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Synchrotron oscillations in high-energy synchrotrons
NUCLEAR
INSTRUMENTS
AND METHODS
72 ( 1 9 6 9 ) 7 9 - - 8 I ; © N O R T H - H O L L A N D
PUBLISHING
CO.
S Y N C H R O T R O N OSCILLATIONS IN H I G H - E N E R G Y S Y N C H R O T R O N S A. PIWINSKI
Deutsches Elektronen-Synchrotron DESY, Hamburg Received 27 December 1968 At very high energies the synchrotron frequency can be of the same order as the revolution frequency. For this case the synchrotron oscillation is investigated for a ring structure subdivided into discrete sections with acceleration and synchrotron radiation,
respectively. A more correct value for the synchrotron frequency and the bunch dimensions and also a limit for the stability of the frequency of the linearized synchrotron oscillation is obtained.
Computing the frequency and the amplitude of the synchrotron oscillation, one usually assumes that the accelerating electric field is distributed uniformly about the orbit. This assumption is valid only if the period of the synchrotron oscillation is long as compared to the time for passing an accelerating unit and the section between two accelerating units. This assumption can also be written in the form
dA dp/ds = - (k/R)KD(AE/E~), where k = harmonic number, _~ = circumference/2n, D = x/(AE/E~) = dispersion, K = curvature. The motion in a curved section is then given by the expression
f~yn '~ N'f0, where N denotes the number of accelerating units and Jo the revolution frequency. In electron synchrotrons or storage rings for very high energies (m 50 GeV) this assumption is not always satisfied. It is then necessary to take into account the exact distribution of the accelerating voltage. We will, in this computation, assume that changes of the machine parameters are slow as compared to the damping time of the oscillation. When an electron passes an accelerating unit its phase deviation from the equilibrium phase A4~= 4~-qSs remains constant, whereas for the energy deviation from the equilibrium energy AE = E--E~ the following relation holds
The transfer matrix for an accelerating unit and the following curved section has the form
[
1
2p/q ] ,
M = M(s2,0)= [
[ -2pq
1 - 4p 2 J
where
2p/q = e U c o s ¢ , ;
2pq = ~
k f ~,s2KDds.
We now assume that the ring consists of N equal accelerating units and curved sections. Straight sections do not change A E and Aq~ and give no contribution to the transfer matrix. In order to have a finite amplitude of oscillation it is necessary that both eigenvalues,
ddE/ds = e(V/sl) (sin ¢ - sin ~bs)~ e(U/s 0 cos ¢~. A¢, where the coordinate s denotes the length on the orbit and sl and U denote the length and the voltage of the accelerating unit. The matrix notation for the motion in the accelerating unit is
•1,2 = 1 - 2 p 2 +_i2p(1 _ p 2 ) ~ ,
1[i _ost s s ]
do not have an absolute value greater than or equal to one. From this one obtains
p2 - keUcosCs fs* K D d s < l . -4RE ~,
In a curved section of the orbit we will at first neglect the statistical character of the radiation and its dependence on the energy deviation and we will introduce it later as a perturbation. Under this assumption, the energy deviation remains constant, whereas the change of the phase deviation is given by
Assuming a uniformly distributed accelerating voltage, one obtains for the synchrotron frequencyX):
fs2n = f 2 k e N V cosd)~ (~ KOds. 4~2E~_~ 3 79
80
A. P I W I N S K I
dlA2[ = A * d A + A d A *
+ [dAI 2
= -8(A*v*-Av2)/@,v*-v~v2)-
j
-8~lvJli@,v*-~*~2Y. The mean values of all 8 and e2 are given by 2)
1
= W~(1+2CAE/E)ds/c,
~2 = Q~ds/c,
where 3 2, W~ = 23rec~sEsK
Qs = (55/24d3)rec2h)~6EsK 3,
7s = Es/(meC2),
C = 1 +½go + (o/n)aB/Ox,
0
0 Fig. 1. Ratio of the exact synchrotron frequency to the synchrotron frequency computed with the assumption that the accelerating field is distributed continuously.
For this frequency we now have the stability condition 3):
Ly. < (Nl~)fo. This means, that the phase shift for one period, computed for a uniformly distributed voltage, has to be smaller than two. The more exact synchrotron frequency is given by fsyn = fo(N/n) arcsin {(n/N) (fsyn Ifo)}, which agrees with f~y, for large N or slow synchrotron oscillations. The dependence of fsyn/fsyn on fsyn/f0 is shown in fig. 1. Near the limit of the synchrotron frequency also the bunch dimensions are changed, as the following calculation shows. The synchrotron oscillation can be described using the eigenfunctions of the transfer matrix•
aE(s)] : av(s)+ A*,,*(s),
r e -- electron radius. After averaging over all phases of the synchrotron oscillation and over the orbit one obtains ( d l A 2 [) = - 2 ( ( W s / E s ) C ) I A 2 [
ds/c-
- {(Qslv~l>/(vxv*-vl*V2)
}ds/c.
