Fundamentals and Background Mathematics to Powder Diffraction Physics, Geometry and Mathematics

• Elements of X-ray Diffraction by Cullity
• Mathematical Theory of X-ray Powder Diffractometry by A.J.C. Wilson (Gordon and Breach, New York, 1963) Geometry - not structure factors
• Journal of Scientific Instrumentation, Pike, ER, 1957, Vol 34, pg 355-363
• Alexander, L. E., J. Appl. Phys., P155, 25, 1954
• Easterbrook, J. N., Brit. J. of Appl. Phys., p349, 3, 1952
• Klug and Alexander
• Marquardt, D. W. J. Soc. Ind. Appl. Math., vol 11, pp 431-331 (1963)
• Langford, J. I., Journal of Scientific Instrumentation, Vol 39, pg 515-516, (1962)
• Langford, J. I., Cernik, R. J., Louer, D., The Breadth and Shape of Instrumental Line Profiles in High Resolution Powder Diffraction, J. Appl. Cryst., Vol 24, pg 913-919, (1991)
• Compact Numerical Methods for Computers by Nash, J. C. , Adam Hilger, Bristol and New York (1990).
• "Numerical Recipes" for obtaining algorithms and source code. Web page with entire book on-line at http://www.nr.com and http://cfata2.harvard.edu/nr/

Genetic Algorithms

• Benson M. Kariuki, Scott A. Belmonte, Malcolm I. McMahon, Roy L. Johnson, Kenneth D. M. Harris and Richard J. Nelmes, (Email: s.a.belmonte@dl.ac.uk)
  "A new approach for indexing powder diffraction data based on whole-profile fitting and global optimization using a genetic algorithm"
  J. Synchrotron Radiation, (1999), 6, pp 87-92.
• Kin Yip Tam and Richard G. Compton, (Email: richard.compton@chem.ox.ac.uk) [Web] [CCP14 Mirror]
  "GAMATCH - a genetic algorithm-based program for indexing crystal faces
  "Journal of Applied Crystallography", (1995) 28, 640-645
• Wojciech Paszkowicz, (Email: paszk@ifpan.edu.pl)
  "On the concept of the 'dynamic penalty function' applied in the smooth genetic algorithm of solving powder diffraction patterns", Proc. Conf.
  "X-ray Powder Diffraction Analysis of Real Structure of Matter", Liptovsy Mikulas, Slovakia, Sept. 1995, Bull. Czech Slovak Crystallogr. Assoc. 1995, p. 36 (abstract)
• Wojciech Paszkowicz, (Email: paszk@ifpan.edu.pl)
  "Application of the smooth genetic algorithm for indexing powder patterns - tests for the orthorhombic system",
  Materials Science Forum 228-231 (1996) 689-694, (Proc. EPDIC IV, Chester 1995)
• B. M. Kariuki, H. Serrano-Gonazlez, R. L. Johnson, Kenneth D. M. Harris (Email: k.d.m.harris@bham.ac.uk)
  "The application of a genetic algorithm for solving crystal structures from powder diffraction data",
  Chemical Physics Letters, 280 (1997) 189-195, School of Chemistry, University of Birmingham. (Structure was ortho-thymotic acid)
• Kenneth Shankland (Email: ken@viz.nd.rl.ac.uk), Bill David (Email: wifd@isise.rl.ac.uk). T. Csoka
  "Crystal structure determination from powder diffraction data by the application of a genetic algorithm.
  Zeitschrift fur Kristallographie. 212 (1997) 550-552.
  (Bill David (refer Rietveld Mailing List - 28th January 1998) "we should also point out that the structure of ibuprofen, solved in our Z. Krist paper, is considerably more complex in terms of its intramolecular flexibility than ortho-thymotic acid.")
• Landree, E., Collazo-Davila, C., Marks, L. D.
  Multi-solution genetic algorithm approach to surface structure determination using direct methods.
  Acta Cryst. B53, 916. 1997
• Chang, G., Lweis, M.
  Molecular replacement using genetic algorithms.
  Acta Cryst. D53, 279. 1997
  (Synopsis: A new molecular replacement strategy is introduced which features a continuous transform and a genetic algorithm (GA) for search optimization. This strategy uses a GA to simultaneously search the rotational and translational parameters of a test model while maximizing the correlation coefficient between the observed and calculated diffraction data. )
• Bella, J., Rossman, M. G.
  A general phasing algorithm for multiple MAD and MIR data.
  Acta Cryst. D54, 159. 1998
  (Synopsis: A phasing algorithm is presented for combining multiple wavelength anomalous dispersion (MAD) data from multiple types of anomalous scatterers, either in the same or in different derivative crystals, as well as for combining MAD data with multiple isomorphous replacement data from different derivative crystals.)

