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D LS T h e o r y
The goal of the ILT technique is to solve Equation 14. The quantity which is known is the lefthand side, the measured autocorrelation function C(t’) (where t’ is given by the discrete
channels t’ = Δt’, 2Δt’, …, 64 Δt’). The unknowns are the M individual weighting coefficients fi,
buried in the summation on the right hand side of Equation 14. This multi-variable equation must
therefore be “inverted” in order to yield the “answer”, which is the set of weighting coefficients fit
Equation 14 is, in fact, the discrete representation of a more general integral equation that
defines the Laplace transform. To appreciate this, we can define a new “reduced”
autocorrelation function H(t’), obtained from the original C(t’): H(t’) = {(C(t’)-B)/A}1/2 .
We also define a new variable, s, which is simply proportional to the diffusion coefficient
variable D : s = DK2 (with K given by Equation 9). Using these new definitions we can recast
Equation 14 in its integral form, in which there are an infinite number of different particle
diameters. Equation 14 therefore becomes,
∞
H (t ′) = ∫ f ( s ) exp(− st ′)ds
(15)
0
where s (or D) ranges from zero (corresponding to an arbitrarily large particle) to infinity
(corresponding to a particle of diminishing small diameter). The above expression defines the
integral Laplace transform of H(t’), which is the desired weighting function f(s).
Returning to Equation 14. we must use an ILT technique to obtain the best possible estimates
for the weighting coefficients fi. The details of the mathematical procedure used to perform the
discrete Laplace transform inversion of the autocorrelation data in the NICOMP Distribution
Analysis are beyond the scope of this manual -- and, in any case, are proprietary. Suffice to say
that the procedure represents a variation on other ILT methods discussed in the light scattering
literature -- most especially, that developed by S. Provencher. The ways in which the NICOMP
procedure differs from this and other methods relate mainly to the manner and extent of
smoothing carried out within the mathematical “algorithm”.
Nicomp 380 Manual
PSS-380Nicomp-030806
06/06
Page 2 - 46