Simultaneous treatment of betatron and synchrotron motions in circular accelerators

Nuclear Instruments and Methods 212 (1983) 37-46 North-Holland Publishing Company

SIMULTANEOUS TREATMENT CIRCULAR ACCELERATORS C.J.A. CORSTEN

OF BETATRON

37

AND SYNCHROTRON

MOTIONS

IN

and H.L. HAGEDOORN

Eindhoven University of Technology, Eindhoven, The Netherlands Received 26 November 1982

In this paper a general theory for the description of coupling and resonance effects in circular accelerators is developed. Therefore, a simultaneous treatment of all three oscillation modes has been set up with the use of the Hamilton formalism taking curvilinear coordinates as canonical variables and the time as independent variable. The theory - which takes into account the rf accelerating electric field and the time dependence of the magnetic field - shows various transverse-longitudinal coupling effects and sources that can excite synchro-betatron resonances.

1. Introduction Until recently the s y n c h r o t r o n oscillations have usually been studied separately from the b e t a t r o n oscillations. The H a m i l t o n formalism has proved to be especially well-suited to study coupled b e t a t r o n oscillations a n d non-linear p h e n o m e n a (see e.g. ref. 1). Schulte a n d H a g e d o o r n have developed a theory for the non-relativistic description of accelerated particles in cyclotrons [2].They used Cartesian coordinates which turned out to b e convenient for the description of the acceleration process and of the m o t i o n in the central region of the cyclotron, although the representation of the magnetic field is rather complex in this system. G o r d o n and M a r t i have studied coupled m o t i o n using relativistic mechanics a n d polar coordinates [3]. Very recently Suzuki gave a canonical formulation of s y n c h r o t r o n and b e t a t r o n oscillations, using the orbit length.as independent variable [4]. A similar t r e a t m e n t has also been given by Mills [9]. In this p a p e r a general theory for the simultaneous description of longitudinal and transverse motions will be presented including the acceleration process due to rf electric fields as well as a t i m e - d e p e n d e n t magnetic field ( b e t a t r o n acceleration). F o r this t r e a t m e n t we use a relativistic H a m i l t o n formalism with curvilinear coordinates and the time as i n d e p e n d e n t variable. The choice enables an easy representation of time varying parameters such as magnetic induction a n d rf frequency. In case of time c o n s t a n t parameters it is convenient to have as i n d e p e n d e n t variable the orbit length. In the next two sections we will first bring the H a m i l t o n function in a p r o p e r form to develop the theory. Afterwards we will start with the description of the particle motion in a cylindrical-symmetric magnetic 0 1 6 7 - 5 0 8 7 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 0 3 . 0 0 © 1983 N o r t h - H o l l a n d

field (section 4 and 5) a n d then extend the theory to magnetic fields with an alternating gradient structure (section 7). The final theory demonstrates the existence of coupling effects in circular accelerators such as (synchro-)cyclotrons, C.G. a n d A.G. synchrotrons, b e t a t r o n s a n d storage rings. Several of these transverse-longitudinal coupling effects are the subject of the following article [5].

2. The general Hamilton function A H a m i l t o n i a n for the m o t i o n of a charged particle with relativistic energy in a time d e p e n d e n t magnetic a n d rf electric field can be represented by

n ( p , q, t) = ~22 + [ p _ eA(q, t ) ] 2 c 2 ,

(1)

where p and q are the vectors of the canonical m o m e n t a a n d coordinates, A(t) is the time varying magnetic vector potential, c is the velocity of light, E r and e are respectively the rest energy a n d the charge of the particle. The time t is the i n d e p e n d e n t variable a n d the value of H equals the total energy of the particle. It is convenient to describe the m o t i o n of the particle in terms of coordinates related to the so-called reference orbit. This orbit is defined for a fixed time t = t o a n d the reference particle which has the n o m i n a l energy moves on this orbit. The reference orbit has the same symmetry as the u n p e r t u r b e d guide field a n d lies in the m e d i a n plane. In a cylindrical-symmetric magnetic field the reference orbit is a circle. In case of a synchrotron the reference orbit is the "design orbit". F o r the deviation of the motion from the reference orbit, we define a curvilinear orthogonal coordinate

C.J.A. Corsten, H.L Hagedoorn / Motions in circular accelerators

38

system x, z, s with x and z as the horizontal and vertical deviations from the reference orbit and s as the coordinate along the reference orbit. In this system a positively charged particle rotates in the s-direction in a magnetic field pointing in the positive z-direction and the length of the infinitesimal vector d o is given by do 2 = dx 2 + dz 2 +

1+

X

ds 2,

(2)

in which p(s) is the local radius of curvature of the reference orbit. The H a m i l t o n i a n of eq. (1) can be written as P

H = IE2r + (1~ - eA~) 2c2 + ( P : - eA.)2c 2

+

(

Ps

1 +x/p(s)

2 ] I/2 eAs) C2

(3)