The first term yields the damping constant ~s, which is not changed due to the discontinuous distribution of the accelerating field. The second term, divided by the first term, is the stationary value of [A z [ caused by the quantum nature of the radiation. To calculate this ]A z [ with ( d [A z [) = 0 it is suitable to assume within the magnets K = const, and D = const., which does not mean an essential restriction. The averaging along the orbit can then easily be performed. The mean squares of the energy and phase deviation at a fixed point s on the orbit are obtained averaging A2E(s) and A2q~(s) over all phases of the synchrotron oscillation. tr2e = (¼(Qs) / as) {(1 - ] p 2 ) / ( 1 _p2)}. • {1--4p2(s/s,)(1--s/s1)},
0 ~-- s ~-- s i ;
tr2E = (¼(Qs)lo~s)(1-~p2)[(1-p2),
s1 = < s= < s2;
tr2~, = q 2 ( ¼ ( Q ~ ) [ e ~ ) ( 1 - ] p 2 ) / ( 1 - p 2 ) ,
0 <- s <- s 1 ;
a2, = q2( ¼( Q~) / cq) {(1 - ] p 2 ) / ( 1 _p2)}.
where
v(s) = M(s,O)v(O),
M. v(O) = ,Iv(O).
The constants A and A* are determined such that quantum fluctuations and damping are in equilibrium. If a photon of energy e is emitted A E j u m p s by - 8 . With a = (/)~AE --I)~A~D)/(1)IV ~ -- I)~V2),
we obtain for the change of the absolute value of A 2
• {1--4p2(S--Sl)(S2--S)/(S2--Sl)2),
Sl <2 S ~ S2.
Here p is given by
p = (n/N) (f~y,~fo), whereas q does not depend on the distribution of the accelerating field. The last equations show that, when fsy. tends to the limit, the width, as far as D is not zero, and the length of the bunches increases infinitely at all
S Y N C H R O T R O N O S C I L L A T I O N S IN H I G H - E N E R G Y S Y N C H R O T R O N S
points of the orbit with two exceptions: In the centre of a curved section the length has a minimum, and in the centre of an accelerating unit the energy deviation has a minimum and consequently also the part of the bunch width that is determined by the energy deviation. Both minima are approaching one third of the value
81
one would obtain assuming a continuously distributed accelerating voltage. References 1) D. Bohm and L. Foldy, Phys. Rev. 70 (1946) 249. 2) M. Sands, Phys. Rev. 97 (1955) 470. a) A. Piwinski, DESY Bericht 67/7 (1967).
INSTRUMENTS
AND METHODS
72 ( 1 9 6 9 ) 7 9 - - 8 I ; © N O R T H - H O L L A N D
PUBLISHING
CO.
S Y N C H R O T R O N OSCILLATIONS IN H I G H - E N E R G Y S Y N C H R O T R O N S A. PIWINSKI
Deutsches Elektronen-Synchrotron DESY, Hamburg Received 27 December 1968 At very high energies the synchrotron frequency can be of the same order as the revolution frequency. For this case the synchrotron oscillation is investigated for a ring structure subdivided into discrete sections with acceleration and synchrotron radiation,
respectively. A more correct value for the synchrotron frequency and the bunch dimensions and also a limit for the stability of the frequency of the linearized synchrotron oscillation is obtained.
Computing the frequency and the amplitude of the synchrotron oscillation, one usually assumes that the accelerating electric field is distributed uniformly about the orbit. This assumption is valid only if the period of the synchrotron oscillation is long as compared to the time for passing an accelerating unit and the section between two accelerating units. This assumption can also be written in the form
dA dp/ds = - (k/R)KD(AE/E~), where k = harmonic number, _~ = circumference/2n, D = x/(AE/E~) = dispersion, K = curvature. The motion in a curved section is then given by the expression
f~yn '~ N'f0, where N denotes the number of accelerating units and Jo the revolution frequency. In electron synchrotrons or storage rings for very high energies (m 50 GeV) this assumption is not always satisfied. It is then necessary to take into account the exact distribution of the accelerating voltage. We will, in this computation, assume that changes of the machine parameters are slow as compared to the damping time of the oscillation. When an electron passes an accelerating unit its phase deviation from the equilibrium phase A4~= 4~-qSs remains constant, whereas for the energy deviation from the equilibrium energy AE = E--E~ the following relation holds
The transfer matrix for an accelerating unit and the following curved section has the form
[
1
2p/q ] ,
M = M(s2,0)= [
[ -2pq
1 - 4p 2 J
where
2p/q = e U c o s ¢ , ;
2pq = ~
k f ~,s2KDds.
