RECEIVED. Negative Transverse Impedance

Transcription

1 RECEVED SEP OSTt > LS- 4 O C a f L W. Chou March 2, 989 (Rev. June 2, 9S9) Negatve Transverse mpedance ntroducton n Ref. ( we report an observaton that the horzontal and the vertcal loss factors have opposte sgns for several types of geometres. Recently, measurements n the SPS show that the coherent tune shft n the horzontal drecton has postve values whereas that n the vertcal drecton has negatve ones. 23 Thus, the exstence of negatve transverse mpedance gets confrmed n a real machne. Ths stmulates us to start a new round of systematc studes on ths nterestng phenomenon. The results obtaned from our computer smulatons are presented n ths note. t s known that, for a crcularly symmetrc geometry, the transverse wakefeld has a postve frst peak. Ths has been dscussed n detal by A. FV. Chao. [3] After havng studed a couple of examples, Chao concludes, among others, n general, one fnds that the polarty of the transverse wake forces s such that t always hurts a short beam. However, the proof gven n Ref. [3] s lmted to the resstve wall case and the case where the boundary s nfntely perodc and has rotatonal symmetry. The generalty of ths concluson s questonable. n fact, n the partcular example that we wll dscuss below, the above concluson s no longer vald. Fg. s reproduced from Ref. [3], whch llustrates that the frst peak of the transverse wake s postve and that the transverse wake force further deflects ( hurts ) the test charge that closey follows the source charge. Ths s a correct pcture nsofar as the structure has rotatonal symmetry. However, when ths symmetry s broken, the whole pcture may change. The frst peak may become negatve and the transverse wake force may help the test charge stay closer to the beam tube axs, as we fnd n the SPS case. 2 Smulatons of SPS Adaptors Fgure 2 shows the structures of two types of adaptors used n the SPS, whch are lkely to be the contrbutors to the negatve mpedance measured n the horzontal drecton. n our smulatons, these adaptors are approxmated by a geometry that conssts of a crcularly symmetrc cavty and two beam tubes of a rectangular cross-secton, as shown n Fg. 3. The detaled structures of the adaptors are consdered not to be essental for our studes and are thus gnored. When an off-axs Gaussan bunch of an r.m.s. length of 5 cm traverses ths geometry, the transverse wake potentals and the

2 loss factors are computed by the 3-D code MAFA. (Ref. [4]) n order to see how a postve peak of the transverse wake potental could become a negatve one when the rotatonal symmetry embedded n a structure s broken, the followng procedure s taken. The mddle part of Fg. 3,.e. the crcularly symmetrc cavty, s kept unchanged, wth a radus of 9 cm, whle the cross-secton of the beam tubes on both sdes vares n the followng way.. The cross-secton s approxmated by a crcle of a radus of 3 cm. The results can then be compared wth that calculated by the 2-D smulatons. 2. The cross-secton s deformed to a square, of whch the half-wdth of the horzontal sde, x, s equal to the half-heght of the vertcal sde, y, and both are equal to 3 cm. The results should be close to that obtaned from the frst run above. J' 3. Keepng the vertcal dmenson y fxed at 3 cm, we vary the horzontal dmenson x from 3 cm to 7 cm. 4. Decreasng the vertcal dmenson y to 2 cm, repeat step 3. /. > The results of these runs are summarzed n Fgs. 4(a) and (b). n Fg. 4(a) t s seen that, as the dmenson x ncreases, the frst peak of the horzontal wake decreases. When x equals 4.5 cm (3.5 cm) and y equals 3 cm (2 cm), ths peak s almost zero. When z ncreases further, the peak becomes a negatve one. To vsualze ths transton, Fg. 5(a) exhbts a postve peak of the horzonta wake potental when x and y are both equal to 3 cm, and Fg. 5(b) a negatve one when x s ncreased.to 7 cm. Fg. 4(b) shows the behavor of the horzontal loss factor. t s smlar to that seen n Fg. 4(a). As a comparson the results from 2-D TBC calculatons, n whch the rectangular beam tubes are replaced by crcular ones, are also ncluded n Fgs. 4(a) and (b). They are n good agreement wth that obtaned from MAFA. We have tred dfferent mesh szes, beam tube lengths and cavty lengths n the smulatons and found that these are rrelevent to our results, just as expected.. n contrast to the horzontal drecton, the frst peak of the vertcal wake, as well as the vertcal loss, are all postve n these runs. Ths s obvously due to the specfcaton of the ratos between the two transverse dmensons n our smulatons. 2

3 DSCLAMER Portons of ths document may be llegble n electronc mage products. mages are produced from the best avalable orgnal document..

4 DSCLAMER Ths report was prepared as an account of work sponsored by an agency of the Unted States Government. Nether the Unted States Government nor any agency thereof, nor any of ther employets, makes any warranty, express or mpled, or assumes any legal lablty or responsblty for the accuracy, completeness, or usefulness of any nformaton, apparatus, product, or process dsclosed, or represents that ts use would not nfrnge prvately owned rghts. Reference heren to any specfc commercal product, process, or servce by trade name, trademark, manufacturer, or otherwse docs not necessarly consttute or mply ts endorsement, m m- menddton, or favorng by the Unted States Government or any agency thereof. The vews and opnons of authors expressed heren do not necessarly state or reflect those of the Unted States Government or any agency thereof.

