Particle trapping during passage through a high-order nonlinear resonance

NUCLEAR

INSTRUMENTS

AND

METHODS

I2I

(I974)

I29-I38;

©

NORTH-HOLLAND

PUBLISHING

CO.

PARTICLE TRAPPING DURING PASSAGE THROUGH A HIGH-ORDER NONLINEAR RESONANCE* ALEXANDER

W. C H A O t

and MELVIN MONTH

Brookhaven National Laboratory, Upton, New York 11973, U.S.A. Received 14 J u n e 1974 A theory o f particle t r a p p i n g a n d t r a n s p o r t during a single passage t h r o u g h a high-order nonlinear resonance is developed. The m a i n result is an expression for the trapping efficiency as a function o f two scaling variables which are related to the resonance excitation width, the nonlinear detuning a n d the speed

o f passage t h r o u g h the resonance. T h e question o f w h a t phasespace region trapped particles are d r a w n f r o m a n d the question o f adiabaticity are discussed. T h e theory is then verified with a c o m p u t e r simulation for crossing a fifth order resonance.

1. Introduction

ticle loss. We are referring to the well-known fact that although nonlinear detuning stabilizes high-order resonances in a system with static tune, in a dynamic system with changing tune, small amplitude particles can "lock into" a resonance and be carried to large amplitudes3). However, the "lock in", or " t r a p p i n g " effect sensitively depends on the speed of resonance crossing. If the resonance strength is sufficiently weak and the resonance crossing speed is fast enough, then the trapping process would substantially decrease in significance. This is perhaps a good description of the accelerator field before storage rings. Dominated by relatively fast tune variations and weak high-order resonances, conventional accelerators could not have been expected to manifest a significant amount of particle trapping. With the advent of storage rings, we must re-examine our preconceptions of fast crossing and weak resonances. Take, for example, the ISR. l f w e consider highorder resonances excited by the b e a m - b e a m interaction and tune variation caused by the momentum diffusion due to intrabeam scattering 7' s), it is clear that the fast crossing-weak resonance combination does not apply. In fact, it is the attempt to respond to the slow crossingstrong resonance implication in this case that is the primary reason for this study. We review in section 2 the theory of isolated resonances. To be specific, we have considered a fifth order resonance. We study the phase-space topology of the dynamic system, laying the groundwork for a dynamic approach to trapping which results in a beam loss mechanism. In section 3, we first introduce the concept of trapping efficiency, simply the fraction of particles carried outward during a single resonance passage. We then consider the various factors which influence the trap-

Beam loss in the stacked coasting beam of the ISR has been correlated with high-order nonlinear resonances~). In specific instances, 5th and 8th order resonances have been correlated with beam lossZ). The evidence is strikingly simple. The beam density across the width of the stack is directly related to the tune. The observation is simply that beam loss occurs preferentially at positions corresponding to these resonances. Although it is clear that high-order resonances are a significant factor in ISR beam loss, how their effect becomes manifest is not clear at all. There are, in effect, two diverging lines of thought. First, it is argued that an explanation be sought using isolated resonance characteristics in the traditional sense3). In the second place, there is the suggestion of a multi-resonance effect, in which many high-order resonances combine to produce a quasirandom force inducing Arnold diffusion of the beam 4' 5). Of course, the main impetus for the second line of thought is the belief that the traditional resonance picture cannot provide a beam loss mechanism, that is, cannot explain the transport of particles from a beam position in the center of a chamber to the aperture limit. It thus seems that the multiple overlapping resonance model is a response to the apparent inability of a single isolated resonance model to describe particle transport to large amplitudes. This being the case, we have restudied the singleres;onance model. We first observe that the traditional concept of single, isolated, high-order nonlinear resonances does indeed possess a mechanism for par* W o r k performed u n d e r the auspices o f the U.S. A t o m i c Energy C o m m i s s i o n . ? ]?resent address: State University o f N e w Y o r k at Stony ]Brook, Stony Brook, N e w York, U.S.A.

