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Accelerator Physics Oak Ridge National Laboratory
Jeff Holmes, Slava Danilov,and Sarah Cousineau
SNS/ORNL
October, 2004
ICFA Advanced Beam Dynamics WorkshopBensheim, Germany
Painting Self-Consistent Beam Painting Self-Consistent Beam Distributions in RingsDistributions in Rings
Accelerator Physics Oak Ridge National Laboratory2
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
Accelerator Physics Oak Ridge National Laboratory3
Self-Consistent Space Charge Distributions
1) To solve beam degradation problems related to space charge, we would like to create self-consistent distributions.
2) 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).
Accelerator Physics Oak Ridge National Laboratory4
How Many Analytic Solutions Have Been Found?
1) 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.
Accelerator Physics Oak Ridge National Laboratory5
Basic Math of Self-Consistent Distributions
equationVlasovtosolutiontheis
nHamiltoniaoffunctionanyHf
UyxkmppH scyx
)(
2/)(2/)( 2222
ii
iirr
iiisc YdXdf
YYXX
dYdXRRrUYXF
))()((
)(2),(
22230
F(X,Y) is the space charge force, f is the distribution function; 2D case is taken just for example.
The first example: linear 1D case – the beam density is constant.
,2/
)()(
2/2
2
bH
kx
xkxH
dHHfconstdpHf
where the distribution function f doesn’t depend on phase.The integral equation is called Abel’s Integral Equation.
x0
Accelerator Physics Oak Ridge National Laboratory6
Math for Self-Consistent Elliptical Distributions
222/)2/(,2
222222 )2/(22/)(
xxpPXxSubst
nxnxxnpxxTransf
x
PX
xpxppxpx
nxxn
xn
solutionsnoD
ondistributiKVHHC
HfCdpdpHfD
HH
CHfC
kxH
dHHfdpHfD
byx
b
H
x
x
b
3
);1959()()(,)(2
;)(,2/
)()(1
2/2
2
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
x
y
constdPPXfNewconstdppxfOld xxxx )(2,)( 2222
Accelerator Physics Oak Ridge National Laboratory7
Envelope Equations
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 :
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.
./4
),/(/)(
),/(/)(
320
3
3
rwhere
babbsKb
baaasKa
yy
xx
a
b
Accelerator Physics Oak Ridge National Laboratory8
New Solutions – 2D set
caseotherwiseRRXYYXCf b }2,2{)0(),()( 0000
The difference with previous cases- the distribution depends on other invariants, not only on Hamiltonian.
RbRotating disk – arrows show the velocities. In all xy, pxpy, pxx,pyy projections this figure givesa disk – different topology then one of the KV distribution x
y
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
.}0,2{,}1,2{);(
...)(.dim.},{
00 caseKVcaseYXHH
Cf
timesmHgfformtheinfSeekncasemn
b
The principle – delta-functions reduce the dimension in the density integral. Remaining eqn. for g(H) is same as in time-independent case
Accelerator Physics Oak Ridge National Laboratory9
New Solutions – 3D set
.}0,3{
;}1,3{)0(...),()(
;}2,3{)0(...),(...)(
;}3,3{)0(...),(...)(...)(
00
0000
000000
caseinsolutionno
caseotherwiseHHaXXHHCf
caseotherwiseHHbXYaXXHH
Cf
caseotherwiseHHcXZbXYaXXCf
bb
b
b
b
3 new 3D cases found. All have ellipsoidal shape in xyz projection. The density inside is always constant.
Accelerator Physics Oak Ridge National Laboratory10
SNS Injection – How the Distribution is Formed
Accelerator Physics Oak Ridge National Laboratory11
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.
1) Present injection – movingthe closed orbit from the foil, no injected particle angles
foil
Growing tails because of space charge-is a typical dilution effect
2) Modified injection – needsparticle angle on the foil, roundbetatron modes (e.g. equal betatron tunes).
This {2,2} distribution has outstanding property – it retains its self-consistent shape at any moment of injection
Accelerator Physics Oak Ridge National Laboratory12
No-Chromaticity Low-Dispersion Case cont.
Painting without angle Painting with angle
Recall sharply peaked longitudinal density profile...
4.371013 protons
Finally, self consistent distributionis one having transverse elliptical shape, and its size should correspond to envelope parameters along longitudinalcoordinate – the last condition can be met whenwe introduce injected beam beta function variation along the beam. It needs an additionalfast kicker or RF quadrupole.
tail
Tr. size:center
Accelerator Physics Oak Ridge National Laboratory13
Paint {2,0} Distribution in SNS:Nonlinear Effects Destroy Self-Consistency
Linear Fringe Fields
Sextupoles Fringe Fields + Sextupoles
Accelerator Physics Oak Ridge National Laboratory14
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.
x-x'
y-y'Tunes
Accelerator Physics Oak Ridge National Laboratory15
{2,0} Distribution: Effects ofSolenoids and Space Charge
Fringe Fields and Sextupoles Included Unless Noted
AddSolenoids
AddSolenoids
Add Space Charge Add Space Charge
Accelerator Physics Oak Ridge National Laboratory16
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 because the transformation x -> 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 uniformellipsoid in projection to3D real space
htoth thangreatertlysignificanp
pD
Chromaticity is equal to zero. Chromatic cases not investigated
Accelerator Physics Oak Ridge National Laboratory17
{3,2} Distribution With Linear TransportIncluding 3D Uniform Focusing (Shown After 100 Turns)
x-y Φ-y
Φ-dE
x-y'
y-x'
Accelerator Physics Oak Ridge National Laboratory18
{2,2} or {3,2} rotating disk case
ff XYYX /,/ ff XYYX
RYX
/,/
,222
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
Rb
x
y
skew quad0,/2
,222
YYX
RYX
f
before
x
x’
y
y’
after
x
x’
y
y’
Amazing detail: regularly looking beam (in x and y phase spaces)transformed into zero emittance beam. Reason – xy correlations.
Accelerator Physics Oak Ridge National Laboratory19
{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
Accelerator Physics Oak Ridge National Laboratory20
{3,2} Distribution: Create for SNS Lattice (Linear Transport) and Manipulate to Make Flat Beam
Uniform Focusing SNS Lattice
x-y x-y
x-y x-y
Add Space Charge Add Space Charge
Accelerator Physics Oak Ridge National Laboratory21
Summary
• 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.
Accelerator Physics Oak Ridge National Laboratory22
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.