Douglas, D R

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Particle Accelerators, 1986, Vol. 19, pp. 119-1230031-2460/86/1904-0119/$10.00/0© 1986 Gordon and Breach, Science Publishers, S.A.Printed in the United States of AmericaTRACKING CODES IN ACCELERATORS: TYPESAND LIMITATIONStDAVID R. DOUGLASLawrence Berkeley Laboratory, Berkeley, CA 94720(Received March 7, 1985)The motivation for tracking simulations of single-particle motion in circular accelerators is outlined.Three mathematical models, which are employed in many tracking programs describing charged-particle behavior, are presented, and the advantages and disadvantages of each model are discussed.I. INTRODUCTIONThe goal of "tracking" is to simulate directly single-particle motion in circularaccelerators, in the presence of nonlinear magnetic fields and errors of varioustypes, to determine those regions of phase space that are stable. "Stability" hastwo meanings in this context. The first is absolute physical stability within theaccelerator (which must be determined). The second is particle stability withinthe mathematical model employed by a particular tracking program. Such studiesassume that if a test particle with initial condition at a specified location in phasespace remains within a bounded region (respresenting a beam pipe) for aspecified interval of simulated time, then all real particles in the vicinity of thesimulated particle will be absolutely stable in the physical accelerator. Allrelevant physical effects must be included in the model, and the single-particlemotion must not exhibit pathological features (such as extensive stochasticity) forthis assumption to be valid.The geometry commonly employed in tracking simulations is as follows. A"cut" is made at a convenient location along the accelerator, and a test particle islaunched into the machine at that point using initial conditions which deviatefrom the phase-space location of an ideal particle, which is assumed to remain atthe origin of phase space as it circulates within the accelerator. The motion of thetest particle is then simulated by solving Hamilton's equations for the positionand momentum of the test particle, relative to the ideal particle, after one turnabout the machine. (The deviations of test-particle motion from the motion of theideal particle are small, so that approximation schemes may be used.) Thelong-term behavior of the test particle may be simulated by iterating this process,using as initial conditions for each iterate the test-particle, phase-space location atthe end of the preceding iterate.The key feature of the process is the Hamiltonian nature of the motion. Thelocation of the test particle at the end of one turn may thereby be viewed as thet Supported in part by the U.S. Department of Energy.119120 DAVID R. DOUGLASimage of the initial condition under a nonlinear canonical transformation called a"transfer map,"l which completely specifies the motion of the test particle for anentire turn around the accelerator. Particle stability over many turns is thusgoverned by the stability of high powers of the transfer map. If the one-turntransfer map is denoted by M, the particle motion for n turns is governed by Mn •The accelerator itself is then viewed as an analog computer, which evaluates theaction of the map M on the single-particle phase space. Tracking codes aredistinguished only by how they represent and evaluate the nonlinear transfermap.Alternate representations of the transfer map prove more or less convenient forinclusion of different physical effects. We shall outline the types of nonlineareffects that influence particle stability in accelerators and that are thus of interestin tracking simulations and then discuss three models employed to describe sucheffects. In each model it is assumed that proper care has been taken in the choiceof Hamiltonian and canonical variables and that the models are obtained througha consistent set of approximations.II. PHYSICAL EFFECTS OF INTEREST IN TRACKING SIMULATIONSAt least two effects are of interest in tracking simulations. The first is theinfluence of the nonlinear spatial dependence of magnetic fields within theaccelerator upon particle stability. In conventional accelerators, the primarysources of such fields are the chromatic-correction sextupoles, within which aparticle experiences a magnetic field (and hence, a magnetic force) that variesquadratically with displacement. In superconducting accelerators, the majorsources of magnetic nonlinearities are systematic and random magnetic multipoleswithin the superconducting magnets. In contrast to conventional magnets (whichhave very weak high-order nonlinear field components), superconducting magnetspossess a rich dependence of magnetic field upon position. In either case, thedependence of magnetic field upon position is a major source of nonlinear effectsand is thus of great interest.A second effect of importance in tracking simulations is that of energyoscillations. Although the energy variation of a particle within an accelerator isusually well described by linear mathematics, this 'synchrotron" oscillationcouples strongly to the nonlinear motion of a particle. As the particle energyvaries, the natural frequency of transverse oscillations is slowly modulateq. As aconsequence, the structure (in time) of the nonlinear transverse force applied to aparticle varies, and the particle may repeatedly cross isolated or multipleresonances, causing instability. Treatment of energy oscillations is thus needed.III. A THIN-LENS MODEL FOR THE TRANSFER MAPThe simplest method employed in tracking programs to describe a transfer map isreferred to as a thin-lens or "kick" model. In this model, a magnetic element withTRACKING CODES 121nonlinear fields is treated in the impulsive approximation. The motion in all otherelements is assumed to be linear. Mathematically, the accelerator is then modeledby a sequence of interleaved linear matrix and nonlinear impulsive transforma-tions. The nonlinear transforms have the following explicitly canonical form inone degree of freedom (q, Pq are canonical conjugates):q=q,pq =Pq + f(q)·(1)Equations (1) model the effect of a thin magnetic lens on a particle, hence thename 'thin-lens model." The method is simple and the transformations are easilyevaluated. Moreover, nonlinearities of any order may be treated. However, theimpulse approximation must be valid, so the nonlinear elements under considera-tion must be short in length and high in field. The required numerical simulationtime will increase linearly with the number of nonlinear elements, and theaccuracy will be degraded as the nonlinearities weaken. Little analytic informa-tion is provided, but this method is ideal for treating accelerators with a moderatenumber of high-field, strongly nonlinear elements.It is possible to treat energy oscillations in a straightforward manner within thismodel either by using a four-dimensional single-particle phase space (whileadjusting the transformations used for simulation to account for the variation ofmagnetic bending with energy) or by using a full six-dimensional phase space andsimulating the effect of the rf system using an additional family of transformationsof the form of Eq. (1).An example of a widely distributed impulse code is the program PATRICIA. 