the betatron tune has no dependence upon momentum but has a dependence upon its oscillation amplitude. Then the bare tune (the tune of vanishingly small.
Beam BeamDynamics Dynamics of of FFAG FFAG Accelerator Accelerator Beam Dynamics of FFAG Accelerator Masamitsu Aiba Masamitsu Aiba
Department of Physics, University of Tokyo Department of Department of Physics, University University of Tokyo Abstract. Beam dynamics of FFAG Accelerator is described, focusing on the non-linearity. The tracking simulations Abstract. Beam dynamics is motion described, focusing the of non-linearity. The Abstract. Beam of FFAG the Accelerator focusing onfield The tracking tracking simulations simulations were performed, in dynamics order to understand non-linear in the guidingon FFAG accelerator. were performed, performed, in in order order to to understand the non-linear motion in the guiding field of were of FFAG FFAG accelerator. accelerator. 6
10 6 10 10% 5 10 5 10
horiaontalacceptance acceptance horiaontal π mm-mrad.) ( ( π mm-mrad.)
INTRODUCTION INTRODUCTION INTRODUCTION A large acceptance of FFAG is ideal as a phase A large large acceptance of of FFAG FFAG is is ideal ideal as aa phase A asnot phase rotator and a acceptance secondary particle accelerator, only rotator and a secondary particle accelerator, not only rotator and a secondary particle accelerator, not only as a high current proton accelerator. On the other hand, as a high current proton accelerator. On the other hand, as a high current On the other hand, lattice magnets of proton FFAG accelerator. are full of non-linearity. It is lattice magnets magnets of of FFAG FFAG are are full full of of non-linearity. non-linearity. It lattice It is is not obvious why FFAG has large transverse not obvious obvious why why FFAG FFAG has has large large transverse transverse not acceptance and what limits the acceptance. We started acceptance and and what what limits limits the the acceptance. acceptance. We We started started acceptance systematic systematic study study toto to answer answer these these questions, questions, both both systematic study answer these questions, both analytically analyticallyand andby bytracking. tracking. analytically and by tracking.
8 8
Number of Turn = 128turn Number Turn Number ofRadius=9.9m Turn ==128turn 128turn N=16,of n_ N=16, N=16, Radius=9.9m Radius=9.9m N=32, Radius=20.5m o_ N=32, N=32, Radius=20.5m
4
10 4 10 3
10 33 10 io
I 1022 1.0 1.5 2.0 2.5 3.0 |100.5 0.5 1.0 1.5 2.0 2.5 3.0
0.5 phase 1.0 advance 1.5 2.0per 2.5 3.0 cell (rad.) (rad.) phase phase advance advance per per cell cell (rad.) FIGURE vs. Phase Phase Advance Advance FIGURE 1. 1. Horizontal Horizontal Acceptance Acceptance vs.
NON-LINEAR NON-LINEARMOTION MOTIONIN INFFAG FFAG NON-LINEAR MOTION IN FFAG InInFFAG FFAGaccelerators, accelerators,for forthe thesake sakeof ofachieving achievingthe the Inthe FFAG accelerators, for the sake of achieving the zero chromaticity condition, the higher zero the chromaticity condition, the higher order order zero the chromaticity condition, the higher order components componentsare areintroduced introducedinto intothe theguiding guidingfield. field.Then, Then, components are introduced into the guiding field. Then, thethebetatron tune has nonodependence upon momentum betatron tune has dependence upon momentum the betatron tune has no dependence upon momentum but buthas hasa aadependence dependenceupon uponits itsoscillation oscillation amplitude. amplitude. but has dependence upon its oscillation amplitude. Then the bare tune (the tune of vanishingly small Then the bare tune (the tune of vanishingly Then the bare tune (the tune of vanishingly small small amplitude) should be selected, taking into account amplitude) should should be be selected, selected, taking taking into into account accountthe the amplitude) the non-linear non-linearmotion motionatat ata aalarge largeamplitude. amplitude. non-linear motion large amplitude.
