From ac31a39e1c92c24c3986644a93a6d13082687d2d Mon Sep 17 00:00:00 2001 From: Deepak Mallubhotla Date: Mon, 19 Sep 2022 18:23:44 -0500 Subject: [PATCH] feat: adds identifying downturn notes --- Makefile | 13 +- do.sh | 2 +- figures/Im rs-no-weirdness.jpg | Bin 0 -> 536314 bytes figures/chiVsZCleanCaseWithProblem.jpg | Bin 0 -> 385287 bytes figures/downturnExampleClean.jpg | Bin 0 -> 515319 bytes figures/downturnExampleDirty-no-downturn.jpg | Bin 0 -> 375856 bytes figures/downturnIntermediate.jpg | Bin 0 -> 370219 bytes figures/t1VsZDirtyCase-no-problems.jpg | Bin 0 -> 411543 bytes figures/t1VsZcleanCase-bad.jpg | Bin 0 -> 410669 bytes figures/t1vsTexamples.jpg | Bin 0 -> 376995 bytes identifying_downturn.tex | 125 +++++++++++++++++++ notes.tex | 2 +- 12 files changed, 136 insertions(+), 6 deletions(-) create mode 100644 figures/Im rs-no-weirdness.jpg create mode 100644 figures/chiVsZCleanCaseWithProblem.jpg create mode 100644 figures/downturnExampleClean.jpg create mode 100644 figures/downturnExampleDirty-no-downturn.jpg create mode 100644 figures/downturnIntermediate.jpg create mode 100644 figures/t1VsZDirtyCase-no-problems.jpg create mode 100644 figures/t1VsZcleanCase-bad.jpg create mode 100644 figures/t1vsTexamples.jpg create mode 100644 identifying_downturn.tex diff --git a/Makefile b/Makefile index 6dc6bee..dc5bd88 100644 --- a/Makefile +++ b/Makefile @@ -1,6 +1,6 @@ ### Build tools # -LATEXMK := latexmk -pdflatex="luahblatex %O %S" -pdf -dvi- -ps- -quiet -logfilewarninglist +LATEXMK := latexmk -pdflatex="lualatex %O %S" -pdf -dvi- -ps- -quiet -logfilewarninglist WS := wolframscript -f ### Directory variables @@ -11,7 +11,7 @@ CALC_DIR := calc ### Here we go # -OUT_PDF:= $(PDF_DIR)/notes.pdf +OUT_PDF:= $(PDF_DIR)/notes.pdf $(PDF_DIR)/identifying_downturn.pdf .PHONY: all all: $(OUT_PDF) @@ -20,7 +20,7 @@ all: $(OUT_PDF) # ## setup main pdf deps as variable that subdirs can add to -MAIN_PDF_DEPS := bibliography.bib +COMMON_PDF_DEPS := bibliography.bib ## Defining common directory recipes $(PDF_DIR): @@ -36,10 +36,15 @@ $(CALC_DIR): FIGURES := ## Making main.pdf and other pdfs # -$(PDF_DIR)/notes.pdf: notes.tex $(MAIN_PDF_DEPS) | $(PDF_DIR) $(FIGURES) +$(PDF_DIR)/notes.pdf: notes.tex $(COMMON_PDF_DEPS) | $(PDF_DIR) $(FIGURES) $(LATEXMK) $(K}675qJ5AS{D=jJ7VFt~YL(vs>6Zv1_=vnwi_VII|0|Uo&@dwzs=+n_bJ95U?mnzsre;=&xdc@AnatknSftKu&QGenH`3 zXde+V@jen_Qc@BU_`AOF_aPD*(j&*uO7EvtyFtd`NGEhREdBte%-b4z_0A@beN!`Yi(8gf*3K@jZtfnQ zUiTk7^bdFx7!>~Oc|_!km#?A{5|ffsQq$5i^70D`i;7E1%ih=4)i*RYHMewi_w@E* zKYi{W866v+n4FrPnO$C4U0dJS+}hsxJ})AO_^;FYubKU7UNrE$_K}bflaPI%7tucV z?-QpXAw7O}{}E|5vKx-H971;w(8+|wzpWwX6jooNzvYhr(!S3g8aOaw0;F%5)-Hg|Xu?ofaW$N(?^7yt|a1^@$qf&bAAq{g+cUAGC3&t#wia6nvR*_Jy zP`lud-xN)sB*heJca0Alusl~o9e+rtc8L^RKn8#TzyM$XFz`n+;JH~Jx=mt5fYt{I z(6s2Z=+rO+a-1fW0AZcsjjX5Dz9enIQ_`1o#AKUZ1hccQI$9~dJKBi5CqjUB7SL-fYb>jaj}a>e zx(U!