Multipole Magnets from Maxwell’s Equations Alex Bogacz USPAS, Hampton, VA, Jan. 17-28, 2011

Maxwell’s Equations for Magnets - Outline Solutions to Maxwell’s equations for magneto static fields: in two dimensions (multipole fields) in three dimensions (fringe fields, insertion devices...) How to construct multipole fields in two dimensions, using electric currents and magnetic materials, considering idealized situations. A. Wolski, Academic Lectures, University of Liverpool , 2008 USPAS, Hampton, VA, Jan. 17-28, 2011

Basis Vector calculus in Cartesian and polar coordinate systems; Stokes’ and Gauss’ theorems Maxwell’s equations and their physical significance Types of magnets commonly used in accelerators. following notation used in: A. Chao and M. Tigner, “Handbook of Accelerator Physics and Engineering,” World Scientific (1999). USPAS, Hampton, VA, Jan. 17-28, 2011

Maxwell’s equations USPAS, Hampton, VA, Jan. 17-28, 2011

Maxwell’s equations USPAS, Hampton, VA, Jan. 17-28, 2011

Physical interpretation of USPAS, Hampton, VA, Jan. 17-28, 2011

Physical interpretation of USPAS, Hampton, VA, Jan. 17-28, 2011

Linearity and superposition USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields USPAS, Hampton, VA, Jan. 17-28, 2011

Generating multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields from a current distribution USPAS, Hampton, VA, Jan. 17-28, 2011

Superconducting quadrupole - collider final focus USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

Multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

Generating multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

Generating multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

Generating multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

Generating multipole fields in an iron-core magnet USPAS, Hampton, VA, Jan. 17-28, 2011

Maxwell’s Equations for Magnets - Summary USPAS, Hampton, VA, Jan. 17-28, 2011

Maxwell’s Equations for Magnets - Summary Maxwell’s equations impose strong constraints on magnetic fields that may exist. The linearity of Maxwell’s equations means that complicated fields may be expressed as a superposition of simpler fields. In two dimensions, it is convenient to represent fields as a superposition of multipole fields. Multipole fields may be generated by sinusoidal current distributions on a cylinder bounding the region of interest. In regions without electric currents, the magnetic field may be derived as the gradient of a scalar potential. The scalar potential is constant on the surface of a material with infinite permeability. This property is useful for defining the shapes of iron poles in multipole magnets. USPAS, Hampton, VA, Jan. 17-28, 2011

Multipoles in Magnets - Outline Deduce that the symmetry of a magnet imposes constraints on the possible multipole field components, even if we relax the constraints on the material properties and other geometrical properties; Consider different techniques for deriving the multipole field components from measurements of the fields within a magnet; Discuss the solutions to Maxwell’s equations that may be used for describing fields in three dimensions. USPAS, Hampton, VA, Jan. 17-28, 2011

Previous lecture re-cap USPAS, Hampton, VA, Jan. 17-28, 2011

Previous lecture re-cap USPAS, Hampton, VA, Jan. 17-28, 2011

Allowed and forbidden harmonics USPAS, Hampton, VA, Jan. 17-28, 2011

Allowed and forbidden harmonics USPAS, Hampton, VA, Jan. 17-28, 2011

Allowed and forbidden harmonics USPAS, Hampton, VA, Jan. 17-28, 2011

Allowed and forbidden harmonics USPAS, Hampton, VA, Jan. 17-28, 2011

Allowed and forbidden harmonics USPAS, Hampton, VA, Jan. 17-28, 2011

Allowed and forbidden harmonics USPAS, Hampton, VA, Jan. 17-28, 2011

Measuring multipoles USPAS, Hampton, VA, Jan. 17-28, 2011

Measuring multipoles in Cartesian basis USPAS, Hampton, VA, Jan. 17-28, 2011

Measuring multipoles in Cartesian basis USPAS, Hampton, VA, Jan. 17-28, 2011

Measuring multipoles in Polar basis USPAS, Hampton, VA, Jan. 17-28, 2011

Measuring multipoles in Polar basis USPAS, Hampton, VA, Jan. 17-28, 2011

Measuring multipoles in Polar basis USPAS, Hampton, VA, Jan. 17-28, 2011

Advantages of mode decompositions USPAS, Hampton, VA, Jan. 17-28, 2011

Three-dimensional fields USPAS, Hampton, VA, Jan. 17-28, 2011

Three-dimensional fields USPAS, Hampton, VA, Jan. 17-28, 2011

Three-dimensional fields USPAS, Hampton, VA, Jan. 17-28, 2011

Three-dimensional fields USPAS, Hampton, VA, Jan. 17-28, 2011

Three-dimensional fields USPAS, Hampton, VA, Jan. 17-28, 2011

Measuring multipoles in Polar basis USPAS, Hampton, VA, Jan. 17-28, 2011

Multipoles in Polar basis – including x-dependence USPAS, Hampton, VA, Jan. 17-28, 2011

Three-dimensional fields USPAS, Hampton, VA, Jan. 17-28, 2011

Three-dimensional fields - Cylindrical basis USPAS, Hampton, VA, Jan. 17-28, 2011

Three-dimensional fields - Cylindrical basis USPAS, Hampton, VA, Jan. 17-28, 2011

Three-dimensional fields - Cylindrical basis USPAS, Hampton, VA, Jan. 17-28, 2011

Three-dimensional fields - Cylindrical basis USPAS, Hampton, VA, Jan. 17-28, 2011