129

130

A L E X A N D E R W. CHAO AND M E L V I N MONTH

ping efficiency, concluding by obtaining a formula for the trapping efficiency. Section 4 describes our computer simulation experiment for passage through a fifth order resonance. Section 5 contains our data. We then correlate these results with the theoretical framework developed in sections 2 and 3. Although our analysis applies only to one-dimensional resonances, we believe that in the coupling resonance case, by an appropriate redefinition o f tune and a proper interpretation of resonance excitation width and detuning strength, our analysis is actually applicable to the two-dimensional case. In general, we find a strong correlation between the theory developed and the numerical results obtained. The theoretical expression for the trapping efficiency derived in section 3 gives a remarkably g o o d fit to our data. We find that fifth order resonances are capable of inducing trapping efficiencies of the order o f 30%. This is an impressive figure in that, using an isolated resonance-diffusion feeding mechanism, a trapping efficiency much less than this value is sufficient to describe the fifth order resonance loss observed at the ISR 8).

tonian, H~

= 0 ( 0 ) y 4 + h ( O ) yS.

The second term is the fifth order resonance term. The first is the octupole stabilizing term and is introduced so that particles with sufficiently small amplitudes are always stable. The tune, of course, will be near some resonance value v ~ ½ p and will be changing with time. We then follow the usual procedure of substituting eq. (1) into eq. (3) and extracting the slowly varying Fourier components of H (1). The result is H ~1) = v~fl} (3Bo a2 + IApl a ~ cos 5~b) +

+ fast varying c o m p o n e n t s , where -

dO O(0),

167rv o IApl e'"• = (13)~ ; o :~ dO e -ipO h(O), 327rv = (kp-v)

2. One-dimensional nonlinear resonance theory

We include in this section a review o f the wellknown one-dimensional nonlinear resonance theory. We shall briefly summarize the phase-space topology. A more detailed study can be found elsewhereg). The single particle motion described by a perturbation Hamiltonian H ~)(y, y', 0) can be written as y(O) = [a(O)~(O)] ~

cos[u0+~o(0)],

a' -

2 ~?H(~) v/3 ~o

,

p' -

2 63H(1)

(2)

v[~ ~a

Note that in the absence of the perturbation, a and ~o are constants of the motion. In particular, a is simply the particle emittance. For simplicity, we assume the " s m o o t h approximation", i.e. we replace fl(O) by its average value ~fl} over 0. This should not affect any of our conclusions. To be specific, we shall deal only with a fifth order resonance. The generalization to any other high-order resonance is straightforward. We have for the perturbation Hamil-

o-~o+~},1,

and r/is a phase factor. Note that B o is proportional to the zeroth harmonic of the stabilizing term O(0) and Ap is proportional to the pth harmonic of the resonance term h(O). F o r a weak perturbation, we expect a(O) and ~0(0) to be slowly varying functions of 0. In that case, we need only keep the slowly varying components of H I1) in eq. (2). Hence we find

c I)

where y is the displacement from the central orbit, 0 is the " t i m e variable", y ' is the derivative o f y with respect to 0, v is the tune of the system, and fl is the wellk n o w n fl-function 1o). The dynamical variables a(O) and ~0(0) satisfy

(3)

a' = - 10a ~ lap] sin 5~b, qo' = 5a ~ lap] cos 5 ~ + 1 2 B o a .

For later convenience, we define ao

= the average emittance of the initial particle distribution, = a/ao = the relative emittance, A c = ½ p - v = the linear tune shift, ANt = -- 12Boao = the nonlinear tune shift at emittance ao, A c = - 5 l A p ] a o = the excitation width at emittance ao, = 5AL/2A e , K = 5ANL/4Ae.

The above equations for a and ~0 then become (~)'

= Ae0~2 sin 5t/,

ffl' = AL-'}-O~ k A e cos 5~-~-ANL~.

(4)

PARTICLE

It is easy to show by direct differentiation that C = ~ + ~ 2 + ~ cos 5 qJ

(5)

is a constant of the motion. A particle whose initial conditions correspond to a given constant C is constrained to lie on the trajectory described by eq. (5). ]Having obtained the invariant C, the topology of particle trajectories in the phase space is completely determined. For example, the fixed points are those for which tVC = 0, ~-a f

tVC = 0. ~--~f

The nature of the fixed points is determined as follows:

t32C .OZ C 0--~T If 0~ 2 f > 0 for stable fixed points, t~2C

~2 C

~-a7 If" ~

f

< 0 for unstable fixed points.

