# Presentation: 014181

##   Overview on Resonant Space Charge Effects in Linacs

*Oak Ridge, June 25/26, 2001*

- Non-equipartitioned Beams
- 2. Resonance Charts
- 3. Mismatch
- Acknowledgment:
-  G. Franchetti, GSI,  J. Qiang, LANL, R. Ryne, LBL
- D. Jeon, ORNL, F. Gerigk, CERN

-   **Overview on Resonant Space Charge Effects in Linacs**

## Types of Resonant Motion

- Emittance growth and halo result
- mainly from resonance
- other negligible sources (% or less):
-  intrabeam scattering
-  conversion of nonlinear field energy
-  filamentation in nonlinear rf potential

- Single particle
- resonance with core  -> halo

- Coherent core resonance with itself                 -> rms growth

- Coherent core resonance with periodic structure

- Avoid: 0 > 900

- 3 kinds:

## Most studies are on isotropic beams

- Anisotropy brings new features

- _Core_: non-equipartition issue – *energy transfer*
- _Halo_: halo size and density depend on tune ratios
- 	see: Ikegam in 2D; Fedotov & Gluckstern in 3D,
-           	Qiang, I.H., Ryne (PAC01) – 2:1 halo *extent to infinity*

- Isotropic cores are stable (non-KV)
- Numerous halo studies (2:1 halo extent to 4-5 )

**We have explored this systematically in 2D to enlarge physics basis for SNS studies **

## Core eigenmodes are multipole oscillations in density

- x

- z

- y

- 3D

- y infinite

- x

- z

- 2D

- strong in initial WB

- only even modes have r-z symmetry

## SNS nominal emittance ratio

## Doubled longitudinal emittanceas part of error study 

## Examples of emittance exchange in 2D

- Approach to equipartition depends on where on stopband

- in 3D shared by x,y

## Cuts in 2D/3D and constant focusing

- Relative emittance growth

## Anisotropy resonance occurs in core – ‘no’ halo 

- Anisotropy resonance occurs in core – ‘no’ halo

- Resulting rms mismatch is negligible,
- hence only 20-30% halo found

## AG –focusing (FODO with x=800  kept constantvaried zero-current phase advance in z

- Envelope instability z > 900

- Relative emittance growth

- In x phase space evidence
- of structure resonance driven
- by periodic focusing

- x-x‘

- x-y‘

- x‘-y‘

- y-y‘

## Two crossings found in Parmila simulations for SNS 

- Two fast crossings may lead to almost maximum exchange (30-40%)
- for doubled longitudinal emittance

- DTL

- SRF

## Mismatch: halo build-up time (isotropic beams)for symmetric focusing

- Mismatch: halo build-up time (isotropic beams)for symmetric focusing

- betatron periods

- k/k0

## MM effect enhanced by anisotropyexample MMz=1.3 / MMx=1.0 

- Relative emittance growth

- doubled initial
- longitudinal emittance

- in 3D ~ half for x,y

## Fixpoints / emittance ratio 1

- core

- fix-points

- fix-points at infinity

- fix-points in core

- breathing mode

- quadrupolar mode

## Maximum size

## Conclusions

- Space charge resonance model now well confirmed by simulation in 2D and  3D – develops unchanged for periodic focusing
- Confirmed also in PARMILA simulations for SNS linac: no concern for design emittances (emittance ratio ~ 1) ; 30-40% transverse emittance growth (in x and y) would arise for doubled input longitudinal emittance
- Generic 2D mismatch studies have shown that rms and halo growth is a function of the tune ratio due to „floating“ fixpoints for 2:1 resonance; worse for mix of mismatch modes or initial halo (from RFQ)
- Next: need to combine initial mismatch with error-driven mismatch and study effect of emittance growth