SUSSIX: A Computer Code for Frequency Analysis of Non Linear Betatron Motion

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1 EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CERN SL DIVISION CERN SL/Note (AP) updated June 29, 1998 SUSSIX: A Computer Code for Frequency Analysis of Non Linear Betatron Motion R. Bartolini and F. Schmidt CERN, Geneva, Switzerland Abstract The SUSSIX program has been developed to postprocess tracking or experimental data via frequency analysis. It allows to evaluate relevant dynamical quantities such as detuning, resonance driving terms and smear directly from the recorded turn by turn data. Geneva, Switzerland March 5, 1998

2 1 Introduction The application of perturbative techniques for the analysis of tracking and also of experimental data has been proven to be difficult since it requires a detailed knowledge of all the magnetic elements in the accelerator lattice. This knowledge is not needed when one applies a frequency analysis of turn by turn data to derive the driving terms of resonances. This has first been attempted by J. Bengtsson [1]. Recently, high precision algorithms [2, 3] for the tune measurements have been developed. It has been shown that the frequency analysis of the betatron motion provides all the relevant information to reconstruct the conjugating function and ultimately the Normal Form directly from tracking data [4]. The postprocessing of turn by turn data via frequency analysis has been implemented in the FOR- TRAN code SUSSIX. In what follows we briefly describe the theoretical background and the features of the SUSSIX program. 2 Frequency Analysis of the Betatron Motion The time series of tracking (or experimental) data is analysed with the algorithms for the precise measure of the betatron tunes [2, 3]. They provide the fundamental frequencies (i.e. the tunes) and the corresponding Fourier coefficients. By subtracting from the time series these tune lines, using the proper amplitudes and phases, we obtain a new signal of equal length which can be reanalysed in the same way. This iterative procedure provides the set of frequencies contained in the betatron motion. In case the motion is regular the frequencies can be expressed as linear combinations of the fundamental tunes. As an example, in the 4D case the evolution of the linearly normalised horizontal variable h, x (N )=^x(n),i^p x(n)after N turns reads: MX h, x (N )= a j e i[2(mx j Qx+my j Qy)N + j ] mx j ;my j 2Z; (1) j=1 where the a j and j are amplitude and phase of the corresponding spectral line. 3 Relation to Normal Form Coefficients In the spectral decomposition of Eq. 1, the line with the largest amplitude is in most cases the tune while the secondary lines are the effect of resonance terms in the Hamiltonian of the motion. Applying the conjugating function to the coordinates, like h, x (N ), we arrive in first order [4] at: h, x (N ) p 2I x e i(qxn +x 0 ) X,2i jf jklm (2I x ) j+k,2 2 (2I y ) l+m 2 e i[(1,j+k)(qxn +x 0 )+(m,l)(qyn+y )] 0 ; jklm (2) where Q x and Q y are the tunes, I x and I y are the nonlinear invariants as given by the normalisation procedure and f jklm are the coefficients of the conjugating function expressed in the resonant basis (see. Ref. [4] for details). This expression is equivalent to the spectral decomposition of Eq. 1. The motion appears as a superposition of spectral lines given by the tune (first row) and the contribution from the resonant terms (second row). Rewriting Eq. 2 as X h, x (N )= HSL jklm e 2i[(1,j+k)Qx+(m,l)Qy]N (3) jklm the relation with the coefficients of the generating function f jklm can be expressed as in Tab. 1. It has to be pointed out that several coefficients of the generating function are feeding the same spectral 2

