Adds figure calcs

.gitignore

@@ -8,7 +8,6 @@ dist
 *.toc
 *.log
 *.pdf
-*.jpg
 *.bbl
 *.bcf
 *.blg

Makefile

@@ -7,6 +7,7 @@ WS := wolframscript -f
 #
 PDF_DIR := pdfs
 FIG_DIR := figures
+CALC_DIR := calcs
 
 ### Here we go
 #
@@ -26,10 +27,23 @@ $(PDF_DIR):
 	mkdir $(PDF_DIR)
 $(FIG_DIR):
 	mkdir -p $(FIG_DIR)
+$(CALC_DIR):
+	mkdir -p $(CALC_DIR)
+
+## Figures
+#
+
+FIGURES := $(FIG_DIR)/Cond1Im.jpg
+$(FIG_DIR)/Cond1Im.jpg: scripts/Cond1ImFigure.wls
+	$(WS) scripts/Cond1ImFigure.wls
+
+FIGURES += $(FIG_DIR)/Cond1Re.jpg
+$(FIG_DIR)/Cond1Re.jpg: scripts/Cond1ReFigure.wls
+	$(WS) scripts/Cond1ReFigure.wls
 
 ## Making main.pdf and other pdfs
 #
-$(PDF_DIR)/paper.pdf: paper.tex $(MAIN_PDF_DEPS) | $(PDF_DIR)
+$(PDF_DIR)/paper.pdf: paper.tex $(MAIN_PDF_DEPS) | $(PDF_DIR) $(FIGURES)
 	$(LATEXMK) $(<F)
 	cp $(@F) $@
 

paper.tex

@@ -114,7 +114,7 @@ The Lindhard dielectric function reflects the important property of having an im
 This is a very generic feature of these types of response functions, and occurs because there are no points on the assumed spherical Fermi surface further than $2 q_{\mathrm{F}}$ apart.
 Thus, there are no available quasiparticle-hole excitations available for energy dissipation (cf discussion in \cite{AGD}, \cite{FetterWalecka} or \cite{SolyomV3}).
 This is a general argument, and it should be expected that a superconducting dielectric function should also have zero imaginary part above some momentum on the order of the Fermi momentum.
-Because the Lindhard dielectric function's imaginary part vanishes above a cutoff $q_c\left(\omega\right)$ and has real part that goes as $\frac{1}{q^2}$, all of the integrals in $\eqref{eq:chi}, \eqref{eq:zp} and \eqref{eq:zs}$ converge.
+Because the Lindhard dielectric function's imaginary part vanishes above a cutoff $q_c\left(\omega\right)$ and has real part that goes as $\Re \epsilon_{\mathrm{Lindhard}} - 1 \sim \frac{1}{q^2}$, all of the integrals in $\eqref{eq:chi}, \eqref{eq:zp} and \eqref{eq:zs}$ converge.
 
 We use the expressions from Nam in~\cite{Nam1967} to represent the superconducting response function.
 This extends the previous models by Mattis and Bardeen~\cite{Mattis} and Abrikosov, Dzyaloshinskii and Gorkov\cite{AGD} to give expressions that allow for broader ranges of impurity values.\todo{Including the full expressions from Nam here is a bit space-prohibitive, but it may be important to show exactly what our assumptions encode to in his notation? ex: by assuming no magnetic impurities, our renormalisation factor becomes simpler.}

scripts/Cond1ImFigure.wls

@@ -0,0 +1,67 @@
+Needs["namConductivity`"];
+Needs["namAsymptoticLowKConductivity`"];
+
+(* Defines lindhard functions *)
+epsL[qP_?NumericQ, omegaP_?NumericQ, vfP_?NumericQ, omegapP_?NumericQ,
+    tauP_?NumericQ] :=
+  With[{u = (vfP*qP)/omegaP, s = 1/(tauP*omegaP),
+    prefactor = 3*(omegapP^2)/(omegaP^2)},
+   1 + ((prefactor)/(u^2))*(1 + ((1 + I*s)/(2*u))*
+         Log[(1 - u + I*s)/(1 + u + I*s)])/(1 + ((I*s)/(2*u))*
+         Log[(1 - u + I*s)/(1 + u + I*s)])];
+epsSeries[qP_?NumericQ, omegaP_?NumericQ, vfP_?NumericQ,
+   omegapP_?NumericQ, tauP_?NumericQ] :=
+  With[{u = (vfP*qP)/omegaP, s = 1/(tauP*omegaP),
+    prefactor = 3*(omegapP^2)/(omegaP^2)},
+   1 + ((prefactor)) ((I/(3*(s - I))) +
+       u^2*(-9*I + 5*s)/(45*(-I + s)^3))];
+epsEf[q_?NumericQ, omega_?NumericQ, vf_?NumericQ, omegap_?NumericQ,
+   tau_?NumericQ] :=
+  epsEf[q, omega, vf, omegap, tau] =
+   Piecewise[{{epsSeries[q, omega, vf, omegap, tau],
+      q < .01 * omega / vf}, {epsL[q, omega, vf, omegap, tau],
+      q >= .01 * omega / vf}}];
+
+(* Nam stuff *)
+makeDimensionlessParams[\[Omega]_, \[Sigma]n_, \[Tau]_, vf_, T_, Tc_] :=
+  With[{\[CapitalDelta] =
+     3.06*Sqrt[
+       Tc*(Tc - T)]}, <|\[Xi] -> \[Omega]/\[CapitalDelta], \[Nu] ->
+     1/(\[CapitalDelta]*\[Tau]), A -> \[Omega]*vf/(1*\[CapitalDelta]),
+     t -> T/\[CapitalDelta], B -> \[Sigma]n/\[Omega],
+    C -> vf/\[CapitalDelta]|>];
+
+
+omega = 1;
+tau := .5;
+omegaPlasma := 10;
+sigmaN := omegaPlasma^2 * tau / (4 * Pi);
+vf := 1
+tempCritical := 3
+temp := .9999 * tempCritical
+params := makeDimensionlessParams[omega, sigmaN, tau, vf, temp, tempCritical]
+Print[params];
+
+epsNam2[q_, ps_] := With[
+	{k = ps[C] * q},
+	1 + 4 * Pi * I *
+		ps[B] * \[CapitalSigma][ps[\[Xi]], k, ps[\[Nu]], ps[t]]
+	];
+
+(* Populates the figures directory as ../figures *)
+figuresDirectory = FileNameJoin[{
+	ParentDirectory[
+		DirectoryName[
+			FileNameJoin[{
+				Directory[],
+				$ScriptCommandLine[[1]]
+			}]
+		]
+	], "figures"
+}];
+
+figure[filename_] := FileNameJoin[{figuresDirectory, filename}];
+
+plot1 = LogLinearPlot[ {Im@epsEf[q, omega, vf, omegaPlasma, tau], Im@epsNam2[q, params]} , {q, 0, 10^2}, ImageSize -> Large, AxesLabel -> {"q", StringForm["Re[\[Epsilon](q, \[Omega] = 1)]"]}, PlotRange -> All, PlotLegends -> {Lindhard, Nam}, ImagePadding -> {{50, Automatic}, {Automatic, Automatic}}];
+
+Export[figure["Cond1Im.jpg"], plot1, ImageResolution -> 1200];

