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SOME NOTES ABOUT CSR AND RINGS

Delve into the realm of collective effects in rings, considering bunch dynamics and CSR effects. Explore equilibrium solutions, coherence length, and Vlasov treatment. Analyze stability, stationary solutions, Haissinski evolution, and equilibrium study by numerical simulation.

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SOME NOTES ABOUT CSR AND RINGS

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  1. SOME NOTES ABOUT CSR AND RINGS

  2. Study of bunch dynamics in rings usually done considering a COASTING BEAM model This is usually justified, studying the stability of the equilibrium solution, by supposing perturbations characterized by wavelength much smaller than the bunch length Is this assumption justified?

  3. COHERENCE LENGTH • Rigid line bunch • We are not interested in the distribution in momentum • r(z’) = r0+r1cos(k z’), where r0and r1 are constants; l=2p/k • Let’s calculate the fields at a fixed z (test point) How many periods l enter in the calculation of dE/d(ct)? Truncate at z-nl; z+nl and look at dE/d(ct) as n increases We know:

  4. We have to study: Change to variable x=z-z’. Finally we have where: Put z=0; k=1.0; r1=0.1r0 and study F1, F2, F3 (F4=0) graphically:

  5. Plot of F1 as function of z + Plot of F2 as function of z + Plot of F3 as function of z

  6. Then: • F1 is the most important term • F2 keeps on oscillating • F3 and F4 give oscillations which fade to 0 We can’t forget about finite bunch dimension but We can’t talk about a well-defined coherent length

  7. All contributions toghether (F1+F2+F3): It is also interesting to plot the asymptotic behavior (ninfinity) and compare with F1+F2+F3+F4

  8. BASIC EQUATIONS Coordinates: Equation of motion: Equation of motion from Hamiltonian:

  9. z-z’ j

  10. Behavior of the potential: Does not go to infinity! Approximation: Error ~ (32/3-2/3)er/g2

  11. VLASOV TREATMENT In general, Vlasov Equation is: In our initial coordinates: How to solve the equation ? What coordinates are better ?

  12. STATIONARY SOLUTION METHOD • Find out a stationary (equilibrium) solution of Vlasov Equation • Perturbation of the equilibrium • Study of stability Does this apply to our case? How long does the system take to evolve to the equilibrium? Consider and the Hamiltonian

  13. If there are no incoherent radiation effects any is solution of the Vlasov equation: But when, as in our case, there are incoherent radiation effects, Fokker-Plank term must be added and, in a time long with respect to the characteristic SR damping time we reach Equilibrium (Haissinski solution) in the storage ring regime Initial condition (might be f(H0)) Haissinski

  14. Possibly we are not going to reach the Haissinski solution! Yet, given any initial condition, the system will evolve to Haissinski So it’s anyway useful to understand the system evolution STUDY OF HAISSINSKI SOLUTION IF a solution exists it can be shown that with BUT NO EXISTENCE NOR UNICITY IS GUARANTEED!

  15. After redefinition of A we can write: Start from the asymptotic, known behavior of Y0: One can build Y0(z,A); this does not insure yet existance. We must check that there is A such that Y0(z,A) is well normalized:

  16. Sketch of a proof for existence and unicity Moreover, asymptotically: and also Let us now show that • Y0 (z) is positive or null • when z equals, e.g., 0:

  17. To prove the latter limit: since we have And, for example when A=1, we have T is non-zero, strictly, since and As a result

  18. Physically: the head accelerates, the tail decelerates (overall CSR loss) Thus, the CSR decreases during the evolution. Intuitively we can have equilibrium solution BUT The equilibrium solution is physically interesting ONLY when The bunch length stays within one half of the RF wavelength As this limit approaches, however the linear approximation for The RF term is no longer valid

  19. Study of equilibrium by means of numerical Simulation is important Since solution exists and is unique one can simply iterate starting from a Gaussian

  20. STUDY OF THE STABILITY The choice of variables must be done in order to simplify the treatment of Vlasov equation as much as possible; the equation is better simplified introducing new coordinates (Q,P) so that H=f(P) only. This is done through Hamilton-Jacobi theory We have some freedom in the choice of f(P)

  21. Starting from H0(z,d) we write the H-J equation Since H0 does not depend on s explicitly then Hamilton’s principal function is which gives the canonical transformation Then, also W can generate the canonical transformation

  22. Then the new H is H=E(P) It is possible to make the choice E(P)=P, then H=P In case of (multi-)periodic systems one can use action-angle variables In general the frequency (1/T) is w(z,d) Notation: use (J,q) instead of (P,Q) Looking for (J,q) such that w =w(J) means E(J)=f(J) with Physically J is the area (over 2pr) enclosed by the trajectory in (z,d): Use of action-angle variables is a matter of taste. The important point is H=H(J) (or P in the other notation)

  23. In the new variables (J,q) the study of stability is simpler. Start from Perturb the equilibrium: Then also But now H0 is simple: H0=H0(J) …looks horrible but in the end:

  24. Now Therefore By substitution, Vlasov equation looks like where We still have to characterize H1 though; in old coordinates: New coordinates:

  25. Then Now expand. The orthogonal set is somehow arbitrary. After manipulations:

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