HORIZONTAL HORIZONTALMOTION MOTION HORIZONTAL MOTION Relation Relationbetween betweenthe theHorizontal Horizontal Relation between the Horizontal Acceptance and the Phase Acceptance and the Phase Advance Acceptance and the PhaseAdvance Advance The Thetracking trackingsimulations simulations(Runge-Kutta (Runge-Kuttaintegration) integration) The tracking simulations (Runge-Kutta integration) using two-dimensional field of “hard edge model”[2] [2] using two-dimensional field of “hard edge model” using two-dimensional field of "hard edge model" [2] were performed to calculate the horizontal acceptance. were performed to calculate the horizontal acceptance. were performed to calculate the horizontal acceptance. Sucha aasimulation simulationisis isone oneofof ofthe theeffective effectivemethods methods to to Such Such simulation one the effective methods to analyzethe thenon-linear non-linearmotion. motion. Figure Figure 11 shows shows the the analyze analyze the non-linear motion. Figure 1 shows the simulationresults resultsfor forthe thecases casesofofthe theperiodicity periodicityofof16 16 simulation simulation results for the cases of the periodicity of 16 and3232 with with triplet triplet (DFD) (DFD) lattice. lattice. InIn Fig.1, Fig.1, the the and and 32 with triplet (DFD) lattice. In Fig.l, the horizontal axiscorresponds corresponds to thephase phase advance per per horizontal horizontalaxis axis correspondstotothe the phase advance advance per cell. Comparing Comparing the the rings rings having having the the different different cell. cell. Comparing the rings having the different periodicity, it is generaland and convenient toto use use the periodicity, periodicity, ititisis general general and convenient convenient to use the the phase advance per cell. Two common tendencies in phase phaseadvance advanceper percell. cell.Two Two common common tendencies tendencies inin tworings ringscan canbebeseen. seen.One Oneisis that that the the acceptance acceptance two two rings can be seen. One is that the acceptance becomes smallerasasincreasing increasing thephase phase advance.The The becomes becomessmaller smaller as increasingthe the phaseadvance. advance. The otherisisthat thatthe theacceptance acceptance becomes becomes small small rapidly rapidly other other is that the acceptance becomes small rapidly around thestructure structure resonancecorresponding corresponding to the around aroundthe the structureresonance resonance corresponding toto the the phase advanceofof2π/3, 2π/3, 2π/4,2π/5. 2π/5. phase phaseadvance advance of 2n/3,2π/4, 2n/4, 2n/5.
FIGURE 1. Horizontal Acceptance vs. Phase Advance First tendency is explained by the large large kk value. value. First tendency is by the First tendency is explained explained by the large k value. The k value is main knob of the horizontal phase The is main knob horizontal phase The kk value value isphase main advance knob of of the the horizontal phase advance; the becomes larger as advance; the phase advance becomes larger as advance; the phase advance becomes larger as increasing the k value. Equation 1 shows the radius increasing the kk value. Equation 11 shows the radius increasing the value. Equation shows the radius dependence of FFAG guiding field and its its expansion. expansion. dependence of FFAG guiding field and dependence offield FFAG guiding fieldradius and its Here, BB00 isis the strength at the r00.. expansion. Here, the field strength at the radius r Here, B0 is the field strength at the radius r0. k k ( k − 1) rr k kk xx + B 00 k ( k − 21) xx 22 ++ ⋅⋅⋅⋅⋅⋅ BBzz == BB00 == B +B B00 ++ B B00 B=B\—\ = rr00 rr00 2!!rr00 2z r0 " 22!r 0 ((Taylor around r = + x).). (1) (1) Taylor Expansion Expansion (Taylor Expansion around r000,, rr == rr000 ++ xx). (1) As shown in Eq.1, a large k value enhances the As As shown shown in in Eq.1, Eq.l, a large k value enhances enhances the the non-linearity. It seems a good solution to take a small non-linearity. non-linearity. It It seems seems a good solution to take take aa small small phase advance so that the acceptance becomes becomes large; phase phase advance advance so so that that the acceptance becomes large; large; however, the phase advance, in other word the value, however, the phase advance, however, the phase advance, in other word the the kkk value, value, should keep the orbit should be kept certain should be be kept kept certain certain large large in in order order to to keep keep the the orbit orbit excursion in a feasible value. excursion in a feasible excursion in a feasible value.
