/fig-stiffnesses.pdf b/fig-stiffnesses.pdf Binary files differindex 4b1d98e..2cc839f 100644 --- a/fig-stiffnesses.pdf +++ b/fig-stiffnesses.pdf @@ -476,9 +476,9 @@ corresponding modulus. shows the location of the \ho\ phase. (a) $\Btg$ modulus data and a fit to the standard form.\cite{Varshni_1970} (b) $\Bog$ modulus data and a fit to \eqref{eq:static_modulus} (magenta, dashed) and a fit to \eqref{eq:C0} (black, solid). The fit gives - $C^0_\Bog\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, $D_\perp - q_*^4/b^2\simeq0.16\,\GPa^{-1}$, and - $a/b^2\simeq6.1\times10^{-4}\,\GPa^{-1}\,\K^{-1}$. Addition of a quadratic + $C^0_\Bog\simeq\big[73-(0.012\,\K^{-1})T\big]\,\GPa$, $D_\perp + q_*^4/b^2\simeq0.12\,\GPa^{-1}$, and + $a/b^2\simeq3.7\times10^{-4}\,\GPa^{-1}\,\K^{-1}$. Addition of a quadratic term in $C^0_\Bog$ was here not needed for the fit.\cite{Varshni_1970} (c) $\Bog$ modulus data and the fit of the \emph{bare} $\Bog$ modulus. (d) $\Bog$ modulus data and the fits transformed by @@ -723,7 +723,7 @@ The order parameter term relies on some other identities. First, \eqref{eq:eta_s \end{equation} and therefore that the functional inverse $\eta_\star^{-1}[\eta]$ is \begin{equation} - \eta_\star^{-1}[\eta](x)=\frac b{2g\eta(x)}\Bigg(1-\sqrt{1-\frac{4g\eta(x)}{b^2}\frac{\delta F_\op[\eta]}{\delta\eta(x)}}\Bigg). + \eta_\star^{-1}[\eta](x)=\frac{b}{2g\eta(x)}\Bigg(1-\sqrt{1-\frac{4g\eta(x)}{b^2}\frac{\delta F_\op[\eta]}{\delta\eta(x)}}\Bigg). \end{equation} The inverse function theorem further implies (with substitution of \eqref{eq:dFodeta} after the derivative is evaluated) that \begin{equation} diff --git a/referee_comments.txt b/referee_comments.txt index c246631..87f2116 100644 --- a/referee_comments.txt +++ b/referee_comments.txt @@ -17,7 +17,22 @@ > look plausible as the disagreement does not appear to develop slowly > as T decreases, but appears immediately below TN, where eta is small. -The disagreement between the theory at low temperature is largely resolved by the addition of an additional interaction in the mean-field free energy of the form ε²η². 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 thin black line. +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 +of the form ε²η². 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. + +Though fine features of the low-temperature behavior are not reproduced, the +addition of further terms in the mean-field free energy supply finer +corrections. For instance, another correction at order η_*² (roughly linear in +ΔT) is produced by a term of the form ε⁴, while myriad terms at yet higher +order provide corrections of order η_*⁴ and up (quadratic and up in ΔT). +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. An inset with a zoom on the critical region has been added to the figure. @@ -26,18 +41,36 @@ An inset with a zoom on the critical region has been added to the figure. > parameters, or does the agreement result from a fine-tuning? (e.g., > 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 temperature window that cuts off at T_max. Our fit's parameters are x_i for i = 1, …, 5 for C₀ = x₁ - x₂ (T / K), x₃ = a/b², x₄ = b²/Dq⁴, and x₅ = b √(-g/u). The variation of these parameters as a function of T_max are shown in referee_response_cutoff_parameters.pdf. The parameter x₁ is fairly stable at all temperature cutoffs, while the rest vary by at most 20–60\% of their 275K value down to cutoffs of ~90K. - -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 these have uncorrelated uncertainties. For the fit including all temperatures (up to 275K), these are (in order of fit uncertainty): - - y₁ = -0.00198126 x₁ + 2.16869 10⁻⁶ x₂ - 0.99998 x₃ - 0.00227756 x₄ - 0.00560291 x₅ - y₂ = -0.0151198 x₁ + 0.0000415145 x₂ - 0.00552438 x₃ - 0.0205384 x₄ + 0.999659 x₅ - y₃ = 0.635138 x₁ - 0.00196902 x₂ - 0.00315925 x₃ + 0.77197 x₄ + 0.0254495 x₅ - y₄ = 0.772222 x₁ - 0.00663886 x₂ - 0.0000753204 x₃ - 0.635317 x₄ - 0.00137316 x₅ - y₅ = 0.00637806 x₁ + 0.999976 x₂ - 4.32279 10⁻⁶ x₃ - 0.00269696 x₄ - 4.93718 10⁻⁷ x₅ - -The variation of these parameter combinations as a function of T_max are shown in referee_response_cutoff_eigenvectors.pdf. The parameter y₁, which is principally x₃ = a/b², varies the most with the cutoff, at most around 60\% of -its value until ~90K. The parameter y₂, which is principally x₅ = b √(-g/u), varies at most around 15\% of its value until ~90K. The other three parameters are stable at any cutoff, and are likewise mixed combinations of x₁, x₂, and x₄. +In order to check the agreement in our fit, we preformed the fit with a moving +temperature window that cuts off at T_max. Our fit's parameters are x_i for i = +1, …, 5 for C₀ = x₁ - x₂ (T / K), x₃ = a/b², x₄ = b²/Dq⁴, and x₅ = b √(-g/u). +The variation of these parameters as a function of T_max are shown in +referee_response_cutoff_parameters.pdf. The parameter x₁ is fairly stable at +all temperature cutoffs, while the rest vary by 2—5x their 275K value down to +cutoffs of ~90K. + +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 +uncorrelated uncertainties. For the fit including all temperatures (up to +275K), these are (in order of fit uncertainty): + + y₁ = -0.0016 x₁ + 1.6 10⁻⁶ x₂ - 1.0 x₃ - 0.0020 x₄ - 0.0044 x₅ + y₂ = -0.017 x₁ + 0.000043 x₂ - 0.0043 x₃ - 0.023 x₄ + 1.0 x₅ + y₃ = -0.62 x₁ + 0.0018 x₂ + 0.0027 x₃ - 0.78 x₄ - 0.029 x₅ + y₄ = -0.78 x₁ + 0.0068 x₂ + 0.000041 x₃ + 0.62 x₄ + 0.0012 x₅ + y₅ = 0.0064 x₁ + 1.0 x₂ - 3.3 