# Presentation: 135463

## Halo formation in Linacs 

**A.V. Fedotov and R.L. Gluckstern**

**ORNL/SNS Workshop**
**June 25-26, 2001**

## Acknowledgments

**This work summarizes studies of beam halo done at DSAT group at University of Maryland (1997-99).**
**Halo studies were driven by the APT project at LANL.**
**The studies were done  together with                              R. Ryne and  S. Kurennoy (LANL). **
**Collaborators:      T. Wangler, J. Barnard, S. Lund,                 ****	****	****	****J. Qiang, N. Pichoff.**

## Definition of Beam Halo

**ICFA Workshop on Beam Halo (Wisconsin 1999):**
- In general, one refers to “**HALO**” as long as there are tails outside the beam core.
- Halo extent can be of any size – still “**HALO**”
- Definition of Halo as particles outside several sigmas is not good.
**Definition is NOT important**
**What really matters – is the source of halo**
**Parametric Halo (P.H.) :**    **parametric resonance between                ****	****	****	****	****           individual particles & collective modes**
**P.H.** is believed to be an important source of halo **in Linacs**
**P.H.** may exist **in circular machines.** **Its existence will be strongly machine dependent** – not necessarily the main source of halo

## Parametric Halo

**     **

**Q – depressed tune**
**m – mismatch parameter**
**k -  space-charge perveance**
**a – beam radius**
**p – frequency of beam oscillations**

**Key mechanism:**** ****parametric resonance**
- The dominant resonance (2:1) : **p=2Q**

**Uniform density:**

## P.H. – Transverse (phase space)

- Typical stroboscopic plot in the
**transverse phase space** for 125
- particles with low angular momenta
- for a **KV beam.**
- (tune depression: 0.8)
- (mismatch parameter: 0.7)

## P.H. – 2D & 3D

**2D:**
**     ****Analytic studies for a KV beam**** were found to give  excellent ****agreement with**** corresponding ****computer simulations**** which were then ****extended to other 2D distributions.**
**     ****The location of P.H. radius gave some confidence that a beam pipe wall could be placed far enough from the beam to avoid intercepting the halo particles.**
**3D:**
**     ****Attention then shifted ****to short**** 3D beam bunches of ellipsoidal shape with ****c/a = length/width**** ratio in the range ****2 - 4.**
**     ****Both transverse and longitudinal modes were capable of generating P.H. **

## P.H. - Longitudinal

- The signature of the **longitudinal halo** was the same as that of the
- transverse halo (a “peanut diagram” in the phase-space projection).
**The transverse and longitudinal mismatch and tune depression**
**parameter space was extensively explored with numerical simulations.**

## P.H. - Longitudinal (phase space)

- Phase-space diagram of a
**longitudinal halo.**
**(c/a=3, ********x****=.65, ********z****=.49, ********z****=1.4)**

## Longitudinal  halo extent for different mismatches (tune depression dependence)

## Longitudinal halo extent for different beam sizes (tune depression dependence)

## Dependence of halo intensity on mismatch

**(c/a=3, ********x****=.65, ********z****=.49)**   ******=1.1,  ********=1.2,  ********=1.3,  ********=1.4**

## Dependence of halo intensity on tune depression

**(c/a=3, ********=1.2)**   ******z****=0.87, ********z****=0.65, ********z****=0.49, ********z****=0.32**

## Dependence of halo onset on mismatch

**(c/a=3, ********x****=.65, ********z****=.49)**   ******=1.1,  ********=1.2,  ********=1.3,  ********=1.4**

## Dependence of halo onset on tune depression

**(c/a=3, ********=1.2)**  **a)** ******x****=0.79, ********z****=0.65 b) ********x****=0.65, ********z****=0.49 **
-                            **c)** ******x****=0.53, ********z****=0.39 d) ********x****=0.45, ********z****=0.32 **

## Low tune depression region – dependence onmismatch

**(c/a=3, ********x****=.93, ********z****=.87, t=900)**   ******=1.2,  ********=1.3,  ********=1.4,  ********=1.6**

## Time evolution

**(c/a=3, ********=1.6, ********z****=.87)** **t=250, t=350, t=450, t=550**

## Halo structure – low density cut

**a) no cut  b) with low density cut – ****allowed to study onset**
**of various order resonances and halo structure**

## Main features – halo extent

**Transverse  & Longitudinal**
**    ****An important quantity is the ratio of the halo radius to that of the matched distribution**
**Halo** **extent ****primarily governed** **by mismatch**
**Small dependence on tune depression does exist (different for KV and non-uniform distributions).**
**Longitudinal halo**
**In addition to features similar to transverse halo, there is significant** **increase of halo extent for severe tune depressions**

## Extent of transverse halo 

**Extent of the transverse halo for the 6D stationary distribution**

## Extent of longitudinal halo

**Extent of the longitudinal halo for the 6D stationary distribution**

## Extent of longitudinal halo – non stationary distributions 

**Extent of the longitudinal halo for the 6D uniform distribution**

## Halo extent

**Halo extent is minimized by matching control**
**Transverse halo extent is larger than that of longitudinal halo**
**For very large mismatch parameter:**
**Transverse (x****max****/x****rms****) can be as big as  5 – 8.**
**Longitudinal  (z****max****/z****rms****) can be as big as  4 – 6.**
**Non-uniform distributions – largest extent**
**Gaussian – largest extent**
**     ****No mismatch threshold on halo formation  for non-uniform distributions.**

