Estimation Problems in Crystallography

  • Robert Schaback
Part of the The IBM Research Symposia Series book series (IRSS)


One of the most important problems in crystallography is the determination of the unit cell of a crystalline substance and the relative positions of atoms within that cell. This contribution is mainly concerned with the estimation of unit cell dimensions from data obtained by x-ray diffraction of a polycrystalline substance. Compared to the number of parameters to be estimated and to the desired accuracy, the given information turns out to be rather limited and relatively noisy. Therefore some deeper insight into the underlying problem is necessary. This reveals a nonlinear mixed-integer approximation problem and gives some hints for the numerical solution of the problem. By combination of common approximation methods and combinatorial strategies of branch- and bound-type a numerical estimation method for a high-speed computer can be developed. In practice this method provides the user with a set of estimates which fit into the noise limits of the data, as is shown by a series of test examples. A reduction of the number of possible estimates can be made by introducing additional restrictions involving more crystallographic information. The purpose of this contribution is not to give a purely crystallographical or a purely mathematical description of the topic but to try to arouse some interest for crystallographical questions among mathematicians working with high-speed computers.


Estimation Problem Phase Relationship Phase Problem Phenyl Ester Polycrystalline Substance 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahmed, F.R.(ed.): Crystallographic Computing, Copenhagen: Munks-gaard 1970Google Scholar
  2. 2.
    Ahmed, F.R.(ed.): Crystallographic Computing II, Copenhagen: Munksgaard (to appear)Google Scholar
  3. 3.
    Azaroff, L.V. and Buerger, M.J.: The Powder Method in X-ray Crystallography, New York: Mc Graw-Hill 1958Google Scholar
  4. 4.
    Babenko, N.F. and Brusentsev, F.A.: Indexing X-Ray Powder Patterns for a Crystal of an Arbitrary System, Kristallografiya 19 (1974), 506–510Google Scholar
  5. 5.
    Barabash, I.A. and Davydov, G.V.: On the Indexing of Powder Patterns for Polycrystalline Materials of the Orthorhombic System, Acta Cryst.A 24 (1968), 608CrossRefGoogle Scholar
  6. 6.
    Bradley, C.J. and Cracknell, A.P.: The mathematical theory of symmetry in solids. Representation theory for point groups and space groups. Oxford: Clarendon Press 1972Google Scholar
  7. 7.
    Braess, D.: Die Konstruktion der Tschebyscheff-Approximierenden bei der Anpassung mit Exponentialsummen, J. of Approx.Th. 3 (1970), 261–273CrossRefGoogle Scholar
  8. 8.
    Braess, D.: Nichlineare Anpassung von MeBdaten nach der Methode der kleinsten Quadrate, Schriftenreihe des Rechenzentrums der Universitat Miinster 6(1974)Google Scholar
  9. 9.
    Braess, D.: Nonlinear Approximation in Lorentz, G.G.(ed.): Approximation Theory, London - New York: Academic Press (to appear)Google Scholar
  10. 10.
    Buerger, M.J.: Crystal-Structure Analysis, New York - London: Wiley, Sons 1960Google Scholar
  11. 11.
    Buerger, M.J.: Vector Space and its Application in Crystal-Strue-ture Investigation, New York: Wiley, Sons 1959Google Scholar
  12. 12.
    Busing, W.R., Martin, K.0. and Levy, H.A.: 0RFLS/0RFFE Oak Ridge National Laboratory Reports 0RNL-TM-305/306, 1962/1961.Google Scholar
  13. 13.
    Cochran, W.: Relations between the phases of structure factors, Acta Cryst. 8 (1955), 473–478CrossRefGoogle Scholar
  14. 14.
    Cochran, W. and Woolfson, M.M.: The theory of sign relations between structure factors, Acta Cryst. 8 (1955), 1–12CrossRefGoogle Scholar
  15. 15.
    Coiro, V.M. and Mazza, F.: The Crystal Structure of the Conformational Analysis of Carbobenzoxy - L - lencyl - p - nitro-phenyl Ester, Acta Cryst.B 30 (1974), 2607Google Scholar
  16. 16.
