A time-of-flight correction procedure for fast-timing data of recoils with varying implantation positions at a spectrometer focal plane

Fast-timing measurements at the focal plane of a separator can suffer from poor timing resolution. This is due to the variations in time-of-flight (ToF) for photons travelling to a given detector, which arise from the changes in the implantation positions of the recoil nuclei emitting the $\gamma$ rays of interest. In order to minimise these effects on timing measurements, a procedure is presented that improves fast-timing data by performing ToF corrections on an event-by-event basis. This method was used to correct data collected with an array of eight LaBr${}_3$ detectors, which detected $\gamma$ rays from spatially distributed ¹³⁸Gd recoil-implants at the focal plane of the Recoil-Ion-Transport-Unit (RITU) spectrometer. The Generalised Centroid Difference (GCD) method was used to extract a lifetime from data in conjunction with a new procedure to calibrate the time walk. The lifetime of the first 2${}^{+}$ state in ¹³⁸Gd, populated by the decay of the $K^{\pi }=8^{-}$ isomeric state, was measured to be 229(24) ps using the ToF-corrected data, which is consistent within three standard deviations to the literature value. The results together with Monte-Carlo simulations show that the ToF correction procedure reduced the uncertainty in the measured lifetimes by 3 % in the case of the spatially distributed nuclei at the focal plane of RITU. However, $\sim$12 % has been estimated for a similar experiment when using a larger focal plane i.e. the Super-FRS at the FAIR facility.