On the shaking of ions in electron storage rings

NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH

Nuclear Instruments and Methods in Physics Research A 340 (1994) 259-271 North-Holland

SectionA

On the shaking of ions in electron storage rings

1

Eva Bozoki

a°*, David Sagan b ° National Synchrotron Light Source, Brookhaven National Laboratory, Upton, NY 11973, USA n Laboratory of Nuclear Studies, Cornell University, Ithaca, AT 14853, USA

Received 20 September 1993 We describe the creation and trapping of ions by an electron beam and the coherent and incoherent instabilities that the trapped ions can cause. The transverse oscillations and longitudinal motion of the tons and the different methods of beam shaking are discussed. The movement of the ions are simulated with and without shaking. We explore the effect of shaking on the stability of ions, as well as the dependence of the size of the effect on the amplitude and frequency of shaking. The effect of neglecting the longitudinal and/or horizontal motion of the ions is studied. A short review 1s given of experimental work by others on the effects of shaking. 1. Introduction Ion trapping is a phenomena in high-energy storage rings in which ionized molecules of the residual gas are trapped within the attractive force created by any beam of negatively charged particles [1]. The presence of trapped ions can lead to beam instabilities . To avoid the unwanted buildup of ions, several methods have been devised to remove them from the beam . Such methods include using an electric field to attract the ions to stripper electrodes [6], and uneven bunch filling [2] in which a time modulation of the beam-ion force resonantly destabilizes the ions . In this paper, we concentrate on a method of extracting ions by shaking the ions via shaking the beam . Shaking is only effective under certain conditions, and thus it is necessary to understand the physics behind the buildup and movement of ions to understand why shaking works in some cases and not in others . The behavior of ions will be discussed in general, but will be illustrated specifically for the compact XLS ring at Brookhaven National Laboratory [3] . The outline of this paper is as follows: Section 2 discusses ion production . Section 3 discusses the coherent and incoherent beam instabilities that can result by an interaction with the ions . Sections 4 and 5 discuss the transverse motion of an ion without shaking and the longitudinal motion of ions . Sections 6 and 7 dis-

* Corresponding author. I Work performed under the auspices of the U.S . Department of Energy under contract no . DE-AC02-76CH00016.

cuss the methods and the effects of shaking the ions, the effect of neglecting the longitudinal and/or horizontal motion of the ions and also neglecting the effect of potential wells at the ends of dipoles. The experimental results of others on shaking are also reviewed . 2. Ion production The main mechanism for ion production is the near collision of a relativistic electron with a residual gas molecule . If the energy imparted to any of the electrons in the molecule is larger than the binding energy, then the molecule will be ionized . The average time T + for a given relativistic electron to ionize a molecule is inversely proportional to the molecular density po : 1/co-0

T +=

r

Po

where Qo is the ionization cross-section and c is the speed of light. The ionization cross-section is only logarithmically dependent on the electron energy (Bethe formula, [1]). For H2, NZ, and CO and for 5 GeV relativistic electrons one has I HZ : = 1 .1 x 10 14 s/m3 , (2) co-o NZ , CO :

1

co-o

= 1 .8 x 1() 13 S/m -3 .

(3)

At room temperature a particle density can be related to an equivalent partial pressure via the equation P[Torr] =3 .035 x 10-z3p [ 1/m3] . (4)

0168-9002/94/$07.00 © 1994 - Elsevier Science B.V . All rights reserved SSDI0168-9002(93)E1041-U

260

E. Bozoki, D. Sagan /Nucl . Instr. and Meth. in Phys. Res. A 340 (1994) 259-271

For example, a background partial pressure of 5 X 10 9 Torr for CO gives an ionization time of 0.1 s/electron . A singly ionized ion may be doubly ionized through another near collision with a relativistic electron and this ionization process may continue . Eventually the ion will disassociate if it is molecular. In any case the ion density will reach an equilibrium where the creation through ionization is balanced by the destruction through instabilities caused by the beam-ion or ion-ion forces . The rate at which singly ionized ions will be doubly ionized can be calculated with Eq . (1) with the singly ionized ion density p + substituted for the molecular density po #1 . In equilibrium the creation and destruction rates for singly ionized ions must be equal. If there were no other rate processes present besides ionization (i .e . there are no other mechanisms to destroy the ions), and if any net spatial flow of the ions is neglected, then in equilibrium For a pressure of 5 x 10 -9 Torr the equilibrium value for p + alone would be 1.65 X 10 14 /m 3 . This density alone is typically large enough to destabilize the beam and ways must be found to lower the ion density. If other rate processes are present (and again neglecting any net spatial flow) the ion density for singly ionized ions p + can be calculated if the ion lifetime t 4fe is known: P

+~

Pbeam

tlife T+

where Pbeam is the time-averaged (not peak) density of the beam . At the center of the beam, P beam is : Pbeam -

N

3. Ion-induced beam instabilities 3 .1. Incoherent instabilities

The main incoherent effect on the electron beam due to the presence of ions is an amplitude-dependent tune shift. In this respect, the effect of the ions on the beam is similar to the effect of one beam upon the other during collisions (beam-beam interaction) . However, any comparison between the beam-ion interaction and the beam-beam interaction must be viewed with some skepticism . There are fundamental differences between the two processes: The ions are spread out over the ring, while the opposing positron beam meets the electron beam in one localized place. As a result, the resonance strength for a given resonance should be lower for the beam-ion interaction . Therefore, in terms of the incoherent instability, we would expect that for equal tune shifts the beam-beam interaction would be more destabilizing than the beam-ion interaction . The amplitude dependence of the ion-induced tune shift is markedly different from the amplitude dependence of the beam-beam interaction tune shift since the ion's transverse distribution differs from that of the beam . To calculate the amplitude dependence of the tune shift from a given ion density we first calculate the kick on an electron due to the ions as a function of position . From this the effective focusing strength and hence the tuneshift can be calculated . The vertical kick per unit length Gy (x, y, s)=dy'/ds due to the ions is given by

