Transition from isotropic to anisotropic beam profiles in a uniform linear focusing channel

Wilson Simeoni Jr

In space-charge dominated beams the nonlinear space-charge forces produce a filamentation pattern, which results in a 2-component beam consisting of a inner core and an outer halo. The space charge driven fourth order difference resonance may lead to emittance coupling. The coupling resonaces depending only on the relative emittances and average focusing strengths. In anisotropic beams, the emittance and/or external focusing force strength are different in the two transverse directions. Most of the halo studies so far have considered round beams with axisymmetric focusing. Some news aspects caused by anisotropy-with the ratio of focusing strengths and/or emittances as additional free parameters- demonstrating an influence of the mismatch on halo size. Simeoni recently, worked a nonlinear analyses of the transport of beams considering nonaxisymmetric perturbations. It is shown that large-amplitude breathing oscillating of an initially round beam couple nonlinearly to quadrupole-like oscillations, such that the excess energy initially constrained to the axisymmetric breathing oscillating is allowed to flow back and forth between breathing and quadrupole-like oscillations. This work deals with the coupled motion between the two tranverse coordinates of a particles beam arising from the space-charge forces, and, in particular, with the effects of the coupling in beam which have ratio / 1 where and are the envelope semi-axes-rms . A beam with non-uniform charge distribution always gives rise to coupled motion. However, it is only when the ratio / 1 and large beam size-rms mismatched on the order of 100% that the coupling can produce an observable effect in the beam as a whole. This effect arises from a beating in amplitude between the two coordinate directions for the single-particle motion and from the coupling between oscillations modes beam, -resulting in grow and transfer of emittance from one phase plane to another, in the beam develops an elliptical shape and therefore; in the transition from isotropic to anisotropic beam profiles.

The model equations

Instituto de Física – UFRGS – Porto Alegre

Transition from Isotropic to Anisotropic beamIntroduction

FIG.2: Evolution of the emittance obtained of the equation (8) and a are represented by red and green lines, respectively. Initial transversal emittance is equal .

FIG.1: Evolution of the envelopes obtained by direct integration of the envelope equation (6). and are represented by red and green lines, respectively.

Conclusions

The author would like to thanks CNPq and, PAC07 Local Organizing Committee (LOC), mainly, Tsuyoshi Tajima for the financial support.

The coupled motion between the two tranverse coordinates of a particles beam is arising from the sapce-charge forces. However, it is only when the ratio and large beam size-rms mismatched that the coupling produces an observable effect in the beam as a whole. This effect arises from a beating in amplitude between the two coordinate directions for the single-particle motion and from the core-core resonances, resulting in growth and transfer of emittance from one phase plane to another, in the beam developing an elliptical shape along directions preferential and therefore in the fast transition from isotropic to anisotropic beam profiles. The anisotropy leading to change of oscillatory energy between phase plane in the presence of nonlinear space-charge forces suggesting as a possible approach to the equipartitioning question, since collisions cannot be made responsible for energy transfer this model.

Acknowledgments

Apoio: CNPq

xr

yr

xr

yr

xr

yr

0

2nbq

r

y

r

x

rr

Nn

yxyx

b

b

2

2

2

2

12

2

3

4

22

0 322

rrr

rrx

rrr

y

rrr

xN

yxx

yx

yxyyxx

b

in

q

2

22

23

41

32

rrrryx

rrr

rry

yxyxyxy

xy

xyq

yxout 22log4

22

0

22222

20

xyqxy

rr yx22 yxyx

222

222 4 222 yx

xmqc

xbb

22

20

'' 02

3

2

20

''

rrr

rrj

j

yx

jj

K

2'2'2 rrrr jjjjj

21

444

2

2

2

22

1

12

3601

1

17207

1

12

4321

r

r

r

r

r

r

r

r

r

r

r

rhk

y

x

y

x

y

x

y

x

y

x

y

x

x

FIG.3: Evolution of the dynamical test particle and initial.The orbits x and y of test particle are represented by blue and pink lines, respectively. The beam envelopes, and are represented by red and green lines, respectively.

FIG.4: The ratio of oscillations energies and anisotropic ratios, and in beam propagation. is represented by red line, and and are overlapping represented by blue line.}

(1) (2)

(3)

(4)

(5) (6)

(7)(8)

We consider an axially long unbunched beam of ions of charge q and mass m propagating with average axial velocity along an uniform linear focusing channel, self-field interactions are elestrostatic. The beam is assumed to have an elliptical cross section centered at x=0=y and vanishing canonical angular momentum where x and y are the positions of the beam particles. We consider nonuiform density beam in space-charge dominated regime and emittance, initially.

where is the axial line density, and are ellipsis semi-axes rms.

where

with an analogous equation for orbit y(s). s is the axial coordinate of a beam , primes denote derivates with respect to s and is represented constant focusing force. With rms radii (henceforth, j ranges over both x and y).

As demonstrated by Sacherer and Lapostolle, enevelope equations for a continuous beam are not restricted to uniformly charged beams, but are equally valid for any charge distribution with elliptical symmetry, provided the beam boundary and emittance are defined by rms (root-mean-square) values.

cmNq

b

bk23

02

2

The can analytical been calculated following a model proposed to Lapostolle for nonlinear space-charge forces, and a sympletic mapping proposed to Simeoni for large mismatch beam. For a beam with large mismatch amplitudes, Simeoni has demonstrated that mismatch is like an impulse so fast that particle's positions not change, only momentum suffers a discontinuous variation. From eletrostatic potential and mismatch amplitudes on the order of 100%, the tranverse momentum impulse can be calculated. This results in a new phase-space distribution and new rms emittance. For example, in the x plane; the change in the momentum component is , where is electric field, h is the effective amplitude of oscillations modes beams, and is the beam velocity. The impulse can also be expressed as change in the divergence angle, given non-relativistically in the paraxial aproximation by t the final second moments of the particle distribution can be evaluted from the expression for x and x’ the final rms emittance can be obtained, knowing that divergence is

.

The result (8) is easily transformed to the y plane interchanging and .

There is a particular solution of the envelope equation (6) for which , where is ratio emittance. We transform the equations to a dimensionless form introducing the following dimensionless variables as parameters: for the independent variable, and for envelope beam, a and for test-particle, and for the scaled space-charge perveance. We introduce the following anisotropy parameters: the ratio emittance , ratio of the envelope beam and the mismatch factor . We launch the beam with and initially, and integrate firstly the envelope equation (6) up to s=50.0.Assuming which beam is usually equipartitioned in x and y ( ), where is the ratio of oscillations energies in the x and y directions, but has different l and large beam size-rms mismatched. For large beam size-rms mismatched and ratio envelopes beam small initially, the ratio of oscillations energies in the x and y directions remaines constant, .

êzb c

0,' xyyxP

1 yx

N b xr x 26 yr y26

0 r j

is the dimensionless perveance of the beam.

r x r y

j

vEp bxx hqxin

xxE vb

vmhEqx bbx2'

vmhEqrxx bbx x 2'

21

020

220

2 24)(

kkrsr bj

yx

s 0 rr xyx 0

~

rr yyy 0

~

xx y 0

~

yy y 0

~

0

~

ykK

rr yx rrrr boybox rr bj 00 ,4.2,1,3~

1

22 yxxy rr 1

1

r x r y

x y

28.0 yx

rx x5.0 ry y5.0

r x r y

1rr yx