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Space charge effects in a heavy ion cyclotron
Nuclear Instruments and Methods in Physics Research 219 (1984) 279-283 North-Holland, Amsterdam
279
SPACE CHARGE EFFECTS IN A HEAVY ION CYCLOTRON C. C H A S M A N a n d A.J. B A L T Z Brookhaven National Laboratory, Upton, New York 11973, USA
Received 14 July 1983
The effect of space charge forces in a heavy ion cyclotron is examined. A simplified model which allows the coupled longitudinal and radial motions to be treated together is used. Numerical calculations, an approximation method that provides physical insight into the motion, and an exact solution are obtained. An upper bound on the dynamic excursion of the beam is obtained analytically.
1. Introduction Recently the proposal for a cyclotron heavy-ion booster for the BNL three-stage tandem facility has been extended to include the possibility of injection of heavy-ion beams into the AGS and the proposed CBA [1]. Unusual though it is to consider cyclotron injection into a synchrotron, the project appears feasible because of the very good emittance properties of the cyclotron with in turn are a result of the exceptionally small emittance of the ion source and the preservation of the emittance after acceieration by the tandem: The proposed cyclotron booster is a heavy-ion isochronous ~yclotron .which increases the energy of the injected tandem bea.ms by about a factor of 30, for example to about 70 M e V / a m u for iodine. Recent achievements in negative ion sputter sources imply that the tandems can provide -1012 pps of 2 M e V / a m u iodine in charge state 31 for injection into the cyclotron. Since the beam is bunched into - 1 / 2 ns pulses, the charge density in the cyclotron is quite high, ~ 2 × 10~l/cm 3, and it is important to consider the effect of the space charge forces in the cyclotron accelerated beam. The effects of space charge in particle beams have been considered in great detail for large synchrotrons but to a much lesser extent for smaller cyclotrons [2,3]. The essential differences between the effects in a synchrotron and cyclotron are that the cyclotron has a small beam dimension in the azimuthal direction and a lower energy. The latter difference implies that the electromagnetic forces are almost entirely internal Coulomb forces in a cyclotron, with image and internal magnetic forces negligible. The small azimuthal dimension means that the continuous charge ribbon used to analyze the synchrotron space charge is not appropriate, and does not present a realistic evaluation of the motion. 0167-5087/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
In order to estimate the space charge effect we consider the simplified problem of a test charge on the periphery of a charged sphere 0.2 cm in diameter expanding under Coulomb forces and contained by a uniform magnetic cyclotron field. By doing this we ignore the well known effect that the space charge has in reducing the radial focusing frequency; we concentrate not on the particle's focusing but rather on central particle's excursion in the radial and transverse directions [4]. In particular, we consider the radial and logitudinal motion, assuming it to be decoupled from the vertical motion. We intentionally overestimate the force on the test charge by further assuming that the whole sphere of charge can be reduced to a point which the test charge can never penetrate. In this way we hope to get an upper limit on the radial and transverse beam size. 2. Equations of motion The nonrelativistic equations of motion of the test charge may be immediately written for the inertial Cartesian frame as 5i + toy = ( e 2 Q q / m p 3 ) x , - to~ = ( e 2 Q q / m p a ) y ,
(1)
where x and y are rectangular coordinates relative to the center of motion of the charged sphere in the plane of its orbit perpendicular to the cyclotron field, to is the cyclotron frequency, q B e / m c , and the internal Coulomb force per unit mass e e Q q / m p 3 is given in terms of the radial separation, p, of the test charge qe and the center of the charged sphere (charge Qe). This set of equations has been solved numerically on a computer for cyclotron parameters suitable for the iodine beam. The numerical technique is simple and straightforward [5], and we will use sample computations for comparison with the following analytical results.
('. (?hasman. A.J. Baltz / Space charge effects
280
3. Approximate method We now have a second order equation in u and a first order equation in q~ both of which have an inconvenient term in 1/u 2. In order to gain some insight into the nature of the motion due to the space charge effect, a linear approximation has been made for the difficult terms in 1/u 2. We set u = 1 + e where e is now the ratio of the increase in separation to the initial separation. After substitution into the two equations we set
f.q//~/
¢
XI
l/u 2 = 1 -- ae,
(6)
where a is left as a free parameter to be adjusted to take in account higher order corrections to our linearization, but which takes on the value 2 in the small e linear case. The linearized equations then become
Fig. 1. Rotating coordinate system (x ', y ') or (p, q~) utilized for description of the motion of test charge q.
