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Dependence of Lattice Parameters on Various Angular Measures of Diffractometer Line Profiles*

Published online by Cambridge University Press:  06 March 2019

W. Parrish ,
J. Taylor  and
M. Mack
W. Parrish
Affiliation:
Philips Laboratories Irvington-on-Hudson, New York
J. Taylor
Affiliation:
Philips Laboratories Irvington-on-Hudson, New York
M. Mack
Affiliation:
Philips Laboratories Irvington-on-Hudson, New York
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Abstract

When the accuracy of a lattice parameter determination is carried beyond about 0.01 % it becomes of special importance to consider the equivalence of λ and θ in solving the Bragg equation for d. A reference angle on the observed diffractometer profile must be identified with the corresponding wavelength, of the incident X-ray spectral distribution. Exact identity is not possible because the diffractometer profiles are broadened, distorted asymmetrically, and displaced from their correct positions by amounts dependent on the shape of the incident spectral lines, the angular separation of the Kα1,2 doublet lines, and the specimen, instrumental, and geometrical aberrations innerent in the experimental method. The aberration functions vary with the experimental conditions and are Bragg-angle-dependent, thereby introducing systematic errors which are not eliminated by extrapolation procedures.

Most of the published diffractometer measurements of lattice parameters have used as the reflection angle, 28, of the diffractometer profile the peak P(2θ) or the midpoint of chords at various heights above background M1/2(2θ), M2/3(2θ), etc., of the Kα1 line. The relationship between these various angular measures of the line profile is not constant; P(2θ) may be equai to, greater than, or less than M1/2(2θ), depending on the asymmetry of the line profile. The X-ray wavelengths currently used in diffractometry refer to the peak P(λ) of the spectral distribution. The use of P(λ) with different angular measures of the diffractometer profiles results in a range of d's from which different values of the lattice parameters are calculated. The selection of arbitrary methods of defining 2θ does not take into account the significant aspects of the diffraction process, nor does it facilitate the correction of the data for systematic errors inherent in the experimental measurements.

Type
Research Article
Information
Advances in X-Ray Analysis , Volume 7: Twelfth Annual Conference on Applications of X-ray Analysis, August 7-9, 1963 , 1963 , pp. 66 - 85
Copyright
Copyright © International Centre for Diffraction Data 1963

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Footnotes

*

Sponsored in part by U.S. Air Force Office of Scientific Research under Contract No. AF49(638)-620.

