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Determination of crystallite size and lattice distortions through X-ray diffraction line profile analysis

Vorschriften, Methoden und Bemerkungen

Bestimmung von Kristallitgrößen und Gitterverzerrungen durch Röntgenbeugungs-Linienprofilanalyse

Recipes, methods and comments

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Zusammenfassung

Die aufeinanderfolgenden Schritte der Messung, notwendige Korrekturen und Datenverarbeitung werden erörtert und Alternativen beschrieben. Besonders betont wird die Analyse der Linienprofile mit Hilfe der Fourier-Beschreibung sowie auf Basis der Integral- und Halbwertsbreiten. Die letztere Methode beruht auf der Beschreibung der Linienprofile mit Voigt-Funktionen. Die Bestimmung der Kristallitgröße und der Gitterverzerrung sowie die Einzel-Linien-Methoden werden kommentiert. Ein praktisches Beispiel für den Einfluß nicht-idealer Standard-Linienprofile und unterschiedener Untergrundschätzungen wird für den Fall der Fourier-Entfaltung und anschließender Analyse der strukturellen Linienverbreiterung nach Warren und Averbach gegeben.

In Zukunft ist zu erwarten, daß die Linienprofilanalyse sich zu einer automatisierten Routinemethode entwickelt, da die Bausteine verfügbar sind: billige (Klein)Rechner, Fehlerberechnungen und kommerzielle Rechenprogramme.

Summary

Methods for the determination of crystallite size and lattice strain from X-ray diffraction line broadening are discussed. The subsequent steps of measurement, data correction and evaluation are elucidated; alternatives are indicated. Emphasis is laid on the rigorous analysis of line profiles in terms of Fourier coefficients. For the analysis in terms of integral breadth and full width at half maximum a powerful method exists which adopts a Voigt function for describing the shape of the profiles. Size broadening, strain broadening and single-line methods are commented. A practical example is given of the influence of a non-ideal standard line profile and of different background estimates when a Fourier deconvolution and a Warren-Averbach size-strain analysis are performed.

It is expected that line profile analysis will become an automated routine-like analytical method soon, since the tools are available: non-expensive computers, error calculations and commercially available software.

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Abbreviations

A :

Cosine Fourier coefficient

a :

period on a sin θ scale

a 3 :

lattice constant of orthorhombic cell, perpendicular to reflecting planes

B :

sine Fourier coefficient

〈D 3〉v :

λ/β S cos θ, average crystallite size obtained from integral breadth

d :

a3/l, interplanar spacing of reflecting planes

e (n):

Z(n)/n, average strain in a column between two unit cells, n cells apart

e :

β D/4t an θ, microstrain obtained from integral breadth

F :

A + iB complex Fourier coefficient of profile f

f :

structurally broadened profile

G :

complex Fourier coefficient of profile g

g :

profile due to the instrumental aberrations and the X-ray spectrum used

H :

complex Fourier coefficient of profile h

h :

f * g, profile measure from specimen to be investigated

k :

β c /gb g√π

L :

na 3, distance perpendicular to the reflecting planes

l :

a 3/d, order of reflection

〈N 3〉:

average crystallite size (column length) in the direction perpendicular to the reflecting planes

n :

harmonic number of Fourier coefficient; separation distance in number of unit cells

p :

crystallite size distribution function

r :

parameter used in distribution function, Eq. (24)

s :

2 dl sin θ/〈λ〉-l-distance to reciprocal lattice point in units 1/a 3

t :

parameter used in distribution function, Eq. (24)

u :

parameter used in distribution function, Eq. (24)

2w :

full width at half maximum

Z(n):

difference of displacements of two unit cells, n cells apart

β :

integral breadth (in degrees or radians 2 θ)

θ :

Bragg angle

λ :

wavelength of X-rays used

〈 〉:

denotes an average (centroid)

g, h, f :

Denotes that the parameter refers to the g, h of f profile

S, D :

denotes that the parameter refers to the size or distortion broadening.

c, g :

Denotes that the parameter refers to the Cauchy or Gaussian component of a profile

a, v :

denotes that the parameter refers to an area-weighted or volume-weighted size distribution