g - (42 + 4~0)(1 + a ) - 122(1 - as) = 0
(7a)
~-(~0+2)(1
(7b)
If we substitute the second equation into the first and again keep only linear terms, then analytical solutions are readily obtainable for the pair: e=
Alternatively the equations can be written in a more suitable form for exact solution and for later approximation by transforming to rotating polar coordinates in a system rotating with the beam (fig. 1). We define the dimensionless quantity, u, in terms of the initial separation between the test charge and the center of the charged sphere u = P/P0
122 = e2Qq/mo3.
q~= [ ~ ' 0 -
u~ + 2/~,~ +/tw = 0.
(4)
Inspection of these equations makes two properties of the solutions already apparent. First, the equations have rotational symmetry about the center of the charged sphere and thus longitudinal and transverse motion are inherently coupled. Secondly, there is no dependence on the energy of the particle bunch comprising the charged sphere, or equivalently no dependence on the radius of the sphere's motion in the cyclotron for a given frequency. As stated earlier, for the purpose of this calculation we assume a nonrelativistic cyclotron with uniform magnetic field throughout. The second of the equations (2) can be integrated once immediately to give
1
/ 24° + °:
(8a)
12'121
j'
( 2£ko2+~° ]( ~7, ]2 a ] ~ k ] ~-7 s i n k ' t
+~
[ 2 q'0 + ~0 ) ~0 cos k't -a t ~---,
k'
k'
+e0°'
(8b)
where k ' = ~q,0(4o + ~0)(2a - 1) + (~02/2 + 122) a
(8c)
12, = ~ 2 + +2° + 4,0~o.
(8d)
(3)
The equations of motion then become
6_(6o+~,
(~') '0 ~7 ( 1 - c o s k ' t ) + ~ ; sink't
(2)
and a new frequency, 12, in terms of th,. ~oulomb force strength and the initial separation, O0:
-ae) +2=0.
,~
(5)
If the initial angular velocity, '~0, and the initial radial velocity, /~o, are zero for the test charge in the rotating frame, then these equations take on a simpler form 12
e=(~)
2
(l-coskt)
aw[12~2[
(9a) 1 sinkt
]
l
(9b)
with k = ~a(to2/2 + t22) = t o v / ~ l / 2 + 122/,o2). Solutions of the exact equations done by computer can be transformed to this rotating coordinate system and compared with our approximate analytical results for a physical case. We have set the cyclotron frequency ¢o to 45 x 106 Hz and the frequency 12 arising from the Coulomb force to 28.64 × 106 Hz to correspond to a
281
C. Chasman, A.J. Baltz / Space charge effects
I y’
Y’
0.
i
Fig. 2. Motion of the test charge in the rotating coordinate system for initial conditions specified in the text. Equations of motion were integrated numerically. bunched iodine beam required for AGS currents of the order of 10” pps. The parameter that measures the strength of the effect of space charge L?/o = 0.636. Fig. 2 shows the motion of the test charge begun at rest in the rotating frame as computed numerically. It is clear that the effect of the space charge is contained within acceptable limits. The beam does not blow up indefinitely: the test charge is constrained to execute a clover leaf or daisy petal oscillation about the center of the charged sphere. The point at which the bunch has made one cyclotron revolution is marked 2n; two revolutions, 4n, etc. For our parameters the radial separation increased from 0.1 cm to a maximum of 0.175 cm for the computer calculation. This must be considered an overestimate due to our approximation that the test charge always lies outside the main body of the bunch. The beam quality remains suitable for injection into the AGS. Variation of the initial conditions leads to different oscillation patterns [as might be predicted by eq. (8)] but the maximum excursion is still contained at acceptable size. Exploratory computer calculations for larger values of the total charge Q led to the beam size being con-
Fig. 3. As in fig. 2 except that linearlized analytical solutions were utilized with a = 1 (solid line) and a = 1.2 (darhed line).
tained at a larger maximum radius of oscillation: 2Q led to Pm, = 0.23; 1OQ led to Pmax = 0.53. Fig. 3 shows the approximate analytical results for the same initial conditions computed from eqs. (9a) and (9b) with a = 1 and a = 1.2. The a = 1 agrees in phase with the exact result while the maximum expansion is 0.189 vs 0.175 exactly; the a = 1.2 has the right maximum expansion while the phase is slightly shifted. Our approximate analytical solution clearly describes the nature of the physical solution. Fig. 4 shows the comparison of the radial term with its expansion to first order with a = 2, a = 1, and a = 1.2. It is clear that a = 1 or 1.2 is a more reasonable choice to represent l/u2 in this case. To get a further insight into the nature of the curve shown in fig. 3 we may eliminate time, t, from eqs. (9a) and (9b) to obtain the transcendental equation relating E and C#B
(10) Clearly E is periodic
in $I.
