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

==0mm 1 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

==0mm SIS-100 proton operation 2 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: N p = 2 . 0 · 10 13 . • Two scenarios: 1. High- γ t operation (S. Sorge): Optics change during acceleration to shift γ t above extraction energy: γ t = 45 . 5 vs. γ = 32 → no transition crossing, but sensitive optics. 2. γ t jump scenario (S. Aumon, D. Ondreka): – γ t jump with high intensity bunch, need for installation of special jump quadrupoles. – Large dispersion → small tunes: Q x , Q y ≈ 10 . 3 . Optics with rather nice behaviour. New requirement: slow extraction of protons for HADES and CBM → topic of this talk.

==0mm SIS-100 proton slow extraction 3 • Low intensity: N p /t ∼ 10 11 / s , where basically 1 s is duration of a cycle. • Up to high energies: E max = 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 Q x = 17 . 31 , Q y = 17 . 45 , and γ t = 14 . 17 . – Injection at E = 4 GeV . → Beam energy has to cross transition energy. – γ t jump impossible because dispersion at positions of dedicated fast γ t 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.

==0mm Outline and model 4 Replace this scenario 1. Analytic estimate of growth of momentum spread. 20 → use conditions of tracking model. γ t =const 15 2. Tracking study using MAD-X: 10 γ → use constant energy and φ s = 0 , π . 5 0 • Assume longitudinal dynamics to depend only on phase 0 0,05 0,1 0,15 0,2 t / (s) slip factor. with – Keep γ = const and move γ t across γ by changing 30 quadrupole settings. Start from γ t > γ . 20 γ =const – Apply ˙ γ t = − ˙ γ nom , ( nom inal change rate of γ ), where γ t γ nom corresponds to ˙ ˙ B 0 of dipole magnets. 10 • Unperturbed lattice. 0 0 0,05 0,1 0,15 0,2 t / (s)

==0mm Conditions 5 Conditions chosen as example case. • Ramp rate of dipole magnetic field: ˙ γ nom = − 67 . 4 s − 1 . B 0 = 4 T / s which yields ˙ γ t = − ˙ • Gamma-t: set Q x , Q y , and γ t with three quadrupole families. – initial: γ t,ini = 25 – final: γ t,fin = 14 . 17 – determined by the usual slow extraction optics. – resulting ramp time: t ramp = 0 . 16 s = 44000 T 0 . • 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: η 0 ,ini = − 0 . 00187 vs. η 0 ,fin = 0 . 00151 . • Initial bunch area: A b = 1 eVs (D. Ondreka)

==0mm Analytic estimate 6 Expected effect: 1. Maximum momentum deviation of bunch in adiabatic and linear approximation, see S. Y. Lee “Accelerator Physics”, Eq. (3.58): � 1 / 4 ω 0 � heV | cos φ s | 1 � δ m = A b ∝ πβ 2 E 2 πβ 2 E | η | | η | 1 / 4 δ m,ini = 1 . 7 · 10 − 3 → initial maximum momentum deviation: 2. Phase slip factor during transition: η → 0 ⇓ Strong δ m growth resulting in growth of horizontal beam width due to x m = D x δ m • On the other hand, synchrotron motion very slow near transition energy (frozen bunch). → Momentum deviation can not infinitely grow if time is too short.

==0mm Analytic estimate 7 • Maximum momentum deviation during transition crossing, see S. Y. Lee “Accelerator Physics”, Eqs. (3.184) � γ t A b δ m = mc 2 ˙ 3 1 / 6 βτ ad Γ (2 / 3) γ with “adiabatic time” (Eq. (3.171) (in fact it is rather the non-adiabatic time because particles behave non-adiabatically during τ ad ): � 1 / 3 πβ 2 mc 2 γ 4 � t τ ad = . γω 2 ˙ 0 heV | cos φ s | • Finite maximum momentum deviation during transition: δ m = 4 . 4 · 10 − 3 with adiabatic time τ ad = 9 . 5 · 10 − 3 s = 2600 T 0 . – About 3 times initial momentum spread: δ m,ini = 1 . 7 · 10 − 3 – Maximum dispersion D m = 4 . 51 m → maximum dispersion orbit deviation x m = D m δ m = 20 mm , is less than aperture.