Three-dimensional fields - Cylindrical basis USPAS, Hampton, VA, Jan. 17-28, 2011

Three-dimensional fields - Cylindrical basis USPAS, Hampton, VA, Jan. 17-28, 2011

Summary – Part II Symmetries in multipole magnets restrict the multipole components that can be present in the field. It is useful to be able to find the multipole components in a given field from numerical field data: but this must be done carefully, if the results are to be accurate. Usually, it is advisable to calculate multipole components using field data on a surface enclosing the region of interest: any errors or residuals will decrease exponentially within that region, away from the boundary. Outside the boundary, residuals will increase exponentially. Techniques for finding multipole components in two dimensional fields can be generalized to three dimensions, allowing analysis of fringe fields and insertion devices. In two or three dimensions, it is possible to use a Cartesian basis for the field modes; but a polar basis is sometimes more convenient. USPAS, Hampton, VA, Jan. 17-28, 2011

Appendix A - The vector potential A scalar potential description of the magnetic field has been very useful to derive the shape for the pole face of a multipole magnet. USPAS, Hampton, VA, Jan. 17-28, 2011

Appendix A - The vector potential USPAS, Hampton, VA, Jan. 17-28, 2011

Appendix A - The vector potential USPAS, Hampton, VA, Jan. 17-28, 2011

Appendix A - The vector potential USPAS, Hampton, VA, Jan. 17-28, 2011

The vector potential USPAS, Hampton, VA, Jan. 17-28, 2011

The vector potential USPAS, Hampton, VA, Jan. 17-28, 2011

The vector potential USPAS, Hampton, VA, Jan. 17-28, 2011

The vector potential USPAS, Hampton, VA, Jan. 17-28, 2011

The vector potential USPAS, Hampton, VA, Jan. 17-28, 2011

Appendix A - The vector potential USPAS, Hampton, VA, Jan. 17-28, 2011

Appendix A - The vector potential USPAS, Hampton, VA, Jan. 17-28, 2011

Appendix B - Field Error Tolerances Focusing ‘point’ error perturbs the betatron motion leading to the Courant-Snyder invariant change: Beam envelope and beta-function oscillate at double the betatron frequency USPAS, Hampton, VA, Jan. 17-28, 2011

Appendix B - Field Error Tolerances Single point mismatch as measured by the Courant-Snyder invariant change: Each source of field error (magnet) contributes the following Courant-Snyder variation here, m =1 quadrupole, m =2 sextupole, m=3 octupole, etc USPAS, Hampton, VA, Jan. 17-28, 2011

Appendix B - Field Error Tolerances Cumulative mismatch along the lattice (N sources): Standard deviation of the Courant-Snyder invariant is given by: Assuming weakly focusing lattice (uniform beta modulation) the following averaging (over the betatron phase) can by applied: USPAS, Hampton, VA, Jan. 17-28, 2011

Appendix B - Field Error Tolerances Some useful integrals …. : will reduce the coherent contribution to the C-S variance as follows: Including the first five multipoles yields: USPAS, Hampton, VA, Jan. 17-28, 2011

Appendix B - Field Error Tolerances Beam radius at a given magnet is : One can define a ‘good fileld radius’ for a given type of magnet as: Assuming the same multipole content for all magnets in the class one gets: The first factor purely depends on the beamline optics (focusing), while the second one describes field tolerance (nonlinearities) of the magnets: USPAS, Hampton, VA, Jan. 17-28, 2011

Appendix B - Field Error Tolerances Standard deviation of the Courant-Snyder invariant: where: USPAS, Hampton, VA, Jan. 17-28, 2011

Appendix B - Field Error Tolerances multipole expansion coefficients of the azimuthal magnetic field, Bq - Fourier series representation in polar coordinates at a given point along the trajectory): multipole gradient and integrated geometric gradient: USPAS, Hampton, VA, Jan. 17-28, 2011

Appendix B - Field Error Tolerances The linear errors, m =1, cause the betatron mismatch – invariant ellipse distortion from the design ellipse without changing its area – no emittance increase. By design, one can tolerate some level (e.g. 10%) of Arc-to-Arc betatron mismatch due to the focusing errors, df1 (quad gradient errors and dipole body gradient) to be compensated by the dedicated matching quads The higher, m > 1, multipoles will contribute to the emittance dilution – ‘limited’ by design via a separate allowance per each segment (Arc, linac) (e.g. 1%) USPAS, Hampton, VA, Jan. 17-28, 2011

Appendix B - Field Error Tolerances Here one assumes the following multipole content for the dipoles and quads: Quads: sextupole (m = 2), octupole (m = 3), duodecapole (m = 5) and icosapole (m = 9) Dipoles: sextupole (m = 2) and decapole (m = 4) The values of multipoles are calculated in the extreme case – a given order (m) multipole by itself exhausts the emittance dilution allowance of 1%. One can use the above analytic formalism to set the magnet error tolerances for specific groups (types) of magnets (dipoles and quads) within each lattice segment (Arc, linac) For each group of magnets within each segment one needs to evaluate: USPAS, Hampton, VA, Jan. 17-28, 2011

Cumulative C-S invariant change due to magnet errors Assuming the same multipole content for all magnets in the class one gets: C-S mis-match emittance dilution ‘Beam region’ defined by : The first factor purely depends on the beamline optics (focusing), while the second one describes field errors of the magnets USPAS, Hampton, VA, Jan. 17-28, 2011