David Sagan. In a colliding beam storage ring the Twiss parameters are aæected by the quadrupolar

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1 ëdcs.papers.dynamic BETAëDYNAMIC BETA.TEX CBN 94í06 April 25, 1995 The Dynamic Beta Eæect in CESR David Sagan Introduction In a colliding beam storage ring the Twiss parameters are aæected by the quadrupolar focusing of the beamíbeam interaction. Likeany quadrupole error this `dynamic beta' eæect is enhanced by running near a halfíinteger or integer resonance. The recentchange in the operating point of CESR so that the horizontal tune is just above a halfíinteger resonance has resulted in the dynamic beta eæect having a noticeable eæect on CESR operation. Under current colliding beam conditions the resulting change in horizontal beta has exceeded ææ x =æ x =0:5. In order to keep the notation as simple as possible while minimizing possible confusion, the following conventions have been adopted: æ Unless explicitly noted otherwise all Twiss parameters and beam sizes refer to the IP. æ A subscript of `0' denotes quantities calculated without the beamíbeam interaction. For example, æ 0 refers to the `unperturbed' beta. æ Since the electron and positron bunch currents or sizes can be diæerent, their eæect on one another will not be symmetric èalthough in practice this asymmetry is usually smallè. Through the beamíbeam interaction, this will result in diæerent Twiss parameters and diæerent beam sizes for each beam. `+' and `,' subscripts will be used to refer to the positron and electron beams respectively. æ `Repeating' subscripts will be dropped from equations. For example, if Eq. è1è is applied to the horizontal plane for positrons, then all the variables get the subscript `x+'. Notice that without the beamíbeam interaction it is assumed that the electrons and positrons have identical properties so that, for example, æ x0 ç æ x0+ ç æ x0,. 1

2 Analysis Following Chaoë1ë the dynamic beta eæect can be analyzed by writing the 1íturn transfer matrix from IP to IP as è! è!è!è! cos ç æ sin ç 1 0 cos, 1 æ sin ç cos ç =, 1 ç0 æ 0 sin ç , 1 sin ç 2f æ 0 cos ç 0, 1 ; 1 è1è 0 2f where the beamíbeam interaction strength of 1=f is given by 1 f x+ = 2N, r e æç x, èç x, + ç y, è ; è2è with analogous formulas for f x,, f y+, and f y,. The beamíbeam parameter ç is deæned by ç ç æ 0 4çf : ç is just the focusing strength of one beam on the other normalized by æ 0. sometimes convenient to deæne another beamíbeam parameter ç by è3è It is ç ç æ 4çf = æ æ 0 ç: è4è Combining Eqs. è1è, è2è, è3è, and è4è gives cos ç = cos ç 0, 2çç sin ç 0 = cos ç 0, 2çç æ 0 æ sin ç 0 ; sin ç = æ 0 æ sin ç 0 : è5è è6è è7è Eliminating ç from Eqs. è5è and è7è gives Alternatively, in terms of ç, one ænds æ æ 0 = ç 1, è2ççè 2 +2è2ççè cot ç 0 ç,1=2 : è8è æ æ 0 = s 1+ è2ççè2 sin 2 ç 0, è2ççè cot ç 0 : è9è Figure 1 shows æ=æ 0 as calculated from Eq. è8è as a function of Q 0 ç ç 0 =2ç for three diæerent values of ç. As can be seen from the ægure, the dynamic beta eæect is enhanced èlarge deviation of æ=æ 0 from 1è when ç 0 is near an integer or halfí integer as expected. Furthermore, from the ægure, it is seen that for tunes just above 2

3 Figure 1: æ relative toæ 0 as a function of tune for three diæerent values of ç. The top scale shows the tune in khz. ainteger or halfíinteger resonance the dynamic beta eæect causes a reduction in æ. This, of course, is what is desired for increased luminosity. As an example, the current operating point has a design horizontal tune of Q x0 =10:52 è204 khzfractional tuneè. Under the assumption that ç is in the vicinity of 0.03 èsee belowè, this implies that there is a large reduction in beta of æ x =æ x0 ç 0:5. Additionally, with the present vertical tune of Q y0 =9:60 è233 khzfractional tuneè, the reduction in vertical beta is æ y =æ y0 ç 0:8. Along with the change in æ at the IP there will also be a betaíwave throughout the ring. The calculation of the betaíwave is straightforward. Given the unperturbed Twiss parameters at some point s in the ring, the unperturbed transfer matrix from IP to s isë2ë ç M m0;11 ç m 0 èsjipè = 0;12 m 0;22 = 0 m 0;21 r æ 0 èsè æ 0 èipè cos ç 0èsè,èsin ç 0 èsè+æ 0 èsè cos ç 0 èsèè p æ 0 èsèæ 0 èipè r æ 0 èipè æ 0 èsè p æ 0 èsèæ 0 èipè sin ç 0 èsè ècos ç 0èsè,æ 0 èsè sin ç 0 èsèè where ç 0 èsè is the unperturbed phase advance from the IP to point s and the fact that æ 0 èipè = 0 has been used. Given any transfer matrix M, the transfer matrix for 1 C A ; è10è 3