24ro,,o-yLl)

where N is the total number of electrons in the beam, o-,, and o-x are the transverse beam sizes, and L o is the ring circumference. Eq . (7) is appropriate when the ion density is limited by instabilities not due to any ion-ion interactions. In this case the ion density can be called "rate-limited" because it is limited by the rate at which ions are created; thus, the ion density is directly proportional to the residual gas pressure . In the case where the ion density is limited by ion-ion interactions, the ion density can be said to be "stabilitylimited" and it will be independent of the residual gas pressure .

*'There is also a change in the value of the cross-section which is roughly equal to (Z-1)/Z, where Z is the total number of electrons. For most gases this change in crosssection can, to zeroth order, be ignored.

ye f

GY (x, y, s) =

~~di ~.dy (X-x)( +(

) Y

y)z

(8)

XPon(x> y, S),

where y is the usual relativistic factor, and p ôn is the ion charge density. The vertical focusing strength per unit length KY is then given by K,(À- ,

Ay,

2 (v A a 'y )

s) Z

2,~ Io

dOy 2 ,rr

I 2~ o

dO y 2 ,rr

xA y o y cos OY GY (A X o-x cos 0., A y o-y cos Oy , s),

(9)

where A x o-,, and A y v-y are the horizontal and vertical oscillation amplitudes of the electron, respectively, and Eo is the electron energy . The total tune shift is then : _ Ov y (A X

, A y ) _ -~ 4,Tr

dsß y (s)K y ( A x ,

A y , s) .

(10)

E. Bozokt, D. Sagan /Nucl. Instr. and Meth. in Phys . Res . A 340 (1994) 259-271 For flat ion distributions the tune shift for low-amplitude electrons is given by combining Eqs . (8), (9) and (10) OPY (0, 0) .-

re ~ ds,6Y(s)p,ô(s) . ye

261

within the potential well created by the beam [6,7] . With no applied magnetic fields the horizontal and vertical oscillation frequencies of the ions in the region near the beam centerline where the beam-ion force is linear is [8-10] :

To put this in perspective, for the XLS ring a tune change of Ov Y (0,0) = 0.01, gives a charge density averaged over the entire ring of p ô/e = 5 X 10 13 m 3 . For a gas of singly ionized ions this density equals the (timeaveraged) beam density when the beam current is 2.4 X 10 -3 A, or, using Eq . (4), this is equivalent to a partial pressure of 1 .2 X 10 -s Torr .

1 2r P C 2Ne ox,Y fr, Y (B = 0) = 21T =21T(ALo-,,Y(o-X+o-Y))

3.2 . Coherent instabilities

f(B) = fx(o)+fB

When trapped ions interact coherently with the beam they can cause coherent instabilities. This process is similar, in many respects to resistive wall instabilities and the ions can be thought of as adding an additional "ion impedance" to the machine impedance [4] . The coherent oscillations between the beam and the ions can be decomposed into dipolar modes, quadrupole modes, etc . For dipolar modes, resonances occur at betatron sidebands of the form fres = (n + P p ) f,, where n is an integer, v ß is the betatron tune, and fo the revolution frequency . Ignoring any ion-ion interaction, the condition for an ion participating in coherent motion of a given order n is [5]

where the cyclotron frequency fB is given by

I fion - fres I ~ "'fres+

(12)

where "fres - VV fresAfbeam >

(13)

and 0 fbeam is the frequency shift that the beam would have if the ions were immobile . Notice that Eqs . (12) and (13) must be solved self-consistently since Afbeam depends upon the number of resonant ions, which is, in turn, dependent upon 'fres . If the frequency of all the ions corresponded to the resonant frequency, then the growth rate 7-1 of the coherent motion would be ,r-1 _ (14) =7rAfres' With a spread of ion frequencies the growth rate will be suppressed due to dephasing effects ("Landau damping") . Neglecting ion-ion forces the condition for complete suppression of the coherent motion is (15) Ofbeam"fon < ( A fres ) 2, where A fbeam and Af,on are the frequency spreads of the electrons in the beam and the ions, respectively.

l /z

(16) In a vertical dipole field the vertical frequency is unchanged [11] while the horizontal frequency is shifted:

1

I

(17)

qB

(18)

fs- 2 , r m

Eq . (16) gives a good approximation for the oscillation frequencies even with a bunched beam as long as the time between bunches is short compared to the oscillation period [12] . With a bunched beam an ion's oscillations in the linear region can be analyzed using matrices . Given a singly ionized ion and assuming uniformly spaced bunches with equal charge in every bunch the horizontal and vertical transfer matrices MX and MY for motion from halfway between kicks to halfway between kicks is [13] Mu-

(0

tb 1 2)( cos(Âu)_

(

-G)~

0)(0

a_

tb1 2)

(11w sln(À .) .) cos(Àu)

sln(À .)

u =x, y, ~1

(19)

where t bb is the time between bunches, a u is the linear kick constant for the kick given the ion by a bunch N

rp c

1

n B o (o x + o'Y ) A

'

(20)

and tu, and À u are given by (Ou

tbb

2 A/A cu - 1

e

(21)

and Àu=cos- 1

( I_

2Aeu ) . A

4. The transverse motion of ions

with the critical masses A,, being

If the electron beam were continuous then, without any ion-ion forces, the ions would oscillate stably

