[1 + (r' )2J (O(eCP**)

Transcription

1 Pceedings f the 1966 Linea Acceleat Cnfeence, Ls Alams, New Mexic, USA Intductin THE "PIERCE"-GEOMETRY AN ACCELERATING COLUMN DESIGN* C. Rbet lilnigh Ls Alams Scientific Labaty Ls Alams, New Mexic The ppula tend in tday's design f acceleating clumns is t make the field gadient nea the extactin aea as lage as pssible. The agument has been that if the ptn beam enegy can be bught up t the acceleat injectin level in a vey sht distance, space chage has little time in which t act t incease the effective emittance by waping the emittance patten. Suppt f this agument has cme fm the success f ecently tested sht clumns in pducing ptn beams with lw nmalized emittances, in the egin f 0.05 cm-milliadian. It is geneally accepted that nly the extactin system f Piece is withut abeatins. TO emphasize these essential featues f the Piece extactin--caxial flw, ze angula mmentum, and a unifm density distibutin--a deivatin based n a ze effective emittance patten thughut the acceleatin is pesented. Thus the ecent successes in geneating lw emittance beams ae pbably the esult f the "Piece"-like gemeties used by the clumn designe t implement his high gadient citeia athe than a esult f the high gadient itself. The high gadient has the advantage f educing the beam diamete t a manageable size f a given cuent. A simple integal equatin f the field equied utside the beam t suppt the Piece gemety is deived. The equatin is exact, based nly n the 4/3 pwe law ptential gadient at the beam bunday and n Laplace's equatin utside the beam. Ze lilni ttance Piece Extactin Fb the mment cnside an idealized appach and ascume that the beam is t be acceleated unde cnditins f ze emittance. In additin, the ze emittance patten in -' phase space shuld emain linea f all values f z t insue that the effective emittance als will be ze duing acceleatin; that is, the emittance patten is nt waped. F example, Fig. 1 shws n the uppe gaph, a ze emittance linea patten; and shws n the lwe gaph, a ze emittance nn-linea patten, the effective emittance being pehaps defined by the aea f an ellipse which will just enclse the ze emittance patten. These tw assumptins (ze amittance and lineaity) ae sufficient t establish that the flw is lamina and the cuent density distibutin in eal space emains the same thughut the acceleatin. This is illustated in Fig. 2. The cntinuus line epesents ne tajecty within the beam, while the dashed line epesents any the tajecty within the beam. It is evident that all tajecties ae elated by simple scaling facts. Thus any equatin epesenting a beam tajecty must have its bunday cnditins such that the equatin is independent f a when a is substituted f. Caxial Beam Tajecties A time-independent spatial desciptin l,2 f individual paticle tajecties within a beam having a given cuent density distibutin and unde the cnstaints f an extenally applied electic and magnetic axially symmetic field is given by whee and C " [1 + (' )2J (O(eCP**) 2( eq>**) O ' O(eCP**») OZ 2 Als e<p* = ecp + (:) is the "effective" electic ptential enegy and e~ is nmalized s that it is equivalent t the kinetic enegy f the paticle. In the abve equatins, z, and 6 ae the usual cylindical cdinates and A is the magnitude f the magnetic vect ptential, including bth beam self fields and extenally applied fields. The pime indicates a diffeentiatin with espect t z. E is the est enegy f the paticle, e the chagg n the paticle, and c the velcity f light. A detailed inspectin f the dependence f the vaius tems in equatins (1) and (2) will shw that the set f cnditins needed t meet the scale change citein develped in(:he )eceding paagaph is that ', ", 6', and eq>ii* IllUst be ze eveywhee in the beam. tajecties must be caxial lute ze emittance patten. Ze Angula Mmentum Thus the beam t insue an abs- Equatin (2) can be expessed in me familia tems by ecgnizing that the angula JlX)mentum is *Wk pefmed unde the auspices f the U. S. Atmic Enegy Cmmissin. (2) 398

