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Experiments in the transverse phase space with the CERN electron storage and accumulation ring (CESAR)
NUCLEAR
INSTRUMENTS
AND
77 (t97 o) 93-1o4;
METHODS
©
NORTH-HOLLAND
PUBLISHING
CO.
E X P E R I M E N T S I N T H E T R A N S V E R S E P H A S E SPACE W I T H T H E C E R N E L E C T R O N S T O R A G E AND A C C U M U L A T I O N R I N G (CESAR) K. H I 3 B N E R , E. J O N E S , H. K O Z I O L , L. M A G N A N I , M. J. P E N T Z and A. R U G G I E R O
CERN, Geneva, Switzerland Received 9 June 1969 The effect of non-linear betatron resonances on the lifetime of an accumulated electron beam was determined. Coherent vertical oscillations o f a coasting beam were investigated. The growth rates and the thresholds for the modes n = 2, 3, 4, 5 were
measured. A time correlation between the different modes was established. It was tried to influence the oscillations by making the clearing field electrodes resonate with the beam.
1. Introduction
mode m4):
The paper describes two sets of experiments: a. the dependence of the lifetime of the circulating particles on the Q-values, b. coherent transverse oscillations of the beam (dipole type oscillations).
"C1/7~m2=~k-hl/
\(m!)2"24'~-3, ]
\nm-mQ/
where b/h is the ratio of beam height to vacuum chamber height (for CESAR, b/h,~ 1/17) and nm is an integer > m Q. The threshold for m > l will not be much different from the threshold for m = 1s). it was therefore decided to use for this experiment a stacked beam with a large spread in energy and consequently in the Q-values, thus suppressing all coherent instabilities. The particle density p(E) was measured as a function of time by subsequent scanning of the stack with empty rf-buckets2). Q(E) was found by radio-frequency knock-out measurements6). The particle density was about 3 × 1 0 9 . I ( c m -3) where I is the circulating current in amperes. The corresponding number of particles in the ring was N ~ 6 × l011 I. The maximum current was 20 mA and the typical current 6 mA. The circulating current was measured by a current transformer with an integration time constant of I000 sec. The parameters of the rf-buckets z) used were
All experiments were performed with a coasting beam. The storage ring and its injector are described in1). A short list of the main parameters can be found in 2). 2. Non-linear resonances in the absence of space charge effects
If the condition n Q v + m Q n ~ p (n, m, p-integers) is fulfilled, perturbations in the magnetic guiding field can lead to a transfer of energy from the longitudinal to the transverse motion causing a growth of betatron amplitudes 3). The Q-values in CESAR depended upon the orbit radius and therefore also on energy. A certain number of resonances may thus occur at various energies in a stacked beam, and some of these may cause significant reductions in beam lifetime. 2.1. EXPERIMENTALPROCEDURE In principle, one could have measured the beam life-time with a single pulse injected and betatronaccelerated to different orbit radii1). This would have been too time consuming, since we wanted to investigate a relatively large region of the QH, Qv diagram. Furthermore, the interpretation of the results would not have been easy, as it would not have been possible to distinguish beam loss due to non-linear resonances from that due to coherent multipole oscillations of the beam. This becomes apparent if one compares the risetimes of the dipole mode (m = 1) to a higher order
a. stacking bucket f = 2.1 x l07 sec-2; F = 0.36; AE = l0 keV; ½A = 4.3 keV; b. scanning bucket f = 2.1 x 107 sec-2; F = 0.38; AE s = 8.4 keV;½A = 3.45 keV. The displacement of the whole stack by the scanning was hardly discernible. The typical number of scanning cycles was 3. Twelve different sets of Qv, QH were used, and are shown in fig. 1. The Q-values were adjusted by changing the currents in the quadrupole windings and in correction windings on the focusing lenses. 93
94
K. H O B N E R et al.
The gas pressure diminished by a factor 1.5 in the course of the experiments, reaching 7 x 10 -1~ Torr towards the end. In spite of the low pressure, the electrons were lost by single scattering only, due to the high percentage of argon in the residual gas. This circumstance was of decisive importance for the experiment, as the life-time would otherwise have been dependent upon the available aperture, which then would have had to be measured across the stack each time, since the aperture was determined by closed orbit distortions which varied very much from experiment to experiment. The half-lifetime was calculated to be 15 s for single scattering at the lowest pressure reached ( 7 × 1 0 -~1 Torr). 2.2. FACTORS LIMITINGTHE ACCURACY OF THE RESULTS The linear, incoherent, Q-shift by space charge forces 7) was always smaller than 3 x 10 -a. Neutralisation was prevented by applying a voltage of 6 V to the clearing field electrodes.
The error in the measurement of the Q-values by radio-frequency knock-out was AQv/Q v < __+3x 10-3; AQH/QH < __+1×
10 -3
.
Electron-electron scattering in the beam 8) could cause a redistribution of p(E) without beam loss, and transport particles to the regions emptied by non-linear resonances. An estimate of the influence of this effect can be obtained by considering a distribution p( E) = D, p(E) = O,
E < El, E < El,
and calculating the initial energy Eo E1 is equal to the probability of its loss through scattering by the residual gas in the vacuum chamber. Neglecting the transfer between the two transverse directions, which is obviously pessimistic, we get for particles that are non-relativistic in the centre of mass system
EI-Eo =(2×10 8"T'i)~.
Fig. 1. Qv as a function of Qn. Number at the crossing points: order of the resonance; other numbers: harmonic number of resonance.
E X P E R I M E N T S IN T H E T R A N S V E R S E P H A S E S P A C E
For i = 1 × 10 -2 A/cm 2 and z = 15 s, E1 - E 0 turns out to be ~_ 5 keV, which is less than the resolution given by the height of the scanning bucket.
Fig. 2. Scanning signal; t = 0: end o f stacking; horizontal scale: 20.4kHz/div. a. 35 pulses; scanning after ts¢ = 0 and 2 s ; b., c. and e. 5 pulses, t~ = 2 s; d. and f. 5 pulses, t~c = 0 and 2s.
95
2.3. EXPERIMENTAL RESULTS
In order to be sure that the dips in the density distribution p(E) were due to effects dependent only upon the Q-values, the following experiment was performed. A stack of 35 pulses was built up, extending over the range of orbit radii corresponding to the Q-trace no. 6 in fig. l. The scanning signal is shown in fig. 2a. The initial particle density is indicated by the first scan. Five pronounced dips can be seen in the second scan. To check whether the dips occur at the same place, i.e. at the same points (QH, Qv) on the Q-trace, when the initial particle density is different, a 5-pulse stack was placed successively at each place and scanned as before. The resulting scanning signals are shown in fig. 2b-2f. It can be seen that the dips occur at the same places as before, independently of the initial particle density. The validity of the basic premise of this experiment is confirmed also by the appearance of a 50-pulse stack (fig. 3), over which the Q-variation is small and where the Q-trace does not cross any non-linear resonance lines of order lower than 7. The Q-variations over this stack were A Qv = 0.01, A Qa ~ 0.005. Having satisfied ourselves that the positions of the dips were not dependent upon particle density, the stacks corresponding to the twelve Q-traces, shown in fig. 1, were each scanned repetitively at intervals of 1 see. All the stacks showed a density modulation. Stack 4 showed the least modulation, since the 6 th order resonances were quite weak. The reduction of lifetime in the dips does not only depend on the strength of the excitation coefficient of this particular resonance but also on the precise form of the Q-trace at the cross-over and on the strength and density of the nearby resonance lines. It was, for example, impossible to stack a beam
Fig. 3. Scanning signal 50 pulses stack. Scanning at intervals ofl s. Hot.: 20 kHz/div.
96
K. H O B N E R et al. TAaL~ 1
No.
