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Frontiers of Particle Beams; Observation, Diagnosis and Correction: Proceedings of a Topical Course Held by the Joint US-CERN School on Particle Accelerators at Anacapri, Isola di Capri, Italy, October 20–26, 1988 PDF

505 Pages·1989·18.79 MB·English
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Preview Frontiers of Particle Beams; Observation, Diagnosis and Correction: Proceedings of a Topical Course Held by the Joint US-CERN School on Particle Accelerators at Anacapri, Isola di Capri, Italy, October 20–26, 1988

Accelerator Physics as a Profession * Hermann A. Grunder Continuous Electron Beam Accelerator Facility 12000 Jefferson Avenue Newport News, Virginia 23606, USA Accelerator physics has become a scientific profession in its own right. Besides its complex and intriguing uses of classical and quantum mechanics, electromagnetism, and statistical me- chanics, this field has a unique combination of intrinsic characteristics: it is both interdisciplinary and international. It is naturally interdisciplinary in that it encompasses a mixture of science and engineering. Through its connections with sciences and advancing technologies around the world, it is naturally international. This essay presents my view of the status of, and outlook for, accelerator physics as a profession. In the six decades since Lawrence first accelerated protons to 80 keV in a device that could be held in the palm of a hand, accelerators have grown and proliferated. So have the opportunities and challenges for those who study them, design them, and build and operate them--not to mention those who use them. Table 1, an informal compilation, 1 gives a sense of the variety and scope of major accelerator initiatives worldwide. Each line of the table represents, roughly, a region of the globe. Table 1 Accelerator Initiatives Worldwide High Enerk, Y Physics LEP, HERA, CLIC, LHC Tevatron, SLC, SSC, TLC TRISTAN, BEPC, JLC UNK, VLEPP Nuclear Physics SIS, Frascati, SIN, Mainz, NIKHEF, ALS CEBAF, MIT/Bates, RHIC, KAON Factory BEP/VEPP2M, Moscow Meson Factory, Kharkov/PSR, Troitsk Dedicated Light Sources Aladdin, NSLS, ALS, LSV, APS BESSY, Daresbury (2 GeV), ESRF, Trieste Photon Factory, Taiwan (1.3 GeV), Korea (2 GeV), Japan (8 GeV) VEPP-3, Moscow One can also gain a sense of the variety and scope of present-day accelerators by noting what this table does not even try to reflect: the many smaller research machines as well as FELs, medical accelerators, and industrial synchrotron radiation sources. * This paper was presented by .M Month A useful distinction can be made between two types of accelerators for physics research. One type includes world-class facilities with unique characteristics and capabilities. At any one time we can have only a few of these, and experimenters may travel great distances to use them. The other type is the regional "workhorse" facility, providing valuable research opportunities to greater numbers of experimenters. Accelerator capabilities determine which experiments can and cannot be done. The diversity of beam requirements for experimental physics offers unique challenges for designers of • storage rings and colliders, • linacs and linear colliders, • lights sources, and • FELs. And just as accelerator capabilities determine possible experimentation, the state of the technological art determines which capabilities are achievable. The challenges for the profession of accelerator physics are therefore not only scientific, but technological. Table 2 is my list of today's technological frontiers for accelerator physics. Table 2 Technological Frontiers Superconducting magnets Beam cooling Superconducting rf cavities Rf power sources Wakefield acceleration Polarized beam sources Instrumentation & control High-lntensity beam sources Lenses Positron production One distant part of the superconductivity frontier--high-To superconductors--illustrates quite well the linkage between accelerator physics technological frontiers and challenges. It also illus- trates how di~cult the needed advances can be. To overestimate the ultimate potential of high-To superconductors for accelerators is hard, and it is also hard to underestimate their short-term difSculties. The ultimate potential could include higher-field magnets, higher-gradient cavities, and warm operation, but in the nearer term this particular technological frontier confronts us with some very rough terrain in terms of material