2 Pceedings f the 1966 Linea Acceleat Cnfeence, Ls Alams, New Mexic, USA Extenal Fields Piece Gemety v Making this substitutin int equatin (2), it can be witten in the fm (4) (\ Fm u citein that (e~) = 0 eveywhee in the beam, equivalently that ecp** must be independent f, it is nted fm equatin (1) that the cnstant C must be ze. If it is nt, equatin (4) shws that a paticle emitted in a magnetic field will attain mechanical angula mmentum when leaving the field. Thus it is imptant that at the plasma suface, whee the ptns ae extacted int the acceleating clumn, thee ae n magnetic fields which will cntibute t the nn-linea emittance patten. Unifm Density Distibutin-Child's Law The fact the ecp** must be independent f, als implies that ~ must be independent f. Fm Pissn's equatin 10 f_ (\ ) (\2 P - ~ (e(fl) + ~(ecfl) = - : v Z it is quite clea that the chage density ( cuent density) als must be independent f. Theefe equatin (5) educes t (6) whee nly the elativistic mass cectin is assumed t be negligible. This diffeential equatin can be slved easily f CfI, the equied ptential distibutin within the beam, and yields the familia Child-Langmui space chage law. 4 --;l/2~ 2/3 CfI = Q'Z /3, whee ex = [~~e) J (7) The ideal appach t an acceleating clumn is t acceleate a unifm density beam accding t the 4/3 pwe law, as utlined in the peceding paagaphs. Unde these cnditins the ptential within the beam is independent f adius and as a esult thee ae n fces pesent intenal t the beam t wap the emittance patten. The field necessay utside the beam t suppt this 4/3 ptential gadient alng the beam has nt been calculated theetically except f the sem1- infinite ectangula beam and f the infinitely thin line cuent. The gemetical design citeia f the cicula beam "Piece gun" is usually develped expeimentally fm electlytic tank measuements. Theefe, it is f bth academic and f pactical inteest t attempt a theetical appach t this pblem. Cnside a unifm density beam f adius P fmed by a paallel flw f paticles being emitted fm a plane plasma suface at ze enegy and being acceleated by a seies f equiptential sufaces fmed in such a manne that the enegy gain beys the equied 4/3 pwe law. In de that the adial fces at the edge f the beam be ze, the adial ptential gadient at the edge f the beam als must be ze. Thus u bunday cnditins ae a. V(R,Z) = z4/3 at R = 1 at R = 1 (8) whee Rand Z ae dimensinless units / and z/ espectively and whee is the ac]jus f the bee... Outside f the bunda~, whee R > 1, the ptential eveywhee must bey Laplace's equatin 1 (\ { 0 } 0 2 R OR R OR V(R,Z) + ~ V(R,Z) =. OZ The fm f the slutin f this equatin which is apppiate t u bunday cnditins can be btained by a sepaatin f vaiables; that is, U(R) W(Z) is a slutin t Laplace's equatin whee and U(R) = a(k) J(kR) + b(k) Y(kR) (10) W(R) = e- kz and whee k is an abitay vaiable. Hee J (kr) is a Bessel functin f the fist kind f d~ ze, and Y (kr) is a Bessel functin f the secnd kind f ~~ ze. We nw ask that U(R) = l~ R = 1 and ~ U(R) = 0 f R = 1. Theefe a(k) J (k) + b(k) Y (k) = 1 0 and -a(k) k Jl(k) - b(k)k Yl(k) =. Slving these tw equatins Simultaneusly, we find 399

3 Pceedings f the 1966 Linea Acceleat Cnfeence, Ls Alams, New Mexic, USA We ae nw able t wite the entie slutin f Z > 0 in tems f u abitay vaiable V(R,Z) = J c(k) U(R) e- kz dk... and V"(R,Z) = J C(k)k 2 U(R) e- kz dk (12) The abitay cefficient can nw be evaluated because V"(R,Z) is knwn alng the bunday, R = 1- Theefe V"(l,Z) J c(k)k 2 e- kz dk = ;a z-2/3. This equatin is well behaved and is cnveniently in the fm f a Laplace tan~fmatin, s that it is easily slved f c(k)k (14) Attempts t slve this expessin analytically have nt been pductive, and althugh slutins in the fm f numeus infinite seies ae pssible, the cnvegence is ften s slw as t make the pcess untenable. Hweve, as the integand is well-behaved f all values f k, the fmulatin is subject t easy analysis by cmpute. Figue 3 is the esult f cmpute uns shwing the plt f R vesus Z f vaius equiptential values f V(R,Z)/c. Als, the values as measued many yeas ag by Piece ae shwn by the dak lines n this plt. The