4 the Twiss parameters isë2ë 0 1 æès 2 è æès 2 è C A = æès 2 è 0 m 2 11,2m 11 m 12 m 2 12,m 11 m m 12 m 21,m 12 m 22 m 2 21,2m 22 m 21 m C A 0 1 æès 1 è æès 1 è C A : æès 1 è è11è Using the beamíbeam kick matrix from Eq. è1è along with Eqs. è10è, and è11è the beta function with the beamíbeam interaction is è! 2 2çç æèsè =æèipè m 0;11, m 0;12 + m2 0;12 æ 0 èipè æèipè : è12è If the beamíbeam interaction is small enough one can use ærst order perturbation theory ècf. Sandsë3ë Eq è to obtain more simply Evaluating Eq. è13è at the IP gives Combining Eqs. è13è and è14è then gives ææèsè æ 0 èsè = æ 0èIPè cosè2ç 0 èsè, ç 0 è : è13è 2f sin ç 0 ææèipè æ 0 èipè = æ 0èIPè cos ç 0 : è14è 2f sin ç 0 ææèsè æ 0 èsè = ææèipè æ 0 èipè æ cosè2ç 0èsè, ç 0 è cos ç 0 : è15è As an example, with the present horizontal tune of Q x0 =10:52, the horizontal betaíwave is roughly given by ææ x èsè æ 0x èsè ç,0:5 cosè2ç x0èsè, ç x0 è : è16è One consequence of this betaíwave is that it changes the emittance function H x èsè èsandsë3ë Eq. 5.71è H x èsè = 1 æ x èç 2 x + ç æ x ç 0, 1 ç 2 è 2 æ0 x ç x and this will aæect the horizontal emittance èsee belowè. ; è17è With the present vertical tune of Q y0 =9:60, the vertical betaíwave is not as large as the horizontal and can be approximated by ææ y èsè æ 0y èsè ç,0:2 cosè2ç y0èsè, ç y0 è : è18è 4

5 Synch Light Luminosity Calculation One result of the dynamic beta eæect is that it throws oæ the luminosity as calculated from the vertical beam heights observed at the electron and positron synch light monitors. As will be shown, this can be used to verify the presence of the dynamic beta eæect. The luminosity is calculated from the vertical beam size by using the formula f rev L = 4çç x ç y n bunch X i=1 N i+ N i, ; where f rev is the revolution frequency, n bunch is the number of bunches, and N i, and N i+ are the number of positrons and electrons respectively in the i th bunch. To take into account the fact that the beam sigmas will be diæerent, ç x and ç y in Eq. è19è are deæned by the overlap integral between the two beams. For example, ç x is deæned by 1 2çç 2 x Z 1 dxe,x2 =2ç 2 x æ =2ç 2 e,x2 x ç 1 Z 1 dxe,x2 =2çx+ 2 æ e,x2 =2çx, 2 :,1 2çç x+ ç x,,1 Eq. è20è is easily integrated. The general result for either the horizontal or vertical planes is s èç+ ç = 2 + ç,è 2 : è21è 2 The beam sigmas ç x+, ç x,, ç y+, and ç y, are calculated from the equations q ç x = æ x æ x ; è22è and ç y = ç y èlsè æ vu u t æ yèipè æ y èlsè ; where èlsè stands for the light source point. If the dynamic dynamic beta eæect is ignored then the unperturbed values for æ x, æ x, and æ y are used in Eqs. è22è and è23è. Taking the dynamic beta eæect into account complicates the calculation since a closed formula does not exist for the ç x or the ç y.fortunately, as outlined below, a relatively simple iterative method is eæective for obtaining the beam sigmas. Since the synch light monitors only gives beam heights averaged over all the bunches, the dynamic beta calculation outlined below calculates a single value for the beam sigmas based on the average electron and average positron bunch currents. With dynamic beta the beam size calculation starts with the horizontal plane. The procedure is as follows: Step 0: Initial values are chosen èguessed atè for the ç x and the ç y, and the dependence of æ x as a function of ç x is calculated using the detailed lattice for the ring. 5 è19è è20è è23è