production, anlsotropy, and high rf losses. For the foreseeable future, the materials of choice must remain Nb, NbTi, and NbsSn. As with the other technological frontiers, overcoming the di~culties is not easy. But this does not mean that the effort should not go forward. There are many frontier areas today where accelerator physics and technology are being challenged. One example is the desire to build linear accelerators with higher gradients, greatly reduced cost per unit energy, and substantially higher energy ef~ciency. With such devices, the hope is to make feasible colliders capable of attaining super-high (N l0 ss) luminosity and very high phase-space density. Attaining super-high luminosity implies challenges in several areas: high-intensity positron beams, beam stability limits, wakefield problems, pinch enhancement, and angstrom-sized beams. Achieving very high phase-space density has implications especially for FELs, and would mean 10 -6 m normalized emittances and 10 -3 energy spreads at high currents. These challenges to accelerator physics are of course only important if we have the prospect of continuing to build and upgrade accelerators. All of my evidence tells me that we do have such prospects; the field of accelerator physics is flourishing. I would like to close by first summarizing what I think are the important things for accelerator builders to remember, and then by offering a suggestion for strengthening the accelerator physics community. My experience suggests that accelerator builders should be keenly aware of how much expan- sion they are demanding on the technological frontiers. It is good to push one or two technologies to the limit, but not more. At the same time, it is good for an accelerator initiative to strive for significant improvements over the performance of previous machines. The focus must remain fixed on the physics to be done--that is, on the requirements of the user--and every effort should be made to build a machine that is understandable. By "understandable" I mean a machine of manageable intellectual complexity. With these general requirements setting the context, I would suggest the following as the specific elements needed for successful accelerator development: • Bright, enthusiastic people who know what they are doing. • Close collaboration between accelerator designers and experimentalists. • Innovative technologies. • A direction and a plan. • Adequate funding through the commitment of government. The last of these is least in our own hands as accelerator physicists and engineers. At the same time, however, it is not at all beyond our reach to influence as members of a scientific profession with a growing record of significant successes and contributions. The way to ensure that these successes and contributions continue is to strengthen acceler- ator physics as a scientific profession. By deciding in principle to establish a Division of Beam Physics, the American Physical Society is recognizing the importance of moving in this direction. As with any science, however, a strong professional community must also have formal training programs and independently funded research. It is encouraging to see accelerator physics curric- ula now being initiated in the universities; one hopes to see more. Independently funded research programs--in the universities and major laboratories, as well as in the smaller laboratories, pos- sibly in cooperation with industry in some cases--will also strengthen the profession. With new Ph.D.-level specialists and with strong programs of basic research in accelerator physics, the field will continue to flourish. REFERENCE 1. For information on design and status of initiatives, consult L. Teng, ~Accelerator Projects Worldwide," and H. Winick, "Synchrotron Radiation," both in Physics o/Particle ,srotareleccA AIP Conference Proceedings 184, a 1989 publication of the U.S. Particle Accelerator School. SINGLE PARTICLE MOTION Richard Talman Laboratory of Nuclear Studies, Cornell University ABSTRACT 1. A general three dimensional description of single particle motion is given. The 6×6 transfer matrix describing motion close to a reference particle is explicitly diagonalized to find the three eigentunes and eigenplanes. Within each of these eigenplanes a generalised Twiss parameter description is given. After specialising from six to four dimensional phase space, these formulas can describe coupled transverse motion or synchrobetatron motion. Within eigenplanes the pseudoharmonic description is just like the well-known description of uncoupled motion in a single transverse plane. 