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Storage ring longitudinal instabilities
Nuclear Instruments and Methods in Physics Research A 393 (1997) 18-22
NUCLEAR INSTRUMENTS 8 METHODS IN PHYSICS RESEARCH SectIonA
ELSVIER
Storage ring longitudinal instabilities K. Wille Institute for Accelerator
Physics and $vnchrotron
Radiation,
University of Dortmund, D-44221 Dortmund,
German?,
Abstract The FEL gain is proportional to the particle density n cc l/as. Both the beam current and the bunch length are influenced by electromagnetic fields, generated by the bunched beam. The longitudinal oscillations of single electrons or the whole bunch can be excited by these fields. This effect causes a blow-up of the bunch dimensions or even the loss of the beam. The only cure against these instabilities is the design of a vacuum chamber with extremely low impedance.
1. Introduction An undulator with the period length A,, the strength parameter K, and the number of periods N, provides in FEL operation the gain
G cc K2N312 83u 2.3 Y
Besides the beam energy y the gain is mainly determined by the particle density nh in the bunch. In addition to low transverse emittance, high beam currents and short bunches are required. Unfortunately, above a particular beam intensity, the bunches become unstable, which limits the current and causes bunch lengthening. Both effects limit the gain of the FEL. In the following, some principal characteristics of instabilities in the longitudinal phase space are discussed. An instability needs in the most simple case two elements: a resonator and an amplifier, which picks up the vibration signal and feeds it back to the resonator, as shown in Fig. 1. G(s) is the transfer function of the resonator after applying a Laplace transform and V(s) = v the constant amplification. The oscillation U(s) =
G(s) 1 - vG(s) ’
(2)
grows exponentially if v exceeds a sharp threshold. The &function is necessary to start the oscillation. Usually it is the signal noise. In the case of longitudinal instabilities in storage rings the resonator is a single electron or the bunch focused in the potential of the accelerating RF voltage. The focusing
Fig. 1. Simplest model of an instability.
causes phase and energy according to AJ?(t) + 2a,AE(t) with the damping quency
Q, =
w(J
SO168-9002(97)00421-X
Aul(t) and AE(t)
+ SZfAE(t) = 0, constant
eU,qctcos
-
27cE
a, and the synchrotron
(3) fre-
ul,
’
where U0 is the peak voltage of the RF, q the harmonic number, e the charge of an electron, CIthe momentum compaction factor, ul, the RF phase and w,, the revolution frequency. The synchrotron radiation leads to the natural bunch length
(5) with the velocity of light c and the momentum of the particles in the bunch.
Ap/p
0168-9002/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved PII
J
oscillations
width
K. Wille 1 Nucl. Instr. and Meth. in Ph_vs. Res. A 393 (1997) 18.~22
19
2. Robinson instability
Coherent bunch oscillations rent according to I,(t) = 21,expi(qo,t
modulate
- Y(t))
the bunch cur-
with Y(t) = YOsinO,t. (6)
Here only the first harmonic of the bunch spectrum (i.e. the cavity frequency) is of interest. The phase oscillation Y(t) generates synchrotron sidebands with the frequencies w+ = qoO + Q, and o- = qw, - 52,. If the beam frequency qwO is not exactly at the maximum of the cavity resonance, this phase oscillations cause an amplitude modulation of the accelerating voltage with the synchrotron frequency Q,. Depending on the phase of the modulation it can excite or damp the coherent bunch oscillation. This effect is called Robinson instability or Rohimm
dmping
[ 11.
In a more
formal way one can describe this effect by the complex cavity impedance Z,,, = Z, + iZ,. The bunch current induces the voltage ii,@) = I,(t) z,,,
(7)
The impedance depends on the frequency. define the following selected impedances:
Thus,
we
01
0. 40
45
Fig. 2. Robinson detuned cavity.
50
instability
55
in a storage
60
CL)
ring with an inductively
For electron beams one has to take the additional radiation damping into account which has to be added to the damping constant (12). It is easy to get rid of this instability. According to Eq. (12) and Fig. 2 a detuning to the capacitive region inverts the sign of uR and provides additional damping. Another method to fight against coherent instabilities is the use of an active feedback system.
Z+ at (0 = CLODt- Q,, Z” at (0 = LIW,,. Z
(8)
3. Potential-well
distortion
at 0, = ~ito~ - Q,
With these impedances we can write the average power transferred from the beam to the cavity in the form (U,(r)l,(t))
= 21;
zo + ; (Z’ - z;)
1
Y(f)
(9) The power of one elementary
(U,I,) ---Lzz To Ih
A&
(U,I,) jPzz---
charge is Ai?
