magnet design for neutron interferometry by: rob milburn

Magnet Design for Neutron Interferometry

By: Rob Milburn

Mathematical Motivation

Derived from two of Maxwell’s Equations

Inside cylinder hollow, second equation will see J as zero

As a result H can be expressed as a gradient of a scalar potential

0 BJH

Derivation for Simulation

MH

0 HB

02 M

Interpretation

Solving Laplace’s equation for magnetic potential

Analogous to complex analytic function w(z)– w=u+iv, z=x+iy

If map scalar potential in complex plane, the equipotential lines (const u) and lines of flow (const v) will be orthogonal

Boundary Conditions

Input into COMSOL:1. Inner Cylinder – expect no change

in B-field flux across boundary2. Outer Cylinder – expect no B-field

outside cylinder

Interpretation of COMSOL output:1. Expect surface current j to flow

along equipotentials of ϕ.2. The current between and two

equipotentials is: I= ϕR-ϕL, where ϕR and ϕL are on the on right and left sides, facing downstream

€

∂∂n

= H0 cos(θ)

0n

€

ˆ n × Δr H =

r j

€

ˆ n • ΔB = 0

€

k = Φ0 + k ⋅ I

Initial Design (What it should look like)

Magnet is composed of two cylinders, one encompassed within the other.

Innermost – constant B Field Region between two – Don’t Care Outside outer – Zero B Field

Initial Simulation

Given by COMSOL Primarily just a fancy PDE solver Solved Laplace’s equations with boundary

conditions above to map the equipotentials

02

2

2

22

yx

Results with 40 Lines

Checking the Results

Use Biot Savart law to verify results from PDE

Blue Lines – magnet potential/current lines Export points on these lines to make into

current elements

Checking continued

Need an algorithm to arrange points to follow path

Need some physics to calculate B Field vector at a given point

Need method to histogram and compare results

db

r

rxdlIB

30

4

Connecting the Dots

Obtained points from COMSOL but not path Very Disorganized Front face Only real worry, Can base rest of

geometry/path of cylinder off this Require different methods for elements

inside/outside inner circle

In between Region

Notice that lines take radial path Start with first given point Look through all given vectors in list Create displacement vector and look for point

which has smallest displacement magnitude This is point closest to it, bubble sort Rinse and repeat for next point telling it to

ignore points before it in list

Don’t connect different lines

Don’t want dl between lines. How do we avoid this?

If we have n lines in upper half of circle, and all are discrete lines wrt angle then expect angular separation

For n lines define difference

n

180

2

1

Relevance?

Create a parallel Boolean array If angular displacement exceeds or is equal

to previous definition, then we flag this position

Flags will be used to indicate start of a new line, will tell computer to not compute dl from previous point to flag

Sort again

Perform another bubble sort If y component greater than zero, sort from

smallest magnitude to greatest Vice versa for negative y component

Lines in inner circle

This time what marks line segments is xvalue Since vertical lines, expect very little/no

variation in x component create flag where this doesn’t occur

Then just sort from highest y value to lowest

How is the back created?

Back face is created in a reverse manner, making the last element in the front face the starting point in the back

Flags are made in a similar manner Then all that’s needed is the addition of a z

component

The lines?

All that’s needed is the point on the face where the line starts

Always the last point in a line segment or the position before a flag

Then just add an increment in the z direction. (400 total dl segments transversing z direction in my simulation)

Actual physics

As stated earlier we use biot-savart law No integral just sum of a lot of infinitesimal

current elements Forces any dl between flags to be zero so no

contribution between lines

Vector Field

Calculated field on a 3-d grid, using the Biot Savart Law

can plot field on a line, plane, or 3d space

Displaying Results

A tree is created displaying the BField Results

The following variables are saved to make histograms from

X coordinate Y coordinate Z coordinate Rho (cylindrical

coordinates) Bx By Bz |B|

Components against space

3x3 plots

Histogrammed Results in Inner Cylinder (Bx:Rho) (20,40,100 Lines)

Interpreting the Results

Mountain range where peaks occur represents most frequent Bx value

Hard to see but as number of lines increase, range gets closer to predicted theoretical value of 1.26 gauss

Also less deviation from main mountain range as number of lines increase, shows greater precision as the number increases

Outside Region – magnitude of B Field(20,40, then 100 lines)

Interpreting results outside of magnet

All results show typical exponential decay as you get further outside the coil

Difference between them is A in the equation Slight differences in lambda but main

difference is initial value of magnitude becomes lower as number of lines increase

expA