JAERI-Conf 95-021

PB39
EFFECT OF FIELD VARIATION ON BEAM PARAMETERS IN RIKEN RFQ LINAC
Y.Batygin, A.Goto, O.Kamigaito and Y.Yano
The Institute of Physical and Chemical Research (RIKEN), Saitama 351-01, Japan

Abstract

2. Beam Transmission Efficiency

Beam dynamics study in an RFQ linac using 3D
particle-in-cell code BEAMPATH was performed toanalyze beam parameters due to variation of RFQ field.
The study shows that linear approximation of RFQ
parameters adopted in P ARMTEQ code along every cellrequires correct treatment of vane cutting in order toreproduce the original P ARMTEQ design. Our RFQlayout I) has non-adiabatically changing region frombuncher to accelerating section. For that we noticed that
inappropriat e treatment of P ARMTEQ design canprovide mismatching of the beam with the channel andparticle loss. Among the linearly varied functions ofaperture and electrode modulation along every cell weclioose the pair of that parameters which represent theinitial PARMTEQ design in the most appropriate way.
Final selection of RFQ structure is characterized bystable values of beam transmission efficiency (around90%) both for zero-current mode and for space charge
dominated regime. Effect of vane errors on beam
parameters was studied as well to define the engineering
tolerance for RFQ vane machining and alignment .

Initial design of the RFQ linac was made withstandard approach using P ARMTEQ code3). In
PARMTEQ code every cell is generated withinterpolation of input parameters: normalized quadrupole
gradient B, synchronous phase <ps. vane modulation mand intervane voltage U 0 • Those initial dependencies aretransformed in geometry parameters (cell length Li,
aperture ai and electrode modulation mi) by iterative
procedure to adjust cell length with local electric field
and required phase shift of synchronous particle per cell.
Output listing of PARMTEQ contains cell lengths L as
well as aperture a and electrode modulation m at the endof each cell. During the vane tip machinery themonotonous functions m(z), a(z) could be approximatedin different ways which result in deviation of parameters
A(z), X(z) from original design. Additional reason for
deviation from original design is that coefficients X andA should be considered as z-independent values at every
cell due to their definition as Fourier-Bessel coefficients:
X= _1_2_
Uo 1t

1. Introduction
In many experimental RFQ tests the observed
values of beam transmission efficiency in RFQ linac issmaller than it is expected from theoretical predictions.
Possible sources for it were found to be higher orderterms in RFQ potential and image charges2>. However,
our study of beam dynamics indicates that transmission
efficiency and other beam parameters can be affectedeven in the case of two lowest order RFQ potential terms
U (r,e z) = Uo [X(.!. )2 cos 2e +A lo(kr) sin (kz)] , (1)a2'due to different treatment of P ARMTEQ vane tip
design. It comes from the fact that in PARMTEQ
design the quadrupole gradient XUofa2 and the axial
voltage AU0 vary linearly over each cell and the final
listing of RFQ parameters contains the geometry design
values ( aperture a and electrode modulation m) at theend of each cell. Different interpretation of output list
parameters results in systematic deviation from original
design. We examined this effect as well as effect of vanetip manufacturing errors on beam transmission efficiency
for RFQ structure designed for upgrading the RIKEN
Linear Accelerator (RILAC)l).

( Z1t ( 2lt U(a,e,z) cos 2e deJo lof. f.
2'1t

A=

21tZUJo(ka)021t

d (kz)

U(a,e,z) sin kz de d (kz) . (2)0In order to verify the effect of different representation ofX(z), A(z), a(z) and m(z) we checked the following
approximations of original P ARMTEQ design:1) linear variation of acceleration efficiency A andfocusing efficiency X along each cell (PARMTEQ type field);2) a step-wise function of above parameters assumingthat the values of <Xi>= 0.5 (Xi-1 + Xi) and <Ai> =0.5 (Ai-l + Ai) equal to mean values of thatparameters over each cell;3) a step-wise function of A,X assuming that at every
cell aperture and electrode modulation are equal totheir average values <ap = 0.5 (ai + ai+ 1), <mi> =0.5(mi +mi+ 1);4) a step-wise function of A,X assuming thatparameters equal to the P ARMTEQ designed values
at the end of each cell Ai , Xi.
Calculations were done using general-purpose particlein-cell code BEAMPATH4). Beam was represented as acollection of 104 model particles. Trajectories were-359-JAERI-Conf 95-021

calculated using 20 steps per cell in the field which is acombination of RFQ potential (1) and self field of
the train of bunches. Space charge field was found ateach time step from the 3-dimensional Poisson's equationon the NX x NY x NZ = 64 x 64 x 256 mesh with theDirichlet boundary conditions for potential on the surfaceof square pipe and periodic conditions in longitudinaldirection.
Results of calculations for the beam with
mass/charge ratio A!Z = 5 are presented at Figs. 1-3 andin Table 1. For linear - varied field at every cell
(PARMTEQ - type field) the beam transmission
efficiency calculated from BEAMPATH code is close to
PARMTEQ predictions (see results 1,2 in Table 2).
Almost the same values of beam transmission efficiency
were observed with step-wise field representation ofRFQ field where parameters were supposed to be equal
to the average values <Xi>,<Ai> at each cell (result 3in Table 1). Smaller value of beam transmission
efficiency for I = 1mA can be explained by different
technique in space charge calculations: in BEAMPATH
the 3D space charge field is renewed every time step
while in P ARMTEQ 2D space charge field appear as akick once per cell.
The other approximations of original design(results 4,5 in Table 1) show decreasing of originally
expected beam transmission efficiencies, especially forthe case where a,m at each cell were supposed to beequal to the designed values at the end of the cell ( result
5 in Table 1). Systematic changes of parameters A, X at
every cell due to different representation of RFQ field
have the effect of permanent error in vane geometry
design which results in deviation in synchronous phaseof the bunch and mismatch of the beam with the channel.
This deviation is most serious in the region between
buncher and accelerating section. This region ischaracterized by a sharp increase of electrode modulation
which results in a quick change of acceleration gradientalong the structure. Most of the particle losses areobserved in this area. Beam losses in the longitudinal
direction are much smaller (typically 2% for zero beam
current and 4% for beam current I= 1 rnA).
Table 1. Beam transmission efficiency in RFQ
structure (A!Z = 5, f =40MHz).