## Halo intensity

**Transverse and Longitudinal**
**Strong dependence on mismatch**
**Very weak dependence on tune depression (except for a very severe tune depression, below 0.5).**

## Halo onset

**Strong dependence on tune depression**
**For elongated bunches Longitudinal Halo develops earlier than Transverse since longitudinal tune is more depressed than transverse.**
**Strong dependence on mismatch parameter**

## Coupling effect – stationary distribution

**Coupling effect** for the 6D stationary distribution with **zero transverse**
**mismatch** and 50% longitudinal mismatch (**c/a=2**, z=.45, x=.55)

## Coupling effect – uniform distribution

**Coupling effect** for the 6D uniform distribution with **zero transverse mismatch** and 50% longitudinal mismatch  (**c/a=2**, z=.45, x=.55)

## Coupling effect – Gaussian distribution

**Coupling effect** for the 6D Gaussian  distribution with **zero transverse mismatch** and  50% longitudinal mismatch  (**c/a=2**, z=.45, x=.55)

## Coupling – halo extent

**Mismatch in one direction leads to both longitudinal and transverse halo.**
**With relatively small mismatch in all directions one can get noticeable halo.**
**Coupling effect becomes important for almost spherical beam with c/a < 3.   **
**    ****Example: Beam (c/a=2) is perfectly matched transversely but has strong mismatch longitudinally  - quick transverse halo and significant beam loss if beam pipe acceptance is not big enough.****     **

## Other issues

**	****Loss from the RF bucket due to longitudinal halo –**** ****stabilization of P.H. by non-linear RF.**
**    ****Loss from RF bucket could not be a problem. However, careful longitudinal matching is necessary to minimize impact on transverse halo when coupling is important.**
**     ****Periodic focusing channel –**** ****apart from the instabilities due to the structure driven resonances, a****  ****close resemblance**** ****to the**** continuous**** ****focusing**** ****channel**** ****results.**
**    **

## Coulomb scattering – halo extent

**Some numerical studies suggested that small angle single Coulomb scattering may not be negligible in high-intensity linacs**
**Because of the importance of this question of high-intensity linacs under design, rigorous treatment of this effect was required.**
**Self consistent 3D distribution was used to estimate the extent and rate of halo formation analytically.**
**N. Pichoff – numerical simulations for non-stationary distributions****.**
**     ****Shell thickness/extent**** ****around the beam (populated due to scattering) depends on tune depression and could be significant even for matched equipartitioned beams. When the beam is nonequipartitioned or the beam with stationary distribution is rms mismatched, halo extent can be big and comparable to the one of the Parametric Halo.**

## Coulomb scattering - rate

**Rate of this process was estimates for various  stationary        ****f=(H****0****-H)****n**** and non-stationary (N. Pichoff) distributions****.**
**Simple expression in terms of beam parameters were obtained for various**** ****n****:**
**     ****For typical beam parameters (APT)**** ****the fraction of ions which leaves the beam per unit length:**
**Relatively singular distribution ****n=-1/2  ->  10****-11****/km**
**Distribution with                           ****n=0      ->  10****-14****/km**
**Distributions with                         ****n >0      ->  10****-15****/km**
**                                             ****Conclusion:                                                                     **
**     ****Effect of Coulomb collisions on halo development in high-current              ion linear accelerators is not important.**

## Practical Note

**Parametric Halo – is a resonant mechanism**** **
**Thus ****at least few**** betatron oscillation (depressed by space charge)   are required. As result, this mechanism is more applicable for transport systems with ****sufficient/large**** number of zero current betatron periods. ****Large**** - will depend on tune depression and mismatch parameter ****from 100’s to just a few****.**
**One has to check how phase advance is changing along the system length and whether this change is adiabatic enough to support the resonant mechanism****.**

## Parametric Halo in Linacs & Rings

**Difference of halo formation between linacs and rings**
**Linacs:**	 ** P.H. could be an important source of halo**
**Rings:**	1. **P.H. is possible** – should be studied for each 			     individual machine
- 			2. **Machine resonances**
**Modest mismatches:**
**P.H. (Linac):**  **strong tune depressions** - **very quick development**
**P.H. (Rings):**  **low tune depression** – **very slow development**
**In Rings**: 	**Rate of P.H. development is the most important   ****	****	****question.**

## Summary

**Development of parametric halo in ideal Linacs is well understood.**** ****Developed models in conjunction with multiparticle simulations give confidence in predictions of halo formation. **
**Predictions & applications to real accelerators are being tested in LEDA experiment.**
**If**** ****developed – P.H. could be  the largest source of halo formation in Linacs since it can lead to a large emittance increase.**
**An important question to understand is formation of Parametric Halo in the vicinity of space-charge coupling resonances (I. Hofmann et al.).**