Survey
* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project
Download Particle Accelerator Physics
Document related concepts
Standard Model wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Density of states wikipedia , lookup
Renormalization wikipedia , lookup
Newton's theorem of revolving orbits wikipedia , lookup
Classical mechanics wikipedia , lookup
Time in physics wikipedia , lookup
Work (physics) wikipedia , lookup
Equations of motion wikipedia , lookup
Thomas Young (scientist) wikipedia , lookup
Elementary particle wikipedia , lookup
History of subatomic physics wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Matter wave wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Transcript
506 14 Hamiltonian Nonlinear Beam Dynamics In treating higher order perturbations we make an eï¬ort to include all perturbations that contribute to a speciï¬c aberration to be studied or to deï¬ne the order of approximation used if higher order terms are to be ignored. A careful inspection of all perturbation terms close to the order of approximation desired is prudent to ensure that no signiï¬cant term is missed. Conversely such an inspection might very well reveal correction possibilities. An example is the eï¬ect of chromaticity which is generated by quadrupole ï¬eld errors for oï¬momentum particles but can be compensated by sextupole ï¬elds at locations where the dispersion function is ï¬nite. Here the problem is corrected by ï¬elds of a diï¬erent order from those causing the chromaticity. To become more quantitative we discuss the analytical solution of (14.1). Since in our approximation this solution is the sum of all partial solutions for each individual perturbation term, the problem is solved if we ï¬nd a general solution for an arbitrary perturbation. The solution of, for example, the horizontal equation of motion x + K(z) x = pµνÏÏÏ xµ x y Ï y δ Ï Î½ Ï (14.4) can proceed in two steps. First we replace the oscillation amplitudes on the r.h.s. by their most signiï¬cant components xµ â (xβ + x0 + xδ )µ , xν â (xβ + x0 + xδ )ν , y Ï â (yβ + y0 + yδ )Ï , y Ï â (yβ + y0 + yδ )Ï . (14.5) As discussed before, in a particular situation only those components are eventually retained that are signiï¬cant to the problem. Since most accelerators are constructed in the horizontal plane we may set the vertical dispersion yδ = 0. Decomposition (14.5) is inserted into the r.h.s of (14.4) and again only terms signiï¬cant for the particular problem and to the approximation desired are retained. The solution xµνÏÏÏ can be further broken down into components each relating to only one individual perturbation term. Whatever number of perturbation terms we decide to keep, the basic diï¬erential equation for the perturbation is of the form P + K(z) P = p (xβ , xβ , xc , xc , xδ , xδ , yβ , yβ , yc , yc , yδ , yδ , δ, z) . (14.6) The second step in the solution process is to derive the actual solution of (14.6). Demonstrating the principle of the solution process we restrict the r.h.s. to linear and quadratic, uncoupled terms. In addition orbit distortions and vertical dispersion are ignored. These assumptions reduce the complexity of formulas without limiting the application of the solution process. We use the principal solutions on the r.h.s. of the simpliï¬ed equation (14.6) and set for the betatron oscillations xβ = Cx (z) xβ0 + Sx (z) xβ0 , xβ = Cx (z) xβ0 + Sx (z) xβ0 , (14.7)
