# Presentation: 575105

## Transverse beam instability in the SNS due to the extraction-kickers impedance

** ****Fedotov, M.  Blaskiewicz, J.  Wei (BNL)     S. Danilov, J. Holmes, A. Shishlo (ORNL)**

**May 14, 2002**

## Acknowledgment

**We thank D. Davino, N. Malitsky, Y.Y. Lee and **
**N. Tsoupas for many useful discussions.**

## Numerical implementation

**In order to study collective beam dynamics, computational models for the impedance and 3D space charge have been developed and implemented in ORBIT            ****(S.Danilov,J.Holmes,J. Galambos)**
**After  these new algorithms were successfully benchmarked they were implemented in UAL ****(A. Shishlo)**
**These new models allow us comprehensive study of collective beam dynamics**

## Simulations

**Preliminary studies with full-intensity beams indicated that 2MW  SNS beam is near the instability threshold                                       ****(S. Danilov, J. Homes, M. Blaskiewicz)**
**We then proceeded with realistic multi-turn injection scenario.**
**In order to avoid numerical diffusion with the 3D space charge model we first obtained saturation of numerical parameters.**
**This resulted in implementation of the code on BNL/ITD parallel cluster  (up to 40 CPU) which allowed this time consuming calculations ****(N. Malitsky)**
**Presented here simulations are done with half a million particles on 20 CPU. **
**Important feature of instability thresholds with 1060-turn injection is the fact that final intensity is reached only at the end of accumulation right before the extraction.**

## Impedance model

**Recent measurements of the transverse coupling impedance of one full-size model of the 14 extraction kickers led to the realistic estimate of this impedance contribution ****(D. Davino and H. Hahn)**
**This impedance was significantly reduced with the 25 ********termination.**
**All 14 kickers are represented by a single impedance node using the average beta-function approach.**
**Such approach overestimates magnitude of the impedance by approximately ****15%.**** As a result, our  threshold estimates are slightly ****conservative****.  **

## Re(Z) for full 14-kicker system 

**Impedance model used in simulations**

## Outline

**Instability thresholds**
**Study topics:**
**Open vs  25 ********termination impedance**
**Low frequency (1-2 MHz) and space charge stabilization**
**Stabilization with b/a (effective tune spread along the bunch)**
**Small b/a , quadrupole effect**

## N=2.0*10^14 at the end of accumulation

**Blue**** –    no SC, nat. chromaticity (-7)**
**Green – SC, zero chromaticity**
**Pink ****–    SC, nat. chromaticity**

## Resulting beam halo for N=2.0*10^14

**Blue**** –    no SC, nat. chromaticity (-7), b=11cm   Green – SC, zero chromaticity, b=11cm                  ****Pink ****–    SC, nat. chromaticity, b=11cm             ****Red ****-     SC, zero chromaticity, b=20 cm**

## N=1.5 *10^14 at the end of accumulation

**Green – SC, zero chromaticity**
**Pink****   -  SC, nat. chromaticity (-7)**

## Resulting beam halo for N=1.5*10^14

**Green – SC, zero chromaticity**
**Pink****   -  SC, nat. chromaticity (-7)**

## Intensity limitation

**N=2.0*10^14**** – ****Unstable****, using natural chromaticity is not enough, additional damping with non-linear spread due to octupoles but most likely unsufficient for this intensity. ****Needs feedback system or minimization of impedance.**
**N=1.5*10^14**** – ****Unstable**** with zero chromaticity. Marginally ****stable**** with natural chromaticity. Additional small damping with octupoles can help. ****Feedback system is not required (when working with nat. chromaticity) but it is recommended.**
**N=1.0*10^14**** – ****Stable**** with natural chromaticity. With zero chromaticity no significant halo is observed by the end of accumulation.    **
** **

## 25 Om vs Open termination

**Open termination:**** Large  impedance around 30 MHz where Landau damping may be effective. Very small impedance till 10 MHz where Landau damping is not effective.**
**25 Om termination**** – removed pick around 30 MHz but introduced much stronger impedance at low frequencies with pick around 5 MHz.**
**       ****With SC ,  open termination case is more unstable.**

## Instability growth due to open termination impedance (6, 18 and 22 MHz)

**N=2*10^14, SC, **
**natural chromaticity**
**Strong instability at **
**high-frequencies**

## Beam halo due to open termination impedance

## Low frequencies and space-charge stabilization

**25 Om termination case has large impedance at very low frequencies (1-2 MHz). If we set chromaticity to zero then dp/p spread is not enough to damp unstable low-frequency harmonics. For these harmonics s[ace-charge has stabilizing effect due to the tune spread associated with b/a variation along the bunch.**
**Analytic estimates shows that without SC and zero chr. 2*10^14 beam is approximately factor of  2 above the threshold, with stable beam below 0.8*10^14.**
**Simulation:  (No space charge, zero chromaticity)**
**                            ****2.0*10^14 – unstable  **
**                            ****1.5*10^14 –unstable**
**                            ****0.8*10^14 - stable****                                                               **

## Stable case vs the case without stabilization

-  **red – zero  chromaticity, no space-charge – no stabilization at low**
**            ****frequency – complete blow-up of beam by the end of accumulation**
**            ****(2*10^14).**

## Instability below 5MHz(2*10^14)

**1.     No SC, chromaticity=0 –unstable at very low freq. 1-6MHz**
**2.     SC, chr0maticity=0, fringe fields, - stable at low freq. Because of space charge longitudinal detuning, unstable for frequencies above 5MHz. **

## b/a dependence and stabilization(b - pipe raius, a- beam radius)

**b/a variation along the bunch introduces effective  tune spread and thus plays stabilizing role.**
** ****Making ratio b/a larger decreases stabilization and makes beam more unstable.**
**If b/a is too small one gets into the region where image effects play the dominant role – “quadrupole” effect.**

## Instability for different b/a ratio

**b=11 cm**

**b=20cm**

**1MHz**

**2MHz**

**6MHz**

**9MHz**

**1MHz**

**6MHz**

**9MHz**

**11MHz**

## b=6cm (with beam radius “a” approaching 4cm by the end  of accumulation)

**1MHz**

**6MHz**

**9MHz**

**11MHz**

## Beam halo for different b/a ratio