The magnetic field of the shim coils. Simon Schroeder Septimiu Balascuta (09/18/2011) 1. Introduction. We report the calculation of the dependence between the direction of the magnetic field and one of the shim coil current starting from the measurements of the three field components for increasing current in one shim coil from 0 (A) to 1.2 (A). This was done with and without the main guide coils turned off. In the absence of the main coils, the magnetic field in the cave is due to the SMP, CM and the Earth magnetic field. With the current in the main guide coils turned on the field in the center of the guide coils is not completely aligned with the vertical Y axis. The angle between the field and the vertical axis can be decreased by changing the current in the lateral shim coils. The magnetic field measured at the two probes placed on top and bellow the spin flipper is measured for different currents in the shim coils. 2. The experimental setup. The field components were measured on September 10, 2011. The field was measured using the LabView interface and the two Barrington magnetic probes MAG1 and MAG2 located above and below the spin rotator like in figure 1. The directions of the X1, Y1 and Z1 fluxgate axes in the first magnetic probe MAG1 are along the beam axis (Z), opposite to the vertical axis (-Y) and along horizontal X axis respectively. The directions of the three fluxgates in the second magnetic probe (X2, Y2, Z2) are aligned with the axes of the lab X, Y and Z. The names of the measured field components is given by the names of the three fluxgates of the two magnetic probes placed above the spin rotator (X1, Y1 Z1) and bellow it (X2, Y2, Z2). The B y1 component measured by MAG1 is always negative while B y2 is positive. The angle between the direction of the fluxgate axis and the lab axes (X, Y and Z) is smaller than 0.02 degrees for MAG1 and smaller than 0.1 degree for MAG2. In biggest misalignment angle is between the Z2 axis and the Z axis of the coils.
SMP and CM 80.62 MAG1 5.2 Spin Rotator x 1 2.6 y 1 MAG2 Detector shield Y 28.5 Z Liquid Hydrogen target 13.4 58 y 2 14 55.5 z 2 47 70 Figure 1. The position of the magnetic probes (MAG1, MAG2) is seen relative to the steel floor, the spin rotator, the four main coils, the liquid hydrogen target and the detector shield. In this picture the different components are projected on the vertical YZ plane passing through the center of the detector, spin rotator and liquid hydrogen target. The directions of the axes of coordinates Y and Z are along the direction of gravity and the beam axis respectively. The field was measured versus the current in each of the four shim coils and main coil to relate the change in the current with the change in the orientation and magnitude of the field. The orientation of the field is given by the two angles theta and psi defined in figure 2. The angle theta is between the direction of the B field vector projected in the XY plane (normal to the beam axis) and the vertical axis Y. Y θ B x, y Y ψ B y, z X Z Figure 2. The angles theta and psi define the direction of the magnetic field relative to the axes of the NPDGamma experiment. The beam axis is along Z, while Y is the vertical axis.
To calculate the change in the angle theta with the current in one of the shim coil, one can derivate tanθ with respect to the current. If the fields are measured at the first magnetic probe then one can Bx Bz1 write: tan θ = B = B (1) y y1 dθ cos db B B db db = + db = z1 z1 z1 z1 y1 2 2 y1 θ By1 By1 B y1 Bz1 By1 (2) d db db z1 y1 θ = sinθ cosθ B B z1 y1 d θ By1Bz1 1 db 1 1 db z y1 = 2 2 di Bz1 + B y1 Bz1 di By1 di (3) (4) With the approximation Bz1/By1<<1 equation 4 can be written: d θ B 1 db 1 db 1 db B db = = + di B B di B di B di B di z1 z1 y1 z1 z1 y1 2 1 y1 z1 y1 y1 y1 (5) The change in the angle with the current can be calculated also from the field components measured by the second magnetic probe: Bx Bx tan B B θ = = 2 (6) y y2 d θ B 1 db 1 db 1 db B db = = di B B di B di B di B di x2 x2 y2 x2 x2 y2 2 2 y 2 x2 y2 y 2 y2 (7) The vertical component is in general more than 10 times bigger than the horizontal components such that the second term on the right hand side of equation 5 or 7 can be neglected. The change in the angle ψ can be calculated from the field components measured at the first and second magnetic probes: dψ db B db = + di B di B di 1 x1 x1 y1 2 1 y1 y1 (8) dψ db = (9) di B di B di 1 dbz 2 Bz 2 2 y2 2 y2 y2
In section three, the field components are measured at the two magnetic probes when the current in the main coils and auxiliary coils are zero, and the current in only one of the four shim coils is increased from 0 (A) to 1.2 (A) with a 0.2 (A) increment. In section 4 the fields are measured for the same increase in the current in one shim coil, for the nominal currents of 23 (A) and 3.3 (A) in the main and auxiliary coils respectively. 3. The magnetic field of the four shim coils for zero current in the Main coils. The field of the current in one shim coil was measured while the current in the other three shim coils, the four main and auxiliary coils were zero. The dependence of the three field components on the currents in the lateral shim coils is presented in figure 2 and 3. The measurements were repeated twice in different days. Due to the change in the remnant magnetization of the shield there is a small systematic error in the field measurements. The two parameters of the linear fit are presented in table 1. The magnetic field for zero current in the coils is not zero because of the external Earth magnetic field, the fringe field of the SMP and CM, and the magnetic steel shield.
