The 11th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

FAST BETATRON TUNE CONTROLLE R FOR CIRCULATIN G BEAM IN ASYNCHROTR ON
Takuyuki ENDO, Kichiji HATANAKA, Kenji SATO, Takane SAITO, Shinji ANO, Hitoshi TAMURA,
Kouji NODA*, Mitsutaka KANAZAWA, Satoru YAMADA,
Hiromitsu INOUE**, Osamu TAKEDA** and Masataka MIZOBATA**
Research Center for Nuclear Physics, Osaka Univ.,
Osaka 567, Japan,*National Institute of Radiological Sciences,
Chiba 263, Japan,**Energy and Industrial Systems Center, Mitsubishi Electric Corporation,Kobe 652, Japan

2 An analytical expression of Mathieu'sequation

Abstract
When rf quadrupole(RFQ) electric field is applied
to the circulating beam in a synchrotron, an equation ofmotion is reduced to Mathieu's Equation. A new analytical method to obtain an approximate solution has beendeveloped, while a numerical computation was usuallyapplied. Translating the behavior of approximate solution into terms of an RFQ electric field and betatronoscillation, a fast tune controll can be achieved by rapid
tuning of both amplitude and frequency of rf voltage.
This process could be applied to suppress a tune shift
caused by a space charge effect and to controll a slow
beam extraction with a low ripple. We have started
another analytical computation using Hamiltonian with
perturbation ofRFQ and the results of this computation
also suggest that it is applicable to slow beam extraction. The fast tune controller has been constructed andthe beam test will be performed at HIMAC synchrotron
in cooperation of RCNP and NIRS.

2.1 The equation of motion for rf quadrupole
When RFQ electric field is applied to the beam in asynchrotron, an equation of motion is given by:qe.
d2:c- E:e(s,:c,t),
ds2 +Ko(s)x= mo-yv2(1)

where s(= vt) is a longitudinal distance, and Ko(s) is aperiodic function of s.
We can express the effect of RFQ electric field bysummation of step function u(s) as given by:

E,(s, :c, t) = Gox sin(wt + ~)Ixl

Loo [u(s+2+nL)-u( s- 2 +nL)],
n=-oo

1 Introduction(2)

An rf knockout slow extraction method [1) has beendeveloped at HIMAC synchrotron constructed at theNational Institute of Radiological Sciences(NIRS). Thisprocess utilizes a transverse rf electric field resonatedwith horizontal betatron tune, where horizontal and vertical kicker electrode is applied to the beam in a synchrotron. Thus, a similar slow extraction method canbe expected by adopting RF Quadrupole(RFQ) electricfield.
Although RFQ is usually used as a linear accelerator
in the low-energy range, we studied applying it to thecirculating beam in a synchrotron. When RFQ electrodes have no modulation, a field potential of RFQ
corresponds to that of quadrupole magnet whose current varies periodically at radio-frequency. Since a betatron tune value of beam depends on the strength ofquadrupole magnet in the ring, a betatron tune value
is modified due to RFQ electric field. If the stability ofmotion in a synchrotron is preserved, some variations ofbetatron frequency can be expected.
In this paper, we introduce an equation of the motion which is given by Mathieu's Equation, and describe
a new analytical method to obtain the approximate solution of the equation.

where Go is a field gradient; 1is an RFQ electrode length;and L is a circumference of the ring.
Assuming thatAcos(K<P+~k)= sin(wt + ~)L [u(s + 2I + nL)- u(s- 21 + nL)],00

n=-oo(3)the equation of motion is reduced to Mathieu's Equation
as given by:~
2qd2 ycos(2z)}y = 0,dz 2 + a{1-awhereX,2z = K<P +~A:,

y= $o

4

2a= K2"o•2q = __!l!__Gof3 2 Aomo-yv2a11•ds·<P-' -,o o Po;and vo is a betatron tune which depends on {30 •-89-(4)The 11th Symposium on Accelerator Science. and Technology, Harima Science Garden City, 1997

z.o1\:.
~~\~

1.2~E;::::I\•J
r-f-:V:~~v!".;

!:?~

~

r' = -~fa (- 2aq) rsin{(2- 2Q)z + 20}~r;.

