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\begin{document}


\begin{slide}
\begin{center}
\Large VORTEX SHEET ON THE A-B INTERFACE OF SUPERFLUID $^3$He

\normalsize
\includegraphics[width=0.32\linewidth]{vortexlayer03.eps}\par
Talk in QFS, Albuquerque 4.-8.8.2003

Erkki Thuneberg\\
Department of physical sciences, University of Oulu

R. H\"anninen, R. Blaauwgeers, V.B. Eltsov, A.P. Finne, M. Krusius, 
and G.E. Volovik \\
Low temperature laboratory,
Helsinki university of technology\\
\end{center}
\end{slide} 

\begin{slide}
{\Large\bf Content}

Introduction to superfluid $^3$He-A

Bending of vorticity near the interface

Structure of vorticity at the interface

Experiments

\end{slide}

\begin{slide}
{\Large \bf Superfluid $^3$He-A}\par
\includegraphics[width=0.8\linewidth]{Avortex123.eps}\par
1, 2 and 3 dimensional vortex structures\par

large core size, $\sim 10\ \mu$m

low critical velocity, $\approx 1$ mm/s
\end{slide} 


\begin{slide}
{\Large \bf Cooper pairs in $^3$He-A}

\includegraphics[width=0.50\linewidth]{AphasOP.eps}

A phase factor $e^{i\chi}$ corresponds to rotation of $\hat{\bf m}$
and $\hat{\bf n}$ around $\hat{\bf l}$:
\begin{eqnarray}
e^{i\chi}(\hat{\bf m}+i\hat{\bf n})&=&(\cos\chi+i\sin\chi)(\hat{\bf
m}+i\hat{\bf n})\nonumber\\
&=&(\hat{\bf m}\cos\chi-\hat{\bf n}\sin\chi)
+i(\hat{\bf m}\sin\chi+\hat{\bf n}\cos\chi).\nonumber
\end{eqnarray}

Superfluid velocity 
\begin{equation} 
{\bf v}_{\rm s}={\hbar\over 2m}\mbox{\boldmath$\nabla$}\chi= 
{\hbar\over 2m}\sum_j\hat m_j\mbox{\boldmath$\nabla$}\hat n_j. 
\label{supvel}
\end{equation}
\end{slide} 

\begin{slide}
{\Large \bf Continuous vortices}

Consider the structure

\includegraphics[width=0.70\linewidth]{CUV0.eps}

Here $\hat{\bf l}$ sweeps once trough all orientations (once a unit
sphere).

$\Rightarrow$ $\hat{\bf m}$
and $\hat{\bf n}$ circle twice around $\hat{\bf l}$ when one goes
around this object. 

$\Rightarrow$ This is a two-quantum vortex. It is called {\em
continuous}, because
$\Delta$ (the amplitude of the order parameter) vanishes nowhere. 
\end{slide} 

\begin{slide}\baselineskip=0.9truecm
{\Large \bf Hydrodynamic theory of $^3$He-A}

Assume the order parameter ($\hat{\bf m}$, $\hat{\bf
n}$,
$\hat{\bf l}$,
$\hat{\bf d}$)
changes slowly in space. Then one can make gradient expansion of the
free energy
\begin{eqnarray}
F&=& \int d^3r\Bigg[-{\textstyle{1\over 2}}\lambda_{\rm D}(\hat{\bf
d}\cdot\hat{\bf l})^2 +{\textstyle{1\over 2}}\lambda_{\rm H}(\hat{\bf
d}\cdot{\bf H})^2
\nonumber \\
&&
+{\textstyle{1\over 2}}\rho_\perp{\bf v}^2
+{\textstyle{1\over 2}}(\rho_\parallel-\rho_\perp)(\hat{\bf l}\cdot{\bf v})^2
+C{\bf v}\cdot\nabla\times\hat{\bf l}
-C_0(\hat{\bf l}\cdot{\bf v})
(\hat{\bf l}\cdot\nabla\times\hat{\bf l})\nonumber \\  &&
 +{\textstyle{1\over 2}}K_{\rm s}(\nabla\cdot\hat{\bf l})^2
+{\textstyle{1\over 2}}K_{\rm t}\vert\hat{\bf l}\cdot\nabla\times\hat{\bf l}\vert^2
+{\textstyle{1\over 2}}K_{\rm b}\vert\hat{\bf
l}\times(\nabla\times\hat{\bf l})\vert^2
\nonumber \\
&&+{\textstyle{1\over 2}}K_5 \vert(\hat{\bf l}\cdot\nabla)\hat{\bf d}\vert^2+
{\textstyle{1\over 2}}K_6[(\hat{\bf l}\times\nabla)_i\hat{\bf d}_j)]^2\ 
\Bigg].\label{e.grad}
\end{eqnarray}

$\Rightarrow$ Theory of continuos vortex structures (including the
core)

Possible to calculate 
structures, (intrinsic) nucleation, and dynamics of vortices
 
\end{slide} 

\begin{slide}
{\Large \bf Vortex lines and sheets}

\includegraphics[width=1.0\linewidth]{CUV&VS.eps}
\end{slide}

\begin{slide}
{\Large \bf Superfluid B phase}\par
For present purposes the B phase is rather conventional superfluid:\par
- vortex lines: single quantum, small core size ($\sim 10$ nm)

