THE TRANSVERSE NONLINEAR TUNE SHIFT AS STABILIZING. FACTOR IN HALO CREATION IN SPACE CHARGE DOMINATED BEAM*. N.Vasyukhin, Yu.
Proceedings of 2005 Particle Accelerator Conference, Knoxville, Tennessee
THE TRANSVERSE NONLINEAR TUNE SHIFT AS STABILIZING FACTOR IN HALO CREATION IN SPACE CHARGE DOMINATED BEAM* N.Vasyukhin, Yu.Senichev and R.Tölle, Forschungszentrum Jülich, Germany. Abstract One of the most important problems for space charge dominated beams in the low energy part of superconducting linacs is halo creation. Many authors show one of the key effects in halo creatiation is parametric resonance due to the mismatched betafunction oscillation (between core and particle). To estimate parametric resonance conditions the nonlinear tune shift for binomial distributed beam is described theoretically in this article. Simultaneously the beam dynamics simulation 3D PIC code was developed. The transverse oscillation frequencies are compared with parametric resonance criteria. As a result the recommendation for space charge shift is concluded to minimize halo creation.
EQUATION OF MOTION There are two different concepts explaining the halo creation: mechanical and thermodynamical. In linear accelerators the acceleration time is much shorter than the relaxation time of the system. Therefore a mechanical concept is more appropriate than a thermodynamical one. The development of this concept was attempted by Yu. Senichev [1]. The point was to describe qualitatively nonlinear parametric resonances due to mismatched RMS size of bunch and forced envelope oscillations in alternate gradient focusing structure. Obviously, this estimation can be done only for the “frozen core” of bunch or for the case, when the redistribution in halo does not affect on the core, and the resonance conditions cause increasing the transverse oscillations of particles in halo only. We used the isolated single resonance approach. To solve the equation analytically the following approximations are taken: axial symmetry, binomial distribution of particles, smooth approximation and “frozen” space charge forces. In order to clarify the overall picture of halo formation mechanism shortened derivations of equations are carried out. Firstly one can write the general equation of radial motion for arbitrary particle considering axial symmetry of RF field: e⋅k ⋅E d r ecG ( z ) x+ =− ⋅ sin ϕ ⋅ r + dz mc β γ 4m c γ β 2
1
2
2
0
1
3
2
qs
+
qs
F mc β γ
(1)
0
2
2
2
0
0
0
2
2
0
2
2
2
0
2
___________________________________________
*Work supported by the European Community-Research Infrastructure Activity under the FP6 "Structuring the European Research Area" program (CARE, contract number RII3-CT-2003-506395)
1
4m c γ β
2πε m c β γ l 2
0
2
3
3
0
qs
m cβ γ
qs
2
qs
0
qs
one can derive from (1): C r (−1) m! r&& + Q( s)r = r 1 − 1 − = C ∑ r r (m − n)!n!r As soon as Q(s) is periodical function the solution should be periodical too. Following Courant-Snyder formalism one can represent the solution as: 2
m
n −1
m
sc
sc
2
2 n −1
2n
n =1
0
0
r ( s ) = Aβ 1 / 2 ( s )η ( s )
(3)
where A – initial amplitude of oscillation, for envelope A = ε , β – beta function, η – oscillating function. It is important to note that if the motion equation in dimensionless coordinates s then beta function should be also dimensionless and emittance dimension is m2. The envelope can be represented as r0 ( s ) = ε 1 / 2 β 1 / 2 ( s ) . Substituting this and (3) to the equation of motion and (−1) m! A one can derive: designating c = (m − n)!n!ε n −1
2 n−2 n
C m β&η& d 2 ( β 1 / 2 ) + + Q ( s ) β 1 / 2 − 1sc/ 2 β 1 / 2 ds 2 β ε =
1
str
2
1
sc
qs
where k = 2π , E1 – accelerating harmonic amplitude, β λ G(z) – function describing quadrupoles gradient, Fsc – space charge force, φ qs – particle phase, βqs – particle
c 0-7803-8859-3/05/$20.00 2005 IEEE
2
sc
β 1 / 2η&& +
sc
2
m −1
enm r 1 − πr l r where m – order of binomial distribution, r – envelope of the beam which is connected with RMS deviation by the r . From this it follows that space charge ratio σ = 2m + 2 force acting at arbitrary particle F = e( E − vB) can be found from Poisson equation: m e 2 n r 2 (2) Fsc (r ) = 1 − 1 − 2 2 2πε 0γ lr r0 where n – number of particles in the bunch, l – length of the bunch. Due to the fact that the equation of motion includes quadrupole focusing term it is convenient to pass to new longitudinal coordinates s = z / S normalized to focusing period S. Substituting (2) in (1) and designating e nS and Q ( z ) = e ⋅ S ⋅ k ⋅ E ⋅ sin ϕ − e ⋅ S ⋅ G ( z ) C =
ρ (r ) =
n
2
0
qs
velocity, βstr –accelerating structure phase velocity. In case of binomial distribution the particle density can be written as:
C sc
β
1/ 2
m
∑ c nη
η = 2 n −1
n=2
Here the linear part of the space charge force is transported to the left-hand side of equation. In order to solve this inhomogeneous differential equation firstly it is necessary to solve the homogeneous equation.
