Preliminary Design of the n2edm Coil System

Preliminary Design of the n2edm Coil System Christopher Crawford, Philipp Schmidt-Wellenburg 2013-07-03 1 Introduction This report details progress towards the design of an electromagnetic coil package for the second phase (n2edm) of the experiment at Paul Scherrer Institut (PSI) to measure the electric dipole moment of the neutron (nedm). A non-zero moment would violate the fundamental symmetry (CP ) of particle physics, which would satisfy one of the three conditions required to explain the matter-antimatter asymmetry of the universe. Mechanisms of the standard model are unable to account for the magnitude of this baryon asymmetry, so observation of an electric dipole moment in a hadronic system would signify a new channel of CP -violation, and a significant departure from the standard model. The search for the nedm is an international effort with collaborations in Switzerland, Germany, France, Russia, England, United States, Japan, Canada, and elsewhere. The goal of a recent visit of one of us (CC) to PSI was to foster greater communication and collaboration between independent experimental collaborations focusing on extending the limits of this technically demanding precision measurement. Phase I of the PSI nedm experiment used the shield and coil package of a previous experiment which ran at the Institut Laue Langevin (ILL). The Phase II magnetic shielding package was completely redesigned to achieve higher shielding of external magnetic fields. The major change was a shift from cylindrical to rectangular geometry, which allows uniform degaussing of all six faces, increasing the effective permeability. The coil package must fit inside this multilayer cubic shield, with 2.2 m inside length, and a 2.0 m clearance. Our immediate goal was the preliminary design of a coil to fit around the n2edm measurement cell and inside this shielding package subject to the following specifications and constraints: 1. The package should include three coils: a uniform vertical field B 0 coil, a uniform horizontal field B RF coil, and a gradient B xy coil. It should also include compensation coils for all spherical magnetic moments up to L = 6. 2. The gradient of the B 0 coil should be less than 2 pt/cm within the area of the measurement cell, a cylinder 60 cm diameter by 20 cm long. 3. Magnetic field lines should be absent from all permeable material to avoid magnetic field dependence on the magnetization state of the material, allow for precise field reversals, and avoid RF eddy-currents. 4. The usable volume inside the coil package should be as large as possible. 5. Several penetrations are required into the coils for passage of ultracold neutrons, high voltage, 3 He, and other magnetometry. Typically, cylindrical or spherical cosine-theta coils are used for these applications, which are optimal for cylindrical or spherical geometry. These can be constructed as field cancellation coils by embedding one such coil of smaller radius inside another, with larger field, but equal and opposite external magnetic dipole moment. Spherical cosine-theta coils have hermetic windings, impeding access to the interior volume. Standard cylindrical cosine-theta coils have fringe fields due to finite length, which must be compensated for to get uniform fields inside. The new design reported herein is a generalized cosine-theta coil with two layers for external field cancellation, with a rounded rectangular profile optimized to the cubic shielding geometry, and with removable end-caps which completely cancel end-fringe effects. Special windings will be designed around each aperture to minimize the resulting field perturbations, such that the gradients will be designed 1

Figure 1: Solution of Laplace boundary value problem for one quarter of a double-rounded-rectangularcosine-theta-coil (DRRCTC). The other sections can be recovered by symmetry. The red lines show the magnetic flux return lines, the continuation of horizontal field lines in the inner rectangle. The blue lines trace equipotentials, along which current flows on the end caps. A straight wire perpendicular to the plane of the figure is situate at each end of each equipotential, along the inner and outer rectangles. as small as physically achievable. Random residual field perturbations due to construction tolerances will be systematically eliminated up to high orders with a set of harmonic cancellation coils. 2 Design Method The well-known analytic formulations of the standard spherical and cylindrical cos-theta coils can be derived as two special cases using a general method. This method can be used to calculate the winding geometry for any coil based on geometrical constraints and field specifications. It is based is based on a practical physical interpretation of the magnetic scalar potential U and its connection to current I in an analogous way that the electric potential V is tied to voltage V. Just as electrodes are designed using the Poisson equation ɛ V = ρ, we calculate coil windings using the Laplace equation µ U = 0. In fact, equipotentials of the solution U form the exact calculated coil windings needed to produce the resulting field B, specified on the boundary of the coil by flux (Neumann) boundary conditions to the Laplace equation. As a result of employing this method, we are not tied to any specific geometry (spherical or cylindrical), but can adapt the coils to their environment, in our case a cubic geometry. In addition, this method prescribes a straightforward procedure for designing optimal end-caps, which preserve the field uniformity and axial symmetry of the infinitely long cos-theta coil, even in very short volumes. While the µ in the Laplace equation above takes magnetic materials in account, we prefer to omit magnetic materials inside the coil package, if possible, to avoid complicated hysteresis effects and dependence on poorly known values of µ. The design for the current coil was broken into simpler problems by proceeding in the following stages: 2