Without loss of generality, let us assume ~ > 0. We summarize the results for a dynamic system (i.e. changing ~): l) When ~ > 0, there are five unstable fixed points in the polar coordinate phase space (c¢4, ~) located at COS 51/If =

~f* = t + ( - x + z )

--

1

} + ( x + z ) ÷,

where we have defined t = 4x/15, x = } [ ~ ( ~ + 107)]12, Z ~

1-~+ t3 5

As ~ diminishes to 0, ~f~ decreases to 3 t. 2) When ~ becomes negative but ~ > - 1 0 / 3 , the unstable fixed points in (1) persist, with their positions given by cos5~f = -1,

ct~ = t - 2t cos ½(nrr+r&), where tp~ = c o s - a ( - z / t 3 ) , and n is chosen to be either 0, 2 or 4 such that cos ½(nrc +tpx) < - ½ . The amplitude decreases from 3t to 2t as ~ changes from 0 to - 10t a. Aside from this set of unstable fixed points, there are two sets of fixed points, one stable, another unstable, created at origin as ~ passes through 0. The positions of the stable ones are COS 5 ~ / f =

--

1,

~f = t - - 2 t cos l(nzc+qol),

(6)

131

TRAPPING

with n chosen from 0, 2 and 4 so that - ½ < cos ½ x x ( n n + q h ) < ½ . The new unstable fixed points are located at COS 51~f = + l , ~ = - t - 2t cos ~ [ ( n + 1) ~ - ~ o , 3 ,

with n chosen to be either 0, 2 or 4 such that cos ½[(n+l) ~ - t p l ] < - ½ . It is e a s y to show that the stable fixed points are created at the origin as soon as ~ passes through 0, and then start to migrate outward until they reach the amplitude 2 t when ~ decreases to - 10 t 3. At this amplitude, the stable fixed points collide with the inward moving unstable fixed points. The newly created unstable fixed points will move from the origin to the amplitude t as ~ changes from 0 to - 1 0 t 3. 3) When ~ < - 1 0 t 3, the colliding stable and unstable fixed points described in (2) annihilate each other. The only remaining fixed points are the unstable ones created at origin when ~ = 0. Their locations now are COS 5 ~tf =

~

:

-

t -

-}- l ,

(x + z) ~ -

(-

x + z) + .

It can be shown that this amplitude will increase from its magnitude t as ~ decreases from - 1 0 t 3. The entire process is pictured in fig. 1. It is clear that the stable islands created at the origin can carry beam particles and cause a beam growth as they travel outward in the phase space. Beam is lost either when the trapped particles hit the walls of the vacuum chamber or when the stable islands are annihilated by the incoming unstable fixed points. In the following, we derive an approximate expression for the total area of the islands when the stable fixed points have an amplitude ~s. Since we are interested in the trapping process, cq is chosen to be inside the beam distribution. We then note that ~: is usually large, of the order of 100. The case of small x corresponds to an immediate particle explosion (i.e. fast amplitude increase along an unstable trajectory) after crosses the resonance. This will be further discussed following eq. (21) in section 5. With these approximations, eq. (6) gives ~+. COs~ 2~c Similarly, eq. (7) gives the amplitude of the outward moving unstable fixed points at the same moment:

~u~ I~--~as'2~c

132

ALEXANDER

LENGTH SCALE:

~

W. C H A O A N D M E L V I N M O N T H

I,:=100 ~=-5×104

K=IO0 ¢ = 3 xlO 2

mately

26.7

L

The factor 5 is because there are five islands. Expand the equation of the separatrix, keeping terms up to 62 . Then it can be shown that

Thus, we have 2 (e) BEFORE CREATION K =100 ~¢=-1.5x105

~ =100 ,~=-2×105

(c) BEFORE ANNIHILATION

(d) AFTER ANNIHILATION

Fig. 1. The phase-space topology for resonance crossing. Each curve represents a c o n s t a n t - C curve according to eq. (5). T h e variable K = 5M~L/4Ae is kept constant, while the tune crosses a 5th order resonance. Blackened dots represent fixed points. (a) Before the tune crosses the resonance (~e= 5AL/2Ae>0). (b) After ~e becomes negative. A set o f stable and unstable fixed points are produced at the origin, m o v i n g o u t w a r d as~: decreases. (c) Before ~ reaches the fixed point collision limit, - 1 0 t 3. (d) After the annihilation o f the colliding fixed points.

