Hamiltonian- based evaluation of the longitudinal acceptance of a high power linac

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1 Brookhaven National Laboratory February 4rd 2014 Hamiltonian- based evaluation of the longitudinal acceptance of a high power linac Emanuele Laface Physicist European Spallation Source

2

3 Motivations:

4 Motivations: The common way to evaluate the longitudinal acceptance is to run a simulation where the particles have large momentum deviation.

5 Motivations: The common way to evaluate the longitudinal acceptance is to run a simulation where the particles have large momentum deviation. This approach can be slo w, e specially in operations; moreover the longitudinal acceptance is a property of the accelerator, not the beam.

6 Motivations: The common way to evaluate the longitudinal acceptance is to run a simulation where the particles have large momentum deviation. This approach can be slo w, e specially in operations; moreover the longitudinal acceptance is a property of the accelerator, not the beam. So, why not try to extract this information directly from the Hamiltonian of the machine?

7

8 Let s start with something simple

9 Let s start with something simple

10 Let s start with something simple

11 Let s start with something simple

12 Let s start with something simple a ring with one accelerating cavity.

13 The Hamiltonian for the longitudinal variables, E! 0 is H = 1 2 h! E E! ev 2 [cos( ) cos( s)+( s)sin( s )]

14 H = 1 2 h! E E! ev 2 [cos( ) cos( s)+( s)sin( s )] where h is the harmonic number = c 12 is the phase-slip factor! 0 is the angular revolution frequency E is the variation in energy is the phase of the particle s, is the synchronous phase are the relativistic quantities

15 H = 1 2 h! E E! ev 2 [cos( ) cos( s)+( s)sin( s )] this Hamiltonian can be solved with a turn-by-turn map as [1]: E n+1 = E n + ev [sin( n ) sin( s )] n+1 = n + 2 h 2 E E n+1 where the index n refer to the nth turn

16 E n+1 = E n + ev [sin( n ) sin( s )] n+1 = n + 2 h 2 E E n ev = 100 MeV s = 30 h =1 c = E k = 45 MeV E/ 2 E [Deg]

17 En+1 = n+1 En + ev [sin( n ) 2 h = n+ 2 En+1 E sin( s )] ev = 100 MeV 0.08 s h= c = Ek = 45 MeV 0.04 ESS TDR 0.02 E/ 2 E = ongitudinal acceptance of the linac, re [Deg]

18

19 And now something less simple

20 And now something less simple

21 And now something less simple

22 And now something less simple

23 And now something less simple can we use the same solution for a linac imaging it as a straight ring? Short answer is no.

24

25 The main difference between a ring and a linac, in terms of RF, is the possibility in the linac to variate the phase and the voltage of each cavity independently.

26 The main difference between a ring and a linac, in terms of RF, is the possibility in the linac to variate the phase and the voltage of each cavity independently. There is also an additional problem with respect to our simple model: the solution used for the ring is valid for a single accelerating gap while a cavity is in general a combination of multiple gaps (3 or 5 in ESS).

27 A possible solution for a linac is [2]: E n+1 = E n + ev cos( s + n) n+1 = n + 2 Lf c n+1 where f L c n+1 is the cavity frequency is the cavity length is the speed of light is the relativistic coefficient calculated after the increase of energy. the -π is for π-mode cavity.

28 E n+1 = E n + ev cos( s + n) n+1 = n + 2 Lf c n+1 the index n here runs from cell to cell in the cavities. It means that if a cavity is composed by 3 cells there will be 3 steps. T h e eff e ct i v e v o lt ag e i s s cale d according to the Transit Time Factor.

29 E/ 2 E [Deg]

30 1 0.8 Energy [MeV] ESS TDR Normalized density E/ 2 E Phase [degrees] Figure 4.14: Longitudinal acceptance of the linac, referred to the entrance of the entering the DTL inside the grey area will reach the end of the linac. The actual bea is represented by the coloured distribution at the centre [Deg]

31 This is an ongoing study. For large phase or energy deviation the equations may not perform well. The phase advance between cavities is not correctly calculated because the source of configuration, TraceWin, uses relative phase while the absolute phase should be used in this study. A more detaile d re su lt w ill be presented at IPAC14 [3].

32 When this study will be mature enough it should be possible to calculate the equation of the separatrix and its evolution from cell to cell.! This will lead to a model that gives immediately the information about the stable and unstable region of the phase space.

33 References [1] S.Y. Lee, Accelerator Physics Seco nd Edition, World Scientific, 2004.! [2] T.P. Wangler, RF Linear Accelerators 2nd and completely revised and enlarged edition, Wiley-VCH, 2008.! [3] E. Laface et al., Longitudinal acceptance evaluation from Hamiltonian., Proceedings of IPAC 2014, Dresden, Germany.

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