How is the channel number coverted into angle?

Mythen II modules are composed by 1280 pixels, each having width p=0.05 mm, and numbered with j=0,..,1279. Angles are counted counterclockwise from the beam direction. For the m-th module, the angle $\alpha_{jm}$ of its j-th pixel center can be determined using the three geometric parameters $R_m$  [mm], $\Phi_m$  [deg], $D_m$  [mm], as in Fig. . The detector group uses instead the 3 parameters center $c_m$  [ ], offset $o_m$  [deg], conversion $k_m$  [ ]. The law with the 3 geometric parameter is

$$\alpha_{jm}=\Phi_m-{\ensuremath{\left({{\ensuremath{\displaystyle... ...remath{\displaystyle{D_m-pj}}}}{{\ensuremath{\displaystyle{R_m}}}}}}}}\right)}}$$ (5.1)

The corresponding law using DG's parameters is

$$\alpha_{jm}=o_m+{\ensuremath{\left({{\ensuremath{\displaystyle{\f... ...t)}}\arctan{\ensuremath{\left[{{\ensuremath{\left({j-c_m}\right)}}k_m}\right]}}$$ (5.2)

One can convert the two forms by equating separately the term out of the arctan and the argument of arctan for two different values of j. It results

$$c_m = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{D_m}}}}{{\ensuremath{\displaystyle{p}}}}}}};$$ (5.3)

$$k_m = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{p}}}}{{\ensuremath{\displaystyle{R_m}}}}}}};$$ (5.4)

$$o_m = \Phi_m-{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyl... ...\frac{{\ensuremath{\displaystyle{D_m}}}}{{\ensuremath{\displaystyle{R_m}}}}}}}.$$ (5.5)

Conversely,

$$\Phi_m = o_m+{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{180}}}}{{\ensuremath{\displaystyle{\pi}}}}}}}c_mk_m;$$ (5.6)

$$R_m = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{p}}}}{{\ensuremath{\displaystyle{k_m}}}}}}};$$ (5.7)

$$D_m = c_m p.$$ (5.8)