Acu

Lz rP I AnBo'u(o',r+o'Y)

(22)

(23)

262

E Bozoky D. Sagan lNucl. Instr. and Meth . in Phys . Res. A 340 (1994) 259-271

From Eq . (22) the ion motion in a plane is unstable if

10

motion determines stability . In this case, the critical mass A, is given by A,=A, Note that d u /t bb is the resonant oscillation frequency of the ion. For A >> A,Y,

0.8 ô II 0 6

beam size varies with azimuthal position, ions of a

0.4

A
AMY >A,,

so the vertical

Eq . (22) reduces to Eq . (16), as expected . Because the

given mass will have varying frequencies and, in fact,

ions can be stable in some regions of the ring and unstable in others . Since

f

and

f,

0 .2

vary with varying

beam size and magnetic field, and since these quanti-

00

ties vary with longitudinal position around the ring, a spectrum of oscillation frequencies is present. This is shown in

00

1 0

2.0 3.0 Ay / ay

4.0

5.0

00

10

20

40

50

Fig. 1, where the variation in

the beam sigmas as well as the variation in horizontal and vertical oscillation frequencies are plotted around the XLS ring for a beam current of 500 mA and an ion mass of varies only 20% 28 AMU. The figure shows that fY

around the ring, compared with over a factor of 2 for f,(0). This greater variation of the horizontal resonant

0.s ô II

0.6

frequency is typical for machines with flat beams (i .e.

virtually all electron machines) and this is one reason

why vertical shaking is generally more effective than

0 2

horizontal shaking.

For ions with an oscillation amplitude larger then about 1a the nonlinearities in the beam-ion force

00

must be taken into account. Fig. 2(a) shows the normalized vertical

oscillation frequency as a function of vertical oscillation amplitude A, for one-dimensional vertical motion when o,, = loo-y., and the beam can be considered to be continuous . The data points corre-

spond to 300 ions generated uniformly with initial conditions x, = 0 and yo in the range 0 < y, < 5o-Y . The figure shows that the frequency decreases with increasing oscillation amplitude.

30

Fig. 2. Vertical oscillation frequency f, normalized to the linear oscillation frequency as a function of the A, lo,, normalized vertical oscillation amplitude, assuming (a) 1D (y only) motion, and (b) 2D (x and y) motion . The calculations were performed with Nb = 1, 1, = 0.5 A and A = 28 for 300 ions, generated at xO, = 0 with uniform distribution m 0 < yu < 5o,, at a longitudinal position corresponding to 1.80 MHz linear oscillation frequency.

If horizontal motion is added, except in the linear

region where the motion

Ay/ay

is decoupled, the vertical

oscillation frequency is no longer simply related to the

vertical oscillation amplitude. Fig. 2(b) illustrates this difference showing

fy

plotted as a function of A Y for

300 ions generated uniformly with initial conditions (x c, yo) in the range 0
The variations in the linear oscillation frequency along the circumference in the ring, added to the spread due to the nonlinear coupled horizontal-vertical motion of the ions at any one location, results in Fig. 1. (a) The horizontal o,x and vertical o-Y beam size, and (b) the linear (small amplitude) oscillation frequencies f(0), f, as well as the cyclotron frequency fa as a function of longitudinal position . In both plots Nb = 1, Ib = 0.5 A, and A=28 .

the presence of a wide range of ion frequencies in the ring . Furthermore, an ion's longitudinal movement will create

which,

a time variation in its oscillation frequency as explained below,

can lead to a resonant

amplification of the ion's transverse amplitude .

-Ion Longitudinal ions undulators, magnetic longitudinal beam ion for (fBxo the velocity amplitudes chamber on the ano by field, acaoln(O'x+Qyl pipe is fields shaking side so of low energy isdipole that dipole fringe may [15] avelocity motion depends the ion is0) initial can frequency beam greatly shaking 0Qx is continuous beam centerline negligible amplitude that horizontally +with or1ais beam an the be to +be longitudinally If field Us,o) In( periodic eV pushed or fields imparted sigmas oy field point longitudinal due ion will the Slow formed movement by centered dielectric distribution potential reduced the will the shaken ison beam [1] fi(B) For axis of forming longitudinal is called isto have the the and beam ions ions, of preferentially of the (0) ions beam born This [14] the towards fixed ~~, electron the and yarray of just dimensions minimum electric the to initial by for so increased dipole Bozoki, will "resonant on created potential constant with trapped the of energy Thus, periodicity with pipe aaisnear velocity, is The outside newly dipole shaking At ions molecule the ions in barrier of be given beam the coordinates large machines beam-ion aroom field give Dion radius magnitude opposite centerline attracted aone with characteristic is horizontal dipole by created Sagan of in be on longitudinal magnets [6,7,14] by betatron small the mode axis the transverse pockets the variations of for temperature free of ain the reflected while large [10,11] Rdrift the dipole potential ions /Nucl the then this ions the beam directions compared much force shaking" centerline space to ions of of of bunch For will itvelocity beta transverse effects sideband, an generally the might points the atrying the aisInstr oscillahave kinetic in is pocket In mobilby examround E If larger effect being Near magions, This such vacwell drift this xand fact the [8] on be and to of to Ba is

E.

.

.

.

Meth. n

5. The ionized thermal implies energy the uum ple, beam than given V

. .04

. .

2 =(60

where the minimum corresponds effect oscillation shaking near ., ity. In the drift velocity and

. .

. .

. : fx

Us With zero either The the enter tion fringe that of in netic formed

(24)

-x ~

(25)

. . . . .

.

.

