chris swank, bob golub - oak ridge national laboratory · 2012. 12. 31. · [1] c. swank and r....
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Chris Swank, Bob Golub
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Geometric phase from magnetic gradients
Gradients natural to coil geometry
Magnetic impurities in cell walls, cell coatings, magnetic dust on coatings.
SQUID pick up leads distort d-c fields
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Geometry Rectangular cell Assume Spin Starts on the Left wall.
( )y y yB L B(0)y yB B
•y rotation from small field in y •x rotation from v×E field.
Slower Rotation about y
Faster Rotation about y
0 1/ wall x
z
y
y
z
z
y
x
yB
v E
yB
E
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yB
E
Spin Starts
Phase after one length
Spin Reflects and Travels Back
Phase after 1 round trip.
z
0.0005
0.001
0.0015
0.002
0.0025
30
210
60
240
90
270
120
300
150
330
180 0
x
y spin
0.001
0.002
0.003
0.004
0.005
30
210
60
240
90
270
120
300
150
330
180 0
yB
E
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Derived from 2nd order density matrix formalism Spectrum of the field-position correlation function
field-position correlation function
field-position volume average
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When the diffusion equation does not accurately describe the correlation function
Accurate description of correlation function necessary.
Simulation of the continuous time random walk
OR
Exact solution of the continuous time random walk.
λ~L
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Solution to the telegraphers equations satisfies isotropic Boltzmann transport in 1D
[1] C. Swank and R. Golub. arXiv:1012.4006 2010.
Valid for Diffusion Valid for Ballistic motion
𝑝(𝑥, 𝑡|𝑥0, 0) =1
𝐿 𝑒−𝑡2𝜏𝑐 cos𝑛𝜋𝑥0𝐿−𝑖𝑠𝑛𝑡
2𝜏𝑐+𝑖
𝑠𝑛sin𝑛𝜋𝑥0𝐿−𝑖𝑠𝑛𝑡
2𝜏𝑐
∞
𝑛=−∞
𝑠𝑛 = 1 − 4𝜔𝑛2𝜏𝑐2, 𝜔𝑛 =𝑛𝜋𝑣
𝐿
𝐻𝑥(𝑡)𝑥(𝑡 + 𝜏) = 𝑑𝑥0𝐺𝑥𝑥0𝑝(𝑥0, 𝑡) 𝑥𝑝(𝑥, 𝑡 + 𝜏|𝑥0, 𝑡)𝑑𝑥𝐿
0
𝐿
0
𝐻𝑥𝑥(𝜔) =2𝐺𝑥𝑣2𝜏𝑐3
𝜋2
1
12+ 𝑛2
𝜔2 − 𝜔𝑛2 2𝜏𝑐4 +𝜔2
𝜏𝑐2
∞
𝑛=−∞
http://arxiv.org/abs/1012.4006
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Spectrum for the infinite region continuous time random walk in 2D
The infinite region 3D spectrum
3D
2D
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Therefore the solution to the restricted conditional probability is defined from
And 3D (will be used.)
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Define the spectrum.
Written in terms of the 3D conditional probability.
One can take advantage of the periodicity of the variables via Fourier series, Simplifying the integration.
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The integration is a sum over the Fourier components of the field.
The position autocorrelation function spectrum in different dimensions
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X=10.2 cm Y=40 cm
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FID Measured Relaxation is Given by:
Where the Pulse loss is calculated from
The error propagates like
1
𝑇1𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡=𝛾2𝐺𝑧2
2
2𝑆𝑥𝑥(𝜔0) + 𝑆𝑦𝑦(𝜔0
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Assume v² distribution
165 neV maximum energy
Completely ballistic motion
xwallv
𝑣𝜏𝑦 𝑤𝑎𝑙𝑙
Specular wall reflections?
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𝑏 = 1 𝑛𝑚
𝑤 = .1 μ𝑚
From AFM setting
[M. Brown, R. Golub, J. M. Pendelbury. Vacuum Vol. 25 num 2, 1974] [V. K. Ignatovich, JINR, p4-7055, 1973] [A Steyerl, Z Physic, 254, 1972]
𝑃(⊥) ∼ 1%
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For (small) magnetic impurities the correlation time is small
𝛿𝜔 = −𝛾2𝐸
𝑐𝐵𝑥𝑥 + 𝐵𝑦𝑦
≈ 0
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M= 0.2 𝑇
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https://nedm.bu.edu/twiki/pub/NEDM/Collab021011Agenda/dipole_detection.pdf
40nG sensitivity is required to detect micron sized
magnetic saturated particles. (Byung Kyu Park)
Cs NMOR magnetometer for testing.
https://nedm.bu.edu/twiki/pub/NEDM/Collab021011Agenda/dipole_detection.pdf
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3mm OD round wire
(Bob Golub)
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Thin film deposition
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Deuterium Source at NCSU PULSTAR reactor
Correlation function studies via helium scintillation from
Critical Dressing Studies.
Effects of magnetic impurities/gradients
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C. Crawford Magnetic Mirror
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Geometric phase from magnetic gradients
Gradients natural to coil geometry
▪ PULSTAR Experiment
Magnetic impurities in cell walls, cell coatings, magnetic dust on coatings.
▪ Cs NMOR cells prior to fabrication
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3D FEM
D(0.4)=977 cm²/s D(0.45)=428 cm²/s
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