+DFN!RHZLC3beD*!y0?VE1x4xXd8MvkVr`}cDF{$gdNLZDf#&V>;cuU3Sq~Y_ z*7v2Z#fm@7lUu3N*>_E8<((U6Uy;p=tpEbVp|#nE+@sWX`c$&uK!BDCU5B^Vx~mrl 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zdnL8qCu|Mo>9xD?Y$#pHnHJ7$kS=((Kq*t-2y~=M`k&pZ_X(+pH=|8Ve{|;-gEnX- l|9NVl%s;{R@6Z2jto-*`{CyUG-+{mH!2j=e0R3a^e*xHuVuSzy literal 0 HcmV?d00001 diff --git a/identifying_downturn.tex b/identifying_downturn.tex new file mode 100644 index 0000000..d98cb4f --- /dev/null +++ b/identifying_downturn.tex @@ -0,0 +1,125 @@ +\documentclass{article} + +%other packages +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{physics} +% +% \usepackage[ +% style=phys, articletitle=false, biblabel=brackets, chaptertitle=false, pageranges=false, url=true +% ]{biblatex} + +\usepackage{graphicx} +\usepackage{todonotes} +\usepackage{siunitx} + +\usepackage{cleveref} + +\title{Identifying the downturn issue with clean case} + +% \addbibresource{./bibliography.bib} + +\graphicspath{{./figures/}} + +\newcommand{\vf}{v_{\mathrm{F}}} +\newcommand{\qf}{q_{\mathrm{F}}} + +\begin{document} + +\maketitle + +\section{The problem} + +As discussed earlier there was an issue with a sharp drop in $T_1(z)$ as temperature approached $T_c$. +The dirty case looks fine in \cref{fig:exdirtynodownturn}, but the clean system in \cref{fig:excleandownturn} shows the problem. +Both of those are at $z$ around $20 \lambda_F$ ($\lambda_F$ is $\approx \SI{0.4}{\nm}$.) + +\begin{figure}[htp] + \centering + \includegraphics[width=\linewidth]{downturnExampleDirty-no-downturn} + \caption{$T_1(z)$, everything looks appropriately good for a dirty system.} \label{fig:exdirtynodownturn} +\end{figure} + +\begin{figure}[htp] + \centering + \includegraphics[width=\linewidth]{downturnExampleClean} + \caption{$T_1(z)$, for a clean system we have the downturn.} \label{fig:excleandownturn} +\end{figure} + +The obvious question is whether this matters, because it's $T$ so close to $T_c$ that cutting it off is probably not an issue. +But the effect matters because in the intermediate values of $\tau$ both the turning up to the normal state and the problematic downturn are present, and that's not great. +\Cref{fig:downturnintermediate} shows what that looks like. +I \emph{think} that that's still probably explainable as having a rise up to the normal state for $T \rightarrow T_c$, masked by the downturn because it only rises close to $T_c$. +But it's not really a convincing graph, so I proceed under the assumption that it's worth investigation. + +\begin{figure}[htp] + \centering + \includegraphics[width=\linewidth]{downturnIntermediate} + \caption{$T_1(z)$, for a clean system we have the downturn.} \label{fig:downturnintermediate} +\end{figure} + +Interestingly, this problem is dependent on the specific $z$. +$T_1$ plotted for different zs in \cref{fig:varioust1vsT}. +The smaller $z$ doesn't have that effect. +But that's the direction that's less helpful, because it's already a very small $z$. + +\begin{figure}[htp] + \centering + \includegraphics[width=\linewidth]{t1vsTexamples} + \caption{$T_1(T)$, for different $z$, some showing the downturn others not.