The Touschek Module
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The Touschek Module in MAD-X. Catia Milardi (LNF-INFN) Frank Zimmermann (CERN) Frank Schmidt (CERN). Topics. Theoretical approach User instructions Differences with MAD-8 Examples Limitations Further developments. Generalities.

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The Touschek Module in MAD-X

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The Touschek Module

in MAD-X

Catia Milardi (LNF-INFN)

Frank Zimmermann (CERN)

Frank Schmidt (CERN)


  • Theoretical approach

  • User instructions

  • Differences with MAD-8

  • Examples

  • Limitations

  • Further developments


  • Particles in a relativistic bunch undergo Coulomb scattering involving:

    • multiple hits leading to increase of all beam sizes (IBS)

    • two particles transforming a small transverse momentum in to a large longitudinal one with consequent loss of both particles (Touschek effect).

  • The Touschek effect depends on beam energy and on the bunch volume so it is much more concerning for:

    • low-energy machines

    • small emmittance machine

    • high bunch current

Piwinski’s approach

Piwinski, TheTouschek effect in strong focusing storage ring, Desy 98-179, Nov. 1998.

The particle loss rate 1/T due to the Touschek effect is computed:

  • considering two arbitrary particles in their center of mass system

  • writing the Møller scattering cross section taking into account:

    - two-dimensional distribution of transverse momenta (due to both betatron and synchrotron oscillation)

    - beam envelopes variation

    - gaussian distribution in all beam coordinates

Particle loss rate due to the

Touschek effect

• rp classical particle radius

• Np number of particles per bunch

• g particle energy in rest mass unit

• ssbunch length

• sx,z standard deviation of the beam transverse size

• sp relativemomentum spread

• Dx,z dispersion function

Piwinski’s integral

I0 is the 0th order Bessel function

… in order to simplify numerical integration

RF momentum acceptance

computed from the bucket size taking into account the energy loss per turn due to synchrotron radiation U0

V0RF voltage

fsRF synch phase

h harmonic number

q overvoltage factor


If U0 = 0

Momentum acceptance

in presence of several RF systems

  • The Touschek module can handle machines with more than one RF systems with different frequencies and/or voltages.

  • The momentum acceptance is computed assuming:

    • • The maximum phase on the separatrix is defined by the reference RF having the lowest frequency and voltage V ≠ 0.

    • • All systems are synchronized with the reference RF.

  • Different RF arrangements are not supported and would require a modification of the module !!

  • Energy loss per turn due to SR

    is computed from I2returned by theTWISS_summtable when the TWISS chrom option is invoked

    • U0 = 0 when I2 is:

      • vanishing,undefined or not computed

    About ss and sp

    The BEAM command computes the synchrotron tune qsfrom the RF cavity parameters.

    Then ssand sp for leptonscome from:

    Et longitudinal emittance

    L machine circumference

    Defining Et < 0 in the BEAM command allows the user to assign the ssand sp values

    For a generic element in the DAFNE lattice

    Using a more accurate routine (CJYDBB) for I0 calculation produces negligible differences in the evaluation of the Piwinski G function, but removing the discontinuity results in a reduction by a factor 2 in the overall execution time.

    Touschek lifetime evaluation in MAD-8

    • T is computed when the BMPM TOUSCH option is invoked.

    • The BMPM module could not handle:

      • combined function dipoles

      • machine elements with TILT ≠ 0

    • It was based on some simplifying assumptions about:

      • beam emittance

      • damping partition numbers

    • Computation formalism see Gygi, Keil, Jowett, BeamParam, CERN/LEP-TH/88-2

    Input Instructions


    RFCAV: rfcavity, l = lcav, volt = 0.09, lag = 0.49, harmon = 120,

    tfill = 1., shunt = 0.2, pg = 0.1;


    BEAM, PARTICLE = positron, energy = .51,sequence = wholekf ,

    npart = 2.E+10, ex = .4E-6 , ey = 1.1964E-9 ,

    et = -6.E-6 , sige = .0003, sigT = .02, radiate = true;


    use, period = wholekf;

    twiss, chrom, sequence = wholekf, centre, table, file =wholekf0.txt, save;

    touschek, tolerance = 1.e-07, file = name;


    Output File

    • name beamline element

    • s azimuth of the element center

    • tli elementcontribution to 1/T

    • tliw tli*Le/L Le element length, L ring circumference

    • tlitot sum of loss rates tliw through the beamline elements

    • The Touschek lifetime T is:

      • T = 1/tlitot @ the end of the beamline

    Contributions tliw to the Loss Rate

    from the elements of the DAFNE lattice

    Touschek lifetime, emittances etc. vs. RF voltage

    for CLIC (DR)

    beam parameters were computed

    by Maxim and include IBS effect

    Touschek lifetime evaluation

    • DAFNE experiences limitations of the transverse aperture due to the presence of:

      • COLLIMATORS affecting physical aperture

      • WIGGLERS and SEXTUPOLES having non-linear terms affecting dynamic aperture

    • As a consequence the maximum stable relative momentum deviation evaluated from RF acceptance is larger than the real one leading to an optimistic evaluation of the Touschek lifetime by a factor ≈2.

    Further developments

    • Introduce the dynamic momentum Acceptance Ap

    • in the evaluation of the Touschek Lifetime (if smaller than RF acceptance)

      • Since Ap depends on:

        • geometrical aperture

        • tune shift on amplitude from non linear terms

        • tune proximity to resonances

      • it can be evaluated by tracking processes only and then passed as an input parameter to the Touschek module.

    • Introduce the exact evaluation of the equilibrium energy spread, in order to take into account rings with SRFF

    • Add Uo to the input parameters in order to deal with high order modes

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