Improved approximation for the variance of the photon emission process in scintillation counters

A determination of the exact statistical limits in the accuracy of time measurements in scintillation counters is presented for the case where the total number of photons in the emission process is randomized in accordance with Poisson statistics. Simple asymptotic formulae for the first two statistical moments are derived for the transient emission process for the case in which both the total and the emitted number of photons are asymptotically large and their ratio is finite. Comparison with formulae obtained in [R.E. Post, L.I. Schiff, Phys. Rev. 80 (1950) 1113] and later in [M. Kelbert, I. Sazonov, A.G. Wright, Nucl. Instr. and Meth. A 564–1 (2006) 185] shows the formulae derived in the present work describe the moments more accurately and over a wider range of parameters.