AY
E =-’2rcqcY
ToIbE
(10)
Here basic equations describing the longitudinal particle motions have been applied. From Eqs. (9) and (10) we get the differential equation (h = const.) AY(t) + 2+AY(t)
+ o;AY(t,
It describes the coherent damping constant UR =
Much more complicated are unstable incoherent particle oscillations driven by higher order modes (HOM) of the bunch spectrum. These higher-order fields are superimposed to the accelerating field and cause a distortion in the bunch region (Fig. 3). This effect is called potentiulwell distortion [2]. The synchrotron frequency is determined by the gradient of the RF voltage. At very low currents the gradient is
= h.
synchrotron
2ry;sys(Z; - Z)Ih.
(11) oscillation
u’ 05 0
with the -0 5
(12)
-1 0
Since in a storage ring cos Ys < 0, the oscillation becomes unstable for Z: > Z;. This occurs in a ring with an inductively detuned cavity as shown in Fig. 2.
1
2
Fig. 3. Potential-well distortion fields of the bunch spectrum.
4
3
caused
'
by higher
II. STORAGE
phase ’ harmonic
RING
FELs
K. Wille / Nucl. Ins&. and Meth. in Ph_vs.Rex A 393 (1997) 18-22
20
4. Wake fields
simply given by (13) At higher beam currents, however, a significant amount of the total voltage acting on the beam is induced in the cavity and the vacuum chambers by the beam itself. In a good approximation one can assume a purely inductive impedance Z(w),, = ioZ,, The induced voltage is (14) For many applications
it is convenient
to define the
normalized impedance
-Zll = lo,,Z,,
with
n
n= w
(mode number)
The field of a charged relativistic particle is in the laboratory frame Lorentz contracted. The field lines spread out in a uniform vacuum chamber radially with a small opening angle of l/y. The image current of the charge moves along without losses. At sudden changes of the cross section, however, a significant fraction of the field is reflected and the particles loose energy. These so-called wake fields are able to act on the particles depending on their position in the bunch [3]. Because of causality each particle sees only fields of charges ahead of it. Fig. 4 shows a bunch passing through a pillbox cavity simulated with the numerical code MAFIA [4]. With the longitudinal wake field Eii of a particle at location s acting on a test charge at s’ we can define the wakefunction in the form
(18)
(15)
WO
and the induced voltage becomes
The total induced voltage at the location s’ for a bunch with the longitudinal particle distribution n,(s) is given by
(16)
= U HOM
This voltage is added to the RF voltage and we can define an effective voltage Uef, = U,,,
sin 'II,+
(Ucavcos
U: +
U,)
= U,,, sin !PS+ U,,, cosLY,+$-Im cav (
Y(t) : ii
2
Y(t). >
=
n,(s) W,,(s
--e
In this formula dIb/dY depends on the longitudinal bunch shape. It says that a short bunch causes much stronger distortions than a long bunch. This behavior is particularly important for FEL operation. The modified voltage changes the gradient in the bunch region and leads to a frequency shift of the synchrotron oscillation. Since the distortion is strongly correlated to the bunch position, it oscillates coherently with the center of the bunch. Therefore, the bunch center only feels the normal gradient of the RF voltage and the coherent synchrotron frequency is unchanged. Completely different is the situation for the single particles in the bunch. They perform incoherent oscillations with quite different frequencies. The incoherent frequency shift is
s')
ds
(19)
and is usually called the wake potential. By integration we get the energy loss of the bunch AEHOM= e 1 n(s’) UHo&‘)ds’ J
+ -e21 1 co
(17)
-
j 5’
(20)
+oO
nb’)
--Ix1
n(s) W , (s - s’) ds ds’
s’
With the bunch current I(s’,t) = I, exp i(ks - wt) we can replace the particle distribution n,(s) and the wake potential becomes UHOM(s’,t)= -;
j,;I(,.;t+%)W,,(s-s’)d(s+ (21)
(18) with a formfactor K depending on the bunch shape. If Im (Z,,/n) > 0 the synchrotron frequency decreases and according to Eq. (5) the bunch length cr, increases.