Figure 2: 2-dimensional fieldmap of a discrete single-rounded-rectangle-cos-theta-coil (SRRCTC), showing the fractional deviation from a uniform field with respect to the center of the coil. The wire positions were optimized to achieve the most uniform field profile at each point on the interior grid marked with x s (left); or to minimize the first 15 interior multipole moments m = 3, 5, 7,..., 31 (right). 1. 2-d infinitesimal windings: The first stage was to calculate exact windings for an infinitely long coil with z-symmetry. In the special case of additional ρ-symmetry, the analytic formula is well-known: the double-cosine-theta coil. We extended the solution to the geometry of inner and outer rounded rectangles (instead of circles) as shown in Fig. 1 by solving the Laplace equation numerically with the finite element method, using the commercial software package COMSOL. The field lines are shown in red, which are orthogonal to the equipotentials of U, shown in blue. In two dimensions, the dot at the end of each equipotential at the inner and outer rounded rectangles represents a wire extending normal to the plane of the figure. The figure shows only a few equipotentials, but the field will be identically uniform inside the inner rectangle and identically zero outside the outer rectangle in the limit of infinite equipotentials (wires), each carrying an equal amount of infinitesimal current, so the entire coil can be wound in series. 2. 2-d discrete windings: A physical coil is limited to n windings, but we expect the field uniformity to improve with higher n. Treating this as an n-dimensional optimization problem, one can expect to minimize up to n 1 spurious internal magnetic moments and n 2 external magnetic moments by adjusting the positions of n = n 1 + n 2 inner and outer wires (not necessarily n 1 and n 2, respectively). This problem is exactly soluble for a cos-theta coil with equally spaced wires, in which case the current in each wire goes as I 0 cos(θ), and hence the name. However for our geometry or even for a cosine-theta coil wound in series, there is no analytic solution. However this problem can also be solved numerically in one of two different ways: a) calculate the field H(x, y) = i 2I i ŷ(x xi) ˆx(y yi) (x x i) 2 +(y y i) 2, where the current I i flows through the wire at (x i, y i ). We then form χ 2 = j (H(x j, y j ) H j ) 2, where H j is the desired field value at each position (x j, y j ) of a grid of response points located inside (H j = ˆxH 0 ) and/or outside (H j = 0) the double coil. Then we minimize χ 2 as a function of the position of each wire, starting from the nominal positions calculated in stage 1. b) calculate the set of internal and external moments of the coil, and solve for the wires positions which zero all moments except for the desired internal dipole (uniform interior field). A preliminary optimization of the coil for n windings is shown in Fig. 2 using methods a) and b), respectively. 3. 3-d finite cylinder: For a finite-length cylinder, individual wires must be connected at each end to form a complete circuit. This is done by connecting inner and outer windings to printed circuit board traces on the front and back as, shown in Fig. 3. As there are more inner than outer wires, the 3

Figure 3: Cartoon illustrating the construction of a double-rounded-rectangular-cosine-theta-coil (DR- RCTC). The current-carrying rods are supported and aligned by a series of identical lamellas, each machined by the same CNC mill for uniformity. The series circuit is continued by a traces etched onto the two end faces, which are soldered onto the adjacent wires. End-caps (not shown) are added to exactly compensate end fringes. extra inner wires are connected between the left and right sectors, which have current of the opposite polarity. While this completes the circuit, it does not deal at all with fringe effects. This is done with a separately wound end-cap at each end of the coil. The currents in the end-caps are also calculated exactly using the scalar potential. From stage 2, we have an exact calculation of the potential U and field lines H for the infinitely long coil. Now we calculate the windings of end caps which will preserve the exact shape of the z-symmetric field lines over a finite length. As before, equipotentials of U determine the exact windings needed to do this. From this potential, we must subtract a potential U representing the end winding tracess above, to obtain the final end-cap windings which will completely cancel fringes. 3 n2edm Coil A attractive feature of the coil designed above is that it maintains z-symmetry of the infinitely long coil. This means that all side windings are perfectly straight, and all curved windings are restricted to the end-caps. 4

This design lends itself to a particularly simple method of laminar construction: a) the support structure for the coil is machined as a series of identical lamellas with holes to insert copper rods for each winding of the coil. Each sheet is milled with a computer-numerical-controlled (CNC) machine, so that all tolerances are free of human error. The lamellas are mounted onto side frames and laser-aligned into position. b) traces for the return flux on the front and end lamina are CNC-routed according to calculations. c) the the end-caps also have one layer of CNC-milled circuits for each coil, similar to standard printed circuit boards (PCB). All outer faces of the frame will be fitted with a series of independent correction coils to provide all moments up to L = 6. A schematic of the construction of the coil without the end-cap is shown in Fig. 3. 4 Conclusion An optimal design for the construction of a generalized cos-theta coil has been presented. A plan for design of the coil has been laid out in detail, and has been implemented to first order. Final details of the design, such as the optimal number of wires, inner and outer rectangle dimensions and end-cap winding geometry have yet to be determined. Our next step will be to design and build a small quarter-scale single-coil prototype. This project, DISCO (Double Iso-Scalar potential COil), will be carried out in collaboration with the lead institution LPC-CAEN, and others in the PSI nedm collaboration. Field uniformity measurements will determine the ultimate uniformity achievable with this type of coil. This design illustrates the power of using the scalar magnetic potential to directly calculate the windings of custom electromagnetic coils. 5