ageement is quite gd cnsideing the accuacy which might be expected fm the expeimental electlytic tank measuements. Ze Ptential Cne Angle The shape f the ze equiptential suface f lage values f R becmes cnical in shape. The cne angle has been 6 calculated by Spangenbeg,5 71 ; and by Lapstlle, 74 10'. In the latte case, the field distibutin was calculated f a line f chage having the 4/3 pwe ptential gadient and shuld cespnd t u case f VIR Z' R > > 1. Setting ~ equal t ze in equatin (17) whee f(~) is the gamma functin f 2/3. One may nw ew~te the equatin f bth V"(R,Z) and V"(l,Z) and subtacting ne fm the the V"(R,Z) = 4 a z-2/3 + J 4a 1 lu(r) (1) k l/3 1 } e -kz dk. F values f R» 1, z» 1, U(R) can be appximated by letting k ~ 0 in U(R), but nt kr. Thus, U(R) ~J (kr) and ne can ewite equatin (18) Integating this expessin twice, ne btains QO 4/3 J 4c 1.. V(R,Z) = cz + 9OO J13 lu(r) } -kz 9f - 3 k - 1 e dk (16) The last tw tems ae geneated as integatin cnstants. As V(R,Z) must bey Laplace's equatin f evey R > 1, Z > ;then F (R) = K2nLR and F (R) = IS. LnR. Hwe~e, u ~unday cnditins als stipulate that ~ V(R,Z) = 0 at R = 1 f all Z; theefe K2 and ~ must bth equal ze. Ou final equatin f the ptential distibutin utside f the beam needed in de t suppt the 4/3 pwe ptential gadient is Again, this expessin has nt yielded t an analytical slutin and ne is fced t make futhe appximatins t put it n the cmpute. Ou appach has been t d bth. The angle as measued ff the cmpute field pl~ts is 74 0 If J (kr) - 1 is appximated by (kr), the fist tgm in the pwe seies expansf~, ne can slve the esulting equatin and finds ~ = '. Hweve, if ne ties t include me tems in the pwe seies, the answe vaies adically depending n the numbe f tems, indicating the integal f the seies des nt cnvege. One is led t the types f appximatins. F example if ne lets kr~ kry J(kR) - 1 "" (kr)2 2' e -"""""2 - ~ "(kr)3 2' e -"""""2 (20) QO z4/3 + J 4 9(~) ~ {U(R) - l}e- kz dk. k (17) R ~ 1, Z > 0 whee. /3 --/2 I' = [:2{2-73j and then slves equatin (19), the esulting 400

4 Pceedings f the 1966 Linea Acceleat Cnfeence, Ls Alams, New Mexic, USA tanscendental equatin can be evaluated and B = 3.42 a cne angle f '. Equatin (20~ is equivalent t a tw-tem appximatin, cnveges, and is Laplace-tansfmable. As it tuns ut, equatin (20) is nt valid f lage kr, but then J(kR) des nd cntibute t the integatin f lage kr. At this stage f theetical develpment vaius analytical slutins will be subject t diffeent appximatins, which can vay slightly the esulting cne angle. Equatin (20) appeas t be in ageement with the cmpute measued angle f 74 ± 1/4 0. Acceleating Clumn As an exmnple f the "Piece" Gemety, Fig. 4 illustates a css sectin f the extactin egin f the peliminay design f an acceleating clumn f the Ls Alams Mesn Facility. The clumn is designed f 50 milliampees and a beam adius f 0.7 em. The equiptential electdes ae t be built f Ti-6Al-4V ally and the maximum ptential gadient, which is at the exit end f the clumn, is appximately 30 KV / em. The ceamic ings and Ti ally electdes will be vacuum bazed in a manne simila t that emplyed by Klystn tube manufactues, that is, the ceamic and Ti ally will have matched tempeatue cefficients and will be sealed at high tempeatues (950 0 C) by eact i ve metal bazing. The suppting stuctue f the equiptential electdes ae tuncated cnes f the same Ti ally and ae pefated by hles t aid in the hydgen pumping pblem. The hles will be staggeed fm suppt t suppt t pevent a cntinuus path f secnday electns thugh the entie clumn. Visual access fm the beam t the ceamic spaces is denied by the ientatin f these suppt stuctues. Finite Emittance It is extemely difficult t include a finite emittance int the beam and t cay ut a simila type f analysis t detemine the ptimum cuent density distibutin and ptimum acceleating ptential distibutin t minimize the gwth f effecti ve emittance. Assumptin f finite emi t tance pattens implies knwledge f phase space distibutins and its inteplay with eal space density distibutins. This knwledge is