What is a Correlation Function (Correlogram)?

Correlation is a statistical method for measuring the degree of non-randomness in an apparently random data set. When applied to a time dependent intensity trace, as measured with a dynamic light scattering instrument, the correlation coefficients, G(τ), are calculated as shown below, where τ is the delay time.

160718FAQcorrelogram1 1  

For direct application, the correlation equation can be expressed as the summation shown below and detailed in Figure 1.



Figure 1: Schematic detailing the summations used to generate the correlation coefficients from an intensity trace in a dynamic light scattering measurement.

Typically, the correlation coefficients are normalized, such that G(∞) = 1. For monochromatic laser light, this normalization imposes an upper correlation curve limit of 2 for G(t0) and a lower baseline limit of 1 for G(∞). In practice, the theoretical upper limit can only be achieved for carefully optimized optical systems. Typical experimental upper limits are around 1.8 to 1.9. The baseline value of 1 is subtracted from the various correlation coefficient values such that the Y-axis is scaled from 0 to 1.

In dynamic light scattering instrumentation, the correlation summations are performed using an integrated digital correlator. Examples of correlation curves measured for two sub-micron particles (20 and 220nm polystyrene latex) are given in Figure 2. For the smaller and hence faster diffusing particle (60nm polystyrene latex), the measured correlation curve has decayed to baseline within 200μs, while the larger and slower diffusing particle (220nm polystyrene latex) requires nearly 2000μs before correlation in the signal is completely lost.


Figure 2: Example correlation functions for 20nm and 220nm polystrene latex measured with a Malvern Panalytical Zetasizer Nano ZSP.