Correlation of two functions (in our case discrete, non-continuous functions
arising from analog-to-digital data sampling) is simply a time-averaged
function of products of intensities of the signals subjected to various
temporal shifts (delta t).
The intensities fluctuations are replaced, in practice, by potential fluctuations
(directly proportional to the intensity fluctuations) gotten from a photodiode
(current and voltage amplified), and in the case of autocorrelation, I2=I1.
Cross-correlation, when applied to a physical system like
ours, yields a time-difference dependent intensity expression of the form
where R(i) is the square intensity function given by cross-correlation.
It is common to consider the inverse Fourier transform of the correlation
function; in this case a Lorentzian, which is known as the power spectrum:
In the case of the power spectrum , the decay constant (Gamma) is described
where K is a constant related to the frequency of the scattering light (lambda),
scattering angle (theta) and index of refraction (n), and D is the diffusion
constant unique to the scattering solution.
The diffusion constant D is given by the Stokes-Einstein relation, and is
dependent upon temperature (T), viscosity (eta), and scattering particle
diameter (d). When S is plotted with the respect to frequency, the decay
constant can be obtained, and from there it is a simple process to derive
the diameter of the scattering particle.