References

  1. Adler T, Houska CR (1979) J Appl Phys 50:3282–3287

    Google Scholar 

  2. Baker TW, George JD, Bellamy BA, Causer R (1968) Adv X-Ray Anal 11:359–375

    Google Scholar 

  3. Cheary RW, Grimes NW (1972) J Appl Cryst 5:57–63

    Google Scholar 

  4. Chipman DR (1969) Acta Cryst A25:209–214

    Google Scholar 

  5. Cohen JB, Dölle H, James MR (1980) In: Block S, Hubbard CR (eds) Accuracy in Powder Diffraction (NBS Special Publication 567). National Bureau of Standards, Washington, pp 453–477; see also special issue Härterei-Techn Mitt 31:1-124 (1976)

    Google Scholar 

  6. Croche R, Gatineau L (1977) J Appl Cryst 10:479–485

    Google Scholar 

  7. Delhez R, Keijser ThH de, Mittemeijer EJ (1978) J Phys E. Sci Instrum 11:649–652

    Google Scholar 

  8. Delhez R, Keijser ThH de, Mittemeijer EJ, van der Pers NM (August 1978) In: Bojarski Z, Bold T (eds) Proc Conf on Applied Crystallography, Kozubnik, Poland, Vol 1, pp 323–336. Katowice: Silesian University in Katowice and Institute of Ferrous Metallurgy in Gliwice (1979)

    Google Scholar 

  9. Delhez R, Keijser ThH de, Mittemeijer EJ (1980) In: Block S, Hubbard CR (eds) Accuracy in Powder Diffraction (NBS Special Publication 567), National Bureau of Standards Washington, pp 213–253

    Google Scholar 

  10. Delhez R, Mittemeijer EJ (1975) J Appl Cryst 8:609–611

    Google Scholar 

  11. Delhez R, Mittemeijer EJ (1975) J Appl Cryst 8:612–614

    Google Scholar 

  12. Delhez R, Mittemeijer EJ, Keijser ThH de, Rozendaal HCF (1977) J hys E Sci Instrum 10:784–785

    Google Scholar 

  13. Delhez R, Mittemeijer EJ (1978) J Appl Phys 49:4770–4775

    Google Scholar 

  14. Eastabrook JN, Wilson AJC (1952) Proc Phys Soc London B65:67–75

    Google Scholar 

  15. Ergun S (1968) J Appl Cryst 1:19–23

    Google Scholar 

  16. Ergun S (1976) Carbon 14:139–150

    Google Scholar 

  17. Gangulee A (1974) J Appl Cryst 7:434–439

    Google Scholar 

  18. Göbel H (1979) Adv X-Ray Anal 22:255–265

    Google Scholar 

  19. Göbel H (1981) Adv X-Ray Anal 24:123–138

    Google Scholar 

  20. Hosemann R (1950) Z Phys 128:1–35, 465–492

    Google Scholar 

  21. Hulst HC van de, Reesinck JJM (1947) Astrophys J 106:121–127

    Google Scholar 

  22. International Tables for X-ray Crystallography (1963) Vol III, Macgillarry CH, Rieck GD (eds) The Kynoch Press, Birmingham, pp 194–200

    Google Scholar 

  23. Jennings LD (1980) In: Block S, Hubbard CR (eds) Accuracy in Powder Diffraction (NBS Special Publication 567), National Bureau of Standards, Washington, pp 73–83

    Google Scholar 

  24. Keijser ThH de, Mittemeijer EJ (1980) J Appl Cryst 13:74–77

    Google Scholar 

  25. Keijser ThH de, Langford JI, Mittemeijer EJ, Vogels ABP (1982) J Appl Cryst, in press

  26. Klug HP, Alexander LE (1974) X-Ray Diffraction Procedures, John Wiley, New York

    Google Scholar 

  27. Langford JI (1978) J Appl Cryst 11:10–14

    Google Scholar 

  28. Langford JI (1980) In: Block S, Hubbard CR (eds) Accuracy in Powder Diffraction (NBS Special Publication 567), National Bureau of Standards, Washington, pp 255–269

    Google Scholar 

  29. Langford JI, Wilson AJC (1963) In: Ramachandran GN (ed) Crystallography and Crystal Perfection, Academic Press, London, pp 207–222

    Google Scholar 

  30. Langford JI, Wilson AJC (1978) J Appl Cryst 11:102–113

    Google Scholar 

  31. Mignot J, Rondot D (1975) Acta Met 23:1321–1324

    Google Scholar 

  32. Mignot J, Rondot D (1977) Acta Cryst A33:327–333

    Google Scholar 

  33. Mittemeijer EJ, Delhez R (1980) In: Block S, Hubbard CR (eds) Accuracy in Powder Diffraction (NBS Special Publication 567), National Bureau of Standards, Washington, pp 271–314