It is c o m m o n use to express A(t) in the c o m p o n e n t s of the magnetic field via B = curl A, where B is fixed by the relations d i v B = 0 a n d c u r l B = 0 . A slow time variation of the magnetic field is represented by a simple multiplying factor. The rapidly varying part of the vector potential, due to rf accelerating structures, will be discussed in section 3. Since A is defined in terms of B by B = curl A, the vector potential is arbitrary to the extent that the gradient of some scalar function can be added as long as the magnetic field does not change in time. However, in case of a time-dependent magnetic field, where the b e t a t r o n acceleration along the reference orbit is given by E = - O A / O t , the simple multiplying factor for B is extended to the vector potential and A is fixed by ~Asds = ~, the enclosed magnetic flux. In case of a s y n c h r o t r o n or storage ring with a cylindrical-symmetric magnetic field, which might be time-dependent, a related vector potential is A,~ = A z = 0,

A,= -Box+

+ 6

O-

~ - b 1 x 2 + ½bl z2

p2

(4)

the reference orbit and its average gradient. F o r the b e t a t r o n we find:

A~ = A z = O,

+ ~7+~

(5)

so that • = ¢A,ds = 2~rBoO2 (on the reference orbit). Substituting the vector potential into eq. (3), the H a m i l t o n i a n can be expressed in a power series of the canonical variables. The coordinates x and z are considered to be small quantities: they are assumed to be m u c h smaller than the local radius of curvature of the trajectory. In the following sections we will develop the Hamilton theory so that it becomes more suited to study coupled orbit motion. First we will discuss the representation of a time-dependent magnetic and electric field.

3. Time-dependent magnetic and electric field The acceleration of charged particles to higher energies is done by rf electric fields (except in case of the betatron). In cyclotrons and synchrocyclotrons the accelerating structures involve one or more Oees, whereas synchrotrons and storage rings have at least one cavity a r o u n d the ring. For simplicity we will assume an accelerating gap with an infinitesimally small width, equivalent to stepwise acceleration. However, the extension of the acceleration gap can be included in the general theory (see ref. 2). In general the electric field will depend upon the coordinate s and u p o n the time t. It has a fast oscillating time dependence a n d its period is comparable with the period of revolution and the periods of the b e t a t r o n oscillations. As a rule the change of the magnetic field occurs extremely slowly with respect to the motion along the reference orbit. The characteristic times are e.g. 10 6 larger can the revolution period. In general we write E = -grad

so that f~Asds = 0 on the reference orbit. B 0 is the magnetic induction on the reference orbit and b 1 is related to the field gradient. In synchrotrons ~Asds is not necessarily equal to zero, hut the vector potential can be slightly fitted to reality. In principle the C.G. s y n c h r o t r o n does not have an exact cylindrical-symmetric field. But the total p a t h length in the straight sections is usually much smaller than the path length in all magnets and as far as the derivation of the theory is concerned, we might represent the magnetic field - as being cylindrical-symmetric - by its average value on

x'+...,

V - OA/Ot.

(6)

In machines with a Dee structure an rf potential function V(r, t) is possible inside the acceleration region and the effect of the accelerating voltage is represented by adding this V(r, t) to the H a m i l t o n i a n of eq. (3) (see ref. 2). This potential function has a non-zero value in the Dee a n d equals zero in the d u m m y Dee. However, this procedure is less evident when dealing with, for instance, one cavity in a ring-shaped accelerator. In case of cavities, a fast oscillating vector potential A(t) can be introduced, corresponding to the rf magnetic flux of the cavities. Note that in case of an odd

C.J.A. Corsten, H.L. Hagedoorn / Motions in circular accelerators number of cavities there is a net flux through the surface enclosed by the reference orbit i.e. ~Eds =x 0 (at a fixed time). To deduce a Hamiltonian in which the time-dependent electric and magnetic fields are visible separately, we split the chosen vector potential in a fast and a slowly varying part. For convenience we o m i t the fringing fields of the electric field and consequently there are no transverse components of the electric field. For the longitudinal component of the vector potential we write

A,(t) =As.f(t ) +A,,~(t),

(7)

where As, f(t ) is the fast oscillating part due to the rf electric field E~:

E,(t) = - ~--A ot ~ ( t )

(g)

and A~,,(t) is the slowly varying component originating from the magnetic field which is discussed in section 2. As the Hamiltonian should be expanded in a power series of the canonical variables, these should all be small quantities. We define a longitudinal reference m o m e n t u m ps0(t) by the relation

pso(t)=P(x=z=O;t)+eA,,,(x=z=O;t),

(9)

in which the kinetic m o m e n t u m in the time-dependent magnetic field is P ( x = z = O ; t)=eB(x=z=0; t)p. We recall the note concerning the choice of the vector potential and the relation between ~Asds and the enclosed magnetic flux in the previous section. In case of an A.G. synchrotron or storage ring the cavity is placed in a straight section. Studying the Hamiltonian of eq. (3) it is convenient to apply a transformation generated by the function • G = f i ~ x +p~z + fi~s +P~o(t)s

