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Theoretical aspects of the Soviet 10 GeV synchrotron
J. Nuclear Energy II, 1958, Vol. 6. pp. 351 te 369. Pergamon Press Ltd., London
THEORETICAL ASPECTS OF THE SOVIET 10 GeV SYNCHROTRON* t M. S. RABINOVICH Abstract-Methods are developed for analysing the behaviour of weak-focusing synchrotrons and, Free oscillations, orbit distortions, resonances in in particular, for calculating design parameters. free and phase oscillations, and injection, are considered. The theory is applied to the 10 GeV synchrotron of the U.S.S.R. Academy of Sciences. INTRODUCTION THE
highest energies so far attained in the acceleration of charged particles have been achieved by the synchrotron method. This depends basically on the “automatic focusing” principle (developed by VEKSLER)which applies to particles moving in a changing magnetic field and on a constant-or almost constant-equilibrium orbit, the frequency of the accelerating potential increasing with the particle velocity. Here we consider acceleration in a system comprising magnet sectors separated by straight sections. The design of an accelerator requires the formulation of a theory of sufficient accuracy for the basic design and for development. Such a theory has been developed at the Academy of Sciences Physical Institute during the years 1948-1952,(l) and it has been applied in the design of a 10 GeV proton synchrotron.(2) However, the methods employed have a much wider field of application (e.g. they may be used in calculations on strong-focusing accelerators). In this paper the theory of particle dynamics in synchrotrons is considered, and the application of this theory to the Academy of Sciences synchrotron is discussed. In an accelerator there exists a stable synchrotron orbit, whose position is defined by the accelerating field frequency and the magnetic field strength H(t). There also exists a family of instantaneous orbits which oscillate about this equilibrium orbitthese synchrotron or radial phase oscillations being connected with the phase of the accelerating field at the time when the particles cross the gap. However, the particles do not for the most part move along these orbits, but execute free vertical and radial oscillations about them. Corresponding to these three types of motion there are three periods which we will consider, using the 10 GeV synchrotron as an example : (1) The change in position of the equilibrium orbit occurs essentially during the acceleration time, i.e. for 3.3 X lo6 psec. (2) The synchrotron oscillations with periods in the range 800-1800 psec. (3) Free radial and vertical oscillations with periods of between 1 and 7 psec. In general these three types of motion are independent, but important exceptions are resonance phenomena and transient modes, which will be discussed later. In particular the independence of these motions permits us to consider the free oscillations without taking into account the actual acceleration process, and also to consider * Translated by A. J. MEADOWS from Atomnuyd Energiyu 2,431, f The synchrotron is at the Joint Institute for Nuclear Research. 351
(1957).
[Reprint
No. AEl25.1
352
M. S.
RABINOVICH
the acceleration process without the oscillations. The coupling between these motions may be considered as a correction, e.g. in the so-called “slit oscillations”. FREE
OSCILLATIONS
OF PARTICLES
The particle motion will be considered in a’co-ordinate system consisting of the instantaneous orbits and their normals. It can then be shown@pg)that assuming the energy of the particles to be constant, the oscillations satisfy the equation: XV+
K2&)X
=
0
(1)
where the primes denote differentiation along the trajectory (i.e. with respect to a); x is the deviation from the instantaneous orbit; 1 - n(a) for radial oscillations, K2= n(a) for vertical oscillations;
(
g(o) = 1/[R2(o)], where R is the radius of curvature of the trajectory at the point cr,and R(u) 6H n(o) = - y z
.
In an ideal accelerator (i.e. one without field distortion in the magnet sectors and where there is no penetration of field into the straight sections): $
(in the magnet sectors)
g(a) =
(3) I 0 (in the straight sections) 1.
We first consider such an ideal accelerator. An examination of equation (1) shows that it has periodically varying coefficients, of period u0 = ROv+ 1, where v is the angular extension of a sector, and I is the length of a straight section. Then it may be shown that the solution of equation (1) for m periodic elements has the form : x = Deirm y(a) + complex conjugate
(4)
where y(o) is a solution of equation (1) satisfying the boundary conditions:
(5) (6) For an ideal accelerator with straight sections: dcos:
sin:+
(in the magnet sectors) 0
0
!
K(ao R RoY)(c - ds) + s + dc (in the straight sections) 0
J
(7)
Theoreticalaspects of the Soviet 10 GeV synchrotron c
where
d=
&p
_
-; s
Kl
; cosp=c-pps;
s=sinKv;
P=2R,
Consider a particle emitted from the injector at an angle y = 2
353
C =
COS KY.
to the orbit.
We denote the distance from the injector (which is at the point oi) to the orbit by xinit. Given these initial conditions a value can be found immediately for the constant D and hence for the displacement x: D = Xinit- i@(Oi)Ro(Y- Yopt) 27&i) where
@(fJJ = -2i
(8)
W(c,)W*(c,) BR
(9)
0
is a real function, B is the Wronskian of the functions 1/,and y* at the point ci (B is independent of o), R, is any given radius, and the angle
Substituting equation (8) into equation (4): X&t +
&‘@Yoi)(r-
1
[pm+ d(c)1
l/opJ2 COS
where a(o) = arg
lP&~l.
(11) is the fundamental expression in the study of the free oscillations. amplitude of oscillation F(o) is given by: = 1~ [wa” = @(ad) .mt. + Ro2@2(d(y - yopJ21.
The
(13)
The minimum value of F(oJ at the injector azimuth is equal to Xinit. This occurs when y = yopt, i.e. when the particle is ejected at the optimum value yopt. Given the values of Xinrtand y, the maximum and minimum displacement P(o) occurs at the azimuth where the function a(o) has a maximum and minimum values. Taking m as the parameter, equation (11) is the equation of a family of curves x = x(o) in this parameter. Further, if m is a continuous variable, then x = &IF(a) is the equation of the envelope to this family of curves. In fact, m is a variable which takes integral values for any given periodic element, a multiple of N where N is the total number of periodic elements. In practice, if the motion is far from a resonance, there will be a suthcient number of different families of curves in each periodic element for us to be able to consider the envelope x = &F(o). The actual value of K@(U) is inversely proportional to the amount of focusing and depends on the accelerator design. In the 10 GeV synchrotron ~@(a) varies from l-05 to 1.10 for radial motion and from 1.03 to 1.12 for vertical motion. Hence the presence of the straight sections diminishes the focusing by less than 10 per cent and it is therefore evident that the
M. S.
354
RABINOVICH
limitation on length of these straight sections does not arise from this cause. As will be shown below the limitation is, in fact, due to the perturbation of resonances inside the region of stability. If Yopt= 0 for circular accelerators then in an accelerator with straight sections: K&nit
-‘?-
sin2 (20 - &Y) in the magnet sectors
0
R
(14)
Yopt =
in the straight sections The length o’is measured from the beginning of a straight section. The magnitude of Yoptin the 10 GeV synchrotron can be as much’as 3’-this is rather high since the permissible error in the angle of injection is only of the order lo’-15’. If vopt were to become too great it would be impossible to inject an almost parallel beam for any length of time? a situation which is indeed observed(9p10) in strong-focusing accelerators. However, by use of the method described by RABINOVICHet u1.,t4) we can investigate oscillation of the particles and the influence of straight sections. THE INFLUENCE OF MAGNETIC FIELD DISTORTION ON PARTICLE MOTION When field distortion is present, the equation of motion can, to a first approximation, be written as equation (1) but with an additional term: Xc + ‘+&)X = q(o).
(15)
We will suppose that ~~ = constant, and q(u) is a periodic function of period No,, where N is the number of periodic elements around the machine, and hence, No,, is the perimeter of the instantaneous orbit. To find a particular solution of equation (15) we vary the constant in equation (4) and find immediately that: dD
(+/JW
-z=-qBe
ifim_ --
q wJ(4 B
.
(16)
Any non-resonance perturbation can be described with sufficient accuracy by finding the closed perturbed orbit about which oscillations occur as they would do about the calculated orbit. Hence, in the presence of perturbation, we replace the calculated path through magnets by the new distorted path. Thus we consider only the solutions of equation (15) which are periodic, with period No,: &g)ei”N
~+~~,
x(4 = - B(1 - eipN) d s
q(t)rj(E) dt + complex conjugate.
(17)
In calculations, equation (17) must be analysed either into a Fourier series or into a series of eigenfunctions of equation (1). The former method has the advantage that it uses known methods of harmonic analysis, whilst the latter gives very simple expressions which have an immediate connexion with resonance theory. Since the eigenfunctions for the Academy of Sciences synchrotron are not very different from harmonic functions it is possible to use both methods in this case. We do not consider
355
Theoretical aspects of the Soviet 10 GeV synchrotron
the detailed solution here, but give only a qualitative discussion of the results. Our first task is to analyse the magnetic field distortions (i.e. the variations from the average value) inside the magnet sectors, into a Fourier series-we will consider only the first harmonics. In a cyclic accelerator the maximum distortion is proportional to the amplitude and frequency of the harmonic:
%s
I 2-
K2
1 2 Kres
FIG. l.-Diagram
2 -
1 1 = ___ = - for vertical oscillations, l(1 - n) y1 = &n
for radial oscillations.