lacking. Hweve, it is quite evident that a finite emittance will eventually degade an initially unifm cuent density distibutin unless the emittance has a vey specific, nn-ealistic, phase space distibutin as detemined, f example, by Kapchinskij and Vladi~Skij7 f the nn-acceleated beam, and by Ohnuma f the acceleated beam. If we make the assumptin that the nmalized emittance is me less cnstant duing the acceleatin thugh the clumn, we can fllw the pcedue f Walsh9 and add a tem t the ighthand side f equatin (1) t include the emittance, e, whee is nw changed t R, the ute adius f the beam, and the density distibutin is assumed t be cnstant. The added tem is Equatin (1) with the abve additinal tem gives the mtin f the envelpe f the beam thugh the acceleating clumn. It is easy t cmpae the added tem f the equatin, cntaining the emittance fact, with the the tems f equatin (1) cntaining the cuent fact. The emittance tem is usually vey small (tw des f magnitude) as cmpaed t the cuent tem f mst f the ecent acceleating clumn designs. Thus the emittance tem can be neglected in detemining the acceleating ptential distibutin and equatin (7) can be used with cnfidence knwing that ne is nt ding any geat vilence t the cncept f maintaining a vey lw effective emittance. Summay The Child-Langmiu law has been slved in a athe undabut manne, including the extenal fields necessay t maintain the 4/3 pwe ptential gadient; but in s ding the facts ae established that in de t maintain a ze effective emittance duing acceleatin, the beam must be: a) fmed in a magnetic field fee egin at the plasma bunday; b) extacted fm the plasma suface with a unifm density distibutin; and c) acceleated accding t the 4/3 pwe ptential gadient. It may be well t pint ut that even if ' and " ae nt ze (but still " «1, giving a "Piece"-like gemety) equatin (1) will still allw the scale change citeia ~ a withut changing equatin (1) t much. Hweve, Slight changes in the density distibutin have a much me pnunced effect and will lead t me seius emittance waping. It wuld appea that a unifm density distibutin shuld be the pimay bjective in a lw emittance clumn design with as nea a "Piece" gemety extactin system as pactical. In additin, the beam shuld be extacted fm the plasma suface in as nea a magnetic field fee egin as pssible. Acknwledgments I wuld like t acknwledge the cntibutins f vaius staff membes f Ls Alams Scientific Labaty f thei cntibutins: Blai Swatz f his advice cncening the theetical develpment, paticulaly in pinting ut the cnvenient fm f equatin (11); Ken Candall f tanslating the extenal field pblem t the cmpute; and Tny Damian f the mechanical design f the acceleating clumn. 401

5 Pceedings f the 1966 Linea Acceleat Cnfeence, Ls Alams, New Mexic, USA Refeences 1. D. H. l!enzel, Fundamental Fmulas f Physics, Pentice-Hall, Inc. 1955, p J. W. Beal, Beam Tajecties Unde the Influence f Self-Fields and Extenal Fcusing Elements, Lawence Radiatin Labaty Rept UCRL (May 1965) J. R. Piece, They and DeSign f Electn Beams, D. Van Nstand C. Inc. (1949), pp P. Lapstlle, A. Blanc-Lapiee, and G. Gudet, ~lectnique G~n~al, Editins Eylles (1959) pp K. R. Spangenbeg, Vacuum Tubes, McGaw-Hill Bk C., Inc. (1948), p P. Lapstlle, Pivate cmmunicatin and als efeence 4 abve. 7. I. M. Kapchinskij and V. V. Vladimiskij, Cnfeence f High-Enegy Acceleats and Instumentatin, CERN, Geneva (1959), p Univesity Intenal 1966). 9. T. R. Walsh, Plasma Physics Junal f Nuclea Enegy Pat C, Vl. 5, pp DISCUSSION l R ZERO EMITTANCE-LINEAR PATTERN ZERO EMITTANCE-NONLINEAR PATTERN BEAM a Figue 1. TRAJECTORIES z Figue 2. C. R. EMIGH, LASL LEFEBVRE, Saclay: I just want t say that, expeimentally, we have fund the density distibutin t be an extemely sensitive fact f the beam quality. I"e fund that slight mdificatins f the expansin cup gemety geatly influence the density distibutin at the extactin level. FOSTE"'TE CERAMIC RINGS T. ELECTRODE SPACERS 6 RING 9 8 R ~~g~~ II 12 Fig. 3. Z Piece gemety extenal field patten. Fig. 4. EXPANSION CUI' Patial acceleating clumn design. 402