a pillbox
a relativistic
K. WilielNucl. Instr. and Merh. in Phvs. Rex A 393 (1997) I&?? or after applying
Fourier
u HOM(t,O4 = - f(t.w)
transform
sexp (
-L PC
+-x
w(s - s’) -i----_
-_z,
PC
)
x N’,(.s - s’)d(s - s’)
LI-
+ A
50 -
(22)
l
x 0
The wake potential UHOM(f.011= - I(t,tu)Z,,(“) induced coupling
21
by the current impedance
(23)
1(t) depends
on the
Eo
us
1.55
0.032
1.55
0.042
2.2 I 0.033 3.0 0.032 from d$/dI
L) Q 2 20 -
hgitudinal
-L
IO Thus, we can write the wake function
(18) in the form
(25) A storage ring has a large number of elements installed in the vacuum system with more or less unavoidable steps of the cross section. It is usually impossible to measure in situ the different contributions to the total chamber impedance. The most convenient and accurate quantity to describe the total resistive impedance of the machine integrated over all frequencies is the lossfactor defined by k HOM
=
AEHOM I_ tJN2
’
where N, is the number of particles in the bunch. Using the wake function, the loss factor can be written in the form
or with the accelerator
(28)
The total power loss in a circular machine with h bunches distributed along the beam orbit becomes P HOM
=
1;
kHoM,foh
with
I,, = bqN&.
5.2 4.5 CT
4 3.5 -
[cm]
3 2.5 -
(2%
The bunch length has a strong influence on the loss factor as measured in many storage rings (Fig. 5). The wake fields pull or push the electrons depending on their location on the beam axis. This changes the equilibrium and the particle distribution in the bunch. The consequence is bunch lengthening starting above a certain current threshold. This effect has been carefully measured in many storage rings. As an example a measurement from BESSY [6] is presented in Fig. 6.
hleaswfflt 247 kV b104 SIhiIRAC-Rechnun~caSW SMtRAC-Rechnun~en loo0
7_
2 1.5 -
I 0.1
impedance
Z,,(ru)lI(o)12 do.
k HOM
Fig. 5. Loss factor of the SPEAR storage ring measured as a function of bunch length [S].
_.J 1
10
/
[WA]
too
Fig. 6. Bunch lengthening as a function of the beam current measured at BESSY including SIMTRAC calculations.
It is remarkable that the disturbed bunch can reach three to four times the length of an undisturbed bunch and the particle density decreases correspondingly. This behavior is of major importance for FEL operation in a storage ring.
5. Conclusion The electro-magnetic fields generated by an intense bunch influence drastically the longitudinal dynamics of a single particle as well as the collective dynamics of the whole bunch and are able to make the bunch unstable.
II. STORAGE RING FELs
22
K. Wille 1 Nucl. Instr. and Meth. in Phys. Res. A 393 (1997) 18-22
A coherent instability is comparatively harmless and simple standard cures are available to suppress it. A real problem are, however, the incoherent instabilities caused by higher-order-mode fields. According to Eqs. (12), (18) and (28), the effect of the higher-order-mode fields on the bunch grows with the resistive impedance Z/n of the vacuum chambers. Because of the large number of single impedances located at different places along the beam orbit it is impossible to compensate their effect by adding complementary impedances. Therefore, the only way to reduce longitudinal instabilities is to keep the chamber impedance as small as possible. The best would be a uniform vacuum system with constant cross section and no gaps and interruptions. This, however, is impossible because of the cavities, the kicker and septum magnets, the bellows, pick-up electrodes etc. necessary for storage ring operation. It is obvious that every vaccum vessel has to be designed and manufactured under strict low-impedance aspects. If the cross section has to be changed one has to do it smoothly without sudden steps. Here one should
note that even steps of some mm have a non-negligible effect. Particularly critical are kicker magnets. Therefore, novel techniques have been developed in order to keep the impedance of these devices at a low value [7]. Applying presently available techniques, chamber impedances of Z/n < 1 iz are possible.
References Cl1 K.W. Robinson, Stability of beam in radiofrequency system, Cambridge Electron Accelerator, Int. Reports CEA-11,1956. PI H. Wiedemann, Particle Accelerator Physics I & II, Springer, Berlin, 1993. [31 P.B. Wilson, Introduction to wake fields and wake functions. SLAC PUB-4547, 1989. c41T. Weiland, Nucl. Instr. and Meth. 212 (1983) 13. r51 P.B. Wilson, R.V. Servranckx, A.P. Sabersky, J. Gareyte, G.E. Fischer, A.W. Chao, IEEE Trans. NS-24 (1977) 1211. 161W. Anders, Ph.D. thesis, University of Dortmund, 1992. c71G. Blokesch, M. Negrazus, K. Wille, A slotted-pipe kicker for high-current storage rings, Nucl. Instr. and Meth. A 383 (1994) 151.