K
showing
4th sector k
I
the regions of the magnet
3rd
sector
,
Direction 2nd sector
2
of motion 1st sectar
In
I-6 L FIG. 2.-Orbit
unused due to orbit distortion.
I”gk
this gap the magnetic field differs from the calculated value
in a field of 515 oersteds, calculated from magnetic of the Academy of Sciences synchrotron.
measurements
on a model
Hence taking the frequency of rotation as the unit, the first multiple corresponds to a radial resonance for n = 0, and the second to a vertical resonance for II = 1. In a circular accelerator these resonances occur on the boundaries of the stability region, but they occur inside this region for a machine with straight sections. For example, resonance values in the Academy of Sciences synchrotron are n,,, = 0.16 for radial oscillations, and n,,, = 0.84 for vertical oscillations. The resonance multiples take the corresponding forms: 1 0.84 - (1 - n)
1 n - 0.16 for radial oscillations,
1 for vertical oscillations. 0.84 - n Over the effective aperture of the synchrotron magnet: O-55 < n <
0.75.
356
M. S. RAB~NOVICH
Thus in this synchrotron the effect of the first harmonic (due to the straight sections) on orbit distortion is 30-40 per cent greater, and its effect on distortion of the average field value 2-8 times greater, than in a corresponding circular accelerator without straight sections. Hence it can be seen that straight sections longer than 8 m would be of little use; with a length of 12 m, II,,, has a value of the order O-75. The effect of higher harmonics on orbit distortion or on the average field value decreases rapidly. Fig. 2 shows the orbit of a particle for a model of a 180 MeV synchrotron, calculated on the basis of magnetic variation, RESONANCE
IN
BETATRON
OSCILLATIONS
The resonance effects in an accelerator with a segmented magnet are essentially different from resonance effects in circular accelerators. Firstly, certain resonance values of the magnetic field alter the index IZ considerably even for short straight sections. Fig. 3 shows the dependence of n at resonance, for vertical oscillations (referred to in the preceding section) and for radial oscillations, on the length of the straight sections. The second difference is that an accelerator with straight sections resonates in a series of harmonics which can be analysed into a Fourier series.
L R FIG. 3.-Variation
of resonance values of n with the length of the straight 1. me9 for vertical oscillations; 2. nres for radial oscillations.
sections.
To calculate the resonances, we first write down the equation of oscillation in the form:
x”+ x2Cg(4 +_fo1x = q(4,
(18)
where the perturbations q(a) and!(o) are functions of period No, (q has the period ao). Then, from equation (16), the solution of equation (4) is:
dD,-do
?n
-
q --fK2[D,
exp (is p&y(u) + complex conj.]
I
(
w*(a)exp -it X
B
0
1
ruk
.
(19)
We will integrate this equation for a single element of period oo, taking D, on the right-hand side to be constant. We add a subscript m to q and f since the functions q and f depend on m, the element of periodicity. The value of o is measured from the beginning of an element.
Theoretical aspects of the Soviet 10 GeV synchrotron
-exp Dm+l -
D,
357
fc2exp(-2i$pK)D..*
=
qmy*dc+
[f,N2dg
B
+;D,
rfmw* do=&&.
(20)
We apply the method of averaging to this differential equation to obtain an equation for resonance (RABINOVICH(~J~) gives a more general method for the calculation of the damping of oscillations, etc.): D m+l -DD,=-.
SL” 2i sin /l
(21)
For the sake of brevity we will consider only one case here. Let the average distance (or value of the magnetic field H) in different sectors be perturbed by an amount AZ, (or AH,), and the magnetic field index by an amount Anm, then
fm=$n; 0
qm=%;
(orq,=_-.A+~)_ 0
Example 1. If AZ, = AZ, = a; Az3 = AZ, = -a; ,u~+~- ,& = 2.5 X 10-3, then the amplitude of vertical oscillations, F, after going through resonance (at n = 0.84) is almost 60 times greater than a. Example 2. Take qnz = 0, (f,,,# 0). As compared with the previous case, we have now only parametric resonance. When p = rr/2 (n = 0*84), the second harmonic of the field perturbation produces resonance in the vertical oscillations and when ,u = r/4 (n = O-79), it produces resonance in the radial oscillations. A particle performing radial phase oscillations can go through the resonance region many times so that the point We will now consider some numerical
Let
of maximum deviation may lie in this region. results for resonance with PZ= 0.79 (,u = n/4).
An, = An, = ho;
An, = An, = -ho;
then D, < Do exp (15h,). Since ho in the Academy of Sciences synchrotron is less than O-5 per cent, during a passage through resonance at the injection point, D, increases by 5-6 per cent. Hence if ,u~+~- plc N 2.5 X lop3 or greater, then the resonance we have been considering has no appreciable effect on the injection process. However, this resonance is much more important during the period of acceleration. If the amplitude of radial phase oscillations increases so much that the instantaneous orbit falls in the resonance region, then, as a result of the numerous passages through resonance, it is possible to have a slow increase in the oscillations. In order that this increase should be inhibited it is sufficient that the amplitude of the oscillations dies away during a period of phase oscillation, owing to the quicker growth of the magnetic field. Let the point of reversal of the radial phase oscillations (of amplitude O.O2R,) lie in the centre of the resonance region; R (&z/aR) ci 100, the energy of the particles W, = 10 MeV, the accelerating potential 6 kV, then D, < Do exp (32h,). Here the index is higher than in the previous case, since in this example, the radial
358
M. S. RABINOVICH
velocity changes direction at the resonance point, so that the particle orbit is in the resonance region for a relatively long time. However, at a distance of 3 cm from the resonance point the index has fallen by a factor of 16. An analogous expression for parametric resonance can also be obtained for the case when n = 0.84 (between the second harmonic and the vertical oscillations). It is evident from the foregoing discussion that it is possible to have an increase in undamped oscillations due to resonance in the injection process, in the acceleration, and in the ejection of particles from the chamber. However, it is possible to avoid these resonances, given the permissible limitations of the magnetic field, and this has been done in the 10 GeV synchrotron. RESONANCE
IN SYNCHROTRON
OSCILLATIONS
The derivation of the phase equations depends basically on the use of a co-ordinate system formed by the instantaneous orbit and its normals. The accelerating field can be analysed into a series of travelling waves along the equilibrium orbit. We thus obtain an equation which enables us to consider the coupling between betatron and phase oscillations, slit oscillations, etc. In small accelerators the phase oscillation frequencies ol, are the most important. In the Academy of Sciences synchrotron the frequency of phase oscillations 0,/27r varies from 2000 to 700 c/s. Oscillations of such frequency can occur in the magnetic field and in the frequency and amplitude of the accelerating potential. The danger of these resonances is due to the low values of the phase oscillation frequencies. This increase of oscillation is dangerous, particularly in the injection process, as it can lower the number of particles captured into the accelerating cycle by a factor of about ten. During the acceleration, the oscillations are not dangerous if the particles remain in the region of stability, but scattering of particles towards the edges of this region leads to a loss. In this way the particles oscillate in resonance and are subsequently lost owing to the large oscillations of H at a frequency of the order 600 c/s. The greatest danger is resonance with magnetic field oscillations and at the frequency of the accelerating field. We will consider, as an example, the resonance increase due to magnetic field harmonics. The phase equation has the form :
(22) where E is the energy, o. the frequency of the accelerating field, I$ (= 4. + IX)is L n 1 the phase of the accelerating field, k = 1 f - , F,=ll-np /P( 1 - n)(27~R+ L)k is a coefficient to allow for the straight sections, V, is the sum of all the accelerating potentials, and Q is the resonance energy:
Q=
eRoPRo+ L)
hn
cos
Qt ,
TX!
