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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