# Presentation: 532508

## Simulation of Collective Effects in High Intensity Proton Rings

**J. A. Holmes****	****	****ORNL**
**S. Cousineau****	****	****ORNL**
**V. Danilov****	****	****ORNL**
**A. Fedotov****	****	****BNL**
**S. Henderson****	****	****ORNL**
**J. D.  Galambos****	****	****ORNL**

**Fermilab, September 11, 2001**

## Motivation

- High intensity proton rings such as FNAL Booster, AGS Booster, PSR, and SNS are characterized by low energy, high beam intensity, and low beam loss requirements for high availability.
- These requirements of high intensity and low losses necessitate a detailed understanding of beam dynamics in this regime.
- Under these conditions collective effects due to space charge and wakefields will dominate the beam behavior, and single particle models will not apply.
- Because of the complexity of collective phenomena for bunched beams in high intensity rings, a computational approach is productive.

## ORBIT

- In response to this need, the SNS AP group at ORNL, with help from BNL colleagues, developed the ORBIT code.  ORBIT started as a C++ rewrite of ACCSIM, but has since undergone extensive independent development.
- ORBIT is a particle-tracking code in 6D phase space.
- ORBIT is designed to simulate real machines: it has detailed models for injection foil and painting, transport through various types of lattice elements, RF and acceleration, ***space charge***, longitudinal and ***transverse impedances***, conducting wall beam pipe, apertures, and collimation.
- ORBIT has an excellent suite of routines for beam diagnostics.

## Applications

- To illustrate the type of analysis possible with ORBIT, I will present two applications:
- Beam broadening in PSR.  This is a detailed simulation of a high intensity PSR experiment using the actual injection scenario, lattice and settings, RF waveforms, and extraction.  Computational results are compared with measured beam profiles at a wire scanner in the extraction line.  These calculations make use of a 2.5D transverse + 1D longitudinal spacecharge module.
- Calculations with the transverse impedance and 3D spacecharge modules.  These include two applications to SNS and a benchmark with analytic theory.

## PSR Simulations:

**Beam profile measurements were performed in the extraction line for a range of intensities at PSR.**
**The experiment was carefully simulated using ORBIT, including lattice settings, injection, painting, and RF scenarios.  A 2.5D transverse + 1D longitudinal spacecharge model was used.**
**The calculated and measured beam profiles were compared, with reasonably good agreement, particularly regarding systematic behavior.**
**The calculations were then analyzed to extract the physics responsible for the observed behavior.**

## Comparison of Measured and Calculated Profiles:

**Beam profile measurements were performed in the extraction line and subsequently simulated in detail for a range of intensities at PSR.**
**Measurements and simulations both yield vertical beam broadening at high intensity and reasonable systematic detailed agreement.**

## Analysis of Simulation Results:

**Analysis of the simulations indicates beam broadening at high intensity.  This is associated with the incoherent tunes of a substantial fraction of the beam crossing Q****y**** = 2.**

## Analysis of Simulation Results:

**In the simulations, longitudinal peaking (low bunch factor) increases the local beam intensity and tune depression.  2.5D transverse spacecharge is critical here.  PSR now obtains higher beam intensities with an inductive insert and “notched” injection scheme.**

## Analysis of Simulation Results:

**Spectrum analysis of the beam second moments in the simulations shows a large N=4 component at high intensity, consistent with a coherent half integer resonance at Q****y**** = 2.**
**For a number of cases, the final RMS emittance increases linearly with the N=4 amplitude above a threshold value.**

##  Analysis of Simulation Results: Summary

**Beam profile measurements and simulations show reasonable systematic and detailed agreement, and both yield vertical beam broadening at high intensity.**
**Analysis of the simulations indicates that the beam broadening at high intensity is accompanied by the incoherent tunes of a substantial fraction of the beam crossing Q****y**** = 2.**
**In the simulations, longitudinal peaking (low bunch factor) increases the local beam intensity and tune depression.  We have just obtained new data from PSR for cases in which the longitudinal profiles are manipulated.  In addition to transverse measurements, the new data includes the longitudinal profiles.**
**Spectrum analysis of the beam second moments in the simulations shows a large N=4 component at high intensity, consistent with a **_**coherent half integer resonance at Q**__**y**__** = 2**_**.**
**For a number of cases, the final RMS emittance increases linearly with the N=4 amplitude above a threshold value, also consistent with a **_**coherent half integer resonance at Q**__**y**__** = 2**_**.**

## PSR Half Integer Resonance: An Envelope Equation Analysis

**A detailed analysis of half integer resonances with space charge and magnet errors was presented using the envelope equations in the thesis of Sacherer (1968).**
**Envelope Equation for y (similar equation for x):**