(23)
where (h sib fit) is the oscillating part (or one of the Fourier components) of the magnetic field. The phase equation is non-linear so that consideration of resonance phenomena is extremely complicated. It is, however, possible to resolve the equation
Theoretical aspectsof the Soviet 10GeV synchrotron into a linear approximation tions may easily be found: $n
from which the amplitude of the additional phase oscilla-
y/%r R,&,
h
_ 4;
max -
359
H(l - n)(2nR, + L)
d&
where W1is the rate of change of the phase oscillation frequency. If special measures are not taken (they are usually anticipated) then the resonance with the second harmonic c/s of the magnetic field (1200 c/s) is very dangerous. (The first harmonic at 600 c/s is due to the rectification of the 50 cycle 12 phase a.c.) Hence h, mitxY 3 x 102h, (h, in oersteds). At resonance W = ---103rr and alin must be considerably less than 0.01 oersted. The value of h, (the amplitude of the magnetic field oscillations) which produces these violent perturbations is a million times smaller than the basic field. However, the rapidly varying part of the field h&2 dH
has a rate of change 50 times smaller than x,
and for resonance, it is the rate of
change which is important rather than the amplitude, which explains the small value of Iz,. It can be seen from this example that for values of h, as small as 0.01 oersted If the the non-linearity of the phase equation must be taken into account. The non-linearity is small, the equation can be solved by normal methods. basic method used is that introduced by KRILOV, BOGOLJUBOVand MITROPOLSKI. For small non-linearity we cannot consider the particle transition through the boundary of the stability region, but we can obtain a more precise value for the amplitude of oscillation. The rate of transition through resonance is small in the Academy of Sciences synchrotron so that even a small non-linearity changes the results markedly. Indeed, in the linear approximation for the amplitude of phase oscillations: h,, -+ 00 when W1-+ 0, whereas correcting for a small non-linearity gives a finite amplitude even for &, = 0. The physical reason for this is clear-the particles are removed from resonance by the frequency decrease which follows the growth of the phase oscillation amplitude. This is called active non-linearity (KRILOVand BOGOLJUBOV), as distinct from passive non-linearity, which does not limit the amplitude in the absence of damping. A very important factor here is the direction of the resonance transition. If the time and amplitude dependence of oscillation frequency act in the a#J aq same direction i.e. 1 *> 0 then the non-linearity acts as an effective damp1 at a~,,, ing. We look for the solution of equation (22) in the form:
cr=acos(7+s/))-; 1 -$OS(‘tY) [
s
1
cot#J,-~2cos3(r+W)
7
where T =
co1dt (,a1 is the phase oscillation frequency); a and w give the amplitude
0
and phase of the oscillations. In general, a and w are time-dependent, case I,U= & n/2 when a satisfies the algebraic equation: a3 -
16(1- 5% * I-;
Cot2+0
except in the
160 (1 +r)(l
;_$OP&))
=Oy
(24)
M.
360
S. RABINOVICH
where the minus sign corresponds to the phase y = rr/2, and the plus sign to the phase y = --n/2. The constant D is given by:
(where h is in oersteds). Here 6 is the ratio of the perturbation frequency Q to the frequency of small phase oscillations wi. Equation (24) will have three different real roots if: - 16(1 - 5)3(1 + 5)2 < _6.5 (25)
D2 1 +;cot2&,
(
1
If the inequality equation (25) is not fulfilled, then there is only one root. If the inequality sign changes to equality then we have the condition for the existence of one multiple and one simple real root. The multiple root is: I.609
lamI=
+’ (
1 +$oPf&
)
The simple root is: a, = 21a,l.
Fig. 4 shows the dependence of amplitude on the ratio of the perturbing frequency to the phase oscillation frequency. An analysis of the stability of the stationary solution.gives the following results. The upper branches of the curves in Fig. 4, corresponding to the phase y = 3~12, are stable, whilst of the two lower branches, corresponding to the phase y = -7r/2, one is stable and the other unstable. The boundary between the two regions is the point a,. If E increases in the 10 GeV synchrotron, then the characteristic point moves along the curve to the lower left, almost to the boundary between the stable and unstable regions. Further consideration of the behaviour of the characteristic point can only be qualitative, however slowly the phase oscillation frequency changes. Near the boundary of the stable region the amplitude of the oscillations increases rapidly and the point crosses to the upper, stable branch. As the transition takes place quickly the characteristic point no longer moves along the fixed curve but performs oscillations near it. The final amplitude will be between 2a, and 3a,. We can see that the maximum amplitude is attained when E < 1, i.e. resonance occurs earlier than the point of equality between small phase and forced oscillations, and the resonance effect increases with the magnitude of the perturbation. These qualitative considerations have been supported by numerical analysis of the equation for D = 0.04 and 2 = 104 = 6. The initial conditions are so chosen that at a considerable distance from resonance the amplitude of oscillation is equal to zero, i.e. it coincides with the fixed curve. The result of the integration is shown in Fig. 5 by the continuous line: the case which we have selected corresponds to a large
Theoretical aspects of the Sovret 10 GeV synchrotron
361
perturbation (the amplitude may be as great as 5 rad). Hence, although the amplitude is much less than in the linear case, it approximately corresponds to the greatest possible value. If the particle performs phase oscillations up to resonance, then it is impossible in the non-linear case to sum over the oscillations. This is illustrated in Fig. 5 by
FIG. 4.-Resonance curves for phase oscillations with varying values of cos & and for the amplitude of disturbance D as a function of &the ratio of the disturbance frequency to the phase oscillation frequency. l-upper
stable branch;
2-lower
unstable
branch;
3-lower
stable branch.
the broken curve, which was obtained by numerical integration. At some distance from resonance the amplitude of the oscillations is equal to O-2 rad. The resonance growth of amplitude sets in earlier but the final value obtained differs very little from the value obtained before, so that a strong resonance effect drives all particles to the edge of the stable region. As we shall see, if the characteristic point is situated on the upper curve at some distance from resonance then instead of the resonance growth of amplitude we have instead a monotonic decrease. Here, however, the
362
M. S. RABINOVICH
initial phase is important as well as the amplitude. It is easier to treat the linear and non-linear cases together, comparing 2a, with G$&
It can then be shown that the effect of non-linearity enables a reduction to be made in the magnitude of the field harmonic. In the calculations on the Academy of
FIG. 5.-Phase oscillation amplitude for a slow change in frequency (in the resonance region) for two cases:-the continuous curve gives the growth of amplitude from zero, the dotted curve gives the growth from aintt -0.2, D = 0.04, cos f& = 0.5, 6 = o*ooo1. l-resonance curve; 2-upper stable branch; 3-unstable branch; 4-lower stable branch.
Sciences synchrotron, consideration was also given to the increased rate of transition through resonance due to a rapidly fluctuating V,, (the accelerating potential) in the resonance frequency region. In this region, in addition to the normal perturbations, noise fluctuations are important. This question can also be analysed by a linear approximation and the requirements for the noise characteristics calculated. INJECTION
In the ii?jection process(2) a beam of particles, of energy 9 MeV, traverses a long path through various types of apparatus before entering a chamber enclosing one of the quadrants. We will consider here the relation between the beam parameters at the entrance to the magnet and the effectiveness of the injection process, i.e. the number of particles captured into the accelerating region. If the magnetic field configuration and the beam parameters were known, then consideration of the present problem would enable us to calculate the orbit. In actual fact, our information is restricted to the known magnetic field and beam parameter tolerances. We must therefore average over all permissible configurations which enable us to calculate the intensity of the accelerated ion beam only to within a factor of two. A detailed analysis shows that we &an average over the motion in the following way. In general, in a constant magnetic geld, the injected particles undergo free oscillations of such
Theoretical aspects of the Soviet 10 GeV synchrotron
363
phases that 50 per cent of the particles are lost on the third turn and the remainder on the sixth. (Previously we have not permitted any error greater than 10-20 per cent.) We will suppose that the beam enters the magnet chamber with a uniform density distribution, a width 2A, and a uniform angular dispersion of fa,. The axis of the beam is at an angle E to the optimum direction. We divide the injection process into two parts. Firstly, before the accelerating potential is applied the particles move along displaced orbits in the varying magnetic field. While the orbit remains fixed, there is a high probability that the particles will strike the injector or the chamber walls-we will call the portion of the particles not striking the injector or chamber walls the capture coefficient of the particles in the first stage of the injection (qI). Then: (27)
“171 = Yp%
where qp is the probability of the particles not striking the injector and Q is the probability of the particles not striking the chamber walls. Together with the probability qp, we must calculate the number of revolutions during injection. Pl VT
=~PG
(28)
’
where pr is the distance from the injector to the centre of the chamber and AR is the change in orbit of the particles (AR = 1.1 cm for Wi = 9 MeV). pi/AR is therefore the total change in the first stage of injection. When the orbit reaches the middle of the chamber, the accelerating potential is applied and a portion Q (transition coefficient) of the particles captured in the first stage are accelerated. The remaining particles (1 - qz) either do not fall within the region of phase stability and are not accelerated, or else strike the injector due to disturbance from radial pb,ase oscillations. The product of the two coefficients QQ is called the effectiveness of injection. We now introduce a series in a where
WWJ
a=-
(29) \ I
Pi
where a is the injection angle of a given particle and @(ai) is defined by equation (9). a can vary between a1 and a2 (aI > aB). We denote the angle between the axis of the beam and the optimum angle by a,. We also take: a,=-;
3AR Pi
a, = 2a,
(30)
Then : Pi 5’T =
4AR *O . a,,,i=l
i:
f P(4, i-1
ad -
@(h,
Ql,
a1 -a2 , _o - pi is the particle beam width, and @(a, a) is a function shown where amax = 2 graphically in Fig. 6.
364
M.
The family Fig. 7. From limits ensuring stable capture
S. RABINOVICH
of curves of qr for the Academy of Sciences synchrotron is given in these, the limits of accuracy governing the injection process (i.e. the -100 per cent capture) can be found immediately. Obviously, for (particularly during the period of adjustment), it is desirable to have
0.1 0.2 0.3 0.4 O-5 0.6 O-7 0.8 O-9 1-O
0
rY.