2. Lattice defects and their correction are discussed. A universal detection-adjustment for- malism is introduced and applied to the examples of tune adjustment, orbit flattening and decoupling. 3. Various examples of "map dynamics" are discussed, the most important being the betatron response to external excitation. Also coupled motion is analysed; in one important case the motion in one plane can be regarded as being driven by the motion in the other plane. 1. THREE DIMENSIONAL LINEARIZED MOTION 1.1 INTRODUCTION It is assumed that the reader is familiar with the conventional 2 x 2 transfer matrix description of motion in a single dimension, and the Twiss parameter formalism, described, for example, by Courant and Snyder l~t or Sands j21 . That description will be rederived, but it will be as a special case in a rather general formalism. Motion of a charged particle in an accelerator can be described by relativistic Hamiltonian mechanics. A particular motion is that of a "reference particle", having the central energy and traveling on a closed orbit. Motion of a particle close to this particle can be described by linearized equations for the displacements of the particle under study, relative to the reference particle. In setting up the linearized formalism it is not necessary to specify the problem more explicitly than that. In fact, the equations will describe motion close to any known conservative motion, not necessarily a closed orbit. The issue of symplecticity is much discussed. Considerable pains will be taken in the beginning to assure that the treatment is manifestly symplectic, (which is only to say that it is Hamiltonian, or conservative.) This is not particularly praiseworthy however, since it is no great trick to maintain symplecticity with linear equations. As a result there will be little discussion of the issue. I hope though that the reader will be astonished at the remarkable algebraic simplifications which are possible only because the theory is symplectic. At least that much appreciation of the elegance of the subject can be achieved with the elementary methods described here, and without getting into the deeper subjects of Lie Algebra and Differential Geometry. Many of the formulas in this paper were worked out for inclusion in TEAPOT, a thin ele- ment accelerator program for optics and tracking, in collaboration with Lindsay Schachinger, to whom I am grateful. 1.2 MOTION CLOSE TO A KNOWN MOTION Hamilton's Equations. The Hamiltonian for a particle having charge e in an electromagnetic field is (using MKS units) / : ~/rn2c 4 + c2(p -- eA) 2 + e¢ (1.1) where the canonical momentum p is related to the mechanical momentum P by* p = P + eA. )2.1( In terms of the potentials(C,A), the electric and magnetic fields are given by OA V¢ E . . . . t0 )3.1( B=VxA. * tI si elbissop that eht symbols p and P era reversed from what the reader si accustomed .ot I made eht choice os that only small srettel would appear ni Hamilton's .snoitauqe The ordinary momentum si P = ,Trtnr where v si eht yticolev and 7" si eht usual citsivitaler .rotcaf nI eerf-dleif regions P = ,p fo course. and the particle equation of motion is dP --=eE+ev ×B. (1.4) dt The mechanical energy of the particle is mc 2 $M -- -- mc2~; (1.5) X/1 - v2 / c 2 it satisfies Med - eE- v (1.6) dt The Longitudinal Coordinate as Independent Variable. The triplet (x, y, s) consists of hori- zontal transverse, vertical transverse, and longitudinal displacements from the reference par- ticle. It is customary, in accelerator physics, to use s rather than the time t, as independent variable. That means that the triplet (x, y, t) is to be regarded as being the quantities whose evolution is to be described. The motion of the reference particle is assumed to be known; it is given by (Xo (s), Pzo (s), Yo (s), Pro (s), to (s), Pro (s)), where pt will be defined shortly. For writing linearized equations we could define small differences, 5x(s) = z(s) - Zo(S) etc., but instead, at a certain point below, we will simply redefine the coordinates (x, pz, y, py, t,pt) as small deviations from the reference orbit. For now, they are absolute, exact particle coordinates of a general particle. The transformation from t to s, as independent variable, in Hamiltonian language, is straightforward but confusing. For the moment suppress (y,p~), since they enter just like (x,p~). The