Le Bail method - Using Extracting Structure Intensities from Powder Data

• Refer Le Bail Method Page
• Bill David (refer Rietveld Mailing List - 28th January 1998) "fitting to the extracted intensities (with the associated covariance matrix) is mathematically equivalent to fitting to the pattern. Ted Prince showed this a few years ago - (E. Prince in "Structure and Statistics in Crystallography" pp. 95-103 ed. A.J.C. Wilson, Adenine Press (1985))"

Maximum-Likelihood

• ABFFIT - Peak Profiling of Powder Diffraction data using Maximum-Likelihood by Anestis Antoniadis, Jacques Berruyer and Alian Filhol
• Email Contact: Alian Filhol (Alain.Filhol@ill.fr)

• References:
  • "Maximum-Likelihood Methods in Powder Diffraction Refinements"; by Anestis Antoniadis, Jacques Berruyer and Alian Filhol; Acta Cryst. (1990). A46, 692-711
  • Estimation semi-parametrique dans les familles. Doublement poissonneinnes et application aux spectres de diffraction"; (Semi-parametrique estimate in the families. Doubly poissonneinnes and application to the spectra of diffraction) by Anestis Antoniadis, Jacques Berruyer and Alian Filhol; Rev. Statistique Appliquee, 1999, XLVII (3), 57-80
  • "4rd Issue Maximum Likelihood Refinement of Powder Diffraction Patterns; by Anestis Antoniadis, Jacques Berruyer and Alian Filhol; May 1990, 90AN08T. Revised versions of report: 86ANO2T, 87AN20T and 87AN20T(b). Corresponds to version 4.0 of the Program ABFfit. 90AN08T
  • "ABFfit for the Macintosh version 4: A Quick Tour"; by Anestis Antoniadis, Jacques Berruyer and Alian Filhol; Institut Laue Langevin, Departement informatique, SCCT AF-006/93,; 15 Oct 1993.

Maximum Entropy

• Web example of Bayesian/Maximum Entropy methods for powder diffraction Fourier Maps:
  • http://www.isis.rl.ac.uk/isis99/science/dataAnalysis.htm

  • Bayesian Analysis (Pointer to the Oxford University Press book: "Data Analysis: a Bayesian Tutorial"): http://www.isis.rl.ac.uk/dataanalysis/dss.htm#bayesian
  • ISIS Data Analysis & Visualisation: http://www.isis.rl.ac.uk/dataanalysis/
  • PGXTAL (3-D plotting with PGPLOT): http://www.isis.rl.ac.uk/dataanalysis/dsplot/

  • MaxEnt - Maximum Entropy Data Consultants Ltd (MEDC) (commercial company): http://www.maxent.co.uk/
    • Bayesian Data Analysis - Hamlet: http://www.maxent.co.uk/bayes.htm
    • More Examples: http://www.maxent.co.uk/examples.htm
    • Including "Deconvolution of motion blurring": http://www.maxent.co.uk/example_1.htm

Some References (taken from "Maximum Entropy & Powder Diffraction", Lecture Notes, Robert J. Papoular, papou@llb.saclay.cea.fr, Les Houches 2000 - recent Developments in NEutron Scattering - 2 -12 May 2000):

• Christopher J. Gilmore, "Maximum Entropy and Bayesian Statistics in Crystallography: a Review of Practical Applications", Acta Crys. Section A. (1996). A 52, 561-589
  "You should all it 'entropy' and for two reasons: first, the function is already in use in thermodynamics under that name; second, and more importantly, most people don't kow that entropy is, and if you use the word 'entropy' you will win every time!
  von Neumann's advice to Shannon as to what to call the function -"

• S.F. Gull and J. Skilling, "Maximum entropy method in image processing" IEE Proceedings, Vol. 131. Pt. F, No 6, October 1984, pg 646-.

• J. Skilling and R.K. Bryan, "Maximum entropy inmage reconstruction: general algorithm" Mon. Not. R. astr. Soc. 1984) 211, 111-124.

