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Analytical & Exact calcs. of EDM B-fields

Analytical & Exact calcs. of EDM B-fields. Analytical studies (simple expansions) To understand limitations to field uniformity May allow simple optimization of field profile Exact studies: Evaluate average gradients over fiducial volume

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Analytical & Exact calcs. of EDM B-fields

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  1. Analytical & Exact calcs. of EDM B-fields • Analytical studies (simple expansions) • To understand limitations to field uniformity • May allow simple optimization of field profile • Exact studies: • Evaluate average gradients over fiducial volume • relevant for Geometric phase contributions to false EDM

  2. y y x x z z Analytical Studies • Evaluate leading-order contributions to main sources of field non-uniformity • Considered two geometries: Open Cosq coil Pure Cosq coil

  3. B-field 1st order expansion • Main field has first-order quadratic dependence on direction due to symmetry • Bx(x) = B0(1 + axx2) – geo. phase contribution • Bx(y) = B0(1 + ayy2) – (not studied yet) • Bx(z) = B0(1 + azz2) – largest non-uniformity

  4. L R y z x Bx(z) study • Field due to “barrel” + “endcap” wires eg: qn

  5. Bx(z) = B0[1 + (abz +aez )z2] • For R>>L ,abz ~ - ½ aez • Now add cylindrical open Ferromagnetic shield • creates mirror currents for barrel wires • this doubles abz givingaez ~ -abz • thus Bx(z) gradient cancelled to 1st order • Sign of gradient depends on details of cancellation • Consistent with measured coil

  6. Bx(x) Study (R=35cm, L=300cm) • Simple expression for Bx(x) for pair of loops • Btotx(x) = B0S (1 + anx x2) • Can try to cancel atotx with single additional loop pair • Also find atotx is minimized for particular value of nloops axtot = 5.3 x 10-6

  7. Pure Cosq coil Open Square Cosq coil y y x x z z Exact calcs. of EDM B-fields • Biot-Savart for straight filaments • Full 3-d profile of B-field • Evaluate <|dBx/dx|> over fiducial volume • Considered two geometries

  8. Comparison with prototype

  9. Fiducial Volume 10 cm 50 cm 7.5 cm 10 cm

  10. False EDM from Geometric phase • Pendlebury et al PRA 70 032102 (04) • Lamoreaux and Golub nucl-ex/0407005 • Geometric phase contribution to false EDM’s depends strongly on radial fields perpendicular to B0 • These result from dBx/dx in our exp. x

  11. Geometric phase contributions • Gradients + vxE field gives dnf • Magnetometers (199Hg & 3He) pick up phase at different rate due to higher velocities • Can be reduced by frequent collisions with buffer gas or phonons (if ) • Neutron can also receive false EDM

  12. Using Pendlebury et al estimates and our basic field profile: • 20 turns of cosq coils without ferromagnetic shield (R=35cm L=300cm) • <dBx/dx> = 0.1 nT/m with B0 = 1 mG • Gives dfn ~ 5 x 10-26 e-cm • But…

  13. With B0 = 10 mG and 42 turns of open (square) cosq coil (R = 61cm, L= L=392 cm – Jan’s ref.) get: • <|dBx/dx|> = 0.15 nT/m • Gives dfn ~ 8 x 10-28 e-cm Note: New Sussex-ILL exp. plans to use • 25 mG • Claims <|dBx/dx|> = 0.1 nT/m “after trimming • Gives dfn ~ 8 x 10-29 e-cm

  14. To Do • Optimize “trimming” of field • Additional coils to minimize <|dBx/dx|> • Check optimum R vs L • Ferromagnetic enclosure with various penetrations

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