- high critical velocity for vortex nucleation (a few cm/s)
\end{slide}

\begin{slide}
{\Large \bf A-B interface}\par
A and B phases are phase coherent $\Rightarrow$ Vortices cannot
terminate at the interface \par
Difference in core size ($r_A\sim 1000 r_B$) and in quantization
($\kappa_A=2\kappa_B$)\par 
$\Rightarrow$ A-phase vortices do not easily penetrate through the
interface\par
\includegraphics[width=0.40\linewidth]{vortexlayer03.eps}\par
\end{slide} 

\twocolumn
\begin{slide}
\scriptsize
{\Large \bf Bending of vorticity at the A-B interface} 


Simple model for bending vortex sheet: Minimize
\begin{eqnarray}
F=\int d^3r{\textstyle{1\over 2}}\rho_sv^2
+\sigma A.
\end{eqnarray}
with constraints \begin{equation}
\nabla\cdot{\bf v}=0,\  
\nabla\times{\bf v}= 2\mbox{\boldmath$\Omega$}.
\label{eq.nablav}\end{equation}
Here ${\bf v}\equiv{\bf v}_s-{\bf v}_n$.
$A$ is the
area of the sheet and
$\sigma$ its surface tension. ${\bf v}$ has tangential discontinuity
at the sheet.

\includegraphics[width=1.0\linewidth]{vortexbending.eps}\\
\end{slide} 


\begin{slide}
\scriptsize
Solution
\begin{equation}
{\bf v}=2\Omega x\hat{\bf y}
\end{equation}
\begin{eqnarray}
\frac{F}{L_xL_y}=\int _0^\infty
dz\bigg(\frac{1}{b}\int_{-b/2+\zeta}^{b/2+\zeta}dx\frac{1}{2}\rho_sv^2
\nonumber\\ +\frac{\sigma}{
b}\sqrt{1+\left(\frac{d\zeta}{dz}\right)^2}\bigg)\nonumber\\ =\int
_0^\infty dz\Bigg[\frac{1}{6}\rho_s\Omega^2(b^2+12\zeta^2)
\nonumber\\
+\frac{\sigma}{b}\sqrt{1+\left(\frac{d\zeta}{dz}\right)^2}\Bigg]
\end{eqnarray}
$\Rightarrow$
$$
 \frac{z}{a}=1-\sqrt{2-\frac{\zeta^2}{a^2}}-
\frac{1}{\sqrt{2}}\ln\frac{\sqrt{2}-\sqrt{2-(\zeta/a)^2}}
{(\sqrt{2}-1)\zeta/a}\nonumber
$$
where $a=b/\sqrt{6}$.

Surprisingly the Bekarevich-Khalatnikov model gives exactly the same
form for vortex lines.
\end{slide} 

\begin{slide}
{\Large \bf Structure of the surface sheet}\par
Numerical calculation\par
\includegraphics[width=1.0\linewidth]{calcarea.eps}
\end{slide} 

\begin{slide}
Boundary conditions at the A-B interface
\begin{eqnarray}
(\hat{\bf{m}}+ {\rm i}
\hat{\bf{n}}) \cdot \hat{\bf{s}} =   e^{{\rm i} \phi}\\
\hat{\bf{l}}\cdot \hat{\bf{s}} = 0\\
\hat{\bf{d}}=   {R} \cdot \hat{\bf{s}}
\end{eqnarray}
\par
Penetration of superflow into B phase essential. Vortex layer on a
solid wall is not stable.
\end{slide}

\onecolumn 
\begin{slide}
{\Large \bf Results}\par
Structures independent of initial conditions (lines and sheets)\par
Two different structures found\par
They consist of units of one or one half of quantum of circulation
\end{slide} 

\begin{slide}
{\Large \bf Low density texture}\par
\includegraphics[width=1.0\linewidth]{u2d2texture.eps}
\end{slide}
 
\begin{slide}
{\Large \bf High density texture}\par
\includegraphics[width=1.0\linewidth]{d2texture.eps}
\end{slide} 

\begin{slide}
{\Large \bf Current}

\includegraphics[width=0.9\linewidth]{currU2D2.eps}

Low density texture

\includegraphics[width=0.9\linewidth]{currD2.eps}

High density texture
\end{slide} 

\begin{slide}
{\Large\bf
Experiments}

\includegraphics[width=0.6\linewidth]{vortexlayer96.eps}\par
No NMR signal directly from the vortex layer

Can count vortices in both A and B phases\\
$\Rightarrow$ vortices penetrate through the interface in multiples of
one circulation quantum.

This supports that the interface sheet has smaller units than vortex
lines, which have circulation 2.
\end{slide}

\begin{slide}
\includegraphics[width=0.7\linewidth]{Steps+CountFlow.eps}\par
\end{slide}


\begin{slide}
{\Large\bf Conclusions}

Two types of vortex sheets in $^3$He-A:\\
 in bulk and on  the A-B interface

The interfacial sheet separates into units of one or one half
circulation quantum. This is consistent with experiments.


\vspace{2cm}
publications and lecture transparencies available via\\
\href{http://boojum.hut.fi/research/theory/}
{http://boojum.hut.fi/research/theory/}

\end{slide}


\end{document}
 