Simonyan, K A

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Particle Accelerators, 1990, Vol. 28, pp. 101-106Reprints available directly from the publisherPhotocopying permitted by license only© 1990 Gordon Cl!ld Breach, Science Publishers, Inc.Printed in the United States of AmericaNEW ASPECTS OF THE THEORY OF RESONANT BEAM EXTRACTIONFROM SYNCHROTRONS (STRETCHERS)Kh.A. SIMONIAI'!Yerevan Physics Institute~ Yerevan Armenia, USSRAbstract The work is devoted to a detailed study ofparticles radial motion in synchrotrons near thethird integral resonance. The well-known generalizedequations describing the radial motion of a particlefrom cycle to cycle (one cycle equals threerevolutions) in the general case are shown todescribe the process of particle ejection fromsynchrotrons. It has been proved that in order todescribe the process of slow ejection on the thirdintegral resonance correctly, it is necessary tofollow the particle position at a given aZimuth, atleast after each its revolution. From the derivedlinearized equations of the particle r-motion withrespect to revolutions two new criteria have beenobtained for the ejection system optimization in theform of two additional conditions which the ejectionsystem must satisfy so that the well-known phasepattern (the triangular separatrix, etc.) of particlemotion would really take place.The theory of extraction at the third integralresonance is known for a long time (1955, Bartont ) and theestimates of the parameters of the extracted beam arebased on the corresponding pattern of ther-motion on the normalized phase plane (n,n);n=f3'tt/2(dr/ds - 1/2 df3/ds .. (3-tr }). However, ourparticle-1./2<n=rf3 ;detailedinvestigations show that the generally accepted theory ofresonant extraction (Bartont , Mashke2 , Hemmie9 ) is basedon incorrect assumptions. Indeed, while the existence of atriangular separatrix on the plane <n,n) really followsfrom the linearized "cycle eqLlations", the application ofthis theory to explain the process of the particlesthrowing into the septLlm-magnet contains a wrongassumption, namely, during a cycle (three revolutions),after each revolution, the phase pattern is rotated sothat the separatrix branches cyclically interchange theirplaces. But in reality (see below), for this to take place[579]/101102/[580] KH. A. SIMONIANit is necessary that the ejection system (the number ofsextupoles, their location, etc.) satisfies certainconditions. Only under such conditions the separatrixbranches will be really "identical" at the moment of thebeam throwing into the septum-magnet. That is why we callthem conditions of the beam extraction systemoptimization.The investigations are based on the equation"r + K(s) r = - g(s)r2 (1)swhere g is a small magnitude.AssLlme that there are j II thin II sex tupo I es wi th I eng th~s. (i=1,2, ••• ,j) on the azimuths s .• Let us introduce the1. 1.small parameter £, 2nG=(2nm+£)/3, which characterizescloseness of the oscillation frequency 9 to its resonancevalue m/3 and take three revolutions (a cycle) as unittime of changing of particle coordinates. Then in thepolar coordinate frame (R,a) with n=cosa, n=R sina, and inthe linear approximation of g and £ one can write thewell-known equation of motion in the following recurrentform:R =R 1u2+ V2n+t n n nan+tVnotn - arctg UnU 1-3ER sin(~+3a). V = £ + 3ER cos(~+3a)n n n· n n nwhere n is the cycle number andC/22E =YC-+D-; 'Pg. fJ.s. (3~/2cos3¥,.; D1.. 1.. 1.. 1..arctg C/D;-1/4 ~ g. t.s.fl~/2sin3lJ1..l ~ 1. 1. 1.1. =1~i and ¥'i are amplitude and phase functions on the azimuthof the i-th sextupole, respectively. From (2) follows thewell-known triangular separatrix with rectilinear brancheswhich allow to visualize the particles throwing into theseptum. Far from the resonance the tr~angle's area isconsiderably larger than the emittance of the circulatingbeam. The closer to the resonance (£~O), the smaller thetriangle's area:RESONANT EXTRACTION THEORY [581]/103s - 3-{3 2 - M~ - ~ no no - 3EThe triangle "squeezes" the emittance in three directions.In a certain instant of time the accelerated beam"occupies" the triangle. Then, when £ ...0, part of particlesis "forced OLtt .. of the triangle through all its verticessimultaneously. Therefore, the particles turn out to bedistributed near the separatrix branches, and moving alongthem depart from the beam core. Choosing the number ofsextupoles, j, and their location sites, si' relative tos one can achieve that on the septum azimuth s one~ep ~QPof the separatrix branches is parallel to the n axis. Thisrequires satisfaction of the condition c=o. Then,patternthirdfrom cycle to cycle, however, itfollows that only adepending on what side of the orbit the