## PSR Half Integer Resonance: An Envelope Equation Analysis

**Sacherer’s analysis simplifies the space charge term by replacing the oscillating beta functions by their average values.  The resulting equations are then solved perturbatively for periodic solutions, assuming that the magnet error and space charge terms are small perturbations.**
**Sacherer’s results that pertain to the present situation are:**
**Because the tunes are separated (********x****= 3.17, ********y****= 2.14), x-y coupling is weak, and a 1-D envelope analysis in y is appropriate.**
**Strong x-y coupling occurs only when the tunes are nearly equal.**
**For circular beams this occurs only when**
**Because magnet errors are ignored here, the periodic solutions to the envelope perturbation equation are constants of unit amplitude except for a very narrow resonance at **

## PSR Half Integer Resonance: An Envelope Equation Analysis

**To apply the results of Sacherer’s analysis to PSR, we calculate periodic solutions of the full, coupled, envelope equations under the following assumptions:**
**We use the same detailed description of the lattice as was used in the ORBIT particle simulations.**
**We present the results in normalized form as shown above.**
**Because of the simplifying assumptions made by Sacherer in his analysis, we expect some differences in the details of our results.  Sacherer**
**neglects the rapid oscillations of the lattice functions in the space charge term, and**
**treats the magnet errors and space charge as perturbations.**
**In light of these similarities and differences, the full envelope equation solutions**
**will show essentially independent, decoupled x and y motion,**
**will reveal resonant behavior for incoherent tunes near Q****y**** = 1.916, but**
**will give non-constant solutions off the resonance, particularly at high intensities where Sacherer’s assumptions are violated.**

## PSR Half Integer Resonance: An Envelope Equation Analysis

**Because our picture of PSR beam broadening suggests sensitivity to peak beam intensity, it is this quantity which should be compared to the beam intensity in the envelope equations.  This is obtained from the simulations by dividing the average intensity by the bunch factor.**
**The amplitude of the periodic solution of the envelope equations peaks strongly at a beam intensity corresponding to that at about 1750 turns in the high intensity particle simulation.  This corresponds to the half-integer coherent resonance.**

- Envelope Amplitude Peak

- Half-Integer Resonance

## PSR Half Integer Resonance: An Envelope Equation Analysis

**The incoherent tune of test particles in the envelope potential decreases with increasing intensity.  However, rather than a uniform decrease, the tune is elevated in the region where the envelope amplitudes peak.  This is a nonlinear effect: the large amplitudes weaken the space charge force, thus reducing the tune shift.  Because of this, plotting the envelope amplitude versus the incoherent tune places the apparent location of the resonance above the predicted value.  However, “removing the bump” in the tune plot places Sacherer’s prediction of 1.916 just above 6*10****13**** protons, in the center of the resonance.  The peak at 7.5*10****13**** protons occurs just before dropping out of resonance on the high intensity side.  Even so, strong nonlinearities persist at intensities above the resonance.**

- Tune at Resonance

- Tune Elevated When Envelope
- Amplitudes Peak

- Half-Integer Resonance

## PSR Half Integer Resonance: An Envelope Equation Analysis

**Periodic envelope functions are shown for a number of different beam intensities, parameterized by the incoherent tunes.  As expected, at low intensity (Q****y**** = 2.14), the envelope function is nearly constant.  After passing through the resonance peak (Q****y**** < 1.916) the phase of the envelope function oscillations shifts by 180 degrees.**

## PSR Half Integer Resonance: Conclusions

**Periodic solutions to the full, coupled, envelope equations using the PSR ring lattice with tunes (Q****x****, Q****y****) = (3.17, 2.14)**
**display essentially independent, decoupled x and y motion,**
**reveal resonant behavior in the y equation amplitudes for incoherent tunes near Q****y**** = 1.916 and envelope intensities of (5-7.5)*10****13****, which correspond to PSR peak beam intensities from 1000 to 1750 turns in the simulation of a high intensity case,**
**give non-constant solutions off the resonance, particularly at high intensities where the sign of the oscillations flips.**
**These results**
**further corroborate the half integer resonance at Q****y ****= 2 as cause of the observed beam broadening in PSR,**
**agree with Sacherer’s analysis in the independence of the x and y solutions and the location of the resonance in incoherent tune,**
**disagree with Sacherer’s analysis in the amplitude and variation of the envelope functions off resonance, particularly at higher intensities.  This is understandable in terms of the simplifying assumptions used in Sacherer’s analysis.**

