# Brian N. Martin

Physics 449: Honour's Thesis
## Vortex Interactions in a Type II Superconductor

Supervisors:
Dr. Choptuik
and Dr. Affleck

### Final paper
PS
PDF

### Overview

When superconductors at low temperatures are put in a weak magnetic field
they exhibit the Meissner Effect: the complete expulsion of magnetic flux
lines. As the magnetic field strength is increased, at some critical value
Hc the flux lines do penetrate. In Type II superconductors, there is an
intermediate phase. For H < Hc1 there is complete Meissner Effect. For
Hc1 < H < Hc2, there is not a perfect total flux expulsion. The flux is
less than its value for the solid in the non-superconducting state, but
it is also not zero. This is called the Schubnikov Phase, or vortex state.
It was discovered by Schubnikov in 1937. This is the region being studied.
(graph 1*)
This property can also be shown in a plot of inductance B vs H.
(graph 2*)
In this phase it is energetically favorable to have the flux lines form
in filaments. The structure of a filament is shown in
(graph 3*)

We now turn to the problem of interacting vortices. My project right now
involves modeling the interaction of two vortices in a thin slab
(picture 4*).
In order to solve the boundary conditions on the surface of the slab we
introduce image vortices, analogous to the image charges used in E&M. The
potential between two vortices is known and to find the total potential we
sum up the potential for one pair interactions. We then map this problem
via a Feynman path integral into a quantum mechanical problem with the
potential V(x) as calculated. The function looks like
this.

The goal of the project is to determine the sign on the parameter
*a*. This parameter is involved in an equation for another
parameter *g* (see part I problem statement:
PS, PDF).

* - graphs to come soon.

## Part I

We first make the simplification that the two vortices are confined to move
only in the middle of the slab. This makes it a one dimensional problem.
In actuality, the vortices can (and will) move in the y direction as well,
but they are most likely to be in the middle.
If we modify the equation for the potential V(x)
(see problem statement:
PS, PDF) so that all the constants
equal 1, then we get the potential function shown
here.

See program code.

solution

Plots

Here is a plot of the fit of the data to a sine wave. The
form of the sine wave is shown on the graph. The data fits only to the positive
x as we are not using absolute value of x in the argument. Here is a
plot of the
error (sine form minus actual form).

## Part II

In part II, we relax the assumption that the vortices move only in the
x-direction and allow them to move in the y-direction as well.
See the problem statement
(PS,
PDF) for details.

The algorithm I am using to solve this problem is called Gauss-seidel
relaxation (details:
PS, .
PDF).

Part II plots
Part II solution plots

## writeup

Outline (PS, PDF)

Plot of 1D solution with x and y set to 0 and z is varying in 3D problem for various
multiples of the potential. The plots are labeled 've(n)' where n
is the number used in the multiple (10^n) of the potential.
(PS)

Convergence checks for the same multiples of v.

n = 0: PS

n = 1: PS

n = 2: PS

n = 3: PS

n = 4: PS

n = 5: PS

n = 6: PS