Proceedings of the 20th Linear Accelerator Meeting in Japan ( September 6—8,1995tOsaka Japan)

Beam Dynamics Study in RIKEN RFQ LinacY.Batygin, A.Gotoy O.Kamigaito and Y.Yano
The Institute o f Physical and Chemical Research (RIKEN), Saitama 351-01，JapanABSTRACT
Beam dynamics study in an RFQ linac using 3D particle-in-cell code BEAM PATH was performed to analyze
beam parameters due to variation o f RFQ field. The study shows that linear approximation o f RFQ parameters adopted
in PARMTEQ code along every cell requires correct treatment o f vane cutting in order to reproduce the original
PARMTEQ design. Our RFQ layout has non-adiabatically changing region from buncher to accelerating section. For
that we noticed that inappropriate treatment o f PARMTEQ design can provide mismatching o f the beam with thechannel and particle loss. Among the linearly varied functions o f aperture and electrode modulation along eveiy cell wechoose the pair o f that parameters which represent the initial PARMTEQ design in the most appropriate way. Finalselection o f RFQ structure is characterized by stable values o f beam transmission efficiency (around 90%) both fo r
zero-current mode and fo r space charge dominated regime. Effect o f vane errors on beam parameters was studied aswell to define the engineering tolerance fo r RFQ vane machining and alignment .

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 charges^). However,
our study of beam dynamics indicates that transmissionefficiency and other beam parameters can be affectedeven in the case of two lowest order RFQ potential terms

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:
2kU(r,0,z) = Uo.[X(X)2 cos 20 + A I0(kr) sin (kz)], ( 1 )2adue to different treatment of PARMTEQ vane tip
design. It comes from the fact that in PARMTEQ
design the quadrupole gradient XU0/a2 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 vane
tip manufacturing errors on beam transmission
efficiency for RFQ structure designed for upgrading theRIKEN Linear Accelerator (RILAC)2).

2. Beam Transmission Efficiency
Initial design of the RFQ linac was made withstandard approach using PARMTEQ code3). In
PARMTEQ code every cell is generated withinterpolation of input parameters: normalized
quadrupole gradient B, synchronous phase (ps, vane
modulation m and intervane voltage U0. Those initialdependencies are transformed in geometry parameters(cell length Li, aperture ai and electrode modulation m[)
by iterative procedure to adjust cell length with local
electric field and required phase shift of synchronous
particle per cell.

X：r 2kU(a，0，z) cos 20 d0 d (kz)
U0 n2

A .
：

. 2k r 2k2

7C2U0Io(ka)U(a,0,z) sin kzd0cl(kz) • (2)In order to verify the effect of different representation ofX(z), A(z), a(z) and m(z) we checked the following
approximations of original PARMTEQ 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 <Xj>= 0.5 (X[.i + Xf) and <A[> 0.5 (Ai_! + 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 <rd{> = 0.5 (ai + ai+ i), <mp> =0.5(mi +mi+i);
4) a step-wise function of A,X assuming thatparameters equal to the PARMTEQ designed values
at the end of each cell A [ , X hCalculations were done using general-purpose paiticlein-cell code BEAMPATH4). Beam was represented as acollection of 10^ model particles. Trajectories werecalculated using 20 steps per cell in the field which is acombination of RFQ potential( 1 ) and self field of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:< 8 w > 2= 2N[<5r0>2 + 4 r mi< ^ -> 2]

Ro(3)

，

where 8r0 is an axis displacement and 8R0 is an error inaverage radius of the structure: R0 = a / VX. To study
this effect via computer simulation the following
parameters were randomly distributed at every cellwithin the max error 土8: cell lengths Li, aperture radiusai, max distance from axis to electrodes m ai，and axis
displacement 8r0i. Results of simulation (see Table 2and Fig. 3) demonstrate that the error of 50 microns does
not create any serious degradation of the beam
parameters while error of 100 microns could cause
notable decreasing of beam transmission efficiency. Theengineering tolerance of 50 microns was thus adopted
for vane tips fabrications.
Table 2 . Beam transmission efficiency due to errors invane fabrication.

8 ，microns

1=1 mAooo

7 58 8o

4 2
9 9070850001= 0oo

1103Table 1 . Beam transmission efficiency in RFQ
structure (A/Z = 5, f = 40MHz).

