Incoherent pion photoproduction on ${}^{12}$C

We present the first detailed measurement of incoherent photoproduction of neutral pions to a discrete state of a residual nucleus. The ${}^{12}$C$(\gamma ,{\pi }^0{)}^{12}$C${}_{4.4MeV}^{\ast }$ reaction has been studied with the Glasgow photon tagger at MAMI employing a new technique which uses the large solid angle Crystal Ball detector both as a ${\pi }^0$ spectrometer and to detect decay photons from the excited residual nucleus. The technique has potential applications to a broad range of future nuclear measurements with the Crystal Ball and similar detector systems elsewhere. The data are sensitive to the propagation of the $\mathrm{\Delta }$ in the nuclear medium and will give the first information on matter transition form factors from measurements with an electromagnetic probe. The incoherent cross sections are compared to two theoretical predictions including a $\mathrm{\Delta }$-hole model.