~max %'O
Tmax/TO
¥/ro Order crossed
1
2
3
4
5
6
7
8
15 8 11.0 11.3
16 10.6 10 8.5 3.8 4 11.3 11.4 11.6 11.9 12.0 12.6 1.5 2.9 2.2 3.2 1.36 0.71 1.40 0.86 0.72 0.32 0.32 0.32 0.13 0.25 0.18 0.25 4(3) 5(6) 5(2) 6(5) 4(4) 4(4) 4(3) 4(2) 5(1) 6(4) 6(5) [1] 5(3) 5(3) [3] 5(3)
9
10 13.6 6.5 0.74 0.48 3(1) 5(2)
10
11
12
15 10 10 15.0 16.5 21.0 4.6 2 . 9 4.9 1.00 0.60 0.48 0.30 0.18 0.23 4(3) 4(3) 4(3) 5(n*)5(3)
6(2) [3l * Note that Q-trace 11 is practically coincident with a 5th order horizontal resonance over most of its length. This is an example of multiple traversal.
close to the point of confluence of 4 th order or 5 th order resonances. Large beam losses were observed near crossing points of single 4 th and 5 th order resonance lines. The results of the twelve experiments lend themselves to a semi-quantitative interpretation, which is summarized in table 1. The following quantities are given: "Cmax the maximum life-time observed anywhere in the stack; = the average life-time of the whole stack, obtained from the integrating beam transformer signal; Zo = the life-time calculated from single scattering. All the life-times are the time constants of exponential decay. The table shows also the order of resonance crossed by the stack. The number in ( ) indicates the number of crossings, and the number in [ ] the order of a resonance in proximity to the stack, but not crossed. It can be seen that the average lifetime in the stack is particularly reduced in case no. 4 and also, though not so much, in cases 7 and 8. This can be explained by the proximity of the integral resonance in case 4 and of the third-order resonance in cases 7 and 8. The quantity ¥/% is of interest for intersecting storage rings, because this would express the ratio of the practically achieved number of interactions to the theoretical number, in periods of observation time longer than half of the beam lifetime. =
Rapid beam losses: In the course of this series of experiments, particularly in experiments 7, 8, and 9, rapid beam losses were observed with the integrating current transformer which could not be satisfactorily explained. These three experiments were distinguished
from all the others by the fact that the Q-traces were near third-order resonance lines. It seems reasonable to assume that sextupole perturbations were rather strong in CESAR since we had built-in sextupole fields. The rapid beam loss was probably always triggered off by space charge forces driving part of the stack into resonance. In experiment 8 the part of the stack for which 1.8< Q v < 1.87 was sometimes lost 0.2 s after the last stacking cycle. The loss occurred if the total current I fulfilled the condition: (3.25 × 10-3 U - 1 7 x 10-3) <~ I ~< ~< (3.25× 10 -3 U - 2 . 5 × 1 0 - 3 ) , where U is the voltage applied to the clearing field electrodes. The two limits define two parallel lines I = #i(U), (i = 1, 2), spaced by A U = 4.5 V. The dependence of the Q-values upon U was measured to be
O < OQn/dU ~< 3 x 1 0 - 4 ;
0 ~ OQv/gU <~ 4 . 5 x 1 0 -4.
This yields the width of the stopband A Qs of the resonance 3Qn = 8,
AQs = (OQ/~3U)AU; AQ~ < 1.3x10 .3 . Another interesting quantity could be obtained from these inequalities, namely the upper limit of OQH/~?I, by assuming that the slope of I = #i(U) is due to the fact that the space charge forces have to compensate ~?Qn/OU if one changes the voltage U. In that case, the total differential of QH(I,U) should be zero along 9~.
~QH/~I + OQ,/~U" ~U/OI = O,
EXPERIMENTS
IN
THE
TRANSVERSE
SPACE
97
for the sextupole forces 7 x 10-3~>R3~>3 × 10 -3, for the octupole forces 2.3 x 10 -3/>.R4~>0.
which yields 0Q./3I ~ -10-'
PHASE
[a-q.
The value of (dQH/t3I)m~x calculated from 7) for 30 pulses stacked is
As a comparison we give the values for the I S R assuming Q = 9, x = I c m , v30 = 1 0 - 1 , OQH/3E=O R 3 = 3.7 x 1 0 - 3 .
(OQ./OI),... = - 6.5 x 10 -2 [ A - ~ ] . It was observed, further, that the beam loss could be suppressed in most cases by scanning the stack immediately after the last stacking cycle. We were unable to explain this effect. 2.4.
CONCLUSIONS
In order to discuss the above results, we consider the motion of a particle with m o m e n t u m p, described by the Hamiltonian H
=
l(x,2..[.
G2x )2+ ~ tI zz
,2_
~Q~z ~)+
+ E ~ vn,.p xnzr"e-lp° p
(n+m > 2).
rGm
The term t,n,~p characterises the driving forces o f the resonance nQ~,+mQz =p. The structure of the machine exhibited 12 equal periods. Thus any harmonic p < 12 of Vm, could only arise from misalignments and closed orbit distortions. F r o m fig. I it can be seen that all Vm,p, [ml + In[ < 5, should have been zero if the machine had been ideal. Only V3oo and V4oo can be estimated f r o m Q(E). T a k i n g i n t o account that Q is also a function o f m o m e n tum, and writing Q for Q,, we obtain /3400 = ¼ [Q( d2 Q/df 2) 1.6 x 109 +
The sextupole nonlinearities could perhaps have the same influence as in C E S A R even with the tighter tolerances in the C E R N Intersecting Storage Rings (ISR)9), as the number of revolutions in the I S R is about 100 times larger than in C E S A R . It would seem advisable, especially in view o f the experiments 7, 8 and 9, to stay well away from third order lines. It must be stressed that a comparison between these machines seems nearly impossible as the type and strength of the non-linear forces in both machines can only be conjectured. However, one had seen in C E S A R for the first time that the lifetime of the electrons at a pressure o f 7 x 10- 11 Torr seemed to enter already in the realm where it is no longer determined by gas scattering but by non-linear forces perturbing the motion of the particles. 3. Space-charge neutralisation driving the beam into a non-linear resonance 3.1. INTRODUCTION In the course o f measurements on the beam lifetime at a pressure of 4 . 5 x 1 0 -1° Torr, it was observed that if the intensity of a single pulse [betatronaccelerated to central orbit1)] exceeded a certain threshold value, then the current decayed very
+ (dQ/df) 21.6x lO 9 - 2 Q 4 / R 2 ] , and /3300
~---
--
g/)400Sx+½ [Q(dQ/df) ( - 4 x
104)q-Q4/R]
neglecting all V,oo, n > 4 . We used for convenience f, the radio-frequency, as parameter because it was used also in all our Q-measurements. We restrict ourselves to give the limits as the values changed from experiment to experinaent. Further one can say to,,0 = - Goo.
3 × 10 2 ~< Iv~ool [ c m - ' ]
~ 6 x 10 -2 ,
0 ~< IV4ool [ cm-2"] ~< 0.1. The ratio, R,, o f non-linear to linear force is about nv.o o x"-2/Q 2. Taking x = b = 0.15 cm, the beam radius, we get, with Q ~ 2
Fig. 4. Rapid beam loss on 4th-order resonance, Upper trace: 1 V applied to clearing field electrodes; lower trace: no clearing fields. Vert.: 1 mA/div; hor.: 0.25 ms/div.
98
K. HOBNER et al.
rapidly during the first 100 ms or so, after which the decay followed approximately the exponential law expected from gas scattering. Furthermore, it was noticed that this effect of rapid initial beam loss depended upon the position at which the beam was left circulating after acceleration, in other words, upon the values of QH and Qv, which also depended upon beam position. In these measurements, no voltage was applied to the clearing field electrodes. As a first step, a 12-pulse stack was built up into a region of the radial aperture in which the instability was observed to occur if the circulating current was above a threshold value, and the effect of applying a voltage to the clearing field electrodes was investigated. Fig. 4 shows a typical signal from the integrating beam transformer, with and without the application of a small voltage (1 V) to the clearing field electrodes. It was immediately obvious that the instability was due to space-charge neutralisation driving the beam into a non-linear resonance.
tended to increase with e, and at v = 12.291 MHz and e = 0.0030 it was already above the largest current that could then be stacked (about 1 mA). 3.3. DEPENDENCEOF BLOW-UP TIME UPON CLEARING-FIELDVOLTAGE With only two pulses stacked per cycle, it proved to be impossible to achieve sufficient reproducibility from cycle to cycle to permit a smooth correlation between the blow-up time and the clearing-field voltage. The operating point for these measurements (v = 12.282 MHz, a = 0.0004) was, unfortunately, too close to e = 0, so that the blow-up could only be suppressed with as much as 8 V applied to the clearing field electrodes. However, in the earlier measurements with a 12-pulse stack, the operating points covered by the stack lay further away from the resonance, and it was possible to correlate the minimum blow-up time observed over a series of cycles with the clearing field voltage, as shown in fig. 6.
The horizontal and vertical Q-values were accordingly measured, by radio-frequency knock-out, over the range of orbit radii available for stacking. The orbit radius was defined in terms of the revolution frequency taken to be the sum of the two vertical or the two horizontal knock-out frequencies. The operating point (Qv, QH) lay just below the fourth order sum resonance 2 Q v + 2 Q H = 9. The effect of space-charge neutralisation was to reduce the difference between the electrostatic defocusing and the magnetic focusing action of the beam upon itself, thereby increasing both Q-values and shifting the beam into the non-linear resonance. This interpretation of the beam loss was verified by measurements on the dependence of the blow-up time upon the beam current and upon the clearing field voltage, at operating points with different values of e = 9 - (2Q~ + 2Qv). In these measurements, a small stack of only 2 pulses was used, in order to define more precisely the orbit position and the corresponding operating point.
XBu[rns] 150
10C
50
limA]
0
0.5
1D
Fig. 5. Beam current vs blow-up time.
0.75
V
3.2. DEPENDENCE In all cases, the blow-up time (the time at which beam loss starts, measured from the completion of the stack) decreased with the beam current in the manner illustrated by fig. 5, for which v = 12.286 MHz and e = 0.0013. In this case the threshold current for beam blow-up was 0.25 mA. The threshold current could not be measured with sufficient precision to permit a smooth correlation with the Q-shift, but it was observed that the threshold
a50
0.25
/ 0~
l
I
I
I00
200
300
Train [m sl], 400
Fig. 6. Clearing field voltage vs minimum blow-up time.