## PSR Beam Broadening: Acknowledgments

**We wish to acknowledge Bob Macek and the PSR staff for encouraging this work and for providing access and assistance in carrying out high intensity measurements at PSR.**
**For theoretical guidance and interpretation we used the following PhD thesis:**
**	****	****Transverse Space-Charge Effects in Circular Accelerators**
**	****	****by F. J. Sacherer**
**	****	****Lawrence Radiation Laboratory**
**	****	****University of California, Berkeley, 1968**

## Transverse Impedance and 3D Space Charge Topics

- ORBIT Model Development
- Transverse Impedance Model
- 3D Space Charge Model
- Applications to SNS
- Ring extraction kicker
- Resistive Wall Impedance
- Benchmark of Models
- Transverse Impedance Benchmark
- 3D Space Charge Model
- Closed Orbit Instability as Transition Process

## Transverse Impedance Model

- Transverse impedance treated as localized node in ORBIT
- Element length must be short compared to betatron oscillation wavelength
- If physical impedance is not short, multiple impedance nodes are required
- Impedance representation
- User inputs Fourier components of impedance at betatron sidebands of the ring frequency harmonics
- Velocities less than light speed included in formulation
- Particle kicks
- Convolution of beam current dipole moment with impedance
- Current evaluation assumes dipole moment evolves from previous turn according to simple betatron oscillation

## 3D Space Charge Model

- Particles are binned in 3D rectangular grid
- 2nd order momentum-conserving distribution of charges to grid (see Hockney and Eastwood)
- Typically, for rings, longitudinal spacing greatly exceeds transverse spacing
- Potential is solved on transverse grid for each longitudinal slice
- Fast FFT solver is used
- Conducting wall boundary conditions (circular, elliptical, or rectangular beam pipe) “tie together” the transverse solutions
- Particle kicks are obtained by interpolating the potentials in 3D
- 2nd order momentum-conserving interpolation scheme is used (see Hockney and Eastwood)

## SNS Extraction Kicker: Instability Threshold (Calculation Based on Old Measurements)

- The transverse impedance module was used to simulate instability thresholds due to the extraction kicker for an actual SNS injection scenario.
- The impedance for the extraction kicker was taken from old measurements (by J. G. Wang).
- Calculations were carried out for open termination and for 50 Ohm termination.
- 50 Ohm termination yields lower instability thresholds than does open termination.
- Large halos are generated for unstable cases.
- The inclusion of additional effects, such as chromaticity and 3-D space charge forces will modify the present results.
- These calculations have been superceded by new impedance measurements, and are presented for illustration only.

## SNS Extraction Kicker: Transverse Impedance Measurement

## SNS Extraction Kicker: Transverse Impedance Simulations, Instability Threshold

- Energy Spreader On

- Energy Spreader Off

## Impedance-Driven Instabilities Lead to Significant Halo

## SNS Resistive Wall Impedance : Instability Threshold

- The transverse impedance module was used to simulate instability thresholds due to the resistive wall impedance for an actual SNS injection scenario.
- The impedance was estimated by applying the standard resistive wall formulation to the SNS ring.
- Without chromaticity or space charge, the beam becomes unstable, dominated by n = 5-7, at around 2*1014 protons
- The inclusion of chromaticity shifts the threshold upward to 13*1014 protons.
- The simulation is yet to be carried out with 3D space charge.  On a single CPU the calculation would require about a month.  We are working on a parallel implementation of 3D space charge in UAL/ORBIT.

## SNS Resistive Wall Impedance: Evolution of n = 4-8 Harmonics.

- 2*1014 protons, No Chromaticity

- 13*1014 protons, Chromaticity

## SNS Resistive Wall Impedance: Particle Distributions.

- 1060 Turns

- 1400 Turns

## Transverse Impedance Studies: Benchmark

**Benchmark ORBIT with analytic calculation using solvable model:**
**Straight uniform focusing lattice.**
**Periodic length 40 m, tunes (1.1,1.05).**
**Longitudinally localized vertical impedance (b/a = 2, single n=2 harmonic, Z = 2*10****5**** Ohm in detailed results shown below).**
**No direct space charge force in benchmark.**
**Coasting “pencil” beam with**
**1 mm displacement in y (n = 2 harmonic);**
**Lorentz energy distribution (1 GeV, RMS width 1%, cutoff at 10%);**
**(0-5)*10****13**** particles.**
**Use 2*10****5**** macroparticles.**
**Analytic calculation with Vlasov equation and Landau damping.**
**Solve for evolution of dipole moment using averaging method assuming single mode and given energy distribution.**

## Benchmark: Stability Threshold is Somewhat Above 2*1013 Particles

- Analytic calculation places threshold at 2.24*1013 particles.

