STUDY ON LONGITUDINAL DYNAMICS OF TRANSITION CROSSING FOR SIS-100 - - PowerPoint PPT Presentation

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STUDY ON LONGITUDINAL DYNAMICS OF TRANSITION CROSSING FOR SIS-100 PROTON SLOW EXTRACTION CONDITIONS

Stefan Sorge, GSI Darmstadt

Work is supported by

Beam dynamics mini-workshop, J¨ ulich, November 18, 2016

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SIS-100 proton operation

Reminder: initially proton operation foreseen only for anti-proton production

• Fast extraction of a single short bunch with high intensity at high energy:

– Injection of 4 bunches and merging them to one in two steps. – Length: ∆t ≈ 50 ns, final energy: E = 29 GeV, intensity: Np = 2.0 · 1013.

• Two scenarios:
• 1. High-γt operation (S. Sorge):

Optics change during acceleration to shift γt above extraction energy: γt = 45.5 vs. γ = 32 → no transition crossing, but sensitive optics.

• 2. γt jump scenario (S. Aumon, D. Ondreka):

– γt jump with high intensity bunch, need for installation of special jump quadrupoles. – Large dispersion → small tunes: Qx, Qy ≈ 10.3. Optics with rather nice behaviour. New requirement: slow extraction of protons for HADES and CBM → topic of this talk.

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SIS-100 proton slow extraction

• Low intensity: Np/t ∼ 1011/s, where basically 1 s is duration of a cycle.
• Up to high energies: Emax = 29 GeV.
• On the other hand, beam has fit alignment of slow extraction devices.

⇒ Necessity to use ion slow extraction optics settings. Resulting problem: – Slow extraction: doublet optics with Qx = 17.31, Qy = 17.45, and γt = 14.17. – Injection at E = 4 GeV. → Beam energy has to cross transition energy. – γt jump impossible because dispersion at positions of dedicated fast γt jump qua- drupoles in slow extraction optics not large which would be necessary for efficient application of these quadrupoles. ⇓ Slow transition crossing during ramp.

• Aim of present study: first, simple attempt to estimate particle loss.

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Outline and model

• 1. Analytic estimate of growth of momentum spread.

→ use conditions of tracking model.

• 2. Tracking study using MAD-X:

→ use constant energy and φs = 0, π.

• Assume longitudinal dynamics to depend only on phase

slip factor. – Keep γ = const and move γt across γ by changing quadrupole settings. Start from γt > γ. – Apply ˙ γt = − ˙ γnom, (nominal change rate of γ), where ˙ γnom corresponds to ˙ B0 of dipole magnets.

• Unperturbed lattice.

Replace this scenario

0,05 0,1 0,15 0,2 t / (s) 5 10 15 20 γ γt=const

with

0,05 0,1 0,15 0,2 t / (s) 10 20 30 γt γ=const

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Conditions

Conditions chosen as example case.

• Ramp rate of dipole magnetic field: ˙

B0 = 4 T/s which yields ˙ γt = − ˙ γnom = −67.4 s−1.

• Gamma-t: set Qx, Qy, and γt with three quadrupole families.

– initial: γt,ini = 25 – final: γt,fin = 14.17 – determined by the usual slow extraction optics. – resulting ramp time: tramp = 0.16 s = 44000 T0.

• Beam energy E = 15 GeV which corresponds to γ = 17.0, β = 0.998, Bρ = 53 Tm.

→ Initial and final zero order phase slip factors with similar moduli: η0,ini = −0.00187 vs. η0,fin = 0.00151.

• Initial bunch area: Ab = 1 eVs (D. Ondreka)

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Analytic estimate

Expected effect:

• 1. Maximum momentum deviation of bunch in adiabatic and linear approximation, see S.
• Y. Lee “Accelerator Physics”, Eq. (3.58):

δm =

• Ab

ω0 πβ2E heV | cos φs| 2πβ2E|η| 1/4 ∝ 1 |η|1/4 → initial maximum momentum deviation: δm,ini = 1.7 · 10−3

• 2. Phase slip factor during transition: η → 0

⇓ Strong δm growth resulting in growth of horizontal beam width due to xm = Dxδm

• On the other hand, synchrotron motion very slow near transition energy (frozen bunch).

→ Momentum deviation can not infinitely grow if time is too short.

