# Presentation: 534422

##   Resonances and Equipartition in High Intensity Beams

*Oak Ridge, May 17, 2001*

- 1. “Thermodynamics” and Linacs
- 2. Collective Resonance Model
- 3. Application to high-current linac projects               (CERN, SNS...)
- Acknowledgment:
-  J. Qiang, R. Ryne,  Los Alamos,
-  O. Boine-Frankenheim, G. Franchetti, GSI
-  F. Gerigk, CERN

-   **Resonances and Equipartition in High Intensity Beams**

## Acceleration leads to anisotropy

- Beams from source plasma are isotropic and ~ Maxwell-Boltzmann
- Longitudinally colder by acceleration
- Distribution changes by scraping, capturing in RF buckets, change of focusing, mismatch  etc.
- Anisotropic beams may result - do they persist as such under space charge?

## Does the system return to equipartition?

-  1) Collisional relaxation

*Reiser’s book: “this equipartitioning process is particularly strong in high-current linacs ...”  -> design linac as equipartitioned!             *
*Kishek, O’Shea, Reiser, PRL ****85, ****2000:“ ... beams naturally want to equipartition”*

***Why and how?***

- _Linacs are too short!_

## Is “thermodynamics” applicable to beams?

-  “Thermodynamic limit” for systems with long-range forces questionable – need size>>range;
-  for Coulomb interaction *only* Debye screening can lead to a short-range force
- Debye /a ~  0 << 1 only at space charge limit, hence usual bunch does not fall into this category
-  real “one-component” plasmas with thermodynamics valid in Penning traps (O’Neill, Physics Today, February1999)

**Coulomb range**

## 2) Nonlinear resonances

- Fermi-Pasta-Ulam’ s classical “experiment” on MANIAC-I:
*chain of N-1 (periodic) mass points*
*to study equilibrium statistical mechanics* :

- E .Fermi, J.Pasta, S. Ulam and M. Tsingou, LASL-report, 1955

- recurrence

*Arnol’d:*
*“**The only computer experiments worth doing our those that yield a surprise”  *

-  why recurrence,
-  and “no” energy
-  in modes 6...32 ?
- _Chirikov_:
- resonance overlap
- needed for chaos

## 3) Collective resonance model of anisotropic beams

- I.H., IEEE Trans. Nucl. Sci. NS-28, 1981 (borrowed idea from Harris modes in infinite plasmas)
- R.A. Jameson confirmed in linac simulations (>1981)
- I.H., Phys. Rev. E **57** (1998),	 I.H., J. Qiang, R. Ryne,  PRL **86** (2001)

-  Resonances evolving from collective motion & space charge non-linearity
-  Analytical perturbation model in 2D (KV-distribution)
-  “Predictions” confirmed by 2D and 3D simulation 		               	closed gap to “real world” linacs
-  Recently also discussed in context of halo formation in rings (2D)

## What is an anisotropic equilibrium solution?  

- Isotropic equilibria can be expressed as function of the 2D or 3D Hamiltonian (constant focussing):  f = f0 (H0)
- With anisotropy we use as solution in 2D the “anisotropic KV-distribution” (the only one known!):
- f0  ~  (H0x +T H0y - mx2a2/2)
- with
- H0x = (px2 + m2 2x2x2)/(2 m)
- H0y = (py2 + m2 2y2y2)/(2 m)
- which implies uniform density within (x/a)2 + (y/a)2 <1 and
- the anisotropy parameter T = E x/ E y = x2a2/ y2b2 = xx / yy
- The unperturbed equilibrium distribution is described by 3 parameters, which can be chosen as x/ y , x/y, y/y0 (in 1D only /0; in fully 3D need 5!)

## Vlasov equation to describe evolution of collisionless beams in 2D

- The time-dependent Vlasov equation df/dt = 0 can be linearized and solved with Poisson’s equation for the perturbed charge densities  (space charge potentials as polynomials)

## Eigenmodes are multipole oscillations in density

-   for l=3,4 densities are non-uniform and space charge potentials correspond to sextupole or octupole  magnets
-   the theory predicts exponential growth of infinitesimal initial perturbations for certain parameters

## Dispersion Relation has been solved for l = 2, 3, 4 

- Example:
- l=3, even

- The coherent frequency /x is solved as function of the 3 parameters:
- y/x, =a/b, pp/x with p2=q2N/(0m3 ab)                  Im instability

