1. Magnetic shielding calculations and also Petr Volegov Andrei Matlashov Michelle Espy LANL, P-21, SQUID team Igor Savukov

2. Requirements for shielding To reduce the gradients and field variation that can shorten helium coherence time To minimize noise in the SQUID gradiometer To avoid instabilities in SQUID operation Gradients are shielded differently than uniform field! SQUID gradiometers are not sensitive to uniform component

3. SQUID detector SQUID pickup loops Helium cell

4. Large gradiometer Extremely Sensitive, Very Long Baseline Planar SQUID Gradiometer R. Cantor and J. A. Hall, STAR Cryoelectronics, Santa Fe, NM USA A. N. Matlachov and P. L. Volegov, Los Alamos National Laboratory, Los Alamos, NM USA We have developed an extremely sensitive, low-Tc first-order planar gradiometer with a very long baseline of 9 cm. The thin-film gradiometric pickup loop consists of two series-configured 3 cm x 6 cm pickup loops with center-to-center spacing of 9 cm fabricated on a 150 mm Si wafer. The pickup loop is connected to the input circuit of a separate dc SQUID chip using superconducting wire bonds. The measured magnetic field sensitivity of the gradiometer referred to one pickup loop is 0.105 nT/  0, and the intrinsic noise of the dc SQUID coupled to the thin-film pickup loop typically is 3 µ  0 /Hz ½. This results in a magnetic field noise of 0.3 fT/Hz ½ and a magnetic field gradient noise of 0.033 fT/cm-Hz ½. Measurements were carried out in our magnetic shielded room.

5. Ambient noise in a lab Rev. Sci. Instr. 71, 1529 (2000) 100 pT/Hz1/2 at 3 Hz For nEDM experiment We need 0.1 fT/Hz^1/2 So shielding factor 1 million is required

6. Drawing of the nEDM shield

7. Estimates of Johnson noise Table compiled by Brad Filippone Estimates based on the theory of Ref.[1] for an infinite cylinder 0.28 0.24 3.7 [1] S.-K. Lee and M. V. Romalis, arXiv:0709.2543v1 A factor of 2 difference?

8. Transverse shielding factor by mu-metal Shielding by a single Infinite cylindrical shell Shielding by two shells Shielding by 4 shells Axial shielding factor will be much smaller and much more uncertain

9. Transverse shielding by SC cylinder 3.92 m 2a a=0.62 m In the center, reduction is by 350 times However, we need to consider proximity of the SC cylinder to mu-metal and that the equation assumes a semi-infinite cylinder So the shielding factor can be about 100 Overall factor fill be about 4x10^5

10. Axial shielding factor by mu-metal I checked numerically that for two shells with L= 4m and spacing 5 cm, by adding the 2 nd shell, the shielding factor only doubled in agreement with the analytical prediction. Thus, for 4 layers, the expected shielding factor will be only 120 This equation contains unknown parameters so instead we can do numerical calculations. For a cylinder with L=4 m, S1=30. For 4 mu-metal cylinders, as in nEDM shield design The condition when shielding factors are only added, not multiplied, T. J. Sumner, J. M. Pendlebury, and K. F. Smith, J. Phys. D: Appl. Phys. 20, 1095 (1987)

11. Axial shielding with a superconducting cylinder Superconducting cylinder Mu-metal shield Solenoid Additional axial shielding by the superconductor is 1000 times, and overall axial shielding becomes 1.2e5 Analytical shielding factor: Exp(-3.832 z/R)~10^4

12. Superconducting endcaps? Superconducting cylinder, axial-component shielding Superconducting cylinder, transverse-field shielding Hi-mu cylinder, axial-component shielding Hi-mu cylinder, transverse-field shielding 1.J. R. Claycomb and J. H. Miller, Jr., IEEE Trans. on Magnetics 42, 1694 (2006) 2.K. H. Carpenter, IEEE Trans. On Appl. Supercond. 6, 142 (1996) It was shown in [2] that the equation is the same for the hole, but instead of the cylinder radius the hole radius needs to be used. The above analytical expressions were derived in Ref.[1] Therefore, shielding can be dramatically improved by adding endcaps on the SC cylinder This might require consideration of boundary conditions. However, since we have ferromagnetic shell just outside the cosine coil, magnetic lines will be closed through this shell and not much will be reflected by superconducting endcaps. Nevertheless, one has to check it! a a End cap a

13. Cosine coil vs solenoid Cosine coil metglass cells Noise from metglass 0.2 fT What about non-uniformity? What about instabilities? Superconducting shield solenoid cells We can “freeze” the field in the superconducting cylinder which is expected to be very stable? Noise level can be lower Electrodes Superconducting shield Electrodes

14. Conclusions Aluminum support structure produces noise 1.8 fT/Hz^1/2, but it will be reduced by superconducting shield Noise from inner metglass layer is acceptable, 0.2 fT/Hz^1/2? SQUID gradiometer noise is 0.3 fT/Hz^1/2 at 4K, but at 0.5K… External laboratory noise level at 3Hz is about 100 pT/Hz^1/2 Required shielding is about 1e6 For a given shield, transverse shielding 4e5 and axial 1.2e5 Extra noise reduction by gradiometer, about 10-100 Overall shielding is satisfactory, although some improvement can be achieved: endcaps, spacing, thicker mu-metal Perhaps, 3D simulations are needed? Some redesign of the shield?