NUCLEAR INSTRUMENTS 8 METHODS IN PHYSICS RESEARCH SectIonA
ELSVIER
Storage ring longitudinal instabilities K. Wille Institute for Accelerator
Physics and $vnchrotron
Radiation,
University of Dortmund, D-44221 Dortmund,
German?,
Abstract The FEL gain is proportional to the particle density n cc l/as. Both the beam current and the bunch length are influenced by electromagnetic fields, generated by the bunched beam. The longitudinal oscillations of single electrons or the whole bunch can be excited by these fields. This effect causes a blow-up of the bunch dimensions or even the loss of the beam. The only cure against these instabilities is the design of a vacuum chamber with extremely low impedance.
1. Introduction An undulator with the period length A,, the strength parameter K, and the number of periods N, provides in FEL operation the gain
G cc K2N312 83u 2.3 Y
Besides the beam energy y the gain is mainly determined by the particle density nh in the bunch. In addition to low transverse emittance, high beam currents and short bunches are required. Unfortunately, above a particular beam intensity, the bunches become unstable, which limits the current and causes bunch lengthening. Both effects limit the gain of the FEL. In the following, some principal characteristics of instabilities in the longitudinal phase space are discussed. An instability needs in the most simple case two elements: a resonator and an amplifier, which picks up the vibration signal and feeds it back to the resonator, as shown in Fig. 1. G(s) is the transfer function of the resonator after applying a Laplace transform and V(s) = v the constant amplification. The oscillation U(s) =
G(s) 1 - vG(s) ’
(2)
grows exponentially if v exceeds a sharp threshold. The &function is necessary to start the oscillation. Usually it is the signal noise. In the case of longitudinal instabilities in storage rings the resonator is a single electron or the bunch focused in the potential of the accelerating RF voltage. The focusing
Fig. 1. Simplest model of an instability.
causes phase and energy according to AJ?(t) + 2a,AE(t) with the damping quency
Q, =
w(J
SO168-9002(97)00421-X
Aul(t) and AE(t)
+ SZfAE(t) = 0, constant
eU,qctcos
-
27cE
a, and the synchrotron
(3) fre-
ul,
’
where U0 is the peak voltage of the RF, q the harmonic number, e the charge of an electron, CIthe momentum compaction factor, ul, the RF phase and w,, the revolution frequency. The synchrotron radiation leads to the natural bunch length
(5) with the velocity of light c and the momentum of the particles in the bunch.
Ap/p
0168-9002/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved PII
J
oscillations
width
K. Wille 1 Nucl. Instr. and Meth. in Ph_vs. Res. A 393 (1997) 18.~22
19
2. Robinson instability
Coherent bunch oscillations rent according to I,(t) = 21,expi(qo,t
modulate
- Y(t))
the bunch cur-
with Y(t) = YOsinO,t. (6)
Here only the first harmonic of the bunch spectrum (i.e. the cavity frequency) is of interest. The phase oscillation Y(t) generates synchrotron sidebands with the frequencies w+ = qoO + Q, and o- = qw, - 52,. If the beam frequency qwO is not exactly at the maximum of the cavity resonance, this phase oscillations cause an amplitude modulation of the accelerating voltage with the synchrotron frequency Q,. Depending on the phase of the modulation it can excite or damp the coherent bunch oscillation. This effect is called Robinson instability or Rohimm
dmping
[ 11.
In a more
formal way one can describe this effect by the complex cavity impedance Z,,, = Z, + iZ,. The bunch current induces the voltage ii,@) = I,(t) z,,,
(7)
The impedance depends on the frequency. define the following selected impedances:
Thus,
we
01
0. 40
45
Fig. 2. Robinson detuned cavity.
50
instability
55
in a storage
60
CL)
ring with an inductively
For electron beams one has to take the additional radiation damping into account which has to be added to the damping constant (12). It is easy to get rid of this instability. According to Eq. (12) and Fig. 2 a detuning to the capacitive region inverts the sign of uR and provides additional damping. Another method to fight against coherent instabilities is the use of an active feedback system.
Z+ at (0 = CLODt- Q,, Z” at (0 = LIW,,. Z
(8)
3. Potential-well
distortion
at 0, = ~ito~ - Q,
With these impedances we can write the average power transferred from the beam to the cavity in the form (U,(r)l,(t))
= 21;
zo + ; (Z’ - z;)
1
Y(f)
(9) The power of one elementary
(U,I,) ---Lzz To Ih
A&
(U,I,) jPzz---
charge is Ai?
