INVESTIGATION AND OPTIMIZATION OF LOWINVESTIGATION AND OPTIMIZATION OF LOW--ENERGY HEAVYENERGY HEAVY--ION BEAM DYNAMICS IN RF ION BEAM DYNAMICS IN RF

PERIODIC AXISYMMETRICAL STRUCTURES WITH PERIODIC AXISYMMETRICAL STRUCTURES WITH ELECTROSTATIC FOCUSINGELECTROSTATIC FOCUSING

V.S. Dyubkov and E.S. Masunov

Department of Electrophysical Facilities, Moscow Engineering Physics Institute (State University), Kashirskoe shosse 31, Moscow 115409,

Russian Federation, +7(495)323-9292

http://www.mephi.ru

ABSTRACTABSTRACT

It is well known that nonsynchronous harmonics of RF field can focus particles. In order to focus ultralow-energy beam particles a spatially periodic electrostatic field can be used too. In this case a transverse focusing is created by a periodic array of electrostatic lenses. This article is dedicated to questions of two types focusing use of high-current low-energy beams in an injector-buncher of an ion linear accelerator, an application of a spatially periodic electrostatic focusing being examined for the first time. A three-dimensional nonlinear motion equation is derived in the Hamiltonian form in smooth approximation. It allows us to study correlation between longitudinal and transverse ion beam dynamics with low energies. Investigation of phase and transverse beam dynamics stability conditions in an initial part of linac are given consideration. Conditions of the beam dynamics stability were found and analyzed. An approach to a field optimization problem is described. Method of geometric parameter choice is worked out for realization of required field distributions in each structure. For results of beam dynamics stability analysis and optimized structure parameter choice to be verified numerical simulations of a low-energy ion beam with a value of charge-to-mass ratio is equal to about 0.1 were carried out for the structures involved.

INTRODUCTIONINTRODUCTIONThe problem of the effective low-energy linac design is of interest to many fields of science,

industry and medicine (e.g. nuclear physics, surface hardening, ion implantation, hadrontherapy). The most significant problem for low-energy high-current beams of charged particles is the question of the transverse stability because of the influence of Coulomb repelling forces. Low-energy heavy-ion beams transport is known to be realized by means of the periodic sequence of electrostatic lenses (electrostatic undulator) [3]. It offers to constructively join a periodic RF cavity and the electrostatic undulator (ESU) in the same device. In order to accelerate the low-energy ion beams one of the following RF focusing types can be used: alternating phase focusing (APF), radio frequency quadrupoles (RFQ), focusing by means of the nonsynchronous wave field as well as the undulator RF focusing. The main principles of APF were described in Refs. [4, 5]. The modification of APF was suggested in later studies [6]. A reached threshold beam current in RFQ [7] is about 100 – 150 mA and the further current rise leads to severe difficulties. A particle focusing with the use of the nonsynchronouswave in the two-wave approach was considered in Ref. [8]. The methods used to describe RF focusing in Ref. [8] have a number of shortcomings. For example, there is no correct relationship between longitudinal and transverse beam motion, the averaging method which was used is not quite valid. Detailed analysis of the focusing by means of nonsynchronouswaves of RF field shows that the focusing by the fast harmonic field in periodical ordinary Wideröe type and Alvarez type structures is not effective as the acceleration rate is very small. Increasing the low harmonic amplitude leads to appearance of the longitudinal beam instability which quickly disrupts the resonant conditions. The RF focusing by the nonsynchronousharmonics was studied in [2] particularly.

Alternatively, the acceleration and the focusing can be realized by means of the electromagnetic waves which are nonsynchronous with a beam (the so-called undulatorfocusing) [9]. Systems without the synchronous wave are effective only for light-ion beams. For the low-energy heavy-ion beams to be accelerated it is necessary to have the synchronous wave with the particles. In the slow wave frame, the bunch is acted on by a field which is similar to the electrostatic undulator one. The goal of this paper is the advantage investigation of the focusing by means of the periodic array of electrostatic lenses in comparison with the RF focusing one.