N=16, k value=13 & 15 N=16, N=16,kvalue=13&
5.6 5.6 5.4 5.4 5.2 5.2
phase advance =2ππ/3 phase /3 phase advance =2c/3
O
5.0 5.0 u»
horizontal tune horizontalune tune ori
Second separation Second tendency Second tendency tendency isis explained explained by by the the separation separation between the bare tune resonance between the bare tune and the structure resonance line. between the bare tune and the structure resonance line. line. Figure 2 shows the the cases that Figure 2 shows the tune shifts for cases that the Figure 2 shows the tune shifts for the cases that the the phase advance is around phase advance is around 2π/3. phase advance is around 2;c/3.
000
10 20 20 30 30 40 50 60 10 40 10 initial 20 ∆30 40 50 50 60 60 r (mm) (mm) initial ∆ r initial Ar (mm)
FIGURE 2. 2. Horizontal Tune Tune Shift Shift FIGURE FIGURE 2. Horizontal Horizontal TuneAdvance Shift of 2π/3 around Phase Phase around Advance π/3 around Phase Advance of of 22n/3
CP642, High Intensity and High Brightness Hadron Beams: 20th ICFA Advanced Beam Dynamics Workshop on High Intensity and High Brightness Hadron Beams, edited by W. Chou, Y. Mori, D. Neuffer, and J.-F. Ostiguy © 2002 American Institute of Physics 0-7354-0097-0/02/$ 19.00 204
The direction of tune shift faces to the structure The direction of tune shift faces toof the structure resonance line. Therefore, “the valley acceptance” resonance line. Therefore, "the valley of acceptance" appears and naturally the acceptance becomes zero on appears and resonance. naturally the acceptance becomes zero on the structure the structure resonance.
the periodicity. Therefore, using Fig.3, it is possible to the periodicity. Therefore,acceptance using Fig.3, possible to estimate the horizontal of ita is ring with any estimate the horizontal acceptance of a ring with any parameters. parameters.
VERTICAL MOTION VERTICAL MOTION Soft Edge Model of FFAG Focusing Field Soft Edge Model of FFAG Focusing Field
Normalization of Horizontal Acceptance Normalization of Horizontal Acceptance
In order to obtain more general result on the In order to obtain more general result on the relation, a normalization of horizontal acceptance is relation, a asnormalization of horizontal acceptance is introduced follows. introduced as follows. With an approximation of the large k value, the low an of approximation of the large kinto value, the low orderWith terms Eq.1 can be transformed Eq.2. order terms of Eq.l can be transformed into Eq.2. 2 3 1 k 1 k k (2) x + ⋅⋅⋅ . x + x + Bz ≅ B0 1 + (2) __,,+! 1,,+ 2 ! 3 ! r r r.ji+^i 0 0 0
In order to study the vertical motion including its In order to study“soft the vertical its non-linear term, edge motion model”including of threenon-linear edgein model" of threedimensionalterm, field is "soft introduced the followings. dimensional field is introduced in the followings. First, three components of the cylindrical First, three components of polynomial the cylindrical coordinate are expressed by the of the coordinate are expressed by the polynomial of the vertical coordinate as shown in Eq.6. vertical coordinate as shown in Eq.6.
