Transcript Magnet Design for Neutron Interferometry

Magnet Design for Neutron
Interferometry
By: Rob Milburn
Mathematical Motivation
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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
B  0
 H  J
Derivation for Simulation
H   M
  B    H  0
 M  0
2
Interpretation
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Solving Laplace’s equation for magnetic
potential
Analogous to complex analytic function w(z)
–
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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.
2.
The current between and two
equipotentials is: I= ϕR-ϕL,

 H0 cos( )
n

0
n

k  0  k  I
where ϕR and ϕL are on the on right and left
sides, facing downstream


nˆ  B  0
r r
nˆ  H  j
Initial Design (What it should look like)
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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
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Given by COMSOL
Primarily just a fancy PDE solver
Solved Laplace’s equations with boundary
conditions above to map the equipotentials
2
2




2
  2  2 0
x
y
Results with 40 Lines
Checking the Results
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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
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Need an algorithm to arrange points to follow
path
Need some physics to calculate B Field
vector at a given point
 0 I  dlx r 
B
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
 3 db
4  r 
Need method to histogram and compare
results
Connecting the Dots
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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
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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
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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
1  180
  
2 n


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Relevance?
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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
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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
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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?
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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?
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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
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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
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Calculated field on a 3-d grid, using the Biot
Savart Law
can plot field on a line, plane, or 3d space
Displaying Results
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A tree is created
displaying the BField
Results
The following variables
are saved to make
histograms from
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X coordinate
Y coordinate
Z coordinate
Rho (cylindrical
coordinates)
Bx
By
Bz
|B|
Components against space
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3x3 plots
Histogrammed Results in Inner
Cylinder (Bx:Rho) (20,40,100 Lines)
Interpreting the Results
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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
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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
A exp  