Benchmark and Study of PSR Longitudinal Beam Dynamics

Sarah M. Cousineau,

Jeffrey A. Holmes, Slava Danilov

MAP Workshop

March 18, 2004

Longitudinal Instability Benchmark

Part I – Benchmark of PSR longitudinal microwave instability.

ORBIT Benchmark of Microwave Instability in PSR

Background:

1999 – 3 ferrite inductive inserts placed in PSR to provide longitudinal space charge compensation.

Inserts lead to unacceptably large microwave instability, were removed.

1999, 2000 – Heating of the inserts was shown to cure instability while still providing space charge compensations. Two heated inserts now used in PSR.

2003 – Chris Beltran models impedances of both sets of inserts using MAFIA (PhD thesis). Results allow for detailed simulations of instability (ESME, ORBIT, etc).

Goal: Benchmark ORBIT’s longitudinal impedance algorithm with experimental data. (Partial response to ASAC request for benchmark of ORBIT impedance capabilities).

Experimental Evidence of Microwave Instability

Wall current monitor signal

End injection Peak instability

• 3 inductive inserts @ 25º C. • Beam intensity = 650nC (4×1012 protons)• Instability peak @ 200 s (150 s after injection)

Figure courtesy C. Beltran, doctoral thesis.

2 Turn Wall Current Monitor Signal (Experimental)

Signal for 2 turns at end of injection Signal for 2 turns at peak of instability

Instability Frequency = 72 MHz (harmonic = 26)

Figures courtesy C. Beltran, doctoral thesis.

Impedance of Inductive Inserts

Impedance of room temperature inductive inserts (C. Beltran, thesis 2003)

ORBIT Simulations of Microwave Instability

ORBIT Simulation Parameters:

• 650 nC (~4×1012 protons).• 150 us accumulation time (~400 turns), + 200 us storage (~600 turns).• Z/n as computed by C. Beltran. p/p as bi-Gaussian, 66% with =6.9×10-4, and 34% with = 2.8×10-4.

• Longitudinal tracking only.• From a numerical convergence study performed, used:

– 256 longitudinal bins.– 8×106 macroparticles.

ORBIT Benchmark Results

End injectionPeak of instability

• Instability peak 150 s after injection (Same as experiment).

Simulated One-Turn Wall Current Monitor Signal

Signal for 2 turns at end of injection Signal for 2 turns a peak of instability

Experimental Experimental

Simulated Simulated

Evolution of Dominant Harmonics

• Exponential growth of harmonics observed.

• Dominant harmonic is h=26, same as experiment.

• Growth time of instability, 42 s; Experiment result is 33 s

Slope=1/ 1/42 s

Analysis of Instability Threshold

• Data set taken in 2002 to understand threshold; 2 inductors at room temp.

• Define threshold by beam intensity at which relevant harmonics rise coherently above noise level. Experimental threshold=80 nC; Simulated threshold=60-70 nC.

Experimental Data

Simulated Data

70 nC (noise level)80 nC (threshold)460 nC (strong instability)

500 nC (strong instability) 70 nC (threshold) 50 nC (noise level)

Linac Microbunch Dynamics

Part II – Linac microbunch dynamics in the PSR.

The PSR 201 MHz Phenomenon

• 201 MHz structure in PSR should disappear in 30 turns

• Microwave instability data shows this structure sticking around for ~1000 turns.

End of Injection End of Injection

Chopped beam Coasting beam

End of Injection

Experimental Analysis of the 201 MHz Structure

• Analysis of 201 MHz harmonic shows structure increasing after injection. Longitudinal profile 300 turns after

end of injection.

70 nC

210 nC

• Analysis also shows 201 MHz structure is stronger at higher intensity

End of injection

End of injection

Chopped beam

Coasting beam

Simulations of the 201 MHz Structure

• 1D tracking simulations with ORBIT show same long-lived 201 MHz microstructure; structure present with or without impedance.

• Structure quickly decoheres in simulations without space charge.

With Space Charge No Space Charge

End of Injection End of Injection

• Long-lived “bubble” structures noticed in CERN PS Booster ring; suspected due to linac micro-structure. Much theoretical work published to explain long-lived structure (resistive wake, etc).

• In 2000 CERN PSB experiment used RF to insert larger, lower frequency “holes” and observe structure during acceleration. Paper by Koscielniak et al argues that longevity of holes due to space charge.

• The PSR machine a special case for which the ring frequency is an exact sub- harmonic of the linac frequency (72nd sub-harmonic).

• See clear formation of separatrix and “anti-buckets” at frequency of 201 MHz.

Formation of 201 MHz “anti-buckets”

fast

slow

fast

slow

fast

Self-consistent, stationary analytical solution can be found for simple 2-state system.

Observations of 201 MHz Structure Dynamics

• Near steady-state condition for certain balance of p/p and intensity.

• Rate of injection also an important condition for establishing steady state.

• We are in the process of investigating these dynamics for the PSR case.

End of injection… …250 turns after injection …650 turns after injection

End of injection… …250 turns after injection …650 turns after injection

200 nC

100 nC

A Vlasov solver for one bucket

• Set up fast Vlasov solver to look for steady-state solutions.

• Solve:

with,

(self-consistency)

and periodic boundary conditions in .

• For steady state, should have:

dt

df

d

df

d

df

d

dH

d

df

dt

df s

dtft

tbatH

),,(),(

),,(2

1),,( 2

)(Hff

Hamiltonian contours

Can anti-buckets live forever?

At least 10000 turns, under right conditions

Long-lived solutions0 turn longitudinal phase space

10,000 turn longitudinal phase space

Density profiles for 0 and 10000 turn distributions

Distribution function

dE

Phi