Bi = Bi 0 (r ,θ ) + Bi1 (r ,θ ) z + Bi 2 (r ,θ )2z 2 + ⋅ ⋅ ⋅ (6) =Bi0(r,0) + B + Bi2(r,0)z (6) i = r ,θ , z i = r,09z
In Eq.2, the quantity kx/r0 can be treated as In Eq.2,coordinate the quantity kx/rdistance. as 0 can be normalized of the Ontreated the other normalized coordinate of the distance. On the other hand, the emittance is expressed in Eq.3, hand, the emittance is expressed in Eq.3, (3) W = x 22 / β ,
W=x /]3,
Inserting these equations into Maxwell’s Equation, these into Maxwell's Equation, the Inserting coefficients of equations the polynomial can be solved, using the coefficients of the polynomial can be solved, usingby the boundary condition on the median plane shown the boundary condition on the median plane shown by Eq.7 and Eq.8, Eq.7 and Eq.8, (7) Br 0 ( r , θ ) = Bθ 0 ( r , θ ) = 0 ,
(3)
where β is the beta-function. Using approximations of where p is the beta-function. Using approximations of the beta-function and the betatron tune, the beta-function and the betatron tune,
(7)
k k ×N = , 2 — N N
β ≅ r0 / ν h , ν h ∝
(4)
k
(4)
r B z 0 ( r , θ ) = B0 E (θ ) , = B0\ —r0 \ E(ff), t0(r9ff)
(8) (8)
where νh is the horizontal tune and N is the periodicity, where v is the horizontal tune and N is the periodicity, Eq.3 canh be transformed into Eq.5, which includes Eq.3 can be transformed into Eq.5, which includes normalized normalized coordinate. coordinate.
W ∝x
r k k = 0 x r0 N kN r0
where B0 is the field strength at the radius r and E(θ), where B0 is the field strength at the radius r and E(0), which changes changes from from 11 to to 0,0,represents representsthe thedistribution distribution which of the fringing field. of the fringing field. Finally, equations equations from from 99 toto 11 11 are are obtained obtainedasas"soft “soft Finally, edge model”, edge model",
2
2
.
(5)
(5)
Nor. acceptance * kN/r0
Then, and that that is is the the Then, the the remaining remaining factor factor is is only only kx/r kxlr00 and normalization factor of the acceptance. As the result of normalization factor of the acceptance. As the result of the thenormalization, normalization, Fig.3 Fig.3 is is obtained. obtained. 12000 12000 10000 r10000
−
N=16 --n—N=16 N=32 -o—N=32
8000
z
(k − 2) B 0 r 1 ( Ek 2 + E ( 2 ) ) 3 r0 3! r0
Bθ (r , θ , z ) = E
B0 r r0 r0
1.5 1.5
2.0 2.0
2.5 2.5
−
3.0 3.0
phase phase advance advance per cell (rad.)
"
"
205
z3 + ⋅⋅⋅
%
&
. k −2 '
−
,(10) ,(10)
k −3
"#$
%
(
Three the Three parameters parameters in in the the normalization factor factor are the main phase main parameters parameters of of the the ring. ring. And the horizontal phase advance advance isis almost almost determined determined only only by the k value and
z 3 + ⋅⋅⋅
k '
r = EB ( r ,θ , z ) = B2z(r,0,z) EB0 0\B r0
z
B r 1 (1) 2 ( E k + E ( 3) ) 03 3! r0 r0 ()*
FIGURE FIGURE 3. 3. Normalized Normalized Horizontal Horizontal Acceptance vs. Phase Phase Advance Advance
k −1
"#$
!
1.0 1.0
k-3
(1)
, (9) ,(9)
k −3
!
§ 6000 6000 4000 £ 4000
2000 2000 0o I 0.5 0.5
k −1
kB r B r ( r ,θ , z ) = E 0 r0 r0
B r 1 ( Ek 2 + E ( 2 ) ) 02 2! r0 r0 ()*
(
%
%
&
z2 ⋅⋅⋅
(11) (11)
2
y
k
!" r k k k ( k − 1) 2 (13) (13) EB £B00 -I =,E !" B0 + B?0 k xv+-1- BR0 k ( k −21) xv2 ++⋅ ⋅ ⋅ ,,« (13) r = !r02 x +⋅ ⋅ ⋅ EB0 r0 = E B0 + B0 r0 x + B0 22.r r0 r0 2!r0 where the notation is changed; the coordinate coordinate yy where the the notation is changed; y represents coordinate z. Then the the coordinate natural result represents the coordinate z. Then the natural result appears; the guiding field of FFAG accelerator can be appears; the field of of FFAG accelerator can be expressed by guiding the summation multipole components. components. expressed by the summation of multipole components. !"