} \label{fig:varioust1vsT} +\end{figure} + +I plotted $T_1$ as a function of $z$ at very high temperatures ($T = 0.99999 T_c$), in the downturn, and again for $T = 0.99 T_c$, which is above the downturn. +In \cref{fig:t1vszdirty}, the dirty case, everything looks sensible. +However, in \cref{fig:t1vszclean} we can see that for $z > \lambda_F$ the $T_1(z)$ curve has a very different flat shape at the very high temperature, which ultimately leads to the downturn. + +\begin{figure}[htp] + \centering + \includegraphics[width=\linewidth]{t1VsZDirtyCase-no-problems} + \caption{$T_1(z)$, at two different temperatures.} \label{fig:t1vszdirty} +\end{figure} + +\begin{figure}[htp] + \centering + \includegraphics[width=\linewidth]{t1VsZcleanCase-bad} + \caption{$T_1(z)$, at two different temperatures.} \label{fig:t1vszclean} +\end{figure} + +I looked at where that might come from. +We have that $T_1 \propto \frac{1}{\chi}$, and +\begin{equation} + \chi_{B,zz} \propto \int_0^\infty \dd{u} u^2 e^{-2 u z} \Im r_s(u), +\end{equation} +in a regime where $u \gg \frac{1}{\lambda}$ for vacuum wavelength $\lambda$. +This is always the regime we care about. +So $\chi(z) \approx \Im r_s(\frac{1}{z})$ or so. +I plotted $\Im r_s(u)$ in \cref{fig:imrs}, and it doesn't really have any of the unusual features we might exepect. +That figure is pretty ugly, but the takeaway is that for $u > 0.2 \lambda_F$, all that matters is the temperature. +However, for $u < 0.2 \lambda_F$ the dirty and clean lines start to split, and for small enough $u$ the difference between the higher and lower temperature cases is less important than the clean-dirty difference. +That makes sense, and doesn't at all explain why $T_1$ is problematic. + +\begin{figure}[htp] + \centering + \includegraphics[width=\linewidth]{Im rs-no-weirdness} + \caption{$\Im r_s$, showing no weirdness.} \label{fig:imrs} +\end{figure} + +So that leaves the integral as a problem. +I haven't been able to excise the issue yet though, either by increasing precision or rewriting the integral different ways to pull $z$ out. +Using something like +\begin{equation} + \chi_{B,zz} \propto \frac{1}{z^3} \int_0^\infty \dd{u'} u'^2 e^{-2 u'} \Im r_s(\frac{u'}{z}) +\end{equation} +should be numerically nicer, but I get about the same answers (with Mathematica's integral handler internally doing something similar to that transform anyway?). +The problem is definitely still apparent in the $\chi$ calculation, which is plotted in \cref{fig:chi}. +So I'm currently confused on this. + +\begin{figure}[htp] + \centering + \includegraphics[width=\linewidth]{chiVsZCleanCaseWithProblem} + \caption{$\chi(z)$, showing the same issue as $T_1$ (as expected).} \label{fig:chi} +\end{figure} + +% \printbibliography + +\end{document} diff --git a/notes.tex b/notes.tex index c4e32e6..5f8da83 100644 --- a/notes.tex +++ b/notes.tex @@ -15,7 +15,7 @@ \usepackage{cleveref} -\title{Noise on ewjn mag noise} +\title{Notes on ewjn mag noise} \addbibresource{./bibliography.bib}

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