6 Step 1: Calculate ç x+ and ç x, from Eqs. è2è and è3è. Step 2: Calculate æ x+ and æ x, from Eq. è8è. Step 3: Calculate æ x+ and æ x, from the curve ofæ x verses ç x. Step 4: Calculate ç x+ and ç x, from Eq. è22è. Step 5: If the ç x have not changed by more than 0.1è then continue with step 6. Otherwise loop back to step 1. Step 6: Calculate ç x from Eq. è21è. Notice that only an inexact value for ç y is used by the horizontal calculation. This is acceptable since the horizontal dynamic beta is only weakly aæected by errors in ç y. After the horizontal calculation the vertical calculation proceeds as follows: Step 1: Calculate ç y+ and ç y, from Eqs. è2è and è3è. Step 2: Calculate æ y+ and æ y, from Eq. è8è. Step 3: Calculate æ y èls+è and æ y èls,è from Eq. è12è Step 4: Calculate ç y+ and ç y, from Eq. è23è. Step 5: If the ç y have not changed by more than 0.1è then continue with step 6. Otherwise loop back to step 1. Step 6: Calculate ç y from Eq. è21è. Since æ is only weakly dependent uponç ècf. ægure 1è the convergence for both the horizontal and vertical calculations is quite rapid and usually takes only a few iterations. Note that while ç has been used in the above procedure, with minor modiæcations ç could have just as easily been used instead. One minor detail: In the results presented below the values for N i+ and N i, as obtained from the beam button monitors have been scaled by anoverall factor in order that their sum agrees with the CERN current monitor. Figures 2 through 8 show the results of a dynamic beta calculation using data from two days of HEP running. In order to minimize possible abnormalities in the data the criterion for selecting the days for analysis was that there was relatively good luminosity on the day chosen and on the days just previous to the chosen day èe.g. days just after a startíup were avoidedè. The ærst day chosen was April 14, 1994 èjulian day 104è. The lattice used on this date was C9A18A000.GE92S 4S. The second day chosen was August 1, 1994 èjulian day 213è. The lattice used on this date was N9A18A600.FD92S 4S. The relevant parameters for both lattices are given in tables 1 and 2. The C9A18A000.GE92S 4S lattice has relatively high fractional tunes compared to the N9A18A600.FD92S 4S lattice and thus would not be expected to be aæected as much by dynamic beta. For each day the luminosity and related parameters were calculated every 5 minutes and in the ægures the parameters are plotted as 6

7 Table 1: C9A18A000.GE92S 4S Parameter Value Q x è225 khz frac. tuneè Q y è246 khz frac. tuneè æ x m æ y m æ x çmærad ç x0 476 çm æ 1:035 æ 10 4 N bunches 7 f rev 2.56 çs Table 2: N9A18A600.FD92S 4S Parameter Value Q x è204.2 khz frac. tuneè Q y è233 khz frac. tuneè æ x m æ y m æ x çmærad ç x0 467 çm æ 1:035 æ 10 4 N bunches 7 f rev 2.56 çs a function of the total positron plus electron current. Figure 2 shows the luminosity as a function of total current. The three sets of data shown correspond to: Aè Data from the CLEO detector which èpresumablyè gives the correct luminosity, Bè The luminosity as calculated from the synch light monitors neglecting the dynamic beta eæect, and Cè The luminosity as calculated from the synch light monitors including the dynamic beta eæect. In each graph there is a noticeable diæerence between neglecting and not neglecting the dynamic beta eæect. For the C9A18A000.GE92S 4S run the three curves are too close together èi.e. the magnitude of the dynamic beta eæect is too smallè to make any conclusions. For the N9A18A600.FD92S 4S run, however, it is clear that one must take the dynamic beta eæect into account. For the N9A18A600.FD92S 4S run the remaining diæerence between the CLEO luminosity and the calculation with the dynamic beta eæect can be attributed to a number of factors: è1è The actual tunes will diæer somewhat from the design tunes but it is the design tunes that are used in the calculation. è2è The hourglass eæectë6ë will reduce 7