EQUATION OF BEAM MOTIONEQUATION OF BEAM MOTION

The analytical investigation of the beam dynamics in a polyharmonical field is a difficult problem. Rapid longitudinal and transverse oscillations as well as a strong dependence of field components on transverse coordinates does not allow us to use the linear approximation in the paraxial region for a field series. Nevertheless the analytical beam dynamics investigation can be carried out by means of the averaging method over the rapid oscillations period (the so-called smooth approximation) in the oscillating fields, which was suggested by P.L. Kapitsa [10] for the first time. Let us express RF field in an axisymmetric periodic resonant structure and ESU field as an expansion by the standingwave spatial harmonics assuming that the structure period is a slowly varying function of the longitudinal co-ordinate z:

( ) ( )

( ) ( )

( ) ( )

( ) ( ),sin

;cos

;cossin

;coscos

01

00

01

00

∑ ∫

∑ ∫

∑ ∫

∑ ∫

∞

=

∞

=

∞

=

∞

=

=

=

ω=

ω=

n

un

un

un

ur

n

un

un

un

uz

nnnnr

nnnnz

dzkrkIEE

dzkrkIEE

tdzkrkIEE

tdzkrkIEE

where , are the nth harmonic amplitudes of RF and ESU fields on the axis; is thepropagation wave number for the nth RF field spatial harmonic, is the factor of the nth ESU field harmonic; D, Du are the geometric periods of the resonant structure and ESU; θ, θu are the phase advances per period D, Du respectively; ωis the circular frequency; I0, I1 are modified Bessel functions of the first kind of orders 0 and 1.

(1)

We shall assume the beam velocity (the one-particle approximation) υ does not equal one of the spatial harmonic phase-velocities υn = ω/kn except the synchronous harmonic of RF field, the geometric period of RF structure being defined as

where s is the synchronous harmonic number, βs is the relative velocity of the synchronous particle, λ denotes RF wavelength. Thus, the solution of the motion equation (the particle path) in the rapidly oscillating field (1) we shall search as a sum of a slowly varying beam radius-vector component and a rapidly oscillating one. We assume that the amplitude of the rapid velocity oscillations is much smaller than the slowly varying velocity component for the smooth approximation to be employed.

( )πθ+λβ= 2sD s

Nondimentional motion equation

( ) ( ) ( )0~2

2ˆ,ˆ~

~,ˆ~=

τ∂∂

+τ=+τ Y

QeY

YYeYYdd

With the aid of the averaging over the rapid oscillations one can readily obtain the motion equation, in the bunch frame, in the form

( )QeQ ˆ,ˆˆ2

2

τ=τd

d ( ) YYQ ~ˆ +=τ( )ρξ,ˆ =Qλπ= RQ 2ˆ

( ){ } ( ) ( )( ){ }YYYeYeY ,~,ˆ21,ˆ ττ∇+τ=

ττMM&&

effUdd

−∇=τ2

2Q( )ηζ= ,Q

or

210 UUUUeff ++=

The effective potential function (EPF) describes the tridimensional low-energy beam interaction with the polyharmonical field of the system and determines the beam-wave system Hamiltonian

( ) ( )QPPQ effUH += 2

21,

WIDERWIDERÖÖEE TYPE STRUCTURETYPE STRUCTUREIf we put θu = θ = π, Du = D and take the point ζ = ρ = 0 as the reference point

of the EPF, then terms of the EPF can be presented as

( ) ( )[ ]

( ) ( ) ( )

( ) ( ) ][

( ) ( )[ ];2cos2sin222cos81

2cos2sin222cos161

;41

161

161

;sincossin21

,,2

2

2,

2

,,12

,2

,02

,

2,

02,

2,

02,

2

1

00

ssspsn

kkksn sn

pn

kkksn

ssspsn

sn

pn

sn

n sn

unsn

n sn

nsn

sn sn

n

ssss

fee

fee

U

fefefeU

IeU

spn

spn

ϕ−ϕζ+ϕ+ζην

+

+ϕ−ϕζ+ϕ+ζην

=

ηι

+ημ

+ην

=

ϕ−ϕζ−ϕ+ζη−=

∑

∑

∑∑∑

=−≠

=+≠

≠

where

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ),

;

