# Presentation: 072714

## Solitons in Synchrotrons

- Longitudinal, nonlinear, stable waves have been observed in both coasting (CERN 70s) and bunched (SPS, Tevatron, RHIC ) beams.
- Transverse stable oscillations have been observed in electron rings (KEK 1999)
- In all cases the filamentation caused by incoherent frequency spread is frustrated by collective forces.

## Review of Collective forces

- The force on the trailing particle is created by the leading particle.
- The customary definition is to use units of voltage to describe the kick.
- My personal preference is to use the relative arrival times of the leading and trailing particles as the longitudinal coordinate.
- The Green functions are called wake potentials and the integrated kicks are called wake fields.

- The kick on the trailing particle due to the leading particle is

## Longitudinal equations of motion

- Arrival time and energy after reference azimuth
- Voltage per turn
- Hamiltonian for short range wakes

## Longitudinal Solitons in Coasting Beams

- Theory developed by Sacherer and Raka (70s, hand written notes)
- See Schamel, PRL 1997,  and Kolscielniak et al PRSTAB, 2001, PRSTAB 044402
- Take constant Z/n. >0  is the line density.  For initially bunched beam >0 increases debunching rate. In bunched beams >0 reduces incoherent synchrotron tune.
- Assume a solution exists and constrain the distribution (potential WELL)
-                                                                        see a hole with SC below transition

## Inclusion of wakes  in transverse equations of motion

- The horizontal equation of motion for a particle is
- The resonance terms are important for half integer tunes or large synchrotron tunes.
- The final equation is equivalent to

## Transverse solitons

- Ohmi and Kobayashi  Phys Rev E, V 59 # 1, 1999
- A bright electron bunch is kicked horizontally with strong octupoles on.
- Depending on the sign of the detuning the bunch damps or oscillates at fixed amplitude.
- For short bunches and a > 0 get stable oscillations.
- Fewer symbols when time is rescaled

## Hamiltonian theory 

- Hamiltonian
- Centroid is a solution
- Slow Hamiltonian
- Go back to cartesian coordinates
-                               is  a stable fixed point
- Aspect  ratio from curvature near SFP

## Did simulations with constant dQ/dJ (left)  and constant coherent tune shift (right).  The agreement is quite good.

- Did simulations with constant dQ/dJ (left)  and constant coherent tune shift (right).  The agreement is quite good.

## Longitudinal Solitons in Bunched Beams

- See persistent, longitudinal hotspots (humps) in proton (and deuteron) bunches.
- Data on right after 40 minutes
-  > 0  here so sign disagrees with coasting beam result.

## Longitudinal Soliton Theory for Bunched Beams

- Spread in synchrotron frequency with amplitude implies a collective effect.
- See hotspots with a defocusing impedance (Z/n measurements in EPAC02)
- Take a constant Z/n and rescale time.
- Line density is related to phase space density by
- Switch to canonical coordinates that rotate with the soliton
- For a canonical transforrmation
- Average over s assuming that the phase space density rigidly rotates

## For delta wake the integral is doable and the averaged Hamiltonian is

                           is the variation in synchrotron tune with amplitude
Notice that the collective part is like the gravitational potential in the plane of a flat galaxy.  Lots of old potential theory to take advantage of.
To get a stable fixed point consider

The collective part gives no net force  so we need 

The Vlasov equation is just the statement that contours of constant f and K coincide

-                            is the variation in synchrotron tune with amplitude
- Notice that the collective part is like the gravitational potential in the plane of a flat galaxy.  Lots of old potential theory to take advantage of.
- To get a stable fixed point consider
- The collective part gives no net force  so we need
- The Vlasov equation is just the statement that contours of constant f and K coincide

## Go back to the A and B coordinates

                                is a SFP 
The unperturbed Hamiltonian looks
like a ring of mountains.
A local hump in the Hamiltonian
is easier to make than a local 
hole!  This explains the sign 
discrepancy between bunched
and coasting beams.
Assuming a parabolic mini-bunch
one can get a good analytic solution. 
General bunch shape requires 
numerical solution.

-                                 is a SFP
- The unperturbed Hamiltonian looks
- like a ring of mountains.
- A local hump in the Hamiltonian
- is easier to make than a local
- hole!  This explains the sign
- discrepancy between bunched
- and coasting beams.
- Assuming a parabolic mini-bunch
- one can get a good analytic solution.
- General bunch shape requires
- numerical solution.

## Analytically can have either sign of  
but >0  is more stable in simulations.
For Vlasov solver >0 was calculated.
See Blaskiewicz, Wei, Luque, Schamel
PRSTAB 044402 (2004)

- but >0  is more stable in simulations.
- For Vlasov solver >0 was calculated.
- See Blaskiewicz, Wei, Luque, Schamel
- PRSTAB 044402 (2004)

## Conclusions and Questions

- Stable, coherent oscillations exist in both the longitudinal and transverse planes.
- Fairly good agreement can be found using first order perturbation theory and simple models.
- How does the simple transverse model translate to our machines?
- dQ/dJ > 0 augments space charge nonlinearity which appears to help Landau damping (Mohl and Schonauer, or PRSTAB 044202).  This is the sign leading to stable oscillations even for large detuning.
- “Dressed” IBS and limits of cooling.
- Thank you for your attention!