Particle Tracking Studies Using Dynamical Map Created from Finite Element Solution of the EMMA Cell

The un­con­ven­tion­al size and the pos­si­bil­i­ty of trans­verse dis­place­ment of the mag­nets in the EMMA non-scal­ing FFAG mo­ti­vates a care­ful study of par­ti­cle be­hav­ior with­in the EMMA ring. The mag­net­ic field map of the dou­blet cell is com­put­ed using a Fi­nite El­e­ment Method solver; par­ti­cle mo­tion through the field can then be found by nu­mer­i­cal in­te­gra­tion, using (for ex­am­ple) OPERA, or ZGOUBI. How­ev­er, by ob­tain­ing an an­a­lyt­i­cal de­scrip­tion of the mag­net­ic field (by fit­ting a Fouri­er-Bessel se­ries to the nu­mer­i­cal data) and using a dif­fer­en­tial al­ge­bra code, such as COSY, to in­te­grate the equa­tions of mo­tion, it is pos­si­ble to pro­duce a dy­nam­i­cal map in Tay­lor form. This has the ad­van­tage that, after once com­put­ing the dy­nam­i­cal map, mul­ti-turn track­ing is far more ef­fi­cient than re­peat­ed­ly per­form­ing nu­mer­i­cal in­te­gra­tions. Also, the dy­nam­i­cal map is small­er (in terms of com­put­er mem­o­ry) than the full mag­net­ic field map; this al­lows dif­fer­ent con­fig­u­ra­tions of the lat­tice, in terms of mag­net po­si­tions, to be rep­re­sent­ed very eas­i­ly using a set of dy­nam­i­cal maps, with in­ter­po­la­tion be­tween the co­ef­fi­cients in dif­fer­ent maps*.