Holding Magnetic Field

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1 Holding Magnetic Field Abstract The design of a solenoid to produce the required rotating/holding magnetic field in the n- 3 He experiment is presented The solenoid is constructed using 19 currents which are 15 cm apart from each other When calculating the magnetic field using the same current in each coil, it was noticed that the field produced, even it is in the z direction, the shape was far from the pursued magnetic field; in order to remedy this problem an optimization was performed: given that the magnetic field in each point in space is the superposition of the magnetic fields of each coil in that particular point, therefore it is advantageous to write the total magnetic field as a vector, which is equal to the product of a magnetic field matrix constructed with the magnetic field of each current times a current vector (the magnetic field vector dimension is n, the number of points where it is desired to establish the magnetic field, the matrix dimension is n m, m is the number of coils, finally the current vector dimension is m) It is easy to see that solving the matrix equation for the current vector, proposing an ideal magnetic field vector, will give the magnitude of the currents which would produce this desired magnetic field Note that n m, consequently to obtain the inverse of this matrix, the singular value decomposition technique was used A weight function was introduced in each side of the matrix equation to help to solve readily the equation, this weight function is constructed by physical insight: in regions where it is of crucial importance to have the proposed magnetic field, the weight function acquires larger values than it acquires in regions where this importance is smaller Using these techniques, the currents which produce the desired magnetic field were successfully found 1 Introduction The set up of the n- 3 He experiment will require some of the technology used in the NPDGamma experiment, particularly the super mirror polarizer which polarizes the neutron spin in the transverse direction of the beam motion; besides, the super mirror produces a residual magnetic field in the y direction few centimetres after the end of the lead wall In order to suppress parity conserving systematic effects, it is needed that the neutron beam polarization rotates to have the same direction of the beam motion, said in other words, the neutron spin σ n and neutron momentum k n must be parallel; moreover, after achieving the rotation, it is required to hold the neutron polarization through the whole experimental configuration The most effective way to achieve these characteristics, is using magnetic fields, therefore a device which produces such a magnetic field was developed It is straightforward to produce magnetic fields using steady electric currents carried by copper wires, and taking advantage of the magnetic scalar potential method it is easy to determine the current geometry [1, 2, 3] 1

2 2 Required Magnetic Field 21 Adiabaticity To rotate the neutron spin by π/2, the change of the magnetic field needs to be slow in order that the spin adiabatically follows the field direction; then it is needed a uniform magnetic field to hold the neutron polarization To establish more precisely that this condition is fulfilled, several fields were proposed and an adiabaticity parameter λ was calculated for each of them: λ = ω L ω B = γb 1/B 2 B (db/dt) = γb ( 3 v B y db z dz B z dby dz ), (1) this equation represents a comparison between the Larmor precession frequency ω L of the neutron spin in a magnetic field and the frequency of change of the magnetic field; B is the magnetic field magnitude, γ is the neutron gyromagnetic constant and v is the neutron speed If λ 1, it can be shown that the neutron spin will adiabatically follow the magnetic field direction [4] It is clear that the magnetic field vector has two components: one in the z direction, B z (z), produced by the rotating/holding field and another that corresponds to the residual magnetic field of the super mirror polarizer in the y direction, B y (z) The proposed fields to manipulate in this way the neutron spin are shown in Figure 1 and have the form B z (z) = (A/2)[ tanh((z z 0 )/α) + 1], where A = 10 G [3] is the maximum field amplitude, α = 10, 15, 20, 25, 30 cm and it helps to set how fast the function reaches A; z 0 = 10 cm and indicates where the function starts to grow The frame of reference adopted in this technical note is the one where the end of the shielding lead wall is at 250 cm [3] The calculated adiabaticity parameter for a 10 mev neutron is shown in Figure 2 From this, the decision to use the case α = 15 cm was taken, because it is expected that this field will be easy to produce in practice In the most adverse situations λ 20 1 [5]; then, λ grows very fast for higher z, consequently the field meets the adiabatic requirements 22 Magnetic Scalar Potential Although it is evident that the geometry of the currents will be similar to a solenoid, it is illustrating to establish this geometry in an analytical way The magnetic scalar potential has proven to be a valuable tool in obtaining the currents when the magnetic field is given beforehand [6, 8, 7] This makes sense when the region of interest has no currents, which is the case of the n- 3 He experiment The magnetostatic equations with J = 0 are: therefore, B = 0, H = 0, (2) H = Φ M, 2 Φ M = 0, (3) where Φ M is the magnetic scalar potential and the magnetic boundary conditions are 2