AY
E =-’2rcqcY
ToIbE
(10)
Here basic equations describing the longitudinal particle motions have been applied. From Eqs. (9) and (10) we get the differential equation (h = const.) AY(t) + 2+AY(t)
+ o;AY(t,
It describes the coherent damping constant UR =
Much more complicated are unstable incoherent particle oscillations driven by higher order modes (HOM) of the bunch spectrum. These higher-order fields are superimposed to the accelerating field and cause a distortion in the bunch region (Fig. 3). This effect is called potentiulwell distortion [2]. The synchrotron frequency is determined by the gradient of the RF voltage. At very low currents the gradient is
= h.
synchrotron
2ry;sys(Z; - Z)Ih.
(11) oscillation
u’ 05 0
with the -0 5
(12)
-1 0
Since in a storage ring cos Ys < 0, the oscillation becomes unstable for Z: > Z;. This occurs in a ring with an inductively detuned cavity as shown in Fig. 2.
1
2
Fig. 3. Potential-well distortion fields of the bunch spectrum.
4
3
caused
'
by higher
II. STORAGE
phase ’ harmonic
RING
FELs
K. Wille / Nucl. Ins&. and Meth. in Ph_vs.Rex A 393 (1997) 18-22
20
4. Wake fields
simply given by (13) At higher beam currents, however, a significant amount of the total voltage acting on the beam is induced in the cavity and the vacuum chambers by the beam itself. In a good approximation one can assume a purely inductive impedance Z(w),, = ioZ,, The induced voltage is (14) For many applications
it is convenient
to define the
normalized impedance
-Zll = lo,,Z,,
with
n
n= w
(mode number)
The field of a charged relativistic particle is in the laboratory frame Lorentz contracted. The field lines spread out in a uniform vacuum chamber radially with a small opening angle of l/y. The image current of the charge moves along without losses. At sudden changes of the cross section, however, a significant fraction of the field is reflected and the particles loose energy. These so-called wake fields are able to act on the particles depending on their position in the bunch [3]. Because of causality each particle sees only fields of charges ahead of it. Fig. 4 shows a bunch passing through a pillbox cavity simulated with the numerical code MAFIA [4]. With the longitudinal wake field Eii of a particle at location s acting on a test charge at s’ we can define the wakefunction in the form
(18)
(15)
WO
and the induced voltage becomes
The total induced voltage at the location s’ for a bunch with the longitudinal particle distribution n,(s) is given by
(16)
= U HOM
This voltage is added to the RF voltage and we can define an effective voltage Uef, = U,,,
sin 'II,+
(Ucavcos
U: +
U,)
= U,,, sin !PS+ U,,, cosLY,+$-Im cav (
Y(t) : ii
2
Y(t). >
=
n,(s) W,,(s
--e
In this formula dIb/dY depends on the longitudinal bunch shape. It says that a short bunch causes much stronger distortions than a long bunch. This behavior is particularly important for FEL operation. The modified voltage changes the gradient in the bunch region and leads to a frequency shift of the synchrotron oscillation. Since the distortion is strongly correlated to the bunch position, it oscillates coherently with the center of the bunch. Therefore, the bunch center only feels the normal gradient of the RF voltage and the coherent synchrotron frequency is unchanged. Completely different is the situation for the single particles in the bunch. They perform incoherent oscillations with quite different frequencies. The incoherent frequency shift is
s')
ds
(19)
and is usually called the wake potential. By integration we get the energy loss of the bunch AEHOM= e 1 n(s’) UHo&‘)ds’ J
+ -e21 1 co
(17)
-
j 5’
(20)
+oO
nb’)
--Ix1
n(s) W , (s - s’) ds ds’
s’
With the bunch current I(s’,t) = I, exp i(ks - wt) we can replace the particle distribution n,(s) and the wake potential becomes UHOM(s’,t)= -;
j,;I(,.;t+%)W,,(s-s’)d(s+ (21)
(18) with a formfactor K depending on the bunch shape. If Im (Z,,/n) > 0 the synchrotron frequency decreases and according to Eq. (5) the bunch length cr, increases.