_~_

f s

[

- e f ' E s (s')d s'. cosf,~rf (t)dr,

in which wrf(t) is the angular frequency of the rf accelerating system. We introduce a terra i n / ~ ,

e V ( s ) "cosf,~rf ( t ) d t , with

V(s) = - f%(s')ds',

(133

where V(s) is a unique function of s but, in contrast with V(r) in case of a cyclotron, not necessarily of the position (note: s has increased with the circumference after one turn, whereas the position has not changed). In case of a cyclotron V ( s ) = V is the voltage in the Dee and V(s) = 0 in the dummy Dee. High energy electron accelerators differ from other machines by the radiation losses. The effects of these losses play a role on a time scale which is very large compared with the period of the betatron and synchrotron oscillations and will not be discussed here. Expanding the Hamiltonian of eq. (11) into a power series of the variables, it can be used to treat the synchrotron and betatron motions simultaneously. It is convenient to eliminate the constants in eq. (11) by a scale transformation. Therefore we define new relative variables and a new dimensionless time unit. The variables are normalized on quantities belonging to the reference orbit and the reference particle. We emphasize that the reference orbit is a solution of the Hamiltonian H with A(q, t) replaced by A(q, to). The new variables are defined by:

z= a/R,

s= g/R,

fix = fix/ Po , l~ = fiz/ Po ,

= x,

z = z,

S~S,

(lO)

Ps =ps - p , o ( t ) --CAs,f(s,t

) •

All variables remain unchanged except the longitudinal canonical m o m e n t u m and the Hamiltonian of eq. (3) becomes (with A x = A~ = 0)

(

E} + fi~c2 + fi}c2 +

d +g~P~o(t)

(

(12)

d

with

H=

and the "potential-like" function can be written as

x=./R,

e l A~,f~s t , t ) d s ' ,

39

Ps + P~° 1 + ~/p

- e f Es(~',t)ds'.

)'

'

"r=Wot

with

wo = ~

and Pff c2 = ( l - 1 ]

eA~ ~ c 2

fi~ = fis/ Po .

(14a)

The quantity R is the circumference of the reference orbit divided by 2~r, the momentum P0 is the kinetic m o m e n t u m of the reference particle. The new time unit is based on the revolution period of the reference particle:

c(

1

1

y2 '

W~ = Wff - E~,

(14b)

T0 / (11)

The influence of the slowly varying magnetic field is incorporated via As.~(t ) and Pso(t) and the acceleration by the rf electric field via the "potential-like" function -ef~E~(s ', t')ds'. In general the variation of the rf electric field with time has the form of a cosine function

with w0 the angular frequency and W0 the total energy of the reference particle. In order to maintain Hamilton's equations, the Harniltonian must be adjusted accordingly =

/ff

1-1

H

Wo

(14c)

40

C.J.A. Corsten, ILL. Hagedoorn / Motions in circular accelerators"

and this Hamiltonian will serve as the starting-point of the further theory. We notice that an equivalent angular variable 0 along the reference orbit is defined by 0 = s / R , which is exactly the variable s of eq. (14a). For this reason s and its canonical conjugate/~ are written as t~ and Po in future. Finally a remark on the method used: since we consider x / o as a small quantity - in order to be able to expand the term (1 + x / p ) I in the Hamiltonian of eq. (11) - this theory may not be generally suitable for studies at very small radii i.e. for central region studies in (synchro-)cyclotrons. Schulte and Hagedoorn developed a theory for the non-relativistic description of accelerated particles in central regions of cyclotrons by using Cartesian coordinates and splitting the horizontal motion into a circle motion and a centre motion [2]. To obtain a deliberate preamble we start with the expansion of the Hamiltonian of eq. (14c) in its variables, neglecting the electric fields. A coupling term between the radial and the longitudinal motion appears in second degree in this Hamiltonian. A canonical transformation is applied to separate the two modes of oscillation in the linearized case and the meaning of the new canonical variables will be explained (section 4). Afterwards, the acceleration by the rf electric fields is taken into account (section 5). As stated before we start with the description in a cylindrical-symmetric magnetic field. The extension of the theory to accelerators with an alternating gradient (A.G.) magnetic field structure is carried out in section 7. This treatment is simplified by the advance knowledge of the preceding sections. In this paper we will show that several transverse-longitudinal coupling terms arise, which may lead to specific sync h r o - b e t a t r o n resonances. These effects will be discussed in more detail in the following article [5].

4. Time-dependent cylindrical-symmetric magnetic field, no rf accelerating structures The reference orbit in a cylindrical-symmetric field is a circle with radius R equal to the radius of curvature p. In general the magnetic field B varies slowly in time and we write B(t)=Bo[I

+b(t)],

(15)

with B 0 the initial magnetic field on the reference orbit. We recall that the time dependence of the magnetic field is merely represented by a multiplying factor. For times characteristic of the transverse and longitudinal motions, the quantity b(t) is very small, in case of the (synchro-)cyclotron b (t) = 0. Before evaluating the Hamiltonian (11) or (14c) we notice that the quantity p,o(t) - defined in eq. (9) and of particular interest in case of time-dependent magnetic fields - depends on the constant term in the vector

potential. This term is determined by the enclosed magnetic flux and we get for the different machines [see eqs. (4) and (5)]: synchrotron