3 Figure 1: Proposed magnetic fields The hyperbolic tangent shape of the fields comes in a natural way because it is a simple analytical function that satisfies the desired field features Here, the end of the lead wall is at z = 250 cm ˆn (B 1 B 2 ) = 0, µ Φ M1 n µ Φ M2 n = 0, (4) ˆn (H 2 H 1 ) = K (5) Using the magnetic boundary conditions is possible to give a simple and powerful interpretation of the magnetic scalar potential: the equipotential curves on each boundary specify the geometry of the wires because the equipotential curve Φ M is by definition perpendicular to Φ M and eq (5) implies that K is also perpendicular to Φ M, since both lie on the boundary surface, K flows along the equipotential lines [7] The present problem is reduced to solve the Laplace equation for Φ M with boundary conditions consistent with the proposed magnetic field either imposing the Neumann or Dirichlet conditions where needed The problem was numerically solved in two dimensions with Comsol, the reason of this is that at first it was not well established the final 3D geometry and having in mind that in the ideal case of an infinite solenoid, it is possible to show that the magnetic field runs parallel to the axis regardless of the cross-sectional shape of the coil [8] The region of interest is a volume which starts just after the end of the lead wall Without surprise it was found that the equipotentials surfaces are planes perpendicular to the field, plotting 19 surfaces they are separated 15 cm from each other, accordingly the electric current geometry has the shape of a solenoid 3

4 Figure 2: Adiabaticity parameter λ for the several proposed magnetic fields in regions for z after the end of the lead shielding wall 23 Magnetic Field Calculations To calculate the magnetic fields generated by the currents, the software Comsol was used again, specifically in the electromagnetic interface the Multi-Turn Coil Calculation domain is the most adequate for this problem, ie it allows to model the current by an ensemble of copper wires [9, 10]; 19 currents separated 15 cm from each other were modelled following the final shape of the coils which is a square with rounded corners, Figure 3 shows this model, the first coil is just after the end of the lead wall To compute the magnetic field two main equations were used: ( µ 1 0 µ 1 r B ) = J e, J e = NI coil A e coil, (6) with the constitutive relation B = µ 0 µ r H, J e is the external current density, this way to introduce J e in the model by the second expression of eq (6) is intended to reproduce the actual winding with copper wire; here N is the number of turns, I coil is the current magnitude in Ampères which passes through the wire, A is the cross sectional area given by the wire gauge and e coil is the direction followed by the current The calculated magnetic field of the 19 coils is shown in Figure a, a comparison between the calculated field and the proposed or ideal one is made, evidently the two fields are pretty different and the produced field by the coils does not satisfy the requirements of the experiment 4

Figure 3: zy view of the solenoid The dimensions shown are the real dimensions given by the n- 3 He experiment requirements; furthermore the environmental conditions are the usual ones: pressure of 1

5 Figure 3: zy view of the solenoid The dimensions shown are the real dimensions given by the n- 3 He experiment requirements; furthermore the environmental conditions are the usual ones: pressure of 1 atm and temperature of K The modelled volume is filled with air and is big enough to enclose the magnetic field lines Figure 4: Comparison between the calculated magnetic field and the ideal magnetic field There is a significant discrepancy between the two of them; the calculated magnetic field is far from satisfying the n- 3 He experiment requirements 5

6 3 Optimization of the Magnetic Field The problem that the calculated field does not give the correct shape could be attributed to the fact that the scalar potential was solved in two dimensions and that the magnitude of the currents was set uniform and somewhat arbitrarily To account for this issue, an optimization is necessary It is advantageous to notice that the magnetic field in a given spatial point is the superposition of all the contributions of the field of each coil in that point, finally is useful to have in mind that the relation between the magnetic field magnitude and the magnitude of the electric current is linear (Biot-Savart Law) Mathematically this ideas can be expressed by the matrix equation: Bz 0,T (z 1 ) Bz(z 1 1 ) Bz(z 2 1 ) Bz m (z 1 ) I0 1 Bz 0,T (z 2 ) Bz(z 1 2 ) Bz(z 2 2 ) Bz m (z 2 ) I 2 0 = (7) Bz 0,T (z n ) Bz(z 1 n ) Bz(z 2 n ) Bz m (z n ) I0 m The notation is as follows: each current is named as I0 k and it is the magnitude of the k th coil, the first I0 1 and the second I0 2 are coplanar and are located just after the end of the lead wall, the third one I0 3 is 15 cm apart from the first ones, then the next I0 4 is 15 cm apart from the third and so on Hence, Bz 0,T (z i ) is the total calculated field in the z direction on the i th point; the subscript/superscript 0 means that are the initial parameters without any optimization Each column of the matrix is the contribution of the field by one specific current in all the spatial points; then, each row of the matrix stands for the contributions to the magnetic field by all the coils in one particular point It is important to point out that the units of the fields used to write the matrix are Gauss, also the units of the magnetic field vector Bz 0,T (z i ) are Gauss, in consequence the current vector which naturally is written as I0 k = N0 k I coil, here it is changed to I0 k = 1 and simply represents the dimensionless magnitude of the currents In other words the number of turns N, the electric current which carries the wire I coil and the cross sectional area of the volumetric current A are all included in the matrix To construct the matrix, the calculation of the field produced for each coil was performed, Figure 5 Now, it is not difficult to note that the optimization consists in changing the initial total vector field Bz 0,T (z i ) by the ideal magnetic field vector Bz id,t (z i ), use the same matrix and solve the matrix equation for the new current ideal vector Iid k, this would be the fraction of I0 k needed to produce the desired magnetic field: each current will be multiplied by Iid k Effectively the change of the current is just the change of the number of turns N or the magnitude of I coil in a specific winding On the typical case the magnetic field matrix will not be square because the number of coils that are being constructed in practice is small compared to the number of spatial points where the field is required to be established, ie n m On that account it is necessary to come up with special techniques: 6

