Powder Diffraction: Many materials (often those of major industrial or pharmaceutical importance including zeolites, polymers, pharmaceuticals etc.) cannot be crystallized and exist only as powders. In powder diffraction a three-dimensional diffraction pattern is collapsed onto one-dimension by spherical averaging, and as a result, reflections, which would otherwise be separately measured, overlap. The degree of overlap increases with Bragg angle and unit cell dimensions, thus often reducing the effective resolution of the data set to 1.5Å or less - in may cases this is closer to 2Å. In general, overlaps arise from the accidental coincidence of Bragg angle, but they can also arise systematically as a consequence of space group symmetry. Synchrotron radiation with its high intensity, and a highly collimated incident beam can also be of great assistance in resolving overlaps experimentally, but many samples exhibit an intrinsic line broadening or non accidental overlaps can be present, so there is a need for a systematic approach to this problem that is capable of utilising overlap information in an active way. The use of the ME formalism has been extended to the powder case. In particular, the overlaps are used in the normalisation procedure, and the use of likelihood to test phase hypotheses is extended to use both overlapped and non-overlapped reflections. In addition, the latter can be included in the basis set as constraints for entropy maximisation using the concept of hyperoctant phase permutation, and the overlaps are also used in the final centroid maps, so they play an active, and essential role in the entire phasing procedure.

The structure of the NU-3 zeolite (left) and the electron density maps from the maximum entropy method (right)

Research funded by Eastman-Kodak.

References for Powder Diffraction

  1. 'The maximum entropy method of solving crystal structures from electron diffraction data.' C.J. Gilmore in Electron Crystallography: Novel approaches for structure determination of nanosized materials. Weirich, Labar & Zou (eds) Chapter C6, (2005).
  2. 'Electron crystallography of Zeolites - the MWW Family as a Test of Direct 3-D Structure Determination' D.L. Dorset, W.J. Roth & C.J. Gilmore. Acta Cryst. (2005). A38. 516-527.
  3. 'Comparison of Electron Diffraction data from Non-linear Optically Active Organic DMABC Crystals Obtained at 100kV and 300kV' I.G.Voigt-Martin, H.Kothe, A.V.Tenkovtsev, H.Zandbergen, J.Jansen and C.J.Gilmore Ultramicroscopy (2000), 83, 33-59.
  4. 'The maximum entropy approach' C.J.Gilmore, K.Shankland & W. Dong, (2001), In press.
  5. 'The Ab Initio Solution of Structures from Powder Diffraction Data: the use of Maximum Entropy and Likelihood to Determine the Relative Amplitudes of Overlapped Reflections Using the Pseudophase Concept' W.Dong & C.J.Gilmore. Acta Cryst. (1998), A54, 438-446.
  6. 'Ab initio Determination of Molecular Crystal Structures Using Powder Diffraction Data from a Laboratory X-ray Source', P.Lightfoot, M.J.Tremayne, K.D.M.Harris, C.Gildewell, K.Shankland C.J. Gilmore, & P.G.Bruce. (1993), Material Science Forum, 133-136, 207-212.
  7. 'The ab initio Determination of Crystal Structures from their Powder Diffraction Patterns Using a Combination of Entropy Maximisation and Likelihood Ranking', K.Shankland & C.J. Gilmore, (1993), Material Science Forum, 133-136, 189-194.
  8. 'Ab initio Structure Determination of LiCF3SO3 from X-ray Powder Diffraction Using Entropy Maximisation and Likelihood Ranking', M.Tremayne, P.Lightfoot, M.A.Mehta, P.G.Bruce, K.D.M.Harris, K.Shankland, C.J.Gilmore, & G.Bricogne, J.Solid State Chem. (1992), 100, 191-196.
  9. 'Application of the Combined Maximum Entropy and Likelihood Method to the ab initio Determination of an Organic Crystal Structure from X-ray Powder Diffraction Data', M.Tremayne, P.Lightfoot, C.Glidewell, K.D.M.Harris, K.Shankland, C.J.Gilmore, G.Bricogne & P.G.Bruce, J.Mater. Chem. (1992), 2, 1301-1302.
  10. 'A Multisolution Method of Phase Determination by Combined Maximisation of Entropy and Likelihood. IV The Ab-initio Solution of Crystal Structures from X-ray Powder Diffraction', C.J. Gilmore, K.Henderson & G.Bricogne, Acta Cryst. (1991), A47, 830-841.
  11. 'The Challenge of X-ray and Neutron Powder Diffraction', C.J.Gilmore & K.Henderson in Bayesian Methods and Maximum Entropy Ed. J.Skilling, Kluwer, (1989), 233-239.