Code

Field Variation
over each cell

1=0

1=1mA

3. Vane Manufacturing Errors
Random errors in vane tips manufacturing resultin amplitude growth of transverse and longitudinaloscillation. Analytical treatment5> shows monotonous
enlargement of transverse oscillation amplitude rmax
after passing through the RFQ section with N cells:where ~r 0 is an axis displacement and ~R0 is an error inaverage radius of the structure: Ro = a I YX. To study
this effect via computer simulation the following
parameters were randomly distributed at every cellwithin the max error ± 5: cell lengths Li. aperture radius
ai. max distance from axis to electrodes mai and axis
displacement &oi· Results of simulation (see Table 2 andFig. 3) demonstrate that the error of 50 microns does notcreate any serious degradation of the beam parameters
while error of 100 microns could cause notabledecreasing of beam transmission efficiency. The
engineering tolerance of 50 microns was thus adopted forvane tips fabrications.
Table 2 . Beam transmission efficiency due to errors invane fabrication.

~,microns

1.

0

2.

50
100

3.

1=0

I= 1mA

0.94
0.92
0.80

0.87
0.85
0.70

Disturbance of quadrupole RFQ symmetry result
in appearance of dipole field component. This effect wasstudied for the case when one of the RFQ electrodes isshifted from its ideal position at the distance A. It meansthat extra terms containing cose, cos29, cos39, ... appear
in the potential expansion. We restrict our consideration
by adding two lowest order terms in RFQ potential:U(r,9,z) = - Uo [X (Ii cos 29 + Alo(kr) sin (kz)

2a+ Aot !.cos 9+Au l1(kr) cose sin (kz)] . (4)l.PARMTEQ

Linear

0.94

a0.90

0.94

0.87

Coefficients Ao1. A 11 are defined by new boundary
conditions where potential is kept constant at the surfaceof electrodes U(r,e,z) =- U0 (2:3. BEAMPATH Step-wise (<Xi>,<Ai>) 0.94

0.86U(r,S,z) =- U0 /2

x =-a (z)4. BEAMPATH Step-wise (<aj>,<mi>) 0.92

0.78

U(r,9,z) =- U0 /2

x = a (z) +A

5. BEAMPATH Step-wise (Xi Ai)•

0.70

where a(z) is the vane-tip profile of the modulated
electrodes defined by equation:

2. BEAMPATH

Linear(PARMTEQ-type)0.86-360-,(5)

JAERI-Conf 95-021

X [ a(z) ]2 + A lo[k a(z)] sin (kz) = 1 .

4. Conclusions(6)aIn summary performed study showed the significant
sensitivity of beam dynamics on different representationof RFQ field even in the case of two lower terms in RFQ
potential. The vanes of our RFQ were designed in theway to adopt values X and A as average values fromoriginal linear function in P ARMTEQ design. Finalversion of RFQ structure is characterized by the initially
expected value of beam transmission efficiency. Dipole
component originating from shifting of the RFQ
electrode results in displacement of stability oscillation
point and does not influence beam dynamics when thevalue of electrode shift is much smaller than apertureradius of the channel.

From boundary conditions (5) the expression for RFQ
potential with dipole component is given by:V(r,9,z) = - Uo [X (.I) 2cos2e + A Io(kr) sin(kz)] +a2+ Uo [x + a(z)] fi [X + A_ (ka)2 sin kz)]
4
2 . a2. (7)Dipole component decreases the focusing strength of pure
RFQ field if electrode is shifted outside the channel fi > 0( and vice versa if fi < 0):(8)

Referencesl).O.Kamigaito, A.Goto, Y.Miyazawa, T.Chiba,
M.Hemmi, M.Kase and Y.Yano: Jpn. J. Appl. Phys. 33(1994) L537.
2) O.R.Sander et. al.: Proceedings of the 17 International
Linac Conference, Tsukuba,Japan (1994), 308. ·3) PARMTEQ code, Los Alamos National Laboratory.
4) Y.Batygin, Proceedings of the 3rd European Particle
Accelerator , Berlin, Germany (1992), 822.
5) I.M.Kapchinsky, Preprint IHEP 72-29, 72-30,
Protvino (1972), in Russian.

Dipole component results in shifting of stable oscillation
point from the axis . From equation (7) the new stability
point Ex (Xo) =0 is defined as x0 =ll/2. Beam dynamicscalculation performed for electrode displacement of fi =50 microns shows that center mass of the beam oscillates
around the new stability point x0 = 25 microns. In thiscase no significant changes of beam parameters wereobserved.

ba

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Fig. 2. Phase trajectories in RFQ
structure.

Fig. 1. Different approximation of PARMTEQ axial voltage design:
a- average step-wise approximation at each cell;b - step-wise approximation at the end of each cell.

.

60cba75z. cw

100

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100126160265075z. C>lFig 3. Particle trajectories in RFQ structure with various value of error in vane fabrication: a) 0 microns·'b) ±50 microns; c)± 100 microns.

-361-100

126160