Hamiltonian-based evaluation of the longitudinal acceptance - - PowerPoint PPT Presentation

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

Emanuele ¡Laface ¡ Physicist ¡ European ¡Spallation ¡Source Brookhaven ¡National ¡Laboratory February ¡4rd ¡2014

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

• perations; 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

• perations; 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 ω0 ◆

H = 1 2 hηω02 β2E ✓∆E ω0 ◆2 + eV 2π [cos(φ) − cos(φs) + (φ − φs) sin(φs)]

where

h η = αc − 1 γ2 ω0 ∆E φ φs β, γ

is the harmonic number is the phase-slip factor is the angular revolution frequency is the variation in energy is the phase of the particle is the synchronous phase are the relativistic quantities

H = 1 2 hηω02 β2E ✓∆E ω0 ◆2 + eV 2π [cos(φ) − cos(φs) + (φ − φs) sin(φs)]

this Hamiltonian can be solved with a turn-by-turn map as [1]: where the index n refer to the nth turn

∆En+1 = ∆En + eV [sin(φn) − sin(φs)] φn+1 = φn + 2πhη β2E ∆En+1 H = 1 2 hηω02 β2E ✓∆E ω0 ◆2 + eV 2π [cos(φ) − cos(φs) + (φ − φs) sin(φs)]

• 0.08
• 0.06
• 0.04
• 0.02

0.02 0.04 0.06 0.08

• 40
• 20

20 40 60 80 100 120 140 160

eV = 100 MeV φs = 30 h = 1 αc = 0.0434 Ek = 45 MeV

∆En+1 = ∆En + eV [sin(φn) − sin(φs)] φn+1 = φn + 2πhη β2E ∆En+1 φ[Deg] ∆E/β2E

∆En+1 = ∆En + eV [sin(φn) − sin(φs)] φn+1 = φn + 2πhη β2E ∆En+1

• 0.08
• 0.06
• 0.04
• 0.02

0.02 0.04 0.06 0.08

• 40
• 20

20 40 60 80 100 120 140 160

eV = 100 MeV φs = 30 h = 1 αc = 0.0434 Ek = 45 MeV

ESS TDR

∆E/β2E 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

• f

RF , is the possibility in the linac to variate the phase and the voltage

• f

each cavity independently.

The main difference between a ring and a linac, in terms

• f

RF , is the possibility in the linac to variate the phase and the voltage

• f

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).

En+1 = En + eV cos(φs + ∆φn) φn+1 = φn − π + 2πLf

cβn+1

where

f c βn+1

is the cavity frequency is the speed of light is the relativistic coefficient calculated after the increase of energy. the -π is for π-mode cavity.

A possible solution for a linac is [2]:

L

is the cavity length

En+1 = En + eV cos(φs + ∆φn) φn+1 = φn − π + 2πLf

cβn+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.

• 0.4
• 0.2

0.2 0.4 0.6 0.8 1

• 50

50 100 150 200 250

∆E/β2E φ[Deg]

• 0.4
• 0.2

0.2 0.4 0.6 0.8 1

• 50

50 100 150 200 250

−50 50 100 150 200 250 −0.2 0.2 0.4 0.6 0.8

Phase [degrees] Energy [MeV]

0.01 0.1 1

Normalized density

ESS TDR

∆E/β2E 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

• f IPAC 2014, Dresden, Germany.