!"
2
vertical tune vertical tune
I ^ o 2 2 3 3
4 horizontal horizontaltune tune 4 horizontal tune (a) (a) Two Two Simulation Simulation Conditions Conditions in in the the Tune Tune Diagram Diagram (a) Two Simulation Conditions in the Tune Diagram
(b) (b) Bounded Bounded motions motions with with avoiding avoidingthe the non-linear non-linear coupling coupling (b) Bounded motions with avoiding the non-linear coupling (A) (A) (A) 8 8 1 6 6 6• 4 554 45 2 ". • ^2 2 : •"""••" 0 0 0 0 0 -2 -2 • B •" • " ~N -4 .4: -5 -4 -5-5-6 m -6 -6-10 _in , , , , , , - , , , , ,-8,-88 -10 4996 49985000 50005002 5002 5004 5006 -15-15 0 5 5510 10 499649985000500250045006 -15 -10-5 -5 -5 10 15 4996 4998 5004 5006 -10-10 0 15 15
0 r(mm) r(mm) z fei m ) r(mm)
r'(mrad) r'(mrad)
10 10-, 10
vererticaltune tune verertical
44 33
'
-J 2
" "--V... /\
FIGURE Non-linear Coupling with “soft edge model” FIGURE5.5. 5.Non-linear Non-linear Coupling with "soft edge model" FIGURE Coupling with “soft edge model”
11 0a 0 00
. " :"jft"I-"
(c) Unbounded motions due the non-linear coupling (c)Unbounded Unboundedmotions motions due to the non-linear coupling (B) (c) due toto the non-linear coupling (B)(B)
periodicity periodicity=8 periodicity ==88
222 •S o
10 10-, 10 555 000 -5 -5
8 8 666 44 4 222 00• 0 -2 -2 -4 -4 -6 -6-6 -8 -10 -10 4996 4996 4998 4998 5000 50005002 50025004 50045006 5006-8 -15 -15 -10 -10 -5 -5 00 55 10 10 15 15 5004 5006 4996 4998 5000 5002 -15 -10 -5 z(mm) 0z(mm) 5 10 15 r f cm ) z(mm)
Vertical Tune Shift Shift Vertical Tune Shift The vertical tune is mainly occurred by the tune shift shift is is mainly mainly occurred occurred by bythe the The vertical vertical tune shift component of normal octupole. The effect of higher normal octupole. The effect of higher component of normal octupole. The effect of higher order components more than octupole is weak, unless octupole is is weak, weak,unless unless order components more than octupole the vertical tune is very close to the value affected by the vertical tune is very close to the value affected by these components. Then, the direction of tune shift tune shift is these components. Then, the direction of tune shift isis one-side. The tracking simulation with “soft edge simulation with "soft edge one-side. The tracking simulation with “soft edge model” for various bare bare tunes. tunes. The The model" for various model” was was carried carried out out for results shown in Fig.4 indicate that the direction of the Fig.4 indicate indicate that that the thedirection ofofthe the results shown shown in Fig.4 vertical tune shift is upward. shift upward. vertical tune shift is upward.