## Charts for giveny/xindicate regions with instability

- 3rd

- 4th

- 2y-2x=0

- y-3x=0

- y-2x=0

*EP*

-  large areas *free from* instabilities,  depending on *tune ratio* y /x
-  for strong tune depression “sea of instability”  (x /0x <0.3)

- y/x =2

- Calculated in I.H., Phys. Rev. E **57** (1998)
- Further studies since – rich in phenomena

- 2:2

- 2:1

- 3:1

- 1:3

## Applied to Linac: fully 3D simulation

- T=4.1 ...2.8

- T=5.9 ...1.6

-  using  IMPACT code (R. Ryne, J. Qiang, LANL) in constant focusing
- (*I.H., J. Qiang, R. Ryne,  PRL ****86**** (2001))*
-  initial rms-matched waterbag distribution (realistic)
-  2-10 million simulation particles on 64x64x64 grid

- z/x =2

- I.H., J. Qiang, R. Ryne,  PRL **86** (2001)

- 2:2

- 2:1

## Initial KV-beam subject to 3rd order resonant instability

## 3D: unexpected departure from initial transverse equipartition

- x

- y

- z

- z

- x,y

##  focusing strength in z and x, y ~ equal  -> sc linac

- T=2.3 -> **1.3**

- We recently started (motivated by SNS) a refined and more systematic scan of parameter space

- z/x =2

-  2:2

## Studied 2:2 in detail using 2D and   y /y0x/y =2

-  x /y

-  x0 /y0

*KV: exponential growth takes time if it grows from noise! – helps in linacs!*

- WB

- y /y0

- KV

- KV-
- theory

- WB

- KV

- WB: overshoot beyond EP

- Montague: 1968 – split tunes in rings to avoid 2Qx-2Qy=0
- (IH et al., PRL acc.)

- Non-oscillating Re  =0

##  Simulation is strongly supporting the thesis: Stopbands of unstable non-oscillating (only!) KV modes characterize behavior of „real“ beams -> basis for our resonance charts 

- y /y0shows ~ _same profile_ for  relative emittance change

- x/y =2

- Non-oscillating

- rates:

- Likely reason: oscillating KV-instabilities are Landau-damped in real beams

- New charts including rates      (0.05~3 betatron periods)

## Slide 18

## Recent „scans“ in parameter space on ~40 Galaxie processors at BNLto check if „hidden“ further resonances (with G. Franchetti)

- Recent „scans“ in parameter space on ~40 Galaxie processors at BNLto check if „hidden“ further resonances (with G. Franchetti)

- 2:2

- 2:1

- 1:3

- Clear evidence that no resonances higher than 4th order!
- (except for KV-beam, where 5th order 3:2 and 2:3 were found )

- x/y =2

##  Ring space charge issues near 2:2 resonanceconstant focusing

-  Ring space charge issues near 2:2 resonanceconstant focusing

- halo

- x/y =1

- KV

- WB

- Actual exchange _cannot_ be derived from a single particle approach
- (ring: Montague/1968, linac: Lagniel/1999)

- y /y0

-  Discussed in paper by D. Jeon et al.,
-  Recently explained by A. Fedotov et al. in terms of these KV modes in SNS lattice

##   In 2:2 resonance different emittances cause resonant exchange

- x/y =1.2

- WB

-   In 2:2 resonance *different* emittances cause resonant exchange

- KV-theory non-oscillating modes:

- Returns to x/y ~1

- Relates to tune splitting needed to enforce or avoid exchange after injection

## Systematic comparison for SNS-parameters

- EP

- EP

- EP

- Nominal lo/tr emittance ratio: 1.2

## Some questions

-  resonances vanish if equipartitioned - near equipartition largest     	resonance-free regions
-  effect of periodic focusing (probably small)
-  need to check which stop-bands are really dangerous over linac 	length
-  linac design:
-  consider possible upper and lower bounds of emittance

## CERN SPL and the Neutrino Factory

## The SPL Design

- H

- -

- RFQ1 chop. RFQ2

- RFQ1 chop. RFQ2

- RFQ1 chop. RFQ2

- DTL

- CCDTL

- RFQ1 chop. RFQ2

- 

-  0.52

- 

-  0.7

- 

-  0.8

-        LEP-II

- dump

*Source  Low Energy section*

*   **DTL*

*Superconducting low-*

**

- 45 keV                       7 MeV             120 MeV                      1.08 GeV                     2.2 GeV

