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determination_of_angles_and_momentum [2013/12/24 09:14] pereira [Ionoptic coordinates in the S800] 
determination_of_angles_and_momentum [2020/02/14 10:51] (current) pereira [Calculation of the inverse map] 
 
===== Reconstruction method =====  ===== Reconstruction method ===== 
With the exception of the [[Magnets#Spectrograph Sextupolesextupole]] 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 ((M. Berz, K. Joh, J. A. Nolen, B. M. Sherrill, and A. F. Zeller, Phys. Rev. C 47, 537 (1993))) uses the ion optics code COSY Infinity ((Ref)) to calculate the transfer matrix //S// from the target to the focal plane, including aberrations up to 5<sup>th</sup> order. The precision attained with the 5<sup>th</sup> order is comparable to the resolution of the positionsensitive 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 eventbyevent fashion, according to the equation //(a<sub>t</sub> y<sub>t</sub> b<sub>t</sub> d<sub>t</sub>) = S<sup>1</sup> (x<sub>f</sub> a<sub>f</sub> y<sub>f</sub> b<sub>f</sub>)//. The subindexes //t// and //f// refer to the positions behind the target and at the focal plane, respectively. The positions measured with the two [[Detectors#Cathode Readout Drift Chambers (CRDC)CRDCs]] are used to determine the position //x<sub>f</sub>// and //y<sub>f</sub>//, and angles //a<sub>f</sub>// and //b<sub>f</sub>// 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.  With the exception of the [[Magnets#Spectrograph Sextupolesextupole]] 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 ((M. Berz, K. Joh, J. A. Nolen, B. M. Sherrill, and A. F. Zeller, Phys. Rev. C 47, 537 (1993))) uses the ion optics code [[http://www.bt.pa.msu.edu/index_cosy.htmCOSY Infinity]] to calculate the transfer matrix //S// from the target to the focal plane, including aberrations up to 5<sup>th</sup> order. The precision attained with the 5<sup>th</sup> order is comparable to the resolution of the positionsensitive 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 eventbyevent fashion, according to the equation //(a<sub>t</sub> y<sub>t</sub> b<sub>t</sub> d<sub>t</sub>) = S<sup>1</sup> (x<sub>f</sub> a<sub>f</sub> y<sub>f</sub> b<sub>f</sub>)//. The subindexes //t// and //f// refer to the positions behind the target and at the focal plane, respectively. The positions measured with the two [[Detectors#Cathode Readout Drift Chambers (CRDC)CRDCs]] are used to determine the position //x<sub>f</sub>// and //y<sub>f</sub>//, and angles //a<sub>f</sub>// and //b<sub>f</sub>// 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 SpecTclS800 SpecTcl]] analysis software provides all necessary functions and interfaces to perform these calculations. 
 
 
===== Calculation of the inverse map =====  ===== Calculation of the inverse map ===== 
Since the energy coordinate //d// is one of the quantities deduced from the inverse map, the beam position x<sub>t</sub> 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 [[Modes of Operation#Dispersionmatched Modedispersionmatched 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 ([[Introduction#Analysis Lineanalysis line]] + [[Introduction#Spectrographspectrograph]]) 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 magneticrigidity setting of the spectrograph. The inverse transfer maps can be calculated [[http://maps.nscl.msu.edu/~s800maps/online]] from the currents of the S800 [[Magnets#Spectrograph Dipolesdipoles]] and [[Magnets#Spectrograph Quadrupole Doubletquadrupole doublet]], magnetic rigidity, and the particle (A,Z) are necessary. The process takes about 20 seconds.  Since the energy coordinate //d// is one of the quantities deduced from the inverse map, the beam position //x<sub>t</sub>// 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 [[Modes of Operation#Dispersionmatching Modedispersionmatching 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 ([[Introduction#Analysis Lineanalysis line]] + [[Introduction#Spectrographspectrograph]]) 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 magneticrigidity setting of the spectrograph. 
 
 **Instructions on how to calculate the inverse transfer maps can be found [[mapserverhere]]**. 
 