2IV. A TAYLOR'S-SERIES EXPANSION TO MODEL THETRANSFER MAPA Taylor's series may be used for a second representation of the transfer map. Inthis method, each position or momentum-vector component of a test particleafter any element is viewed as a function of all components upon entering theelement. This function is written as a Taylor's expansion and truncated at someorder. Thus, in a four-dimensional phase space we may write the result for atransverse displacement (relative to an ideal particle), accurate to second order,as follows;i = M(x;x)x + M(x;px)Px + M(x;y)y + M(x;py)py+ T(x; X, x)x2 + T(x; x, Px)xpx + T(x; x, y)xy + · . · + T(x;py, py)PXpy. (2)The "matrix elements" M and T may be computed using the equations ofmotion. 3 Relations such as Eq. (2) may therefore be used to model the transfermap by carrying the test-particle initial condition at the beginning of a turn insequence through each element of the accelerator. However, Eq. (2) is atruncated series, and so it is not canonical. This problem may be overcome by122 DAVID R. DOUGLAS(3)replacing Eq. (2) by a transformation of the following form:i = aF(x, y, Px, py)/apx; y = aF(x, y, Px, py)/apy;Px = aF(x, y, Px, py)/ax; py = aF(x, y, Px, py)/ay.The generating function F is chosen to duplicate the Taylor's series through thedesired order of accuracy [order 2 in Eq. (2)] and may be simply written in termsof M and T. Equation (3) is canonical in all orders, thereby providing aHamiltonian representation of the transfer map. Unfortunately, Eq. (3) isimplicit; i, Px, ... may only be obtained numerically in most cases.This representation may be extended to second or third order in mostapplications. It is therefore restricted to the description of low-order non-linearities. However, a full six-dimensional phase space may be treated, and thereare no restrictions to either "short" or "strong" elements (an improvement overthe impulse approximation). Moreover, as a specific series is provided for everyelement in the accelerator, collections of transformations may be combined toprovide a single transform that describes (approximately) a collection ofnonlinear elements. If only low-order nonlinearities are relevant, a reduction inthe number of transformations required to represent the single-turn transfer mapis thereby achieved, and a corresponding increase in tracking speed is obtained.Energy oscillations may be introduced in a natural manner by treating rfcavities as an additional type of element in the accelerator. As an explicitnonlinear expression [such as Eq. (2)] is utilized, significant analytical informationis provided. The program DIMAr is one tracking code which employs aTaylor's-series representation of the transfer map.v. A LIE ALGEBRAIC REPRESENTATION FOR THE TRANSFERMAPThe final transfer-map model we consider makes use of certain group propertiesof Hamiltonian-generated canonical transformations. In this model, the transfermap is represented by use of a product of "Lie transformations."s We denote thismap as follows:M = exp :/2: exp :h: exp :/4: . · · exp :/n:· (4)Here, M is the transfer map for an element, collection of elements, or even anentire turn. The 1m are homogeneous polynomials of degree m in the canonicalvariables. The location and momentum of a particle after entering the element orbeam line described by the map Eq. (4) are obtained by evaluating the action ofM on the phase-space location of the test particle upon entering the element orbeam line. Here, as elsewhere, the transfer map is to be regarded as an activecanonical transformation carrying the test particle through the accelerator.In this formalism, the polynomials 1m are uniquely specified if the Hamiltonianfor motion through each element in the accelerator is given. The representationof Eq. (4) is explicitly canonical and accurate to order n - 1. In practice,third-order (n = 4) descriptions are available, and fourth-order (n =5) descrip-TRACKING CODES 123tions may be possible. The formalism is thus limited to low-order nonlinearities,but it allows the description of finite-length nonlinear elements and is often ofhigher accuracy (order 3 vs. order 2) than the Taylor's-series method. As theformalism is based on Eq. (4), considerable analytic information is available, andit appears possible in some cases to describe a fairly large portion of anaccelerator with a single transfer map. Numerical calculations with this formalismmay be carried out with considerable speed in such cases. However, firmguidelines for when this is possible do not as yet exist.In practice, Eq. (4) is not evaluated exactly (which would require summing aninfinite series) but is expanded in powers of the operators :fm:. The resulting(noncanonical) transform is treated in the same manner as the matrix transformEq. (2); a generating-function representation is obtained and evaluated. A finalstrength of the Lie algebraic formalism is that it, like the Taylor's seriesformalism, is explicitly six-dimensional and thereby allows a natural description ofenergy oscillations. Additional power is provided by the analytic capabilities ofthe formalism, which emphasize the equivalence of all directions in six-dimensional phase space. No distinction is made between longitudinal andtranverse coordinates (a critical feature when simulating large-amplitude energyoscillations). One tracking program making use of the Lie algebraic formalism isthe program MARYLIE.6REFERENCES1. A. J. Dragt, Physics of High Energy Particle Accelerators (Fermilab Summer School, 1981), Conf.Proc. 87, 147 (1981).2. H. Wiedemann, SLAC-PEP-Note 220 (1976).3. K. L. Brown, SLAC Report-75 (1972).4. R. V. Servranckx and K. L. Brown, SLAC Report-246 (1984).5. A. J. Dragt, Ope cit.6. D. R. Douglas and A. J. Dragt, 1.E.E.E. Trans. Nuc. Sci., NS-30, 2442 (1983).

Tracking Codes In Accelerators: Types And Limitations

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