The cos-theta coil re-re-visited Christopher Crawford, University of Kentucky DNP Fall Meeting, Newport News, VA 2013-10-26

Magnetic scalar potential B.C.’s: Flux lines bounded by charge Flux lines continuous Flow sheets continuous (equipotentials) Flow sheets bounded by current Field Equations field potential Boundary conditions field potential Geometrical Gauss -> Ampere’s law U interpretation as boundary currents Statement in terms of boundary conditions Technique for calculating coils

What is a cos θ coil ? Solenoid U=-z Cos φ coil U = -x = - ρ cos φ U = -z = - r cos θ CYLINDER SPHERE Symmetry z φ Wire pos. φi θi Const. surf. ρ0 r0 Moment Topology infinite bound

Single cos θ coil

Single solenoid

Single cos φ coil

Arbitrary angle

Flux containment Three limits of boundary conditions in the return yoke: μ = ∞ (ferromagnetic) Magnetization currents High static shielding factor THIN! Image currents: automatic self-compensation (approximate) (exact) μ = -1 (superconductor) Super- currents Infinite dynamic shielding factor Need space between the coil and shield μ = 1 (wires) Conductor currents No external field distortion Must calculate wire positions! Three topologies: sphere Separate shells cylinder Common surface Restores z-symmetry torus No flux return

Double cos θ coil Dipole Moments μinner = - μouter I = ΔA/A H0

Double cos φ coil n3He Spin Rotator (TEM RF mode): Reverses either longitudinal OR transverse polarized neutrons No fringe fields in the neutron beam Self-shielding – no eddy currents in Aluminum enclosure

Discretization of cos φ coil Standard winding: one wire at center of each slice Optimization: – nominal dipole m=1 – vanishing higher moments m=3,5,7,… Nonlinear equations: – solved iteratively by Newton-Rhapson method for φi – up to m=15 (15 wires) or m=27 (30 wires)

Discretization of cos φ coil Equally spaced wires Fourier cosine series Optimization: – nominal dipole m=1 – vanishing higher moments m=3,5,7,… Linear equations: – unitary matrix – can null N-1 odd moments using N wire pairs – can tune individual currents in situ – use as shim coils for series cos θ winding

Rounded cos φ coil

Rounded cos θ coil B0

Conclusion The Cos θ coil can be classified according to symmetry, moment, and topology Can use double layers for exact field-cancellation Use the scalar potential, one can apply the properties of a symmetric cos θ coil to any geometry