EXPERIMENTS IN THE TRANSVERSE PHASE SPACE
99
In the process o f a subsequent change o f the lens current, a very small a d j u s t m e n t was first m a d e , which increased the Q-values so t h a t the s u m 2 Qv + 2 QH was slightly greater than 9. As expected, no b e a m b l o w - u p then occurred, with o r w i t h o u t clearing fields, since space-charge n e u t r a l i s a t i o n now shifted the o p e r a t i n g p o i n t further away f r o m the resonance. A f t e r further adjustment, the o p e r a t i n g p o i n t arrived fortuitously at the value Qv = 1.736, QH = 2.748. A l t h o u g h the sum 2 Q v + 2 Q H = 8.968 was n o w well a w a y f r o m the sum resonance, the b l o w - u p still occurred as before, due to the fact that, with 4 Q u = 10.992, the o p e r a t i n g p o i n t now lay just below the 4 th o r d e r h o r i z o n t a l resonance 4QH = 1 !. It was necessary to shift the Q-values again to Q8 = 2.755, Qv = 1.725, i.e. with 4 Q , = 11.02, above the resonance, in o r d e r to a v o i d losses. 3.4. THEORETICAL ESTIMATES S o m e calculations were u n d e r t a k e n subsequently and yielded the following results s u p p o r t i n g our hypothesis: a. The shift in the Q-values due to space-charge forces is sufficiently large. W e o b t a i n from 7) p u t t i n g the neutralization t / t o 0 : f o r n = 2: AQv = - 2 x l 0 - 3 , andfor n = 12: AQv = - 6 × 10 -3. n denotes the n u m b e r o f pulses; l/n was 10 - 4 A. b. N e u t r a l i z a t i o n is fast enough. A Q will vanish at the time when ~(t) equals V-2 7). T a k i n g into a c c o u n t the gas pressure a n d the gas c o m p o s i t i o n we f o u n d that this s h o u l d occur ~ 0.3 s after injection to). c. 2 V field field from
on the clearing field electrodes yields a strength which is equal to the m a x i m u m strength at t h e edge o f the b e a m built up 12 p u l s e s l ° ) .
d. A r e m a n e n t field o f 100 m G at the p o l e tips o f one q u a d r u p o l e was calculated to be sufficient to drive the resonance ~~), T h e m a x i m u m r e m a n e n t field m e a s u r e d at the pole tips o f the lenses was 170 r a G . 4. Coherent transverse oscillations of the beam
D u r i n g the running-in period, an intensity d e p e n d e n t b e a m loss was noticed w h e n a single pulse o f high intensity was injected a n d left coasting. T h e rate o f b e a m loss was m u c h faster t h a n expected f r o m gas scattering. A detailed inspection showed that the intensityd e p e n d e n t b e a m loss m i g h t be originated by coherent vertical oscillations o f the particles in the b e a m .
Fig. 7. Signal picked up by one of the clearing field electrodes. a. n= 2 ; l = 2.3mA. Vert.: 2.5 × 10-4 A cm/div; hor.: 1 ms/div. b. n = 2 ; I = 2.2rnA. Vert.: 1.25 x 10-4 A cm/div; hor.: 20 ms/div. c. Terminating inductance L ~ 44/rid, 1 = 1.2 mA. Upper trace: n = 2; lower trace: n = 1; hor. scale: 2 ms/div. d. Terminating inductance L = 2650/tH, 1 = 1.3 mA, n ~ 2. Vert.: 4 x t0 -~ A cm/div; hor.: 100 ms/div.
100
K. H U B N E R et al.
We felt that this instability merited some attention in view of the general trend to increase the intensity in accelerators and of the limited amount of experimental results available. We made two sets of experiments with a coasting beam betatron-accelerated to central orbit. In the first one the clearing field electrodes were connected to their power supply, and in the second the electrodes were terminated by inductances to make them resonant at frequencies f , = ( n - Q~). v, to get some information on the influence of the electrodes on the beam behaviour. Growth-rates and thresholds were determined. The time correlation between the maxima of various modes was investigated. 4.1. EXPERIMENTAL METHODS
The clearing field electrodes, henceforth called the electrodes, were 22 plates placed at the bottom of the curved sections of the vacuum chamber. Their length was 50 cm, their width was 7 era. The electric connection was at one third of their length. One of them was used to pick up the signal arising from the oscillations. Also removing the two cavities did not change the appearance of the oscillations. It was confirmed by analysing the spectrum up to 20 M H z that only the modes n > Qv grew in time. Their frequeneiesf, were equal to ( n - Qv)V within the error of measurement of Qv and v, the revolution frequency of the particles. The oscillations were never artificially induced as this might change the spread in Qv due to betatron amplitudes. They grew up from initial amplitudes given by injection errors and noise. Fig. 7a shows a typical example. The signal in the first 0.5 ms is due to the azimuthal structure of the injected beam. The betatron acceleration was accomplished after 1.5 ms. Deceleration started after 8 ms. A growth-time can be defined: ,
receiver were used to distinguish between the different frequencies. 4.2. EXPERIMENT 1
The pressure in the vacuum chamber was 4 x 10- ~o Torr. The half-lifetime of the beam was around 5 s. The rate of neutralization was calculated to give t7 = 0.16 t, t in (s), where r/in the neutralization factor. The electrodes were connected in parallel to the power supply via a single 50 f~ cable. The output of the power supply was shunted by 910g). "Cross-talk" between the electrodes was not excluded. The external impedances seen by the single electrodes were therefore dependent upon frequency. But we could infer that the electrodes did not form a resonant circuit at the frequencies of interest with these impedances, as the appearance of the signals picked up was not dependent on whether the power supplies were on or off, which implied a big change in impedance. Applying this definition, a normalized growth time -~
I
= tl+T
= usually chosen to be 2 ms = 0: injection.
The sensitivity was 3 V/cm. A. The frequency response of the set-up including the cathode follower was frequency independent in the range from 1 M H z to 100 MHz. Filters of a bandwidth of 0.5 MHz and a frequency analyser which could also work as a superheterodyne
I
[ A.s]
r
I
I
I
I
r~
I
l
(
~_
•
•
•
%;
O0
I0"5
$
+04-
O
+
o+
• 4-
+-'-2_
+
O O
I0 -6
O
o8 r
N --- noise amplitude seen without beam Ai(ti) - amplitude t~ = chosen at a time when A~ _>2N, t~ > 1,5 ms T t
1
"Bl.(2-Qv)'b
-c = T/In ( A z , N / A l - N )
t2
I
o
I [mA] I0"7
I
0
I
]
2
3
Fig. 8. Normalized gr ow t h-t i m e versus beam current; mode n = 2. Electrodes connected to the power supplies: (O); terminating inductance L ~ 2.65 mH: (+); terminating inductance L = 44/tH : (©).
EXPERIMENTS
IN
THE
TRANSVERSE PHASE
r , ' I x / ( n - Q v ) for n = 2, 3, 4 was calculated and is plotted in figs. 8, 9 and 10. It should be remarked that only the mode n = 2 always showed a fairly exponential growth. The threshold currents are listed below:
n I,[mA]
2 0.5
3 4 5 1.4 0.9 2.2
Fig. 7b shows the typical longtime behaviour of the oscillations. 10-5
i
-
i
i
r
]
'
I
'
'
'
'
I
'
'
'
'
I
~r/a-g2:-67
10-6
101
SPACE
We found with the help of a vertical plate target that the amplitude of the oscillations never exceeded 0.1 cm. The frequency and the depth of modulation increased with higher currents. The oscillations always died out after 0.1 s, The rate of loss of particles was as expected from gas scattering. 4.3. EXPERIMENT
II
The pressure was 4 x 1 0 - 9 Tort, the half-lifetime of the beam was 0.6 s. The rate of neutralization was about ~/ = 1.6 t. That we had no clearing field applied, did not matter, as the observations were made during the first 50 ms after injection. After that time the oscillations had ceased in most cases. The mean value of the capacity (7 of the electrodes was 140 pF. Table 2 gives a list of the parameters for the various inductances L which were put in parallel to the capacity C. TABLE 2
o o o
O0
0
0
L
o
p,°,
t
,
,
I
,_l
1
,
j
i
,
2
I 3
I
,
, 4
,
t
(/tH)
fo -1/2:,rv/(LC) R (MHz)
Afo= fo/Q
AfLc
nT
(~) (MHz) (MHz)
(nr- Q,)v V nL (MHz)
nL"
P
(MHz)
I mA
F i g . 9. N o r m a l i z e d g r o w t h - t i m e v e r s u s b e a m c u r r e n t ; m o d e n = 3. Electrodes connected to the power supplies: (O); terminating inductance L = 1 /iH: (O).
10
5-
I
F
i
I
m
o
o
o
O
J
,
J
I
~0.5 1 35 33
1>0.8 0.167 0.125 0.026
4.1 1.3 0.2 0.026
5 3 2 .
