Chris Swank, Bob Golub

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

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

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

Derived from 2nd order density matrix formalism Spectrum of the field-position correlation function

field-position correlation function

field-position volume average

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

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

Spectrum for the infinite region continuous time random walk in 2D

The infinite region 3D spectrum

3D

2D

Therefore the solution to the restricted conditional probability is defined from

And 3D (will be used.)

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.

The integration is a sum over the Fourier components of the field.

The position autocorrelation function spectrum in different dimensions

X=10.2 cm Y=40 cm

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

Assume v² distribution

165 neV maximum energy

Completely ballistic motion

xwallv

𝑣𝜏𝑦 𝑤𝑎𝑙𝑙

Specular wall reflections?

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%

For (small) magnetic impurities the correlation time is small

𝛿𝜔 = −𝛾2𝐸

𝑐𝐵𝑥𝑥 + 𝐵𝑦𝑦

≈ 0

M= 0.2 𝑇

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

3mm OD round wire

(Bob Golub)

Thin film deposition

Deuterium Source at NCSU PULSTAR reactor

Correlation function studies via helium scintillation from

Critical Dressing Studies.

Effects of magnetic impurities/gradients

C. Crawford Magnetic Mirror

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

3D FEM

D(0.4)=977 cm²/s D(0.45)=428 cm²/s