Correlation Function Profile Analysis in Laser Light Scattering. III. An Iterative Procedure

Abstract

In photoelectron correlation function profile analysis, the inversion of the Laplace transform

$$\left| {{g^{\left( l \right)}}\left( \tau \right)} \right| = \int\limits_0^\infty {G\left( \Gamma \right){e^{ - \Gamma \tau }}} d\Gamma $$

(1)

to obtain the normalized linewidth distribution function G(Γ) from the net electric field correlation function g^(l)(τ) is essentially an unresolved ill-conditioned problem, where Γ and τ are, respectively, the characteristic linewidth and the delay time. In practice, g^(l)(τ) contains noise and the integral has upper (b) and lower (a) bounds. Consequently, in order to remove the ill-conditioning, we need to have estimates of both the signal-to-noise ratio and the width, in terms of the support ratio γ(≡ b/a), of the linewidth distribution function. However, asg^(l)(τ) depends upon the delay time range of our experiment, we now encounter a problem whereby our experimental conditions and the results we hope to obtain are interactive. Then, the success of a laser light scattering experiment depends upon (1) a proper choice of experimental conditions, as well as (2) appropriate inversion of the measured g^(l)(τ) to obtain G(Γ). Thirdly, further analysis of G (Γ) is often required to obtain the desired information, such as molecular weight distribution, internal motions, etc. These three requirements are highly interdependent and the experimenter must be aware of the uncertainties introduced at each step. In this article, we propose an iterative procedure that tries to meet the above requirements.

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