THE USE OF MAPPINGS FOR STABILITY PROBLEMS IN BEAM DYNAMICS
E. Todesco August 2, 1996
A review of some stability problems in single-particle dynamics of circular accelerators when the nonlinearities are not negligible is given. The betatron motion is modelized through the discrete formalism of symplectic mappings. Some numerical tools are used to work out the resonance patterns and the relation with long-term stability. A complementary picture is obtained through the perturbative methods of normal forms.
Abstract
1 Introduction: dynamics of circular accelerators
In circular accelerators, charged particles (electron, protons, antiprotons, ions ...) are kept on a circular orbit and accelerated through electromagnetic elds. In modern large machines such as the synchrotron, one has time-independent magnetic elds provided by electromagnets that keep and focus the particles on the reference orbit. The acceleration of the beam is provided by time-dependent electric elds generated by the radiofrequency cavities. The relation between the energy E , the radius r of the machine, and the bending magnetic eld B that keeps the particle on a circular orbit according to the Lorentz force is: E = eBr (1)
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The physics of this type of accelerators involves several stability problems 1, 2]. The beam circulates for a limited time in the machine (typically a few hours), but since it has the speed of light, it makes a very large number of revolutions. For the planned LHC 3], that is 27 Km length, the revolution time is about 10?4 seconds and therefore during a typical run the particle makes about 108 turns. This number is of the same order of magnitude of the number of turns of an inner planet for a relevant fraction of the estimated age of the universe. The stability problems can be separated into two main families: singleparticle e ects and collective e ects. In the rst case one neglects the mutual interaction of the particles inside the beam, that can produce several instabilities. In this paper we will analyse some aspects of the single-particle dynamics in a circular accelerator; the lattice of magnetic elements of such a machine is made up of a regular part where the beam is bent, and a few straight sections where the particles collide in the interaction points, and where the beam is injected and ejected. The particles oscillate around the reference orbit both in the transversal plane and in the longitudinal direction. Since these two frequencies have di erent orders of magnitude, in a rst approximation one can consider these motions as uncoupled. The motion in the plane transverse to the orbit is called betatron motion, and is modelized by an Hamiltonian with two degrees of freedom (i.e. the two transverse coordinates and the relative momenta) with a periodic dependence on the azimuth s along the machine 4, 5, 6]. The cells ensure the stability of the betatron motion. They are made up of elements (magnets) with separated aims. The dipoles provide the constant bending eld B ; the quadrupoles provide a eld linear in the transversal distance to the orbit (as a spring) to stabilize the linear motion. The sextupoles provide a quadratic eld to stabilize particles with di erent energies; such a nonlinear eld also generates unwanted instabilities. Moreover, one has unwanted high order components of the magnetic elds (especially in the dipoles) due to the superconducting technology: the multipolar errors. The systematic part of these errors are corrected through the multipolar correctors, i.e. elements whose eld is intrinsically nonlinear and is powered to compensate the nonlinearities of the machine 7]. A crucial problem is to determine the dynamic aperture, i.e. the dimension of the stability domain around the closed orbit, for a given
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con guration of magnets (dynamic aperture estimate); moreover one has to consider strategies to correct the nonlinearities in the lattice in order to enlarge the dynamic aperture if it is not su ciently large. In order to solve these problems, three complementary approaches are available. Experiments on real machines 8, 9]: e ective instrumentation based on beam position monitors and pick-up give a lot of relevant information on the beam, and can provide several measurements: { a direct measurement of the dynamic aperture; { a check of the agreement of the numerical model with the real machine; { a check of the e ectiveness and feasibility of the correction schemes; { a reconstruction of the phase-space dynamics; { a long-term stability evaluation in real time. Tracking: it is the numerical integration of the lattice model through symplectic integrators 10, 11]. It can give the following informations: { a phenomenological description of the machine; { a numerical search for correction schemes; { short- and medium-term dynamic aperture, Lyapunov exponents, frequency analysis ... Perturbative tools: they can be based either on the Hamiltonian ow 4, 5, 12] or on the Poincare mapping 6, 13, 14]. The perturbative parameter can be xed as either the magnet gradients or the distance to the orbit. These tools provide the following items: { analytical invariants, phase space structures, islands position and widths; { an analytical understanding and optimization of the lattice; { no direct indication on the dynamic aperture.