FIG. 6.-Graph
of function
CD(a, IX)(from equation
Ymm
FIG. 7.-Relation
between
Wi = 10 MeV;
(31);
and reference
degrees
the number of turns, the error in the injection horizontal dispersion of the beam, yms,x.
pi = 50 cm;
6).
dH/dt = 4000 oersteds/sec;
angle and the
beam width = 5 cm.
the beam with a given angular distribution. Under these conditions, the number of captured particles decreases slightly, but depends less critically on variations in the injection apparatus. However in order to calculate the capture coefficient in the first stage we must take into account collisions with the walls of the chamber (Fig. 8). We will now consider the undamped oscillation amplitude distribution at the end of the first stage. Suppose we have a parallel beam, the axis of which is inclined to
Theoretical
aspects
of the Soviet
10 GeV synchrotron
365
the optimum direction at an angle a,. The resultant amplitude distribution is shown in Fig. 9. We assume that, in this case, the distribution is approximately uniform when F = !?!!!fZ> a, (where pmaxis the amplitude of the oscillations, pi is the distance Pi
from the injector to the centre of the chamber, and the distribution function y(F) is
25
L FIG. S.-The
degrees
percentage of particles which avoid striking the chamber walls, given an error in angle of E, and a vertical dispersion of the beam 5,. D, is the chamber height; 20, = 25 cm.
0.8 0.6 0.4 0.2 0 0.2 FIG. 9.-The
amplitude
distribution
0.4
0.6 F for a parallel
0.8
1.0
beam after the first stage of injection.
equal to zero when F < a,). To this approximation the number of particles with small oscillation amplitudes is somewhat decreased for small a, but is increased for large a,. If the beam has a considerable angular scatter, then to a sufficient degree of accuracy the distribution has the form y(F) dF = + d/F dF
(32)
Fig. 10 compares this approximate equation for the amplitude distribution with a more exact expression. The second stage of the injection process is characterized by the development of 7
366
M. S. RABINOVICH
radial phase oscillations. If particles which have missed the injector on the first stage are to miss it again, the following condition must be satisfied: F + (P&i) < 1
(33)
where p+ is the amplitude of radial phase oscillations (which are simply related to the actual phases) and has two corresponding phases: $2 > $0;
A<
$0~
between which phase oscillations occur. The phase oscillation amplitude is a simple function of the ratio pdp, where p is the maximum possible phase oscillation amplitude: ,E= R,
J
eV, sin +,( 1 - & cot c&) L 71(27& + L)
1
(34)
FIG. lO.-Amplitude l-from
When,!?<
distribution after the first stage of injection for a wide beam with a large angular scatter (aI = 0.1, a2 = 0.3, E > 0.3). the approximate equation (32); 2-from the more exact expression (equation (62) of reference 6).
1: (35)
42 - 41 = 27&&p/P)?
Graphs can be drawn for the function s(pJ~) and also for the approximate expression:
4&/P)-
20 - docot60)’ Pd n-p
[1+
o*3(P$m2
1 .
Two possible errors can occur at the beginning of the second stage of injection: either the accelerating field is not applied at the correct time, or the energy of the beam (i.e. the initial frequency of the accelerating field) is incorrect. In practice, both these errors lead to the same result. If the beam of particles is not monoenergetic then it is evidently impossible to find an accelerating potential which will include all particles. Suppose that the position of the instantaneous orbit when the field is applied is different from the mean orbit by an amount p&f (the mean orbit is the
Theoreticalaspectsof the Soviet 10 GeV synchrotron
367
orbit for which the frequency of rotation is equal to the frequency of the accelerating field). M is related to the error in energy AEi by: piA4 = R,
AEi 2(1 -
(37)
rz)Wi’
or to the error At in the time of appearance of the field by: piM = R,
Hi At
(38)
Hi(l - n)
where Hi and Eii are, respectively, the magnetic field and its rate of change at the moment of injection. The error in the initial frequency Aw displaces the mean orbit position (which we are taking as our co-ordinate origin) by an amount: pJ4
Aw(2rR, + L)
= R,
(39)
coo[n(27rR, + L) - L] ’
We now define the initial phase ~init and an initial phase velocity ~init (or an initial radial deviation piM) to correspond to pd, the amplitude of radial phase oscillations.
[(?_J2_ (!g2] = sin
$0 -
sin &it Win
h
-
($0 -
Ad
CoS 40
MO cos 543
’
(40)
As can be seen from equation (40), ~init is a multi-valued function of
[($‘- (Jg2]. It is possible, from equation (40), to find two minimum values (of absolute magnitude o construct a combination (&it,2 - +inrt,r) and to calculate the
(C!) 2- (F)“I**
&it,2 - hit,1 = 2nTTEm [
(41)
Obviously, the only difference between E, and E (defined in equation (36)) is the argument. From equations (40) and (41) we see that ($rnrt,a- &,& is the magnitude of the initial phase region for which, with the given M, the amplitude of radial phase - hnit,l is the proportion of 25. particles with such phases. Hence the number of particles with amplitudes between p6 and &+d+ is equal to oscillations is less than or equal to pd, and
&nit,2
f% dpPd. dP+
(42)
In the second stage, only those particles will be captured which have a free oscillation amplitude: pmax < pi - po. If y(F) is the amplitude distribution, then the number of particles obeying this inequality is:
s
1-(PJPi)
0
Y(F)CJF.
368
M. S.
RABINOVICH
Multiplying equations (42) and (43), and integrating over all p4, we obtain the capture coefficient for the second stage of injection: p2 =
1% dp4~-(@‘$(F) dF.
(44)
Changing the order of integration in equation (44) and integrating with respect to p+: % = l-e-(;
2000
3000
2/(1 - F)2 -
4000
5000
6000
M2)y(F)
dF.
7000
“0
Fro. 1 I.-The
transition coefficient (Q) for a parallel beam with an angular error aE and an error M in the time of appearance of the field. V,, is the amplitude of the accelerating potential in volts.
Equation (45) is a simpler form, and is more useful for calculation of amplitude distributions. Equation (44) also gives the amplitude distribution for the second stage of injection. If the particles have a range of values of M (corresponding to different energy values), with a distribution y&W), then the coefficient becomes:
Graphs of q2 for various values of a, and M are given in Fig. 11, If the particles also have an energy distribution then it is difficult to integrate equation (46) for large angles of injection (see equation (32)), and hence we replace equation (32) by the approximate expression : y(F) = 2F.
(47)
This approximation slightly underestimates i2. If a frequency q times greater than the frequency of rotation is employed, then the equations are unaltered, except that in equation (34), ,5 is reduced by a factor q*. Table 1 gives the maximum value r], for an injection energy wi = 10 MeV, for an equilibrium energy distribution and for two different values of pi. Thus, for q = 1, the maximum value i2 is of the order 25 per cent. To summarize, we have briefly considered theoretically some of the factors affecting the performance of the Academy of Sciences synchrotron (or of any high energy synchrotron). From these calculations it is possible to determine the permissible tolerances on the accelerator parameters for normal operation to be possible.
Theoretical aspects of the Soviet 10 GeV synchrotron
369
TABLE1.-VALUESOF qz( %)
Distance from injector
Form of function Energy distribution (%)
Some important topics such as the problems of ejection, and injection optics, have not been considered here, but it is hoped that this paper gives an overall survey of the theoretical problems which have been considered in the design of the Academy of Sciences synchrotron. The author wishes to thank V. I.
VEKSLER, A. A. KOLOMENSKI and V. A. PETUKHOV
for their assistance in this work.
REFERENCES 1. VEKSLER V. I., KOLOMENSKY A. A., PETUKHOV V. A. and RABINOVICH M. S. Communication to the All-Union Conference on High-Energy Particle Physics, Moscow (1956). 2. VEKSLERV. I. et al. J. Nucl. Energy 4,333 (1957). 3. RABINOVICH M. S., BALDINA. M. and MIKHAILOVV. V. The Motion of Particles in a Synchrotron with Straight Sections. U.S.S.R. Academy of Sciences Physical Institute Report (1949). 4. RABINOVICH M. S., BALDINA. M. and MIKHAIL~V V. V. Ibid. (1950); Zh. eksp. teor. $2. 31,
993 (1956). 5. RABINOVICH M. S. The Theory of Free Oscillations in an Accelerator with Straight Sections. U.S.S.R. Academy of Sciences Physical Institute Report (1947). 6. RABINOVICH M. S. The Theory of a Synchrotron with a Segmented Magnet. U.S.S.R. Academy of Sci’ences Physical Institute Report (1952). 7. MIKHAILOVV. V. The Effect of a Segmented Magnetic Field on Particle Motion. U.S.S.R. Academy of Sciences Physical Institute Report (1949). 8. RABINOVICH M. S. A Theoretical Investigation of Resonance Phenomena in Accelerators. U.S.S.R. Academy of Sciences Physical Institute Report (1952). 9. BALDINA. M. and MIKHAILOVV. V. Periodic Motion of ChargedParticles in an Magnetic Field. U.S.S.R. Academy of Sciences Physical Institute Report (1952). 10. RABINOVICH M. S. A Theoretical Study of Acceleration by a Changing Magnetic Field. U.S.S.R. Academy of Sciences Physical Institute Report (1953).