• Alastair Livesey and John Skilling, "Maximum Entropy Theory" Acta Cryst. (1985). A41, 113-122.

• Shintaro Kumazawa, Yoshiki Kubota, Masaki Takata and Masoto Sakata, "MEED: a program package for electron-distribution calculation by maximum-entropy" J. Appl. Cryst. (1993). 26, 453-457

• W.I.F. David, "Extending the power of powder diffraction for structure determination" Nature, Vol 346, 23 August 1990, 731-

• D.S. Sivia and W.I.F. David, "A Bayesian Approach to Extracting Structure-Factor Amplitudes from Powder Diffraction Data" Acta Cryst (1994), A50, 703-714.

• Wei Dong and Christopher J. Gilmore, "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" Acta Cryst (1998), A54, 438-446.

• M. Sakata, R. Mori, S. Kumazawa and M. Takata, "Electron-Density Distribution from X-ray Powder Data by Use of Profile Fits and the Maximum-Entropy Method" J. Appl. Cryst. (1990). 23, 526-534

• Tomohiro Ikeda, Makoto Sakata, Masaki Takata, Brendan j. Kennedy, David J. Cookson and Christopher J. Howard, "Electron and Nuclear Densities of a Pyrochlore, Y2Sn2O7 Studied by the Maximum Entropy Method" Proc. Int. Conf. SRMS-2, Jpn. J. Appl. Phys. Vol. 38 (1999) Suppl. 38-1, pp. 93-96.

• R. J. Papoular and D. E. Cox, "Model-Free Search for Extra-Framework Cation in Zeolites Using Powder Diffraction" Europhysics Letters, 32 (4), pp. 337-342 (1995).

• K. Knorr, F Madler and R. J. Papoular, "Model-free density reconstruction of host/guest compounds from high-resolution diffraction data" Microporous and Mesoporous Materials 21 (1998) 353-363.

• Masaki Takata, Eiji Nishibori, Buntaro Umeda, Makoto Sakata, Etsuji Yamamoto and Hisanori Shinohara, "Structure of Endhedral Dimetallofullerene Sc2@C84" Physical Review Letters, Volume 78, Number 17, 28 April 1997.

• Masaki Takata, Eiji Nishibori, Buntaro Umeda, Makoto Sakata, Etsuji Yamamoto and Hisanori Shinohara, "Structure of Endhedral Dimetallofullerene Sc2@C84" Physical Review Letters, Volume 78, Number 17, 28 April 1997.

• Masaki Takata, Eiji Nishibori, Buntaro Umeda, Makoto Sakata, Etsuji Yamamoto and Hisanori Shinohara, "Conformation by X-ray diffraction of the endohedral nature of the metallofullerene Sc2@C84" Nature, Vol. 377, No. 6544, pp. 46-49, 7th September 1995.

• B. Bagautdinov, J. Luedecke, M. Schneider and Sander van Smaalen, "Disorder in the Crystal Structure of Cs2HgCl4 Studied by the Maximum Entropy Method" Acta Cryst (1998), B54, 626-634.

• Robert E. Dinnebier, Martin Schneider, Sander van Smaalen, Falk Olbrich and Ulrich Behrens, "Disorder determined by high-resoluton powder diffraction: structure of pentamethylcyclopentadienyllithium" Acta Cryst (1999), B55, 35-44.

• K. Burger and W. Prandl, "A new type of constraint in the maximum-entropy method using ambigous phase information from anomalous-scattering powder data" Acta Cryst (1999), A55, 719-728.

• K. Burger, D. Cox, R. Papoular and W. Prandl, "The Application of Resonant Scattering Techniques to Ab Initio Structure Soluiton from Powder Data using SrSO4 as a Test Case" Acta Cryst (1998), 31, 789-797.

• Bo Brummerstedt, Jens Ledet Jensen and Jacob Danielsen, "Errors in Maximum-Entropy Charge-Density Distributions Obtained from Diffraction Data" Acta Cryst (1997), A53, 376-387.

• R. Y. de Vries, W. J. Briels and D. Feil, "Critical Analysis of Non-Nuclear Electron-Density Maxima and the Maximum Entropy Method" Physical Review Letters, Volume 77, Number 9,26th August 1996.