3 line. Nevertheless, it is has been shown in Ref. [4] how to extract each generating function term from one or several spectral lines. A prerequisite is to choose an amplitude which on one hand is small enough to avoid higher order contributions and on the other hand large enough to obtain sufficiently large amplitudes of the spectral lines. To demonstrate the applicability of the method we show a correlation plot (Fig. 1) between spectral lines and Normal Form coefficients of a fourth order resonance for 60 realizations of the random multipolar errors of the LHC version 4. Table 1: Relation between spectral lines and coefficients of the generating function Generating Function coefficient Spectral Line Amplitude jf jklm j jhsl jklm j =2j(2I x ) j+k,1 2 (2I y ) l+m 2 jf jklm j jvsl jklm j =2l(2I x ) j+k 2 (2I y ) l+m,1 2 jf jklm j PHSL jklm = jklm + x0, 2 Phase jklm PVSL jklm = jklm + y0, Normal Resonance Driving Term ( 2-2 ) 0.05 y = x + 2E-05 R 2 = Normal Form Frequency Analysis of Tracking Data Figure 1: Resonance terms from normal form and from tracking data for 60 seeds of the lhc lattice version 4 3

4 4 Description of the Software The high precision tune measurement is used in SUSSIX to perform the spectral decomposition of tracking data. The data have to be provided in files of unit 90 and below and they can be either SIXTRACK binary files or ASCII files containing two columns for each plane of motion, i.e. x; p x ; y; p y ; ; p. SUSSIX actions are specified in the sussix.inp input file. All the input options are described in detail in Tab. 2. Some options, however, need further clarifications: In case of data from lepton machines the radiation damping itself can be used to scan the phase space amplitude by varying time windows of data in which the amplitudes do not change significantly. This can be accomplished by analysing the signal for various pairs of initial and final turn numbers (see TURNS parameters in Tab. 2). As an example, signals affected by a damping time less than 200 turns have been successfully analysed [5]. In case the motion is regular an estimate of the smear can be given by summing all lines with the exception of the tune line [6]. The flag ISME must be set to 1 and the output is written to file unit IUSME. The calculation of the invariants is performed with the method derived in Ref. [7]. For example, using the notation of Eq. 3, the horizontal invariant of a 4D orbit is given by: I x = X jklm (1 + k, j)jhsl jklm j 2 +(k,j)jvsl jklm j 2 (4) The flag INV must be set to 1 and the output is written to file unit IINV. Concerning the evaluation of the coefficients of the conjugating function according to the Tab. 1, the computation of the third order coefficients is presently available. An extension to the fourth and higher orders is in the planning. The flag ICF must be set to 1 and the third order conjugating function coefficients with their corresponding indices are written to file unit IICF. The output is printed to file unit 300 (on HP unit 30). It contains for each spectral line the corresponding amplitude, phase and frequency as specified in the Eq. 1. The spectral lines are given in a descending order of amplitude and the frequencies are identified as the closest linear combinations of the fundamental frequencies. The error column gives the difference between the calculated spectrum line and this linear combination of the tunes. If the line is not identified within the specified tolerance EPS the error is set to that value. The output file of the SUSSIX analysis of some LHC tracking data are shownintab.3. We would like to remind the reader that in the chaotic regime or close to resonances both Normal Form and the spectral decomposition of Eq. 1 break down. Under these conditions SUSSIX cannot produce meaningful results (see Ref. [4] for details). 4