THEORETICAL ASPECTS OF THE SOVIET 10 GeV SYNCHROTRON* t M. S. RABINOVICH Abstract-Methods are developed for analysing the behaviour of weak-focusing synchrotrons and, Free oscillations, orbit distortions, resonances in in particular, for calculating design parameters. free and phase oscillations, and injection, are considered. The theory is applied to the 10 GeV synchrotron of the U.S.S.R. Academy of Sciences. INTRODUCTION THE
highest energies so far attained in the acceleration of charged particles have been achieved by the synchrotron method. This depends basically on the “automatic focusing” principle (developed by VEKSLER)which applies to particles moving in a changing magnetic field and on a constant-or almost constant-equilibrium orbit, the frequency of the accelerating potential increasing with the particle velocity. Here we consider acceleration in a system comprising magnet sectors separated by straight sections. The design of an accelerator requires the formulation of a theory of sufficient accuracy for the basic design and for development. Such a theory has been developed at the Academy of Sciences Physical Institute during the years 1948-1952,(l) and it has been applied in the design of a 10 GeV proton synchrotron.(2) However, the methods employed have a much wider field of application (e.g. they may be used in calculations on strong-focusing accelerators). In this paper the theory of particle dynamics in synchrotrons is considered, and the application of this theory to the Academy of Sciences synchrotron is discussed. In an accelerator there exists a stable synchrotron orbit, whose position is defined by the accelerating field frequency and the magnetic field strength H(t). There also exists a family of instantaneous orbits which oscillate about this equilibrium orbitthese synchrotron or radial phase oscillations being connected with the phase of the accelerating field at the time when the particles cross the gap. However, the particles do not for the most part move along these orbits, but execute free vertical and radial oscillations about them. Corresponding to these three types of motion there are three periods which we will consider, using the 10 GeV synchrotron as an example : (1) The change in position of the equilibrium orbit occurs essentially during the acceleration time, i.e. for 3.3 X lo6 psec. (2) The synchrotron oscillations with periods in the range 800-1800 psec. (3) Free radial and vertical oscillations with periods of between 1 and 7 psec. In general these three types of motion are independent, but important exceptions are resonance phenomena and transient modes, which will be discussed later. In particular the independence of these motions permits us to consider the free oscillations without taking into account the actual acceleration process, and also to consider * Translated by A. J. MEADOWS from Atomnuyd Energiyu 2,431, f The synchrotron is at the Joint Institute for Nuclear Research. 351
(1957).
[Reprint
No. AEl25.1
352
M. S.
RABINOVICH
the acceleration process without the oscillations. The coupling between these motions may be considered as a correction, e.g. in the so-called “slit oscillations”. FREE
OSCILLATIONS
OF PARTICLES
The particle motion will be considered in a’co-ordinate system consisting of the instantaneous orbits and their normals. It can then be shown@pg)that assuming the energy of the particles to be constant, the oscillations satisfy the equation: XV+
K2&)X
=
0
(1)
where the primes denote differentiation along the trajectory (i.e. with respect to a); x is the deviation from the instantaneous orbit; 1 - n(a) for radial oscillations, K2= n(a) for vertical oscillations;
(
g(o) = 1/[R2(o)], where R is the radius of curvature of the trajectory at the point cr,and R(u) 6H n(o) = - y z
.
In an ideal accelerator (i.e. one without field distortion in the magnet sectors and where there is no penetration of field into the straight sections): $
(in the magnet sectors)
g(a) =
(3) I 0 (in the straight sections) 1.
We first consider such an ideal accelerator. An examination of equation (1) shows that it has periodically varying coefficients, of period u0 = ROv+ 1, where v is the angular extension of a sector, and I is the length of a straight section. Then it may be shown that the solution of equation (1) for m periodic elements has the form : x = Deirm y(a) + complex conjugate
(4)
where y(o) is a solution of equation (1) satisfying the boundary conditions:
(5) (6) For an ideal accelerator with straight sections: dcos:
sin:+
(in the magnet sectors) 0
0
!
K(ao R RoY)(c - ds) + s + dc (in the straight sections) 0
J
(7)
Theoreticalaspects of the Soviet 10 GeV synchrotron c
where
d=
&p
_
-; s
Kl
; cosp=c-pps;
s=sinKv;
P=2R,
Consider a particle emitted from the injector at an angle y = 2
353
C =
COS KY.
to the orbit.
We denote the distance from the injector (which is at the point oi) to the orbit by xinit. Given these initial conditions a value can be found immediately for the constant D and hence for the displacement x: D = Xinit- i@(Oi)Ro(Y- Yopt) 27&i) where
@(fJJ = -2i
(8)
W(c,)W*(c,) BR
(9)
0
is a real function, B is the Wronskian of the functions 1/,and y* at the point ci (B is independent of o), R, is any given radius, and the angle
Substituting equation (8) into equation (4): X&t +
&‘@Yoi)(r-
1
[pm+ d(c)1
l/opJ2 COS
where a(o) = arg
lP&~l.
(11) is the fundamental expression in the study of the free oscillations. amplitude of oscillation F(o) is given by: = 1~ [wa” = @(ad) .mt. + Ro2@2(d(y - yopJ21.
The
(13)
The minimum value of F(oJ at the injector azimuth is equal to Xinit. This occurs when y = yopt, i.e. when the particle is ejected at the optimum value yopt. Given the values of Xinrtand y, the maximum and minimum displacement P(o) occurs at the azimuth where the function a(o) has a maximum and minimum values. Taking m as the parameter, equation (11) is the equation of a family of curves x = x(o) in this parameter. Further, if m is a continuous variable, then x = &IF(a) is the equation of the envelope to this family of curves. In fact, m is a variable which takes integral values for any given periodic element, a multiple of N where N is the total number of periodic elements. In practice, if the motion is far from a resonance, there will be a suthcient number of different families of curves in each periodic element for us to be able to consider the envelope x = &F(o). The actual value of K@(U) is inversely proportional to the amount of focusing and depends on the accelerator design. In the 10 GeV synchrotron ~@(a) varies from l-05 to 1.10 for radial motion and from 1.03 to 1.12 for vertical motion. Hence the presence of the straight sections diminishes the focusing by less than 10 per cent and it is therefore evident that the
M. S.
354
RABINOVICH
limitation on length of these straight sections does not arise from this cause. As will be shown below the limitation is, in fact, due to the perturbation of resonances inside the region of stability. If Yopt= 0 for circular accelerators then in an accelerator with straight sections: K&nit
-‘?-
sin2 (20 - &Y) in the magnet sectors
0
R
(14)
Yopt =
in the straight sections The length o’is measured from the beginning of a straight section. The magnitude of Yoptin the 10 GeV synchrotron can be as much’as 3’-this is rather high since the permissible error in the angle of injection is only of the order lo’-15’. If vopt were to become too great it would be impossible to inject an almost parallel beam for any length of time? a situation which is indeed observed(9p10) in strong-focusing accelerators. However, by use of the method described by RABINOVICHet u1.,t4) we can investigate oscillation of the particles and the influence of straight sections. THE INFLUENCE OF MAGNETIC FIELD DISTORTION ON PARTICLE MOTION When field distortion is present, the equation of motion can, to a first approximation, be written as equation (1) but with an additional term: Xc + ‘+&)X = q(o).
(15)
We will suppose that ~~ = constant, and q(u) is a periodic function of period No,, where N is the number of periodic elements around the machine, and hence, No,, is the perimeter of the instantaneous orbit. To find a particular solution of equation (15) we vary the constant in equation (4) and find immediately that: dD
(+/JW
-z=-qBe
ifim_ --
q wJ(4 B
.
(16)
Any non-resonance perturbation can be described with sufficient accuracy by finding the closed perturbed orbit about which oscillations occur as they would do about the calculated orbit. Hence, in the presence of perturbation, we replace the calculated path through magnets by the new distorted path. Thus we consider only the solutions of equation (15) which are periodic, with period No,: &g)ei”N
~+~~,
x(4 = - B(1 - eipN) d s
q(t)rj(E) dt + complex conjugate.
(17)
In calculations, equation (17) must be analysed either into a Fourier series or into a series of eigenfunctions of equation (1). The former method has the advantage that it uses known methods of harmonic analysis, whilst the latter gives very simple expressions which have an immediate connexion with resonance theory. Since the eigenfunctions for the Academy of Sciences synchrotron are not very different from harmonic functions it is possible to use both methods in this case. We do not consider
355
Theoretical aspects of the Soviet 10 GeV synchrotron
the detailed solution here, but give only a qualitative discussion of the results. Our first task is to analyse the magnetic field distortions (i.e. the variations from the average value) inside the magnet sectors, into a Fourier series-we will consider only the first harmonics. In a cyclic accelerator the maximum distortion is proportional to the amplitude and frequency of the harmonic:
%s
I 2-
K2
1 2 Kres
FIG. l.-Diagram
2 -
1 1 = ___ = - for vertical oscillations, l(1 - n) y1 = &n
for radial oscillations.
