Spectral Analysis of Relativistic Bunched Beams

Transcription

1 SLACPUB7159 because the bea would have to be decelerated for that to happen. The iage current flowing through a bea detector produces a voltage VOut given by Vout () =, (4) where S is the detector longitudinal sensitivity. The sensitivity depends on the detector, on whether it is a cavity, strip line, capacitive pickup, etc., and S(o)can vary rapidly or slowly with frequency. a assuing the variation is slow, and that the spectru of the output voltage is the sae as that of the wall and bea currents. Many easureents can be ade in a narrow frequency band where the sensitivity can be treated as a constant, so this is a good approxiation. f it isn't, corrections ust be ade to account for frequency dependence of the sensitivity. f the bea is offset fro the vacuu chaber center, the iage current is not unifor; instead, it varies around the bea pipe due to the displaceent of the bea. A transverse bea detector responds to the dipole oent Spectral Analysis of Relativistic Bunched Beas R.H. Sieann* Stanford Linear Accelerator Center, Stanford University, Stanford, CA 9439 NTRODUCTON Particles in a storage ring are oscillating in the longitudinal and transverse diensions, and therefore, the frequency doain is natural for analyzing any bea generated signals. nforation ranging fro oscillation frequencies to bea phase space distributions can be extracted fro the spectral content of these signals. t is often necessary to switch between tie and frequency doains using Fourier transfors. f f(t) is a function of tie, F(w) given by (@No) where rl(t) is the displaceent fro the center. The output voltage of a transverse detector is + where SA and D are the transverse sensitivity and the Fourier transfor of the dipole oent, respectively. As with the longitudinal, SA is assued to vary slowly with frequency, and corrections ust be ade if that isn't a good approxiation. W F(o) = jf(t)ejotdt (1) is its Fourier transfor. n this equation j = & and i o is the angular frequency which is the appropriate atheatical variable rather than the frequency v = w/2n that is easured by spectru analyzers. The eaning of the word "frequency" depends on context, but the notation v for frequency and w for angular frequency will be rigorous. The inverse transfor is A SNGLE PARTCLE The spectru of a single particle is like a Green's function, and it is the key to understanding the spectru produced by a bea. Three separate cases are consider in an order of increasing coplexity: 1) constant revolution frequency, 2) Frequency Modulation introduced by synchrotron oscillations, and 3) Aplitude Modulation introduced by betatron oscillations. 1 f(t) = JF(w)ejO'do, 27t Longitudinal Motion & Constant Revolution Frequency These notes are restricted to relativistic beas, and for these beas the iage current flowing in the walls of an accelerator vacuu chaber, i,, has a line density equal to the line density of the bea current, ib, The current of a single, unitcharge particle with a constant revolution frequency Or, Figure 1, is a periodic set of ipulses spaced a tie T = 27r/, apart iw(t) = ib(t) and W(w)= b(o) ib(t) = x 6 ( t nt). (3) with an overall inus sign since it is an iage current. The iage current is an ideal current source; nothing placed in the vacuu chaber wall can affect it * ~ Work supported by the Departent of Energy, contract DEAC376SF515. Paper Presented at the 7th Bea nstruentation Workshop Argonne, L, May 6 9, (7)