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Analytic estimate

• Maximum momentum deviation during transition crossing, see S. Y. Lee “Accelerator

Physics”, Eqs. (3.184) δm = γt 31/6βτadΓ (2/3)

• Ab

mc2 ˙ γ with “adiabatic time” (Eq. (3.171) (in fact it is rather the non-adiabatic time because particles behave non-adiabatically during τad): τad =

• πβ2mc2γ4

t

˙ γω2

0heV | cos φs|

1/3 .

• Finite maximum momentum deviation during transition: δm = 4.4 · 10−3

with adiabatic time τad = 9.5 · 10−3 s = 2600 T0. – About 3 times initial momentum spread: δm,ini = 1.7 · 10−3 – Maximum dispersion Dm = 4.51 m → maximum dispersion orbit deviation xm = Dmδm = 20 mm, is less than aperture.

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Simulation

Particle tracking for

• Working point Qx = 17.31, Qy = 17.45.
• Total: 60000 turns, where γt changes during 45000 turns.
• Three quadrupole families, change focusing strengths k1

during tracking linearly in time.

• Pre-determine k1 to match γt at three points:

start, when γt = γ, and end.

0,05 0,1 0,15 0,2 t / (s) 10 20 30 γt γ=const φs switch

• Sudden switch of synchronous phase φs = 0 → φs = π, when γt = γ.
• Tracking particles along dispersion orbits: x = Dδ, x

′ = D ′δ and y = y ′ = 0,

i.e. transverse emittances ǫx = ǫy = 0.

• 100 particles, initial Gaussian distributions for small adiabatic bunch truncated at 2σ

with σct = 1.7 m and σδ = 8.7 · 10−4.

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Simulation results

Maximum particle momentum deviation:

• analytic estimate: δm = 4.4 · 10−3
• simulation: δm = 7.8 · 10−3

Particle loss: Ploss = 4 %.

20000 40000 60000 turns

• 0,015
• 0,01
• 0,005

0,005 0,01 0,015 δ of particles Start of γt ramp φs switch End of γt ramp

• Particle loss is low due to precisely setting

γt = γ when switching φs.

• If not done so, particle loss much higher.

– No control of working point and γt during ramp. → mismatch between φs and sign of η0 during ∼ 1000 turns possible. – Found Ploss = 44 % in first attempt.

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Discussion

• Particle loss possibly acceptable because of low intensity, but idealized case chosen:

– unperturbed lattice. – particle emittances ǫx = ǫy = 0.

• Scenario used in the simulation is not foreseen but also not unrealistic because slow

extraction optics necessary only after acceleration. → Possible support of γt crossing due to lattice change.

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Application of γt ramp

• Small differences between initial and final focusing strengths of quadrupoles:

k1,def = −0.2023 m−2 → k1,def = −0.2017 m−2 k1,f,1 = 0.1866 m−2 → k1,def = 0.2015 m−2 k1,f,2 = 0.2235 m−2 → k1,def = 0.2015 m−2 and long ramp time result in very low ramp rate: ˙ k1 ≡ |k1,fin − k1,ini|max tramp ≤ 0.137 m−2/s.

• Maximum ramp rate of quadrupoles (Technical Parameter List):

∂ ∂t ∂By,q ∂x

• max

= 57 T/m/s. With Bρ = 53 Tm → ˙ k1 = 1 Bρ ∂ ∂t ∂By,q ∂x

• max

= 57 T/m/s 53 Tm = 1.07 m−2/s. → Ramp time could be reduced to tramp = 0.16 s 0.137 1.07 = 0.02 s = 5700 T0

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Application of γt ramp

Apply tramp = 6000 T0, found in first estimate from ramp rate of dipoles.

• Found maximum momentum deviation

δm = 4 · 10−3, which is greater than that from analytic formula δm = 3.1 · 10−3, but much less than that for longer ramp time.

• No beam loss.

4000 8000 12000 16000 turns

• 0,006
• 0,004
• 0,002

0,002 0,004 0,006 δ of particles Start of γt ramp φs switch End of γt ramp

• Linear change of k1 from start to end without determining time for φs switch.

→ Scenario is more robust.

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Summary

• 1. Estimate growth of bunch area and beam loss for conditions of slow transition crossing

with analytic formula and particle tracking performed with MAD-X.

• Simulation for γt shift across γ due to optics change in realistic time interval.
• Acceptable particle loss of 4 %, but idealized scenario applied:

→ unperturbed lattice, transverse particle emittances set to zero.

• 2. On the other hand, γt shift by optics change much faster than γ shift during acceleration.
• Possible support of transition crossing due to significantly reduced crossing time.
• Reduction growth of bunch size and particle loss.

For more precise study, more proper code should be applied to include acceleration.