K
showing
4th sector k
I
the regions of the magnet
3rd
sector
,
Direction 2nd sector
2
of motion 1st sectar
In
I-6 L FIG. 2.-Orbit
unused due to orbit distortion.
I”gk
this gap the magnetic field differs from the calculated value
in a field of 515 oersteds, calculated from magnetic of the Academy of Sciences synchrotron.
measurements
on a model
Hence taking the frequency of rotation as the unit, the first multiple corresponds to a radial resonance for n = 0, and the second to a vertical resonance for II = 1. In a circular accelerator these resonances occur on the boundaries of the stability region, but they occur inside this region for a machine with straight sections. For example, resonance values in the Academy of Sciences synchrotron are n,,, = 0.16 for radial oscillations, and n,,, = 0.84 for vertical oscillations. The resonance multiples take the corresponding forms: 1 0.84 - (1 - n)
1 n - 0.16 for radial oscillations,
1 for vertical oscillations. 0.84 - n Over the effective aperture of the synchrotron magnet: O-55 < n <
0.75.
356
M. S. RAB~NOVICH
Thus in this synchrotron the effect of the first harmonic (due to the straight sections) on orbit distortion is 30-40 per cent greater, and its effect on distortion of the average field value 2-8 times greater, than in a corresponding circular accelerator without straight sections. Hence it can be seen that straight sections longer than 8 m would be of little use; with a length of 12 m, II,,, has a value of the order O-75. The effect of higher harmonics on orbit distortion or on the average field value decreases rapidly. Fig. 2 shows the orbit of a particle for a model of a 180 MeV synchrotron, calculated on the basis of magnetic variation, RESONANCE
IN
BETATRON
OSCILLATIONS
The resonance effects in an accelerator with a segmented magnet are essentially different from resonance effects in circular accelerators. Firstly, certain resonance values of the magnetic field alter the index IZ considerably even for short straight sections. Fig. 3 shows the dependence of n at resonance, for vertical oscillations (referred to in the preceding section) and for radial oscillations, on the length of the straight sections. The second difference is that an accelerator with straight sections resonates in a series of harmonics which can be analysed into a Fourier series.
L R FIG. 3.-Variation
of resonance values of n with the length of the straight 1. me9 for vertical oscillations; 2. nres for radial oscillations.
sections.
To calculate the resonances, we first write down the equation of oscillation in the form:
x”+ x2Cg(4 +_fo1x = q(4,
(18)
where the perturbations q(a) and!(o) are functions of period No, (q has the period ao). Then, from equation (16), the solution of equation (4) is:
dD,-do
?n
-
q --fK2[D,
exp (is p&y(u) + complex conj.]
I
(
w*(a)exp -it X
B
0
1
ruk
.
(19)
We will integrate this equation for a single element of period oo, taking D, on the right-hand side to be constant. We add a subscript m to q and f since the functions q and f depend on m, the element of periodicity. The value of o is measured from the beginning of an element.
Theoretical aspects of the Soviet 10 GeV synchrotron
-exp Dm+l -
D,
357
fc2exp(-2i$pK)D..*
=
qmy*dc+
[f,N2dg
B
+;D,
rfmw* do=&&.
(20)
We apply the method of averaging to this differential equation to obtain an equation for resonance (RABINOVICH(~J~) gives a more general method for the calculation of the damping of oscillations, etc.): D m+l -DD,=-.
SL” 2i sin /l
(21)
For the sake of brevity we will consider only one case here. Let the average distance (or value of the magnetic field H) in different sectors be perturbed by an amount AZ, (or AH,), and the magnetic field index by an amount Anm, then
fm=$n; 0
qm=%;
(orq,=_-.A+~)_ 0
Example 1. If AZ, = AZ, = a; Az3 = AZ, = -a; ,u~+~- ,& = 2.5 X 10-3, then the amplitude of vertical oscillations, F, after going through resonance (at n = 0.84) is almost 60 times greater than a. Example 2. Take qnz = 0, (f,,,# 0). As compared with the previous case, we have now only parametric resonance. When p = rr/2 (n = 0*84), the second harmonic of the field perturbation produces resonance in the vertical oscillations and when ,u = r/4 (n = O-79), it produces resonance in the radial oscillations. A particle performing radial phase oscillations can go through the resonance region many times so that the point We will now consider some numerical
Let
of maximum deviation may lie in this region. results for resonance with PZ= 0.79 (,u = n/4).
An, = An, = ho;
An, = An, = -ho;
then D, < Do exp (15h,). Since ho in the Academy of Sciences synchrotron is less than O-5 per cent, during a passage through resonance at the injection point, D, increases by 5-6 per cent. Hence if ,u~+~- plc N 2.5 X lop3 or greater, then the resonance we have been considering has no appreciable effect on the injection process. However, this resonance is much more important during the period of acceleration. If the amplitude of radial phase oscillations increases so much that the instantaneous orbit falls in the resonance region, then, as a result of the numerous passages through resonance, it is possible to have a slow increase in the oscillations. In order that this increase should be inhibited it is sufficient that the amplitude of the oscillations dies away during a period of phase oscillation, owing to the quicker growth of the magnetic field. Let the point of reversal of the radial phase oscillations (of amplitude O.O2R,) lie in the centre of the resonance region; R (&z/aR) ci 100, the energy of the particles W, = 10 MeV, the accelerating potential 6 kV, then D, < Do exp (32h,). Here the index is higher than in the previous case, since in this example, the radial
358
M. S. RABINOVICH
velocity changes direction at the resonance point, so that the particle orbit is in the resonance region for a relatively long time. However, at a distance of 3 cm from the resonance point the index has fallen by a factor of 16. An analogous expression for parametric resonance can also be obtained for the case when n = 0.84 (between the second harmonic and the vertical oscillations). It is evident from the foregoing discussion that it is possible to have an increase in undamped oscillations due to resonance in the injection process, in the acceleration, and in the ejection of particles from the chamber. However, it is possible to avoid these resonances, given the permissible limitations of the magnetic field, and this has been done in the 10 GeV synchrotron. RESONANCE
IN SYNCHROTRON
OSCILLATIONS
The derivation of the phase equations depends basically on the use of a co-ordinate system formed by the instantaneous orbit and its normals. The accelerating field can be analysed into a series of travelling waves along the equilibrium orbit. We thus obtain an equation which enables us to consider the coupling between betatron and phase oscillations, slit oscillations, etc. In small accelerators the phase oscillation frequencies ol, are the most important. In the Academy of Sciences synchrotron the frequency of phase oscillations 0,/27r varies from 2000 to 700 c/s. Oscillations of such frequency can occur in the magnetic field and in the frequency and amplitude of the accelerating potential. The danger of these resonances is due to the low values of the phase oscillation frequencies. This increase of oscillation is dangerous, particularly in the injection process, as it can lower the number of particles captured into the accelerating cycle by a factor of about ten. During the acceleration, the oscillations are not dangerous if the particles remain in the region of stability, but scattering of particles towards the edges of this region leads to a loss. In this way the particles oscillate in resonance and are subsequently lost owing to the large oscillations of H at a frequency of the order 600 c/s. The greatest danger is resonance with magnetic field oscillations and at the frequency of the accelerating field. We will consider, as an example, the resonance increase due to magnetic field harmonics. The phase equation has the form :
(22) where E is the energy, o. the frequency of the accelerating field, I$ (= 4. + IX)is L n 1 the phase of the accelerating field, k = 1 f - , F,=ll-np /P( 1 - n)(27~R+ L)k is a coefficient to allow for the straight sections, V, is the sum of all the accelerating potentials, and Q is the resonance energy:
Q=
eRoPRo+ L)
hn
cos
Qt ,
TX!