2 Figure 1. A particle with a constant revolution frequency passing a bea detector located at one point in the ring and the spectru it produces. Fourier transforing this gives x 6 ( t nt)ejwtdt Figure 2. Positive and negative frequency phasors for the sae n1 and the atheatical and easured, physical spectra. The line at n = is onehalf that for n # because there is no corresponding negative frequency line n= n= The last su is an infinite su of unit agnitude phasors. f these phasors are at different angles they will add up to zero because there is an infinite nuber of the, but if they are all at the sae angle the su will be infinite. The latter happens when o/o, equals an integer; every angle is an integer ultiple of 2% in that case. This is written forally as n= n= (9) The spectru is a cob of equal aplitude lines spaced at o,extending fro o = to o =. t is shown in Figure 1. The spectru has two properties the frequencies that are given by the arguent of the &function and the envelope that is given by the factor in front. The frequencies are haronics of or, and the envelope equals w, independent of frequency. There are both positive and negative frequencies in eq. 9, but spectru analyzers easure only positive frequencies. How should the negative frequencies be interpreted? Fourier transforing to the tie doain The positive and negative frequency phasors appear in phase and at the sae at the sae frequency on the spectru analyzer. The resulting spectru is shown in Figure 2. Frequency Modulation ntroduced By Synchrotron Oscillations Synchrotron otion odulates the arrival tie by z= za cos(o,t + cp) where z, is the synchrotron oscillation aplitude, os is the synchrotron frequency and cp is the phase.* nstead of being a series of ipulses spaced exactly T apart, the current is ib(t) = x 6 ( t [nt + 2, cos(o,nt n= + cp)]). (13) Each ter in the su is unit agnitude phasor. The angles of the positive frequency phasors increase with tie while those of the negative phasors decrease with tie. The phasors for the sae n1 are shown in Figure 2. The real parts are in phase, and the iaginary parts are 18 out of phase. The physically eaningful quantity is the real part that is given by where z, is the synchrotron oscillation aplitude, os is the synchrotron frequency and cp is the phase at n =. The synchrotron frequency and synchrotron tune Q s are related by w, = Qsor. The Fourier transfor is 3 4 * The synchrotron frequency and synchrotron tune Qsare related by ws= QSop

3 DSCLAMER Portions of this docuent ay be illegible in electronic iage products. ages are produced fro the best available original docuent..

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5 b(w) = xexp(jw[nt n= + za cos(o,nt + q)]). ( 14) n the liit of sall aplitude or low frequency, a<< 1, the second ter in the exponent can be approxiated with a Taylor expansion n=... / ; j... k = l.6 n.4 s".2 W &..2 The first of the three sus is the sae as for a constant revolution frequency. Take a look at the second one;.. zexp(jnot + jo,nt + jq)) = ej9 1.. zexp(j2nn(w w,)/o,) n= n= The last step follows fro the sae logic used for a constant revolution frequency the su equals zero unless (61 co&o, is an integer. The frequencies differ fro the rotation haronics by +%. The third ter in eq. 15 leads to frequencies that differ fro the rotation haronics Synchrotron otion has lead to two new frequency cobs displaced fro the rotation haronics. Both of these cobs have envelopes that depend linearly on frequency. The approxiation in eq. 15 is valid for sall aplitudes and/or low frequencies. The Taylor series expansion could be continued. The next ter would affect the envelope of the rotation haronics and introduce frequency cobs displaced fro the rotation haronics by f 2 %. However, rather than doing this it is better to perfor a Bessel function expansion 1 (17) where Jk is an ordinary Bessel function of order k. This sae expansion leads to Bessel functions in every analysis of Frequency Modulation. Using this in eq ,...e a. '.*.,a 5 1 Figure 3. The envelopes for different synchrotron sidebands and cp = n/2. k=.4 As before the suation over n restricts the possible frequencies. n this case it requires that (61 k@)/o, equals an integer. Perforing the su Equation 19 gives the general expression for the spectru of a particle undergoing synchrotron otion. t is an iportant result that deserves substantial discussion. 1. The approxiate treatent based on a Taylor expansion corresponds to three of the ters, k = 1,, 1, when the lowest order Taylor expansions for the Bessel functions are used. 2. For each rotation haronic there is an infinite nuber of sidebands. They are displaced fro the rotation haronic by ko,, k = =J,..., and have different envelopes. The envelopes illustrated in Figure 3 have the usual properties of ordinary Bessel functions: i) they are even (odd) functions of a if k is even (odd) and ii) the first axiu of Jk is at a= k. When 11 << l/z, only the rotation haronics, k=o, are present, and as the frequency increases ore sidebands appear. This is illustrated in Figure The synchrotron sideband nuber, k, appears in two different places: i ) the frequency shift fro the rotation haronic is k%, and i i ) the envelope is Jk(oz,). The best frequency region to observe the kth sideband is at o Wz, This connection between frequency shift and frequency region for observation will becoe iportant when the phase space structure of a bea is considered. 5 6