: pso(t)=eBo[1 + b(t)]p,

betatron

: Ps0 ( t ) = 0,

(16)

whereas for the (synchro-)cyclotron the shape of the vector potential is of less importance because of the constant magnetic field [see eq. (11)]. By this particular choice ofps 0 there are no first degree terms containing £ in the square root of eq. (11), which significantly simplifies the work of the expansion of this Hamiltonian. We now consider the case of a cyclotron and a synchrotron, whereas the betatron will be briefly discussed at the end of this section. The Hamiltonian of eq. (14c) is expanded up to third degree in the variables. An expansion to higher degree is more cumbersome but still possible. The Hamiltonian, with the scaled time ~- as the independent variable, is 2P.~ ~-st 1 - n ) f f 2 +

2Po+fio 23'0

+ 2p) + ~ n z

1+

2 ~ Po 1 -

+ b/~-/50~- t(3 - n)~ 3

-

+ 2yzPo z -SPoP~

x

1-

,

(17)

where the dot above a variable stands for d / d r and n = - ( R Z / B o p ) ( O B z / O X ) o is the field index. The canonical variables are defined in eq. (14a) with Po = eBop and as stated in section 3 the variables s and /~ are respectively written as ~ and/~0. The time-dependent magnetic field is written as B O ' ) = B0(1 + b~-) and the time derivative is taken into account in the lowest degree of the Hamiltonian only. A more detailed description gives some correction terms, all being small [6]. We notice that a coupling term between the radial and longitudinal motion appears in second degree of eq. (17), i.e. the rio ~ term. It is convenient to perform a transformation to new variables in order to separate the two modes of oscillation in the linearized case. The generating function is

=

1

G =~xY+~O,i+fioO- ~_nfiOfi ,so that

P0 x=~+l~,

L=PX-l~'

b

b x,

1- n

(18)

C.J.A. Corsten, H . L Hagedoorn / Motions in circular accelerators z=~,

~ =Px,

i f = 0 + ~1 Px _ ,

~ =:0.

The Hamiltonian, expressed in the new variables, is /Zl= ½/~ + ½(1 -n)22+½~O~4-½nY. 2

1 -1 n ) + P o ( l + ~ o 2 ) + b O

' - 2 ( Y° 12 + 5po -~(3-n)

5pop~

23

1-

3n

1 -

n)2)

+ ~/7022( ( 1

(1(,1) +

+:~ -7~o~

vg ~

1- n

y2

1

1 3-5n

~6(l_-n) ~

)

+ 2 y~ Po z

- ½Pop~(1 - e l ) .

(19)

The variables 2,/~x and ,~,/7z describe the horizontal and vertical betatron oscillations and their corresponding oscillation ." frequencies" are Qx = v/1 - n and Qz = ¢nFrom t} = 3(-I/3Po = 1 we find that in first order approximation ti is equal to the scaled time ~-. The new longitudinal coordinate 0 contains the radial motion via the m o m e n t u m / ~ according to eq. (18). As illustrated in fig. 1 the quantity fix is strongly related to the position of the centre of the orbit. The variable t~ is based on the same idea as the so-called Central Position (CP) phase, first introduced by Schulte and Hagedoorn in their treatment of the motion of accelerated particles in cyclotrons using cartesian coordinates [2]. Thus when dealing with coupling between transverse and longitudinal motion [coupling terms exist in eq. (19)] it is neces-

41

sary to perform a slight change of the longitudinal coordinate. In case of an A.G. accelerator the relation between the coordinate 0 and the centre position is much less evident because the reference orbit is no longer a real circle. But, as we will see in section 7, it remains possible to define an equivalent CP phase based on the same idea, namely decoupling of both linear radial and longitudinal motions. The longitudinal momentum P0 is related to the deviation of the kinetic m o m e n t u m P(t) of an arbitrary particle - moving in the time dependent magnetic field on a radius p + x - with respect to the kinetic momentum of a particle moving on radius 0 [7]:

:o

AP(T) eo

aW(~-) AW(~-) Poc(1 - 1/vg - R~oPo

(20)

In case of cylindrical symmetry R = 0The quantity P0/(l - n), subtracted from the x c o o r dinate in eq. (18), is the relative change in the orbit radius due to a relative m o m e n t u m deviation and 1/(1 - n ) is equal to the so-called m o m e n t u m compaction factor a. F r o m / ~ 0 = - 3 I ~ I / 3 0 = - b and eq. (20) we find in case of the C.G. or weak-focusing synchrotron d P ( , ) / d ~ " -- 0,

(21)

in first order corresponding with the fact that the energy or m o m e n t u m remains constant when no rf accelerating structures are present: the betatron action is (almost) zero in synchrotrons.

4.1. Betatron acceleration A certain amount of betatron acceleration is represented in the Hamiltonian by taking a vector potential of the form

A, = - B( t)O(fo + f ( x, z)},

(22)

where fo is a constant (0 < f0 < 1) that "measures" the enclosed magnetic flux. In case of the betatron f0 = 1 [see also eq. (5)]. After having carried out all transformations performed to obtain a Hamiltonian like eq. (19), we find ~'~/~"'~--

~

reference orbit

/(

/2/=~'2+½(1-n)22+½/~~px

02(1y~

l-1 n

+Po( l +bo2 ) - ( f o - 1 ) b # + " . .

centre of reference orbit

Fig. 1. Schematic iUustration of correction angle Px of the azimuth in a homogeneous magnetic field (n = 0).

