Magnet Design for Neutron Interferometry By: Rob Milburn.

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<ul><li><p>Magnet Design for Neutron InterferometryBy: Rob Milburn</p></li><li><p>Mathematical MotivationDerived from two of Maxwells EquationsInside cylinder hollow, second equation will see J as zeroAs a result H can be expressed as a gradient of a scalar potential</p></li><li><p>Derivation for Simulation</p></li><li><p>InterpretationSolving Laplaces equation for magnetic potentialAnalogous to complex analytic function w(z)w=u+iv, z=x+iyIf map scalar potential in complex plane, the equipotential lines (const u) and lines of flow (const v) will be orthogonal</p></li><li><p>Boundary ConditionsInput into COMSOL:Inner Cylinder expect no change in B-field flux across boundaryOuter Cylinder expect no B-field outside cylinder Interpretation of COMSOL output:Expect surface current j to flow along equipotentials of .The current between and two equipotentials is: I= R-L, where R and L are on the on right and left sides, facing downstream</p></li><li><p>Initial Design (What it should look like)Magnet is composed of two cylinders, one encompassed within the other. Innermost constant B FieldRegion between two Dont CareOutside outer Zero B Field</p></li><li><p>Initial SimulationGiven by COMSOLPrimarily just a fancy PDE solverSolved Laplaces equations with boundary conditions above to map the equipotentials</p></li><li><p>Results with 40 Lines</p></li><li><p>Checking the ResultsUse Biot Savart law to verify results from PDEBlue Lines magnet potential/current linesExport points on these lines to make into current elements</p></li><li><p>Checking continuedNeed an algorithm to arrange points to follow pathNeed some physics to calculate B Field vector at a given point </p><p>Need method to histogram and compare results</p></li><li><p>Connecting the DotsObtained points from COMSOL but not pathVery DisorganizedFront face Only real worry, Can base rest of geometry/path of cylinder off thisRequire different methods for elements inside/outside inner circle</p></li><li><p>In between RegionNotice that lines take radial pathStart with first given pointLook through all given vectors in listCreate displacement vector and look for point which has smallest displacement magnitudeThis is point closest to it, bubble sortRinse and repeat for next point telling it to ignore points before it in list</p></li><li><p>Dont connect different linesDont 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 separationFor n lines define difference </p></li><li><p>Relevance?Create a parallel Boolean arrayIf angular displacement exceeds or is equal to previous definition, then we flag this positionFlags will be used to indicate start of a new line, will tell computer to not compute dl from previous point to flag</p></li><li><p>Sort againPerform another bubble sortIf y component greater than zero, sort from smallest magnitude to greatestVice versa for negative y component</p></li><li><p>Lines in inner circleThis time what marks line segments is xvalueSince vertical lines, expect very little/no variation in x component create flag where this doesnt occurThen just sort from highest y value to lowest</p></li><li><p>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 backFlags are made in a similar mannerThen all thats needed is the addition of a z component</p></li><li><p>The lines?All thats needed is the point on the face where the line startsAlways the last point in a line segment or the position before a flagThen just add an increment in the z direction. (400 total dl segments transversing z direction in my simulation)</p></li><li><p>Actual physicsAs stated earlier we use biot-savart lawNo integral just sum of a lot of infinitesimal current elementsForces any dl between flags to be zero so no contribution between lines</p></li><li><p>Vector FieldCalculated field on a 3-d grid, using the Biot Savart Lawcan plot field on a line, plane, or 3d space</p></li><li><p>Displaying ResultsA tree is created displaying the BField ResultsThe following variables are saved to make histograms fromX coordinateY coordinateZ coordinateRho (cylindrical coordinates)BxByBz|B|</p></li><li><p>Components against space3x3 plots</p></li><li><p>Histogrammed Results in Inner Cylinder (Bx:Rho) (20,40,100 Lines)</p></li><li><p>Interpreting the ResultsMountain range where peaks occur represents most frequent Bx valueHard to see but as number of lines increase, range gets closer to predicted theoretical value of 1.26 gaussAlso less deviation from main mountain range as number of lines increase, shows greater precision as the number increases</p></li><li><p>Outside Region magnitude of B Field(20,40, then 100 lines)</p></li><li><p>Interpreting results outside of magnetAll results show typical exponential decay as you get further outside the coilDifference between them is A in the equationSlight differences in lambda but main difference is initial value of magnitude becomes lower as number of lines increase</p><p>******************************</p></li></ul>