(23)
where (h sib fit) is the oscillating part (or one of the Fourier components) of the magnetic field. The phase equation is non-linear so that consideration of resonance phenomena is extremely complicated. It is, however, possible to resolve the equation
Theoretical aspectsof the Soviet 10GeV synchrotron into a linear approximation tions may easily be found: $n
from which the amplitude of the additional phase oscilla-
y/%r R,&,
h
_ 4;
max -
359
H(l - n)(2nR, + L)
d&
where W1is the rate of change of the phase oscillation frequency. If special measures are not taken (they are usually anticipated) then the resonance with the second harmonic c/s of the magnetic field (1200 c/s) is very dangerous. (The first harmonic at 600 c/s is due to the rectification of the 50 cycle 12 phase a.c.) Hence h, mitxY 3 x 102h, (h, in oersteds). At resonance W = ---103rr and alin must be considerably less than 0.01 oersted. The value of h, (the amplitude of the magnetic field oscillations) which produces these violent perturbations is a million times smaller than the basic field. However, the rapidly varying part of the field h&2 dH
has a rate of change 50 times smaller than x,
and for resonance, it is the rate of
change which is important rather than the amplitude, which explains the small value of Iz,. It can be seen from this example that for values of h, as small as 0.01 oersted If the the non-linearity of the phase equation must be taken into account. The non-linearity is small, the equation can be solved by normal methods. basic method used is that introduced by KRILOV, BOGOLJUBOVand MITROPOLSKI. For small non-linearity we cannot consider the particle transition through the boundary of the stability region, but we can obtain a more precise value for the amplitude of oscillation. The rate of transition through resonance is small in the Academy of Sciences synchrotron so that even a small non-linearity changes the results markedly. Indeed, in the linear approximation for the amplitude of phase oscillations: h,, -+ 00 when W1-+ 0, whereas correcting for a small non-linearity gives a finite amplitude even for &, = 0. The physical reason for this is clear-the particles are removed from resonance by the frequency decrease which follows the growth of the phase oscillation amplitude. This is called active non-linearity (KRILOVand BOGOLJUBOV), as distinct from passive non-linearity, which does not limit the amplitude in the absence of damping. A very important factor here is the direction of the resonance transition. If the time and amplitude dependence of oscillation frequency act in the a#J aq same direction i.e. 1 *> 0 then the non-linearity acts as an effective damp1 at a~,,, ing. We look for the solution of equation (22) in the form:
cr=acos(7+s/))-; 1 -$OS(‘tY) [
s
1
cot#J,-~2cos3(r+W)
7
where T =
co1dt (,a1 is the phase oscillation frequency); a and w give the amplitude
0
and phase of the oscillations. In general, a and w are time-dependent, case I,U= & n/2 when a satisfies the algebraic equation: a3 -
16(1- 5% * I-;
Cot2+0
except in the
160 (1 +r)(l
;_$OP&))
=Oy
(24)
M.
360
S. RABINOVICH
where the minus sign corresponds to the phase y = rr/2, and the plus sign to the phase y = --n/2. The constant D is given by:
(where h is in oersteds). Here 6 is the ratio of the perturbation frequency Q to the frequency of small phase oscillations wi. Equation (24) will have three different real roots if: - 16(1 - 5)3(1 + 5)2 < _6.5 (25)
D2 1 +;cot2&,
(
1
If the inequality equation (25) is not fulfilled, then there is only one root. If the inequality sign changes to equality then we have the condition for the existence of one multiple and one simple real root. The multiple root is: I.609
lamI=
+’ (
1 +$oPf&
)
The simple root is: a, = 21a,l.
Fig. 4 shows the dependence of amplitude on the ratio of the perturbing frequency to the phase oscillation frequency. An analysis of the stability of the stationary solution.gives the following results. The upper branches of the curves in Fig. 4, corresponding to the phase y = 3~12, are stable, whilst of the two lower branches, corresponding to the phase y = -7r/2, one is stable and the other unstable. The boundary between the two regions is the point a,. If E increases in the 10 GeV synchrotron, then the characteristic point moves along the curve to the lower left, almost to the boundary between the stable and unstable regions. Further consideration of the behaviour of the characteristic point can only be qualitative, however slowly the phase oscillation frequency changes. Near the boundary of the stable region the amplitude of the oscillations increases rapidly and the point crosses to the upper, stable branch. As the transition takes place quickly the characteristic point no longer moves along the fixed curve but performs oscillations near it. The final amplitude will be between 2a, and 3a,. We can see that the maximum amplitude is attained when E < 1, i.e. resonance occurs earlier than the point of equality between small phase and forced oscillations, and the resonance effect increases with the magnitude of the perturbation. These qualitative considerations have been supported by numerical analysis of the equation for D = 0.04 and 2 = 104 = 6. The initial conditions are so chosen that at a considerable distance from resonance the amplitude of oscillation is equal to zero, i.e. it coincides with the fixed curve. The result of the integration is shown in Fig. 5 by the continuous line: the case which we have selected corresponds to a large
Theoretical aspects of the Sovret 10 GeV synchrotron
361
perturbation (the amplitude may be as great as 5 rad). Hence, although the amplitude is much less than in the linear case, it approximately corresponds to the greatest possible value. If the particle performs phase oscillations up to resonance, then it is impossible in the non-linear case to sum over the oscillations. This is illustrated in Fig. 5 by
FIG. 4.-Resonance curves for phase oscillations with varying values of cos & and for the amplitude of disturbance D as a function of &the ratio of the disturbance frequency to the phase oscillation frequency. l-upper
stable branch;
2-lower
unstable
branch;
3-lower
stable branch.
the broken curve, which was obtained by numerical integration. At some distance from resonance the amplitude of the oscillations is equal to O-2 rad. The resonance growth of amplitude sets in earlier but the final value obtained differs very little from the value obtained before, so that a strong resonance effect drives all particles to the edge of the stable region. As we shall see, if the characteristic point is situated on the upper curve at some distance from resonance then instead of the resonance growth of amplitude we have instead a monotonic decrease. Here, however, the
362
M. S. RABINOVICH
initial phase is important as well as the amplitude. It is easier to treat the linear and non-linear cases together, comparing 2a, with G$&
It can then be shown that the effect of non-linearity enables a reduction to be made in the magnitude of the field harmonic. In the calculations on the Academy of
FIG. 5.-Phase oscillation amplitude for a slow change in frequency (in the resonance region) for two cases:-the continuous curve gives the growth of amplitude from zero, the dotted curve gives the growth from aintt -0.2, D = 0.04, cos f& = 0.5, 6 = o*ooo1. l-resonance curve; 2-upper stable branch; 3-unstable branch; 4-lower stable branch.
Sciences synchrotron, consideration was also given to the increased rate of transition through resonance due to a rapidly fluctuating V,, (the accelerating potential) in the resonance frequency region. In this region, in addition to the normal perturbations, noise fluctuations are important. This question can also be analysed by a linear approximation and the requirements for the noise characteristics calculated. INJECTION
In the ii?jection process(2) a beam of particles, of energy 9 MeV, traverses a long path through various types of apparatus before entering a chamber enclosing one of the quadrants. We will consider here the relation between the beam parameters at the entrance to the magnet and the effectiveness of the injection process, i.e. the number of particles captured into the accelerating region. If the magnetic field configuration and the beam parameters were known, then consideration of the present problem would enable us to calculate the orbit. In actual fact, our information is restricted to the known magnetic field and beam parameter tolerances. We must therefore average over all permissible configurations which enable us to calculate the intensity of the accelerated ion beam only to within a factor of two. A detailed analysis shows that we &an average over the motion in the following way. In general, in a constant magnetic geld, the injected particles undergo free oscillations of such
Theoretical aspects of the Soviet 10 GeV synchrotron
363
phases that 50 per cent of the particles are lost on the third turn and the remainder on the sixth. (Previously we have not permitted any error greater than 10-20 per cent.) We will suppose that the beam enters the magnet chamber with a uniform density distribution, a width 2A, and a uniform angular dispersion of fa,. The axis of the beam is at an angle E to the optimum direction. We divide the injection process into two parts. Firstly, before the accelerating potential is applied the particles move along displaced orbits in the varying magnetic field. While the orbit remains fixed, there is a high probability that the particles will strike the injector or the chamber walls-we will call the portion of the particles not striking the injector or chamber walls the capture coefficient of the particles in the first stage of the injection (qI). Then: (27)
“171 = Yp%
where qp is the probability of the particles not striking the injector and Q is the probability of the particles not striking the chamber walls. Together with the probability qp, we must calculate the number of revolutions during injection. Pl VT
=~PG
(28)
’
where pr is the distance from the injector to the centre of the chamber and AR is the change in orbit of the particles (AR = 1.1 cm for Wi = 9 MeV). pi/AR is therefore the total change in the first stage of injection. When the orbit reaches the middle of the chamber, the accelerating potential is applied and a portion Q (transition coefficient) of the particles captured in the first stage are accelerated. The remaining particles (1 - qz) either do not fall within the region of phase stability and are not accelerated, or else strike the injector due to disturbance from radial pb,ase oscillations. The product of the two coefficients QQ is called the effectiveness of injection. We now introduce a series in a where
WWJ
a=-
(29) \ I
Pi
where a is the injection angle of a given particle and @(ai) is defined by equation (9). a can vary between a1 and a2 (aI > aB). We denote the angle between the axis of the beam and the optimum angle by a,. We also take: a,=-;
3AR Pi
a, = 2a,
(30)
Then : Pi 5’T =
4AR *O . a,,,i=l
i:
f P(4, i-1
ad -
@(h,
Ql,
a1 -a2 , _o - pi is the particle beam width, and @(a, a) is a function shown where amax = 2 graphically in Fig. 6.
364
M.
The family Fig. 7. From limits ensuring stable capture
S. RABINOVICH
of curves of qr for the Academy of Sciences synchrotron is given in these, the limits of accuracy governing the injection process (i.e. the -100 per cent capture) can be found immediately. Obviously, for (particularly during the period of adjustment), it is desirable to have
0.1 0.2 0.3 0.4 O-5 0.6 O-7 0.8 O-9 1-O
0
rY.