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2 Maps of betatron motion
Let (x; y ) be the transversal coordinates and (px; py ) be the conjugated momenta: the one-turn map of betatron motion gives the phase space position of a single particle after one turn as a function of the position of the same particle at the turn before. One can write
x0 = M (x) x = (x; px; y; py):
(2)
We assume that there is a periodic orbit linearly stable: let 0 be the intersection of this orbit with our section. Then, the origin is an elliptic xed point M (0) = 0, and using the coordinates z to diagonalize the motion, the map reads
0 z1 0 z2
= ei!1 z1 + = ei!2 z2 +
X
X
n 2 j1 +j2 +j3 +j4 =n
X
X
j j F1;j1 ;j2 ;j3 ;j4 z11 z1 j2 z23 z2 j4 j j F2;j1 ;j2 ;j3 ;j4 z11 z1 j2 z23 z2 j4
(3)
n 2 j1 +j2 +j3 +j4 =n
The linear part is the direct product of two rotations of angles !1 and !2 , that are the linear tunes. The simplest model of nonlinear betatron motion is given by a linear lattice plus a quadratic nonlinearity (i.e. a sextupole element). The mapping reads
0 z1 0 z2 i = ei!2 z2 + 2 (z1 + z1 )(z2 + z2 ) ; i = ei!1 z1 ? 4 (z1 + z1 )2 ? (z2 + z2 )2
(4)
where is the ratio between the beta functions in the sextupole (see Ref. 6]); we also give the expression in real coordinates x ? ipx = z1, y ? ipy = z2 :
1 0 x0 1 0 cos !1 sin !1 x 0 0 10 C B C B CB B 0 C B CB C B p x C B ? sin !1 cos !1 C B px + (x2 ? y2) C 0 0 CB C B C B C B C=B CB C B C B CB C B y0 C B 0 CB C B C B y 0 cos !2 sin !2 C B C B C B CB A @ A @ A@ 0 py ? 2 xy 0 0 ? sin !2 cos !2 py
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This map is the composition of a rotation plus a nonlinear quadratic kick, and since for = 0 it becomes the conservative 2D Henon mapping times a linear rotation, it has been called the 4D Henon map 6]. We would like to point out some di erences between this mapping and the well-known standard mapping de ned by Froeschle 15]. First of all, contrary to the standard mapping, the Henon map has a non compact phase space; since it is a polynomial mappings, one has that beyond a certain amplitude the high orders are dominant and therefore one has a fast escape to in nity. Second, the standard map has a perturbative parameter that measures its deviation from a twist mapping, whilst the Henon map is the perturbation of a linear rotation, and the parameter is the amplitude. The phenomenology of the phase space for generic Henon-type maps (i.e., one-turn maps of betatron motion) can be summarized as follows. Low amplitudes: one has very small nonlinearities, practically all the initial conditions lie on 2D KAM tori, and only Arnold di usion (that is negligible for all practical purposes) is left. Average amplitudes: one has islands of stability, intermittency, chaotic motion, short-term and long-term particle loss (from 102 to 1 turns); it is still unclear whether there are di usion phenomena 1 . Large amplitudes: the nonlinearities are dominant, and one has a short-term particle loss. We conclude this section by giving a short summary of the main open problems in the comprehension the dynamics of betatron one-turn maps. Concerning the dynamic aperture: { It is still not clear what kind of de nition of the dynamic aperture can be given using tracking: one should decide how to take into account of the deformation in phase space 18], of the relevance of holes of instability, and of long-term phenomena.
frequencies are varying with time 16, 17].
1 One can prove that di usion phenomena occur for betatron maps when the linear
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The agreement between tracking and experiments is not extremely good, but rather satisfactory of the analysed cases (between 20 % - 10 % for the HERA 8] and for the SPS 9]). { Is it possible to give a fast estimate of the dynamic aperture using perturbative tools 19, 20]? { Di usion phenomena: are they relevant ? Which mechanisms are involved ? Concerning the optimization of the lattice and correction schemes, the big question is how to increase the dynamic aperture at low price. The local correction strategy consists in placing corrector magnets whose nonlinearities cancels out the multipolar errors in the same place: this is a trivial solution, and in general is very expensive. The lumped correction strategy consists in placing some correctors around the lattice hoping that their nonlinearities nicely interact with the errors and increase the dynamic aperture. It is still not clear whether this correction strategy is always possible, and in what cases it is e ective 7, 21].
{
3 Numerical tools for maps
In this section we will present some numerical tools to investigate the 4D dynamics of betatron maps, based on numerical integration. One can consider a grid of initial conditions in the (x; y ) space ( xing the momenta to zero), and for each condition one computes the orbit for a given number of iterations. The plot of the stable initial conditions gives the shape of the boundary of stability in phase space. One could argue that the scan over the initial conditions should be carried out over the four phase space variables: this is not necessary, since most of the stable orbits have two nonlinear invariants and take place on 2D tori. On the other hand, scanning over only one variable, as it is done sometimes in simulating complicated lattices to save CPU time, it is not correct. In Fig. 1 we plot such a gure for the Henon map at the linear tune !x =(2 ) = 0:28 and !y =(2 ) = 0:31. Horizontal and vertical axes are the linear invariants of the initial condition x = x2 + p2 = x2 and x 2 2 2 y = y + py = y respectively. The image of these initial conditions in the frequency space gives the tune footprint 22] that shows the distri-
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Figure 1: Plot of initial conditions stable after 2048 iterations in the space of the linear invariants x and y for the Henon map; initial phases have been set to zero.