;1

,1,1,0,0,,

2

,1,1,0,0,,

1

,2

1,20

,0

ηιηι+ηιηι=η

ηιηι−ηιηι=η

−ηι+ηι=η

spsnspsnpsn

spsnspsnpsn

snsnsn

IIIIf

IIIIf

IIf

snsn kk=ι , ( ) ssnsn kkk −=ν , ( ) ssnsn kkk +=μ ,

From these expressions we can see that the term U0 of EPF is responsible for both the beam acceleration and its transverse defocusing. The term U1influences only the transverse motion, always focusing the beam in the transverse direction. The term U2 has an influence not only on the longitudinal motion but on the transverse one. The extreme point of as well as is a saddle point. Therefore, the necessary condition for simultaneous transverse and longitudinal focusing is the existence of the total minimum of EPF. If ESU is absent (i.e. en

u = 0) the term U1 is reduced. In this case the transverse stability is achieved by using only nonsynchronous harmonics of RF field. For the amplitudes of the nonsynchronous harmonics to be increased we should include additional electrodes. Thereby, the utilization of ESU allows us to achieve the transverse stability without the complication channel geometry if the equal in absolute value d-c potentials are supplied on the same electrodes which are used to excite RF field, i.e. we can constructively join the periodic RF system and ESU in the same device.

At first we analyze EPF in the paraxial region. The EPF Ueff is expanded in Maclaurin series

,61

21

21

21 322222 L+δζ+εζη+ηω+ζω= ηζeffU

This expression of the EPF looks like a generalized Hénon-Heilespotential. In this case the equations of motion are integrable only for

.61,611)4

;,61)3

;1,1)2

equation); aldifferenti sMathieu' toreduced isequation (motion ,0)1

22

22

22

22

=ωω=δε

ωω∀=δε

=ωω=δε

ωω∀=ε

ζη

ζη

ζη

ζη

Here

( )

( ) ,

;

;2cos161

2cos321

83

323

323sin

41

;2cos212cos

41sin

21

2

2

2

2

22

2,

2

2

22

2,

22,2

,

22

,2,

22

2

2,

2

2,

2

s

s

kkksn

ss

pnpn

sn

pn

kkksn

ss

pnpn

sn

pn

n

un

nsn

sn

n

snsn

sn

nss

kkksn

ssn

pn

kkksn

ssn

pnss

spn

spn

spnspn

kkkkkee

kkkkkee

eeee

eeeee

ϕ∂ω∂=δ

ϕ∂ω∂=ε

ϕ++

ν+

+ϕ−+

ν+

++ιμ

+ιν

+ϕ−=ω

ϕν

−ϕν

−ϕ=ω

ζ

η

=−≠

=+≠

≠η

=−≠

=+≠

ζ

∑

∑

∑∑∑

∑∑

It is necessary that the parameters of the channel will be chosen in terms of the conditions of nonnegativity of the frequencies squares (for the simultaneous transverse and longitudinal focusing). Furthermore, if we take into account the third-order terms in the anharmonic potential it may lead to appearance of internal parametric resonances (radial phase oscillations). It can destroy the stable beam dynamics due to beam-wave system energy transfer between the two degrees of freedom. In this case the small frequencies of oscillations satisfy the relation ωη/ωζ = l/2, where l = 1, 2, 3 … . Another important restriction on the choice of the spatial harmonic amplitudes can be obtained from the condition of nonoverlappingfor different resonances of waves in the phase space . On the one hand, this restriction defines limits of the applicability of the averaging method. On the other hand, it is the necessary condition of the longitudinal (phase) stability.

Approach to a field optimization problemApproach to a field optimization problemIt is well-known that the “longitudinal” beam limit current value is

proportional to the channel acceptance. So

( ) ( ) ( ) ( )( ) ( ) ( ),

ˆ00ˆ

0,, ξξσξ

σ⋅β=ξβ l

s

seffseffs e

e

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( )

( ) ( )( ),cos21

cos21cos2

4

cos4

sin21 2

ξϕ+ξζ+

+ξϕ+ξζ−ξϕξζ−ξζξζ

+

+ξϕξζ

−ξϕ+ξζξζ−ξζ=ξσ

sl

srsrlr

sl

slrl

By using motion equation one can have

( ) ( ) ( ) ( ) ( ) ( ) 23

23

23

23

29

00ˆ082sinˆ

00ˆ0cosˆˆˆˆ

3,,

3

ll

l

l

σσβϕχ

−σ

σβϕ

−ξσ

σ−

ξ=

ξ seffs

ss

seffs

sssss

ee

ee

dde

dde

ded

for structure with RF focusing and the same equation but without the last term for ESU.

Solving these equations we have the optimal field distribution. It is important to correctly choose the initial acceptance, i.e. ( )0ˆse

On the one hand, it must be rather small to minimized the particle momentum spread. On the other hand, it should be enough to capture particles into acceleration process.