282
C
Chasm@n,d.J. Bahz / Space charge effects
1.0
and k
d u )2 =
(~ I
-a=l.2
(itu)~
4w
260/,0+
/
1 -
(14)
u2
Eqs. (13) and (14) can be used to solve analytically for u as a function of t and ~ as a function of u in terms of elliptic integrals. However, it is known that the solutions of these equations have certain well defined properties: in particular they are bounded and periodic. Of mo~t importance is the upper bound on the value of u. The condition comes from the fact that the square root of the left hand side must be real. Consequently the roots of the quartic on the right side of eq. (13) can be written as: O92
(/~U) 2 = 4-
[ ( @ l - - U ) ( U - - @ 2 ) ( /~ -- 0 / 3 ) ( U -- 0 / 4 ) ] ,
@1 ~" @2 ~> @3 ~ 0/4
and u is bound, @1> u > a 2. To illustrate this b o u n d consider the case for the initial conditions q'0 =/~0 = 0. Then u = 1 is a root: O92
(u/~)2 = ~ - ( a , - u ) ( u - 1 ) ( u - a 3 ) ( u - 0 / 4 ) , 0
L 0.2
I 0.4
"~
where al, 0/3, a4 are roots of the equation:
I _ - - ~ 0.6 0.8
u 3 + u 2 - (1 + 8122/o92)u- 1 = 0,
Fig. 4. Comparison of 1 / u 2 and 1 - au for expansion parameters a = 2, 1.2, 1.
Another first integral of the motion [in addition to eq. (5)] can be readily obtained by observing that the Hamiltonian of the system in the rotating coordinate system is given by:
ylZp3om/p,
8$22u O92
(-~
+ 1) 2}
fu
udu - u)(.
- @3)(u - @,)],/2
= 2~ {a F( O, k ) ---~II(
(12)~
H is a constant of the motion and eqs. (5) and (11) can be used to separate the variables in eq. (4). We obtain/~ and + as a function of u and time; du/dq~ as a function of u. The result of this procedure leads to +1
+ 2.5225),
(11)
Po=2[p2Og(2-2--~-:+l)+pRwsinq~].
-u4+u "~
u)(u + 0.2267)(u
which gives 1 _< Um~, _< 1.749 in agreement with the computer analysis. In fact eq. (13) can be integrated [6] in terms of elliptical integrals F and H to yield:
[(@1 - u ) ( 1
which is independent of time. Of course, the canonical m o m e n t a are not given by mia, mO21k; instead: m Pp = - ~ - [ 2 f f - Roa cos q~],
(/~u)2 = ~ -
which must have three real roots. For our example 122/o92 = (0.636) 2 and % = h 0 = 0. Then ( u u ) 2 = ( u - 1)(1.7489 -
4. E x a c t a n a l y t i c s o l u t i o n
H = 2 [ Ib2 + ( p + ) z ] +
(15)
+1+
(16)
where
F(O,k)=foO t
dq,
-,
1 - k 2 sin2+
//(~b, m, k ) = f 0 °
d~b d + m sinZ~k~l - k 2 sin/~k
with
a=a3(0/1-1);
d=b=@3-0/1;
c=oq-
1:
3 = ( a , - 1 ) ( a I - a3)(1 - 0/3)
+
= (13)
d , k ) } = (og/2)t,
u
0/3)(1 _ @ , ) , a sin20 + b c sin20 + d '
k2= ( @,@'---kl @3ii J\ @3-0/,1-@,11
C Chasrnan,A.J. Baltz / Spacechargeeffects The period of u is given by four times the complete elliptic integrals given in eq. (16). Likewise eq. (14) can be integrated [6] to yield:
eP-q'°=28 ,~cF(O'k)+~H( O' ~ , k ) } 26 a +T(TF(O,k)
+--~II(O, -~,k)}. c
Once again u is periodic in ~ with the period given by the complete elliptic integrals. For our case with ( ~ / w ) 2 = (0.636)2 the period of u with q~ is 84.59 ° and the period of u with t is 1.094 ( 2 ~ / ~ ) in agreement with the computer results.