We now assume that the ring consists of N equal accelerating units and curved sections. Straight sections do not change A E and Aq~ and give no contribution to the transfer matrix. In order to have a finite amplitude of oscillation it is necessary that both eigenvalues,
ddE/ds = e(V/sl) (sin ¢ - sin ~bs)~ e(U/s 0 cos ¢~. A¢, where the coordinate s denotes the length on the orbit and sl and U denote the length and the voltage of the accelerating unit. The matrix notation for the motion in the accelerating unit is
•1,2 = 1 - 2 p 2 +_i2p(1 _ p 2 ) ~ ,
1[i _ost s s ]
do not have an absolute value greater than or equal to one. From this one obtains
p2 - keUcosCs fs* K D d s < l . -4RE ~,
In a curved section of the orbit we will at first neglect the statistical character of the radiation and its dependence on the energy deviation and we will introduce it later as a perturbation. Under this assumption, the energy deviation remains constant, whereas the change of the phase deviation is given by
Assuming a uniformly distributed accelerating voltage, one obtains for the synchrotron frequencyX):
fs2n = f 2 k e N V cosd)~ (~ KOds. 4~2E~_~ 3 79
80
A. P I W I N S K I
dlA2[ = A * d A + A d A *
+ [dAI 2
= -8(A*v*-Av2)/@,v*-v~v2)-
j
-8~lvJli@,v*-~*~2Y. The mean values of all 8 and e2 are given by 2)
1
= W~(1+2CAE/E)ds/c,
~2 = Q~ds/c,
where 3 2, W~ = 23rec~sEsK
Qs = (55/24d3)rec2h)~6EsK 3,
7s = Es/(meC2),
C = 1 +½go + (o/n)aB/Ox,
0
0 Fig. 1. Ratio of the exact synchrotron frequency to the synchrotron frequency computed with the assumption that the accelerating field is distributed continuously.
For this frequency we now have the stability condition 3):
Ly. < (Nl~)fo. This means, that the phase shift for one period, computed for a uniformly distributed voltage, has to be smaller than two. The more exact synchrotron frequency is given by fsyn = fo(N/n) arcsin {(n/N) (fsyn Ifo)}, which agrees with f~y, for large N or slow synchrotron oscillations. The dependence of fsyn/fsyn on fsyn/f0 is shown in fig. 1. Near the limit of the synchrotron frequency also the bunch dimensions are changed, as the following calculation shows. The synchrotron oscillation can be described using the eigenfunctions of the transfer matrix•
aE(s)] : av(s)+ A*,,*(s),
r e -- electron radius. After averaging over all phases of the synchrotron oscillation and over the orbit one obtains ( d l A 2 [) = - 2 ( ( W s / E s ) C ) I A 2 [
ds/c-
- {(Qslv~l>/(vxv*-vl*V2)
}ds/c.
The first term yields the damping constant ~s, which is not changed due to the discontinuous distribution of the accelerating field. The second term, divided by the first term, is the stationary value of [A z [ caused by the quantum nature of the radiation. To calculate this ]A z [ with ( d [A z [) = 0 it is suitable to assume within the magnets K = const, and D = const., which does not mean an essential restriction. The averaging along the orbit can then easily be performed. The mean squares of the energy and phase deviation at a fixed point s on the orbit are obtained averaging A2E(s) and A2q~(s) over all phases of the synchrotron oscillation. tr2e = (¼(Qs) / as) {(1 - ] p 2 ) / ( 1 _p2)}. • {1--4p2(s/s,)(1--s/s1)},
0 ~-- s ~-- s i ;
tr2E = (¼(Qs)lo~s)(1-~p2)[(1-p2),
s1 = < s= < s2;
tr2~, = q 2 ( ¼ ( Q ~ ) [ e ~ ) ( 1 - ] p 2 ) / ( 1 - p 2 ) ,
0 <- s <- s 1 ;
a2, = q2( ¼( Q~) / cq) {(1 - ] p 2 ) / ( 1 _p2)}.
where
v(s) = M(s,O)v(O),
M. v(O) = ,Iv(O).
The constants A and A* are determined such that quantum fluctuations and damping are in equilibrium. If a photon of energy e is emitted A E j u m p s by - 8 . With a = (/)~AE --I)~A~D)/(1)IV ~ -- I)~V2),
we obtain for the change of the absolute value of A 2
• {1--4p2(S--Sl)(S2--S)/(S2--Sl)2),
Sl <2 S ~ S2.
Here p is given by
p = (n/N) (f~y,~fo), whereas q does not depend on the distribution of the accelerating field. The last equations show that, when fsy. tends to the limit, the width, as far as D is not zero, and the length of the bunches increases infinitely at all
S Y N C H R O T R O N O S C I L L A T I O N S IN H I G H - E N E R G Y S Y N C H R O T R O N S
points of the orbit with two exceptions: In the centre of a curved section the length has a minimum, and in the centre of an accelerating unit the energy deviation has a minimum and consequently also the part of the bunch width that is determined by the energy deviation. Both minima are approaching one third of the value
81
one would obtain assuming a continuously distributed accelerating voltage. References 1) D. Bohm and L. Foldy, Phys. Rev. 70 (1946) 249. 2) M. Sands, Phys. Rev. 97 (1955) 470. a) A. Piwinski, DESY Bericht 67/7 (1967).