5 5 Conclusions We have presented the SUSSIX program for the postprocessing of turn by turn data via frequency analysis. The code works on various platforms by running the system dependent executables: /afs/cern.ch/group/si/slap/share/sussix/bin/sussix. The source code file sussix.f and a library called sussix r.f are found in: with an example run using sussix.inp in: /afs/cern.ch/group/si/slap/share/sussix/src, /afs/cern.ch/group/si/slap/share/sussix/example. References [1] J. Bengtsson, Non Linear Transverse Dynamics for Storage Rings with Application to the Low Energy Antiproton Ring (LEAR) at CERN, CERN 88 05, (1988). [2] J. Laskar, C. Froeschlé and A. Celletti, The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping, Physica D 56, pp (1992). [3] R. Bartolini, A. Bazzani, M. Giovannozzi, W. Scandale, and E. Todesco, Tune evaluation in simulations and experiments, Part. Acc. 56, pp (1996). [4] R. Bartolini and F. Schmidt, Normal Form via Tracking or Beam Data, Part. Accel., in press and CERN LHC Project Report 132 (revised December 1997), (1997). [5] R. Bartolini, J. Corbett, M. Cornacchia, M. Giovannozzi, C. Pellegrini, W. Scandale, E. Todesco, P. Tran, A. Verdier, Measurements of the tune variations induced by nonlinearities in lepton machines, Proceedings of the 5 th European Particle Accelerator Conference EPAC96, Sitges, Spain Vol. II, (1996), pp and CERN SL (AP), (1996). [6] R. Bartolini and F. Schmidt, Evaluation of Non Linear Phase Space Distortions via Frequency Analysis, LHC Project Report 98 and in the proceedings of the workshop on: Nonlinear and Collective Phenomena in Beam Physics, Arcidosso, September 1996, [7] A. Bazzani, L. Bongini and G. Turchetti, Analysis of Resonances by Action Map comparing Tracking and Normal Forms, Proceedings of the workshop on: Nonlinear and Collective Phenomena in Beam Physics, Arcidosso, September 1996, 5

6 Table 2: SUSSIX input file Entries per Line Description ISIX flag to choose the analysis of ASCII (0) or SIXTRACK (1) data NTOT total number of files to be analysed IANA analysis flag: (0) postprocessing of previously generated output (1) set up full analysis of tracking data (2) as (1) but write FFT to unit 310 (HP 31) ICONV flag for the conversion of physical data to linearly normalised data: off (0), on (1) (ISIX=1 only) NT1 NT2 initial and final number of turns to be analysed NARM number of spectral lines to be analysed ISTUNE ETUNE(3) flag to choose fundamental frequencies: (0) take frequencies with largest amplitudes (1) use lines found in the interval TUNEX,Y,SETUNE(3), otherwise take largest amplitude lines (2) enforce the ones specified as TUNEX,Y,S ETUNE(3) are the the allowed differences to the guess TUNEX,Y,S TUNEX,Y,S (guess) frequencies for the fundamental ones, requires ISTUNE>1 NSUS if NSUS > 1 the first NSUS next to leading frequencies are subtracted from the signal (ISIX=1 only) IDAM degrees of freedom of the motion NTWIX flag to single particles (1) or treat twin (2) (ISIX=1 only) IR flag to specify if the signal is complex (0) or real (1) IMETH flag to select the time window filter for the evaluation of the spectrum: Hanning filter (1), no filter (2) NRC maximum order of the linear combinations of the tune lines up to which the spectral lines are to be identified EPS tolerance on the identification of the lines as linear combination of the tunes NLINE number of lines with different L,M,K indices each to be written to a separate file L,M,K indices of the spectral lines to be written to separate files. WARNING: NLINE of entries are expected in this input file IDAMX flag to specify which plane must be analysed for the NLINE option: horizontal (1), vertical (2) or longitudinal (3) IFIN offset for unit of separate output file: first file unit (IFIN+1) up to last file unit (IFIN+NLINE) ISME flag for the calculation of smear: off (0), on (1) IUSME unit of output file containing the calculation of smear INV flag for the calculation of invariants: off (0), on (1) IINV unit of output file containing the calculation of the invariants ICF flag for the calculation of conjugating function coefficients: off (0), on (1) IICF unit of output file containing the calculation of the conjugating function coefficients 6

7 Table 3: SUSSIX output file ANALYSIS OF X SIGNAL, CASE: 91 TUNE X = Line Frequency Amplitude Phase Error mx my ms p E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E ANALYSIS OF Y SIGNAL, CASE: 91 TUNE Y = Line Frequency Amplitude Phase Error mx my ms p E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E NO S SIGNAL 7

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH European Laboratory for Particle Physics. Normal Form Analysis of the LHC Dynamic Aperture

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH European Laboratory for Particle Physics Large Hadron Collider Project LHC Project Report 119 Normal Form Analysis of the LHC Dynamic Aperture F. Schmidt, CERN,