6 8 F of longitudinal otion alone. Synchrotron oscillations cobined with dispersion do not introduce new frequencies although they do affect the envelopes.2 Betatron otion produces Aplitude Modulation that leads to new frequencies and new phenoena in the spectru. The next two sections concentrate on betatron otion. Begin with a constant revolution frequency. The displaceent is given by c\ where Ap is the betatron aplitude, and op is the betatron frequency that is related to the betatron tune, Qpo, by Qpo = OP/Or. The dipole oent and its Fourier transfor are Figure 4. The spectra for different o,t, for a synchrotron tune Q s =.1 and cp = d2. The heavy dashed line is k = in each case and J 4. The phase cp is the phase of the synchrotron oscillation when n =. Since the su is infinite, n = is arbitrary, and, therefore, cp is arbitrary. This situation changes when ultiple particles are considered, and the phases of the particles relative to each other has significance. 5. The observed spectru is obtained by cobining the positive and negative frequency coponents. The signals at w = wnk = n o r + k o, and = %k have the sae physical frequency, and as the next line shows, they add coherently in the physical spectru. Fourier transforing back to the tie doain and taking the real part gives %(ib (t)) = %[ ejk(cpwj k(onkta)ejwkt + ejk(cp~/2)jk(onkta )ejonkt Aplitude Modulation ntroduced By Betatron Oscillations Constant Revolution Frequency There are two frequency cobs. One is displaced fro the rotation haronics by +up and the other by cop. Each has a constant envelope. This is illustrated in Figure 5. The betatron tune is alost always greater than one, and, as consequence of easuring the dipole oent at only one point on the orbit, the integer part of the tune cannot be easured fro the spectru. There is also an abiguity in easuring the fractional part of the tune. The spectru when the fractional part of the tune equals q p is the sae as when the fractional part equals 1 qp. Figures 5b and 5d illustrate this. The abiguity arises because AM produces sidebands above and below the rotation haronics. One way to resolve it is to increase the tune by strengthening a focusing quadrupole and observing whether the spectral line oves to lower or higher frequency. The frequency of a line in the region no, < o c (n + 1/2)o, will increase if qp <.5, and it will decrease if qp >.5. The physical spectra in Figures 5b, 5d could have been derived in a different way. Rather than using eq. 21 for the transverse displaceent, r l could have been written A particle can be offset fro the center of a transverse bea detector due to closedorbit errors, synchrotron oscillations cobined with dispersion, or betatron oscillations. A closed orbit offset produces a spectru identical to that This would have resulted in a atheatical spectru with only upper sidebands of aplitude Ab. However, when the real part of the resulting expression was 7 8

7 : l l : l l The energy deviation is 9 out of phase with the tie deviation, z t 4 Qpi 7) 1 n n+l 1 a sin(o,t + cp) where a is the oentu copaction. Substituting this into eq. 26 and integrating gives yp(t) = opt + opacos(ost + cp) + u/ The chroatic frequency is W < = w&/a, and related to the betatron phase at t = by. v is a constant of (29) integration 1 olw, 2 2 :: : 1 : 3 :i / lil a=a1 dz =dt VV, 1 1 The dipole oent is 1 Figure 5. The atheatical (a, c) and physical (b, d) spectra given by eq. 23 when Qpo = 2.33 (a, b) and Qpo = 1.66 (c, d). For purposes of illustration the lines fro one cob are solid and the lines fro the other cob are dashed. d(t) = ib(t)rl(t) = Apejvp(t) x 6 ( t [nt +z, cos(o,nt n= + cp)]). (31) Following the procedure developed above, and identifying the iportant frequencies gives taken, the physical spectra would be the sae as in Figures 5b and 5d. A generalization of eq. 24 is the ore convenient for for the transverse displaceent when the effects of synchrotron oscillations are included. Betatron and Synchrotron Motion The focusing strength of a quadrupole depends on energy, and energy is odulated by synchrotron otion. As a result, the betatron phase does not advance soothly. The generalization of eq. 24 in ters of the betatron phase typ is This result gives the general expression for the transverse spectru of a particle undergoing betatron and synchrotron oscillations and is coparable in iportance to eq There are an infinite nuber of synchrotron sidebands centered about the betatron frequencies. These sidebands are displaced fro the betatron lines by k%, k =,..., 2. The envelopes are ordinary Bessel functions with arguent ( o o p o<)za n contrast to the longitudinal, these envelopes are offset fro o = by WP = 6 since typically op << 5, A positive chroaticity shifts the envelopes to positive frequencies, etc. This is illustrated in Figure The envelope of sideband k is Jk((M<)za). The best frequency region to observe the kth sideband is at w 5 + UT,. Depending on the chroatic frequency, high order sidebands can be seen at low frequencies. 4. The phase of the betatron oscillation when n = only has eaning for ultiple particles.. The tie rate of change of yrp is dv= op(1 +56)+o(62) dt where 6 is the fractional deviation fro the central energy and op = Q p o q is the betatron frequency when 6 =. The chroaticity, 5, easures the variation of betatron tune with energy. t is defined as 9 1