8 the actual luminosity relative to the calculated value. è3è Any vertical dispersion at the synch light ports makes the vertical betatron size seem bigger than actual and this increases the actual luminosity relative to the calculated. è4è Finally, there is always the possibility of a resonant blowup in æ x due to the nonílinear part of the beamíbeam interaction. Figure 3 shows the average beta hæi çèæ + +æ, è=2 normalized by the unperturbed beta as a function of total current. Since the horizontal tune is always closer to the halfíinteger resonance the reduction in hæ x i is always larger than hæ y i. For the N9A18A600.FD92S 4S run the reduction in hæ x i is quite dramatic, being over a factor of 2 at the larger currents. This is opposite of what happens to the average horizontal emittance hæ x i çèæ x+ + æ x, è=2 asshown in Figure 4. For both lattices hæ x i is an increasing function of ç x as shown in ægure 5. The result is that the drop in æ x is partially oæset by the rise in æ x producing a ç x with only a weak current dependence as shown in ægure 6. ënote that for ægure 6 ç y0 is taken to be the sigma as calculated from Eq. è23è using the unperturbed beta. This is not the same as the sigma that one expects without the beamíbeam interaction.ë The diæerence between ç, ç, and æq ç èç, ç 0 è=2ç is shown in ægures 7 and 8 as a function of total current. For the C9A18A000.GE92S 4S run the tunes are far enough away from the halfíinteger resonance so that the dynamic beta eæect is small and ç ç ç ç æq. On the other hand, for the N9A18A600.FD92S 4S run, ç x is clearly closer to æq x than ç x. Notice that for the N9A18A600.FD92S 4S run ç x shows no signs of saturation and reaches a level of 0.05 at the highest current levels. æq x and ç x, on the other hand, show some signs of saturation at the highest current levels and stay within the `normal range' of less than 0.03 or so. This is in accord with the standard argumentë4ë that since a measure of the strength of the beamíbeam force is the shift of the tune æq, and since the beamíbeam induced resonances limit the performance of the machine, that for æq there is a `saturation limit' above which one cannot go without the beams falling out or the beams blowing up èwhich lowers æq back to its limiting valueè. That ç is a better approximation to æq than ç is a general feature as shown in ægure 9. In ægure 9, Eqs. è4è, è5è, and è8è have been used to graph ç, ç, and æq as functions of Q 0 with ç somewhat arbitrarily æxed at avalue of Over the range of Q 0 from 0.5 to 0.6, which is the range where one would like to operate to take advantage of the dynamic beta eæect, ç is clearly closer to æq than ç. 8

9 Figure 2: Luminocity as a function of total current for two days of HEP running. 9

10 Figure 3: Average beta hæi çèæ + + æ, è=2 normalized by æ 0 as a function of total current. 10

11 Figure 4: Average horizontal emittance hæ x içèæ x+ + æ x, è=2 normalized by æ x0 as a function of total current. 11

12 Figure 5: Horizontal emittance as a function of beamíbeam parameter ç x for the N9A18A600.FD92S 4S and C9A18A000.GE92S 4S lattices. 12

13 Figure 6: Average beam sigma hçi çèç + + ç, è=2 normalized by ç 0 as a function of total current. 13

14 Figure 7: Average hç x i, hç x i, and hæq x i as a function of total current. 14

15 Figure 8: Average hç y i, hç y i, and hæq y i as a function of total current. 15

16 Figure 9: ç, ç, and æq as functions of Q 0 with ç somewhat arbitrarily æxed at 0:04. 16

17 Amplitude dependence In terms of single particle dynamics the beamíbeam force is nonlinear beyond 1ç either horizontally or vertically. The fact that the beamíbeam force starts to fall oæ beyond 1ç results in a monotonic decrease of the eæective quadrupolar focusing strength with increasing particle oscillation amplitude. This results in the dynamic beta eæect being amplitudeídependent with large amplitude particles being relatively unaæected by the dynamic beta eæect. This implies that the deleterious eæects of reduced single particle lifetime that are associated with a lower æèipè are not present with dynamic beta. In other words, the dynamic beta eæect is materially diæerent from using a lattice with a lower æèipè. The amplitude dependence of the dynamic beta eæect was explored with a simple 1ídimensional tracking program. The procedure was as follows: The one turn transport map from IP to IP was taken to have the same form as the right hand side of Eq. è1è with the,1=2f kick terms replaced with the full amplitude dependent beamí beam kick as given by the complex error function formula of Bassetti and Erskineë5ë. Particles were seeded at diæerent amplitudes and tracked for 300 turns. For a single particle the resulting motion in phase space was ætted to an ellipse and a value for æ extracted. Figure 10 shows the dependence of æ=æ 0 on oscillation amplitude A for both the horizontal and vertical planes. As can be seen, æ is insensitive to changes in amplitude for the particles with oscillation amplitudes below about 2ç. This implies Figure 10: Relative beta as a function of oscillation amplitude for both the horizontal plane èsolid lineè and the vertical plane èdashed lineè. 17