(23)

For the betatron (f0 = 1) the term b0 is missing. Consequently /~a is constant and for the reference particle Pa = 0. As (2, Px) = (0, 0) is a solution of the Hamiltonian of eq. (23) we find [from eq. (20)]

?(,)= eB(~)o.

(24)

C.J.A. Corsten, H.L. Hagedoorn/ Motions in circular accelerators

42

The momentum (or energy) of the particle increases proportionally with the rate of change of the magnetic field (betatron action). Of course eq. (24) can also be derived directly using A~(t) of eq. (22) and ~ = Pofio.

whereas all other variables remain unchanged. The new coordinate ~ is a deviation from a reference pointer with ~rf(~')/h and we should call # a "phase". The new Hamiltonian becomes

[I = kl + OG/8"r = / 4 5. The influence of longitudinal rf accelerating electric fields on the orbit motion

In general the longitudinal electric fields oscillate in time with a time-dependent frequency ~rf(/). For our description of the acceleration only the number of gaps or cavities and their positions are sufficient. After application of the scale transformation eq. (14), the potential-like function of eq. (13) which represents the rf acceleration in the Hamiltonian becomes

eV,(#)

cos/~Or)"('r) tiT, J

02 0

with

(25)

Rf#F.d# ' v,(#)

= (1-

l/y02) W0 '

where we remind the reader that the variable # of eq. (14a) is written as #. In this paper we restrict ourselves to homogeneous electric fields. In practice this may not be quite true and a substantial extension of the theory might be the description of e.g. radial electric fields or a variation of the accelerating voltage along the gap, resulting a representation of effects such as e.g. phase compression or electric focusing. Generally the frequency of the accelerating voltage is a multiple of the revolution frequency, indicated by the harmonic number h. Its time dependence is incorporated by writing ~0rf (~") = h~0[1 + 6 ( ' r ) ] = h~00(l + 8-r),

- [1 + 6('r)]/~0,

and the potential-like function i n / t is

eV](~+

#x +,c*)

COS

h'r*,

(30)

where we a b b r e v i a t e d / [ 1 + 8(~-)]d~- by ~-*. To evaluate this function V~ we subtract the purely time-dependent function VI(~-*) from Vl(~-* + 0 +/~U1 - n), a similar procedure as given in ref. 2. The result is a function V.f(t~ + / ~ J l - n ; "r*) as sketched in figs. 2 and 3 for the case of only one cavity in a synchrotron and for a one-Dee system, respectively z The width of the pulses of this function f depends on 0 and/~, and as a Hamiltonian must be expanded into its variables, it seems more appropriate to use this function f instead of V~ of eq. (30). This is allowed without further action because we subtract a function depending on the independent variable ~- only. Thus we write h=½/~+

.-. {seeeq.(19))-.. +

-[l + 8(~)]#0 +eV.f(O+

1 #~ - n ; ~'*/ cos h'r*,

(31)

where eV is the maximum fractional energy gain at the

(26)

in which 60") is of the same order of magnitude or even smaller than bO') in eq. (15). The Hamiltonian, including the acceleration term, expanded in the variables of eq. (18) becomes /~ = ~p:, , -2 + .

(29)

vI (~*)

(b)

~ f(°+~o >

(c)

. (see . eq. . . (19)5 .

+eV,(O+ l~n ) cosSh[l+8(.)]d..

(27) I

To study resonance effects we are especially interested in slowly varying terms. Therefore we first subtract the fast time-dependence from /J [note that 0 = ~+ . • •, see eq. (19)] so that the new longitudinal coordinate ~ varies slowly. The transformation is generated by the function

G = xfix+ z.#.+ 0#o-#of[I + 8(¢)]dr, so that

~=o-f[1 +,(.)]a~,

(28)

8f

I I I i

(d) ;

8f (e)

I I

j

l

J

2~k

l~

2~(k+l)

Fig. 2. The potential-like fraction VI of eq. (30) as a function of time "r* for arbitrary value of the variables (a) and for ~'* as an argument (b) in case of one cavity. The difference (c) between these two functions is Vf. The derivatives of f with respect to the variables are shown in (d) and (e).

C.J.A. Corsten, H.L. Hagedoorn / Motions in circular accelerators

I I I I

1

I

1

vI~,~+~'I

(a)

Subsequently, the function f ( ~ ) which is periodic in ~-* can be represented by a Fourier series:

F V

V I (~()

(b)

f(~)=

_

~

~×

v. f c0 * ~ d

(c)

of

(d)

3f

(e)

,in

I

A

43

I I

I ae

~ Ap(~)cosp,*+Bp(~)sinp'r*. p>_0

(35)

After the calculation of these Fourier components and substitution into eq. (34) the acceleration term - and thus the total Hamiltonian - is expanded into the canonical variables. In principle this treatment enables us to examine the influence of any Dee or cavity configuration. As an illustration we give the Fourier components of eq. (35) for the two cases sketched in figs. 2 and 3: only one cavity in the ring -

A°

2~r '

I

2 "~k

2kT~+ 2 "IT.