38.8 14.3 1.9 . .
3 1 -
36.8 12.26 -
.
½fo (AL/L + AC/C).
m
o
2650
42.5 13.4 2.0 0.26
R = r e s i s t a n c e in series w i t h t h e i n d u c t a n c e L ; v = r e v o l u t i o n f r e q u e n c y ; Qe = n u m b e r o f v e r t i c a l b e t a t r o n o s c i l l a t i o n s p e r t u r n ~ 1 . 8 4 ; Q = c i r c u i t q u a l i t y f a c t o r ; nT = t r a n s v e r s e h a r m o n i c n u m b e r ; nL = l o n g i t u d i n a l h a r m o n i c n u m b e r ; IAfLcl =
10"6
lO"7
1 44
I
.Ci//n _ Ov
-
0.1
,
I 1
i
l l l l , t t t l 2
3
F i g . 10. N o r m a l i z e d g r o w t h - t i m e v e r s u s b e a m c u r r e n t ; m o d e n = 4, e l e c t r o d e s c o n n e c t e d t o t h e p o w e r s u p p l i e s : ( O ) ; m o d e n = 5, t e r m i n a t i n g i n d u c t a n c e L = 0.1 ,uH: ( O ) .
With the inductances of 2650/tH, we had no resonances of the electrodes in the region of interest. The capacity was not the same for all electrodes since their position in the vacuum chamber had changed during a bake-out. This is the reason for 3fLC exceeding Af0. a. Growth-rates. It was always the mode (n - Qv)v ~f0 which showed a nice and clean exponential blow-up. The measured growth-times r, of these modes are shown in figs. 8, 9 and 10. b. Frequency spectrum. Keeping the injected current and the voltage drop of the Van de Graaff constant, we scanned the frequency spectrum up to 100 MHz with the analyser.
102
K. H U B N E R
Nearly all frequencies (n+_ Qv)v were present. The longitudinal harmonics nv were only seen in the case o f beam loss. The ratio of m a x i m u m amplitudes of the various modes to the m a x i m u m amplitude of the "excited" mode [ f o ~ ( n - Qv)v] is plotted in fig. 11. It should be remarked that the maxima of different modes were in general not coincident in time. The modes t7 = 1,0, - 1, - 2, - 3 attained their m a x i m u m amplitude always after the m a x i m u m of the "excited" mode. It appears that these modes grew during the damping o f the "excited" mode even when we did not lose particles, Fig. 12a, b may serve as illustration. For n = 1 only the envelope of the signal is shown, as it is the output of the analyser working as a superheterodyne receiver. We convinced ourselves that the bandwidth o f 100 kHz was sufficient. l f t h e excitation of the modes n = 1,0, - 1, - 2 , - 3 (stable modes in the small-amplitude theory) were due to beam loss, the amplitudes o f these modes should increase with higher beam loss. Increasing the beam loss by moving in a vertical plate target, we found, however, that only the signal from the electrode enclosing the beam (e.g. fig. 12a, upper trace) increases, but the signals arising from vertical oscillations diminished. Fig. 7c, d shows the long-time behaviour of the oscillation.
et al.
The threshold for beam loss was a r o u n d 1 m A in the case o f L = 1/~H and L--- 44/~H. Beam loss with L = 0.1/zH was strongly dependent upon the radial position of the beam, as at the time of the experiment with this inductance t3Qv/~gE had a minimum around central orbit. N o beam loss was seen with L = 26501tH for the larg',st currents that could be injected ( I < 4 m A ) . This is in agreement with experiment I, in which no beam loss was observed and where the electrodes were connected to the power supplies. 4.4.
COMPARISON WITH THEORY
It is reasonable to compare the growth-rate and threshold measured with L = 2650/tH with the corresponding values one can calculate from the resistive wail instability~2). The curved sections of the v a c u u m chamber had an elliptic cross-sec'.ion (semi-axes: 4.5 cm, 1.5 cm) and the straight sections a circular one (r = 5 cm). Each type of section covered half of the circumferenc:. Approximating by a rectangular v a c u u m chamber (2h = 3.5 cm) we get: I [A] rn [seC]/x,' ( n - Ov) ~ 7 x 10 -6 . The following parameters have been used:
MHz I
20
30
40
50
0
-IC
-20
-30
db
relative
F i g . 11. F r e q u e n c y s p e c t r u m ; r a t i o o f m a x i m u m
I n d u c t a n c e L: Excited mode:
(o)
CA)
(0)
0.1 I t H n = 5
1 /~H n = 4
4 4 ,uH n = 2
amplitudes.
(+) 2 6 5 0 ~tH --
EXPERIMENTS
IN T H E T R A N S V E R S E
103
PHASE SPACE
and betatron amplitude a. AS = A S e + AS~; AS = 2 n j ( n - O v ) (gv/gE) - (OQ~/~E)vl ½AE + + 2 n I ( n - QO (cv/~3a2) - v (OQ./c~a2)l Aa 2 . Introducing for ~3Q/~3E and c3Q/?a z (calculated f r o m O2Q/OE2) the values we measured at the time o f the experiment and with gv/'~a 2 ~
ori cm2..l Zl , AE ~
3 x 103eV (fullspread),
one obtail~s ASr, = 103 s-I , AS,, = 2 . 3 × 1 0 4 s AS = ASE+ASa
Fig. 12. a. T i m e correlation between two transverse modes, with b e a m loss; t e r m i n a t i n g inductance L = 44 ttH. U p p e r trace: Signal o f the electrode enclosing the b e a m ; middle trace: Signal o f clearing field electrode, n = 2; lower trace: Signal o f clearing field electrode, n = 1; hor. scale: 0.5 ms/div, b. Signal from integrating b e a m t r a n s f o r m e r ; vert. scale: 0.2 mA/div., hor. scale: 2 ms/div.
Q~
= 1.84;
a
= 1.4 × 10 ~6 s- ~: conductivity o f the material;
~,
= ( l - f 1 2 ) - ~ = 4.5;
R
= 380 cm: radius o f the machine;
2a
= 0.2 cm: width o f the beam;
~-~ = 20: function of geometry12); h/b
= 17.
The stability criterion has the form A S > U for U~> 1/~,12). U/2n is the real part o f the shift in ( n - Qv)v originated by the vertical oscillations o f the whole beam. The formulae for rectangular geometry yield:
c[s-'j lO"×/[A].
~,
= ( 1 + 2 3 ) × 1 0 3 s- I .
Thus U exceeds A S by one order of magnitude at the threshold. But one sbould keep in mind the numerous approximations in the calculation of U. 4.5. CONCLUSIONS
With the clearing field electrodes connected to their power supply, it was shown that a. the modes n = 2, 3, 4 were unstable, b. the growth rate o f this instability agreed quite well with the rates calculated assuming a resistive wall instability, c. the oscillations were amplitude limited, showed some sort of beating and disappeared after a certain time without giving rise to discernible beam loss. By terminating the clearing field electrodes with inductances we were able to enhance certain modes. Their growth rate was increased by at least one order o f magnitude. The frequency spectrum was scanned up to 100 M H z and the modes n < Q, were found to be antidamped during the damping of the modes n > Q~.
Introducing the threshold current o f 1.5 m A one gets U = 1.5x 1 0 5 I s - r ] . A S is the spread in the quantity ( n - Qv)2nv b r o u g h t a b o u t by the dependence of Qv and v u p o n energy E
The n u m b e r of people who have contributed to the successful running o f C E S A R and to the experiments described here, either by stimulating discussions or by active help in their preparation, is very large and we
104
K. HUBNER et al.
c a n n o t t h a n k t h e m here i n d i v i d u a l l y . T h e i r n a m e s a n d m a i n c o n t r i b u t i o n s are listed in 1). W e w o u l d like, h o w e v e r , to express o u r p a r t i c u l a r a p p r e c i a t i o n in h a v i n g h a d t h e c o n t i n u o u s i n t e r e s t a n d s u p p o r t o f K. J o h n s e n a n d t h e late A. S c h o c h .
References 1) 2) 3) 4)
The CESAR Group, CERN Report 68-8 (1968). A. Briickner et al., Nucl. Instr. and Meth. 77 (1970) 78. A. Schoch, CERN Report 57-21 (1957). V . I . Balbekov and A . A . Kolomensky, Proc. Int. Conf. High energy accelerators (Frascati, 1965) p. 358.
5) A. M. Sessler, private communication (1967). 6) F . T . Cole et al., Rev. Sci. Instr. 28 (1957) 403. 7) L. J. Laslett, Proc. 1963 Summer Study Storage Rings, Ace. and Exp. at Super-High Energies, Brookhaven National Laboratory Report, BNL 7534 (1963) p. 324. 8) C. Bernardini et al., Phys. Rev. 10 (1963) 407. 9) The CERN Study Group on New Accelerators, CERN Internal Report AR/Int. SG[64-9 (1964) unpublished. 10) K. Hiibner, CERN Internal Report AR/Int. SRM/66-1 (1966) unpublished. it) K. Hiibner, CERN Internal Report, AR/lnt. SRM/66-2 (1966) unpublished. 12) L. J. Laslett et al., Rev. Sci. Instr. 36 (1965) 436.