bution of the frequencies of the stable initial conditions. In Fig. 2 the footprint is shown together with the resonant lines up to order 9: one sees that besides a large fraction of initial conditions that give rise to 2D KAM tori, one has a relevant fraction of conditions whose orbits are locked on single resonances (lines in the footprint). The strength of the resonance can be qualitatively evaluated through the footprint: the strongest resonances give rise to resonant lines with a larger depletion area in their neighbourhood. Changing the phases of the initial conditions, the plot of the stable initial conditions is modi ed in the region closer to the dynamic aperture, whilst the footprint remains the same since the frequencies are good nonlinear invariants for the stable orbits. However, the phase space section (x; y ) is very signi cant because, contrary to the footprint, it provides the dimension and the shape of the stability domain. Another diagram that provides very relevant information on the 4D dynamics is the graph of the network of resonances in the space (x; y ), i.e. the plot of all the initial conditions in the space (x; y ) whose frequencies are locked on single resonances. In Fig. 3 we show this plot for the same model of Figs. 1 and 2. Using this diagram one
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Figure 2: Tune footprint for the Henon map; resonances up to order 9 are marked as dotted lines
can directly evaluate the width of the resonances and their relation with the stability border. For instance, for the Henon map at the considered frequencies, one has rather wide islands for the (6; ?2) and for the (3; ?6) resonances. The resonance (1; ?4) has no phase locking, as it appears from the footprint, since the initial conditions that fall on it are unstable. In the next section we will show how the perturbative tools allow one to analytically evaluate the position and the width of the resonances. The graphs showed in Figs. 1, 2 and 3 were computed for 2048 iterations of the mapping. Indeed, since the beam in the accelerator is stored for a larger number of turns, one has to consider the relation between short-term and long-term stability. In the case of the LHC, the beam stays for 107 turns at the lowest energy; then it is accelerated at the main energy. Since the process of acceleration involves a
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Figure 3: Net of resonances: plot of initial conditions stable after 2048 iterations and whose frequencies satisfy a resonant condition in the space of the linear invariants x and y for the Henon map; initial phases have been set to zero and the number of the resonances are marked close to the resonant lines.
shrinking of the beam proportional to the gain in the energy, after 107 turns the particles that are still stable will be squeezed and stabilized on the closed orbit. In Fig. 4 we show the results of a long-term simulation (107 turns) for the Henon map with the same parameters of the previous gures. Empty circles are conditions that are stable for more than 107 turns; particles lost at medium and long-term are plotted as solid circles with diameter roughly proportional to the number of turns. The main result of this analysis is that there is a full domain of initial conditions that are stable for very long times, even if the net of resonances is rather strong. In this example, there are no holes of
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instability, i.e. unstable particles surrounded by stable ones; in other cases holes can be found, but they are always at high amplitudes and they are rather rare. Moreover, we see that long-term particle loss often occurs where the island mets the region of fast escape to in nity. The analysis of other models shows that long-term losses with lowest amplitudes are located on the sides of large islands, i.e. close to the separatrices. It appears clear from the above gures that one can have very large islands without particles loss, such as for resonances (3; ?6) and (6; ?2), and that no loss along the net of resonances (what is sometimes improperly called as Arnold di usion) is visible up to 107 turns using this coarse-grain scan in the phase space initial conditions. For this model it appears that the long-term stability is very close to the short-term one. This seems to be not the case of complicated betatron mappings, where the particle loss between 103 and 107 can lead to relevant reduction in the dynamic aperture. This kind of behaviour can be obtained also for the Henon map, for instance by setting the linear frequencies close to low-order resonances 20].
4 Perturbative tools for maps
The perturbative theory of normal forms for symplectic mappings is based on a perturbative expansion in powers of the amplitudes. Perturbative series are divergent but appropriate truncation provide good approximations over interesting domains. The original map is transformed through a symplectic function to a simpler map, the normal form, that exhibits explicit invariants and symmetry. The normal form is then written as the Lie series at integer times of an interpolating Hamiltonian, that provides the following relevant informations over the nonlinear dynamics. Nonlinear invariants 1 and 2 that agree with the linear invariants x and y at rst order. Dependence of the nonlinear frequencies on the nonlinear invariants. Position, topology, stability and eigenvalues of the 1D resonant tori that are generated by a single resonance. Moreover, one can de ne a generalization of the island width to the 4D case.