FIG. 6.-Graph
of function
CD(a, IX)(from equation
Ymm
FIG. 7.-Relation
between
Wi = 10 MeV;
(31);
and reference
degrees
the number of turns, the error in the injection horizontal dispersion of the beam, yms,x.
pi = 50 cm;
6).
dH/dt = 4000 oersteds/sec;
angle and the
beam width = 5 cm.
the beam with a given angular distribution. Under these conditions, the number of captured particles decreases slightly, but depends less critically on variations in the injection apparatus. However in order to calculate the capture coefficient in the first stage we must take into account collisions with the walls of the chamber (Fig. 8). We will now consider the undamped oscillation amplitude distribution at the end of the first stage. Suppose we have a parallel beam, the axis of which is inclined to
Theoretical
aspects
of the Soviet
10 GeV synchrotron
365
the optimum direction at an angle a,. The resultant amplitude distribution is shown in Fig. 9. We assume that, in this case, the distribution is approximately uniform when F = !?!!!fZ> a, (where pmaxis the amplitude of the oscillations, pi is the distance Pi
from the injector to the centre of the chamber, and the distribution function y(F) is
25
L FIG. S.-The
degrees
percentage of particles which avoid striking the chamber walls, given an error in angle of E, and a vertical dispersion of the beam 5,. D, is the chamber height; 20, = 25 cm.
0.8 0.6 0.4 0.2 0 0.2 FIG. 9.-The
amplitude
distribution
0.4
0.6 F for a parallel
0.8
1.0
beam after the first stage of injection.
equal to zero when F < a,). To this approximation the number of particles with small oscillation amplitudes is somewhat decreased for small a, but is increased for large a,. If the beam has a considerable angular scatter, then to a sufficient degree of accuracy the distribution has the form y(F) dF = + d/F dF
(32)
Fig. 10 compares this approximate equation for the amplitude distribution with a more exact expression. The second stage of the injection process is characterized by the development of 7
366
M. S. RABINOVICH
radial phase oscillations. If particles which have missed the injector on the first stage are to miss it again, the following condition must be satisfied: F + (P&i) < 1
(33)
where p+ is the amplitude of radial phase oscillations (which are simply related to the actual phases) and has two corresponding phases: $2 > $0;
A<
$0~
between which phase oscillations occur. The phase oscillation amplitude is a simple function of the ratio pdp, where p is the maximum possible phase oscillation amplitude: ,E= R,
J
eV, sin +,( 1 - & cot c&) L 71(27& + L)
1
(34)
FIG. lO.-Amplitude l-from
When,!?<
distribution after the first stage of injection for a wide beam with a large angular scatter (aI = 0.1, a2 = 0.3, E > 0.3). the approximate equation (32); 2-from the more exact expression (equation (62) of reference 6).
1: (35)
42 - 41 = 27&&p/P)?
Graphs can be drawn for the function s(pJ~) and also for the approximate expression:
4&/P)-
20 - docot60)’ Pd n-p
[1+
o*3(P$m2
1 .
Two possible errors can occur at the beginning of the second stage of injection: either the accelerating field is not applied at the correct time, or the energy of the beam (i.e. the initial frequency of the accelerating field) is incorrect. In practice, both these errors lead to the same result. If the beam of particles is not monoenergetic then it is evidently impossible to find an accelerating potential which will include all particles. Suppose that the position of the instantaneous orbit when the field is applied is different from the mean orbit by an amount p&f (the mean orbit is the
Theoreticalaspectsof the Soviet 10 GeV synchrotron
367
orbit for which the frequency of rotation is equal to the frequency of the accelerating field). M is related to the error in energy AEi by: piA4 = R,
AEi 2(1 -
(37)
rz)Wi’
or to the error At in the time of appearance of the field by: piM = R,
Hi At
(38)
Hi(l - n)
where Hi and Eii are, respectively, the magnetic field and its rate of change at the moment of injection. The error in the initial frequency Aw displaces the mean orbit position (which we are taking as our co-ordinate origin) by an amount: pJ4
Aw(2rR, + L)
= R,
(39)
coo[n(27rR, + L) - L] ’
We now define the initial phase ~init and an initial phase velocity ~init (or an initial radial deviation piM) to correspond to pd, the amplitude of radial phase oscillations.
[(?_J2_ (!g2] = sin
$0 -
sin &it Win
h
-
($0 -
Ad
CoS 40
MO cos 543
’
(40)
As can be seen from equation (40), ~init is a multi-valued function of
[($‘- (Jg2]. It is possible, from equation (40), to find two minimum values (of absolute magnitude o construct a combination (&it,2 - +inrt,r) and to calculate the
(C!) 2- (F)“I**
&it,2 - hit,1 = 2nTTEm [
(41)
Obviously, the only difference between E, and E (defined in equation (36)) is the argument. From equations (40) and (41) we see that ($rnrt,a- &,& is the magnitude of the initial phase region for which, with the given M, the amplitude of radial phase - hnit,l is the proportion of 25. particles with such phases. Hence the number of particles with amplitudes between p6 and &+d+ is equal to oscillations is less than or equal to pd, and
&nit,2
f% dpPd. dP+
(42)
In the second stage, only those particles will be captured which have a free oscillation amplitude: pmax < pi - po. If y(F) is the amplitude distribution, then the number of particles obeying this inequality is:
s
1-(PJPi)
0
Y(F)CJF.
368
M. S.
RABINOVICH
Multiplying equations (42) and (43), and integrating over all p4, we obtain the capture coefficient for the second stage of injection: p2 =
1% dp4~-(@‘$(F) dF.
(44)
Changing the order of integration in equation (44) and integrating with respect to p+: % = l-e-(;
2000
3000
2/(1 - F)2 -
4000
5000
6000
M2)y(F)
dF.
7000
“0
Fro. 1 I.-The
transition coefficient (Q) for a parallel beam with an angular error aE and an error M in the time of appearance of the field. V,, is the amplitude of the accelerating potential in volts.
Equation (45) is a simpler form, and is more useful for calculation of amplitude distributions. Equation (44) also gives the amplitude distribution for the second stage of injection. If the particles have a range of values of M (corresponding to different energy values), with a distribution y&W), then the coefficient becomes:
Graphs of q2 for various values of a, and M are given in Fig. 11, If the particles also have an energy distribution then it is difficult to integrate equation (46) for large angles of injection (see equation (32)), and hence we replace equation (32) by the approximate expression : y(F) = 2F.
(47)
This approximation slightly underestimates i2. If a frequency q times greater than the frequency of rotation is employed, then the equations are unaltered, except that in equation (34), ,5 is reduced by a factor q*. Table 1 gives the maximum value r], for an injection energy wi = 10 MeV, for an equilibrium energy distribution and for two different values of pi. Thus, for q = 1, the maximum value i2 is of the order 25 per cent. To summarize, we have briefly considered theoretically some of the factors affecting the performance of the Academy of Sciences synchrotron (or of any high energy synchrotron). From these calculations it is possible to determine the permissible tolerances on the accelerator parameters for normal operation to be possible.
Theoretical aspects of the Soviet 10 GeV synchrotron
369
TABLE1.-VALUESOF qz( %)
Distance from injector
Form of function Energy distribution (%)
Some important topics such as the problems of ejection, and injection optics, have not been considered here, but it is hoped that this paper gives an overall survey of the theoretical problems which have been considered in the design of the Academy of Sciences synchrotron. The author wishes to thank V. I.
VEKSLER, A. A. KOLOMENSKI and V. A. PETUKHOV
for their assistance in this work.
REFERENCES 1. VEKSLER V. I., KOLOMENSKY A. A., PETUKHOV V. A. and RABINOVICH M. S. Communication to the All-Union Conference on High-Energy Particle Physics, Moscow (1956). 2. VEKSLERV. I. et al. J. Nucl. Energy 4,333 (1957). 3. RABINOVICH M. S., BALDINA. M. and MIKHAILOVV. V. The Motion of Particles in a Synchrotron with Straight Sections. U.S.S.R. Academy of Sciences Physical Institute Report (1949). 4. RABINOVICH M. S., BALDINA. M. and MIKHAIL~V V. V. Ibid. (1950); Zh. eksp. teor. $2. 31,
993 (1956). 5. RABINOVICH M. S. The Theory of Free Oscillations in an Accelerator with Straight Sections. U.S.S.R. Academy of Sciences Physical Institute Report (1947). 6. RABINOVICH M. S. The Theory of a Synchrotron with a Segmented Magnet. U.S.S.R. Academy of Sci’ences Physical Institute Report (1952). 7. MIKHAILOVV. V. The Effect of a Segmented Magnetic Field on Particle Motion. U.S.S.R. Academy of Sciences Physical Institute Report (1949). 8. RABINOVICH M. S. A Theoretical Investigation of Resonance Phenomena in Accelerators. U.S.S.R. Academy of Sciences Physical Institute Report (1952). 9. BALDINA. M. and MIKHAILOVV. V. Periodic Motion of ChargedParticles in an Magnetic Field. U.S.S.R. Academy of Sciences Physical Institute Report (1952). 10. RABINOVICH M. S. A Theoretical Study of Acceleration by a Changing Magnetic Field. U.S.S.R. Academy of Sciences Physical Institute Report (1953).