5. Longitudinal motion The longitudinal motion is directly coupled to the radial motion as shown in eq. (4). The beam consequently will not be isochronous but after one turn will cross an inertial axis at a phase ~A t separated from the isochronous phase. To lowest order in the strength of the Coulomb force it can be shown that ~0At-Pocr(fl/to)2/(R+Po). Four our parameters toAt10 -3 rad. at the injection radius (R = 30 cm) and diminishes approximately linearly with increasing radius. Since the phase shift diminishes with radius the increase in the longitudinal energy spread due to space charge is negligible for our parameters.
6. Summary The two dimensional problem of a charge rotating in a uniform magnetic field under the influence of Coulomb
283
forces has been solved analytically by quadrature, by computer and by approximation methods. The maximum radial displacement due to space charge and the period of oscillation are independent of the radius of the orbit. The linear approximation method yields resuits of modest precision but gives a simple picture for the physical nature of the motion; computer calculations demonstrate the region of its accuracy. The analytic solution is complex but the bounds on the radial motion can be obtained simply by the solution to a cubic or quartic equation. Furthermore, we also obtain the period of the oscillation from the (tabulated) complete elliptic integrals of the first and third kinds. This work was supported by the Division of Basic Energy Sciences, U.S. Department of Energy, under Contract No. DE-AC02-76CH00016.
References [1] C. Chasman (ed.), Proposal for a 15 A.GeV Heavy Ion Facility at Brookhaven, Informal Report BNL-32250, (January 1983). [2] L.J. Laslett, BNL-7534, 1963. [3] Henri Brock, Accelerateurs Circulaires de Particules (Presses Universitaires de France, 1966) and refs. therein. [4] A. Chabert, T.T. Luong and M. Prome, IEEE Trans. Nucl. Sci. NS-22 (1975) 3. This paper describes a beam orbit computer code which includes space charge effects used in studies for the GANIL cyclotron. [5] R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1964), Vol. I, p. 9-7. [6] Wolfgang Grobner and Nikolaus Hofreiter, Integraltafel, Erster Teil Unbestimmte Integrale, Wien, (Springer, Berlin, 1961).
279
SPACE CHARGE EFFECTS IN A HEAVY ION CYCLOTRON C. C H A S M A N a n d A.J. B A L T Z Brookhaven National Laboratory, Upton, New York 11973, USA
Received 14 July 1983
The effect of space charge forces in a heavy ion cyclotron is examined. A simplified model which allows the coupled longitudinal and radial motions to be treated together is used. Numerical calculations, an approximation method that provides physical insight into the motion, and an exact solution are obtained. An upper bound on the dynamic excursion of the beam is obtained analytically.
1. Introduction Recently the proposal for a cyclotron heavy-ion booster for the BNL three-stage tandem facility has been extended to include the possibility of injection of heavy-ion beams into the AGS and the proposed CBA [1]. Unusual though it is to consider cyclotron injection into a synchrotron, the project appears feasible because of the very good emittance properties of the cyclotron with in turn are a result of the exceptionally small emittance of the ion source and the preservation of the emittance after acceieration by the tandem: The proposed cyclotron booster is a heavy-ion isochronous ~yclotron .which increases the energy of the injected tandem bea.ms by about a factor of 30, for example to about 70 M e V / a m u for iodine. Recent achievements in negative ion sputter sources imply that the tandems can provide -1012 pps of 2 M e V / a m u iodine in charge state 31 for injection into the cyclotron. Since the beam is bunched into - 1 / 2 ns pulses, the charge density in the cyclotron is quite high, ~ 2 × 10~l/cm 3, and it is important to consider the effect of the space charge forces in the cyclotron accelerated beam. The effects of space charge in particle beams have been considered in great detail for large synchrotrons but to a much lesser extent for smaller cyclotrons [2,3]. The essential differences between the effects in a synchrotron and cyclotron are that the cyclotron has a small beam dimension in the azimuthal direction and a lower energy. The latter difference implies that the electromagnetic forces are almost entirely internal Coulomb forces in a cyclotron, with image and internal magnetic forces negligible. The small azimuthal dimension means that the continuous charge ribbon used to analyze the synchrotron space charge is not appropriate, and does not present a realistic evaluation of the motion. 0167-5087/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
