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ICFA Advanced Beam Dynamics Workshop Bensheim, Germany. Painting Self-Consistent Beam Distributions in Rings. Jeff Holmes, Slava Danilov, and Sarah Cousineau SNS/ORNL October, 2004. Brief Problem Description. SNS example: beam distribution after MEBT 1) S-shape was formed;

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painting self consistent beam distributions in rings

ICFA Advanced Beam Dynamics Workshop

Bensheim, Germany

Painting Self-Consistent Beam Distributions in Rings

Jeff Holmes, Slava Danilov,

and Sarah Cousineau


October, 2004

brief problem description
Brief Problem Description
  • SNS example: beam distribution after MEBT
  • 1) S-shape was formed;
  • 2) Halo strings grew to 10 rms in y-direction.
  • Reasons: tails and core have different frequencies, tails not properly populated. This causes fast dilution of the phase space. Fast dilution and/or core oscillations cause resonances; resonant particles are subject to amp. growth and losses
self consistent space charge distributions
Self-Consistent Space Charge Distributions
  • To solve beam degradation problems related to space charge, we would like to create self-consistent distributions.
  • By self-consistent, we mean beam distributions that

a) are time-independent or periodic in time,

b) support linear space charge forces, and

c) maintain their shape and density under all linear transformations.

3) Ellipsoidal distributions of uniform density will satisfy self-consistency. A distribution which is self-consistent, or nearly so, will produce no loss and preserve rms emittance in the course of accumulation (acceleration).

how many analytic solutions have been found
How Many Analytic Solutions Have Been Found?
  • Time-independent up to 3D (Batygin, Gluckstern…).

Their use is limited, because of the fact that the conventional focusing uses alternating gradient.

2) Time-dependent with nonlinear force – none.

3) Time-dependent with linear force – up to 2D (Kapchinsky-Vladimirsky distribution).

4) New classes of self-consistent distributions have been described in Danilov, et al, PRST-AB 6 (2003) .

  • This presentation will highlight progress in ongoing studies with these newly described distributions.
basic math of self consistent distributions
Basic Math of Self-Consistent Distributions

F(X,Y) is the space charge force, f is the distribution function; 2D case is taken just forexample.



The first example: linear 1D case – the beam density is constant.

where the distribution function f doesn’t depend on phase.

The integral equation is called Abel’s Integral Equation.

math for self consistent elliptical distributions
Math for Self-Consistent Elliptical Distributions



Outstanding fact – any linear transformation of the phase space preserves the elliptical shape. Valid for all-D cases. 1D sample drift transform proof:

After linear drift transformation there exist substitution of variables such that the density integral in new variables is exactly same. It means constant density again

envelope equations
Envelope Equations

General Result – if distribution depends on quadratic form of coordinates and momenta and initially produces constant density in coordinate space, this density remains constant under all linear transformations.

Linear motion – quadratic invariant- solution to Vlasov equation for constant density as a function of this invariant- linear force-linear motion. We get closed loop of self-consistency. One final step – boundaries of the beam determine the force, force determines the particle dynamics, including dynamics of the boundary particles. Boundaries (or envelopes) must obey dynamic equations. In 2D case :



new solutions 2d set
New Solutions – 2D set


Rotating disk – arrows show

the velocities. In all xy, pxpy, pxx,

pyy projections this figure gives

a disk – different topology then

one of the KV distribution



The difference with previous cases- the distribution depends on other invariants, not only on Hamiltonian.

Any linear transformation preserves elliptical shape. The proof: 4D boundary elliptical line remains always elliptical, the projection of elliptical line onto any plane is an ellipse, the density remains constant under any linear transformation

The principle – delta-functions reduce the dimension in the density integral. Remaining eqn. for g(H) is same as in time-independent case

new solutions 3d set
New Solutions – 3D set

3 new 3D cases found. All have ellipsoidal shape in xyz projection. The density inside is always constant.

self consistency no chromaticity low dispersion case
Self-Consistency:No-Chromaticity Low-Dispersion Case
  • SNS case – chromaticity can be eliminated by sextupoles, dispersion exists only in the arcs (no dispersion in the drifts, etc.), but there is strong double harmonic - 2D case is enough (neglect longitudinal space charge force.

2) Modified injection – needs

particle angle on the foil, round

betatron modes (e.g. equal

betatron tunes).

  • Present injection – moving
  • the closed orbit from the foil,
  • no injected particle angles


This {2,2} distribution has

outstanding property – it retains

its self-consistent shape at

any moment of injection

Growing tails

because of space charge-

is a typical dilution effect

no chromaticity low dispersion case cont
No-Chromaticity Low-Dispersion Case cont.