With therefore, icant since resonant quency, those dipole tribution necessary spond between shaking much One quency larger (drops), "lock" ent which of moving amplitude case, change frequency ion An side shaking experimentally the Modulating slow dependence ever, increased. pattern . The mal" are This same ence which resonance .

263

.

. . .

. .

. . . .

. .

7. movement

6. Ions shaking then

i Effect Phys effectiveness analysis Fermilab Ion can shaken from of "cyclotron iswas such amplitude enough shaking ions the at as shaking larger limits results coherent to ion amplitudes (the amovement that Res longitudinally in on same isof and attain this shaking resonant oscillation betatron some of of acaused ato change used that shaking that the resonant that amplitude This tracking the Athe ion of small betatron horizontally ion kick of the "rigid thus, shaking shake AA can rate in 340 analysis will this to other at the are shaking arbitrarily problems the point frequency cyclotron of when resonance to atypically instabilities resonant effectiveness can outside must the be (1994) shaking achieve ring by in sideband greater oscillation simulation the mode ions also resonance situation study primarily their preferentially mode shaking program, atapplied frequency side resonant the CERN sideband be in the on can [6,22] be that beam ions frequencies 259-271 initially can frequency) athe inside resonant longitudinal frequency the with partially results ion voltage shaking shaking" the frequencies, applied large effectiveness becomes be This shaking direction shaking" has will are frequency shows counteract AA to responsible issimulating Depending frequency dipoles program of same effectiveness ofisfrequency offset stability shaking resonantly the has been on oscillation be athe The resonantly near the and increasing in can remove dipole effectively that more frequency case and resonance overcome been that beam frequency significantly tends beam with isasuch process by EPA major applied The produce motion, the saw qualitatively itfor downward and resonant [8,12,16] atthe shaking where the do for upon effective demonstrated isthe that amplitudes caused magnet to excited, shaking main arings In ions amplitudes modulation the of sometimes [5] not excited oscillatory producing difference change amplitude be motion fixed addition, This must to removes can for its shaking athe shaking, the changes and ions higher, In to signifwhose and differcorremode How"norsmall by than at [17] lose ions and disfreloss this one the be on to of ina at

7.1.

. .

An ions,

.

264

E. Bozoki, D Sagan / Nuel Instr and Meth . to Phys. Res. A 340 (1994) 259-271

The beam-ion interaction was modeled as a series of impulsive kicks, the beam was assumed to be Gaussian in shape, and the transverse and longitudinal components of the beam-ion kick were computed using the closed expressions developed by Bassetti and Erksine [18] and Sagan [14]. Evaluation of the complex error function needed for computing the beam-ion kick was obtained using a Padé approximation scheme developed by Okamoto and Talman [19] . If the ion is in a field-free region, its movement between kicks obtained by translating the position coordinates. If the ion is in a magnetic field the appropriate rotation matrix is applied to its velocity and position coordinates . The values of the elements of the rotation matrix are based upon the value of the magnetic field at the start of the rotation . 7.2. Ion movement without and with shaking The amplitude and frequency of an ion's transverse oscillations depends upon its initial position (x o, yo, so) and initial velocity (u .0, uyo, v,o) . In general, in the absence of shaking, the small amplitude oscillations are quite regular, modified by the periodic effect of possi-

0.08 f 0.06

ble gaps in the bunch train (see Bocchetta and Wrulich [15]). In fact, ion tracking simulations made for the XLS ring showed that ions created at the lo- surface of the beam with thermal velocities perform very regular oscillations with an oscillation frequency near to the linear frequency and oscillation amplitude slightly varying around 1 beam sigma (see Fig. 3a). The regular pattern of oscillation of the ions changes considerably with shaking, which modulates both the amplitude and the frequency of their movement . In general, ions located close to the center of the beam, will show an increase in amplitude until the amplitude samples the nonlinear region of the bunch. At this point, a shift will occur to lower frequencies and a modulation of frequency will be seen [15] . As a consequence, neither nor a, are growing monotonically, but fluctuate in time . This change can be seen from ion tracking for the XLS ring (Figs. 3b,c). Also, with the appropriate amplitude and frequency of shaking and initial ion coordinates, the ion will become instable and hit the wall of the vacuum chamber, as shown in Fig. 3(d). In each case, the shaking frequency was chosen to be in the most effective range (cf. Fig . 5) .

f,

0 .100

(a)

no

shaking

(b)

0.075 F

0.04

0.050 F

0.02

0.025

ash- .1 o"y

s ,=1 .25MHz

r0 -0.025

-0.02 -0.0a

-0.050

umlpp

-0.06

0

5

10

15 t (psecl

20

25

F1911 11111 ï~!171 1

-0.075 0

30

0.3

5

10

t

15

20

Iw_l

2

0.2 É

0.1

-0.1 -0.2

-2 0

5

10

15 t rosec]

20

25

30

0

5

10

15 t lt 3-l

20

25

30

Fig. 3. Vertical projection of the 3D movement of an A = 28 ion, created at the la surface of the beam (shown with dotted lines) as a function of time (a) without shaking, and (b)-(d) with shaking with different amplitudes (a sh = 0.1, 0.5, and 2 ory ), in each case at a frequency in the corresponding most effective region . The calculations were performed for 0.5 A beam in 1 bunch.