In order to estimate the space charge effect we consider the simplified problem of a test charge on the periphery of a charged sphere 0.2 cm in diameter expanding under Coulomb forces and contained by a uniform magnetic cyclotron field. By doing this we ignore the well known effect that the space charge has in reducing the radial focusing frequency; we concentrate not on the particle's focusing but rather on central particle's excursion in the radial and transverse directions [4]. In particular, we consider the radial and logitudinal motion, assuming it to be decoupled from the vertical motion. We intentionally overestimate the force on the test charge by further assuming that the whole sphere of charge can be reduced to a point which the test charge can never penetrate. In this way we hope to get an upper limit on the radial and transverse beam size. 2. Equations of motion The nonrelativistic equations of motion of the test charge may be immediately written for the inertial Cartesian frame as 5i + toy = ( e 2 Q q / m p 3 ) x , - to~ = ( e 2 Q q / m p a ) y ,
(1)
where x and y are rectangular coordinates relative to the center of motion of the charged sphere in the plane of its orbit perpendicular to the cyclotron field, to is the cyclotron frequency, q B e / m c , and the internal Coulomb force per unit mass e e Q q / m p 3 is given in terms of the radial separation, p, of the test charge qe and the center of the charged sphere (charge Qe). This set of equations has been solved numerically on a computer for cyclotron parameters suitable for the iodine beam. The numerical technique is simple and straightforward [5], and we will use sample computations for comparison with the following analytical results.
('. (?hasman. A.J. Baltz / Space charge effects
280
3. Approximate method We now have a second order equation in u and a first order equation in q~ both of which have an inconvenient term in 1/u 2. In order to gain some insight into the nature of the motion due to the space charge effect, a linear approximation has been made for the difficult terms in 1/u 2. We set u = 1 + e where e is now the ratio of the increase in separation to the initial separation. After substitution into the two equations we set
f.q//~/
¢
XI
l/u 2 = 1 -- ae,
(6)
where a is left as a free parameter to be adjusted to take in account higher order corrections to our linearization, but which takes on the value 2 in the small e linear case. The linearized equations then become
Fig. 1. Rotating coordinate system (x ', y ') or (p, q~) utilized for description of the motion of test charge q.
g - (42 + 4~0)(1 + a ) - 122(1 - as) = 0
(7a)
~-(~0+2)(1
(7b)
If we substitute the second equation into the first and again keep only linear terms, then analytical solutions are readily obtainable for the pair: e=
Alternatively the equations can be written in a more suitable form for exact solution and for later approximation by transforming to rotating polar coordinates in a system rotating with the beam (fig. 1). We define the dimensionless quantity, u, in terms of the initial separation between the test charge and the center of the charged sphere u = P/P0
122 = e2Qq/mo3.
q~= [ ~ ' 0 -
u~ + 2/~,~ +/tw = 0.
(4)
Inspection of these equations makes two properties of the solutions already apparent. First, the equations have rotational symmetry about the center of the charged sphere and thus longitudinal and transverse motion are inherently coupled. Secondly, there is no dependence on the energy of the particle bunch comprising the charged sphere, or equivalently no dependence on the radius of the sphere's motion in the cyclotron for a given frequency. As stated earlier, for the purpose of this calculation we assume a nonrelativistic cyclotron with uniform magnetic field throughout. The second of the equations (2) can be integrated once immediately to give
1
/ 24° + °:
(8a)
12'121
j'
( 2£ko2+~° ]( ~7, ]2 a ] ~ k ] ~-7 s i n k ' t
+~
[ 2 q'0 + ~0 ) ~0 cos k't -a t ~---,
k'
k'
+e0°'
(8b)
where k ' = ~q,0(4o + ~0)(2a - 1) + (~02/2 + 122) a
(8c)
12, = ~ 2 + +2° + 4,0~o.
(8d)
(3)
The equations of motion then become
6_(6o+~,
(~') '0 ~7 ( 1 - c o s k ' t ) + ~ ; sink't
(2)
and a new frequency, 12, in terms of th,. ~oulomb force strength and the initial separation, O0:
-ae) +2=0.