INSTRUMENTS
AND
77 (t97 o) 93-1o4;
METHODS
©
NORTH-HOLLAND
PUBLISHING
CO.
E X P E R I M E N T S I N T H E T R A N S V E R S E P H A S E SPACE W I T H T H E C E R N E L E C T R O N S T O R A G E AND A C C U M U L A T I O N R I N G (CESAR) K. H I 3 B N E R , E. J O N E S , H. K O Z I O L , L. M A G N A N I , M. J. P E N T Z and A. R U G G I E R O
CERN, Geneva, Switzerland Received 9 June 1969 The effect of non-linear betatron resonances on the lifetime of an accumulated electron beam was determined. Coherent vertical oscillations o f a coasting beam were investigated. The growth rates and the thresholds for the modes n = 2, 3, 4, 5 were
measured. A time correlation between the different modes was established. It was tried to influence the oscillations by making the clearing field electrodes resonate with the beam.
1. Introduction
mode m4):
The paper describes two sets of experiments: a. the dependence of the lifetime of the circulating particles on the Q-values, b. coherent transverse oscillations of the beam (dipole type oscillations).
"C1/7~m2=~k-hl/
\(m!)2"24'~-3, ]
\nm-mQ/
where b/h is the ratio of beam height to vacuum chamber height (for CESAR, b/h,~ 1/17) and nm is an integer > m Q. The threshold for m > l will not be much different from the threshold for m = 1s). it was therefore decided to use for this experiment a stacked beam with a large spread in energy and consequently in the Q-values, thus suppressing all coherent instabilities. The particle density p(E) was measured as a function of time by subsequent scanning of the stack with empty rf-buckets2). Q(E) was found by radio-frequency knock-out measurements6). The particle density was about 3 × 1 0 9 . I ( c m -3) where I is the circulating current in amperes. The corresponding number of particles in the ring was N ~ 6 × l011 I. The maximum current was 20 mA and the typical current 6 mA. The circulating current was measured by a current transformer with an integration time constant of I000 sec. The parameters of the rf-buckets z) used were
All experiments were performed with a coasting beam. The storage ring and its injector are described in1). A short list of the main parameters can be found in 2). 2. Non-linear resonances in the absence of space charge effects
If the condition n Q v + m Q n ~ p (n, m, p-integers) is fulfilled, perturbations in the magnetic guiding field can lead to a transfer of energy from the longitudinal to the transverse motion causing a growth of betatron amplitudes 3). The Q-values in CESAR depended upon the orbit radius and therefore also on energy. A certain number of resonances may thus occur at various energies in a stacked beam, and some of these may cause significant reductions in beam lifetime. 2.1. EXPERIMENTALPROCEDURE In principle, one could have measured the beam life-time with a single pulse injected and betatronaccelerated to different orbit radii1). This would have been too time consuming, since we wanted to investigate a relatively large region of the QH, Qv diagram. Furthermore, the interpretation of the results would not have been easy, as it would not have been possible to distinguish beam loss due to non-linear resonances from that due to coherent multipole oscillations of the beam. This becomes apparent if one compares the risetimes of the dipole mode (m = 1) to a higher order
a. stacking bucket f = 2.1 x l07 sec-2; F = 0.36; AE = l0 keV; ½A = 4.3 keV; b. scanning bucket f = 2.1 x 107 sec-2; F = 0.38; AE s = 8.4 keV;½A = 3.45 keV. The displacement of the whole stack by the scanning was hardly discernible. The typical number of scanning cycles was 3. Twelve different sets of Qv, QH were used, and are shown in fig. 1. The Q-values were adjusted by changing the currents in the quadrupole windings and in correction windings on the focusing lenses. 93
94
K. H O B N E R et al.
The gas pressure diminished by a factor 1.5 in the course of the experiments, reaching 7 x 10 -1~ Torr towards the end. In spite of the low pressure, the electrons were lost by single scattering only, due to the high percentage of argon in the residual gas. This circumstance was of decisive importance for the experiment, as the life-time would otherwise have been dependent upon the available aperture, which then would have had to be measured across the stack each time, since the aperture was determined by closed orbit distortions which varied very much from experiment to experiment. The half-lifetime was calculated to be 15 s for single scattering at the lowest pressure reached ( 7 × 1 0 -~1 Torr). 2.2. FACTORS LIMITINGTHE ACCURACY OF THE RESULTS The linear, incoherent, Q-shift by space charge forces 7) was always smaller than 3 x 10 -a. Neutralisation was prevented by applying a voltage of 6 V to the clearing field electrodes.
The error in the measurement of the Q-values by radio-frequency knock-out was AQv/Q v < __+3x 10-3; AQH/QH < __+1×
10 -3
.
Electron-electron scattering in the beam 8) could cause a redistribution of p(E) without beam loss, and transport particles to the regions emptied by non-linear resonances. An estimate of the influence of this effect can be obtained by considering a distribution p( E) = D, p(E) = O,
E < El, E < El,
and calculating the initial energy Eo
EI-Eo =(2×10 8"T'i)~.
Fig. 1. Qv as a function of Qn. Number at the crossing points: order of the resonance; other numbers: harmonic number of resonance.
E X P E R I M E N T S IN T H E T R A N S V E R S E P H A S E S P A C E
For i = 1 × 10 -2 A/cm 2 and z = 15 s, E1 - E 0 turns out to be ~_ 5 keV, which is less than the resolution given by the height of the scanning bucket.
Fig. 2. Scanning signal; t = 0: end o f stacking; horizontal scale: 20.4kHz/div. a. 35 pulses; scanning after ts¢ = 0 and 2 s ; b., c. and e. 5 pulses, t~ = 2 s; d. and f. 5 pulses, t~c = 0 and 2s.
95
2.3. EXPERIMENTAL RESULTS
In order to be sure that the dips in the density distribution p(E) were due to effects dependent only upon the Q-values, the following experiment was performed. A stack of 35 pulses was built up, extending over the range of orbit radii corresponding to the Q-trace no. 6 in fig. l. The scanning signal is shown in fig. 2a. The initial particle density is indicated by the first scan. Five pronounced dips can be seen in the second scan. To check whether the dips occur at the same place, i.e. at the same points (QH, Qv) on the Q-trace, when the initial particle density is different, a 5-pulse stack was placed successively at each place and scanned as before. The resulting scanning signals are shown in fig. 2b-2f. It can be seen that the dips occur at the same places as before, independently of the initial particle density. The validity of the basic premise of this experiment is confirmed also by the appearance of a 50-pulse stack (fig. 3), over which the Q-variation is small and where the Q-trace does not cross any non-linear resonance lines of order lower than 7. The Q-variations over this stack were A Qv = 0.01, A Qa ~ 0.005. Having satisfied ourselves that the positions of the dips were not dependent upon particle density, the stacks corresponding to the twelve Q-traces, shown in fig. 1, were each scanned repetitively at intervals of 1 see. All the stacks showed a density modulation. Stack 4 showed the least modulation, since the 6 th order resonances were quite weak. The reduction of lifetime in the dips does not only depend on the strength of the excitation coefficient of this particular resonance but also on the precise form of the Q-trace at the cross-over and on the strength and density of the nearby resonance lines. It was, for example, impossible to stack a beam
Fig. 3. Scanning signal 50 pulses stack. Scanning at intervals ofl s. Hot.: 20 kHz/div.
96
K. H O B N E R et al. TAaL~ 1
No.
~max %'O
Tmax/TO
¥/ro Order crossed
1
2
3
4
5
6
7
8
15 8 11.0 11.3
16 10.6 10 8.5 3.8 4 11.3 11.4 11.6 11.9 12.0 12.6 1.5 2.9 2.2 3.2 1.36 0.71 1.40 0.86 0.72 0.32 0.32 0.32 0.13 0.25 0.18 0.25 4(3) 5(6) 5(2) 6(5) 4(4) 4(4) 4(3) 4(2) 5(1) 6(4) 6(5) [1] 5(3) 5(3) [3] 5(3)
9
10 13.6 6.5 0.74 0.48 3(1) 5(2)
10
11
12
15 10 10 15.0 16.5 21.0 4.6 2 . 9 4.9 1.00 0.60 0.48 0.30 0.18 0.23 4(3) 4(3) 4(3) 5(n*)5(3)
6(2) [3l * Note that Q-trace 11 is practically coincident with a 5th order horizontal resonance over most of its length. This is an example of multiple traversal.
