BEAM LOADING EFFECTS IN PROTON LINACS. R. L. G1uckstern Yale University

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1 Octber 21, 1963 BEAM LOADING EFFECTS IN PROTON LINACS R. L. G1ucktern Yale Univerity Intrductin A bunched beam f charged particle paing thrugh a cavity interact with the field in the cavity. It cuple, in additin, t ther renant mde, exciting field which may react n the beam. The preent dicuin i centered n tw effect, the interactin with the externally applied accelerating field and ubequent reactin n the beam, and the generatin f tranvere mde capable f deflecting the beam. Interatin f the Beam with the Accelerating Mde Effect f Beam n Cavity Impedance -- Equivalent Circuit: It can be hwn that the equivalent circuit f the cavity, laded by a peridic beam pule i a hwn in Fig. 1. c L Figure 1 95

2 The quantitie R, L.and C refer t the unladed renant cavity, related t the renant frequency, 00, and Q by 002 = 1. LC ' Q = - R L = CR (1) and the current Ib i an equivalent impreed beam current given by (2) where I i the average beam current (during the pule), fbi i the cmpnent f the beam current pule renant with the cavity and ~b i the phae f the center f the beam pule (aumed ymmetric fr cnvenience) relative t Sme fixed tandard in the pwer urce. The element Land C playa majr rle in the tranient repne f the cavity t the incident pwer. The uual methd f cmpenating fr beam lading in the tranient repne f hrt pule cnit f injecting the beam a the cavity vltage rie. The cmplicatin f thee tranient phenmena will nt be invetigated here. Intead we hall cnfine ur attentin t the teady-tate repne f the cavity t the beam pule. may be mitted. In thee cnideratin the element Land C Tw mdel have been cnidered fr the pwer urce. The firt cnit f a generatr in erie with me rt f internal r matching impedance. 96

3 The ecnd cnit f a generatr ediju~ugh an ilatr. In either cae ne find a rejal:in f the frm: v = V c 0 Z eq. fbie -hdb (3) where V (taken t be real) i the teady tate: cavity vltage in the abence f beam lading and Z i eq. an equivalent impedance cniting f the parallel cmbinatin f the hunt reitance R and the internal impedance in the firt mdel, r the tranfrmed characteritic guide impedance in the ecnd mdel. Beam Dynamic with Beam Lading: The reactin f the beam t the mdified cavity field depend n the magnitudelvcland phae Wc f the cavity accelerating vltage. The beam react in fact nly t the cmbinatin IVcl c (q)b - w ) which decribe the energy c gain f each bunch in the cavity. Frm Eq. (3) ne find IVcl c(w b W ) = Re(V eiwb) = V c W - fir (4) c cb b 0 eq. and R i the real part f Z If the deign eq. eq. vltage gain and ynchrnu phae are V and w ' the deviatin in vltage gain frm ynchrnu i 6.V = V c wb - fir b 0 eq. v c ~ (5) 97

4 The beam therefre act a if there were a new ynchrnu phae given by fir + V c ('O b 0 eg. V (6) and perfrm cillatin abut thi phae, the limit n phae tability being the aciated with ~b' ne want t retain a value m a the laded ynchrnu phae it i then neceary t mdify V a the current build up accrding t the relatin If V = V + fir b 0 eg c ('() (7) In a matched ituatin, R = R /2. In thi cae eg. ne ha 6.V = fir b 0 eg V V c c = 2 2c c I V c V 2/R CI) S (8) In thi frm it i bviu hw the fractinal increae in urce vltage i related t the rati f beam pwer t cavity pwer. (Fr a tightly bunched beam fb = 1.00.) Accrding t the cheme jut preented, ne will have t mdify each pwer urce a the beam increae in a way which depend n fbi and R,quantitie eq. which will vary alng the acceleratr. Clearly, ne huld learn a great deal abut the prpertie f the pwer urce and the way in which it vltage and 98

5 phae are cntrlled. Numerical tudie f the beamladed dynamic will then be pible. Further tudy f thi prblem i bviuly called fr. Interactin f the Beam with the Deflecting Mde The phenmenn f beam blwup in travelling wave electrn acceleratr nw appear t be well undertd. (1,2) The bunched beam cuple t a tranvere band and generate a wave with apprximately the ame phae velcity a the accelerating mde. Thi mde caue a deflectin f the beam. A additinal beam enter the guide the field build up further until they are capable f deflecting the beam int the irie. Accrding t thi undertanding, the pule length befre beam cut-ff i inverely prprtinal t the beam current, a i cnfirmed in bervatin. Cnideratin f thi type appear at firt t be nt directly applicable t tanding wave acceleratr, where there i nly a dicrete pectrum f deflecting mde phae velcitie. In additin, it appear mre deirable t cntruct the timulated field frm the cmplete pectrum f renant mde. Neverthele, the phenmenn f beam blwup al ccur fr tanding wave acceleratr. Our purpe i t explre beam blwup in a tanding wave acceleratr, and t dicver which parameter are relevant, and in what way any damaging mde can be uppreed. Qualitatively, what happen when a beam pule enter a cavity i that it cuple t many mde. When the pule leave the cavity, cillatin have been 99

