INTEGRAL EXPRESSIONS OF THE DERIVATIVES OF THE SMALL-ANGLE SCATTERING CORRELATION-FUNCTION

The integral expressions of the derivatives of the small-angle scattering correlation function are obtained in terms of the parametric equations of the interphase surfaces. For each derivative, its integral expression is the surface average of a loop integral of a vectorial field that is the only quantity depending on the derivative order. The field relevant to the nth derivative is obtained by letting a suitable differential operator act (n-2) times on the field relevant to the second derivative. Both the differential operator and the last vectorial field are given in terms of the interface parametric equations. Starting from these expressions, the limiting values of the derivatives of the correlation function as r → 0 are studied by a procedure similar to the one used by Kirste and Porod [Kolloid-Z. 184, 1 (1962)] and Wu and Schmidt [J. Appl. Crystallogr. 4, 224 (1971)]. In this way, the conjecture put forth by Wu and Schmidt that, when the interfaces are sufficiently smooth, all even-order derivatives of the correlation function are null at the origin is proven to be true. © 1995 American Institute of Physics.