E. Bozoki, D. Sagan /Nucl. Instr. and Meth. a n Phys. Res A 340 (1994) 259-271

7.3. Effect of shaking on the median oscillation amplitude

increases the oscillation amplitude outside the beam, so that static clearing can remove the ions more easily. Indeed, experiments on the XLS ring [20], the CERN EPA ring [6,22], and the UVSOR ring [12] showed that shaking combined with do clearing is more effective then either shaking or do clearing alone. When larger shaking amplitudes are applied (Figs. 4c and 4d), the median oscillation amplitudes increase with time and the ions become unstable . If these higher shaking amplitudes could be delivered, static clearing would not be necessary at all. Another conclusion we can draw from the simulations is that when the beam is vertically "blown up" (due to ions or other reasons), then the same shaking amplitude is less effective because the amplitude of ion oscillation still might be inside the beam . This has been verified in XLS experiments [20] where shaking was found not to be effective at higher currents with significant beam blow-up. Since there is a tune shift for ions with oscillation amplitudes above loY , the most effective shaking frequency depends upon the amplitude of shaking. Indeed, the most effective frequency is different for small-amplitude ions as opposed to large amplitude ions ; it is expected to be lower than the small amplitude linear oscillation frequency and expected to de-

One possible figure of merit for the effectiveness of shaking is the increase in the median amplitudes of oscillation. In the present case, the 10%, 50% and 90% median amplitudes were tracked under various conditions where a% is the amplitude, within which n% of the initial number of ions are found. For the simulation, 200 ions with mass 28 were used . Their initial position was chosen using a random number generator with a Gaussian weighting in the transverse coordinates and a uniform weighting in the longitudinal coordinate . The initial velocities also were generated at random using a Gaussian weighting with the rms velocity being the thermal velocity . As an example, the median amplitudes as a function of time for various shaking amplitudes ranging from O.loY up to 1oY are shown in Fig. 4. In each case, the shaking frequency was chosen to be in the most effective range (cf. Fig. 5) . With very small amplitudes of shaking (Figs. 4a,b) after a fast initial rise the median oscillation amplitude stays practically constant or rises very slowly (90% medians) after a large number of turns; all ions are stable . While the ions are stable this constant amplitude is higher than without shaking; even a small a,

4

(a) ash=' 1 Cr y fsh =1 .25MH

i

0

0

265

f~-.

8

( b ) ash= .2Qy fsh =1 1 .2MHz I-,-,,,g0%

bT

6

50%

0 .5

1 .0

1 .5

N turns

2 .0

[x104]

2.5

3.0

0

30

0 .5

1 .0

1 .5 2.0 Nturns [x104]

2.5

3 .0

40

25

30

T 20 \ 15

T

b

b

\ 20 ô

0

0 .5

1 .0 Nturns

1 .5

2 .0

[x104]

2 .5

3 .0

10 0

0

0 .25

0 .50 Nturns

0 .75

[x104]

1 .00

1 .25

Fig. 4 . aio%, a5o% and ago% median amplitudes as a function of the number of turns (time) for different amplitude of shaking at frequencies in their most effective range: (a) ash = O.1QY , (b) a sh = 0 .2o,, (c) a sh = 0 .5o,, and (d) a,,, = lo-, .

266

E Bozoki, D . Sagan /Nucl. Instr. and Meth. in Phys. Res. A 340 (1994) 259-271

crease with increasing amplitude. Furthermore, the resonance width is expected to be broader than if the beam-ion force was strictly linear . This relationship is shown in Fig. 5, where the median oscillation amplitudes are plotted as a function of the shaking frequency for shaking amplitudes from 0.1o, to 1.00" . " 1 .2 and There is an effective frequency range between 1 .6 MHz, below the 1 .80 MHz linear frequency, where the a,,, median amplitudes have a maximum and this peak shifts towards lower frequencies with increasing shaking amplitude. There is an even more effective and much broader region above 1.8 MHz for the a,,o,, median amplitude (and in the case of larger shaking amplitudes for a 5,,,, as well). This second peak is only a faint indication at very small amplitudes (a s,, = O.lo-, ), but beyond that, it quickly becomes significant. For shaking amplitudes where the n% median has increased but is not growing rapidly with time (see Fig. 4) the shaking resonance width for the corresponding anus, is relatively narrow (a fraction of a MHz) . When the n% median is strongly increasing with time (close to instability), the distribution becomes very wide (3-4 MHz) . There is also a fine structure in this region ; not all frequencies inside it are equally effective.

4

7.4. Effect of shaking on the lifetime and stability of ions When large shaking amplitudes are applied, then the method of using medians to indicate the effectiveness of shaking is of limited usefulness since too many particles are lost and the 90% and 50% medians can not be calculated . Notice, for example, that the curves on Figs . 5c and 5d corresponding to a sh = 0.5o-y. and a sh = 1 .00-y were calculated after only 3000 and 1000 turns, respectively. The similar curves in Figs . 5a and 5b, which correspond to much smaller shaking amplitudes, were calculated after 30 000 and 20 000 turns, thus yielding information on long-term stability. For larger amplitudes, a better measure of the effectiveness of shaking is the lifetime of the ions, that is the relative number of ions still in and around the beam (living ions). Fig. 6 shows the lifetime of the 200 ions up to 40 000 turns, shaken with amplitudes ranging from 0.50- to 2.0o-y in each case, with a number of different frequencies . The figure shows that the long term effect of shaking, i.e . its effect on the lifetime of ions in the beam, depends strongly on the shaking frequency, and for each shaking amplitude there is a wide range of effective shaking frequencies .

6 5

3

1 0

0

1

2

3

f sh

4

5

[MHz]

0

6

15 .0

0

1

2

fsh

3

4 [MHz]

5

6

30

12 .5

25

,T10 .0

br20 15

~~ 10 5 2

f sh

3

[MHz]

4

5

6

0

0

1

2

fsh

4 3 [MHz]

5

6

Fig. 5 a , a50 ,,, and ay,,, median amplitudes as a function of the shaking frequency, with (a) ash = 0.1, (b) ash = 0.2, (c)

a,h = 0.5, and (d) a,h = 1 shaking amplitudes . The median amplitudes were calculated 30000, 20000, 3000 and 1000 turns after creation for cases (a)-(d), respectively.