Hamiltonian, Eq. (1.1), has the form = ~(:~,p~,~,ps,t). (1.7) Of Hamilton's equations, the ones we will refer to below are ds Og . dt 0~ -1 (1.8) -~ : ops' or ~ - (~) d~ O~ and dt - Ot (1.9) Define a new variable (1.10) ~p = -~(~,p~,~,ps,t). This is to be solved for ps, with the answer expressed in terms of a function K, which will turn out to be the new Hamiltonian; P8 = - K (z, p~, t, pt, s). (1.11) From Eqs. (1.1) and (1.2) it can be seen that the numerical value of -Pt is the total energy = SM-k e¢. The differential dp8 can be obtained either directly from Eq. (1.11), or indirectly from Eq. (1.10), using Eq. (1.8). The results are aP'=--a; OK dx ,pT;a-- OK d P~-W OK dt --og OK d P'--a7 OK ds (1.12) =(-ap, - 1~ )@ dx - -Oil - d p, - l~O ds _ t~@ dr'" @g )-1 ' Equating coefficients, and using Eq. (1.9), as well as the other Hamilton equations in the original variables, the equations of motion in the new variables can be written in Hamiltonian form, with derivatives with respect to s being symbolized by primes; KO KO X I =__ pl z zp3c ; -- Ox (1.1a) aK KO t t =--; ~P -- Opt Ot The manipulations which have been described can be performed explicitly, using Eqs. (1.1) and (1.10), with the result = -eA, - V/(p, + e¢),/c2 - m=c= - (p± - eA±)2, (1.14) where components parallel and perpendicular to the reference orbit have been introduced. In field free regions we have (1.15) where the generalized momentum Pt is minus the energy. Linearized Motion. To shorten the formulas, we will use either the notation (xi,pi), i = 1, 2, 3 or xi, i = 1,...,6 for (x,pz,y, py,t,pt). The reader will have to figure out from the context whether a symbol like x2 stands for pz or y. Without writing it explicitly every time, indices will be assumed to run from 1 to 3, or from 1 to 6, as appropriate, and the summation convention will be used for repeated indices. The Hamilton equations are , OK. , K}'o {1.16) xi = cgpi, Pi- Ox i" From this point on, as previously warned, the quantities xi and pi will be regarded as small deviations from the known reference trajectory. The right hand side of Eq. (1.16) can be approximated by the first term in a Taylor expansion. K20 K20 ! xi - x3c I -Opi -xi + --c3pj Pi iPO (1.17) K2a 02K ~p = - --xj = pj OxjOxi OpjOxi The partial derivatives are evaluated on the reference trajectory. These equations correspond to a quadratic Hamiltonian, 1 c32K 02K 1 02K K -- z" XOlZO" m" ZlZm nrPlZ"~ff -q""~mptClXO -q 2 mp3clpgt plpm" (1.18) Introducing a row matrix, X T = (x, Pz, Y, P~, t, Pt) and using matrix notation the Hamil- tonian is K = ~XTKX = ~xiKiixj (1.19) where Kii is a symmetric matrix, dependent on s. Introducing a matrix S, given in one and two dimensions, by 0 0 S ; = , (1.20) 1 0 0 0 1 0 1 with an obvious generalization to three dimensions, Hamilton's equations take the form (1.21) x' = -~KX. Observe that S T = -S and S 2 = -I. (1.22) From any two solutions X1 and X2 of Eq. (1.21) an expression, sometimes called a "Lagrange invariant", xT sxI = -z2pxl -q xlpz2 - y2pyl -4- ylP~2 - t2ptl -{- tlpt2 (1.23) can be constructed whose invariance follows from Eqs. (1.21) and (1.22). Evolution of a vector X from so to s is described by a transfer matrix M, X(s) = M(s, so)X(so). (1.24) The invariance of xTsx1, when 1X and )(2 evolve according to (1.24), yields a relation which the transfer matrix must satisfy, MTSM = S; (1.25) this is called the symplectic condition. * A common source of confusion stluser when the term Kphase space" si applied both to the ,x 'x space and the ,x pz space. The former si common during practical operations but the ,rettal which we will stick ,ot si better rof preserving citsivitaler and Hamiltonian features ni theoretical analysis. When the absolute value of the s'elcitrap momentum si preserved .g.e( because there si no .f.r acceleration) then, ni eerf-dleif ,snoiger the ratio Pz/P si ta( least rof small angles) approximately equal to ;'x ni these circumstances the two quantities are the same, except rof .stinu The 6 × 6 matrix M can be partitioned in terms of 2 × 2 matrices / Bi) (1.26) M= C D G H A useful matrix operation is "symplectic conjugation" defined by fii = - S AT S. (1.27) For a 2 × 2 matrix :( ): (' :) = = = A -I det IA. (1.28) --C The last expression is meaningful only if the determinant is non-zero. When applied to the 6 × 6 matrix M the result is -__/Th D H • (1.29) F J Because M is symplectic, one gets, using Eqs. (1.25) and (1.22), that SMTSM = SS = -I and, as a result h~/'M = I or hT/= M -1. (1.30) It should soon become clear what an enormously powerful relation this is. Periodic Lattices, Eigenfunctions, and Stability. To this point the transfer matrix M(so, s) has represented transport through a general sector from any point so to any other point s. Such a map will be called a "sector map" to distinguish it from a"period map" which represents transport through one period of a periodic lattice. In an accelerator of circumference C, the sector from any point s to s + C is always one period. In this, the most important case, M will be called the "once-around transfer map."