The fact that % is nearly equal to % indicates that the shape of each island will be approximately an ellipse. As ~ decreases, the islands will assume a shape more like triangles as shown in fig. lc. The equation for the separatrix is C = {~u + ~c~ + ~

A ~--~c x/2

(b) A F T E R CREATION

= {~ + ~:cd + ~ cos 5 0 .

Set cos 50 = - 1 . Let e = % + 6 and solve for 6. The difference between the two solutions 6+-(3_ is the width of the island ellipse along the radial direction multiplied by 2 ~ , i.e.

2c~¢. (8)

3. Theory of particle trapping The mechanism for beam loss has been explained in the previous section. In this section, we are concerned with the actual trapping process, deriving an expression for trapping efficiency as a function of the crossing speed, the resonance excitation width and the nonlinear detuning strength, in the case of a Gaussian amplitude distribution. Consider a physical particle in the phase space. The amplitude of the particle, ~, changes according to eq. (4). On the other hand, when the amplitude of the stable fixed points passes the particle amplitude (cq = ~ at this moment) as ~ changes, the amplitude of the stable fixed point is growing at tile rate (c~2), _ d~ 1 dO '2s c~.,- 4K~)

(9)

Comparing eqs. (4) and (9), we obtain a criterion for adiabaticity: A particle at amplitude c~ is considered adiabatic if its amplitude is such that

or Zje~ 2

>~ d{ 1 . do ~ c ~ - 4~:c~~

(10)

When the adiabaticity condition is fulfilled, the particle will possess essentially the same value of C in successive revolutions. If we replace the inequality by an equality in eq. (10), we find the solution 0q ~c%

( 3o) I +--~

,

(II)

where The width across the other principal axis is simply 7r~. Hence the total area of the islands is approxi-

=(. Y

(12)

PARTICLE

and e is the change ofA e per revolution. We have again used the condition K >> 1. In what follows, we assume for simplicity that e is independent of 0. Hence we conclude that when a particle is trapped by an island and brought to an amplitude sufficiently larger than ea, the adiabaticity condition becomes satisfied and the particle will remain trapped. In other words, a particle already trapped will remain trapped. Using the above discussion, we suggest that a necessary condition for particle trapping is Ace2 > d~ 1 dO @ ~ - 4 ~ e +' or

> el-

(13)

The justification for this is that if a particle travels sufficiently slower than the by-passing stable fixed point, the changing topology of the phase space will not influence the particle motion, whereas a particle with sul]iciently rapid amplitude change will feel the impact of the passing fixed points. It is this latter condition that we associate with a finite trapping probability. Since the phase angle 5~ is changing much faster than e [see eq. (4), and recall that ANL >> Ae ifK >> 1], the trapping should be essentially independent of this phase angle. Using the above trapping condition an upper limit to the trapping efficiency, defined by, the number of particles trapped by stable islands the total number of particles is then simply the number of particles with e > ~t in the initial distribution the total number of particles

~ ~

{

~,

if cq > 1,

1,

if ~1 < 1.

(16)

The reason for eq. (16) is as follows: When c¢1 > 1, a typical particle being trapped will most likely have an amplitude close to the lower limit c¢~, simply due to the denser distribution at smaller amplitudes. When ~l < 1, a typical particle satisfying ~¢> c¢1 will be at c¢~ 1 because of the denser distribution there. We thus propose that the trapping efficiency can be expressed by A PT -- - exp(-c¢l), (17)

~(~)~

and satisfying the normalization condition ' ~ D(~ +) dc~+

less of their phases. Eq. (15) is not an estimate of the trapping efficiency, for it assumes that for slow enough crossing, all particles will be trapped. This is clearly untrue. The trapping must depend on the strength of the resonance relative to the stabilizing detuning. Although the process appears complex, we will see that the limitation can be formulated in a rather general way, as a consequence of the well-known Liouville theorem for particle densities. In the (e+, ~9) phase space, the local particle density according to the Liouville theorem, cannot change. This means that the number of particles trapped is limited by the phase-space area of the outward moving islands. In general, since a particle is trapped only when its amplitude c¢+ is greater than e l+, the area of the island should be evaluated when the stable fixed point passes by a typical particle with c~> e l . Let the amplitude of the stable fixed point be ~ at this moment. We expect, for cq not too far away from unity, that the typical particle amplitude can be defined by

(14)

Consider a Gaussian amplitude distribution, given by the density D(~ +)d~ + = 2c~+ d e + e - a ,

j

133

TRAPPING

I.