a pillbox
a relativistic
K. WilielNucl. Instr. and Merh. in Phvs. Rex A 393 (1997) I&?? or after applying
Fourier
u HOM(t,O4 = - f(t.w)
transform
sexp (
-L PC
+-x
w(s - s’) -i----_
-_z,
PC
)
x N’,(.s - s’)d(s - s’)
LI-
+ A
50 -
(22)
l
x 0
The wake potential UHOM(f.011= - I(t,tu)Z,,(“) induced coupling
21
by the current impedance
(23)
1(t) depends
on the
Eo
us
1.55
0.032
1.55
0.042
2.2 I 0.033 3.0 0.032 from d$/dI
L) Q 2 20 -
hgitudinal
-L
IO Thus, we can write the wake function
(18) in the form
(25) A storage ring has a large number of elements installed in the vacuum system with more or less unavoidable steps of the cross section. It is usually impossible to measure in situ the different contributions to the total chamber impedance. The most convenient and accurate quantity to describe the total resistive impedance of the machine integrated over all frequencies is the lossfactor defined by k HOM
=
AEHOM I_ tJN2
’
where N, is the number of particles in the bunch. Using the wake function, the loss factor can be written in the form
or with the accelerator
(28)
The total power loss in a circular machine with h bunches distributed along the beam orbit becomes P HOM
=
1;
kHoM,foh
with
I,, = bqN&.
5.2 4.5 CT
4 3.5 -
[cm]
3 2.5 -
(2%
The bunch length has a strong influence on the loss factor as measured in many storage rings (Fig. 5). The wake fields pull or push the electrons depending on their location on the beam axis. This changes the equilibrium and the particle distribution in the bunch. The consequence is bunch lengthening starting above a certain current threshold. This effect has been carefully measured in many storage rings. As an example a measurement from BESSY [6] is presented in Fig. 6.
hleaswfflt 247 kV b104 SIhiIRAC-Rechnun~caSW SMtRAC-Rechnun~en loo0
7_
2 1.5 -
I 0.1
impedance
Z,,(ru)lI(o)12 do.
k HOM
Fig. 5. Loss factor of the SPEAR storage ring measured as a function of bunch length [S].
_.J 1
10
/
[WA]
too
Fig. 6. Bunch lengthening as a function of the beam current measured at BESSY including SIMTRAC calculations.
It is remarkable that the disturbed bunch can reach three to four times the length of an undisturbed bunch and the particle density decreases correspondingly. This behavior is of major importance for FEL operation in a storage ring.
5. Conclusion The electro-magnetic fields generated by an intense bunch influence drastically the longitudinal dynamics of a single particle as well as the collective dynamics of the whole bunch and are able to make the bunch unstable.
II. STORAGE RING FELs
22
K. Wille 1 Nucl. Instr. and Meth. in Phys. Res. A 393 (1997) 18-22
A coherent instability is comparatively harmless and simple standard cures are available to suppress it. A real problem are, however, the incoherent instabilities caused by higher-order-mode fields. According to Eqs. (12), (18) and (28), the effect of the higher-order-mode fields on the bunch grows with the resistive impedance Z/n of the vacuum chambers. Because of the large number of single impedances located at different places along the beam orbit it is impossible to compensate their effect by adding complementary impedances. Therefore, the only way to reduce longitudinal instabilities is to keep the chamber impedance as small as possible. The best would be a uniform vacuum system with constant cross section and no gaps and interruptions. This, however, is impossible because of the cavities, the kicker and septum magnets, the bellows, pick-up electrodes etc. necessary for storage ring operation. It is obvious that every vaccum vessel has to be designed and manufactured under strict low-impedance aspects. If the cross section has to be changed one has to do it smoothly without sudden steps. Here one should
note that even steps of some mm have a non-negligible effect. Particularly critical are kicker magnets. Therefore, novel techniques have been developed in order to keep the impedance of these devices at a low value [7]. Applying presently available techniques, chamber impedances of Z/n < 1 iz are possible.
References Cl1 K.W. Robinson, Stability of beam in radiofrequency system, Cambridge Electron Accelerator, Int. Reports CEA-11,1956. PI H. Wiedemann, Particle Accelerator Physics I & II, Springer, Berlin, 1993. [31 P.B. Wilson, Introduction to wake fields and wake functions. SLAC PUB-4547, 1989. c41T. Weiland, Nucl. Instr. and Meth. 212 (1983) 13. r51 P.B. Wilson, R.V. Servranckx, A.P. Sabersky, J. Gareyte, G.E. Fischer, A.W. Chao, IEEE Trans. NS-24 (1977) 1211. 161W. Anders, Ph.D. thesis, University of Dortmund, 1992. c71G. Blokesch, M. Negrazus, K. Wille, A slotted-pipe kicker for high-current storage rings, Nucl. Instr. and Meth. A 383 (1994) 151.