Recall sharply peaked longitudinal density profile...

4.371013 protons

Tr. size:


Painting with angle

Painting without angle

Finally, self consistent distribution

is one having transverse elliptical

shape, and its size should

correspond to envelope

parameters along longitudinal

coordinate – the last condition can be met when

we introduce injected beam beta function

variation along the beam. It needs an additional

fast kicker or RF quadrupole.


paint 2 0 distribution in sns nonlinear effects destroy self consistency
Paint {2,0} Distribution in SNS:Nonlinear Effects Destroy Self-Consistency


Fringe Fields


Fringe Fields + Sextupoles

solution include solenoids to split tunes and paint to an eigenfunction
Solution: Include Solenoids to Split Tunesand Paint to an Eigenfunction

For the same SNS case as the previous slides, include two solenoids in straight section to split tunes and paint {2,0} distribution to one of the eigenfunctions.

Results show painted beam in phase space and tune separation and spread of eigenvectors.




2 0 distribution effects of solenoids and space charge
{2,0} Distribution: Effects ofSolenoids and Space Charge

Fringe Fields and Sextupoles Included Unless Noted



Add Space Charge

Add Space Charge



ring high dispersion case
Ring High-Dispersion Case
  • Dispersion effect is stabilizing
  • Not every energy distribution provides linearity of the transverse force
  • The condition for self-consistency – any transformation x -> x + D p/p should preserve the linearity of the force
  • Newly found 3D distributions satisfy the conditions becausethe transformationx -> x + D p/p is linear in 6D phase space.
  • The longitudinal force, which is proportional to the derivative of the linear density is also linear!!!

fully self-consistent

ring distribution is a uniform

ellipsoid in projection to

3D real space

Chromaticity is equal to zero. Chromatic cases not investigated

3 2 distribution with linear transport including 3d uniform focusing shown after 100 turns
{3,2} Distribution With Linear TransportIncluding 3D Uniform Focusing (Shown After 100 Turns)






2 2 or 3 2 rotating disk case
{2,2} or {3,2} rotating disk case

skew quad














The conclusion: the rotating disk self-consistent distribution can be transformed by skew quad into distribution with one (e.g. vertical) zero emittance. This is the ideal choice for collider with flat beams

Amazing detail: regularly looking beam (in x and y phase spaces)

transformed into zero emittance beam. Reason – xy correlations.

3 2 distribution transforming to flat beam with skew quad followed by transport
{3,2} Distribution: Transforming to Flat BeamWith Skew Quad Followed by Transport
  • Take the {3,2} distribution shown in the previously:
    • x-y‘ shown
    • y-y‘ shown
  • Pass through skew quad of appropriate strength:
    • y-y‘ shown
  • Transport through appropriate length:
    • y-y‘ shown
  • Top figure is without space charge. Bottom figure includes space charge, which opposes compression.

Add Space Charge

3 2 distribution create for sns lattice linear transport and manipulate to make flat beam
{3,2} Distribution: Create for SNS Lattice (Linear Transport) and Manipulate to Make Flat Beam

Uniform Focusing

SNS Lattice



Add Space Charge

Add Space Charge



  • This presentation builds on previous work (V. Danilov et al, PRST-AB 6, 094202 (2003)) to explore some of the newly found 2D {2,2} and 3D {3,2} distributions. Specifically, it is shown that:
    • In 2 dimensions:
      • Although we can paint a {2,2} distribution into SNS, nonlinear effects such as fringe fields or sextupoles destroy self-consistency.
      • This can be rectified by adding solenoids to split the tunes, and then painting to one of the resulting eigenfunctions.
      • With or without solenoids, the inclusion of space charge gives round beams.
    • In 3 dimensions:
      • We demonstrated the self-consistency of {3,2} distributions under linear transport over 100 turns both for uniform focusing and for the SNS lattice.
      • We examined the process to create flat beams from the {3,2} distribution with the use of a skew quad followed by transport.
      • We found that this method works with linear transport, but that space charge limits the compression of the beam.
next steps
Next Steps
  • There are several immediate steps to be taken:
    • We need to work on matching with space charge. Although the inclusion of space charge results in round beams, they are not matched and the densities are not precisely linear.
    • We need to study nonlinear effects on self-consistent 3D ({3,2} distribution) beams.
    • We need to determine injection schemes to create self-consistent 3D beams. The 3D distributions studied so far were created externally and injected at once.
    • We need to study space charge limits with self-consistent beams. One of the motivations for such beams is the smaller tune shifts provided by uniform space charge distributions.