where A is the total area of the islands when the stable fixed points have the amplitude given by eq. (16). An expression for A is given in eq. (8). The factor exp ( - cq) describes the fraction of particles available with c¢> c¢1 . The factor 1 / ~ ( ~ ) 2 ensures the correct normalization: rr (c¢~)z is the effective beam area at the "typical trapping amplitude". Substituting eq. (8) into eq. (17), we finally obtain

0 PT

Eq. (14) for this initial distribution is

= - -7r ~:-+c¢s e x p ( - e D ,

,,/2

or

PT ~

D(~ ~) d~ + = e x p ( - c q ) .

(15)

= +.0 l--'+ J+ The trapping efficiency reaches the value e x p ( - e a ) only when all particles with c~> cq are trapped regard-

-

exp(-+,).

(,+)

",ANL] The essential factors in eq. (18) are (1) the square root

134

ALEXANDER

W.

CHAO

AND

of the relative strength of the excitation width and the nonlinear detuning, and (2) an exponential factor which decreases with the crossing speed. Aside from this functional dependence, we have deduced the nontrivial result that the physically relevant parameters, i.e. the proper scaling variables, are in fact, AdANL and

e/A~ A N L .

MELVIN

35

i i i 35 (e) INITIAL II DISTRIBUTION :.'30 H ---EXPECTED J,J-,n DISTRIBUTION 25

30 25 20

MONTH

'

20

'

15

15

l0

IO

95

5

_o

~ l

I

(b) AFTER :50000

I

VOLUTION S

4. The computer simulation i

A computer simulation experiment for the crossing of a fifth order resonance has been carried out. We have assumed &function perturbations

b_ 351 o | ~30~-

z

3

i I i (c) AFTER 5 0 0 0 0 REVOLUTIONS

r]]

§251~ I~q

30 25

h(O) = ho[5(0) - &(0-½~r) + 5(O-re) - 5 ( 0 - 3 7 r ) ] ,

o (0) = Oo [a (0) + a ( 0 - ½~) + a ( 0 - ~) + a ( 0 - ~ ~)],

35

20 15

15

I0

I0

(19) 5

where h(O) and 0 ( 0 ) are defined in eq. (3). We then somewhat arbitrarily la) choose in our numerical computations:

v ~ 102/5, 0.003,

a o = 1.6x10 -6radm. The significance of the constants ho and O o are such that each particle receives a kick represented by

y(O +) - y(O-) = O, y'(O +) - y'(O-) =

4y(0) 3 0 o -- 5y(0) 4 ho,

-

at 0 = 0, 7t and a kick represented by

y(O + ) - y(O-) = O, y'(O +) - y'(O-) = - 4y(0) 300 + 5y(0) 4ho, at 0 = ½7r and 37r. Using formulae derived in section 2, we can find the relationship between O o , h o on the one hand and ANL, A~ on the other. To do this, we first compute Aao 2 and B o. They are

ho(/~) ~ Alo

2 --

_

_

87zv

,

Bo -

Hence we obtain /NL

2

3

3 I

(d) AFTER 75000 REVOLUTIONS

5,; I

2

3

,,G

OoG) 4~rv

We then follow a set of 300 particles, assuming eq. (3) is the only perturbation, and giving kicks according to the above recipe. The free parameters are e, Ae, ANL and the n u m b e r of revolutions N. The initial particle distribution is completely determined by ao. We assume a Gaussian distribution in c~{ with volume element 2~re{dc~~. The random variable technique is used to generate the initial distribution. The procedure is as follows: We first randomly generate two variables zl and z2 in the interval (0, 1). The amplitude and phase of a particle is then simply a = ao ln[l/(l-zl)],

~ = 2rCZ2 .

One possible initial particle distribution calculated for 300 particles is plotted in fig. 2a. The dotted curve is the expected distribution in the ideal case.

