Brookhaven ¡National ¡Laboratory February ¡4rd ¡2014 Hamiltonian-‑based ¡evaluation ¡of ¡the ¡ longitudinal ¡acceptance ¡of ¡a ¡high ¡power ¡linac Emanuele ¡Laface ¡ Physicist ¡ European ¡Spallation ¡Source
Motivations:
Motivations: The common way to evaluate the longitudinal acceptance is to run a simulation where the particles have large momentum deviation.
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 slow, especially in operations; moreover the longitudinal acceptance is a property of the accelerator, not the beam.
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 slow, especially 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?
Let’ s start with something simple
Let’ s start with something simple
Let’ s start with something simple
Let’ s start with something simple
Let’ s start with something simple a ring with one accelerating cavity.
The Hamiltonian for the longitudinal variables is ✓ ◆ φ , ∆ E ω 0 ◆ 2 h ηω 02 H = 1 ✓ ∆ E + eV 2 π [cos( φ ) − cos( φ s ) + ( φ − φ s ) sin( φ s )] 2 β 2 E ω 0
◆ 2 h ηω 02 ✓ ∆ E H = 1 + eV 2 π [cos( φ ) − cos( φ s ) + ( φ − φ s ) sin( φ s )] 2 β 2 E ω 0 where is the harmonic number h η = α c − 1 is the phase-slip factor γ 2 is the angular revolution frequency ω 0 is the variation in energy ∆ E is the phase of the particle φ is the synchronous phase φ s are the relativistic quantities β , γ
◆ 2 h ηω 02 ✓ ∆ E H = 1 + eV 2 π [cos( φ ) − cos( φ s ) + ( φ − φ s ) sin( φ s )] 2 β 2 E ω 0 this Hamiltonian can be solved with a turn-by-turn map as [1]: ∆ E n +1 = ∆ E n + eV [sin( φ n ) − sin( φ s )] φ n +1 = φ n + 2 π h η β 2 E ∆ E n +1 where the index n refer to the nth turn
∆ E n +1 = ∆ E n + eV [sin( φ n ) − sin( φ s )] φ n +1 = φ n + 2 π h η β 2 E ∆ E n +1 eV = 100 MeV 0.08 φ s = 30 � h = 1 0.06 α c = 0 . 0434 E k = 45 MeV 0.04 0.02 ∆ E/ β 2 E 0 -0.02 -0.04 -0.06 -0.08 -40 -20 0 20 40 60 80 100 120 140 160 φ [Deg]
∆ E n +1 = ∆ E n + eV [sin( φ n ) − sin( φ s )] φ n +1 = φ n + 2 π h η β 2 E ∆ E n +1 eV = 100 MeV 0.08 φ s = 30 � h = 1 0.06 α c = 0 . 0434 E k = 45 MeV 0.04 ESS TDR 0.02 ∆ E/ β 2 E 0 -0.02 -0.04 -0.06 -0.08 -40 -20 0 20 40 60 80 100 120 140 160 φ [Deg]
And now something less simple
And now something less simple
And now something less simple
And now something less simple
And now something less simple can we use the same solution for a linac imaging it as a “straight” ring? Short answer is no.
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.
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).
A possible solution for a linac is [2]: E n +1 = E n + eV cos( φ s + ∆ φ n ) φ n +1 = φ n − π + 2 π Lf c β n +1 where is the cavity frequency f is the cavity length L is the speed of light c β n +1 is the relativistic coefficient calculated after the increase of energy. the - π is for π -mode cavity.
E n +1 = E n + eV cos( φ s + ∆ φ n ) φ n +1 = φ n − π + 2 π Lf c β 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. The effe ctive vo ltag e is s cale d according to the Transit Time Factor.
1 0.8 0.6 ∆ E/ β 2 E 0.4 0.2 0 -0.2 -0.4 -50 0 50 100 150 200 250 φ [Deg]
1 0 . 8 ESS TDR Normalized density 0 . 6 Energy [MeV] 1 0 . 4 0.1 0 . 2 0.8 0 0.6 − 0 . 2 0.01 ∆ E/ β 2 E − 50 0 50 100 150 200 250 0.4 Phase [degrees] 0.2 0 -0.2 -0.4 -50 0 50 100 150 200 250 φ [Deg]
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 detailed result will be presented at IPAC14 [3].
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.
References [1] S.Y. Lee, “Accelerator Physics Second 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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