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Proceedings of 2005 Particle Accelerator Conference, Knoxville, Tennessee
derivation term the equation can be written in new
Homogeneous Motion Equation Solution The solution for η of homogeneous equation will be considered in the form a ⋅ e ϕ . Substituting this in homogeneous equation of motion and separating imaginary and real part one can obtain the equation system: dz i∫ β η = e (4) d 2 (β 1/ 2 ) C sc m 1 1/ 2 + − = β Q s ( ) ds 2 β 1 / 2ε β 3 / 2 i
The first equation reflects the Floquet phase dependency on Floquet amplitude and the second equation is the well known envelope equation. Now let’s solve the envelope equation in smooth approximation Aβ 1 / 2 ( s ) = R ( s )(1 + q ( s )) (5) where R(s) – slow function, q(s) – fast oscillating function. After averaging by focusing period one can obtain averaged envelope equation in smooth approximation: 2 4 && + µ 2 R − A − A Csc m = 0 R 0 3 R Rε
coordinate τ = ∫ ds : β 2 m d η + η = βC sc ∑ c nη 2 n −1 n=2 dτ 2 The complexity lies in that the derivatives in this equation are taken by τ but right-hand side is depending on s . Considering periodical solution r ( s ) = r ( s + 1) let’s represent τ in averaged form τ ( s ) = µ ⋅ s , where s
0
dζ – average oscillation frequency. In this case β (ζ ) the motion equation system can be rewritten: C m d 2 (β 1 / 2 ) 1 + Q( s ) β 1 / 2 − 1sc/ 2 = 3 / 2 β ε β ds 2 m η&& + µ 2η = µ 2 βC ∑ c nη 2 n −1 sc n=2 Now let’s substitute (8) in the second equation of this motion equation system: s +1
µ=∫ s
R c2 2 R c rm 2R 2 C m cos(µ~ s ) + 2c q sc2 ∑ c nη 2 n −1 + 2 2 A n=2 A A A
η&& + µ 2η = µ 2
(6)
The solution for this equation can be found using asymptotical methods of Bogolubov-Mitropol’skij [2]. Following this asymptotical method η can be represented as η = a cos(θ ) , where in a and θ are in first approximation defined as:
&& = 0 : The stationary solution will be at R A R = h + h +1 µ 2
c
0
where h = C m – analogue of Kapchinskiy’s parameter. 2εµ Now let’s consider behaviour of mismatched beam using perturbation theory R( s ) = R + ∆r ( s ) . Substituting this to (6) one can find that the solution for perturbation can be represented as: 2 (7) ∆ r = rm cos( µ 0 2 + s ) = rm cos( µ~ s ) 2 h + h2 +1 where r – amplitude of mismatching, µ~ – frequency of
[
da ds = − µM (R C ⋅ A dθ = µ − µ M ds a C ⋅ A
sc
0
sc
c
(
2
c
2
)
sc 2
+ 2 R r cos( µ~ s ) + 2 R q )⋅ 2
c m
m
∑c a n
n=2
[(R
2 c
cos
2 n −1
(θ ) sin(θ )
(9)
+ 2 R r cos( µ~ s ) + 2 R q )⋅ 2
c m
m
∑c a n=2
2 n −1
c
n
2 n −1
cos
c
2 n −1
(θ ) cos(θ )
m
perturbed envelope oscillation. In case of extremely high currents the frequency of envelope oscillations is approaches to µ 2 , while the frequency of particle oscillations in the centre of bunch is approaches to null. Substituting (7) and (6) into (5) one can write the approximate solution for beta function: A 2 β ≈ R c2 + 2 R c rm cos( µ~ s ) + 2 R c2 q (8) 0
Thus, the approximate solution of homogeneous motion equation considering linear space charge force is found.