A B A B
z'(mrad) z'(mrad)
°Q
3 3
r'(mrad) r'(mrad)
Furthermore, first term of the the radial and and vertical first term of thein and vertical fieldFurthermore, can be expanded as Eq.12 and Eq.13 expanded as shown shown inradial Eq.12 and Eq.13 field can be expanded as shown in Eq.12 and Eq.13 respectively. respectively. k£ k£(£-1) (k − 1) j y + B0 k (k − 1) xy B0 k −1 k r rro0 xy kB r B r0 y + B0 (12) ,, (12) E kB0 r k −1z = E 0 ro0 r , (12) 0 0 r − − k ( k 1 )( k 2 ) r E 0 0 z = E +1 B *(*-l)(*-2) xx 2,y ⋅ ⋅ ⋅ 0 k ( k − 1)(k 2 − 2) 2 r0 r0 22.r !r002 x y ⋅ ⋅ ⋅ J + B0 2!r0
CONCLUSION CONCLUSION CONCLUSION 11
22
33
4 44
horizontal horizontaltune tune FIGURE Shift FIGURE 4. Vertical Vertical Tune Tune Shift Shift with “soft edge model” "soft edge edge model” model"
Then, the the criterion criterion of the tune Then, selection isis obtained; obtained; tune selection the vertical vertical tune tune should not be set just bellow the the set just bellow the not be strong resonance resonance line. strong strong resonance line.
Non-linear Coupling Coupling Coupling Non-linear coupling coupling Non-linear induced by normal sextupole Non-linear coupling induced induced by by normal normal sextupole sextupole is observed with the simulation as is shown in Fig.5. is observed observed with with the the simulation simulation as as shown shown in in Fig.5. Fig.5. When the pair of tunes is very close to the non-linear When the pair of tunes is very close to the non-linear When the pair of tunes is very close to the non-linear +2ννyy=q), =q), the resonance line line ((ννxx+2 the motions in each phase resonance resonance line (vx+2Vy=q) the motions motions in in each each phase phase space are are not not bounded. bounded. As9 shown in Eq.2, the guiding space As shown in Eq.2, the space are not bounded. As shown in Eq.2, the guiding guiding field has normal normal sextupole, intrinsically. Especially, field field has has normal sextupole, sextupole, intrinsically. intrinsically. Especially, Especially, the non-linear structure resonance line should be the the non-linear non-linear structure structure resonance resonance line line should should be be avoided. avoided. avoided.
The beam dynamics FFAG accelerator was The The beam beamdynamics dynamicsofof ofFFAG FFAGaccelerator acceleratorwas was studied obtain large acceptance. We defined studied studied toto toobtain obtaina aalarge largeacceptance. acceptance.We Wedefined defined normalized coodinates. Using that horizontal normalized normalized coodinates. coodinates.Using Usingthat thatthethe thehorizontal horizontal acceptance described phase advance acceptance phase advance acceptanceisisisdescribed describedasas asa afunction a function functionofof of phase advance per cell, independently of the periodicity. Furthermore, per cell, independently of the periodicity. Furthermore, per cell, independently of the periodicity. Furthermore, “soft toto study thethe vertical “soft edge model” was introduced study vertical "softedge edgemodel” model"was wasintroduced introduced to study the vertical motion. motion. shows the vertical tune shift and motion. ItItIt shows showsthe thevertical verticaltune tuneshift shiftand andthethe the nonlinear nonlinear coupling FFAG. We found that nonlinear coupling couplinginin inFFAG. FFAG.We Wefound foundthat thatthethe the direction has toto bebe taken direction the tune shift fixed and has taken directionofof ofthe thetune tuneshift shiftisisisfixed fixedand and has to be taken into account to choose bare tune. into into account account to to choose choose bare bare tune. tune. REFERENCES REFERENCES REFERENCES 1.1. inin Japan: Based onon Y. Mori, ‘Neutrino Factory Japan: Based 1.Y. Y.Mori, Mori,‘Neutrino 'NeutrinoFactory Factory in Japan: Based on FFAG Accelerator’, Proc. of EPAC02, p278-p280. FFAG Accelerator’, Proc. of EPAC02, p278-p280. FFAG Accelerator', Proc. of EPAC02, p278-p280. 2.2. M. Aiba etet al., ‘Study ofofAcceptance of FFAG 2. M. M. Aiba Aiba et al., al., ‘Study 'Study of Acceptance Acceptance of of FFAG FFAG Accelerator’, Proc. of EPAC02, p1226-p1228. Accelerator’, Accelerator', Proc. Proc. of of EPAC02, EPAC02, p1226-p1228. pl226-p!228.
206