,~
(5)
If the initial angular velocity, '~0, and the initial radial velocity, /~o, are zero for the test charge in the rotating frame, then these equations take on a simpler form 12
e=(~)
2
(l-coskt)
aw[12~2[
(9a) 1 sinkt
]
l
(9b)
with k = ~a(to2/2 + t22) = t o v / ~ l / 2 + 122/,o2). Solutions of the exact equations done by computer can be transformed to this rotating coordinate system and compared with our approximate analytical results for a physical case. We have set the cyclotron frequency ¢o to 45 x 106 Hz and the frequency 12 arising from the Coulomb force to 28.64 × 106 Hz to correspond to a
281
C. Chasman, A.J. Baltz / Space charge effects
I y’
Y’
0.
i
Fig. 2. Motion of the test charge in the rotating coordinate system for initial conditions specified in the text. Equations of motion were integrated numerically. bunched iodine beam required for AGS currents of the order of 10” pps. The parameter that measures the strength of the effect of space charge L?/o = 0.636. Fig. 2 shows the motion of the test charge begun at rest in the rotating frame as computed numerically. It is clear that the effect of the space charge is contained within acceptable limits. The beam does not blow up indefinitely: the test charge is constrained to execute a clover leaf or daisy petal oscillation about the center of the charged sphere. The point at which the bunch has made one cyclotron revolution is marked 2n; two revolutions, 4n, etc. For our parameters the radial separation increased from 0.1 cm to a maximum of 0.175 cm for the computer calculation. This must be considered an overestimate due to our approximation that the test charge always lies outside the main body of the bunch. The beam quality remains suitable for injection into the AGS. Variation of the initial conditions leads to different oscillation patterns [as might be predicted by eq. (8)] but the maximum excursion is still contained at acceptable size. Exploratory computer calculations for larger values of the total charge Q led to the beam size being con-
Fig. 3. As in fig. 2 except that linearlized analytical solutions were utilized with a = 1 (solid line) and a = 1.2 (darhed line).
tained at a larger maximum radius of oscillation: 2Q led to Pm, = 0.23; 1OQ led to Pmax = 0.53. Fig. 3 shows the approximate analytical results for the same initial conditions computed from eqs. (9a) and (9b) with a = 1 and a = 1.2. The a = 1 agrees in phase with the exact result while the maximum expansion is 0.189 vs 0.175 exactly; the a = 1.2 has the right maximum expansion while the phase is slightly shifted. Our approximate analytical solution clearly describes the nature of the physical solution. Fig. 4 shows the comparison of the radial term with its expansion to first order with a = 2, a = 1, and a = 1.2. It is clear that a = 1 or 1.2 is a more reasonable choice to represent l/u2 in this case. To get a further insight into the nature of the curve shown in fig. 3 we may eliminate time, t, from eqs. (9a) and (9b) to obtain the transcendental equation relating E and C#B
(10) Clearly E is periodic
in $I.
282
C
Chasm@n,d.J. Bahz / Space charge effects
1.0
and k
d u )2 =
(~ I
-a=l.2
(itu)~
4w
260/,0+
/
1 -
(14)
u2
Eqs. (13) and (14) can be used to solve analytically for u as a function of t and ~ as a function of u in terms of elliptic integrals. However, it is known that the solutions of these equations have certain well defined properties: in particular they are bounded and periodic. Of mo~t importance is the upper bound on the value of u. The condition comes from the fact that the square root of the left hand side must be real. Consequently the roots of the quartic on the right side of eq. (13) can be written as: O92
(/~U) 2 = 4-
[ ( @ l - - U ) ( U - - @ 2 ) ( /~ -- 0 / 3 ) ( U -- 0 / 4 ) ] ,
@1 ~" @2 ~> @3 ~ 0/4
and u is bound, @1> u > a 2. To illustrate this b o u n d consider the case for the initial conditions q'0 =/~0 = 0. Then u = 1 is a root: O92
(u/~)2 = ~ - ( a , - u ) ( u - 1 ) ( u - a 3 ) ( u - 0 / 4 ) , 0
L 0.2
I 0.4
"~
where al, 0/3, a4 are roots of the equation:
I _ - - ~ 0.6 0.8
u 3 + u 2 - (1 + 8122/o92)u- 1 = 0,
Fig. 4. Comparison of 1 / u 2 and 1 - au for expansion parameters a = 2, 1.2, 1.
Another first integral of the motion [in addition to eq. (5)] can be readily obtained by observing that the Hamiltonian of the system in the rotating coordinate system is given by:
ylZp3om/p,
8$22u O92
(-~
+ 1) 2}
fu
udu - u)(.