* The relations derived so far, such as Eqs. (1.25) and (1.30) are true for general sectors. We will now derive certain "eigenproperties" of M(so, s). Most of these relations will be algebraicly valid for general sectors, but they will only have important physical interpretations when M is a period map. The point is that eigenfunctions are vectors which transform into themselves. When so and s are separated by one period, they are the same point, either actually or effectively, and they have identical coordinate axes. When the six components at s are the same as the six components at so, the particle positions they represent are unambiguously identical. When so and s are not separated by an integer number of periods, the same cannot be said, since the coordinate axes are not the same. Eventually, unique coordinate axes will be set up at every point so that the eigenfunctions of the map for any sector map are physically significant, but for now we concentrate on period maps. * To avoid circumlocution, no effort will be expended in making the formulas cover superperiodicity other than one. An accelerator always has at least that periodicity; in the presence of errors it has no other. To apply the formulas to a general periodic lattice, C need only be interpreted as the period. 01 For analysing stability the eigenvalues of M are of paramount importance. That they come in reciprocal pairs can be seen from the following equations. Assuming that ,) is an eigenvalue of M and hence also of M T then det M T - ),I} = 0. (1.31) Multiplying by SM and using Eq. (1.25) yields det IS - ),SM I = O. Multiplying by S, using (1.22) and dividing by ,) yields 1 det IM - ~I I -- 0; (1.32) that completes the proof. A definition of stability due to Lyapunov MT requires that a particle which starts close to the reference particle, stays close forever. This definition is usually regarded as being too restrictive in most areas of mechanics. For example, in Celestial Mechanics, even the tiniest perturbation can cause a large orbit shift after a long time. (For example, consider the famous Einstein advance of perihelion of Mercury.) For accelerators, however, the Lyapunov definition is just right. The stability of synchrotron oscillations, discovered independently by McMillan and Veksler, assures that particles stay in the R.F. "bucket" longitudinally, forever, and the transverse motion is similarly stable. To apply this stability criterion, observe that multiplying an eigenfunction by M simply multiplies M by the complex number .X By multiplying M by itself repeatedly it can be seen that stability of a particular eigenmotion requires that the absolute value of the corresponding eigenvalue be less than or equal to one. But then, since, as we have just seen, the eigenvalues come in reciprocal pairs, it is clear stability demands that the eigenvalues all lie on the unit circle. In that case the eigenvalue ,) and )-i are symmetrically placed, above and below the real axis. That makes their sum real, a result that will be important below. 1.3 SPECIALIZATION FROM THREE TO TWO DIMENSIONS At this time we will specialize from three to two dimensions. It is not really necessary to do this -- Appendix B describes how to continue in all generality -- but it is simpler, more explicit, and easier to follow in two dimensions. Furthermore, the formulas will serve for the two most important applications, coupling between transverse planes and synchrobetatron coupling between the longitudinal plane and one of the transverse planes. In what follows the language to be employed will assume that coupling between transverse planes is what is being described, but the symbols could be reinterpreted to describe the other case. The partitioned 4 × 4 matrix M is M= D " When written out explicitly Eq. (1.33) gives relations among A,B,C, and D which follow from

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