Freely Available Maximum Entropy software for X-ray and Neutron Diffraction

• MEED/MEM/Fousyn/mevius/dsurf - Fortran (webpages in Japanese):
  • http://www.bk.tsukuba.ac.jp/~kumazawa/Software/Software.html
  • MEED Manual in English: http://www.bk.tsukuba.ac.jp/~kumazawa/Software/meedman.html

• MEED, MEEDCAB and MEND - Fortran:
  • http://www.uni-tuebingen.de/uni/pki/maxent/maxent.html
  • [CCP14 UK Web Mirror] | [CCP14 Canadian Web Mirror] | [CCP14 Australian Web Mirror]

Fundamental Parameters for Rietveld, Peak Fitting and Powder Diffraction

Click here [BGMN site | CCP14 Automirror] to see plots on total peak profiles modelled using the fundamental parameters approach, Philips Bragg Bretano (IUCr CPD Round Robin PbSO4); thin film with non-infinite thickness; transmission; and capilliary showing splitting of peaks at low angle due to high linear absorption
(From the BGMN site - J. Bergmann, bergmann@rcs.urz.tu-dresden.de)

Monte Carlo

• Is main part of ESPOIR Structure Solution from Powder Diffraction Software - Armel Le Bail
  • Contact: alb@cristal.org
  • Web SITE
  • Executable for Win95 and Source code provided for other operating systems.
  • Original at http://www.cristal.org/sdpd/espoir/index.html
  • Original at http://fluo.univ-lemans.fr:8001/sdpd/espoir/index.html
  • CCP14 UK Web Mirror
  • CCP14 Australian Web Mirror

• Is incorporated into Fullprof Rietveld for structure solution from powder data - Juan Rodriguez-Carvajal
  • Contact: juan@bali.saclay.cea.fr
  • FTP SITE
  • UNIX, VMS, DOS, Mac, etc
  • Original at ftp://bali.saclay.cea.fr/pub/divers/
  • CCP14 Web Mirror | CCP14 FTP Mirror

• Is main part of Endeavour Structure Solution from Powder Diffraction Software - Holger Putz
  • Contact: info@crystalimpact.com
  • Web SITE
  • Executables for Windows/DOS/Unix
  • Original at http://www.crystalimpact.com/endeavour/
  • CCP14 UK Web Mirror

Scatting Factor Calculations

• Sean Brennan (Email: Bren@slac.stanford.edu) and P.L. Cowan.
  "A suite of programs for calculating x-ray absorption, reflection and diffraction performance for a variety of materials at arbitrary wavelengths"
  Rev. Sci. Instrum., 63, 850 (1992).
  • Source Code for UNIX computers (grab the sb-1.0.tar.Z file)
  • Original at ftp://apollo.apsl.anl.gov/pub/cross-section_codes/
  • CCP14 Web Mirror | CCP14 FTP Mirror
• D. Waasmaier and A Kirfel
  
  Acta Cryst. A51, 416 (1995)
  • Incorporated in the latest Math-lab version of Sean Brennan's Scattering Factor software.

Thermal Motion

• Some references:
  • R. J. Nelmes, "Representational Surfaces for Thermal Motion", Acta Cryst, (1969), A25, 523-526

  • R. J. Nelmes, "Anisotropy in Thermal Motionand in Secondary Extinction", Acta Cryst, (1980), A36, 641-653.
    • Note for Fig 2. and Fig 3.: The description 'peanut shell' for the shape of Fig. 2(c) and 'buttoned cushion' for that of Fig 3(c) are not technical! - and may not convey familiar images to all readers. The purpose is simly to provide an easy way to referring to these two very different forms of the prepresentational surface for r.m.s.d.A (and, later, r.m.s.m.A).

Minimum Variance Counting

• Some references (Thanks to Rob Delhez for this):
  • Wilson, A.J.C. "Statistical Variance of Line-Profile Parameters. Measures of Intensity, Location and Dispersion" (1967) Acta Crystallographica A23, 888-898.

  • There are two corrections:
    • Wilson, A.J.C. (1968) Acta Crystallographica A24, 478.
    • Wilson, A.J.C. (1969) Acta Crystallographica A25, 584.

  • The paper in which the method was applied:
    B.J. Thijsse, "The accuracy of experimental radial distribution functions for metallic glasses", J. Appl. Crystallography, (1984), 61-76.