DIAMETER DISTRIBUTION OF SPHERICAL PRIMARY GRAINS IN THE BOOLEN MODEL FROM SMALL-ANGLE-SCATTERING

W.Gille

Martin -Luther- Universitat Halle- Wittenberg, FB Physik, Hoher Weg 7, D -06099 Halle, Germany

A method is presented, which determines the size- distribution density V(D) of small spheres with random diameter D which are arranged, without any interaction between them, at random positions in space and form an isotropic two -phase system. The stereological information is obtained from the angular intensity distribution (Small- Angle Scattering experiment) (SAS) of the particle system concerned. The result still includes, in a first representation, the volume fraction p as a free parameter. In a second step, a method for the determination of p from the set covariance of the random closed set (RACS) is derived.

The basic methods of interpreting the method of SAS in structure research using the correlation function (c.f. $\gamma$(r) are explained in detail in the book by Guinier and Fournet [1]. The c.f. follows from the scattering experiment.

  1. Description of particle systems by the Boolean model Sometimes it is a good idea to approximate a structure by a so -called random set with the following property: Its intersections with a bounded set contain a finite number of distinct points (nucleation points in crystal growth, for example). Here the sphere is often used as a so -called primary particle (p .p .) at a "grain- position" with density $\lambda$. These positions are called primary points and are Poisson- distributed in space. Therefore the p .p ., of course, overlap each other irregularly in space [2,3]. Let be $\gamma_{o}$(r,D) the c.f. of the p .p .(see [1]), which are spheres with a mean volume $\mu$. L is the maximum chord length of all the p .p.
  2. Determination of V(D) and p Starting from an equation, which is based on considerations in [3],

\begin{displaymath} T(r) \equiv \frac{1}{\lambda\mu}ln\lbrack \frac{\gamma(r)p}{1-p} + 1 \rbrack = \frac{1}{\mu} \int_{r=D}^{L} \frac{1}{6} \pi D^{3}\gamma_{o}(r,D)V(D) dD \end{displaymath}

a solution for the function V(D),(numerical aspects are discussed in [4]),

\begin{displaymath} V(D) = - \frac{2\mu}{\pi} \lbrack \frac{T^{"}(D)}{D} \rbrack^{'} \end{displaymath}

is obtained, which still contains p. From the c.f. $\gamma$(r) of the particle system p follows in the form \begin{displaymath} p = \frac{\gamma^{"}(0)}{[\gamma^{'}(0)]^{2}} \end{displaymath}

which contains the first and second derivative of the c. f. This relation generally holds true for p .p. without surface singularities (edges, comers, tips, points). In the special case p$\rightarrow$ 0 all relation collapse to the- well known result for the infinitely diluted system [3].

1. A.Guinier, G.Fournet: Small-Angle Scattering of X-Rays, John Wiley, New York 1955
2. J.P.Serra: Image Analysis and Math. Morphology, Academic Press, London 1982
3. D.Stoyan, W.S.Kendall, J.Mecke: Stochastic Geometry, Akad. Verl. Berlin 1987
4. W.Gille: The integrals of SAS, Journal de Phys. IV,3 (1993) 503-506