determination_of_angles_and_momentum

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determination_of_angles_and_momentum [2013/10/18 14:04] pereira |
determination_of_angles_and_momentum [2020/02/14 10:51] (current) pereira [Calculation of the inverse map] |
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===== Ion-optic coordinates in the S800 ===== | ===== 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<sub>0</sub>=qBρ<sub>0</sub>//, where //ρ<sub>0</sub>// 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<sub>0</sub>//, where //p// is the momentum of the particle. The energy coordinate //d// is defined as //d= (E-E<sub>0</sub>) / E<sub>0</sub>//, being //E// the energy of the particle and //E<sub>0</sub>// the energy equivalent to //p<sub>0</sub>//. | + | 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<sub>0</sub>=qBρ<sub>0</sub>//, where //ρ<sub>0</sub>// 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<sub>0</sub>//, where //p// is the momentum of the particle. The energy coordinate //d// is defined as //d= (E-E<sub>0</sub>) / E<sub>0</sub>//, being //E// the energy of the particle and //E<sub>0</sub>// the energy equivalent to //p<sub>0</sub>//. |

===== Reconstruction method ===== | ===== Reconstruction method ===== | ||

- | With the exception of the [[Magnets#Spectrograph Sextupole|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 ((Recontruction 1)) 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 5th order. The matrix is then inverted and applied to the coordinates measured at the focal plane on an event-by-event 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. | + | With the exception of the [[Magnets#Spectrograph Sextupole|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 ((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.htm|COSY 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 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<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|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<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#Dispersion-matching Mode|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 ([[Introduction#Analysis Line|analysis line]] + [[Introduction#Spectrograph|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. | ||

+ | |||

+ | **Instructions on how to calculate the inverse transfer maps can be found [[mapserver|here]]**. | ||

- | The corrections of the aberrations introduced by the 5th order in the transfer matrix are comparable to the resolution of the position-sensitive detectors in the focal plane. The main advantage of this method is to avoid tracking of each individual particle in the magnetic fields of the spectrograph, and therefore a much faster processing of the data. The standard S800 SpecTcl analysis software provides all necessary functions and interfaces to perform these calculations. | ||

- | |||

- | Calculation of the inverse map | ||

- | The inverse map relates the coordinates xf, yf, af, bf measured in the focal plane to the coordinates dt, yt, at, bt at the target position. Since the energy is one of the quantities deduced from the map, the beam position in the dispersive plane at the target 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 (see previous paragraph). Note that this is still the case even when the S800 is operated in dispersion-matched 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. Users can request these inverse maps for which only the magnet strengths, magnetic rigidity, and the particle (A,Z) are necessary. The calculation of the inverse is not automated (yet!) and therefore a latency of a few days is to be expected before receiving the map file. | ||

determination_of_angles_and_momentum.1382119447.txt.gz · Last modified: 2013/10/18 14:04 by pereira