8 1.o Longitudinal Phase Space Structure.8 When bea intensity is low, particle otion is deterined by agnet and RF cavity fields. Bea generated fields can be neglected, and particles ove independently of each other. The longitudinal phase space density can depend on %a,but it can't depend on cp. f it did, particles would not be independent.* The phase space density is p(za,cp) = p(za)/2n, and the bea generated signal is.6.4 (o) = o r. 6(w ko, nor)jz& =or , Figure 6. Envelopes for the betatron line (k = ) for three different values of chroatic frequency. 5. The observed spectru is obtained by cobining the positive and negative frequencies as has been done above. The atheatical spectru based on eq. 25 has only upper sidebands, but the physical spectru has both upper and lower sidebands. The lower sidebands coe fro negative frequencies in this atheatics. MULTPLE PARTCLES The expressions in eqs. 19 and 32 are the basis for understanding bea generated signals. For exaple, the longitudinal spectru of a bea can be derived by convoluting eq. 19 with the longitudinal phase space density of the bea, p(ta, cp), where ptadtad9 is the charge in phase space area ZadTadq; C CU k,n= CU 2A S(okos nor)jzadza dcpp(z,,cp)e'k'pjk(wz,)* (33) The envelope has changed, but the frequencies haven't. The exaples below will show how inforation about the bea can be extracted fro spectra. These exaples are intended to illustrate ethods that can be applied to any probles. C n= 1 2n d(ppo(za)ejk'pjk(6xa) 2n (34) S(wno,)Sz,dzaPo(za)Jo(~a) 3 Only the k = ter is not equal to zero once the cp integral is perfored. There are rotation haronics but no synchrotron sidebands. f the bea has charge Q and is Gaussian in z with rs bunch length o,, (35) and3 (@)= Qor exp(2~/2) C 6 ( w no,) n=w The spectru is a cob of rotation haronics with a Gaussian envelope with rs width of l/o,. A detector with a flat sensitivity up to o l/o, can be used to easure the bunch length. Alternatively, an aplitude easureent at a fixed frequency of o l/otcould be used to onitor changes in bunch length. The appearance of synchrotron sidebands and aziuthal structure in longitudinal phase space are directly related. Observation of synchrotron sidebands iplies aziuthal phase space structure, and aziuthal phase space structure leads to synchrotron sidebands. Phase space structure arises fro interaction of the bea with its own fields. Since these fields depend on the bunch distribution, that distribution is the solution of a self consistency proble forulated using the Vlasov Equation.4 The phase space density can be written as a Fourier expansion 1 " p(ta,q>=zp(ta)eirn'p. 2' = Substituting into eq. 33 * 11 k,n=m.2 (o) = o r.2 ~ This holds down to the Schottky noise level of the bea. 12 (37)

9 1oo 1l 1o ~ Figure 7. Phase space for the quadrupole perturbation given by eq co 6(o ko, (o) = 2 "k,n,= 2 no,)[tadr,tdqp(ta)e j(k+hjk(a) 2 4 % Figure 8. The envelopes for the rotation haronics and second synchrotron sidebands for the exaple in eq. 39. Longitudinal Schottky Noise There is a direct relation between each sideband and a specific haronic of the phase space structure; the th haronic of the phase space structure produces a signal at the kth sideband. As a specific exaple, suppose that the bea has a quadrupole perturbation illustrated in Figure 7 (39) where po is given by eq. 35. The Ta dependence of the perturbation was chosen for the purpose of this exaple. Substituting eq. 39 into 38 and perforing the integrals3 Particles ove independently of each other when bea generated fields are negligible, and it was argued in the previous section that a consequence is that the longitudinal phase space density, p(za, cp), cannot depend on 9. However, there is a liit to this because a bea consists of individual particles not a sooth distribution. The particles cannot arrange theselves to reove all cp dependence. They wouldn't be independent if they could. There is soe residual cp dependence and soe signal, called Schottky noise, fro it. Taking account of individual particles, the phase space density is P p(za,cp)=qc6(ta TapP(VTp) p=l where q is the particle charge, Zap and cpp are the aplitude and phase of the pth particle, and there are P particles in the bea. Using this distribution, eq. 33 becoes n= There are rotation haronics and sidebands at f 2 ~ The. envelopes are shown in Figure 8. The sideband signal is a axiu at o 1.5/oz and is about a factor of 1 less than the rotation haronics in this frequency range for A2 =

