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

Derivation for Simulation

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 boundary 2. Outer Cylinder – expect no B-field outside cylinder Interpretation of COMSOL output: 1. Expect surface current j to flow along equipotentials of ϕ.  The current between and two equipotentials is: I= ϕ R - ϕ L, where ϕ R and ϕ L are on the on right and left sides, facing downstream

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

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

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

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