## Benchmark: Analytic and Computational Time Evolutions of Beam Moments Agree

- At 3*1013 particles, computational growth rate exceeds analytic by ~18%.

## Benchmark: Unstable Case - Distribution Width and Fraction in Halo Agree

## Benchmark: Unstable Case - Analytic and Computed Phase Space Evolutions Agree

## Benchmark: Stable Case - in Spite of Stability, the Beam Grows and Halo Forms

## Benchmark: Stable Case - Beam Grows and Halo Forms, Phase Space Evolutions Agree

## Transverse Impedance Benchmark: Results Summary

- Instability threshold at ~2.2*1013 particles: good agreement.
- Evolution is in good agreement both for stable and unstable cases, not only for averaged quantities, but also for detailed phase space distributions.
- Halo can form even for “stable” cases.  This is found both analytically and computationally, and the results of the two approaches are in good agreement.
- These results neglect chromatic effects and direct space charge forces.

## Without Direct Space Charge, Evolution of Distributed and Pencil Beams Agree

- To study the effects of space charge, a distributed beam must be used.  Here, distributed beam solutions without space charge are compared to pencil beam solutions.  The results agree.

## Correct Simulation of Space Charge with Transverse Impedance Requires 3D Model

- This plot illustrates the necessity of a 3D space charge model when transverse impedances are studied.  The harmonic evolution using a 2D space charge model is significantly different from that obtained using a 3D model.  Transverse impedance kicks depend on the longitudinal distribution of the dipole moments.

## 3D Space Charge Benchmark: 3D Model Agrees with 2.5D + Longitudinal Models

- The 3D and 2.5D space charge models were compared for a case with uniform focusing lattice, circumference 248 m, tunes (6.4,6.3), triangular longitudinal distribution, KV transverse distribution, circular beam pipe with b/a = 2, 3*1014 protons, and 2*105 macroparticles.  Good agreement was obtained for kicks and for tunes (which also agree with the Laslett formula).  The fuzziness of the 3D tune distribution results from fewer particles per longitudinal slice.

## Transverse Impedance and Space Charge:  Stable Case

- For the stable transverse impedance benchmark case with distributed beam, space charge has little effect on the evolution.

## Transverse Impedance and Space Charge:  Unstable Case

- The unstable transverse impedance benchmark case is further destabilized by the 3D space charge.

## Transverse Impedance and Space Charge:  Marginally Stable Case

- The unstable transverse impedance benchmark case is destabilized by the 3D space charge.  These results were obtained for fixed transverse impedance and varying space charge densities (consistent with the beam current density).  More general results would require varying both impedance and current, and including other effects, such as chromaticity.

## Closed Orbit Instability as Transition Process

- A new transverse instability mechanism has been studied analytically using a simple wake function analysis.
- If there is a significant transverse impedance at very low frequency (~kHz), this can act over many turns to amplify any closed orbit distortion.
- Such an impedance may occur in the case of a thin metallic vacuum chamber surrounded by magnets with much higher surface impedance.  The impedance of such an element has been estimated by Danilov, using the Panofsky-Wenzel formula, to be proportional to frequency-1.
- An estimation for the SNS ring, assuming the impedance in question is generated by ceramic injection kicker chambers coated with TiN, gives an inverse growth rate of 236 turns.
- This mechanism is effective only while the fields live inside the vacuum chamber.  If the penetration time is several thousand turns, this could impact the closed orbit during injection.

## Summary

**New transverse impedance and 3D space charge models have been developed in the ORBIT code.**
**The transverse impedance model has been applied to previous measurements of the SNS extraction kicker impedance, resulting in instabilities and halo at SNS beam intensities.  Although the results have been superceded, they led to significant efforts to verify and reduce the extraction kicker impedance.**
**The transverse impedance model has been applied to the resistive wall impedance of the SNS ring.  At full beam intensities the ring is seen to be marginally stable if chromaticity is neglected, but quite stable when chromatic effects are included.  The effects of space charge have not yet been studied for this case.**
**Benchmarks of the transverse impedance and 3D space charge models with analytic calculations have been conducted, showing good agreement.  The transverse impedance benchmark illustrates a stable case in which halo is generated even as the harmonics decay.**
**A new closed orbit transverse instability mechanism has been studied analytically using a simple wake function analysis.  This mechanism indicates that the closed orbit may be transitionally unstable to impedances generated by thin metallic vacuum walls surrounded by larger impedances.**