10 The current depends on the phase space coordinates of P particles, and it is different for every enseble of particles. This is just like a rando walk proble. The result of a rando walk is unknowable, but statistical quantities based on an enseble of rando walks have eaning. The relationship between the phase space coordinates of different particles changes with tie due to nonlinearities and rando processes such as eission of synchrotron radiation photons. Therefore, while the current at a particular tie is unknowable, the ean and rs currents are eaningful statistical quantities. They can be deterined by averaging over all possible saples of P particles. This averaging is denoted by angular brackets, < Each of the ters in the expression for nk (eq. 42) is a phasor with agnitude given by the Bessel function and direction given by the arguent of the exponential. When k f, the phasors point in all directions, and when averaged over all possible saples c nk > =. The phase which had no eaning for a single particle is critically iportant for a bea. When k = all the phasors point in the sae direction, and the su does not vanish. The discrete particle nature of the bea is critical for evaluating the phasor su, but once that is done the su over particles can be evaluated with an integral using the phase space density po(ta). The ean current, < n >, is given eq. 34. The square of the rs current is given by 2nA E 15 Y u 1 >s. The ters in the double su are phasors. n general they point in arbitrary directions aking the su equal to zero when the average is taken. However, for any saple of P particles the phasors line up and add coherently when pp = cp;; i.e. when particle p is the sae as particle s. This reoves one of the sus, and The su over particles has been replaced by an integral over 751 in the last step. When the bea is Gaussian and po is given by eq rno, 6 8 Figure 9. Envelopes for the rotation haronics, a coherent structure with A2 = O 4 (eq. 4), and the k = 2 rs Schottky current for 1 particles and Vr = 5 khz. Equations 44 and 45 have several interesting features. First, since Q = qp, the rs current is proportional to dp as expected fro shot noise. n contrast to eq. 38 that depends on unknown constants (e.g. A2 in eq. 39), the noise calculation is absolute. Equation 44 gives the noise current, and currents above this value are due to phase space structure. This is illustrated in Figure 9 where the current fro a quadrupole structure with A2 = is larger than Schottky noise for obt< 3.5. f A2 = 15the cross over point would ove to obz 2.. Transverse Phase Space Structure Analyses of Schottky noise and signals associated with phase space structure can be perfored for transverse otion also. The Schottky analysis is an extension of the previous section and is not repeated here. t can be found in Ref. 2. The signals fro transverse phase space structure introduce new ideas and are treated in this section. n general, a 4diensional phase space density, K, ust be used to calculate the transverse signal because eq. 32 depends on the betatron aplitude and phase and the synchrotron aplitude and phase 16

