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SCSeminarReport/README

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+Report for the topical seminar on muSR (Uemura)

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+\documentclass[12pt,journal,onecolumn]{IEEEtran}
+
+\usepackage{graphicx}
+\usepackage{amsmath}
+
+\begin{document}
+
+%\title{Report on Uemura's lectures\\ Rotons in superfluid helium}
+\title{Learning about Rotons}
+\author{Tran Hoai Nam}
+
+\maketitle
+
+%\begin{abstract}
+%This is abstract text: This simple document shows very basic features of \LaTeX{}.
+%\end{abstract}
+
+
+%\chapter{First Chapter}
+\section{Theoretical framework: Landau and Feynman}
+%Explain contribution of Landau and Feynman on studies of rotons
+In 1938, P. Kapitza \cite{kapitza}, and simultaneously, J.F. Allen and  A.D.
+Misener \cite{allen} reported  experimental evidence
+of an anomalous behavior of liquid helium below 2.2 K, which is later called ``superfluidity''. 
+In the same year, F. London \cite{london} suggested that superfluidity is
+consequence of Bose-Einstein condensation (BEC), and L. Tisza proposed a ``two fluid
+model'' to explain the observation.  Both of these ideas connected to BEC.
+
+The two fluid model was subsequently developed by L.D. Landau \cite{landau1}, however without
+reference to BEC. In his theory, Landau considered energy states of the liquid instead of
+individual atoms, it means without giving
+an underlying microscopic basis (such as Bose statistics of the $^4$He atoms).
+%(c.f. Bose condensed gases at finite temperature, p. 329) 
+Landau assumed that single
+particle states were replaced by collective modes of two different kinds:
+phonons, i.e. sound quanta, and quantized vortices called ``rotons''.
+Phonons have basically a linear dispersion relation: 
+\begin{equation}
+	\omega = ck,
+	\label{eq:phonon}
+\end{equation} and rotons
+have a quadratic one with a gap:
+\begin{equation}
+	\epsilon = \Delta + p^2/2\mu,
+	\label{eq:roton1}
+\end{equation} 
+Later \cite{landau2}, Landau
+modified the dispersion relation of rotons in order to obtain better agreement
+with thermodynamic properties. The phonon branch evolved continuously into a
+roton branch: 
+\begin{equation}
+	\epsilon = \Delta + (p-p_0)^2/2\mu,
+	\label{eq:roton2}
+\end{equation} 
+where $\Delta$ and $p_0$ are the energy and  momentum of the minimum, and
+$\mu$ is effective mass of excitation. 
+
+M. Cohen and R.P. Feynman \cite{cohen} then proposed to verify Landau's theory by neutron
+scattering experiments. These experiments were strong support to Landau's
+theory.  Nevertheless, Feynman thought that there are weaknesses in Landau's theory. Notably,
+Landau's quantum hydrodynamical approach treated helium II as a continuous
+medium, which right from the beginning sacrificed the atomic structure of the
+liquid and thus forestalled the possibility to calculate various characteristics
+of the system.  By writing
+partition function as an integral over trajectories, using the space-time
+approach to quantum mechanics, Feynman \cite{feynman1953} could derive a Landau-type energy
+spectrum and further demonstrate how phonon-like excitation evolved into
+roton-like ones at large momenta. 
+
+\section{Experimental evidence}
+%Choose one experiment probing rotons, explain principle, describe what results
+%indicate.
+In this section, I want to introduce about one neutron scattering experiment to
+probe rotons. The experiment was done by D.G. Henshaw and A.D.B. Woods in
+1960 \cite{exp}. 
+\subsection{Principle of the experiment}
+%It is refered to Cohen and Feynman's paper --> should look at it before
+%proceeding.`
+In this experiment, inelastically scattered neutrons from superfluid
+$^4$He were measured. Changes in energy and momentum of scattered neutrons
+are equal to the energy and momentum of an excitation produced in the liquid.
+By measuring wavelength distribution of
+scattered neutrons, the authors could infer a dispersion curved for liquid
+$^4$He at 1.12 K. The dispersion curved was compared with Landau's theory of
+rotons.