7 Figure 5: Magnetic fields generated for individual of the coils a) this plot shows the magnetic field of a single coil in the region of interest, it is clear that is a function of z, Bzk (z), it was calculated as mentioned in the text, using the Ampe re s law; b) this graph plots the individual fields of each coil, it is impractical to name them; nevertheless, the field of the first coil which is just after the end of the lead wall, is at the beginning of the graph in the left side, the next is 15 cm apart and so on 1 The singular value decomposition (SVD), could be used to calculate the pseudoinverse of a matrix [11] The SVD for a matrix has the form C=USVT ; C is a matrix with dimensions of m n, U is a unitary matrix with dimension of m m, S is a diagonal matrix with dimension m n and V is a unitary matrix with dimensions of n n It is assumed that the S diagonal entries, the singular values, are not negative The problem is to solve: Iid = B 1 Bid, (8) where Iid is the ideal current vector, B is the calculated magnetic field matrix and Bid is the ideal magnetic field vector Then, the SVD pseudoinverse computation: 1 T B 1 = BT B B, B = USVT, BT B = VST UT 1 1 BT B = VS2 VT = VS 2 VT, 1 T BT B B = VS2 VT VSUT = VS 1 UT USVT (9) (10) (11) Notice that even S may has zero values on the diagonal, when calculating S 1 the matrix is not singular and the zero values stay the same Therefore the ideal current vector is calculated with Iid = VS 1 UT Bid (12) 2 Unfortunately, the way that Iid is calculated, for using a pseudoinverse, does not give the expected result, to solve this situation a weight function w(z) was introduced 7

8 which needs to be consistent with n- 3 He experiment s features, in other words this weight function will establish in what regions it is more important to have the ideal field, leaving other regions where the requirement of generating the proposed magnetic field is smaller The weight function w(z) tries to relax the imposed conditions and allow eq (12) to work accurately; it is highly useful to consider two particular points: a) When taking a closer look to the calculated field produced by I 0, where z < 250 cm is evident that the two fields, the ideal and initial are close, therefore it is suggested that the weight function should be small here For z > 250 cm it becomes more distant one field from the other, furthermore the importance to obtain the ideal magnetic field is greater, consequently the weight function would need to gradually grow b) It is helpful to note that in regions near z > 500 cm where there will not be currents, it is physically complicated to approximate to the ideal field (which is constant and equal to 10 G) in these regions, because the magnetic field generated by I 0 decreases naturally when moving away from the currents, so ask that w(z) gradually decreases, seems to be a good option Stated in different terms, adding the weight function attempts to bring the ideal field to the real capabilities of the device, always fulfilling the n- 3 He experiment s requirements It is worth to mention that the dimensions of w(z) are arbitrary To see on firm grounds how the weight function is going to be introduced, the matrix equation including the weight function is: w 1 Bz id,t (z 1 ) w 1 Bz(z 1 1 ) w 1 Bz(z 2 1 ) w 1 Bz m (z 1 ) Iid 1 w 2 Bz id,t (z 2 ) w n Bz id,t (z n ) = w 2 Bz(z 1 2 ) w 2 Bz(z 2 2 ) w 2 Bz m (z 2 ) w n Bz(z 1 n ) w n Bz(z 2 n ) w n Bz m (z n ) I 2 id I m id (13) The definition w(z i ) = w i was adopted It is clear that at the end of the calculation, I id will be found, given that w(z) is introduced in both sides of the matrix equation The shape of w(z) is subtle, the weight function which gave the best results is plotted in Figure 6 In the SVD calculation of the pseudoinverse, when S 1 is constructed, it is possible to have the freedom to choice how many of these singular values may be used, ie when an election of how many singular values is made, the diagonal matrix just have this quantity of singular values and the remaining diagonal entries are filled with zeros The more singular values are used, the more information is introduced in the calculation of I id ; this freedom proves to be useful, the reason of this is that sometimes using all the singular 8