11 The phase space density can be expanded in a Fourier series in y and cp. The only coponent of the u~ expansion with a nonzero dipole oent is the one with syetry exp(jy). All of the others equal zero when the \v integral in eq. 46 is perfored. Each of the coponents of the cp expansion can produce a signal, and, as in eq. 38, there is a relation between phase space structure and sidebands, The new aspect is that the betatron aplitude can depend on the synchrotron aplitude, AP = AP(Ta). n general, coherent transverse odes have such dependencies.4 As a specific exaple consider all of the particles oscillating with the sae aplitude, Ao, and a Gaussian distribution in T~ (eq. 35) with no cp dependence. The phase space density is apt =n/2,,, L,..,:..Le' *, ' 1 2 3, (47) and The envelope is a Gaussian centered at o = op + 5 = 5 and shifts as the chroaticity changes. The transverse displaceent at t = is (eqs. 25 and 3) Figure 1. The dipole oent at different ties for different values of chroaticity. Coupled Bunch Signals The connection between the transverse displaceent and the shift of envelope with chroaticity can be understood by plotting the dipole oent, p ~ l for, different chroaticities and at different ties. This is done in Figure 1. When E, =, the transverse otions of all the particles are in phase. The signal has an average value, Le. a coponent at o = ; the tie scale of the dipole oent is the bunch length, C Y ~and, the characteristic frequency is o 1/7. As the chroaticity increases a headtotail phase shift is introduced, and the head and tail are out of phase. The average value of the signal decreases thereby decreasing the signal at o =. n addition, the dipole oent varies ore rapidly along the bunch. This shortens the tie scale and increases the characteristic frequency; the envelope has shifted to higher frequency as given by eq The exaples above have all involved convolution of single particle signals with the bea phase space density. The final one goes back to the derivation leading to eq. 19. Assue that the bea has 2 equally spaced bunches each of which can be treated as a acroparticle and that these bunches are coupled together by a long range wakefield. The phase shift between bunches, Acp, can have only discrete values fro the following arguent. f the phase of the first bunch is cp, the phase of the second bunch is cp + Acp, and the phase of the third bunch is cp + 2Acp. The third bunch is the sae as the first bunch, so the possible values for Acp are Acp =, n. The first bunch current is given by eq. 19. Assuing the second bunch has the sae aplitude, its current is given by eq. 18 with a phase factor explj(kacp +jnot/2)] ultiplying it. The first ter is the bunchtobunch phase shift, and the second is due to the arrival tie delay of the second bunch. Adding these two currents together gives 18

12 These paper was intended to introduce ideas and ethods for understanding bea spectra. hope it serves that purpose. REFERENCES Figure 11. The two bunch spectra for coupled bunch odes with and n phase shift between bunches b(a)=@r x ej k ( V W j k ( U M a ) ( l + e j ( k 4 + W 2 ) ) q o ko, ",) k,n=. (5) For rotation haronics the phase difference between bunches doesn't atter, the current equals zero when n is odd, and it is double the single bunch current when n is even. The latter two follow fro nort/2 = nn. When k = +1, the sidebands of the odd rotation haronics are present if Acp = n and are issing (neglecting a factor O(osT/2)) if Acp =. The sidebands of the even rotation haronics are issing if Acp = n. and present if AT =. This is illustrated in Figure 11. The presence or absence of particular sidebands tells the relative otion of the bunches. This analysis can be generalized to any bunches. For B bunches the possible values for the phase shift between bunches are Acp = 27c/B where =,..., B 1 is called the coupled bunch ode nuber. Only every Bth rotation haronic appears, and if one sees synchrotron sidebands of the nth rotation haronic it is fro a coupled bunch ode with ode nuber n = od(n, B). This follows because the phasors representing the current of the individual bunches have the sae phase and add up coherently only when nort2n 27cn Acp + + = 27cp; p = integer. B B B S. Gradshteyn and. M. Ryshik, Tables of nteyrals, Series. and Products (New York: Acadeic Press, 1965), eq R. H. Sieann, AP Conference Proceedings 184 (New York, 1989, edited by M. Month and M. Dienes), p. 43. Reference 1, eq J. L. Laclare, 1 1th nternational Conference of HiPhEnergv Accelerators (Basel: Birkhauser Verlag, 198, edited by W. S. Newan), p Reference 1, eq DSCLAMER This report was prepared as an account of work sponsored by an agency of the United States Governent. Neither the United States Governent nor any agency thereof, nor any of their eployees, akes any warranty, express or iplied, or assues any legal liability or responsibility for the accuracy, copleteness, or usefulness of any inforation, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific coercial product, process, or service by trade nae, tradeark, anufacturer, or otherwise does not necessarily constitute or iply its endorseent, recoendation, or favoring by the United States Governent or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Governent or any agency thereof. f bunches are unequally spaced or have different charges or aplitudes, these siple results ay not hold, but the sae phasor addition can be used to identify doinant lines for different couple bunch odes. CONCLUDNG REMARK There is a wealth of inforation in the bea spectru. The techniques used in the exaples above can be applied and/or extended to chroaticity easureents, coherent frequency shifts, etc. n addition, there is a close relationship between bea generated signals and bea stability because it is the current and dipole oent that drive accelerator ipedances and produce forces 19 2