Figure 2: The magnetic field component along X, -Y and Z axis, is measured by the Z1, Y1 and X1 fluxgate of the magnetic probe MAG1, versus the current in one of the four shim coils while the currents in the other three shim coils, the main and auxiliary coils is zero.
Figure 3: The field components measured at the magnetic probe MAG2 (bellow the spin rotator) as a function of the current in the four shim coils. The current in the main and auxiliary coils is zero. The above results indicate that the magnetic field components depend linearly on the current in one of the four shim coils. The coefficients of the fitting function B k =a + b*i are presented in the table 1 bellow. The three axis of the first magnetic probe MAG1 (X1, Y1, Z1) are almost aligned along the Z, X and Y directions respectively. For the second magnetic probes the three axes (X2, Y2, Z2) are almost aligned with the X, Y and Z axes respectively.
Table 1. The two coefficients of the linear fitting function are calculated from the experimental data measured at the two magnetic probes. The slope of the linear fit of the magnetic field components versus the current in the shim coil is b1 and b2 for magnetic fields read at MAG1 and MAG2 respectively. I(shim) Varied (A) a1 b1 Field read by MAG1 Fluxgat e directi on a2 b2 Field read by MAG2 Fluxgate direction Left -0.006689-0.01534 Bx1 X1=Z 0.03027-0.097626 Bx2 X2=X Right -0.006716 0.012689 Bx1 X1=Z 0.030291-0.09909 Bx2 X2=X Front -0.007557-0.068429 Bx1 X1=Z 0.030259-0.79980E-4 Bx2 X2=X Back -0.006755-0.015429 Bx1 X1=Z 0.030292 0.0004244 Bx2 X2=X Left -0.009352-0.026328 By1 Y1=-Y 0.032024-0.014088 By2 Y2=Y Right -0.009306 0.018883 By1 Y1=-Y 0.031922 0.012252 By2 Y2=Y Front -0.009356 0.012352 By1 Y1=-Y 0.032075 0.00911 By2 Y2=Y Back -0.009359-0.026328 By1 Y1=-Y 0.032007-0.003549 By2 Y2=Y Left 0.01877-0.11187 Bz1 Z1=X -0.003385-0.010916 Bz2 Z2=Z Right 0.018686-0.083713 Bz1 Z1=X -0.003398 0.01477 Bz2 Z2=Z Front 0.018647-0.008753 Bz1 Z1=X -0.003435-0.06602 Bz2 Z2=Z Back 0.0181656 0.00255 Bz1 Z1=X -0.003456-0.016416 Bz2 Z2=Z The increase in the current in the left or right shim coil decreases the field component Bx because the slopes dbz1/di[left] and dbz1/di[right] (with Bz1 = Bx) and the slopes dbx2/di(left) and dbx2/di(right) (with Bx2=Bx) are negative. Therefore the fields of the right and left shim coils are both along the negative X axis. The field component along the Z axis (Bz2 or Bx1) also decreases with the increase in the current in the front or back shim coils (dbz2/di(front)=-0.6602 and dbz2/di(back)=-0.016416), (dbx1/di(front)= -0.06843, dbx1/di(back)=-0.01543) proving that the Z components of the fields of the front and back shim currents are along the negative Z axis. 4. The magnetic field of the shim coils in the presence of the guide coils. The field components measured by the first magnetic versus the current in one of the four shim coils, while the current in the main and auxiliary coils are 23 (A) and 3.3 (A) are presented in figure 4. The first magnetic probe located above the Spin Rotator for an increasing current from 0 to 1.2 A in one of the four shim coils. The component of the field in the Y direction is measured by the Y1 fluxgate of the first magnetic probe located above the spin flipper, for increasing current in one of the shim coil (left, right, back or front) from 0 (A) to 1.2 (A). The fields were measured four times. The average and standard deviation are presented in figure 4.