E . Bozokc, D. Sagan /Nucl. Instr. and Meth . in Phys. Res. A 340 (1994) 259-271

To see more clearly the frequency dependence of the long term effects of shaking, the relative number of living ions after N number of turns (N = 100-30 000) and for shaking amplitudes from 0.2o-ß to 5o y are plotted as a function of the shaking frequency (Fig. 7) . This figure, together with Fig. 5, shows that (i) there is a narrow, effective shaking frequency region below the linear oscillation frequency, and this shifts toward lower values as the shaking amplitude increases; (ii) there is also a much wider resonance harmonic at higher frequencies above the linear oscillation frequency; (iii) this second frequency region gets narrower and shifts toward lower frequencies as the shaking amplitude

0.8

0.2 0

1

2 Nturns

[x104]

3

4

0.8 0 0.6 0 0.4 0.2 0

increases, and above a certain shaking amplitude and after many turns, the two effective ranges blend together ; and (iv) there is a "structure" within the second frequency range, and not all frequencies inside it are equally effective (this structure can be seen more clearly after a small number of turns) . After establishing the effective frequencies, we can now compare the effects of shaking with different shaking amplitudes . Fig. 8 shows the life-curve of the ions for shaking amplitudes ranging from 0.2o-Y to 2o-,, when shaken at their most effective shaking frequency. We see (Fig . 4), that the ions are completely stable with ash S 0.2o-y . Beyond this value, some ions are unstable . As the shaking amplitude is increased a larger percentage of the ions will become unstable and so the ions could be cleared out without the need for electrostatic clearing elements . 7.5. Effect of different approximations

0 0.6 ô 0.4

0

267

0

2

1 Nturns

Nturns

[x104]

3

4

Fig. 6. Relative number of living ions as a function of the number of turns (time) at difterent shaking frequencies and with (a) a sh = 0.5Q, (b) ash = LOQY, and (c) ash = 2.0o-Y.

7.5 .1. Potential wells and dipole fields All our results were obtained taking into account the effect of the 1 .1 T magnetic field in the dipoles. However, we neglected the effect of the potential wells [21] at the ends of the dipole, where, in the XLS, a funnel-shaped vacuum chamber piece connects the 4 cm radius round part to the 2 em x 4 cm rectangular part . Fig. 9 shows the lifetimes of ions shaken at the same frequency and amplitude with and without the effect of the dipole magnetic field including fringe-field effects and with and without the beam pipe potential; their effect on the lifetime of ions is negligible - they do not modify the effectiveness of shaking. 7.5.2. ID, 2D and 3D approximation As described in previous sections, the ions move in three dimensions ; they oscillate transversely (x and y) and drift longitudinally (s). In some papers the ions have been considered fixed at an azimuthal location, and their horizontal and vertical motion are treated independently; hence, it is interesting to examine the effect of these assumptions . Consider first the ion's vertical oscillation frequency . In the 1D approximation (y motion only), the oscillation frequency is a simple function of the beam sizes and the ion's initial position (oscillation amplitude) and velocity . In the 2D approximation (x and y motion), the horizontal motion is coupled to the vertical motion of the ions, leading to a wide distribution of vertical oscillation frequencies at a given azimuthal location in the ring, as seen by comparing Figs . 2a and 2b in Section 4. In the 3D case, the longitudinal motion is only weakly coupled to the transverse motion of the ions (there is some coupling through conservation of action). However, the variations in the linear oscillation frequency along the circumference in the ring, and

268

E. Bozoki, D. Sagan /Nucl. Instr and Meth . in Phys. Res. A 340 (1994) 259-271

5,000 turns

0.8

0.8

v

1

.0 .6 ~O

.0 .6

ash = 1 ay

(b)

t

0

0.4

30,000 turns

Sf

0.2

0.4

a (a) ash - .5 y

0 0

1

2

fsh

3

4 [MHz]

5

30,000 turns/

0.2 0 0

6

/

n~

1

1

2

f sh

3

4 [MHz]

5

6

1 .0

C O

0.8

0.8

0.6

.0 .6

0.4

0.4

0.2

0.2

C _O

0

0.5

1 .0

1 .5

2.0 2.5 fsh [MHz]

3.0

3.5

0

4.0

0

1

fsh

2 [MHz]

3

4

Fig. 7. Relative number of living ions as a function of the shaking frequency after N turns (N=100-30000) with shaking amplitudes of (a) ash = 0.2o-Y, (b) ash 0.5o, (c) a sh =1 .OQ. and (d) a sh toy =

a consequent possible resonant amplification of the ion's transverse amplitude will modify the 2D motion (as pointed out in Section 3) . The coupling of the horizontal and vertical motion also will effect the frequency dependence of the median oscillation amplitudes, that is, the most efficient shaking frequency is expected to differ in the different approximations . Two hundred ions (as in Section 7.3) were tracked with 1D, 2D and 3D approximations with

a small amplitude (a sh = O.lo-,,) at several shaking frequencies . Then, the 10%, 50% and 90% median oscillation amplitudes after 30000 turns were plotted as functions of fsh (fig . 10). The figures show that the most effective frequency range broadens, with the peak shifting to slightly lower shaking frequencies due to the horizontal /vertical coupling and to the longitudinal movement of the ions . 1.0

10

ashlay

0.8

0.4

.06 ô 04

. 1 v

0 .2

02 0

0

0 .5

1 .0

-no B-field, Pot off ----B-field, Pot on ---B-field, Pot off

0.8

= 0.2

`. 0.5

y y

~,06

0

=

1 .5

Nturns

2.0

[x104]

2.5

30

Fig. 8. Relative number of living ions as a function of the number of turns (time) when shaken with ash amplitudes ranging from 0.2Q}, to 2ov at the corresponding most effective frequency.

