• charge density;
  • soft X-ray transmission diffraction pattern;
  • two-wavelength inversion;
  • tomography;
  • multiple scattering;
  • multiple scattering inversion problem

A method is described for reconstructing the two-dimensional real-space charge density of an isolated object from measurement of the soft X-ray transmission diffraction pattern when it is affected by strong multiple scattering. The Bloch-wave scattering-matrix approach is used to show that the diffracted amplitude depends only on a simple product of X-ray wavelength and sample thickness (unlike the case of relativistic electron diffraction) under reasonable approximations. The multislice formulation then gives the effect of a small change in wavelength, which involves only single scattering. Dynamical diffraction patterns are recorded at two adjacent wavelengths, phased by iterative methods, transformed to real space and divided to give a single-scattering wavefunction. This can then be used to produce a charge-density map. The extension of the method to tomography is discussed. Consideration is first also given to the possibility that absorption due to the photoelectric effect may be so severe for soft X-rays that multiple elastic scattering becomes so much less probable than photoelectric absorption that it may be neglected entirely. A discussion of signs in soft X-ray, positron and electron multiple-scattering theory is given.