~IN·~{II11'I...

O' = -( Va- Q)- ~Va (- 2aq) cos{(2- 2Q)z + 20}.

1/r~ 1/}1-·14.
I .,..
l'··f..J II 1'1-··~~ !,..I ~II II
rrr"~.

~t-- ..... f.:::::~"~~

neglect rapidly oscillating components which average tozero. TheMathieu's equation can be approximated byPrffl'..:.!!

..

~

I 1.11 )..1

1\t:E: t-:::B:1:::-t-> f--l'-:1%
1\\\ i:\:1:-' t-::1-' 1:;;8:1%~\N Kt--: t-r::-:; p;;;~

1,4

.

:

1.4

"''

~\I'- 1-' !<.rJ
1:1,~['.

II
I~\lALlL4.

(8)Transforming 0 into 1/J, where 1/J = (1- O)z + 0, weobtain approximate solution as given by:

1.&(l'rom 1.1. Z&roodc7, Aa •III-W7 nrt1ew of tlle M•&blwoBIU ~usUoa or lui nti:aW. bued oa DWDtrlc&l tolu-{

doaa. hWido a-tdl Labcan&otl' M•mo. Bopt.ITS,. Abordeen Pro'f'!.Dc Orouu.d, Md.. lNS, wtd1 peraalraoa.)•-•••-cOtWionl,· in 1tn.tl4bl.t r~gior&.f- - - - •-con.taft.l; in. .flAW. rcgioru-.-.- LifNII of condcnl Nluu o/ -q.

Fig. 1 Comparison between numerical solution andanalytical one. Bold lines correspond to lines
of constant values of v.

tan'I/J =-r 2 = A 2 ~ B 2 {A+ 2B cos(2VA2- B2z + k)},(9)where IAI > IBI(stable region),

2.2 Mathieu's Equation{Since the coefficients of Mathieu's equation
2qd2ydz 2 + a{1--;; cos(2z)}y = 0

tan 1/J =.

r2 =~Va (- ~q) .

~--B2+A2
B2 -A 2 coth(VB 2 - A 2 z + k)

2 (B 2 ~ A2) {A+ Bcosh(VB2- A2z + k)}.
(10)

2.4 Comparison between numerical solution and analytical one
Assuming that frequency of oscillation in Eq.(9) corresponds to that in Eq.(8), the expression which represents the realtion between parameter q, a and Q in stableregion is obtained as given by:ei(v-2)z }]{ ei(v+2)z+ ... '4(v + 1) - 4(v- 1)(6)where vis not an integer. If the characteristic exponentv ""' 1, the term with underline, which.has a frequencyof (2- v) "' 1, in the above expansions has main contribution. Utilizing this frequency, we develop a new
analytical study in the next section.

B ={F 11 (z) = e1"z P(z),where v depends on a and q, and is called the characteristic exponent; P(z) is a periodic function, of the same
period as that of the coefficients in Eq .( 5).
A numerical solution of Mathieu's equation has beenobtained (2] for each characteristic exponent as shownin Fig.1 (from HANDBOOK OF MATHEMATICALFUNCTIONS).
For small q, the following expansions may be used

A=fa-1

Similarly, in unstable region (IAI < IBI),(5)

are periodic functions of z, it follows from Floquet 's Theorem that there exists a solution of the form

Fv(z) =co [ e'"z - q+BI¥{--tan(v'A2- B2z + k)
A-B

2VA 2 - B 2 = 2 - 2Q2q=4=> - a(Q=2-v)

J(.ja- 1)2- (Q- 1)2;;: 'ya(11)The expression in. unstable region is also obtained asgiven by:

1rVB2- A2 =InS

2.3 A new analytical method for Mathieu's equation-jay',

we transTaking conjugate momentum p =form y and pinto rand 0 in Mathieu's equation as givenby:y = rcos(Qz- 0){(7)
p = -r sin(Qz- 0),

where Q is a resultant frequency of oscillation.
Assuming that Q"' l(Q = 2-v) as derived from previous section, we remain only slowly-varying terms and(12)where v is complex.
Fig.l shows comparison between numerical solution
and analytical one~ In stable region, analytical expression corresponds to a line of constant value of v, whereQ = 2- v. In unstable region, it corresponds to a linesof S = e•"" =constant.

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Table 1
An example of tune shift for vertical tune at HIMAC
synchrotron

Fig. 2 A photograph of the fast tune controller
3 Design and example of tune shift
With an analytical solution of r and (}, we can express a particle motion as given by:z = coVPoTJ = co$orcos ( Q .rat/>-(}+ ~Wt),(13)

where co is constant.
From Eq.(ll), a variation of q for a fixed a corresponds to a variation of Q, where parameter q is proportional to the amplitude of RFQ electric field; andparameter a is related with frequency of RFQ and thatof revolution as given by:

2volrJg =/rev( ..;a-m)
whereoriginal tune

vo = 3.13 .

resultant tune

II= 3.123

value of tune shift

1111 = 0.007

ratio between the frequency of RFQ
and that of revolution~=2.215

energy par nucleon

6[MeV/u]ratio between charge and mass

~ = 1/2{3 function

Po = 15.4[~]

bore radius

65[mm]

length of electrode0.6[m]

field gradient426[A:V/m2]

voltage

LSfA:V...,lRF Power

l.O[A:W)(m = 0,±1,±2, .. ·), (14)w
v/rjg = 27r,/rev = L'

Eq.(14) is derived from the relation between parameters in the eq.(3). In this case, the original tune v0 isshifted to the resultant tune Q'$.
Table 1 shows an example of tune shift for verticaltune, where the experiment is assumed to be performed
at HIMAC synchrotron(£= 130[m]).
The fast betatron tune controller has been constructed and the high power performance test was done
up to 1 kW rf power. The rf voltage o~er 1.4 kVpp was
obtained at 1 kW rfpower in the range from 0.3 MHz to2.0 MHz using the bridged-T type all-pass network. In
the next stage, the authors plan to install this controller
in HIMAC, and the beam test will be performed soon.
Fig.2 shows the photograph of the fast tune controller.
This controller has the same design as shown in Table 1.
4 Conclusion and discussion
An analytical solution of Mathieu's equation, which
describes a motion of circulating beam perturbed by
RFQ, was obtained. From this solution, a fast tune controll can be achieved by rapid tuning of both amplitudeand frequency of rf voltage. This process could be applied to suppress a tune shift caused by a space charge

effect, although the ordinary tune shift by quadrupole
magnet is not applicable because of its slow response.
On the other hand, our computation of transfer matrix which had an RFQ strength varying with eveey turnsuggests that the betatron tune value oscilates itself withevery turn and its mean value is shifted from the originalone. From this result, the motion of beam perturbed by
RFQ is recognized that it includes a tune shift with sometune spread. This behavior corresponds to the oscilationof r in the eq.(9), (10) and (13). Thus we can utilize it
to controll a slow beam extraction with a low ripple.
Furthermore, We have started another analyticalcomputation using Hamiltonian with perturbation ofRFQ. So far this results also suggest that RFQ is applicable to slow beam extraction.
·These are the problems which need more studies.
References(1] K. Noda et al, Nucl. /nstrum. Methods A 374, 269{1996).
(2] HANDBOOK OF MATHEMATICAL FUNCTIONS, ed. M.Abramowitz and I.A.Stegun(DoverPublications, Inc., New York, 1970).

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