10⁻⁶ x₃ - 0.0028 x₄ - 4.4 10⁻⁷ x₅ + +The variation of these parameter combinations as a function of T_max are shown +in referee_response_cutoff_eigenvectors.pdf. The parameter y₁, which is +principally x₃ = a/b², varies the most with the cutoff, at most around 5x its +value until ~90K. The parameter y₂, which is principally x₅ = b √(-g/u), varies +at most around 1.5x its value until ~90K. The other three parameters are stable +at any cutoff, and are mixed combinations of x₁, x₂, and x₄. + +Notably, x₁ and x₂ are the only parameters involved in transforming the +experimental data in Figure 2(d), and their stability as a function of the data +window means that transformation is likewise stable. Plots of the fits performed between 90 and 275 K are shown in referee_response_cutoff_curves.pdf. @@ -47,7 +80,6 @@ referee_response_cutoff_curves.pdf. The behaviour of c/a is indeed interesting, but our model only considers the coupling to the two in-plane shear strains, since it is one of these that shows the anomalous behaviour. To talk about the c/a ratio we would have to introduce coupling between the order parameter and the A1g strains (\epsilon_xx + \epsilon_yy, and \epsilon_zz). Because the order parameter we consider breaks both translational and (locally) point-group symmetries, this coupling would be quadratic-in-order-parameter, linear-in-strain, and would thus be generic to any order parameter. Put more simply - our model has special coupling to a particular shear strain, whereas the c/a ratio is related to compresisonal strains, which couples to our order parameter in the same way as it does to any other (non-A_1g) order parameter. - > At last, I understand that the model is meant to be purely > phenomenological, but given the plethora of publications on URu2Si2 > over 30 years, where any conceivable order parameter has been proposed @@ -83,11 +115,8 @@ This is a consistent with hexadecapolar order > within the mean-field treatment and does not provide a novel > theoretical advance. -[Jaron will talk about what is new from his results. Below is Brad's more general statement] +What our manuscript provides is a new way of interpreting a very clear experimental signature - that is, nearly perfect Curie-Weiss 1/(T-T_0) in (c11-c12)/2. We show that a staggered nematic order parameter explains this behaviour. We agree that coupling strains and order parameters is not new, 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. -What our manuscript provides is a new way of interpreting a very clear experimental signature - that is, nearly perfect Curie-Weiss 1/(T-T_0) in (c11-c12)/2. We show that a staggered nematic order parameter explains this behaviour. We agree that coupling strains and order parameters is not new, 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. - -> > 2) One can generically expect several sources for softening elastic > constants. For example, the authors in ref.25 also succeeded in the > quantitative fits in the framework of a 4f crystal field model for T > @@ -105,9 +134,29 @@ There are a couple of very important distinctions to be made between our work an > does not lead uniquely to the authors' arguments on the realization of > the B1g order parameter. -The disagreement between the theory at low temperature is resolved by the addition of an additional interaction in the mean-field free energy of the form ε²η², now shown in Fig. 2 as a thin black line. - -While terms like this provide cusp-like features in the modulus for each strain symmetry, they cannot explain the 1/ΔT softening seen in the high-temperature phase, since their contribution to the response function is zero above T_c. +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 +of the form ε²η². 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. + +Though fine features of the low-temperature behavior are not reproduced, the +addition of further terms in the mean-field free energy supply finer +corrections. For instance, another correction at order η_*² (roughly linear in +ΔT) is produced by a term of the form ε⁴, while myriad terms at yet higher +order provide corrections of order η_*⁴ and up (quadratic and up in ΔT). +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. + +While terms that modify the behavior below the transition provide cusp-like +features in the modulus for each strain symmetry, they cannot explain the 1/ΔT +softening seen in the high-temperature phase, since their contribution to the +response function is zero above T_c. Likewise, mean field theories with a +uniform order parameter cannot explain the finite value of the modulus at the +transition. > 4) The most important point in the HO problem is the microscopic > identification of symmetry breaking and the order parameter. In spite diff --git a/referee_response_cutoff_curves.pdf b/referee_response_cutoff_curves.pdf Binary files differindex f9245c9..8c3e7e5 100644 --- a/referee_response_cutoff_curves.pdf +++ b/referee_response_cutoff_curves.pdf diff --git a/referee_response_cutoff_eigenvectors.pdf b/referee_response_cutoff_eigenvectors.pdf Binary files differindex ef69c94..455c060 100644 --- a/referee_response_cutoff_eigenvectors.pdf +++ b/referee_response_cutoff_eigenvectors.pdf diff --git a/referee_response_cutoff_parameters.pdf b/referee_response_cutoff_parameters.pdf Binary files differindex 0ed3d76..cee5bc0 100644 --- a/referee_response_cutoff_parameters.pdf +++ b/referee_response_cutoff_parameters.pdf |