ADVANCED MAGNETIC FIELD DESCRIPTION AND MEASUREMENTS ON CURVED ACCELERATOR MAGNETS P. Schnizer, E. Fischer, A. Mierau, GSI, Darmstadt, Germany B. Schnizer, Technische Universität Graz, Graz, Austria P. Akishin, JINR, Dubna, Moscow Region, Russia Abstract The SIS100 accelerator will be built within the first real- isation phase of the FAIR project. The series production of its superconducting bending magnets was started without any test model in 2013. This time saving strategy requires a careful investigation of the magnetic field quality for the first manufactured dipole. The consequences of the curved magnet design was analysed developing advanced multi- poles for elliptical and toroidal magnet geometries. We present the theoretical results together with measured data obtained for the first of series dipole. A description of the ro- tating coil probe based measurement method will be given together with the achieved field quality as well as an estima- tion of the limits of the chosen field representation and its beam dynamics interpretation. INTRODUCTION The beam pipe aperture of the SIS100 dipole is to a large extent covered by the ion particle beam. Given that the magnet is of small size a proper understanding of the har- monics content is required to forecast the machine perfor- mance. The standard method, the search coil probes, used for qualifing conventional magnets, which measure the field on the mid plane, will not allow calculating harmonics reli- ably. Further the SIS100 magnets are superconducting and thus sliding a coil probe on an air cushion system is not ap- plicable. These demands were tackled extending the rotating coil probe measurement method to elliptical apertures [1] and curved magnets [2]. THEORY Cylindric Elliptic Multipoles In a magnet with a rectangular gap an ellipse as refer- ence curve covers a larger area than a circle [1,3]. So it is advantageous to use elliptic coordinates x = e cosh η cos ψ and y = e sinh η sin ψ, with a , b and e = √ a 2 − b 2 the major, minor semi-axes and the eccentricity of the refer- ence ellipse, which is expressed in the above coordinates by η 0 = tanh −1 (b/a). The Cartesian and the elliptic coordi- nates are connected by a conformal map: z = x + iy = e cosh(η + iψ) = e cosh w. (1) Solving the potential equation by separation leads to hyper- bolic functions in η and trigonometric functions in ψ. The complex field expansion is B(w) = B y (η,ψ)+ iB x (η,ψ): B(w) = B 0        E 0 2 + M  n=1 E n cosh[n(η + iψ)] cosh(nη 0 )        . In view of the transformation (1) expansions for the same field are related. In fact: cosh[n(η + iψ)] = cosh(nη) cos(nψ)+ i sinh(nη) sin(nψ) = ∑ n m=0  Re(t m , n z m )+ i Im(t m , n z m )  (2) with the residue t mn = Res  sinh w cosh(nw)/ cosh m+1 w), w = iπ/2  . (3) Also from the values for the E n values for the C m may be found. Toroidal Circular Multipoles Rotating coil probes integrate the field along their axis. To judge if these can be used for measuring a curved mag- nets, a field description following the curvature but invari- ant to this coordinate is required. Local Toroidal coordi- nates [2] are obtained by rotating off-centre dimensionless polar coordinates ρ,ϑ by an angle ϕ: X + iY = R C he i ϕ , Z = R Ref sin ϑ, h = 1+ ǫρ cos ϑǫ = R Ref /R C R C = major radius = radius of curvature; R Ref = minor radius = reference radius; ǫ the inverse aspect ratio. The Cartesian coordinates X , Y , Z are centred in the torus cen- tre; Z is normal to the equatorial plane. The approximate solutions of the potential equation obtained by the approximate R-separation are: Φ m = h −1/2 ρ m e imϑ , m = 0, 1, 2, .... Introducing Cartesian co- ordinates x ′ , y ′ in the plane ϕ = const: z ′ = x ′ + iy ′ = R Ref ρ e i ϑ (4) we get the approximate circular toroidal multipoles: Φ m ( x ′ , y ′ ) =  z ′ R Ref  m − ǫ 4         z ′ R Ref  m+1 +  z ′ R Ref  m −1 | z ′ | 2 R 2 Ref        . (5) Corresponding (normal and skew) vector fields are (m = 1, 2, ...):  T m ( x ′ , y ′ ) = − R Ref m ∇ ′ Φ m ( x ′ , y ′ ),  T ( n) m ( x ′ , y ′ ) =Re(  T m ( x ′ , y ′ )),  T ( s ) m ( x ′ , y ′ ) =Im(  T m ( x ′ , y ′ )). 5th International Particle Accelerator Conference IPAC2014, Dresden, Germany JACoW Publishing ISBN: 978-3-95450-132-8 doi:10.18429/JACoW-IPAC2014-THPRO057 THPRO057 3002 Content from this work may be used under the terms of the CC BY 3.0 licence (© 2014). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI. 05 Beam Dynamics and Electromagnetic Fields D02 Non-linear Dynamics - Resonances, Tracking, Higher Order