    Google Scholar 

  34. Mitra GB, Misra NK (1967) Acta Cryst 22:454–456

    Google Scholar 

  35. Nandi RK, Sen Gupta SP (1978) J Appl Cryst 11:6–9

    Google Scholar 

  36. Opinski AJ, Orehotsky JL, Hoffman CWW (1962) J Appl Phys 33:708–712

    Google Scholar 

  37. Page Y, le Gabe EJ, Calvert LD (1979) J Appl Cryst 12:25–26

    Google Scholar 

  38. Pike ER (1957) J Sci Instrum 34:355–361

    Google Scholar 

  39. Pines BYa, Sirenko AF (1962) Soviet Physics-Cryst 7:15–21

    Google Scholar 

  40. Ramarao P, Anantharaman TR (1965) Trans Ind Inst Metals 18:181–186

    Google Scholar 

  41. Rothman RL, Cohen JB (1969) Adv X-Ray Anal 12:208–235

    Google Scholar 

  42. Rothmann RL, Cohen JB (1971) J Appl Phys 24:971–979

    Google Scholar 

  43. Schwartz LH, Cohen JB (1977) Diffraction from Materials, Academic Press, New York

    Google Scholar 

  44. Smith DK, Barrett CS (1979) Adv X-Ray Anal 22:1–12

    Google Scholar 

  45. Smith RS (1960) IBM J Res Develop 4:205–207

    Google Scholar 

  46. Smith WL (1976) J Appl Cryst 9:139–141

    Google Scholar 

  47. Stokes AR, Wilson AJC (1944) Proc Camb Phil Soc 40:197–198

    Google Scholar 

  48. Stokes AR, Wilson AJC (1944) Proc Phys Soc 56:174–181

    Google Scholar 

  49. Stokes AR (1948) Proc Phys Soc London 61:382–391

    Google Scholar 

  50. Suortti P, Jennings LD (1977) Acta Cryst A33:1012–1027

    Google Scholar 

  51. Suortti P (1980) In: Block S, Hubbard CR (eds) Accuracy in Powder Diffraction (NBS Special Publication 567), National Bureau of Standards, Washington, pp 1–20

    Google Scholar 

  52. Wagner CNJ, Aqua EN (1964) Adv X-Ray Anal 7:46–65

    Google Scholar 

  53. Wagner CNJ (1966) In: Cohen JB, Hilliard JE (eds) Local Atomic Arrangements Studied by X-Ray Diffraction, Addison-Wesley, Reading, Mass, p 219

    Google Scholar 

  54. Warren BE, Averbach BL (1950) J Appl Phys 21:595–599

    Google Scholar 

  55. Warren BE, Averbach BL (1952) J Appl Phys 23:497

    Google Scholar 

  56. Warren BE (1955) Acta Cryst 8:483–486

    Google Scholar 

  57. Warren BE (1969) X-Ray Diffraction, Addison Wesley, Reading, Mass

    Google Scholar 

  58. Warren. BE (1976) Acta Cryst A32:897–901

    Google Scholar 

  59. Wilkens M, Hartmann RJ (1963) Z Metallkde 54:676–682

    Google Scholar 

  60. Wilkens M (1979) J Appl Cryst 12:119–125

    Google Scholar 

  61. Williamson GK, Smallman RE (1954) Acta Cryst 7:574–581

    Google Scholar 

  62. Wilson AJC (1963) Mathematical Theory of X-Ray Powder Diffractometry, Centrex, Eindhoven

    Google Scholar 

  63. Young RA, Gerdes RJ, Wilson AJC (1967) Acta Cryst 22:155–162

    Google Scholar 

Download references

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Authors and Affiliations

  1. Laboratory of Metallurgy, Delft University of Technology, Rotterdamseweg 137, NL-2628, AL Delft, The Netherlands

    R. Delhez, Th. H. de Keijser & E. J. Mittemeijer

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  1. R. Delhez
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  2. Th. H. de Keijser
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  3. E. J. Mittemeijer
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Delhez, R., de Keijser, T.H. & Mittemeijer, E.J. Determination of crystallite size and lattice distortions through X-ray diffraction line profile analysis. Z. Anal. Chem. 312, 1–16 (1982). https://doi.org/10.1007/BF00482725

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