Statistical analysis of dynamic light scattering data: Revisiting and beyond the Schätzel formulas

We revisited the classical Schätzel formulas (K. Schätzel, Quantum Optics, 2, 1990) of the variance and covariance matrix associated to the normalized auto-correlation function in a Dynamic Light Scattering experiment when the sample is characterized by a single exponential decay function. Although thoroughly discussed by Schätzel who also outlined a correcting procedure, such formulas do not include explicitly the effects of triangular averaging that arise when the sampling time ∆t is comparable or larger than the correlation time τc. If these effects are not taken into account, such formulas might be highly inaccurate. In this work we have solved this problem and worked out two exact analytical expressions that generalize the Schätzel formulas for any value of the ratio ∆t/τc. By the use of extensive computer simulations we tested the correctness of the new formulas and showed that the variance formula can be exploited also in the case of fairly broad bell-shaped polydisperse samples (polydispersities up to ∼ 50−100%) and in connection with single exponential decay cross-correlation functions, provided that the average count rate is computed as the geometrical mean of the average count rates of the two channels. Finally, when tested on calibrated polystyrene particles, the new variance formula is able to reproduce quite accurately the error bars obtained by averaging the experimental data.