close to the point of confluence of 4 th order or 5 th order resonances. Large beam losses were observed near crossing points of single 4 th and 5 th order resonance lines. The results of the twelve experiments lend themselves to a semi-quantitative interpretation, which is summarized in table 1. The following quantities are given: "Cmax the maximum life-time observed anywhere in the stack; = the average life-time of the whole stack, obtained from the integrating beam transformer signal; Zo = the life-time calculated from single scattering. All the life-times are the time constants of exponential decay. The table shows also the order of resonance crossed by the stack. The number in ( ) indicates the number of crossings, and the number in [ ] the order of a resonance in proximity to the stack, but not crossed. It can be seen that the average lifetime in the stack is particularly reduced in case no. 4 and also, though not so much, in cases 7 and 8. This can be explained by the proximity of the integral resonance in case 4 and of the third-order resonance in cases 7 and 8. The quantity ¥/% is of interest for intersecting storage rings, because this would express the ratio of the practically achieved number of interactions to the theoretical number, in periods of observation time longer than half of the beam lifetime. =
Rapid beam losses: In the course of this series of experiments, particularly in experiments 7, 8, and 9, rapid beam losses were observed with the integrating current transformer which could not be satisfactorily explained. These three experiments were distinguished
from all the others by the fact that the Q-traces were near third-order resonance lines. It seems reasonable to assume that sextupole perturbations were rather strong in CESAR since we had built-in sextupole fields. The rapid beam loss was probably always triggered off by space charge forces driving part of the stack into resonance. In experiment 8 the part of the stack for which 1.8< Q v < 1.87 was sometimes lost 0.2 s after the last stacking cycle. The loss occurred if the total current I fulfilled the condition: (3.25 × 10-3 U - 1 7 x 10-3) <~ I ~< ~< (3.25× 10 -3 U - 2 . 5 × 1 0 - 3 ) , where U is the voltage applied to the clearing field electrodes. The two limits define two parallel lines I = #i(U), (i = 1, 2), spaced by A U = 4.5 V. The dependence of the Q-values upon U was measured to be
O < OQn/dU ~< 3 x 1 0 - 4 ;
0 ~ OQv/gU <~ 4 . 5 x 1 0 -4.
This yields the width of the stopband A Qs of the resonance 3Qn = 8,
AQs = (OQ/~3U)AU; AQ~ < 1.3x10 .3 . Another interesting quantity could be obtained from these inequalities, namely the upper limit of OQH/~?I, by assuming that the slope of I = #i(U) is due to the fact that the space charge forces have to compensate ~?Qn/OU if one changes the voltage U. In that case, the total differential of QH(I,U) should be zero along 9~.
~QH/~I + OQ,/~U" ~U/OI = O,
EXPERIMENTS
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SPACE
97
for the sextupole forces 7 x 10-3~>R3~>3 × 10 -3, for the octupole forces 2.3 x 10 -3/>.R4~>0.
which yields 0Q./3I ~ -10-'
PHASE
[a-q.
The value of (dQH/t3I)m~x calculated from 7) for 30 pulses stacked is
As a comparison we give the values for the I S R assuming Q = 9, x = I c m , v30 = 1 0 - 1 , OQH/3E=O R 3 = 3.7 x 1 0 - 3 .
(OQ./OI),... = - 6.5 x 10 -2 [ A - ~ ] . It was observed, further, that the beam loss could be suppressed in most cases by scanning the stack immediately after the last stacking cycle. We were unable to explain this effect. 2.4.
CONCLUSIONS
In order to discuss the above results, we consider the motion of a particle with m o m e n t u m p, described by the Hamiltonian H
=
l(x,2..[.
G2x )2+ ~ tI zz
,2_
~Q~z ~)+
+ E ~ vn,.p xnzr"e-lp° p
(n+m > 2).
rGm
The term t,n,~p characterises the driving forces o f the resonance nQ~,+mQz =p. The structure of the machine exhibited 12 equal periods. Thus any harmonic p < 12 of Vm, could only arise from misalignments and closed orbit distortions. F r o m fig. I it can be seen that all Vm,p, [ml + In[ < 5, should have been zero if the machine had been ideal. Only V3oo and V4oo can be estimated f r o m Q(E). T a k i n g i n t o account that Q is also a function o f m o m e n tum, and writing Q for Q,, we obtain /3400 = ¼ [Q( d2 Q/df 2) 1.6 x 109 +
The sextupole nonlinearities could perhaps have the same influence as in C E S A R even with the tighter tolerances in the C E R N Intersecting Storage Rings (ISR)9), as the number of revolutions in the I S R is about 100 times larger than in C E S A R . It would seem advisable, especially in view o f the experiments 7, 8 and 9, to stay well away from third order lines. It must be stressed that a comparison between these machines seems nearly impossible as the type and strength of the non-linear forces in both machines can only be conjectured. However, one had seen in C E S A R for the first time that the lifetime of the electrons at a pressure o f 7 x 10- 11 Torr seemed to enter already in the realm where it is no longer determined by gas scattering but by non-linear forces perturbing the motion of the particles. 3. Space-charge neutralisation driving the beam into a non-linear resonance 3.1. INTRODUCTION In the course o f measurements on the beam lifetime at a pressure of 4 . 5 x 1 0 -1° Torr, it was observed that if the intensity of a single pulse [betatronaccelerated to central orbit1)] exceeded a certain threshold value, then the current decayed very
+ (dQ/df) 21.6x lO 9 - 2 Q 4 / R 2 ] , and /3300
~---
--
g/)400Sx+½ [Q(dQ/df) ( - 4 x
104)q-Q4/R]
neglecting all V,oo, n > 4 . We used for convenience f, the radio-frequency, as parameter because it was used also in all our Q-measurements. We restrict ourselves to give the limits as the values changed from experiment to experinaent. Further one can say to,,0 = - Goo.
3 × 10 2 ~< Iv~ool [ c m - ' ]
~ 6 x 10 -2 ,
0 ~< IV4ool [ cm-2"] ~< 0.1. The ratio, R,, o f non-linear to linear force is about nv.o o x"-2/Q 2. Taking x = b = 0.15 cm, the beam radius, we get, with Q ~ 2
Fig. 4. Rapid beam loss on 4th-order resonance, Upper trace: 1 V applied to clearing field electrodes; lower trace: no clearing fields. Vert.: 1 mA/div; hor.: 0.25 ms/div.
98
K. HOBNER et al.
rapidly during the first 100 ms or so, after which the decay followed approximately the exponential law expected from gas scattering. Furthermore, it was noticed that this effect of rapid initial beam loss depended upon the position at which the beam was left circulating after acceleration, in other words, upon the values of QH and Qv, which also depended upon beam position. In these measurements, no voltage was applied to the clearing field electrodes. As a first step, a 12-pulse stack was built up into a region of the radial aperture in which the instability was observed to occur if the circulating current was above a threshold value, and the effect of applying a voltage to the clearing field electrodes was investigated. Fig. 4 shows a typical signal from the integrating beam transformer, with and without the application of a small voltage (1 V) to the clearing field electrodes. It was immediately obvious that the instability was due to space-charge neutralisation driving the beam into a non-linear resonance.
tended to increase with e, and at v = 12.291 MHz and e = 0.0030 it was already above the largest current that could then be stacked (about 1 mA). 3.3. DEPENDENCEOF BLOW-UP TIME UPON CLEARING-FIELDVOLTAGE With only two pulses stacked per cycle, it proved to be impossible to achieve sufficient reproducibility from cycle to cycle to permit a smooth correlation between the blow-up time and the clearing-field voltage. The operating point for these measurements (v = 12.282 MHz, a = 0.0004) was, unfortunately, too close to e = 0, so that the blow-up could only be suppressed with as much as 8 V applied to the clearing field electrodes. However, in the earlier measurements with a 12-pulse stack, the operating points covered by the stack lay further away from the resonance, and it was possible to correlate the minimum blow-up time observed over a series of cycles with the clearing field voltage, as shown in fig. 6.
The horizontal and vertical Q-values were accordingly measured, by radio-frequency knock-out, over the range of orbit radii available for stacking. The orbit radius was defined in terms of the revolution frequency taken to be the sum of the two vertical or the two horizontal knock-out frequencies. The operating point (Qv, QH) lay just below the fourth order sum resonance 2 Q v + 2 Q H = 9. The effect of space-charge neutralisation was to reduce the difference between the electrostatic defocusing and the magnetic focusing action of the beam upon itself, thereby increasing both Q-values and shifting the beam into the non-linear resonance. This interpretation of the beam loss was verified by measurements on the dependence of the blow-up time upon the beam current and upon the clearing field voltage, at operating points with different values of e = 9 - (2Q~ + 2Qv). In these measurements, a small stack of only 2 pulses was used, in order to define more precisely the orbit position and the corresponding operating point.
XBu[rns] 150
10C
50
limA]
0
0.5
1D
Fig. 5. Beam current vs blow-up time.
0.75
V
3.2. DEPENDENCE In all cases, the blow-up time (the time at which beam loss starts, measured from the completion of the stack) decreased with the beam current in the manner illustrated by fig. 5, for which v = 12.286 MHz and e = 0.0013. In this case the threshold current for beam blow-up was 0.25 mA. The threshold current could not be measured with sufficient precision to permit a smooth correlation with the Q-shift, but it was observed that the threshold
a50
0.25
/ 0~
l
I
I
I00
200
300
Train [m sl], 400
Fig. 6. Clearing field voltage vs minimum blow-up time.