Inhomogeneous Motion Equation Solution Unfortunately, the only clear way to solve the inhomogeneous motion equation is to abandon the Floquet functions. Considering that the perturbation is small it is possible to substitute second equation of (4) to inhomogeneous equation. In order to exclude first
1T ∫ f (θ )dθ – is the averaging operator. T 0 The first term in the right-hand side of the first equation is proportional to Rc2 and after averaging it is equal zero
Here M [ f (θ )] =
because of orthogonal property for cos 2 n −1 (θ ) and sin(θ ) . The second term in the right-hand side of the second equation contains constant constituent in expansion of cos 2 n (θ ) and this constant can be taken out from averaging operator. Thus, two oscillating components remain under averaging operator in both equations.
Resonances Caused by Beta Function Modulation Examining (9) one can note that in order to perform averaging [2] it is necessary to introduce a slow phase. As soon as oscillating components under averaging operator
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Proceedings of 2005 Particle Accelerator Conference, Knoxville, Tennessee
r
COSY-Injector accelerating channel [3] with 17mA average current and 10% mismatch (see Fig. 1).
halo
have different frequencies, the choice of considered resonance type should be done. First the resonance caused by mismatched beta function oscillation will be considered. Therefore q(s) function can be omitted expecting that frequencies of harmonics from q(s) function expansion are quite far from frequencies of nonlinear components binomial expansion and it does not contribute to the averaged value of right-hand side (9). The slow phase ψ can be introduced as: 2n ψ = (2n − 2k )θ − µ~s Now, considering isolated resonance condition µ~ ≈ (2n − 2k )θ& , passing to the coordinates motion-angle a = I and remaining the biggest item in binomial expansion k=0 one can write updated equation system. Supposing that it is canonical equation system the Hamiltonian can be written: µ~ 1 I − R C A ∑ c I G − H = µ − 2 n n
δImax I0
r
r
s
2
−2
2
sc
c
m
r
0
n
n
n =2
2n
r
µ
Rcrm C sc A −2 c nr I nr d nr 0 cos( 2nrψ ) nr This Hamiltonian describes the averaged transverse motion in vicinity of the nr nonlinear resonance for an arbitrary particle. Taking Tailor in a resonance point I = I 0 one can find for the Hamiltonian perturbation −
ψ Figure 1: Hamiltonian perturbation H on forth order resonance conditions (n=2, 4θ ≈ µs , I ≈ I ). Using the same approach one can obtain Hamiltonian and resonance conditions for resonances caused by beta function oscillation because of the focusing system. These resonances can be clearly expressed in phase space because they are caused by external forces (see Fig. 2).
H (δI ,ψ ) = H (δI ,ψ ) − H ( I ) : r
0
µR C 2
Hr = −
c
2A
0
m
Rc ∑ (k − 1)c k I 0k −2 G2 k (δI ) 2 −
sc
2
k =2
RcC sc rm c nr I 0nr d nr 0 cos( 2nrψ ) 2 nr A and for the separatrix size when particles will be captured in resonance: −
rc I d 4 n R ∑ (n − 1)c I G nr
δI
max
=
m
nr
0
nr 0
m
r
c
n=2
(10)
n−2
n
0
2n
Amplitude of transverse oscillation increases for captured particles, but with increasing of amplitude the nonlinear tune shift leads the particles away from the resonance. Thus, the mechanical approach describes the system in the non-linear space charge resonance with the “frozen core”. However, the particles involved in the resonance are redistributed, and they try to destroy the resonance, taking the more smooth distribution. Nevertheless, this approach describes the halo growth in initial stage transforming into the stationary distribution, and it is the good quantitative and qualitative evaluation of halo growth stabilized on the fixed level.
COMPUTATIONAL RESULTS Using modern calculation platforms it is possible to substitute practical values in order to observe Hamiltonian transformation. For instance, the above described Hamiltonian was calculated in MathCAD® for the
c 0-7803-8859-3/05/$20.00 2005 IEEE
Figure 2: Transverse phase portrait in case of 4th resonance for q-function with particle motion.
CONCLUSIONS Using the isolated resonance approach, the theory allowing to predict the halo stabilization due to non-linear features of the space charge resonance was developed. The analytical formula for the resonant deviation of particles in halo, explaining the numerical calculations, were obtained.
REFERENCES [1] Yu. Senichev, Mechanical and thermodynamical approach to halo creation problem in high intense beam, HEACC-98, Dubna, 1998, p. 123-125. [2] N. Bogolyubov and Yu. Mitropol`skij, “Asymptotic Methods in the Theory of non-linear oscillations”, Hindustan Publ., Delhi, 1961. [3] R.Tölle et al., “A superconducting injector linac for COSY” EPAC 02, Paris, 2002, p. 966.
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