Figure 4: The field components are measured at the first magnetic probe versus the current in one of the four shim coils, while the currents in the other three shim coils are zero and the current in the main and auxiliary coils are 23 A and 3.3 A respectively. The linear fitting parameters are presented in table 2 for each of the 12 sets of data.
Table 2: The two parameters of the linear fit B x, y, z = a + b*i are calculated for the measured fields versus the current in one of the four shim coils, when the current in the other three shim coils is zero and the current in the main and auxiliary coils are 23 (A) and 3.3 (A) respectively. Field Shim coil a b Field Shim coil a b component current (G) (G/A) component current (G) (G/A) Bx1 Left 0.004509-0.015342 Bx2 Left 0.08217-0.09805 Right 0.004462 0.012377 Right 0.082178-0.09936 Front 0.003859-0.068965 Front 0.081486-0.000144 Back 0.0037428-0.015475 Back 0.081436 0.0004267 By1 Left -9.4224-0.026772 By2 Left 9.4061-0.013593 Right -9.4228 0.018764 Right 9.4053 0.012657 Front -9.4235 0.011766 Front 9.4065 0.0096238 Back -9.4241-0.002478 Back 9.4071-0.003126 Bz1 Left 0.11745-0.11195 Bz2 Left 0.055262-0.010844 Right 0.1176-0.08422 Right 0.05567 0.01474 Front 0.1169-0.008846 Front 0.05596-0.066307 Back 0.11663 0.002507 Back 0.05553-0.016389 The slopes of the linear fit of the measured field components versus the current of one of the four shim coils are measured in G/A: dbx1 /di (Left) = -0.015342; dby1/di (Left) =-0.026772; dbz1 /di (Left) = -0.11195; dbx1 /di (Right) = 0.012377; dby1/di (Right) =0.018764; dbz1 /di (Right) = -0.08422; dbx1 /di (Front) = -0.068965; dby1 /di (Front) =0.011766; Bz1/dI (Front) = -0.008846; dbx1/di (Back) = -0.015475; dby1/di (Back) =-0.002478; dbz1 /di (Back) = 0.002507;
Figure 4. The field components measured at the magnetic probe MAG2 below the spin flipper and the shim coil current. The three X2, Y2 and Z2 fluxgate axes are approximately aligned with the X, Y and Z axes of the experiment. The slopes of the linear fit in G/A are calculated for each of the four cases for both magnetic probes. dbx2 /di (Left) = -0.098048; dby2/di (Left) =-0.013593; dbz2 /di (Left) =-0.010844; dbx2 /di (Right) = -0.099356; dby2/di (Right) =0.012657; dbz2/di (Right) = 0.01474; dbx2/di (Front) = -0.00014405; dby2 /di (Front) =0.0096238; Bz2/dI (Front) = -0.066307; dbx2/di (Back) =0.00042676; dby2/di (Back) =-0.0031256; dbz2/di (Back) = -0.016389; The change in the angle θ with the current in one of the shim coils is presented in table 3. In the calculations I considered the field component in the X direction equal with 0.15 Gauss, which is the
maximum value for this field component measured for shim currents between 0 (A) and 1.2 (A). This field component multiplied with the slope dby/di and divided with By 2 is less than 5E-5 (1/A). It is therefore negligible relative to the first term when the current in the left or right shim coils are changed. Table 3: The change in the angles θ and ψ with the current in one of the four shim coils is calculated when the currents in the main and auxiliary coils are respectively 23 A and 3.3 A and the currents in the other three shim coils are switched to zero. Fields read dθ /di(left) dθ /di(right) dθ /di(front) dθ /di(back) at: (rad/a) (rad/a) (rad/a) (rad/a) MAG1-0.01193-0.00898-9.6E-4 2.706E-4 MAG2-0.01043-0.01057-3.156E-5 5.058E-5 Fields read dψ /di(left) dψ /di(right) dψ/di(front) dψ/di(back) at: (rad/a) (rad/a) (rad/a) (rad/a) MAG1 1.72E-3 1.35E-3 7.35E-3-1.65E-3 MAG2 1.174E-3 1.586E-3 7.055E-3-1.745E-3 Since the angle θ has to be smaller than 20E-3 radians, the change in this angle with 5E-3 is accepted. More stringent requirements are placed on the stability of the currents in the left and right shim coils. A change with only 0.1 A in one of left shim coil can increase (or decrease) the angle theta with about 1.2E-3 radians, if the currents in all the other coils are not changed. The current given by the main power supply (Dan Physics) can vary with up to 0.3% in the first 20 minutes after the power supply is switched on. In the next section, the change in the angle theta with the current in the main coils is calculated. 