EXPERIMENTS IN THE TRANSVERSE PHASE SPACE
99
In the process o f a subsequent change o f the lens current, a very small a d j u s t m e n t was first m a d e , which increased the Q-values so t h a t the s u m 2 Qv + 2 QH was slightly greater than 9. As expected, no b e a m b l o w - u p then occurred, with o r w i t h o u t clearing fields, since space-charge n e u t r a l i s a t i o n now shifted the o p e r a t i n g p o i n t further away f r o m the resonance. A f t e r further adjustment, the o p e r a t i n g p o i n t arrived fortuitously at the value Qv = 1.736, QH = 2.748. A l t h o u g h the sum 2 Q v + 2 Q H = 8.968 was n o w well a w a y f r o m the sum resonance, the b l o w - u p still occurred as before, due to the fact that, with 4 Q u = 10.992, the o p e r a t i n g p o i n t now lay just below the 4 th o r d e r h o r i z o n t a l resonance 4QH = 1 !. It was necessary to shift the Q-values again to Q8 = 2.755, Qv = 1.725, i.e. with 4 Q , = 11.02, above the resonance, in o r d e r to a v o i d losses. 3.4. THEORETICAL ESTIMATES S o m e calculations were u n d e r t a k e n subsequently and yielded the following results s u p p o r t i n g our hypothesis: a. The shift in the Q-values due to space-charge forces is sufficiently large. W e o b t a i n from 7) p u t t i n g the neutralization t / t o 0 : f o r n = 2: AQv = - 2 x l 0 - 3 , andfor n = 12: AQv = - 6 × 10 -3. n denotes the n u m b e r o f pulses; l/n was 10 - 4 A. b. N e u t r a l i z a t i o n is fast enough. A Q will vanish at the time when ~(t) equals V-2 7). T a k i n g into a c c o u n t the gas pressure a n d the gas c o m p o s i t i o n we f o u n d that this s h o u l d occur ~ 0.3 s after injection to). c. 2 V field field from
on the clearing field electrodes yields a strength which is equal to the m a x i m u m strength at t h e edge o f the b e a m built up 12 p u l s e s l ° ) .
d. A r e m a n e n t field o f 100 m G at the p o l e tips o f one q u a d r u p o l e was calculated to be sufficient to drive the resonance ~~), T h e m a x i m u m r e m a n e n t field m e a s u r e d at the pole tips o f the lenses was 170 r a G . 4. Coherent transverse oscillations of the beam
D u r i n g the running-in period, an intensity d e p e n d e n t b e a m loss was noticed w h e n a single pulse o f high intensity was injected a n d left coasting. T h e rate o f b e a m loss was m u c h faster t h a n expected f r o m gas scattering. A detailed inspection showed that the intensityd e p e n d e n t b e a m loss m i g h t be originated by coherent vertical oscillations o f the particles in the b e a m .
Fig. 7. Signal picked up by one of the clearing field electrodes. a. n= 2 ; l = 2.3mA. Vert.: 2.5 × 10-4 A cm/div; hor.: 1 ms/div. b. n = 2 ; I = 2.2rnA. Vert.: 1.25 x 10-4 A cm/div; hor.: 20 ms/div. c. Terminating inductance L ~ 44/rid, 1 = 1.2 mA. Upper trace: n = 2; lower trace: n = 1; hor. scale: 2 ms/div. d. Terminating inductance L = 2650/tH, 1 = 1.3 mA, n ~ 2. Vert.: 4 x t0 -~ A cm/div; hor.: 100 ms/div.
100
K. H U B N E R et al.
We felt that this instability merited some attention in view of the general trend to increase the intensity in accelerators and of the limited amount of experimental results available. We made two sets of experiments with a coasting beam betatron-accelerated to central orbit. In the first one the clearing field electrodes were connected to their power supply, and in the second the electrodes were terminated by inductances to make them resonant at frequencies f , = ( n - Q~). v, to get some information on the influence of the electrodes on the beam behaviour. Growth-rates and thresholds were determined. The time correlation between the maxima of various modes was investigated. 4.1. EXPERIMENTAL METHODS
The clearing field electrodes, henceforth called the electrodes, were 22 plates placed at the bottom of the curved sections of the vacuum chamber. Their length was 50 cm, their width was 7 era. The electric connection was at one third of their length. One of them was used to pick up the signal arising from the oscillations. Also removing the two cavities did not change the appearance of the oscillations. It was confirmed by analysing the spectrum up to 20 M H z that only the modes n > Qv grew in time. Their frequeneiesf, were equal to ( n - Qv)V within the error of measurement of Qv and v, the revolution frequency of the particles. The oscillations were never artificially induced as this might change the spread in Qv due to betatron amplitudes. They grew up from initial amplitudes given by injection errors and noise. Fig. 7a shows a typical example. The signal in the first 0.5 ms is due to the azimuthal structure of the injected beam. The betatron acceleration was accomplished after 1.5 ms. Deceleration started after 8 ms. A growth-time can be defined: ,
receiver were used to distinguish between the different frequencies. 4.2. EXPERIMENT 1
The pressure in the vacuum chamber was 4 x 10- ~o Torr. The half-lifetime of the beam was around 5 s. The rate of neutralization was calculated to give t7 = 0.16 t, t in (s), where r/in the neutralization factor. The electrodes were connected in parallel to the power supply via a single 50 f~ cable. The output of the power supply was shunted by 910g). "Cross-talk" between the electrodes was not excluded. The external impedances seen by the single electrodes were therefore dependent upon frequency. But we could infer that the electrodes did not form a resonant circuit at the frequencies of interest with these impedances, as the appearance of the signals picked up was not dependent on whether the power supplies were on or off, which implied a big change in impedance. Applying this definition, a normalized growth time -~
I
= tl+T
= usually chosen to be 2 ms = 0: injection.
The sensitivity was 3 V/cm. A. The frequency response of the set-up including the cathode follower was frequency independent in the range from 1 M H z to 100 MHz. Filters of a bandwidth of 0.5 MHz and a frequency analyser which could also work as a superheterodyne
I
[ A.s]
r
I
I
I
I
r~
I
l
(
~_
•
•
•
%;
O0
I0"5
$
+04-
O
+
o+
• 4-
+-'-2_
+
O O
I0 -6
O
o8 r
N --- noise amplitude seen without beam Ai(ti) - amplitude t~ = chosen at a time when A~ _>2N, t~ > 1,5 ms T t
1
"Bl.(2-Qv)'b
-c = T/In ( A z , N / A l - N )
t2
I
o
I [mA] I0"7
I
0
I
]
2
3
Fig. 8. Normalized gr ow t h-t i m e versus beam current; mode n = 2. Electrodes connected to the power supplies: (O); terminating inductance L ~ 2.65 mH: (+); terminating inductance L = 44/tH : (©).
EXPERIMENTS
IN
THE
TRANSVERSE PHASE
r , ' I x / ( n - Q v ) for n = 2, 3, 4 was calculated and is plotted in figs. 8, 9 and 10. It should be remarked that only the mode n = 2 always showed a fairly exponential growth. The threshold currents are listed below:
n I,[mA]
2 0.5
3 4 5 1.4 0.9 2.2
Fig. 7b shows the typical longtime behaviour of the oscillations. 10-5
i
-
i
i
r
]
'
I
'
'
'
'
I
'
'
'
'
I
~r/a-g2:-67
10-6
101
SPACE
We found with the help of a vertical plate target that the amplitude of the oscillations never exceeded 0.1 cm. The frequency and the depth of modulation increased with higher currents. The oscillations always died out after 0.1 s, The rate of loss of particles was as expected from gas scattering. 4.3. EXPERIMENT
II
The pressure was 4 x 1 0 - 9 Tort, the half-lifetime of the beam was 0.6 s. The rate of neutralization was about ~/ = 1.6 t. That we had no clearing field applied, did not matter, as the observations were made during the first 50 ms after injection. After that time the oscillations had ceased in most cases. The mean value of the capacity (7 of the electrodes was 140 pF. Table 2 gives a list of the parameters for the various inductances L which were put in parallel to the capacity C. TABLE 2
o o o
O0
0
0
L
o
p,°,
t
,
,
I
,_l
1
,
j
i
,
2
I 3
I
,
, 4
,
t
(/tH)
fo -1/2:,rv/(LC) R (MHz)
Afo= fo/Q
AfLc
nT
(~) (MHz) (MHz)
(nr- Q,)v V nL (MHz)
nL"
P
(MHz)
I mA
F i g . 9. N o r m a l i z e d g r o w t h - t i m e v e r s u s b e a m c u r r e n t ; m o d e n = 3. Electrodes connected to the power supplies: (O); terminating inductance L = 1 /iH: (O).
10
5-
I
F
i
I
m
o
o
o
O
J
,
J
I
~0.5 1 35 33
1>0.8 0.167 0.125 0.026
4.1 1.3 0.2 0.026
5 3 2 .
38.8 14.3 1.9 . .
3 1 -
36.8 12.26 -
.