0

0.5 Nturns

1 0 1 .5 [x104]

2.0

Fig. 9. Relative number of living ions as a function of the number of turns (time) without B and pipe potential (solid line), with B and without pipe potential (broken line), and with B and pipe potential (dotted line). fsh = 1 .80 MHz, a sh = 2o  .

269

E. Bozoki, D. Sagan INucl. Instr. and Meth. i n Phys . Res. A 340 (1994) 259-271 4.0 0.8

3.0 N C

0.6 0.4 0.2

0.0

0 .5

1 .0

1 .5 fahake (MHz)

2.0

2.5

4 .0

3.0

0

0.5

1 .0

1 .5

Nturns

2.0

[X104]

2.5

3.0

Fig. 11 . Relative number of living ions as a function of the number of turns (time) m (a) 1D, (b) 2D and (c) 3D approxi mations, when shaken with a sh = 0 .5 amplitude at the corresponding most effective frequencies .

reduces the stability of ions was discussed in Section 6. This effect can be seen in Figs . 11 and 12 . Fig. 11

shows the decreasing lifetimes of the ions (using the

appropriate fsh ) in the 1D, 2D and 3D approximations . Fig. 12 shows the relative number of living ions as a function of fsh for a sh = 0.5oy after large number of 0.0

turns. Besides increasing the effectiveness, the familiar broadening of the most efficient frequency range oc0.5

1 .0

1 .5 fahake (MHz)

2.0

2.5

curs as the longitudinal motion is included . 7.5.3 . Linear and nonlinear approximation

4.0

The oscillation amplitude of ions, located close to the center of the beam, grows linearly until the amplitude samples the nonlinear region of the bunch. At this

3.0

point, a frequency shift occurs to lower frequencies,

and nonlinear frequency decreases with increasing oscillation amplitude (Fig . 2a).

Using similar argument, we see that the ions will

escape the beam more easily if the increase in beamion force is assumed to be slower then linear as the ions get further away from the beam center (but are 0.0

still within the beam). Simulations for the XLS ring 0 .5

1 .0 fahake

1 .5 (MHz)

2.0

2.5

Fig. 10. a1o%, a 50% , and ago% median amplitudes as a function of the shaking frequency, fah for a shaking amplitude of O .loy . The ions are restricted to 1D (y only) and 2D motion (x, y) in (a) and (b), respectively; (c) reflects the 3D motion .

The long-term effects of shaking with a slightly stronger shaking amplitude (a sh = 0 .5oy ) are that the ions are more stable in 2D than in the 3D approximation, and even more stable in 1D calculation. Further, that adding horizontal motion and distribution (not

restricting the ions to the middle of the beam horizontally) to the vertical ones, reduces the stability of the ions . That the longitudinal motion of the ions further

0.8 .0.6 O

0.4 0.2 0 0.5

2.0

fsh

3 .5 [MHz]

5.0

Fig. 12 . Relative number of living ions as a function of the shaking frequency, fsh for a shaking amplitude of 0 .5oy m (a) 1D, (b) 2D and (c) 3D approximations .

270

E Bozoki, D. Sagan

I Nucl.

Instr and Meth . in Phys Res A 340 (1994) 259-271

20

0

1.0

1 .5

2.0

f sh

2.5

[MHz]

3.0

35

4.0

and aen,, median amplitudes as a function of the shaking frequency, with ash = 0.5Q,, after 5000 turns. Linear beam-ion forces were assumed. Fig 13 . aio%, a,,,;

showed that, assuming linear forces, the a, median amplitudes exhibit one strong peak at a lower than linear frequency, and the very effective and broad region at higher frequencies is not present (the nonlinear case is shown in Fig. 5); this is illustrated in Fig. 13 . Consequently, the number of living ions stays approximately constant after an initial rapid loss, even with a shaking amplitude of 0.5o-, (using the corresponding most effective shaking frequency), as shown in Fig. 14 . 7.6. Review of experimental obsercations

At the CERN AA and EPA and at FNAL AA [6,22] shaking slightly above (by few ten kHz) one of the (n ± v) sidebands was found to be the most effective in increasing the stacking rate and reducing the vertical emittance. In addition, at the CERN AA, an increase in the clearing current was also noted. At FNAL, an ion-driven dipolar coherent instability appearing with higher currents at (2 + v) was best suppressed by shaking the beam slightly below the (0 - v) sideband . As discussed in Section 6, the nonlinear theory can explain [6] why frequencies above the fast waves and below the slow waves work best .