2r~k* ~

1 sinp~

Ap

~r

(36)

p

Fig. 3. Similar to fig. 2 but now for a one-Dee system or two equally spaced cavities with 0 ° phase difference between the electric fields.

Be

cavity or gap crossing [see also eq. (14c)]

the one-Dee system or two equally spaced cavities with a 0 ° phase difference between the electric fields

1 (l-cosp

)

P

~

,

p>~l;

-

el; eV

A0=0,

(1 - 1/7~2) We '

(32) ~(1Ap = - --

with /2 the peak voltage in the Dee or in the cavity. The influence of the rf acceleration on the radial and longitudinal motion is now given by

x = e V- ~ Of cos h'r*, ap,

and /~o = - e I ? 3~c°s h~'*

(33)

a#

where the dot above a variable stands for d / d r and 3f/3~:, and 3f/3~ both consist of delta pulses as illustrated in figs• 2 and 3. The next step in the discussion of the influence of the acceleration on the orbit motion is the examination of the function f. The argument of this function f ( ~ + flU(1 - n ) consists of a slowly variable ~ and of the fast oscillating variable/~:,. This means that the width of the pulses of f may vary rapidly over one "period" of T*. The variable /~x is small compared to the usually assumed values of 0 and in order to expand this function f into the canonical variables t) and /~., we write f as a Taylor series: f /}+ 1

n

=f(t})+

~

3~

1 ,6.~ 32f(/} ) + - - - + ..-. 2 (1 - n ) 2 O~ 2

(34)

Be=I{l_

( - 1 ) e } Smpp0 ,

(_l)p)

(37)

(1 - c o s p ~ ) P

p>_l.

Substitution of the Taylor series (34) and the Fourier series (35) into eq. (31) generally yields slowly and fast oscillating terms. In fact, the fast oscillating terms must be transformed to higher degree. We will not carry out this procedure now, assuming they do not give any significant contributions (a procedure to eliminate fast oscillating terms is demonstrated extensively for the one-dimensional betatron motion in e.g. ref. 8). Keeping the relevant terms, the Hamiltonian becomes [see eqs. (19) and (31)]: H='2p~,2+½( 1 - n ) x 2 + ½ p ~ (

lyoz

l-l)n

b

+ b O - ( 8 - y~ )Po + ½eVA,~( O) 1 eV

+2~P~-

OAh + 1

eV

20ZAh

4 (l-,,)2P' oo ~ + . . - ( 3 8 )

For convenience we omitted the vertical motion, the third degree terms of eq. (19) and double tildes ( = ) above the variables. The term p,. 3Ah/O0 has the effect of producing a displacement of the equilibrium orbit. This displaced orbit is sometimes called the "accel-

C.J.A. Corsten, H.L. Hagedoorn / Motions in circular accelerators

44

crated equilibrium orbit" abbreviated by AEO [2]. The motion with respect to the AEO is described by the introduction of new variables, via the transformation G = x ~ + Op~

eV 2(1-n)

~A h x 30'

with eV -2(1- - n )

X'=X,

fix=px+

0=0,

eV Po=Pe'4 2(1-n)

OAh

a0' (39) 32Ah x 302 '

and no first degree terms in x, Px appear in the Hamiltonian: tq='-z+½(l--n)x2+

+b°-(~-~, 1

+4

eV (l-n)

v~ ]]Po

mation of eq. (39)]. For an arbitrary particle the outward shift has an additional component depending on p~. This component is represented by the last term in eq. (40) and results in a change of the radial oscillation frequency due to the acceleration process. This radiallongitudinal coupling effect will be discussed more closely in the following article [5].

y~

1--n

+ ½ e -V A h ( O-)

~o= -½eV3A-_ h ~0

1

yo2

= 0

and

y02

1 -

'

(42)

n

where /~e represents the relative deviation in kinetic m o m e n t u m from a central particle known as the synchronous particle. For the time being we restrict ourselves to the case of a one-Dee system and the new Hamiltonian becomes: 1 1-n

one cavity,

)eV ~"

sinh0 h

b

one-Dee system.

These relations can also be directly obtained from eq. (33) by substitution of the Fourier series of a delta pulse at r* = 2~rk - 0, see fig. 2 and additionally at "r* = (2k + 1)~r - ~ in case of the one-Dee system, see fig. 3. Eq. (41) therefore fits into the physical idea we have of acceleration. No effective acceleration occurs in the case of the one-Dee system (or in case of two equally spaced cavities with 0 ° phase difference between the electric fields) if the harmonic number h is even because the particle then alternatingly accelerated and decelerated. In case of two equally spaced cavities with a 180 ° phase difference between the electric fields, h must be even. The radial oscillation are performed about an equilibrium orbit which shifts outwards in position at each gapcrossing [see third line in eq. (38) and the transfor-