    Cruickshank, D.W.J., Pilling, D.E., Bujosa, A., Lovell, F.M. and Truter, M.R.: Computing Methods and the Phase Problem in X-ray Crystal Analysis, Oxford: Pergamon Press 1961Google Scholar
  17. 17.
    Cullity, B.D.: Elements of X-ray Diffraction, Reading, Mass.: Addison - Wesley 1956Google Scholar
  18. 18.
    Debaerdemaeker, T. and Woolfson, M.M.: On the Application of Phase Relationships to Complex Structures IV: The Coincidence Method Applied to General Phases, Acta Cryst.A 28 (1972), 477CrossRefGoogle Scholar
  19. 19.
    Debaerdemaeker, T. and Woolfson, M.M.: The determination of structure invariants II, Acta Cryst.A 31 (1975), 401CrossRefGoogle Scholar
  20. 20.
    Declerq, J.P., Germain, G., Main, P. and Woolfson, M.M.: On the application of Phase Relationships to Complex Structures V: Finding the solution, Acta Cryst.A 29 (1973), 231CrossRefGoogle Scholar
  21. 21.
    Declerq, J.P. and Woolfson, M.M.: On the Application of Phase Relationships to Complex Structures VIII: An extension to the magic-integer approach, Acta Cryst.A 31 (1975), 367CrossRefGoogle Scholar
  22. 22.
    Gassmann, J.: Least-squares Refinement of Phases in Direct and Reciprocal Spaces, Acta Cryst.A 32 (1976), 274CrossRefGoogle Scholar
  23. 23.
    Gassfnann, J.and Zechmeister, K.: Limits of Phase Expansion in Direct Methods, Acta Cryst.A 28 (1972), 270CrossRefGoogle Scholar
  24. 24.
    Germain, G. and Woolfson, M.M.: Some Ideas on the Deconvolution of the Patterson Function, Acta Cryst.A 21 (1966), 845CrossRefGoogle Scholar
  25. 25.
    Germain, G. and Woolfson, M.M.: On the Application of Phase Relationships to Complex Structures, Acta Cryst.B 24 (1968), 91CrossRefGoogle Scholar
  26. 26.
    Germain, G., Main, P. and Woolfson, M.M.: On the Application of Phase Relationships to Complex Structures II: Getting a Good Start, Acta Cryst.B 26 (1970), 274CrossRefGoogle Scholar
  27. 27.
    Germain, G., Main, P. and Woolfson, M.M.: On the Application of Phase Realtionships to Complex Structures III: The Optimum Use of Phase Relationships, Acta Cryst.A 27 (1971), 368CrossRefGoogle Scholar
  28. 28.
    Haendler, H. and Cooney, W.: Computer Determination of Unit Cell from Powder-Diffraction Data, Acta Cryst. 16 (1963), 1243Google Scholar
  29. 29.
    Hauptman, H. and Karle, J.: The solution of the phase problem I: The centrosymmetrical crystal, Amer.Cryst.Assoc.Monograph No. 3, Ann Arbor: Edwards Brothers 1953Google Scholar
  30. 30.
    Hauptman, H. and Karle, J.: Phase determination from new joint probability distributions: Space group P1, Acta Cryst. 11 (1958), 149–157CrossRefGoogle Scholar
  31. 31.
    Hauptman, H., Fisher, J. and Weeks, C.M.: Phase Determination by Least-Squares Analysis of Structure Invariants: Discussion of This Method as Applied on Two Androstane Derivations, Acta Cryst.B 27 (1971), 1550Google Scholar
  32. 32.
    Hauptman, H.: Crystal structure determination, New York: Plenum Press 1972CrossRefGoogle Scholar
  33. 33.
    Hauptman, H.: A joint probability distribution of seven structure factors, Acta Cryst.A 31 (1975), 671CrossRefGoogle Scholar
  34. 34.
    Hauptman, H.: A new method in the probabilistic theory of the structure invariants, Acta Cryst.A 31 (1975), 680CrossRefGoogle Scholar
  35. 35.