5. Numerical results m

3ao Oo(fl) TOY

5 a~ (/J)~ ho A e --

P' I

I

2 I

Fig. 2. The trapping process: instantaneous particle distributions during trapping. As a typical example, we take de = 2 x 10 5, AZ~L = 2 x 10-3 and e = 0.25 x 10-6. (a) The initial distribution. (b), (c), and (d) The distributions after 30 000, 50000 and 75 000 revolutions respectively. The amplitudes of stable fixed points are indicated by arrows. The dashed curve in (a) represents the expected particle distribution.

fl = (fl) = 21.4 m ,

(AL)initia I =

5

I I

87rv

The trapping process is clearly demonstrated in fig. 2. A typical set of parameters is chosen in this case. Figs. 2 a - d correspond to four successive snapshots of the particle distribution when N = 0, 30 000, 50 000, and 75 000 respectively. Fig. 3 shows the initial (N = 0) and

135

PARTICLE TRAPPING t h e final ( N = 75 000) d i s t r i b u t i o n s in t h e ( ~ , ¢ ) p h a s e s p a c e for the s a m e p a r a m e t e r s . T h r e e h u n d r e d p a r t i c l e s are used. A t r a p p i n g efficiency o f 12% is o b s e r v e d . L i s t e d in tables 1 - 4 are o u r d a t a o f t r a p p i n g effic i e n c y for v a r i o u s sets o f p a r a m e t e r s . W e find t h a t a g o o d fit is g i v e n by PT = 2.4

A~ ~ e}- e x p ( - c q ) ,

(20)

X ~

•

(a)

.

:

TABLE 1 Data of the computer simulation experiment for ~o=0.998.

e(10 -6) Ae(10 -5) ANL(10-~) PT =

2 1.15 0.5 0.25 1 0.57 1 2.3 1.56 0.5 1.14 1

4 3 2 2 4 3 2 3 2.5 4 6 8

number of particles trapped total number of particles

4 3 2 1 2 1.5 4 6 5 1 1.5 1

26/3OO 27/300 29]300 41/300 42/3O0 38/300 20/30O 17/300 54/600 62/300 59/300 79/300

TABLE 2 Data of the computer simulation experiment for c~0 = 0.756.

- -

+

~(10-6) Ae(10-5) A~L(10 -z) PT =

number of particles trapped total number of particles

÷

-2-

0.25 1 0.56 2 0.5 0.78 0.28 0.5 2 1.14 0.25 1 0.5 0.5

+

I N I T I A L PARTICLE DI STR I BUTI ON

X =V~-sin~ X' =,/~- cos~/

+

+

÷

, ++ * ~ 2

(b)

-2

÷

,

+ ~1¢+

~,~ + + ,+ % *~ ++

+

•z

--x

, + +

+++

2 4 3 4 2 2.5 1.5 4 8 6 1 2 1 0.8

35/300 100/900 63/600 46/300 117/900 99/600 43/300 23/300 21/300 25/300 59/300 62/300 88/300 105/300

+

%++¢ }. +++'.

,~+2÷*,,

2 4 3 8 4 5 3 2 4 3 4 8 8 10

+

-2 ¸ DISTRIBUTION AFTER 75000 rev

Fig. 3. Particle distribution in the (~½,~p) phase space. (a) The initial distribution. (b) The distribution after 75 000 revolutions. Typical parameters are used: A e = 2 X l 0 -5, AZ~L=2Xl0 -3, e = 0.25 x 10-6.

w h e r e a s is g i v e n by eq. (16). It is e x t r e m e l y i n t e r e s t i n g t h a t eq. (20) deviates f r o m eq. (18) o n l y slightly in the o v e r a l l n u m e r i c a l c o n s t a n t . N o t e t h a t eq. (20) m a k e s sense o n l y w h e n 2.4 ( A ¢ ~ + < ~ 1, \ A NL/

(21)

w h i c h f o l l o w s f r o m eq. (15). Eq. (21) is v i o l a t e d w h e n eq. (8) o v e r e s t i m a t e s t h e island area, o r w h e n the i s l a n d a r e a b e c o m e s t o o large t h a t the n o r m a l i z a t i o n o f eq. (17) fails. B u t it t u r n s o u t t h a t w e are q u i t e safe in a s s u m i n g eq. (21). It is v i o l a t e d w h e n K < 7, w h i c h in t u r n m e a n s t h a t s o o n after the t u n e crosses t h e reso-

136

A L E X A N D E R W. CHAO AND MELVIN MONTH

TABLE 3 Data of the computer simulation experiment for c~0 = 1.317.

t 15%

[

"

2 3 4 4 6 2 6 3 4 8 6 1 1.5 9 8

2 3 4 2 3 1 2 1 1 2 1.5 4 6 1 1

21/300 22/300 57/600 31/300 35/300 28/300 79/600 40/300 37/300 44/300 38/300 12/300 12/300 66/300 60/300

\If

I

~-ao

i" ",

I ,.o

] ................