+\subsection{Results}
+\begin{figure}[htpb]
+	\begin{center}
+		\includegraphics[width=0.7 \textwidth]{dispersion}
+		\caption{The dispersion curve for helium liquid at 1.12 K}
+		\label{fig:dispersion}
+	\end{center}
+\end{figure}
+The dispersion curve for liquid helium at 1.12 K under its normal vapor pressure
+is shown in Fig. \ref{fig:dispersion}. The open circles are data points, showed
+the energy and momentum of excitations of the liquid. The smooth curve drawn
+through data points is the guide to the eye.
+ The parabolic curve from the origin represents
+the theoretically calculated dispersion relation of free helium atoms at 0 K. 
+
+The dispersion curve showed behavior similar to what is expected from Landau's
+theory: a linear phonon branch evolves to a quadratic branch of roton. In Fig.
+\ref{fig:dispersion}, the part rising linearly from about 0.2 A$^{-1}$ until 0.6
+A$^{-1}$ is the phonon branch. It has the slope of the velocity of sound of
+237 m/s. Beyond 0.6 A$^{-1}$, the curve falls below the phonon branch, has
+one maximum at 1.10 A$^{-1}$ and 13.7 K. The curve has a minimum at 1.91
+A$^{-1}$ and 8.65 K. When fitted with equation \eqref{eq:roton2}, parameters of
+the roton branch at 1.12 K are:
+\begin{align*}
+	\Delta/k &= 8.65 \pm 0.04 \mathrm{K} \\
+	p_0/ \hbar &= 1.91 \pm 0.01 \mathrm{A}^{-1}\\
+	\mu &= 0.16\mathrm{ m_{He}}
+	\label{eq:fittedpar}
+\end{align*}
+
+The authors also investigated temperature variations of above parameters in the
+range from 1.1 K to 4.2 K. All of them showed a remarkable change in slope at
+the superfluidity transition point. The plot of mean energy change of 4.039 A
+neutrons is shown in Fig. \ref{fig:tempvar}.
+\begin{figure}[htpb]
+	\begin{center}
+		\includegraphics[width=0.7\textwidth]{tempvar}
+	\end{center}
+	\caption{The temperature variation of the mean energy change of 4.039 A
+	neutrons scattered through 80$^o$ from liquid helium.}
+	\label{fig:tempvar}
+\end{figure}
+\section{My opinion on employing roton concept for superconductivity}
+I do not have enough background to judge if employing roton in
+superconductivity is good or not. However, I think that superfluidity and
+superconductivity are both manifestations  quantum behaviors in
+macroscopic scale. So, in some limit, idea of roton might be applied to explain
+superconductivity. 
+%Your own view on employing roton concepts for superconductivity.
+%%%%%%%%%
+%Ref
+%%%%%%%%%
+\begin{thebibliography}{9}
+	\bibitem{kapitza}
+		P. Kapitza, \emph{Nature} \textbf{141}, 74 (1938).
+	\bibitem{allen}
+		J.F. Allen and A.D. Misener, \emph{Nature} \textbf{141}, 75 (1938).
+	\bibitem{london}
+		F. London, \emph{Nature} \textbf{141}, 643 (1938).
+	\bibitem{tisza}
+		L. Tisza, \emph{Nature} \textbf{141}, 913 (1938).
+	\bibitem{landau1}
+		L.D. Landau, \emph{J. Phys. USSR} \textbf{5}, 71 (1941).
+	\bibitem{landau2}
+		L.D. Landau, \emph{J. Phys. USSR} \textbf{11}, 91 (1947).
+	\bibitem{cohen}
+		M. Cohen and R.P. Feynman, \emph{Phys. Rev.} \textbf{107}, 13 (1957).
+	\bibitem{feynman1953}
+		R.P. Feynaman, \emph{Phys. Rev. } \textbf{91}, 1291 (1953).
+	\bibitem{exp}
+		D.G. Henshaw and A.D.B. Woods, \emph{Phys. Rev.} \textbf{107}, 13 (1957).
+
+\end{thebibliography}
+\end{document}