    Hoppe, W.: Phaseribestimmung durch Quadrierung der Elektronen-dichte im Bereich von 2 A - bis 1, 5 A - Auflosung, Acta Cryst. A 15 (1962), 13Google Scholar
  36. 36.
    Hoppe, W. and Gassmann, J.: Phase Correction, a New Method to Solve Partially Known Structures, Acta Cryst.B 24 (1968), 97CrossRefGoogle Scholar
  37. 37.
    Ishida, T. and Watanabe, Y.: Analysis of Powder Diffraction Patterns of Monoclinic and Triclinic Crystals, J.Appl.Cryst. 4 (1971), 311CrossRefGoogle Scholar
  38. 38.
    Ito, T.: X-ray Studies on Polymorphism, Tokyo: Maruzen 1950Google Scholar
  39. 39.
    Jeffery, J.W.: Methods in X-ray Crystallography, London - New York: Academic Press 1971Google Scholar
  40. 40.
    Karle, J. and Hauptmann, H.: Phase determination from new joint probability distributions: Space group P1, Acta Cryst. 11 (1958), 264–269CrossRefGoogle Scholar
  41. 41.
    Karle, J. and Karle, I.L.: The Symbolic Addition Procedure for Phase Determination for Centrosymmetric and Noneentrosymmetrie crystals, Acta Cryst. 21 (1966), 8149Google Scholar
  42. 42.
    Kleber, W.: Einfuhrung in die Kristallographie, lO.Auflage, Berlin: VEB Verlag Technik 1969Google Scholar
  43. 43.
    Koch, M.H.J.: On the Application of Phase Relationships to Complex Structures VI: Automatic Interpretation of Electron-Den-sity Maps for Organic Structures, Acta Cryst.B 30(197*0, 67Google Scholar
  44. 44.
    Konnert, J.H.: A Restrained-Parameter Structure-Factor Least- Squares Refinement Procedure for Large Asymmetric Units, Acta Cryst.A 32 (1976), 614CrossRefGoogle Scholar
  45. 45.
    Larisch, E.: Algorithmus zur Indizierung von Rontgen-Pulverauf-nahmen, Diplomarbeit Munster 1975Google Scholar
  46. 46.
    Larisch, E.: Private communication, 1976Google Scholar
  47. 47.
    Lenstra, A.T.H. and Schoone, J.C.: An Automatic Deconvolution of the Patterson Synthesis by Means of a Modified Vector-Verifi-cation Method, Acta Cryst.A 29 (1973), 419CrossRefGoogle Scholar
  48. 48.
    Lessinger, L.: On the Application of Phase Relationships to Complex Structures IX, MULTAN Failures, Acta Cryst.A 32 (1976), 538CrossRefGoogle Scholar
  49. 49.
    Louer, D. and Louer, M.: Methode d1 Essais et Erreurs pour lf Indexation Automatique des Diagrammes de Poudre, J.Appl.Cryst. 5 (1972), 271CrossRefGoogle Scholar
  50. 50.
    Lynch, M.F., Harrison, J.M., Town, W.G. and Ash, J.E.: Computer Handling of Chemical Structure Information, London: Mac Donald, New York: American Elsevier 1971Google Scholar
  51. 51.
    Main, P., Woolfson, M.M. and Germain, G.: MULTAN: A Computer Program for the Automatic Solution of Crystal Structures, Univ. of York 1971Google Scholar
  52. 52.
    Main, P., Woolfson, M.M., Lessinger, L., Germain, G. and Declerq, J.P.: MULTAN 74: A system of Computer Programmes for the Automatic Solution of Crystal Structures from X-Ray Diffraction Data 1974Google Scholar
  53. 53.
    Pawley, G.S.: Advances in Structure Research by Diffraction Methods, New York: Pergamon PressGoogle Scholar
  54. 54.
    Rae, A.D.: The Phase Problem and its Implications in the Least- Squares Refinement of Crystal Structures, Acta Cryst.A 30 (1974), 761CrossRefGoogle Scholar
  55. 55.