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Position, topology, stability and eigenvalues of the xed points (i.e., 0D resonant tori) that are generated by a double resonance (crossing of single resonances). Action-angle variables of the system close to a single resonance. A detailed exposition of the normal form applied to polynomial symplectic mappings can be found in Ref. 6]; an explicit analysis of the resonant structures that arise in the 4D case is given in 24, 25], and the algorithms to construct the normal form in the 4D case based on Lie series are described in 26]. Here we will show the results of a code 27, 28] that automatically analyses the single-resonance interpolating hamiltonian, evaluating the position of the resonances and the island width. In Fig. 5 we plot the net of resonances evaluated through normal forms in the space of the nonlinear invariants 1 2. The agreement with the analogue plot obtained through numerical analysis in the space x y of the linear invariants (see Fig. 3) seems to be excellent. Therefore, it appears that the normal forms allow one to reconstruct the net of resonances in phase space, i.e. the global dynamics of the system. With respect to the numerical approach, in this case one has explicit expressions for these nonlinear indicators and therefore analytical optimizations are possible. On the other hand, no direct information on the dynamic aperture is provided by the normal form series 2 .
5 Acknowledgements
We wish to thank Prof. Turchetti for stimulating this work and stimulating discussions; we also wish to acknowledge A. Bazzani, G. De Ninno, M. Gemmi, M. Giovannozzi and W. Scandale for relevant contributions. A special thank to the organizers of the meeting in Aussois.
References
1] W. Scandale and G. Turchetti, Nonlinear problems in future particle accelerators (World Scienti c, Singapore, 1990).
the normal forms provide the explicit form of the separatrix and therefore very precise estimates of the dynamic aperture can be given.
2 Indeed, when the dynamic aperture is determined by low-order unstable resonances,
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2] S. Chattopadhyay et al., Ed., Nonlinear Problems in Particle Accelerators: theory and experiments (AIP, New York, 1995). 3] The LHC Study Group, CERN 93-03 (1993). 4] A. Schoch, CERN 57{21 (1957). 5] G. Guignard, CERN 76{06 (1976). 6] A. Bazzani, E. Todesco, G. Turchetti and G. Servizi, CERN 94{ 02 (1994). 7] W. Scandale, F. Schmidt and E. Todesco, Part. Accel. 35, (1991) 53{88. 8] O. Bruening, W. Fischer, F. Schmidt, F. Willeke, CERN SL (AP) 95{05 (1995). 9] W. Fischer, M. Giovannozzi, F. Schmidt, CERN SL (AP) 95-96 (1995). 10] F. Schmidt, CERN SL (AP) 94{56 (1994). 11] H. Grote and F. C. Iselin, 6 Users's Reference Manual CERN SL (AP) (90{13/1990). 12] F. Willeke, DESY HERA 87{12 (1987). 13] A. Bazzani, P. Mazzanti, G. Servizi and G. Turchetti, Nuovo Cim., B 102, (1988) 51{80. 14] E. Forest, M. Berz and J. Irwin, Part. Accel. 24, (1989) 91{113. 15] C. Froeschle, Astron. & Astrophys. 4, (1970) 115{128. 16] G. Turchetti, these proceedings 17] A. Bazzani, these proceedings 18] E. Todesco, M. Giovannozzi, Phys. Rev. E 53, (1996) 4067. 19] F. Schmidt, F. Willeke and F. Zimmermann, Part. Accel. 35, (1991) 249{256. 20] E. Todesco, M. Giovannozzi, W. Scandale, Part. Accel. , (1996) in corso di stampa. 21] M. Giovannozzi, R. Grassi, W. Scandale, E. Todesco, Phys. Rev. E 52, (1995) 3093{101. 22] J. Laskar, Physica D 67, (1992) 257{81.
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23] J. Laskar, C. Froeschle and A. Celletti, Physica D 56, (1992) 253{69. 24] E. Todesco, Phys. Rev. E 50, (1994) R4298{301. 25] E. Todesco, Physica D , (1996) in press. 26] A. Bazzani, M. Giovannozzi and E. Todesco, Comput. Phys. Commun. 86, (1995) 199{207. 27] M. Giovannozzi, E. Todesco, A. Bazzani and R. Bartolini, CERN PS (PA) 96-12 (1996). 28] E. Todesco, M. Gemmi and M. Giovannozzi, \NERO: Nonlinear Evaluation of Resonances in One-turn mappings", in preparation.
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Figure 4: Long-term stability: plot of initial conditions in the space of the linear invariants x and y for the Henon map; conditions stable for more than 106 iterations (empty circles), conditions that are lost between 105 and 106 iterations (large black circles), between 104 and 104 iterations (medium black circles), and between 103 and 104 iterations (small black circles); initial phases are set to zero.
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Figure 5: Analytical determination of the network of resonances and of their width through normal forms in the space of nonlinear invariants 1 and 2.
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