3. Vane Manufacturing Errors

1 2the train of bunches. Space charge field was found ateach time step from the 3-dimensional Poisson’sequation on the NX x NY x NZ = 64 x 64 x 256 meshwith the Dirichlet boundary conditions for potential on
the surface of square pipe and periodic conditions inlongitudinal direction.
Results of calculations for the beam with
mass/charge ratio A/Z = 5 are presented in figs 1.2 andin Table 1 . For linear - varied field at every cell
(PARMTEQ - type field) the beam transmission
efficiency calculated from BEAMPATH code is close toPARMTEQ 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 PARMTEQ 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 c e ll(
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 vanegeometry design which results in deviation insynchronous phase of the bunch and mismatch of thebeam with the channel. This deviation is most serious in
the region between buncher and accelerating section.
This region is characterized by a sharp increase ofelectrode modulation which results in a quick change ofacceleration gradient along the structure. Most of theparticle losses are observed in this area. Beam losses in
the longitudinal direction are much smaller (typically
2% for zero beam current and 4% for beam current 1 =1mA).

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 cos0，cos20，cos30，... appear
in the potential expansion. We restrict our consideration
by adding two lowest order terms in RFQ potential:

CodeRFQ Field

I =0I=lmA

1 .PARMTEQ

Linear

0.94

0.90+ A o i[c o s 0+Au Ii(kr) cos0 sin (kz)] • (4)
a0.94

0.87

3. BEAMPATH Step-wise (<Xi>，くAi>) 0.94

Coefficients Aqi,An are defined by new boundary
conditions where potential is kept constant at the surfaceof electrodes U(r,0,z) = - U0 12:0.86

4. BEAMPATH Step-wise (<ai>,<mi>) 0.92

0.78

5. BEAMPATH Step-wise (X i Ai)

0.70

2. BEAMPATH

Linear(PARMTEQ-type)0.86

U(r,0,z) = -[X (i)2 cos 20 + AI0(kr) sin (kz)U(r,0,z) = -U o /2

x = : a (z )U(r,0,z) = -U o/2

x= a (z ) 十 A(5)

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

x [^a]2 + A Io[k a(z)] sill ( kz ) = 1 .

(6)

From boundary conditions (5) the expression for RFQ
potential with dipole component is given by:V(r，0，z) = - ^ [ X (X)2cos26 + A I0(kr) sin(kz)] +2a. (7)+[と え 姻 [x + A_ (ka)2 sin kz)]
2a2
4
Dipole component decreases the focusing strength ofpure RFQ field if electrode is shifted outside the channel
A > 0 ( and vice versa if A < 0):E^ipole = • 仏
24 tX + —a24(ka)2 Sin ⑶ 】 .

（8)

4. Conclusions
In summary performed study showed thesignificant sensitivity of beam dynamics on difTerent
representation of RPQ field even in the case of twolower terms in RFQ potential. The vanes of. our RFQ
were designed in the way to adopt values X and A as
average values from original linear function inPARMTEQ design. Final version of RFQ struclure ischaracterized by the initially expected value of beamtransmission efficiency. Dipole component originatingfrom shifting of the RFQ electrode results indisplacement of stability oscillation point aiul does not
influence beam dynamics when tlie value oi' electrodeshift is much smaller than aperture radius oi' the channel.

References

Dipole component results in shifting o f stableoscillation point from the axis . From equation (7) thenew stability point Ex (x0) = 0 is defined as x0 = A/2. In
fig. 4 the beam axis variation for electrode displacementof A = 50 microns is presented. It is shown that the .
center mass of the beam oscillates around the newstability point x0 ニ 25 microns. In this case no
significant changes of beam parameters were observed.

1 ) . 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) PARMJEQ code, Los Alamos National Laboratory.
4) Y.Batygin, Proceedings of the 3rd European Panicle
Accelerator，Berlin, Germany (1992), 822.
5) I.M.Kapchinsky, Preprint IHEP 72-29 ，72-30,
Protvino (1972), in Russian.

Fig.l. Linear (PARMTEQ-type) and step-wise functionof axial voltage at every cell along the RFQ structure.

Fig.3. Particle trajectories in the RJhQ structure with liieerror in vane fabrication of 8 = 50 microns (see result 2in Table 2). Transmission efficiency is 92%.

Fig. 2. Particle trajectories in RFQ structure (see results
1,2,3 in Table 1).Transmission efficiency is 94%.

Fig. 4. Oscillation of center mass of the beam due tovanes asymmetry of A = 50 microns.