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%\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}
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