1. The magnetic field uniformity and gradient of the B0 coils inside the lower cryostat for the nEDM experiment.
S.Balascuta , dr. Ricardo Alarcon
ASU
Dr. Brad Plaster, Dr. Brad Filippone
Caltech
February 12, 2006

2. The Model of the Lower Cryostat in TOSCA (Post-processor)
R=60 cm, L=396 cm
Saddle “cosine theta coils”
Cryoperm shield (R=62.23 cm, L= cm, thickness cm (.04 inch)).
Superconducting shield (R=62.87 cm, L= cm, thickness cm (.01 inch)).
Four mumetal shields (all inch thick, L=610 cm) with radii: cm, cm cm and cm.

3. The problem of the false EDM of the neutron.
A false nEDM can arise from two sources:
1. The presence of magnetic field gradients.
2. The effective magnetic field of a particle moving in the electric field.
B0=10mG
For ultra cold neutrons If
then

4. Circular cosθ and “modified” coils.Y
N=4 K=0
N=4, K=0.3 I1
N=8, K=0 Y 1 1’ 1 1 2’ 2 2 3’ 3 2 4’ 4
X(2) X
X’(2) X
I2L
I2R R R
Define the ratio K:
K=(x (j) - x’ (j))/ x (j) A B
x‘ (j)= x (j) * (1-K)
J=1,…, N/2
J=1,…, N/2
x(j)=(2j-1)*R/N

5. Field uniformity and average field gradient for a system of 40 cosθ coils (k=0) and modified coils (K from to 0.002) inside the Cryoperm, Superconductor and 4 Mumetal shields.

6. |<dBx/dx>|/B0<1E-6 only if K>1.6E-3 and K<1.88E-3.
The average volume field gradient for a system of 40 modified coils (with different K and no errors in position) placed inside the Cryoperm, Superconductor and 4 Mumetal shields. Y
12 cm X
10 cm
7.6 cm
-25 cm < z < 25 cm
The relative average volume gradient inside the right cell (40 modified coils)
|<dBx/dx>|/B0<1E-6 only if K>1.6E-3 and K<1.88E-3.

7. The average volume field gradient for a system of 40 modified coils (with different K and no errors in position) placed inside the Cryoperm, Superconductor and 4 Mumetal shields.
For 40 Coils (K=1.6E-3) DS= mm (coil1) to mm (coil 15).
The 40 Coils (K=1.88E-3) DS =0.056 mm (coil 1) to 1.27 mm (coil 15).
This requires a precision better then 1 mm in the location of the coil (difficult to achieve in practice).

8. 10 sets of 40 coils (K=0.00174) with random (Gaussian) errors (added to the position of the coils).
If the position of the 40 coils (K= ) have no errors then <dBx/dx>/B0= E-6 (1/cm).
Statistics
<dBx/dx>/B0(right cell)
<dBx/dx>/B0 (left cell)
Max |<dBx/dx>|/B0
7.24E-6
3.839E-6
Standard deviation
3.365E-6
2.218 E-6
Mean
E-6
E-6 (1/cm)

9. N=40 coils (K1) and M=8 coils (K2) with no errors in location of the coils (K1=K2).
I1 and I2 are the electric currents through the N=40 and the M=8 coils respectively.
K1=K2
B0(mG)
I1(mA)
I2(mA)
0.01 10
14.5
50.76
24.88
6.59
0.0
24.9
-10.71
Figure 1(a). The average volume gradient of the field inside the
N=40 coils + M=8 coils (K1=K2) with no errors in position versus the ratio I2/I1. For an optimum ratio I2/I1 , |<dBx/dx>/B0| is zero.

10. 40+8 modified coils with errors in the location of the coils for K=0 (I2/I1=-0.5) and K=0.01 (I2/I1=4). A B
Figure 1 (A, B): For 10 sets of coils with random errors in position of the coils with K=0 (FIG A) and K=0.01 (FIG B).
Fig 1 A: Mean=-0.71 E-6 1/cm
Standard dev=3.213 E-6 1/cm
Fig 1 B: Mean=1.63 E-6 1/cm.
Standard dev=1.954 E-6 1/cm.

11. The sum of the average gradients <dBx/dx>/B0 in the left and in the right cells can be decreased by adjusting one of the electric currents I2L or I2R while the other one is left unchanged.

12. The average volume electric field <dBx/dx> inside the left and right cells can be made smaller then 1E-8 (1/cm) by adjusting the electric currents I2L and I2R through the M=8 coils. Before and after the adjustment B0 remains essentially the same (41.8 mG for E1 and mG).
Set of coils
K=0.01
1) I1=60mA
I2L=I2R=
210mA
I2L/I1=3.5
<dBx/dx>/B0
(1/cm),
left / right
Step 2)
I1=60 mA
I2R=210 mA
( 1/cm)
Left Cell Right Cell
3) I1=60 mA
I2R and I2L are both increased
dBx/dx>/B0
(1/cm)
Left Cell
Right Cell B0
(mG) E1
Left cell
1.879E-6
I2L= mA
2.088E-6
I2L = mA
I2R = mA
E-6
41.8
Right cell
-2.234E-6
-2.079E-6
E-6 E2
5.647E-6
I2L=209.64 mA
3.209E-6
I2R=231.2
I2L=230.8
0.183 E-6
43.33
-0.357E-6
E-6
-0.21 E-6

13. Conclusion
If two set of coils (N=40 andM=8) are built with the same K and with currents I1 and I2 then the electric current I2 (through the M=8 coils) can be adjusted such that the average gradient of the field is minimized
The sum of the average volume gradients in both cells can be made zero if the electric currents through the coils with centers on +OX (I2R) is allowed to be different then the current through the coils with centers on -OX (I2L). This difference depends on the “errors” in the position of the coils. The average value of the relative ratio (I2L-I2R)/I2L is smaller then 1%.