Scattering by inhomogeneous systems with rough internal surfaces: Porous solids and random-field Ising systems

For a two-component inhomogeneous system consisting of compact domains of characteristic size R, I show that if the domain walls are ‘‘rough’’ and their root-mean-square fluctuation w over a distance r obeys a power law w=b(r/a${)}^x$ (a is the lattice constant and x>0), then the geometrical correlation function γ(r) has leading terms proportional to $r^x$ and r for r≪R. In a scattering experiment (neutron, x ray, etc.), the scattered intensity I(q) (or cross section) is proportional to the Fourier transform of γ(r) by the Born approximation and, therefore, has leading terms proportional to $q^{\mathrm{-}(d+x)}$ and $q^{\mathrm{-}(d+1)}$ for wave vector q≫$R^{\mathrm{-}1}$, where d is the dimension of the system. Two possible applications of this result are discussed. (i) In granular porous solids which have a minimum grain size $R_{\min }$, the above result implies that surface roughness can cause I(q) to fall off like 1/$q^{\alpha }$ for q≫$R_{\min {}^{\mathrm{-}1}}$, where α=3+x>3 for d=3. In particular, when x>1, the surface becomes a fractal with dimension D=1+x=α-2, which can be extracted from the scattering data. On the other hand, if the grains are smooth and their size distribution obeys a power law dN(R)/dR∝$R^{\mathrm{-}\beta }$ over a range $R_{\min <\mathrm{R}<R_{\max }}$, where $R_{\max }$ is the maximum grain size and 3<β<4, then the sum of their surfaces forms a fractal with D=β-1, in which case, I(q)∝1/$q^{6\mathrm{-}D}$ for q in the range $R_{\min \ll 1/\mathrm{q}\ll R_{\max }}$. (ii) For random-field Ising systems, I argue that the power-law decays of I(q) in the field-cooled experiments are consistent with the prediction of Grinstein and Ma of x=3-d. Alternatively, the same behavior can also be caused by either nonequilibrium effects or a small value of the nonuniversal length b, i.e., b≪a. .AE