6 - b/yoz 1 1

/~0 =/3s

(41) -~(1 - ( - 1)h)cos h0,

1- n

o2( 1 /2/= ½p 702

=

o,

_ b/v0 ~

1

(40)

in which we neglected terms such as ( e V ) z, BeD, b e d and beVbecause of their assumed smallness. In fact the coefficient eV defined in eq. (32) is not small in the central region of cyclotrons since these machines accelerate ions, starting (in most cases) from nearly zero energy. The increase of momentum (or energy) due to the gap crossing is represented by the term in the second line of eq. (40). This increase is (ff =p~ = 0):

eDt

In this section the uncoupled longitudinal motion which is described by the Hamiltonian of eq. (40) with x =Px = 0 will be discussed and we will end with a general expression for the central position (CP) phase. A few manipulations with the Hamiltonian lead to the classical Hamiltonian for the description of the synchrotron oscillations. The first degree term in f0 is removed by a transformation of the form:

G=Op°+

~2 32A~ 2 x 3t~2 + ' " ,

~ - cos h O,

6. Synchrotron oscillations and the central position phase

+

1

1- n 1

y02

l-n

t~.

(43)

It is customary to define the phase in such a way that the rate of change of this phase is negative for particles rotating faster than the synchronous particle; therefore we change the sign of O. Returning to real time units t = r / ~ o o and to the generalized momentum A W / ~ o o = RPol) o and Po = eBop [see eq. (20)], the Hamiltonian must be changed to H = -~ooPoR121, resulting in H

1 ~o2Ko ( A W l 2 2 W o \ wo ]

Wo ~

,%K0

1 do~ '~0 d t

el/ sin ht~ ~r h a

d~ )

B-0 ~ -

(44) '

C.J.A. Corsten, H.L. Hagedoorn / Motions in circular accelerators

simplify the notation we omit the vertical motion and fringing fields and find: (see for the vector potential ref. 7)

with K0

- 1/~g 1-

l/Y0 z

and

a -

l 1 - n

,

where a is the m o m e n t u m compaction factor. This Hamiltonian is similar to the one derived in the classical way. However, one has to take notice that the phase t7 must be regarded as the CP phase instead of the normally used rf phase. The definition of 0 differs somewhat from that of the CP phase as introduced by Schulte and Hagedoorn [2], but the basic ideas are essentially the same. We recall that in ref. 2 the motion of accelerated particles in cyclotrons with Q~ = 1 (i.e. n = 0) has been studied. We define q~cP = - h O ,

(45)

and the relation between the CP phase and the rf phase is [compare with eq. (18)]

hPx hb + - ~CP = (~HF q'- 1 - - n (1--n) z'

45

(46)

with/3~ = P J P o . The existence of this formula, except for the factor b, has been reported recently by Gordon who derived it in a different way, not using the Hamilton formalism [3]. Thus, when dealing with coupling between the radial and the longitudinal motion, it is useful to extend the definition of the phase. Since the extra terms in eq. (46) are proportional to h, the correction will be most pronounced for high harmonic number and further in the central regions of cyclotrons P:,/Po may have a significant value. In principle this definition of the CP phase is only correct for circular equilibrium orbits. A more general expression will be derived in the following section which deals with the motion in an A.G. magnetic field structure.

7. A l t e r n a t i n g g r a d i e n t m a g n e t i c f i e l d s t r u c t u r e

In this section we extend the preceding theory for the motion in an A.G. synchrotron or storage ring with a separated function lattice. The treatment is fundamentally the same as before and therefore only the most important stages are quoted. Finally we will obtain a Hamiltonian - similar to eq. (38) - which will turn out to be an appropriate start to study coupling effects. We will omit the time-dependence of the magnetic field and of the frequency of the accelerating voltage, i.e. b(~-) = 0 and 8(~-)= 0, as this has no essential consequences for the theory. As the coupling term /~eY is most important we evaluate the Hamiltonian up to this quadratic term. To

i~- _i -_~ p1~2 ; + ~1 t1", 2 ( ~ ) - n ( g ) ] ~

l

+ 2yg/3~ + / ~ - c ( f f ) / ~ x .

(47)

The scaled time r is again the independent variable and and n represent the "normalized" dipole and quadrupole field components defined as e =Rp

and

R 2 (oBz]

n=-B~0p

-ff~-x}0"

(48)

These components depend on the azimuthal position in the machine, i.e. on the coordinate 0 = s / R whi_ch is, after the various transformations, written and 0 (see section 3). The elimination of the term ~(ff)/50.~ will again lead to a new longitudinal coordinate which includes the horizontal motion, similar to eq. (18). The elimination is achieved by a transformation generated by the function

+ ~'(~)/~0x- ½~l(ff)~'(~)fi 2 ,

(49)

with ' = d / d f f , and the relation between the old ( ~ ) and the new (') variables now becomes x=~+np~,

L = L + n'P.,

( . ) )po, and the Hamiltonian up to the second degree in the variables becomes

=~ex+½(,~-,)~+~_¢~ = - ~ n +¢o,

(0' )