9 Figure 6: Weight Function In the adopted reference coordinate system, the importance of the field before the lead wall is small compared with subsequent sectors, at the end of the currents this importance diminish in the same way values could cause inaccuracies, and one needs to check how many of these singular values are the best in each case For this instance: the magnetic field with α = 15 cm, with w(z) plotted in Figure 6, it was chosen to use 16 of the 19 singular values available because with these we are introducing enough information to obtain good results but not too much to force the method It is worth to mention that the computations were performed in MATLAB, because it has a lot of efficient functions intended to handle matrices, particularly it includes an automatic SVD calculation The Table 1 shows I id Figure 10 shows the ideal field and the optimized field, now, they are actually very close, in the regions where it is needed Vector Coil 1 Coil 2 Coil 3 Coil 4 Coil 5 Coil 6 Coil 7 Coil 8 Coil 9 Coil 10 I I id Vector Coil 11 Coil 12 Coil 13 Coil 14 Coil 15 Coil 16 Coil 17 Coil 18 Coil 19 I I id Table 1: I 0 y I id vectors When changing the initial currents with the ideal currents, the ideal field is achieved: NI k coil will be multiplied by Ik id 9

10 Figure 7: Comparison between the ideal field, the optimized field an the calculated in Comsol with the new currents The differences between them are not a problem to the needs of the n- 3 He experiment 4 Practical Issues 41 Wire To construct the solenoid in practice, it was thought as mentioned above, that the current will flow trough copper wire The available power supply could provide a maximum current of 25 Ampères and 50 Volts, therefore the 12 AWG copper wire should be useful, its maximum ampacity is of 25 Ampères, its resistance is 5211 Ω/km For a magnetic field of 10 G= µ 0 NI, NI 796 Ampères/m; for a 25 m large solenoid, with a current of 125 Ampères in the wires, N 8 for each coil, this numbers are the inputs in the Comsol model Because each coil has a square frame has a length of 70 cm for each N is needed 280 cm of copper wire, the total wire needed is about 0365 km With the same number of turns in each coil, the total resistance, if connected in series, is R T = (5211 Ω/km)(0365 km) = 19 Ω, so the needed voltage is V 24 V, this is supported by the power supply 42 Frames To maintain the windings in the correct place, holding frames are needed The material of these frames should be non magnetic, easy to machine and should have a good resistance to the heat and the wire weight; the acrylic plastic meets all of these conditions Hence, the design of such a frames was made; the design takes into account the requirements of the experimental set up: the size needed to envelope all the components and how the frames are going to be mounted, also the design is consistent with the machine shop capabilities The engineering planes are shown in Figure 10, the frames consist in two 10

11 different pieces which are glued together, in middle there is a channel to put the wire 11

12 Figure 8: Several views of the acrylic frame design The lengths are given in mm It is intended to draw how two pieces are constructed to bond together and complete a single frame a) The frontal view is similar for the two pieces; b) the side view of piece 1 shows that this piece has a 5 mm step, moreover this piece has a total height of 35 mm and a 25 mm internal height, the width is 15 mm; c) the side view of the second piece is similar to the piece 1, but here the 5 mm step is inverted, therefore the two pieces perfectly match 12

13 References [1] J D Bowman et al, A Measurement of the Parity violating Proton Assymmetry in the Capture of Polarized Cold Neutrons on 3 He: A Proposal, Submitted to the SNS FNPB PRAC,(2007) [2] J D Bowman et al, Detector Development for an Experiment to Measure the Parity Violating Proton Asymmetry in the Capture of Polarized Cold Neutrons on 3 He, (2008) [3] R Alarcon et al, Proposal Update for the n- 3 He Experiment, (2009) [4] R G Littlejohn, S Weigert, Adiabatic Motion of a Neutral Spinning Particle in an Inhomogeneous Magnetic Field, Physical Review A, VOL 48, 2, (1993) [5] S Balascuta et al, The Implementation of a Super Mirror Polarizer at the SNS Fundamental Physics Beamline, Nuclear Instruments and Methods in Physics Research A, VOL 671, (2012) [6] J D Jackson, Classical Electrodynamics, 3rd Edition, John Wiley & Sons, Inc, (2001) [7] C B Crawford, Yunchang Shin, A Method for Designing Coils with Arbitrary Fields, Technical Note, (2009) [8] D J Griffiths, Introduction to Electrodynamics, 3rd Edition, Addison Wesley, (1999) [9] COMSOL, COMSOL Multiphysics User s Guide, (2012) [10] COMSOL, Single-Turn and Multi-Turn Coil Domain in 3D Tutorial, (2012) [11] L N Trefethen, D Bau, Numerical Linear Algebra, Society for Industrial and Applied Mathematics, (1997)

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