6 built up and cntinue, in the n-l apprximatin. A ubeq!]ent pule will caue additinal cillatin. If the eparatin f pule i jut right, the field frm ucceive pule will be in phae and will lead t a build-up f field which may then interact with the beam pule. If it happen that the beam cuple appreciably t the mde in a tranvere band, the reult may be that the build-up f tranvere field ccur t the pint where the beam i deflected int an iri. Quantitatively ne may etimate the effect a fllw. If the beam cnit f pule eparated in time by 21f/tl) with an average charge in each pule i 21fI /w. 0 current I, the ingle pule f charge thrugh the cavity a build-up f field prprtinal t The mtin f thi lead t i1f!,...,- w E: im.z J v dz' (9) where w. i the frequency f the trubleme mde, t J 0 i the time at which the pule enter the cavity, and E~(Z) are the nrmalized tranvere cavity field. Succeive pule clearly lead t field which are ut f phae by w, 21f/w 0 r by 21f [~ - nj (10) where n i the clet integer t, W./ill J 0 The maximum 100

7 field amplitude will therefre ccur. after the rder f M = ttl w. - ttl n J 0 ( 11) pule and will lead t a field amplitude M time a large a that given in (9). The tranvere deflectin due t thi field will be given by integrating Et(z,t) at value f t given by t = z/v + t I. Thi will lead t a tranvere mmentum f the rder J ~ = 1. dz e E t (z, ~ + t I) P P v v 0 ittl.z' ] Mle I~ Ej (z I) e- t v dz'.. pv 2 ttl 1'IE{(Z') I dv -:::'-- (12) where the denminatr ha been included t allw an arbitrary nrmalizatin f E{. The cmbinatin invlving E{ i a tranvere equivalent t the quantity R /Q = rl/q, where r = R /L i the hunt reitance per unit length. Equatin (12) can be put in a frm fr cmparin with previu reult fr backward wave cillatr thery(2,3). The typical number f pule cntributing t blwup can be written in term f the grup velcity a M = ttl &f3v g ttll "'"'-;;- g (13) 101

8 ince the difference in prpagatin cntant 6~ adjacent mde i w/l. Writing pv ~ ev crrepnding t the beam energy ne ha fr wl v g I V rl Q (14) The "tarting current" I i et by limiting the angular deflectin t f rder A/L. Thi lead t an exprein imilar in frm t that given in bth reference, prviding reaurance that the phenmenn i undertd qualitatively in bth tanding wave and travelling wave term. One can nw return t Eq. (12) fr cnideratin f the eriune f the effect. A crude etimate f the magnitude f the angular deflectin fr a ingle prtn 1inac ectin being cnidered fr the Yale men factry i f the rder 10-5 t 10-6 (w./w ) - n J 0 Since angle f the rder f 10-3 radian are preent in the tranvere fcuing ytem, nly a renance f the tranvere mde frequency with the beam bunch frequency within 0.01 f an integer will lead t difficulty. In view f the fact that the phyical parameter f each ectin will be different due t changing velcity, it i highly unlikely that any renance will be eriu fr mre than cner tw ectin. 102

9 T ummarize, we have frmulated the phenmenn f beam blwup in a tanding wave prtn 1inac in uch a way that the exprein reemble the previuly btained fr beam blwup in travelling wave electrn 1inac. In view f the exitence f (1) a tranvere fcuing ytem and (2) changing parameter within a ectin and frm ectin t ectin, the effect i nt eriu in prtn 1inac. In additin, if an accidental renance huld ccur, methd exit fr uppreing r mving the tranvere band t alter the renance cnditin. Studie are cntinuing t determine the precie frm f the beam-field cupling, the effect f magnetic field and the influence f ther neighbring mde which reflect the diperive character f the tructure. It i hped that the Etimate can be made quantative1y mre reliable. Dicuin: (Nt reprted. Ed.) Reference (1) Tranient and Beam-lading Phenmena in Linear Electrn Acceleratr. J. E. Lei and R. A. Schrack, Internal Reprt, Natinal Bureau f Standard, Wahingtn, D. C. (Octber, 1962). (2) P. B. Wiln, High Energy Phyic Labratry (Stanfrd) HEPL-297. (3) See fr example Kmpfner and William Prc. IRS 41, 1602 (1953). 103