NNUMERICALUMERICAL SIMULATIONSIMULATION RESULTSRESULTS

Initial phase density distribution is similar to ‘water bag’ distribution

We used CIC method to simulate beam dynamics in the polyharmonic field.

4Weight factor χ

42.67Maximal value of acceleration harmonic at the axisE0, max, kV/cm

π/2, π/8Input φs(0)/Output φs(Lgr) value of equlibrium particle, rad

1.75Bunching length Lgr, m1.75Field increasing length Lf, m2.44System length L, m

33.76Frequency f, MHz260Output energy Wout, keV/u2.5Input energy Win, keV/u

0.12Charge to mass ratio Z/APb25+Particle kindValueParameter

System with RF focusing by the first harmonic

± 25.5Output phase spread Δφ, grad12.7Output relative energy spread δW, %85Current coefficient KI, %

8π, 66.8πInput/output transverse acceptance Aηmm·mrad

π, 5.7πRMS input/output transverse emittenceEη, mm·mrad

3.7π, 37.7πInput/output longitudinal acceptance Aφ, keV·ns/u

0.5π, 134.8πInput/output longitudinal emittense Eφ, keV·ns/u

0.5Initial beam radius rb, mm5Beam current I, μA

ValueParameter

Output beam parameters

Current coefficient and beam envelope

Output phase and trace spaces

9.5Weight factor χ

11.95Maximal value of acceleration harmonic at the axisE0, max, kV/cm

0.5π, 0.2π

Input φs(0)/Output φs(Lgr) value of equlibrium particle, rad

95Bunching length Lgr, cm95Field increasing length Lf, cm

250System length L, cm101.28Frequency f, MHz

103Output energy Wout, keV/u2.5Input energy Win, keV/u

0.12Charge to mass ratio Z/APb25+Particle kindValueParameter

System with the focusing by a periodic array of electrostatic lenses

± 33Output phase spread Δφ, grad17.6Output relative energy spread δW, %86Current coefficient KI, %

8π, 122.8πInput/output transverse acceptance Aη mm·mradπ, 17πRMS input/output transverse emittence Eη, mm·mrad

0.62π, 7.5πInput/output longitudinal acceptance Aφ, keV·ns/u0.15π, 15.8πInput/output longitudinal emittense Eφ, keV·ns/u

0.5Initial beam radius rb, mm5Beam current I, μA

ValueParameter

Output beam parameters in the structure with ESU

Current coefficient and beam envelope

Output phase and trace spaces

CHANNELCHANNEL GEOMETRYGEOMETRY CHOICECHOICEFor Wideröe type structures with a simple period the spatial harmonic

amplitudes are a rapidly decreasing function of its number which proportional to , where Emax is the maximal field strength at the aperture a in the electrode-gap, Yn = πng/D, g – the inter-electrode gap width. For the amplitude e1 to be increased up to 10e0 it is necessary to set three electrodes on the period D and either provide the adjacent electrodes (which have equal apertures) with different potentials or periodically vary the internal radius of the adjacent electrodes. Only the latter way can be practically realized in the RF cavity. In this case the additional electrodes with internal radius bmust be shifted to the position D/3 + ∆ and 2D/3 – ∆ respectively, where . By varying the length and corner radii of electrodes one can depress the higher harmonics and obtain only two waves in the structure. The ratio between internal radii b/a versus a/D for different values of e1/e0 is shown in the next figure.

( ) ( ) 110 sin −−

nnnmax YYakIE

10 32 eDe π=Δ

Values of b/a vs a/D for different ratio e1/e0

Note, if the beam injection energy is about 100 keV and λ = 1.5 m the lengths of electrodes are about 3 mm. Thus, the significant amplitude e1 may lead to RF discharge; also it is difficult to produce such small construction units. As it was mentioned above, the additional d-c voltage generator can be used for the transverse focusing in the structure with the simple period. However, in this case we face difficulties which are in the fact that the desired value e0

u must be held along the channel. The question of high-voltage input into RF cavity should be investigated too.

CONCLUSIONCONCLUSION

1. The comparison of the two undulator types was done.

2. The possibility of the electrostatic undulator application to focusing the low-energy high-intensity heavy ions was studied.

3. The computer simulation of the high-intensity ion beam dynamics in Wideröe type structure with RF undulator as well as with ESU was carried out.

4. It was shown that the electrostatic undulator can be a substitution for RF one.

Thus, one can assume that ESU is the unique adequate problem solving of the ultra low-energy high-intensity heavy-ion beams focusing.