P,

2.4 /"

I0%~

0.1

30%

,/

...... t

I

,, ,,,'"~

I

e-: 0

PT

e"

20%

,'" I I ,,-£4, Ae_e.,~" ~- -

,o,1 0.2

0,3

////

/ ..{..

0J

/-

~ ..{-"

~. ~.A, d ~,, ~ °, . .~.'""

I

(b) a o =0.998

0.2

I

0.3

,

,

t.--'2 4 ( ~ e ) ~

10%

,y

I

I

q 30%

,,/""

20%

I(°) ao = 0.7561

5.0

140`

//

,,'~

,"'"'"'"{" I

:2

g e- a,,,,"

20N

4.0

p o n e n t i a l b e h a v i o u r s t r o n g l y reflects t h e e x i s t e n c e o f a critical amplitude below which no particles can be trapped. I n fig. 5 a - c w e h a v e p l o t t e d PT VS (Ae/ANL) ~, f o r d i f f e r e n t v a l u e s o f % . A l i n e a r b e h a v i o r is clear. I n t h e s e figures, t h e d o t t e d c u r v e s a r e o b t a i n e d f r o m eq. (20). T h e d a s h e d c u r v e s a r e e x p ( - c q ) . T h e e r r o r b a r s

-40%

,, , ,}'"

I

3.o

Fig. 4. The trapping efficiency as a function of the crossing speed. c~o=e/4~ A~LAe is the adiabaticity parameter. ANL and A~, are kept fixed. The dashed curve is an upper bound of PT. The dotted curve is our theoretical fit.

----jo. . . . . . . . . . . . . . .

30%

I

ao

\47rAeANL/

I

40%

'", e-%~e-a'

5%

n a n c e , t h e u n s t a b l e fixed p o i n t s l o c a t e d a t ctf ~ -45K a n d cos 5~'r = - 1 a r e so c l o s e t o t h e p a r t i c l e d i s t r i b u t i o n that an explosion immediately occurs. I n fig. 4, w e h a v e p l o t t e d PT VS % . T h e p a r a m e t e r s a r e t y p i c a l , w i t h cq ~ % t o w i t h i n 1 - 2 % , T h e d a s h e d c u r v e s h o w s t h e u p p e r l i m i t a c c o r d i n g t o eq. (15). T h e d o t t e d c u r v e s h o w s o u r fit, eq. (20). T h e o b v i o u s ex-

I

i

ANL = 2x ~0- 3 Ae: 2x10-5

",

24 e-% PT '0%

i

~~

\T

number of particles trapped e(10-6) Zle(10-5) ANL(10 -3) PT = total number of particles

1 2.25 4 2 4.5 0.5 3 0.75 1 4 2.25 1 2.25 2.25 2

i

/."

I

oJ

"

~-~,

/"NL

I

0.2

(c) a o= 1.317

I

0.3

± (A e /ANL )2

Fig. 5. The trapping efficiency as a function of (Ae/ANI3~. (a), (b), and (c) correspond to ~o=0.756, 0.998, and 1.317 respectively. The solid line curves are simply exp (-c~o). The dashed curves are the upper limits exp ( - ~ 1 ) of the trapping efficiency. The dotted curves are the theoretical fit.

PARTICLE TABLE 4 D a t a o f the c o m p u t e r simulation experiment for Ae = 2 x and LINL = 2 x 10-3.

e(10 -6)

PT =

10-s

n u m b e r o f particles trapped total n u m b e r o f particles

20 l0 5 3 1.5 1 0.75 0.5 0.25 0.1

137

TRAPPING

Gaussian amplitude distribution, we find the trapping efficiency also contains an exponential factor which falls off with the crossing speed. In addition to predicting the trapping efficiency, the theory describes the trapping probability distribution in the phase space. As a result we can infer from what phase-space regions trapped particles are drawn. We have verified the

6/600 4/300 10/300 11/300 18/300 21/300 39/600 29/300 35/300 47•300

I

:50

I

I

I

ANL = 2 x 10 -3 A e =2 x l O -5

r': '

25

t

r = 125

= 3 x 10 -6

v'~(= 1.43

20 15 I0

are given by a simple random walk consideration:

5

;I

." L, P T Jr- A P T =

Ira(1 -

m --

n

m/n)[÷

2 I

-t-

#1

where m = the observed number of particles trapped, and n = the total number of particles. To study the problem of which particles are trapped, we have shown the initial and final particle distributions for three cases in fig. 6a-c, arranged in order of decreasing cq (increasing adiabaticity). We clearly see that only particles with ~ > ~ are trapped. We also see that as (AJANL) increases, more particles with c~> cq are trapped. Fig. 6b shows how a large (Ao/ANL)affects those particles with c~> cq. Most of these particles are carried away by the islands. Fig. 6c corresponds to a small cq, but with (AJANL) not so large as in fig. 6b. It is clear that a small cq plus a large (AJAr~L) would sweep out most of the particles in the beam. A closer examination also shows a shrinkage to smaller amplitudes for those particles with ~ > ~ but not trapped, hence filling in the vacancies left over by trapped particles after the islands have passed through.

6. Conclusions Our main result is an expression for the trapping efficiency during passage through a high-order nonlinear resonance, as a function of the crossing speed e, the nonlinear detuning ANL and the excitation width Ae. In particular we find two scaling variables e/AeAyE and A.JANL. Following a rather general argument connected with the invariance of the local particle density, we have shown that the trapping efficiency is proportional to the island area. Also, for an initial

I v"E

I f~ 5 l

Z&NL= 10-3 3O

8A L

--

LtJ

Nrev

°J 25

7 -61 10 r

= 0.5X

(a)

L.]

I 4 I

l¢ = 15.6

0.888

I~,=

A e = 8 x 10-5 ,, J ii ii ii ii I1


'i

~5

rll i l l ,

,,=,

rl

I I I

10 I I

:D

z

5

rJ

~' I I I

50 25 20 15 I0 5

]rl

L~

',II

~

r"n v'E

I 2 3 I I A N L : 2 x 10-3

8/"L

I

=10

tev

4 I

125 =

-7

0.726

rl i I

:,, r~,,

'

I I

(b) r~ I

A e = 2 x 10-5

r

I

I

1

:I

t'l

I

i

i , i

I- 1 I I

~-

rl

'.'] ,

i i.~

[~, ~fF---i 2

~

I

(c)

3

Fig. 6. Influence o f initial p h a s e space position on trapping. Solid curves are initial distributions. D o t t e d curves are the distributions after trapping. (a) 0~t large, (b) A e / A ~ L large, (c) cq small. Note that (1) only particles with amplitude ~ > cq are trapped. (2) A large Ae/LlZCL m e a n s a strong trapping effect for those particles with ~¢> cq.

138

A L E X A N D E R W. CHAO AND MELVIN MONTH

theory with a computer simulation for crossing a fifth order resonance. The agreement between theory and computer experiment is remarkably good. References 1) E. Keil, CERN Report, CERN/ISR-TH/73-38 (1973): Lectures given at the International School of Applied Physics, Erice, Italy (June 5-16, 1973). ~) ISR Performance Reports (CERN, Geneva), R U N 376 (Oct. 30, 1973) and RUN 400 (Dec. 6, 1973). 3) See for example, A. Schoch, CERN Report, CERN 57-23 (1958). 4) G. M. Zoslavskii and B. V. Chirikov, Usp. Fiz. Nauk. 105

5) 6) 7)

s) 9) 10) 11)

(1971) 3 [translation, Sov. Phys. USPEKHI, 14 (1972) 549]; and E. Keil, Proc. 8th Intern Conf. on High energy accelerators, Geneva (1971) p. 372. E. Keil, CERN Reports, CERN/ISR-TH/72-7 and 72-25 (1972). See, for example, M. H. R. Donald, Rutherford Report, Synchrotron-betatron stop bands, R H E L / M / N I M (1973). E. Keil, Ill All-Union National Conf. on Particle accelerators(1972); and H. G. Hereward, CERN Report, CERN/ISR-PI/72-26 (1972). M. Month, BNL Report, CRISP 73-25 (1973). A. W. Chao and M. Month, BN L Report, CRISP 74-9 (1974). E. D. Courant and H. S. Snyder, Ann. Phys. 3 (1958) 1. The values of the parameters used correspond roughly to currently accepted values for proton storage ring design.