septum-magnet isplaced and on the sign of £, one provides such a sign of Dthat the particle motion along the separatrix branchparallel to the n axis turns out to be directed to theseptum. Then, with a definite probability, the particlesextracted from the region of stability will reach theseptum and part of them will appear beyond it.From SLtch a "behaviour" of the beam phaseof the beam will be thrown into the septum (withoutaccount of losses on the IIblade" of the septLtm), since itis not clear how the particles moving along the two otherbranches will "behave" relative to the septum. It is justhere that the generally accepted theory makes baselessassumptions that the separatrix branches interchange theirplaces within a cycle. First, it supposes existence of atriangle separatrix within a cycle, at least after eachrevolution. And second, it supposes that after eachrevolution the separatrix is rotated by an angle equal toexactly 2nm/3. But it is not proved. If it is really so,then it must be followed from the investigations of themotion from turn to turn, just what we have done.In contrast to the generally accepted theory wefollowed the position of particles on the given azimuthafter each revolution. In the polar coordinate frame, and104/[582] KH. A. SIMONIANin the same approximation the motion of particles fromturn to turn is described by the recurrent formulaeVkarctg u;kUk = l-ERk [sin(!p+30tk ) - A cos Otk + 8 sin Otk]; (3)Vk = £/3 + ERk [COS(!P+3Otk ) + 3A sin Otk + 38 cos Otk]where k is the number of revolutions andA = 1/4E t g . .dS.1?3,/2. 8 1/4E ~ A ~9/2. £ 1. 1.1,\. s~n VIi; = . l 9 i U.Si 1.1 i cos Y-\1.=1 1.=1A simple comparison of (2) with (3) shows that the motionfrom turn to turn depends on the parameters A, B, m,whereas the "cycle eqLtations" lack these parameters.Therefore, on the first stage the investigations werecarried out according to (3), assuming A=B=O. It becameclear (see fig.!) that only at A=B=Othere really exists atriangular separatrix with rectilinear branches whichafter each revolution are rotated by 2nm/3 and the beamphase pattern exactly corresponds to the generallyaccepted notion. In all other cases (see, e.g. fig.2).i.e. when A~O, B~O, there is no such separatrix. In otherwords, in the general case t.he "cycle equations" describet.he particles motion near the resonance region incorrectlyand hence, it. is at least. incorrect to compare theexperiment with the predictions of the generally acceptedtheory. The investigations via (3) have shown that whenA~O, B~O t.he quadratic nonlinearit.y leads to deformationof the beam phase volume, which is different. at. differentvalues of A and B. At £~ we get more accurate formulae(without linearization over e)R =R ~2+V2k+l k Jc kA cos Cl), + 8 sin Cl),]' (4)RESONANT EXTRACTION THEORY [583]/105The conditions of existence of quasistationary points onthe plane (n,n), i.e. points cyclically interchangingtheir places after each revolution, are:Rst Rst 9t 9t 2nm (5)= ; Ol = otk - 3 •k+t k lc+tIt is easy to show that at any Ilk" the expressions (5)have solutions only when A=B=O, which have the form:sin M ·6 '9tCll.t.(i=0,1,2)6+'P + M3 18(6)assurewhere (±) is the sign of s.In conclusion let us elucidate the physical meaningof the conditions A=B=O. To do this, come back to theeq.(l), the solution of which we represent as:res) = r 0 (s) + r t (s) (7)where roes) is the solution of (1) at g=O. St.tbsti tLlting(7) into (1), in an approximation linear over g we obtain" [K(S) + 2 g r o]2r + r = - gro. (8)t tWhen deriving the relations (2)+(4), we did not takeaccount of the second term in square brackets which, as isknown, gives correction to the frequency 9. In our case ithas the form:(9)whence it follows that the conditions A=B=Oindependence of G on the amplitude of oscillations.The results of calculations according to (4) havebeen compared with the numerical computations using theparameters of our synchrotron. The smaller the value of'g', the better the agreement between the results.Thus, the new approach developed allows to make theejection system optimization easy and simple.106/[584] KHo A. SIMONIAN.0(0= ~/36 «o"-Jr/3\\ 3D\-20-'-.R =8=C"0; E<OI:1Q'"O.316Dx=E'P': =7.9M- t•REFERENCES:·••• 10••). t·.·:<OOj'.. >:\--_-ft-------1-2P- ~ 0 .» ---L.J t.·:::···l ::)~••••••••:; -G j .:..~: ~/ Fig.lFig.21. M.G.Barton, Proc. of VIII Int. Conf. Q!l High EnergyAce., CERN (Geneva, 1971), p.SS.2. A.Mashke, K.Symon, Proc. of All-Union Workshop onCharged Particle Accelerators, (Moscow, 1970), p.516.3. B.Hemmie und H.Walter, DESY 70/51, October 1970.

New Aspects Of The Theory Of Resonant Beam Extraction From synchrotrons (stretchers)

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