½fo (AL/L + AC/C).
m
o
2650
42.5 13.4 2.0 0.26
R = r e s i s t a n c e in series w i t h t h e i n d u c t a n c e L ; v = r e v o l u t i o n f r e q u e n c y ; Qe = n u m b e r o f v e r t i c a l b e t a t r o n o s c i l l a t i o n s p e r t u r n ~ 1 . 8 4 ; Q = c i r c u i t q u a l i t y f a c t o r ; nT = t r a n s v e r s e h a r m o n i c n u m b e r ; nL = l o n g i t u d i n a l h a r m o n i c n u m b e r ; IAfLcl =
10"6
lO"7
1 44
I
.Ci//n _ Ov
-
0.1
,
I 1
i
l l l l , t t t l 2
3
F i g . 10. N o r m a l i z e d g r o w t h - t i m e v e r s u s b e a m c u r r e n t ; m o d e n = 4, e l e c t r o d e s c o n n e c t e d t o t h e p o w e r s u p p l i e s : ( O ) ; m o d e n = 5, t e r m i n a t i n g i n d u c t a n c e L = 0.1 ,uH: ( O ) .
With the inductances of 2650/tH, we had no resonances of the electrodes in the region of interest. The capacity was not the same for all electrodes since their position in the vacuum chamber had changed during a bake-out. This is the reason for 3fLC exceeding Af0. a. Growth-rates. It was always the mode (n - Qv)v ~f0 which showed a nice and clean exponential blow-up. The measured growth-times r, of these modes are shown in figs. 8, 9 and 10. b. Frequency spectrum. Keeping the injected current and the voltage drop of the Van de Graaff constant, we scanned the frequency spectrum up to 100 MHz with the analyser.
102
K. H U B N E R
Nearly all frequencies (n+_ Qv)v were present. The longitudinal harmonics nv were only seen in the case o f beam loss. The ratio of m a x i m u m amplitudes of the various modes to the m a x i m u m amplitude of the "excited" mode [ f o ~ ( n - Qv)v] is plotted in fig. 11. It should be remarked that the maxima of different modes were in general not coincident in time. The modes t7 = 1,0, - 1, - 2, - 3 attained their m a x i m u m amplitude always after the m a x i m u m of the "excited" mode. It appears that these modes grew during the damping o f the "excited" mode even when we did not lose particles, Fig. 12a, b may serve as illustration. For n = 1 only the envelope of the signal is shown, as it is the output of the analyser working as a superheterodyne receiver. We convinced ourselves that the bandwidth o f 100 kHz was sufficient. l f t h e excitation of the modes n = 1,0, - 1, - 2 , - 3 (stable modes in the small-amplitude theory) were due to beam loss, the amplitudes o f these modes should increase with higher beam loss. Increasing the beam loss by moving in a vertical plate target, we found, however, that only the signal from the electrode enclosing the beam (e.g. fig. 12a, upper trace) increases, but the signals arising from vertical oscillations diminished. Fig. 7c, d shows the long-time behaviour of the oscillation.
et al.
The threshold for beam loss was a r o u n d 1 m A in the case o f L = 1/~H and L--- 44/~H. Beam loss with L = 0.1/zH was strongly dependent upon the radial position of the beam, as at the time of the experiment with this inductance t3Qv/~gE had a minimum around central orbit. N o beam loss was seen with L = 26501tH for the larg',st currents that could be injected ( I < 4 m A ) . This is in agreement with experiment I, in which no beam loss was observed and where the electrodes were connected to the power supplies. 4.4.
COMPARISON WITH THEORY
It is reasonable to compare the growth-rate and threshold measured with L = 2650/tH with the corresponding values one can calculate from the resistive wail instability~2). The curved sections of the v a c u u m chamber had an elliptic cross-sec'.ion (semi-axes: 4.5 cm, 1.5 cm) and the straight sections a circular one (r = 5 cm). Each type of section covered half of the circumferenc:. Approximating by a rectangular v a c u u m chamber (2h = 3.5 cm) we get: I [A] rn [seC]/x,' ( n - Ov) ~ 7 x 10 -6 . The following parameters have been used:
MHz I
20
30
40
50
0
-IC
-20
-30
db
relative
F i g . 11. F r e q u e n c y s p e c t r u m ; r a t i o o f m a x i m u m
I n d u c t a n c e L: Excited mode:
(o)
CA)
(0)
0.1 I t H n = 5
1 /~H n = 4
4 4 ,uH n = 2
amplitudes.
(+) 2 6 5 0 ~tH --
EXPERIMENTS
IN T H E T R A N S V E R S E
103
PHASE SPACE
and betatron amplitude a. AS = A S e + AS~; AS = 2 n j ( n - O v ) (gv/gE) - (OQ~/~E)vl ½AE + + 2 n I ( n - QO (cv/~3a2) - v (OQ./c~a2)l Aa 2 . Introducing for ~3Q/~3E and c3Q/?a z (calculated f r o m O2Q/OE2) the values we measured at the time o f the experiment and with gv/'~a 2 ~
ori cm2..l Zl , AE ~
3 x 103eV (fullspread),
one obtail~s ASr, = 103 s-I , AS,, = 2 . 3 × 1 0 4 s AS = ASE+ASa
Fig. 12. a. T i m e correlation between two transverse modes, with b e a m loss; t e r m i n a t i n g inductance L = 44 ttH. U p p e r trace: Signal o f the electrode enclosing the b e a m ; middle trace: Signal o f clearing field electrode, n = 2; lower trace: Signal o f clearing field electrode, n = 1; hor. scale: 0.5 ms/div, b. Signal from integrating b e a m t r a n s f o r m e r ; vert. scale: 0.2 mA/div., hor. scale: 2 ms/div.
Q~
= 1.84;
a
= 1.4 × 10 ~6 s- ~: conductivity o f the material;
~,
= ( l - f 1 2 ) - ~ = 4.5;
R
= 380 cm: radius o f the machine;
2a
= 0.2 cm: width o f the beam;
~-~ = 20: function of geometry12); h/b
= 17.
The stability criterion has the form A S > U for U~> 1/~,12). U/2n is the real part o f the shift in ( n - Qv)v originated by the vertical oscillations o f the whole beam. The formulae for rectangular geometry yield:
c[s-'j lO"×/[A].
~,
= ( 1 + 2 3 ) × 1 0 3 s- I .
Thus U exceeds A S by one order of magnitude at the threshold. But one sbould keep in mind the numerous approximations in the calculation of U. 4.5. CONCLUSIONS
With the clearing field electrodes connected to their power supply, it was shown that a. the modes n = 2, 3, 4 were unstable, b. the growth rate o f this instability agreed quite well with the rates calculated assuming a resistive wall instability, c. the oscillations were amplitude limited, showed some sort of beating and disappeared after a certain time without giving rise to discernible beam loss. By terminating the clearing field electrodes with inductances we were able to enhance certain modes. Their growth rate was increased by at least one order o f magnitude. The frequency spectrum was scanned up to 100 M H z and the modes n < Q, were found to be antidamped during the damping of the modes n > Q~.
Introducing the threshold current o f 1.5 m A one gets U = 1.5x 1 0 5 I s - r ] . A S is the spread in the quantity ( n - Qv)2nv b r o u g h t a b o u t by the dependence of Qv and v u p o n energy E
The n u m b e r of people who have contributed to the successful running o f C E S A R and to the experiments described here, either by stimulating discussions or by active help in their preparation, is very large and we
104
K. HUBNER et al.
c a n n o t t h a n k t h e m here i n d i v i d u a l l y . T h e i r n a m e s a n d m a i n c o n t r i b u t i o n s are listed in 1). W e w o u l d like, h o w e v e r , to express o u r p a r t i c u l a r a p p r e c i a t i o n in h a v i n g h a d t h e c o n t i n u o u s i n t e r e s t a n d s u p p o r t o f K. J o h n s e n a n d t h e late A. S c h o c h .
References 1) 2) 3) 4)
The CESAR Group, CERN Report 68-8 (1968). A. Briickner et al., Nucl. Instr. and Meth. 77 (1970) 78. A. Schoch, CERN Report 57-21 (1957). V . I . Balbekov and A . A . Kolomensky, Proc. Int. Conf. High energy accelerators (Frascati, 1965) p. 358.
5) A. M. Sessler, private communication (1967). 6) F . T . Cole et al., Rev. Sci. Instr. 28 (1957) 403. 7) L. J. Laslett, Proc. 1963 Summer Study Storage Rings, Ace. and Exp. at Super-High Energies, Brookhaven National Laboratory Report, BNL 7534 (1963) p. 324. 8) C. Bernardini et al., Phys. Rev. 10 (1963) 407. 9) The CERN Study Group on New Accelerators, CERN Internal Report AR/Int. SG[64-9 (1964) unpublished. 10) K. Hiibner, CERN Internal Report AR/Int. SRM/66-1 (1966) unpublished. it) K. Hiibner, CERN Internal Report, AR/lnt. SRM/66-2 (1966) unpublished. 12) L. J. Laslett et al., Rev. Sci. Instr. 36 (1965) 436.