1. Neil Marks; DLS/CCLRC Lecture to Cockcroft Institute 2005/6. © N.Marks MMIV Magnets interactions with beam. Neil Marks, DLS/CCLRC, Daresbury Laboratory, Warrington WA4 4AD, U.K. Tel: (44) (0)1925 603191 Fax: (44) (0)1925 603192

2. Neil Marks; DLS/CCLRC Lecture to Cockcroft Institute 2005/6. © N.Marks MMIV Dipoles Dipoles can be: Parallel ended dipole. Sector dipole These dipoles have different focusing effect on the beam; (the curved nature is to save material and has no effect on beam focusing).

3. Neil Marks; DLS/CCLRC Lecture to Cockcroft Institute 2005/6. © N.Marks MMIV End focusing effects Sector dipoles focus horizontally : The end field in a parallel ended dipole focuses vertically : B beam Off the vertical centre line, the field component normal to the beam direction produces a vertical focusing force. B

4. Neil Marks; DLS/CCLRC Lecture to Cockcroft Institute 2005/6. © N.Marks MMIV Upright quadrupole field. Beam that is radially focused is vertically defocused.

5. Neil Marks; DLS/CCLRC Lecture to Cockcroft Institute 2005/6. © N.Marks MMIV Beam that has radial displacement (but no vertical) is deflected vertically; horizontally central beam with vertical displacement is deflected radially; so skew quadrupole couples horizontal and vertical betatron oscillations. Skew quadrupole field.

6. Neil Marks; DLS/CCLRC Lecture to Cockcroft Institute 2005/6. © N.Marks MMIV Sextupole field. +C -C +C -C +C -C By x off-momentum particles are incorrectly focused in quadrupoles (eg, high momentum particles with greater rigidity are under-focused), so betatron oscillation frequencies are modified - chromaticity; but off momentum particles circulate with a radial displacement (high momentum particles at larger x); so positive sextupole field corrects this effect – can reduce chromaticity to 0; we shall also see that the first allowed error harmonic in a dipole is sextupole field – resulting negative chromaticity requires correction.

7. Neil Marks; DLS/CCLRC Lecture to Cockcroft Institute 2005/6. © N.Marks MMIV Octupole field Effect? By x 3 Octupole field induces Landau damping : introduces tune-spread as a function of betatron amplitude; de-coheres betatron oscillations; reduces couplings.