==0mm Simulation 8 Particle tracking for • Working point Q x = 17 . 31 , Q y = 17 . 45 . 30 φ s switch • Total: 60000 turns, where γ t changes during 45000 turns. 20 γ =const • Three quadrupole families, change focusing strengths k 1 γ t 10 during tracking linearly in time. • Pre-determine k 1 to match γ t at three points: 0 0 0,05 0,1 0,15 0,2 t / (s) start, when γ t = γ , and end. • Sudden switch of synchronous phase φ s = 0 → φ s = π , when γ t = γ . ′ = D ′ = 0 , ′ δ and y = y • Tracking particles along dispersion orbits: x = Dδ, x i.e. transverse emittances ǫ x = ǫ y = 0 . • 100 particles, initial Gaussian distributions for small adiabatic bunch truncated at 2 σ with σ ct = 1 . 7 m and σ δ = 8 . 7 · 10 − 4 .

==0mm Simulation results 9 0,015 Maximum particle momentum deviation: φ s switch 0,01 • analytic estimate: δ m = 4 . 4 · 10 − 3 0,005 δ of particles • simulation: δ m = 7 . 8 · 10 − 3 0 -0,005 Particle loss: P loss = 4 % . -0,01 Start of γ t ramp End of γ t ramp • Particle loss is low due to precisely setting -0,015 0 20000 40000 60000 turns γ t = γ when switching φ s . • If not done so, particle loss much higher. – No control of working point and γ t during ramp. → mismatch between φ s and sign of η 0 during ∼ 1000 turns possible. – Found P loss = 44 % in first attempt.

==0mm Discussion 10 • Particle loss possibly acceptable because of low intensity, but idealized case chosen: – unperturbed lattice. – particle emittances ǫ x = ǫ y = 0 . • Scenario used in the simulation is not foreseen but also not unrealistic because slow extraction optics necessary only after acceleration. → Possible support of γ t crossing due to lattice change.

==0mm Application of γ t ramp 11 • Small differences between initial and final focusing strengths of quadrupoles: k 1 ,def = − 0 . 2023 m − 2 → k 1 ,def = − 0 . 2017 m − 2 0 . 1866 m − 2 → k 1 ,def = 0 . 2015 m − 2 k 1 ,f, 1 = 0 . 2235 m − 2 → k 1 ,def = 0 . 2015 m − 2 k 1 ,f, 2 = and long ramp time result in very low ramp rate: k 1 ≡ | k 1 ,fin − k 1 ,ini | max ˙ ≤ 0 . 137 m − 2 / s . t ramp • Maximum ramp rate of quadrupoles (Technical Parameter List): � ∂ ∂B y,q � = 57 T / m / s . ∂t ∂x max � ∂ � k 1 = 1 ∂B y,q = 57 T / m / s ˙ = 1 . 07 m − 2 / s . With Bρ = 53 Tm → Bρ ∂t ∂x 53 Tm max → Ramp time could be reduced to t ramp = 0 . 16 s 0 . 137 1 . 07 = 0 . 02 s = 5700 T 0

==0mm Application of γ t ramp 12 Apply t ramp = 6000 T 0 , found in first estimate from ramp rate of dipoles. • Found maximum momentum deviation 0,006 φ s switch δ m = 4 · 10 − 3 , 0,004 0,002 δ of particles which is greater than that from analytic formula 0 -0,002 δ m = 3 . 1 · 10 − 3 , -0,004 but much less than that for longer ramp time. Start of γ t ramp End of γ t ramp -0,006 0 4000 8000 12000 16000 turns • No beam loss. • Linear change of k 1 from start to end without determining time for φ s switch. → Scenario is more robust.

==0mm Summary 13 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 γ t 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, γ t 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.