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

Abstract

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 \ensuremath{\gamma}(r) has leading terms proportional to ${r}^{x}$ and r for r\ensuremath{\ll}R. In a scattering experiment (neutron, x ray, etc.), the scattered intensity I(q) (or cross section) is proportional to the Fourier transform of \ensuremath{\gamma}(r) by the Born approximation and, therefore, has leading terms proportional to ${q}^{\mathrm{\ensuremath{-}}(d+x)}$ and ${q}^{\mathrm{\ensuremath{-}}(d+1)}$ for wave vector q\ensuremath{\gg}${R}^{\mathrm{\ensuremath{-}}1}$, 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 ${R}_{\mathrm{min}}$, the above result implies that surface roughness can cause I(q) to fall off like 1/${q}^{\ensuremath{\alpha}}$ for q\ensuremath{\gg}${R}_{\mathrm{min}{}^{\mathrm{\ensuremath{-}}1}}$, where \ensuremath{\alpha}=3+x>3 for d=3. In particular, when x>1, the surface becomes a fractal with dimension D=1+x=\ensuremath{\alpha}-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\ensuremath{\propto}${R}^{\mathrm{\ensuremath{-}}\ensuremath{\beta}}$ over a range ${R}_{\mathrm{min}<\mathrm{R}<{R}_{\mathrm{max}}}$, where ${R}_{\mathrm{max}}$ is the maximum grain size and 3<\ensuremath{\beta}<4, then the sum of their surfaces forms a fractal with D=\ensuremath{\beta}-1, in which case, I(q)\ensuremath{\propto}1/${q}^{6\mathrm{\ensuremath{-}}D}$ for q in the range ${R}_{\mathrm{min}\ensuremath{\ll}1/\mathrm{q}\ensuremath{\ll}{R}_{\mathrm{max}}}$. (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\ensuremath{\ll}a. .AE