4. The magnetic field of the main and auxiliary coils measured for zero current in the shim coils. The three magnetic field components measured by the two magnetic probe, change also with the current of the main coil. The linear dependence is presented in figure 5.
Figure 5: The three field components of the magnetic field measured at the two magnetic probes are plotted versus the electric current in the main coils. The currents in the shim coils are zero. The current in the auxiliary coils is fixed at 3.3 (A). The relation is linear for all three field components: Bx = Bz1=0.034374 +0.0036041*I[main] and Bx=Bx2=0.040162 +0.0018089*I[main] By=By2=0.66004 + 0.38029*I[main] and By= By1=0.64706 +0.38159 *I[main] Bz = Bx1=-0.01225 +0.77385E-3*I[main] and Bz = Bz2=0.0087994 +0.002008*I[main] Since tan(θ) = Bx/By, the angle theta depends also in the current in the main guide coils.
The derivative of the angles θ and ψ with respect to the main coil current is presented in table 4. The change in the field magnitude with the current in the main coils is presented in the last column. The upper limit of the change in the field magnitude is given by the width of the resonance curve of the spin rotator. Table 4: The change of the angles theta and psi with the current in the main coil is calculated from the field measured at MAG1 and MAG2. The change in the field magnitude with the same current is also presented in the last column of this table. Fields read dθ /di(main) dψ /di(main) db/di(main) at probe (rad/a) (rad/a) (G/A) MAG1 1.029E-3 0.7286E-3 0.3816 MAG2 0.836E-3 0.8574E-3 0.3803 For a 1 Amp increase in the main current the two angles change with less than 1.1E-3 radians. The field magnitude changes with 0.4 Gauss for a 1 (A) variation in the main coil current. Since the Larmor frequency of the RF field in the spin rotator is fixed by the field magnitude, a 0.1 (G) change in the field magnitude can change the optimum RF frequency with 1.2 KHz. The resonance curve of the Spin Rotator has a width at half height equal with 0.4 KHz. Since the resonance frequency of the spin rotator is fixed but the Joerger module of the VME3 NIM crate, the variation in the vertical field magnitude has to be less than 0.04 Gauss to assure that the spin rotator works at the optimum frequency. This requires that the maximum variation in the main coil current is 0.1 (A). 5. Conclusion. The magnetic field was measured from May 30 to July 6 th 2011, during the data collection for Chlorine and Aluminum [1]. The field component in the vertical direction changed with less than 0.03 Gauss. This suggest that the main power supply assures that the main coil current is stable enough such that the angle theta changes with less than 1E-3 radians during about 6 weeks of data collection. If the angle theta has to be smaller than 20E-3 radians, a variation with 0.1 A in the intensity of the currents in the right and left shim coils is acceptable. 6. Reference [1] S. Balascuta, Simon Schroeder: The stability of the magnetic field during the Aluminum and Chlorine measuremets (May-July 2011), posted on: http://battlestar.phys.utk.edu/mediawiki/index.php/guide_field