- @3)(u - @,)],/2
= 2~ {a F( O, k ) ---~II(
(12)~
H is a constant of the motion and eqs. (5) and (11) can be used to separate the variables in eq. (4). We obtain/~ and + as a function of u and time; du/dq~ as a function of u. The result of this procedure leads to +1
+ 2.5225),
(11)
Po=2[p2Og(2-2--~-:+l)+pRwsinq~].
-u4+u "~
u)(u + 0.2267)(u
which gives 1 _< Um~, _< 1.749 in agreement with the computer analysis. In fact eq. (13) can be integrated [6] in terms of elliptical integrals F and H to yield:
[(@1 - u ) ( 1
which is independent of time. Of course, the canonical m o m e n t a are not given by mia, mO21k; instead: m Pp = - ~ - [ 2 f f - Roa cos q~],
(/~u)2 = ~ -
which must have three real roots. For our example 122/o92 = (0.636) 2 and % = h 0 = 0. Then ( u u ) 2 = ( u - 1)(1.7489 -
4. E x a c t a n a l y t i c s o l u t i o n
H = 2 [ Ib2 + ( p + ) z ] +
(15)
+1+
(16)
where
F(O,k)=foO t
dq,
-,
1 - k 2 sin2+
//(~b, m, k ) = f 0 °
d~b d + m sinZ~k~l - k 2 sin/~k
with
a=a3(0/1-1);
d=b=@3-0/1;
c=oq-
1:
3 = ( a , - 1 ) ( a I - a3)(1 - 0/3)
+
= (13)
d , k ) } = (og/2)t,
u
0/3)(1 _ @ , ) , a sin20 + b c sin20 + d '
k2= ( @,@'---kl @3ii J\ @3-0/,1-@,11
C Chasrnan,A.J. Baltz / Spacechargeeffects The period of u is given by four times the complete elliptic integrals given in eq. (16). Likewise eq. (14) can be integrated [6] to yield:
eP-q'°=28 ,~cF(O'k)+~H( O' ~ , k ) } 26 a +T(TF(O,k)
+--~II(O, -~,k)}. c
Once again u is periodic in ~ with the period given by the complete elliptic integrals. For our case with ( ~ / w ) 2 = (0.636)2 the period of u with q~ is 84.59 ° and the period of u with t is 1.094 ( 2 ~ / ~ ) in agreement with the computer results.
5. Longitudinal motion The longitudinal motion is directly coupled to the radial motion as shown in eq. (4). The beam consequently will not be isochronous but after one turn will cross an inertial axis at a phase ~A t separated from the isochronous phase. To lowest order in the strength of the Coulomb force it can be shown that ~0At-Pocr(fl/to)2/(R+Po). Four our parameters toAt10 -3 rad. at the injection radius (R = 30 cm) and diminishes approximately linearly with increasing radius. Since the phase shift diminishes with radius the increase in the longitudinal energy spread due to space charge is negligible for our parameters.
6. Summary The two dimensional problem of a charge rotating in a uniform magnetic field under the influence of Coulomb
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forces has been solved analytically by quadrature, by computer and by approximation methods. The maximum radial displacement due to space charge and the period of oscillation are independent of the radius of the orbit. The linear approximation method yields resuits of modest precision but gives a simple picture for the physical nature of the motion; computer calculations demonstrate the region of its accuracy. The analytic solution is complex but the bounds on the radial motion can be obtained simply by the solution to a cubic or quartic equation. Furthermore, we also obtain the period of the oscillation from the (tabulated) complete elliptic integrals of the first and third kinds. This work was supported by the Division of Basic Energy Sciences, U.S. Department of Energy, under Contract No. DE-AC02-76CH00016.
References [1] C. Chasman (ed.), Proposal for a 15 A.GeV Heavy Ion Facility at Brookhaven, Informal Report BNL-32250, (January 1983). [2] L.J. Laslett, BNL-7534, 1963. [3] Henri Brock, Accelerateurs Circulaires de Particules (Presses Universitaires de France, 1966) and refs. therein. [4] A. Chabert, T.T. Luong and M. Prome, IEEE Trans. Nucl. Sci. NS-22 (1975) 3. This paper describes a beam orbit computer code which includes space charge effects used in studies for the GANIL cyclotron. [5] R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1964), Vol. I, p. 9-7. [6] Wolfgang Grobner and Nikolaus Hofreiter, Integraltafel, Erster Teil Unbestimmte Integrale, Wien, (Springer, Berlin, 1961).