Characterization of a tagged $\gamma$-ray beam line at the DA$\mathrm{\Phi }$NE Beam Test Facility

At the core of the AGILE scientific instrument, designed to operate on a satellite, there is the Gamma Ray Imaging Detector (GRID) consisting of a Silicon Tracker (ST), a Cesium Iodide Mini-Calorimeter and an Anti-Coincidence system of plastic scintillator bars. The ST needs an on-ground calibration with a $\gamma$-ray beam to validate the simulation used to calculate the energy response function and the effective area versus the energy and the direction of the $\gamma$ rays. A tagged $\gamma$-ray beam line was designed at the Beam Test Facility (BTF) of the INFN Laboratori Nazionali of Frascati (LNF), based on an electron beam generating $\gamma$ rays through bremsstrahlung in a position-sensitive target. The $\gamma$-ray energy is deduced by difference with the post-bremsstrahlung electron energy \cite{prest}-\cite{hasan}. The electron energy is measured by a spectrometer consisting of a dipole magnet and an array of position sensitive silicon strip detectors, the Photon Tagging System (PTS). The use of the combined BTF-PTS system as tagged photon beam requires understanding the efficiency of $\gamma$-ray tagging, the probability of fake tagging, the energy resolution and the relation of the PTS hit position versus the $\gamma$-ray energy. This paper describes this study comparing data taken during the AGILE calibration occurred in 2005 with simulation.