scripts/Cond1ReFigure.wls

@@ -0,0 +1,67 @@
+Needs["namConductivity`"];
+Needs["namAsymptoticLowKConductivity`"];
+
+(* Defines lindhard functions *)
+epsL[qP_?NumericQ, omegaP_?NumericQ, vfP_?NumericQ, omegapP_?NumericQ,
+    tauP_?NumericQ] :=
+  With[{u = (vfP*qP)/omegaP, s = 1/(tauP*omegaP),
+    prefactor = 3*(omegapP^2)/(omegaP^2)},
+   1 + ((prefactor)/(u^2))*(1 + ((1 + I*s)/(2*u))*
+         Log[(1 - u + I*s)/(1 + u + I*s)])/(1 + ((I*s)/(2*u))*
+         Log[(1 - u + I*s)/(1 + u + I*s)])];
+epsSeries[qP_?NumericQ, omegaP_?NumericQ, vfP_?NumericQ,
+   omegapP_?NumericQ, tauP_?NumericQ] :=
+  With[{u = (vfP*qP)/omegaP, s = 1/(tauP*omegaP),
+    prefactor = 3*(omegapP^2)/(omegaP^2)},
+   1 + ((prefactor)) ((I/(3*(s - I))) +
+       u^2*(-9*I + 5*s)/(45*(-I + s)^3))];
+epsEf[q_?NumericQ, omega_?NumericQ, vf_?NumericQ, omegap_?NumericQ,
+   tau_?NumericQ] :=
+  epsEf[q, omega, vf, omegap, tau] =
+   Piecewise[{{epsSeries[q, omega, vf, omegap, tau],
+      q < .01 * omega / vf}, {epsL[q, omega, vf, omegap, tau],
+      q >= .01 * omega / vf}}];
+
+(* Nam stuff *)
+makeDimensionlessParams[\[Omega]_, \[Sigma]n_, \[Tau]_, vf_, T_, Tc_] :=
+  With[{\[CapitalDelta] =
+     3.06*Sqrt[
+       Tc*(Tc - T)]}, <|\[Xi] -> \[Omega]/\[CapitalDelta], \[Nu] ->
+     1/(\[CapitalDelta]*\[Tau]), A -> \[Omega]*vf/(1*\[CapitalDelta]),
+     t -> T/\[CapitalDelta], B -> \[Sigma]n/\[Omega],
+    C -> vf/\[CapitalDelta]|>];
+
+
+omega = 1;
+tau := .5;
+omegaPlasma := 10;
+sigmaN := omegaPlasma^2 * tau / (4 * Pi);
+vf := 1
+tempCritical := 3
+temp := .9999 * tempCritical
+params := makeDimensionlessParams[omega, sigmaN, tau, vf, temp, tempCritical]
+Print[params];
+
+epsNam2[q_, ps_] := With[
+	{k = ps[C] * q},
+	1 + 4 * Pi * I *
+		ps[B] * \[CapitalSigma][ps[\[Xi]], k, ps[\[Nu]], ps[t]]
+	];
+
+(* Populates the figures directory as ../figures *)
+figuresDirectory = FileNameJoin[{
+	ParentDirectory[
+		DirectoryName[
+			FileNameJoin[{
+				Directory[],
+				$ScriptCommandLine[[1]]
+			}]
+		]
+	], "figures"
+}];
+
+figure[filename_] := FileNameJoin[{figuresDirectory, filename}];
+
+plot1 = LogLinearPlot[ {Re@epsEf[q, omega, vf, omegaPlasma, tau], Re@epsNam2[q, params]} , {q, 0, 10^2}, ImageSize -> Large, AxesLabel -> {"q", StringForm["Re[\[Epsilon](q, \[Omega] = 1)]"]}, PlotRange -> All, PlotLegends -> {Lindhard, Nam}, ImagePadding -> {{50, Automatic}, {Automatic, Automatic}}];
+
+Export[figure["Cond1Re.jpg"], plot1, ImageResolution -> 1200];