18 that the amplitude dependent eæects on the luminosity are small. In the tails of the beam, where A x é ç 10ç x or A y é ç 50ç y, the dynamic beta eæect is seen to be small. Luminosity Considerations From ægure 5 it is seen that for the particular lattices considered æ x is an increasing function of ç x. It can be argued that this increase in æ x with increasing ç x results in an increased ç x with an attendant decrease in luminosity. One possibility then for increasing luminosity is to design a lattice in which æ x decreases with increasing ç x. Unfortunately, this might not be practical in CESR. To see this, consider H x èsè as given by Eq. è17è. The dynamic beta eæect does not aæect çèsè or the damping partition numbers since the IP is in a dispersion free zone. The eæect of dynamic beta upon æ x is thus through the betaíwave that is generated. In order to make æ x be a decreasing function of ç x one would have tochange where in the lattice æ x is increased and where it is decreased. From Eq. è15è, since ç 0 is conæned to be near a halfíinteger èwe want to have a signiæcant dynamic betaè, the only way to shift the betaíwave is to shift around the phase advance ç x0 èsè. To shift ç x0 èsè, however, would change where the pretzel maxima are. It is therefore not clear èat least not at this point in timeè whether a decreasing æ x is compatible with multibunch operation. Another tactic for increasing L might be to run with ç y nearer to the halfíinteger instead of ç x. The reason for this is contained in the equation L = Iæç y è1 + rè 2er e æ y0 ; è24è where r = ç y =ç x. Using the dynamic beta eæect it could be argued that one should be able to increase ç y above the traditional limit in CESR of 0.03 to The problem is that the hourglass eæectë6ë is non-negligible in CESR so with æ y comparable to ç z it is not clear that using the dynamic beta eæect to increase ç y and lower æ y is helpful to the luminosity without simultaneously lowering ç z. Additionally, it is not clear that æq y is the limiting factor with the current conditions. This is especially true since the crossing angle increases the horizontal resonance strengths faster than the vertical resonance strengths. Experimentally it was found awhile back that switching to a lattice where the vertical tune was closer to the halfíinteger made things worse. Operationally, however, conditions have changed since that experiment èfor one we now have a crossing angleè and therefore, like the stock market, past experience is not necessarily indicative of future performance. Simulations are one possible way to try to understand howhaving ç y near the halfí integer will aæect luminosity. Unfortunately, the betaíwave generated by the dynamic beta eæect may aæect coupling compensation, etc. Ultimately machine studies will have to be done if an answer is to be obtained. 18

19 Acknowledgements My thanks to Jim Welch, Dave Rubin, and Dave Rice for useful discussions. My thanks also to Dave Rubin for helping with the emittance calculation. References ë1ë Alexander Chao, ëbeamíbeam Instability," from Physics of High Energy Particle Accelerators, p. 201, AIP Conference Proceedings No. 127, Melvin Month et. al, eds. è1985è. ë2ë C. Bovet, R. Gouiran, I. Gumowski, and K. H. Reich, ëa Selection of Formulae and Data Useful for the Design of A. G. Synchrotrons," CERNèMPS-SIèInt. DLè70è4 è1970è. ë3ë Matthew Sands, The Physics of Electron Storage Rings, An Introduction. SLAC- 121 Addendum è1970è. ë4ë J. Seeman, ëobservations of the BeamíBeam Interaction," Proc. Joint USíCERN school on Part. Acc., Nonlinear Dynamics Aspects of Part. Acc, Sardinia, 121, è1985è. ë5ë M. Bassetti and G. Erskine, CERN-ISR-THè80í06 è1970è. ë6ë Stephen Milton, ëcalculation of How the Ratio æ æ =ç z Aæects the Maximum Luminosity Obtainable: The HourGlass Eæect," Cornell CBN 89-1 è1989è. 19