5 3 Smulatons of Other Geometres n addton to the rectangular-beam-tube/crcular-cavtytype of geometry, we have tested some other types of 3-D structures.. The beam tubes are crcularly symmetrc and the cavty s of a rectangular shape. n ths case, we no longer see any negatve transverse mpedance for varous wdth.to-heght ratos of the rectangle. 2. Both the beam tubes and the cavty have rectangular cross-sectons. The results of ths geometry are more complcated. When we repeat steps 3 and 4 above, the horzontal loss factor decreases when x ncreases, but never becomes negatve. The frst peak of the horzontal wake potental, on the other hand, does cross the zero lne and takes negatve values when z s bg enough. 4 Dscusson When a beam traverses a dscontnuty n a beam chamber, t always loses energy. Negatve transverse mpedance would mply an energy gan of a beam n the transverse drecton. Thus one mght be concerned about energy conservaton. But actually there s no volaton of the conservaton of energy n our case. The energy gan n one jransverse (horzontal) drecton comes from the energy loss n another transverse (vertcal) drecton and the longtudnal drecton. The total energy change of the beam s stll a loss. The wake potentals that we compute are, of course, not the same thng as the wakefelds that are dscussed n Ref. [3]. The wake potental of a fnte bunch s the convoluton ntegral of the wakefeld and the lne densty of the bunch. Therefore, one mght argue that Fg. s stll a correct pcture whereas there may be a quck turn over of the frst wake peak n our case. Ths scenaro s qute unlkely, although we cannot prove t mpossble at ths moment. A qualtatve understandng of negatve transverse mpedance can be gven as follows. The transverse mpedance s proportonal to the dfference of the coherent mage coefkent, (, and the ncoherent mage coeffcent, e. As the horzontal dmenson of a geometry ncreases, ( H would decrease. n the lmt case that the horzontal dmenson becomes nfnty, & would be equal to zero, because a horzontal dsplacement of the beam would not change the forces actng on t. [5] Meanwhle, C H would reman fnte. Therefore, when the horzontal dmenson s large enough, the dfference, - E H, would change ts sgn. So would the horzontal mpedance. e~ For the tme beng, our results are solely obtaned from computer smulatons. A further analytcal study on ths nterestng subject s necessary n order to convnce ourselves the truthfulness of the results. Ths work s under way.

6 n summary, our smulatons demonstrate that the negatve transverse mpedance may appear when the rotatonal symmetry embedded n a dscontnuty s broken, and that the geometres that we have studed may be the sources of the postve horzontal tune shft measured n the SPS. We thank Dr. D. Brandt for brngng Ref. [2] to our attenton and for helpful dscussons. We also thank Dr. J. Cook for encouragement and for many enjoyable dscussons we had wth hm. References [l] W. Chou and Y. Jn, mpedance Studes - Part 3: Transverse-Loss Compensaton, ANL Lght Source Note LS-4 (Aprl 988). [2] D. Brandt et al., Tune Shft Measurements n the SPS,SPS/AMS/Note/88-4 (Decefnber 4, 988). 3 A. W. Chao, Coherent nstabltes of a Relatvstc Bunched Beam, SLAC-PUB2946 (June, 982). [4]T. Weland, EEE Trans. Nucl. Sc., NS-32, 2738 (985). [5 B. Zotter, Nucl. nstrum. Methods, v. 29, 377 (975). 4 /

7 A5.., /J////////////////// * fe.4 Fg.. (a) Sketch of the transverse wake functon, whch exhbts a postve peak. (b) T h e transverse wake force further deflects the test charge. Both fgures are reproduced from Ref. 3.

8 , n 0 0. N n 0 * Fg. 2. Two types of adaptors n the SPS.

9 ' \ "\. Fg. 3. The geometry used n the 3-D smulatons. (a) 3-D vew and (b) Front vew.

10 ~.. - Transton of H o r z o n d Wake Potental a 4-0 d Horzontal Dmenson x [cm) Transton of Horzontal Loss Factor 2.0 F L b b b L o t t Horzontal D'menson x (cm) Fg. 4. (a) The frst peak of the horzontal wake potental vs. x, the half-wdth of the horzontal sde of the rectangular cross-secton of beam tubes. (see Fg. 3) (b) The horzontal loss factor vs. x. The sold lnes correspond to a half-heght of the vertcal sde, y, of 3 cm, and the dot-dashed ones to an y of 2 cm. Note that all the curves cross the zero lne. - ' e

11 d-! t -4 D # Fg. 5. The bunch ( B ),the horzontal wake (W ) and the longtudnal wake (W ) for an z of (a) 3 cm and (b) 7 cm. The scales on the vertcal axs are normazed. The horzontal axs s n unt a,the r.m.s. ength of the bunch.