0.8 0.6

linear ', nonlinear s

= 1 .5 MHz ---------------

0.4

Acknowledgement

0.2 0

At EPA a study was made of the effects of shaking with and without static clearing . The reduction of the vertical beam size was much greater (though a possible head-tail effect expanded the beam horizontally) when shaking and static clearing were used together . Since they applied resonant shaking, near one of the betatron sidebands, the effect of the shaking was both, virtually independent of the rf voltage beyond a minimum, and more pronounced for large enough beam currents . These experiments showed that shaking removed or diluted the ions . However, because of the variations of the beam size around the ring, the ion oscillation frequencies also varied, and only a part of the ions responded to any shaking frequency [22] . At UVSOR, an earlier experiment showed that when the beam was shaken near the calculated ion oscillation frequency there was a significant reduction in the vertical beam size [23] . In a second experiment shaking was applied close to the betatron frequency [12] . Tune shift measurements showed, that (with all the bunches filled in the ring) rf shaking alone is insufficient to suppress the tune shift, but it becomes satisfactory when do clearing is also used . do clearing alone required required very high fields, leading to strong desorption and high local pressure . At TERAS [24] using resonant beam shaking, the vertical tune shift and vertical beam size were reduced with a shaking voltage of 100-200 V. Small amplitude "rigid mode" shaking of the XLS compact storage ring beam [20] showed that there is a frequency which clears the ions from the beam . By slowly sweeping the shaking frequency, the beam size suddenly changed from round or vertically larger beam to a much flatter (vertically smaller) beam . At the same time, the Touchek limited beam lifetime suddenly became shorter. There was no change either in beam size or in lifetime when shaking alone was used, without clearing electrodes . However, with higher currents, shaking with the same (small) rf voltage was ineffective because the beam size is big enough that the oscillation amplitude of the ions is still inside the beam . Further, since the beam's potential well is proportional to the beam current, the ions need more kinetic energy to get out of it .

f sh

0

1

2 Nturns

=

[x104]

1 .7

3

MHz

4

Fig 14 Relative number of living ions as a function of the number of turns, with a,h = 0.5o,, assuming linear and nonlin ear beam-ion forces . In each case, the most effective shaking frequency was applied.

One of us (David Sagan) would like to thank the National Science Foundation for financial support. References [1] Y. Baconnier, CERN 85-19 (1985) . [2] M. Barton, Nucl . Instr . and Meth . A

243 (1983) 278.

E. Bozoki, D . Sagan /Nucl. Instr. and Meth. in Phys. Res. A 340 (1994) 259-271

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[6] [7] [8]

[10] [11]

J .B . Murphy, L.N . Blumberg, E . Bozoki, E . Desmond, J . Galayda, H . Halama, R. Heese, H . Hsieh, S. Kalsi, J . Keane, S. Kramer, P . Mortazavi, H.O . Moser, M . Reusch, J . Rose, J . Schuchman, S . Sharma, O . Singh, L . Solomon, M . Thomas and J .M . Wang, Proc . EPAC, Nice, Vol . 2 (1990) p . 1828 ; J .B . Murphy, R . Biscardi, J . Bittner, L.N . Blumberg, E. Bozoki, E. Desmond, H . Halama, R . Heese, H. Hsieh, J . Keane, S . Kramer, R . Nawrocky, T . Romano, J . Rothman, J . Schuchman, M . Thomas, J .M . Wang, J . Krishnaswamy, W . Louie and R . Rose, Proc . IEEE PAC, San Francisco, Vol . 2 (1991) p . 1107 . P . Zhou, P .L . Colestock and S.J . Werkema, Proc . IEEE PAC, Washington, DC, (1993) in press. R . Alves- Pires, D . Möhl, Y . Orlov, F. Pedersen, A . Poncet and S . van der Meer, Proc . IEEE PAC, Chicago, Vol. 2 (1989) p . 800 . A. Poncet, CERN/MT/90-1 (ES) (1989) . Y . Gomei K. Nakayama, K. Fukushima and S . Sukenobu, Nucl . Instr . and Meth . A 278 (1989) 389 . R. Alves -Pires and R . Dilao, Instituto Superior Technico IST-DF-6/91, Lisboa, Portugal, 1991 ; R . Alves- Pires, Proc . Fermilab III Instabilities Workshop (1990) p. 18 . D. Sagan, Cornell University CBN 91/2, 1991 ; D . Sagan and Y. Orlov, Proc . PAC, San Francisco, Vol . 3, (1991) p . 1839 . Y. Miyahara K . Takayama and G . Morikoshi, Nuc] . Instr. and Meth . A 270 (1988) 217 . R.D . Kohaupt, DESY H1-71/2 (1971) .

[12] [13] [14] [15] [16] [17] [181 [19] [20] [21] [22]

[23] [24]

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T. Kasuga, Jpn . J . Appl . Phys. 2 5 (1986) 1711 . Y . Baconnier and G . Brianti, CERN/SPS/80-2 (DI) . D . Sagan, Nucl . Instr . and Meth . A 307 (1991) 171 . C .J . Bocchetta and A . Wrulich, Synchrotrone Trieste ST/M-88-26, Italy (1988) . Y . Orlov, CERN /PS/89-01 (AR) (1988) . P . Zhou and J .B . Rosenzweig, Proc . PAC, San Francisco (1991) p . 1776 . M . Bassetti and G . Erksine, CERN-ISR-TH/80-06 (1970). Y. Okamoto and R . Talman, Cornell University CBN 80-13 (1980) . E . Bozoki and S . Kramer, Proc . EPAC, Berlin, Vol . 1 (1992) p . 789 . E . Bozoki and H . Halama, Nucl . Instr . and Meth . A 307 (1991) 156 ; H . Halama and E . Bozoki, PAC, San Francisco, Vol . 4 (1991) p . 2313 . J . Marriner and A. Poncet, Fermi National Laboratory p note 481, 1989 ; A . Poncet and Y . Orlov, CERN/PS/ML/Note 98-1 ; J. Marriner, D . Mdhl, J . Orlov, A . Poncet and S . van der Meer, CERN /PS/89-48 (AR) . T . Kasuga, H . Yonehara, T . Kinoshita and M . Hasumoto, Jpn . J . Appl . Phys . 24 (1985) 1212 . S . Sugiyama, T. Noguchi, T. Nakamura, T . Mikado, M . Chiwaki, T . Tomimasu and M. Ogura, Proc . 6th Symp . on Accelerator Science and Technology, Tokyo, 1987 .