EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH

TOUSCHEK: Touschek Lifetime and Scattering Rates

The TOUSCHEK module computes the Touschek lifetime and the scattering rates around a lepton or hadron storage ring, based on the formalism of Piwinski [A. Piwinski, "The Touschek Effect in Strong Focusing Storage Rings," DESY-98-179; see also Piwinski's article on Touschek lifetime in the Handbook of Accelerator Physics and Engineering (A. Chao, M. Tigner, eds.), World Scientific, 1999] .

The syntax of the TOUSCHEK command is: 

TOUSCHEK, TOLERANCE=real, FILE=filename;

TOUSCHEK should be called after fully qualified BEAM command and a TWISS command. One or several cavities with rf voltages should be defined prior to calling TWISS and TOUSCHEK.

[Warning: Calling EMIT between the TWISS and TOUSCHEK commands leads to TOUSCHEK using wrong beam parameters, even if the BEAM command is reiterated.]

The momentum acceptance is taken from the bucket size taking into account the energy loss per turn U0 from synchrotron radiation. The value of U0 is computed from the second synchrotron radiation integral synch_2 in the TWISS summ table (synch_2 is calculated only when the TWISS option 'chrom' is invoked), using Eq. (3.61) in Matt Sands' report SLAC-121, which was generalized to the case of several harmonic rf systems. If synch_2 is zero, not defined or not calculated, zero energy loss is assumed.

In the case of several rf systems with nonzero voltages, it is assumed that the lowest frequency system defines the phase of the outer point on the separatrix when calculating the momentum acceptance, and that all higher-harmonic systems are either in phase or in anti-phase to the lowest frequency system.
(Note: if a storage rings really uses a different rf scheme, one would need to change the acceptance function in the routine cavtousch for that ring.)

The arguments have the following meaning:

Example: 

BEAM, PARTICLE=PROTON, ENERGY=450, NPART=1.15e11, EX=7.82E-9, EY=7.82E-9, ET=5.302e-5;
...
USE,PERIOD=LHCb1;	
...
VRF=400;
...
SELECT, FLAG=TWISS, CLEAR;
TWISS, CHROM, TABLE;
TOUSCHEK, FILE, TOLERANCE=1.e-8;
...

The first command defines the beam parameters. It is essential that the longitudinal emittance is set. The command USE selects the beam line or sequence. The next command assign a value to the cavity rf voltage vrf (example name). The select clear previous assignments to the twiss module, twiss calculates and saves the values of all twiss parameters for all elements in the ring; the touschek command computes the Touschek lifetime and writes it to the file 'touschek' (default name).

The results are stored in the TOUSCHEK table, and can be written to a file (with the default name 'touschek' in the example above), or values can be extracted from the table using the value command as follows 

value, table(touschek,name), table(touschek,s), table(touschek,tli), table(touschek,tliw), table(touschek,tlitot);

where 'name' denotes the name of a beamline element, s the position of the center of the element, tli the instanteneous Touschek loss rate within the element, and tliw the instantaneous rate weighted by the length of the element divided by the circumference (its contribution to the total loss rate), and tlitot the accumulated loss rate adding the rates over all beamline elements through the present position. The value of tlitot at the end of the beamline is the inverse of the Touschek lifetime in units of 1/s.

Also, all results can be printed to a file using the command 

write, table=touschek, file;

The MADX Touschek module was originally developed by Catia Milardi and Frank Zimmermann.

The MAD-X Touschek module was partially rewritten in November 2013 by Ghislain Roy after the discovery of a few bugs in the original code:

The first bug concerned a numerical instability in the computation of the B2 parameter as listed in Eq. 34 in the paper of Piwinski referred above. The initial alogorithm implemented the calculation of square root of the difference between two expressions. It turned out that the numerical values of both expressions could sometimes be very large and nearly equal. Because of limited precision in floating point calculations, the difference could sometimes lead to negative values and the square root returned an undefined value (NaN). The integrator then failed to compute the integral and returned a value of zero, with the printing of a faintly related message that too high accuracy was required for integrator DGAUSS. The algorithm didn't stop there and the end result was that the summation over all elements in the range was wrong and the end results were also wrong.


This bug was eliminated by evaluation of the first expression in equation 34 which calculates directly the B2 factor by taking the square root of the sum of two squares, hence ensuring that an instability of the same kind cannot happen.
Another problem was that in the original algorithm the inverse Touschek lifetime was calculated by taking the average of the twiss parameters at both ends of the element as input. The resulting set of parameters was no longer consistent, resulting also in poor calculation. This has been changed by calculating the inverse Touschek lifetime at specific points, always considering as input the Twiss parameters given by Twiss at a single location. This provides at least very accurate results for the TLI parameters.
The integration over the length of the element is now done in different ways, depending whether the preceding TWISS command calculated the Twiss parameters at the end of the element or at the centre (CENTRE option of TWISS).
In the first case (calculation at the end of the element, CENTRE=false), the inverse Touschek lifetime is calculated at the end of each element. The weighted contribution of element i to the total inverse Touschek lifetime is then given by 
TLIW[i] = 0.5 * (TLI[i] + TLI[i-1]) * L[i] / CIRC

In the second case (calculation at the center of the element, CENTRE=true) the inverse Touschek lifetime is also calculated at the center of each element. The weighted contribution of element i to the total inverse Touschek lifetime is then given by 
TLIW[i] = TLI[i] * L[i] / CIRC

Another bug that was uncovered in the original algorithm was that the vertical dispersion was wrongly taken into account and mostly ignored: DY and DPY were set uniformly to half the initial values for lack of updating in the loop over elements.
The new algorithms have been inserted in MAD-X as of version 5.01.04, a development release dated early december 2013.