3 he spin dephasing in the nedm cell due to b-field gradients
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3 He Spin Dephasing in the nEDM Cell due to B-field Gradients. Steven Clayton University of Illinois. Contents Arbitrary gradients: Monte Carlo calculation Linear gradients: analytic solution Arbitrary gradients: numerical solution Dressing field gradients. - PowerPoint PPT PresentationTRANSCRIPT
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3He Spin Dephasing in the nEDM Cell due to B-field Gradients
Steven ClaytonUniversity of Illinois
nEDM Collaboration Meeting at Duke, May 21, 2008
Contents1. Arbitrary gradients: Monte Carlo calculation2. Linear gradients: analytic solution3. Arbitrary gradients: numerical solution4. Dressing field gradients
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From last collaboration meeting…
H0
x
y • long times can be simulated because collision time is much longer• requires field B(t,x,y,z) at all points in the cell
N 1000T2 = 4202 s
Here, the optimized, 3D field map was (poorly)parameterized by 4th order polynomialsin x, y, z.
Long T2 can be simulated.Dressing effect can be simulated.
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Diffusion Monte Carlo Simulation• Geant4 Framework• isotropic scattering from infinite-
mass scattering centers.• monoenergetic. particle velocity
v3 = sqrt(8 kBT/( m))• mass m = 2.4 m3
• mean free path = 3D/v3,• D = 1.6/T7 cm2/s• Spin evolved via “quality-
controlled” RK solution to Bloch equation (Numerical Recipes)
• “Lambertian” reflection from walls– Scattering kernel:– cos necessary to satisfy
reciprocity– results in uniform density
throughout cell
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N=34, l/r = 6.4 field profile
B. PlasternEDM November 2007 Collaboration Meeting
With ferromagnetic shield at 300 K
Known for some time that N=34 uniformity worsens in presence of ferromagnetic shield
Hence, reason for design of “modified” cos θ coils with wire positions offset from nominal
ASU, S. Balascuta TOSCA
Caltech, M. Mendenhall
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T2 due to dressing field gradients
A deviation in B1 can be mapped to an equivalent deviation in B0:
B0
x
y
B1
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Uniform Dressing Field, n34 B0 field (no FM shield)
T2 from Fit 8615 386 s
1674 77 s
3896 175 s
7913 563 s
12839 815 s
B1
off
B1
on
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Dashed 3He
Solid UCN
0 0ω γω ωγω
= =
=
d d
d
d
By
Bx
P. Chu (collab. meeting at ASU)
Effective γ (Y < 1)
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Nonuniform dressing field (B0 uniform)T2 from fit 641 +- 39 s 1.37 +- 0.2 s 871 +- 98 s3268 +- 332 s
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Redfield theory for T1, T2
Spectral density of field at spin location:
ensemble average:
McGregor (PRA 41, 2631) solves this for a linear gradient of H_z:
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“Generalized McGregor”
Diffusion equation solution in 1-D for
Spectral density in terms of 3-D cosine transform overthe rectangular prism cell with arbitrary B0(x,y,z):
cosine transform amplitudes of Hq
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Weight factors of cosine transform components
• ~ 2
• 3D phase space ~ 2
For T2
For T2
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Contributions to T2-1 of components of n34
B0 field (no FM shield)
(nx,ny,nz) = (0,0,2)
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Redfield theory calc. vs. MC simulation
MC:1. arbitrary geometries
can be simulated.2. arbitrary fields OK.3. OK for arbitrary
dressing fields, including Y ~ 1.
4. ~1 CPU-hour per particle simulated to 1000 s. ~100 CPU-hours to get T1,T2.
Redfield theory calculation:• practical only for simple
geometries• can be applied to
arbitrary (small) field non-uniformities.
• can be used for dressing field gradients, if B1y is mapped to B0x
• computationally fast for nEDM cell (using fast discrete cosine transform). ~1 CPU-second to get T1 and T2.
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Relaxation times for different cell lengthsB0: optimized n34 coil, no FM shield. B1 off.
(Redfield theory calculation)
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Constraint on dressing field uniformity
For a uniform gradient (à la McGregor):
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Diffusive edge enhancement?
z0 z0
cell wall
Initial positions are distributed uniformly
particle diffuses over distance Ld~sqrt(D tm) during measurement time tm.
Particles initially near a wall do not sample as much z as particles initially far from walls, if Ld is not big enough.
Smaller z smaller B longer T2
At T = 450 mK,D = 500 cm2/s.If tm = 1500 s,Ld ~ 866 cm >> Lz,but, signal decays during tm…
z/Lz
Particle distributions after some elapsed time:
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Relaxation times for different cell lengths
400 mK