- 2 MeV

- 18MeV

-  237MeV 389MeV

- 13m

- 78m

- 334m

- 357m

- PS / Isolde

*Stretching and*

*collimation line*

- Accumulator Ring

*Superconducting *

**

*1*

- 785 m

##  Chart for the SPL (emittance ratio of 2)

**growth rate**

**emittance ratio=2**

**case 1**

**case 2**

**SPL IIb**

**case 3**

- F. Gerigk/CERN

## 3D IMPACT full linac simulation confirms 2:2 resonance 

******z**

******z**

******z**

******z**

******x,y**

******x,y**

******x,y**

******x,y**

## Mismatch 1.3 rx, 0.7 ry

******z**

******z**

******z**

******z**

******x,y**

******x,y**

******x,y**

******x,y**

## Tested on SNS (preliminary)

- Have raised nominal lo/tr emittance ratio from 1.2 to 2

- Found predicted „exchange“ of emittances:
- ratio returns to ~ 1.4, but ~ 30% increase of both transverse emittances in DTL + further effects (?)

## To explore mismatch we conduct “scans” on charts for different emittances first in 2D

- x/y =1 MM=1.3/1.3

**breathing mode**

- X-max/X-rms

- Need to explore also „resonant“ behavior of emittance and halo growth due to mismatch
- 2D-scans -> 3D -> Linac code

## Conclusions

- Collective resonance model & KV-based charts well confirmed by simulation in 2D _and _3D
- Only low-order resonances of concern (3rd and 4th) – free areas of non-equipartitioned beam in between!
- For sc linacs mainly 2:2 resonance (z0 / x0 ~ 1)  of concern – suppressed by EP, but also weak longitudinal focusing in DTL
- exchange not necessarily reversible!
- need to include mismatch driven emittance growth
- Require „realistic“ modeling of mismatch

## Linac space-charge resonancesmain candidate is avoided in SNS for nominal emittance ratio 1.2

- Emittance exchange

- Analytical Resonance Chart

- x

- z

-  IMPACT code (Qiang/Ryne, LANL) in constant focusing

- Linac space-charge resonancesmain candidate is avoided in SNS for nominal emittance ratio 1.2

-   kz/kx

-   kx/kox

- z/x =2

- z/x =1.2
- Nominal SNS

- EP

## Examples by R. Jameson

*longitudinal*

*transverse*

## 3D stability charts

- z /x

- Found very similar topology of stable and unstable regions
- confirm resonance pattern
- all unstable in”sea of instability
- checked some vs. semi-Gauss

## Choice of parameters

- .80 cm

- 2.2 ms

- 5.60 cm

- 13.3 ms

- .80 cm

- 14.2 ns

- 1.30 cm

- 22.7 ns

- {

- 5 bunches

****** ****linac macropulse**

**linac micropulse ******

****** ****11mA pulse current, 18 mA bunch current, 16.5% duty cycle**

- 10   high energy muons/year  4 MW beam power on target
- 2.2 GeV: above pion production threshold, few space charge problems in accumulator
- 2.2 ms pulse length  660 injection turns for accumulator (no instabilities)
- 75 Hz repetition rate ( 13.3ms)  below mechanical resonances of SC LEP cavities
- 5/8 micropulse structure  lossfree ring injection

## emittance for 1.3 rx,y, 0.7 rz

******x,y**

******x,y**

******x,y**

******x,y**

******z**

******z**

******z**

******z**

## tune ratios

**full current**

**full current**

**full current**

**full current**

**zero current**

**zero current**

**zero current**

**zero current**

## Discussion

- T = E z/ E x= z,nz / x,nx = 1:
-  equipartition - can this be satisfied easily?
- Reiser’s book:
- z0 ~ (00) -3/2                    x0 ~ (x0 / 0) ~ 0
- z0 / x0 ~ 0-1/2 0 -3/2           decreases with acceleration
- -> beam gets transversely “hotter”
- maintaining equipartition requires reducing x0 with energy
- as ~ 1/(003 ) 1/2
- -> bore radius should increase - undesirable
- -> z0 should increase (frequency doubling)?
- don’t enforce equipartition!

##  focusing in z and x, y equal -> sc linac

- T=2.3 -> **1.3**

- z/x

- z/x =2