(501

with ~ the so-called off-momentum or dispersion function, which is the reduced displacement of the closed orbit per unity momentum deviation. Due to the use of reduced coordinates [see eq. (14a)] the T-function is also a reduced dispersion function, i.e. the usual dispersion function divided by R. The ~-function is defined as that solution of the differential equation

<' + ( , 2 - , , ) n =,,

(51)

which has the same periodicity as the linear magnetic guide field. We recall that in case of the cylindrical-symmetric guide field e = 1 and n = constant so that ~ = 1/1 - n and eq. (49) reduced to eq. (18). Subsequently the acceleration must be included. This is achieved by following the procedure as outlined in section 5, resulting in a Hamiltonian which is in first

46

C.J.A. Corsten, H.L. Hagedoorn / Motions in circular accelerators

order approximation given by __1_2}

IZ

H-2p x

2

~,

,t~ - n ) x

where x a n d / 5

+ ~ p o2[| ~ 12 - ~ ]

2 !

•

~ Yo

canonical variable to describe coupling effects between the transverse a n d longitudinal motions in an A.G. machine. The h a r m o n i c n u m b e r h may be very large (e.g. h = 100) but x a n d / ~ are usually very small.

/

+ ½eCAh(O ) + eV((~cp~ - W'~x) \

×

[/ ~o-A OP cos p ' r cos h'r + ~-OBP sin P ~"cos h • ,[t I /

+~e~

x

are the reduced variables of eq. (14a):

x = x / R and/~-~ = px/Po. This very CP phase is a proper

I

8. Conclusion

(n~-n~)

a ~ 5- c o s p ~ c o s h • + ~

s m p r cos h •

(52) The variables are defined by eq. (28) but for convenience we omitted the marks above the variables. This H a m i l t o n i a n is the analogue of eq. (38). The brackets ( ) indicate that we are only interested in the resonant or slowly varying terms. 7/c and 7t~, are the values of respectively the o f f - m o m e n t u m function and its derivative at the position of the cavities. We notice that the introduction of an " A E O " by a transformation similar to eq. (39) has no direct consequences for our further discussion and therefore we will omit the t r a n s f o r m a t i o n [compare eq. (40) with eq. (38)]. As the horizontal tune Q~ is usually not too close to an integer in an A.G. synchrotron, the coupling term between the first brackets ~ ) in eq. (52) may give resonant terms only in case the oscillations in 0 are not too slow, i.e. the synchrotron n u m b e r Q~ is not too small. F r o m the fourth and fifth lines in eq. (52) we will only m e n t i o n the term with p = h, leading to a change in the radial tune, similar to that mentioned in section 5. In the following article [5] we will return to the r a d i a l - l o n g i t u d i n a l coupling due to a non-zero value of the o f f - m o m e n t u m function and its derivative in the cavity. Then we will also discuss the synchro-betatron resonance Q~ - Q: + Q, = 0 which might be relevant in, e.g., electron storage rings which often operate at nearly equal tunes Q~ a n d Q~. It will be shown that this coupling resonance is not excited in an " i d e a l " machine but it can be excited by a so-called skew or rotated quadrupole field. Finally we recall that an equivalent CP phase can be defined for the motion in an A.G. accelerator with a separated function lattice. Analogously to eqs. (45) and (46) we get: ~cP = 4~.v + h ~

- h~' x ,

(53)

In order to develop a simultaneous treatment of the b e t a t r o n and synchrotron motions in circular accelerators the rf accelerating electric field and a time-dependent magnetic field have been incorporated into the general Hamilton formalism via a time-dependent vector potential consisting of a fast and slowly varying part. In the H a m i h o n i a n we used curvilinear coordinates as canonical variables and the time as the indep e n d e n t variable. A " c e n t r a l position" phase - instead of the well-known rf phase turns out to be a proper canonical variable for the description of tile synchrotron motion. The difference between the CP phase and the rf phase is most p r o n o u n c e d in central regions and in case of high harmonic acceleration. The results correspond with results derived by Schulte and Hagedoorn [2] in the case of cyclotrons with Q~ -~ 1 a n d with results derived by G o r d o n [3], b o t h with different treatments. The definition of the CP phase is extended to the case of motion in a time-dependent magnetic field and an alternating gradient structure. The following article contains some applications of this theory [5],

References [1]

G. Guignard, CERN 76-06 ISR/1976 and CERN 78-

11/1978. [2] W.M. Schuhe and H.L. Hagedoorn, Nucl. Instr. and Meth. 171 (1980) 409. [3] M.M. Gordon and F. Marti, Part. Accel. 12 (1982) 13. {4] T. Suzuki, KEK Report 82-10 (July 1982) Japan. [5] C.J.A. Corsten and H.L. Hagedoorn, Nucl. Instr. and Meth. 212 (1983) 47. [6] C.J.A, Corsten, Internal report Eindhoven University of Technology V D F / N K 82-08 (September/October 1981). [7] C.J.A. Corsten, Thesis, Eindhoven University of Technology (September 1982). [8] H.L. Hagedoorn and N.F. Verster, Nucl. Instr. and Meth. 18/19 (1962) 201. [91 F.E. Mills, Nucl. Instr. and Meth. 23 (1963) 197.