    Rollett, J.S.(ed.): Computing Methods in Crystallography, Oxford: Pergamon Press 1965Google Scholar
  56. 56.
    Runge, C.: Die Bestimmung eines Kristallsystems durch Rontgen-strahlen, Physik.Z. 18 (1917), 509–515Google Scholar
  57. 57.
    Sayre, D.: The Squaring Method: A New Method for Phase Determination, Acta Cryst. 5 (1952), 60CrossRefGoogle Scholar
  58. 58.
    Sayre, D.: The double Patterson -function, Acta Cryst. 6 (1953), 430CrossRefGoogle Scholar
  59. 59.
    Sayre, D.: On Least-Squares Refinement of the Phases of Crystal-lographic Structure Factors, Acta Cryst.A 28 (1972), 210CrossRefGoogle Scholar
  60. 60.
    Sayre, D.: Least-Squares-Refinement II: High-Resolution Phasing of a Small Protein, Acta Cryst.A 30 (1974), 180CrossRefGoogle Scholar
  61. 61.
    Schaback, R.: Ein Optimierungsproblem aus der Kristallographie, in Collatz, L. und Wetterling, W.(ed.): Numerische Methoden bei Optimierungsaufgaben, ISNM 23, Basel–Stuttgart: Birkhauser 1974, 113–123CrossRefGoogle Scholar
  62. 62.
    Stewart, J.M., Kundell, F.A. and Baldwin, J.C.: The X-RAY 70 system, Computer Science Center, Univ.of Maryland, College Park, Maryland 1970Google Scholar
  63. 63.
    Stewart, J.M., Kruger, G. J., Amnion, H.L., Dickinson, C. and Hall, S.R.: X-RAY system, Report TR-192, Computer Science Center, Univ.of Maryland, College Park, Maryland 1972Google Scholar
  64. 64.
    Vand, V. and Johnson, G.G.: Indexing of X-Ray Powder Patterns, Part I. The Theory of the Triclinic Case, Acta Cryst.A 24 (1968), 543CrossRefGoogle Scholar
  65. 65.
    Viswanathan, K.: A Systematic Approach to Indexing Powder Patterns of Lower Symmetry Using De Wolff’s Principles, The American Mineralogist 53 (1968), 2047Google Scholar
  66. 66.
    Visser, J.W.: A Fully Automatic Program for Finding the Unit Cell from Powder Data, J.Appl.Cryst. 2 (1969), 89–95CrossRefGoogle Scholar
  67. 67.
    Viterbo, D. and Woolfson, M.M.: The Determination of Structure Invariants I. Quadrupoles and Their Uses, Acta Cryst.A 29 (1973), 205CrossRefGoogle Scholar
  68. 68.
    White, P.S. and Woolfson, M.M.: The Application of Phase Relationships to Complex Structures VII: Magic integers, Acta Cryst.A 31 (1975), 53CrossRefGoogle Scholar
  69. 69.
    Wilson, A.J.C.: Mathematical Theory of X-Ray Powder Diffracto-metry, Eindhoven: Philips Technical Library 1963Google Scholar
  70. 70.
    de Wolff, P.M.: On the Determination of Unit-cell Dimensions from Powder Diffraction Patterns, Acta Cryst. 10 (1957), 590–595CrossRefGoogle Scholar
  71. 71.
    de Wolff, P.M.: Indexing of Powder Diffraction Patterns, Advances in X-ray Analysis 6, 1–17Google Scholar
  72. 72.
    Woolfson, M.M.: Direct methods in Crystallography, Oxford: Per-gamon Press 1961Google Scholar
  73. 73.
    Zachariasen, W.H.: A new analytical method for solving complex crystal structures, Acta Cryst. 5 (1952), 68–73CrossRefGoogle Scholar
  74. 74.
    Zachariasen, W.H.: Interpretation of Monoclinic X-ray Diffraction Patterns, Acta Cryst. 16 (1963), 784CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1977

Authors and Affiliations

  • Robert Schaback
    • 1
  1. 1.Lehrstühle für Numerische und Angewandte MathematikGöttingenGermany

Personalised recommendations