TOUSCHEK TRACKING

Touschek lifetime is approximately¹ given as function of momentum acceptance (MA) and bunch volume integrated over the lattice structure [6,12].

While the RF MA is given by the cavity voltage and almost constant along the lattice, the lattice MA depends on where the scattering event occurred and varies along the lattice. In particular we have to distinguish between

• non-dispersive sections, where a scattered particle will just follow the dispersive orbit, and local MA is determined by the momentum range of closed orbit existence or physical aperture, and
• dispersive sections, where a scattered particle will start an oscillation around the dispersive orbit, and local MA may be determined by dynamic aperture or by mismatch of dynamic to physical aperture.

Usual calculations [13] assume a perfectly linear and chromaticity corrected lattice and obtain the local MA from

$$\delta_{acc}^{L}(s_o)=\min_{i=1\dots N} \left\{ \frac{a_{xi}}{\sqrt{H_o\beta_{xi}}+ \eta_i}\right\}$$ (1)

with $\delta := \Delta p/p_o$, H_o the lattice invariant (dispersion's emittance) at scattering location and $\beta_{xi}, \eta_i, a_{xi}$ horizontal beta function, dispersion and vacuum chamber half width at other lattice locations.

In modern light sources, designed for lowest emittance (at limited circumference), strong sextupoles for correction of large chromaticities generated by the required focusing, introduce significant nonlinearities into the lattice that have to be considered in MA calculations:

• Momentum dependency of linear optics parameters: Calculations on light source lattices optimized for large MA [1] have to consider momentum deviations up to $\pm$10%. Within this wide range the $2\nu_x, 2\nu_y$ chromatic resonance drive terms from the sextupoles' Hamiltonian cause momentum dependent beta beats and second order chromaticities [2].
• Nonlinear variation of closed orbit with momentum, i.e. higher order dispersion as included earlier in calculations at SOLEIL [11].
• Nonlinear betatron motion: Momentum dependent dynamic apertures smaller than the physical apertures or distortions of the transverse eigenfigures and mismatch to the physical apertures as illustrated in Fig. 1 lead to a reduction of local MA.
• Synchrotron oscillation: Due to higher order chromaticity scattered particles walk over wide regions in the tune diagram crossing several betatron resonances. Hence we observe a significant reduction of MA when including synchrotron oscillations compared to fixed-$\delta$ calculations.
• Magnet alignment errors: Touschek lifetime depends on the emittance coupling factor $\kappa~=\varepsilon_y/\varepsilon_x$. Including alignment errors generating nonzero $\kappa$ in a flat lattice is required to predict numbers for average lifetime and its variation for different error distributions (''seeds'').
• Mini gap insertions: The beam halo has larger coupling than the beam core, as observed for example at ESRF [10], since large amplitude particles from scattering suffer more from higher order coupling resonances. For the performance of light sources using undulators with full gap heights as small as 4 mm it is essential to know how this affects the lifetime.

  

Figure 1: Nonlinear betatron motion: A Touschek scattered particle starting to oscillate around the off-momentum orbit would be accepted by the linear separatrix (ellipse) but not by the nonlinear separatrix.

In order to include all these effects from the Touschek lifetime point of view we take a brute force approach by starting particles from the beam core with some momentum deviation, i.e with the 6D initial vector $(x,p_x,y,p_y,\delta,\Delta s)=(0,0,0,0,\pm\delta,0)$ as it will be immediately after a Touschek scattering event [12]. Tracking and binary search for the maximum accepted $\delta$ gives the local MA. The resulting stepwise function of lattice MA $\delta_{acc}^L(s)$ then is entered into the Touschek integral. If misalignments are to be included the calculation is repeated for a number of random seeds. This procedure was implemented into the program TRACY [5].