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Analytical Estimation of Dynamic Apertures Limited by the Wigglers in Storage Rings
| Content Provider | Semantic Scholar |
|---|---|
| Author | Gao, Jennifer |
| Copyright Year | 2003 |
| Abstract | By applying the general dynamic aperture formulae for the multipoles in a storage ring developed in ref. [1] (J. Gao,Nucl. Instr. and Methods A451 (2000), p. 545), in this paper, we give the analytical formulae for the dynamic apertures limited by the wigglers in storage rings. INTRODUCTION Wigller as an insertion device finds many applications in damping rings [2], synchrotron radiation facilities [3] [4], and storage ring colliders [5]. Intrinsically, as a nonlinear device, together with the perturbations to the linear optics it brings additional limitations to the general performance of the machines, such as reducing dynamic apertures. In this paper, we will estimate in an analytical way the dynamic apertures limited by wigglers. Firstly, in section 2, we make a brief review of the beam dynamics inside a wiggler, and secondly, in sections 3 a wiggler is inserted into a storage ring as a perturbation. By applying the general dynamic aperture formulae of multipoles in a storage ring developed in ref. [1], in section 4 we derived analytical formulae of the wiggler limited dynamic aperture. Finally, in section 5 some numerical examples will be given. PARTICLE'S MOTION INSIDE A WIGGLER Considering a wiggler of sinusoidal magnetic field variation, one can express the wiggler's magnetic fields, which satisfies Maxwell equations, as follows: Bx = kx ky B0 sinh(kxx) sinh(kyy)cos(ks) (1) By = B0 cosh(kxx) cosh(kyy)cos(ks) (2) Bz = − k ky B0 cosh(kxx) sinh(kyy) sin(ks) (3) with k x + k 2 y = k 2 = ( 2π λw )2 (4) whereB0 is the peak sinusoidal wiggler magnetic field, λw is the period length of the wiggler, and x, y, s represent horizontal, vertical, and beam moving directions, respectively. The Hamiltonian describing particle's motion can be written as [3]: Hw = 1 2 ( pz + (px − Ax sin(ks)) + (py − Ay sin(ks)) ) (5) where Ax = 1 ρwk cosh(kxx) cosh(kyy) (6) Ax = − kx ky sinh(kxx) sinh(kyy) ρwk (7) andρw is the radius of curvature of the wiggler peak magnetic fieldB0, andρw = E0/ecB0 with E0 being the electron energy. After making a canonical transformation to betatron variables, averaging the Hamiltonian over one period of wiggler, and expanding the hyperbolic functions to the fourth order inx andy, one gets: Hw = 1 2 (px + p 2 y) + 1 4kρw (k xx +k yy )+ 1 12kρw (k xx +k yy +3kk xx y) − sin(ks) 2kρw ( px(k xx 2 + k yy ) − 2k xpyxy ) (8) After averaging the motion over one wiggler period, one obtains the differential equations for particle's transverse motions [6]: dx ds2 = − k 2 x 2kρw ( x + 2 3 k xx 3 + kxy ) (9) dy ds2 = − k y 2kρw ( y + 2 3 k yy 3 + k xk 2 k2 y xy ) (10) Considering the wigglers are built with plane poles, one has kx = 0. WIGGLER AS AN INSERTION DEVICE IN A STORAGE RING Now we insert a “wiggler” of only one period (or one cell) into a storage ring located at sw. The total Hamiltonian of the ring in the vertical plane can be expressed as follows: H = H0 + 1 4ρ2 y + k y 12ρ2 yλw ∞ ∑ |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://accelconf.web.cern.ch/accelconf/p03/PAPERS/RPPG041.PDF |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
