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# Determination of Angles and Momentum (inverse map)

## Ion-optic coordinates in the S800

The optical coordinates used in the S800 are described in relation to a central trajectory passing through the center of the S800 magnets, and with the reference momentum given by *p*_{0}=qBρ_{0}, where *ρ*_{0} is the central bend radius for the dipole, *B* is the dipole field and *q* is the ionic charge. These are the longitudinal distance *z* along the reference path, and the transversal distances *x* and *y* (perpendicular to z) with respect to the reference trajectory in the dispersive and non-dispersive directions, respectively. (Note that since the S800 bending dipoles are oriented vertically, the coordinate *x* corresponds to the vertical direction, while *y* refers to the horizontal direction.) The dispersive and non-dispersive angles *a* and *b* are referred with respect to the *z* axis in the *xz*-plane and *yz*-plane, respectively. The momentum coordinate *δ* is defined according to the equation *δ + 1 = p / p*_{0}, where *p* is the momentum of the particle. The energy coordinate *d* is defined as *d= (E-E*_{0}) / E_{0}, being *E* the energy of the particle and *E*_{0} the energy equivalent to *p*_{0}.

## Reconstruction method

With the exception of the sextupole in front of the S800 spectrograph and the trim coils, the aberrations—introduced mostly by the fringe fields of the magnets—are calculated and corrected analytically based on measured field maps. This method ^{1)}) uses the ion optics code COSY Infinity to calculate the transfer matrix *S* from the target to the focal plane, including aberrations up to 5^{th} order. The precision attained with the 5^{th} order is comparable to the resolution of the position-sensitive detectors in the focal plane. The tracking algorithm inverts the matrix *S* before applying it to the coordinates measured at the focal plane on an event-by-event fashion, according to the equation *(a*_{t} y_{t} b_{t} d_{t}) = S^{-1} (x_{f} a_{f} y_{f} b_{f}). The subindexes *t* and *f* refer to the positions behind the target and at the focal plane, respectively. The positions measured with the two CRDCs are used to determine the position *x*_{f} and *y*_{f}, and angles *a*_{f} and *b*_{f} at the focal plane. The main advantage of this method is the fast processing of the data as it avoids the tracking of each individual particle in the magnetic fields of the spectrograph. The standard S800 SpecTcl analysis software provides all necessary functions and interfaces to perform these calculations.

## Calculation of the inverse map

Since the energy coordinate *d* is one of the quantities deduced from the inverse map, the beam position *x*_{t} cannot be calculated and is assumed to be zero. This assumption implies that the final resolution is obtained by folding the finite size of the beam spot in that direction with the size obtained from the reconstruction, itself depending on the detector resolution and the order to which the calculation is performed. Note that this is still the case when the S800 is operated in dispersion-matching mode, where the incoming beam is momentum dispersed—and therefore rather large—on the target. In that case it is the size at the object location that matters since the entire S800 (analysis line + spectrograph) is achromatic. Because the shapes of the fringe fields vary significantly with the absolute strength of the magnets, inverse maps need to be calculated for each magnetic-rigidity setting of the spectrograph. The inverse transfer maps can be calculated online from the currents of the S800 dipoles and quadrupole doublet, magnetic rigidity, and the particle *(A,Z)* are necessary. The process takes about 20 seconds.