Scattering on triply periodic minimal surfaces - the effect of the topology, Debye-Waller, and molecular form factors

We compute scattering patterns for four triply periodic surfaces (TPS). Three minimal-Schwarz P (Im(3) over bar m), Schwarz D-diamond (Pn(3) over bar m), Schoen G-gyroid (Ia(3) over bar d), and one nodal S1 (Ia(3) over bar d). Simple approximations are adopted to examine the influence of the molecular form factor, and the Debye-Waller factor on the scattering pattern. We find that the Debye-Waller factor has a much smaller influence on the scattering intensities of TPS than on the intensities of the lamellar structure consisting of parallel surfaces. This is caused by an almost spherelike distribution of normal vectors for TPS. We give a simple formula that allows a comparison of the experimental scattering data with the data for the P, D, G mathematical surfaces. Finally, the spectra of the two surfaces G and S1 of the same space group symmetry and different topologies are compared. It is found that in the case of the more complex S1 structure the intensities of the first two peaks are very small. (C) 2000 American Institute of Physics. [S0021-9606(00)50633-3].