References

1. Parrish, W., “Results of the I.U.Cr. Precision Lattice-Parameter Project,” Acta Cryst. 13: 838, 1960.Google Scholar
2. Ladell, J., Mack, M., Parrish, W., and Taylor, J., “Comparison of Various Measures of Line Position in Powder Diffractometry,” Abstract F-2, American Cristallographic Association, Washington, D.C., January 28, 1960.Google Scholar
3. Donnay, G. and Donnay, J. D. H., “The Symmetry Change in the High-Temperature Alkali-Feldspar Series,” Am. J. Sa. BowenVol. (Pt, 1); 115, 1952.Google Scholar
4. Chayes, F. and MacKenzie, W. S., “Experimental Error in Determining Certain Peak Locations and Distances Between Peaks in X-Ray (Powder) Diffractometer Patterns,” Am, Mineralogist 42: 534, 1957.Google Scholar
5. Tournarie, M., “Methode Generale de Correction des Effets Instrumentaux Appliquée a l'Interprétation des Diagrammes de Rayons X,” Bull. soc. franc, minéral, et criât. 81: 278, 1958.Google Scholar
6. Neflf, H., “Über die Präzisionsbestimmung von Gitterkonstanten mit dem Zählrohr-Interferenz-Goniometer,” Z. angew. Phys. 8: 505, 1956.Google Scholar
7. Smakula, A. and Kalnajs, J., “Precision Determination of Lattice Constants with a Geiger-Counter X-Ray Diffractometer,” Phys. Rev. 99: 1737, 1955.Google Scholar
8. Merrill, J. J. and DuMond, J. W. M., “Precision Measurement of the L X-Ray Spectra of Uranium and Plutonium,” Phys. Rev. 110: 79, 1958.Google Scholar
9. Beu, K. E., “An Evaluation of Geiger Counter X-Ray Techniques for Measuring Stresses in Hardened Steels,” Proc. ASTM 57: 1282, 1957.Google Scholar
10. Adler, R. P. I. and Wagner, C. N. J., “X-Ray Diffraction Study of the Effects of Solutes on the Occurrence of Stacking Faults in the Silver-Base Alloys,” J. Appl. Phys. 33: 3451, 1962.Google Scholar
11. Berthotd, R. and Gerard, V., “Ein neues Verfahren zum Ausmessen letzter Debye-Scherrer- Linien mit dem Zahlrohr,” Z. Metallk. 46: 599, 1955.Google Scholar
12. Pike, E. R. and Wilson, A. J. C., “Counter Diffractometer—The Theory of the Use of Centroids of Diffraction Profiles for High Accuracy in the Measurement of Diffraction Angles,” Brit. J. Appl. Phys. 10: 57, 1959.Google Scholar
13. Ladell, J., Parrish, W., and Taylor, J., “Interpretation of Diffractometer Line Profiles,” Acta Cryst. 12: 561, 1959.Google Scholar
14. Zevin, L. S., Umanskii, M. M., Kheiker, D. M., and Panchenko, Yu. M., “The Question of Diffractometer Methods of Precision Measurements of Unit Cell Parameters,” Soviet Phys. Cryst. 6: 277, 1961.Google Scholar
15. Delf, B. W., “The Practical Determination of Lattice Parameters Using the Centroid Method,” Brit. J. Appl. Phys. 14: 345, 1963.Google Scholar
16. Taylor, J., Made, M., and Parrish, W., “Evaluation of Truncation Methods for Accurate Centroid Lattice Parameter Determination,” Acta Cryst. 14 (in press), 1964.Google Scholar
17. Bearden, J. A. and Shaw, C. H., “Shapes and Wavelengths of K Series Lines of Elements Ti 22 to Ge 32,” Phys. Rev, 48: 18, 1935.Google Scholar
18. DuMond, J. W. M., Cohen, E. R., and McNish, A. G., “X-Ray Wavelength and Crystal-Spacing Units of Measurement in Relation to the Standard Metre,” International Tables for X-Ray Crystallography, Vol. III, The Kynoch Press, Birmingham, England, 1962, p. 41.Google Scholar
19. Parratt, L. G., “Kα Satellite Lines,” Phys. Rev. 50: 1, 1936.Google Scholar
20. Bearden, J. A., Personal communication. October 1960.Google Scholar
21. Brogreri, G., “Relation Between the Width of an X-Ray Line and the Resolving Power of the Double-Crystal Spectrometer,” Phys. Rev. 96: 589, 1954.Google Scholar
22. Rachinger, W. A., “A Correction for the α1α2 Doublet in the Measurement of Widths of X-Ray Diffraction Lines,” J. Sci. Instr. 25: 254, 1948.Google Scholar
23. Keating, D. T., “Elimination of the α1, α2 Doublet in X-Ray Patterns,” Rev. Sci. Instr. 30: 725, 1959.Google Scholar
24. Mack, M., Parrish, W., and Taylor, J., “Methods of Determining Centroid X-Ray Wavelengths : Cu Kα and FeKα,” J. Appl. Phys. 35 (in press), 1964.Google Scholar
25. Ladell, J., Mack, M., Parrish, W., and Taylor, J., “Dispersion, Lorentz and Polarization Effects in the Centroid Method of Precision Lattice Parameter Determination,” Acta Cryst. 12: 567, 1959.Google Scholar
26. Ladell, J., “Interpretation of Diffractometer Line Profiles Distortion Due to the Diffraction Process,” Acta Cryst. 14: 47, 1961.Google Scholar
27. Pike, E. R. and Ladell, J., “The Lorentz Factor m Powder Diffraction,” Acta Cryst, 14: 53, 1961.Google Scholar
28. de Wolff, P. M., “Diffractometer Measurements of Low-Order Reflections,” Acta Cryst. 13: 835, 1960.Google Scholar
29. Wilson, A. J. C., “Some Problems in the Definition of Wavelengths in X-Ray Crystallography,” Z.Krist. 111: 471, 1959.Google Scholar
30. Wilkens, M., “Uber den Einfluss asymmetrischer Verteilungsfunktionen bei Präzisionsgitterkonstantenmessunger,” Z. angew. Phys. 10: 433, 1958.Google Scholar
31. Wilson, A. J. C., Mathematical Theory of X-Ray Powder Diffractometry, Philips Technical Library, Eindhoven, The Netherlands, 1963.Google Scholar
32. Parrish, W., “Advances in X-Ray Diffractometry of Clay Minerals,” Seventh National Conference on Clays and Clay Minerals, Pergamon Press, London, 1959.Google Scholar
33. Parrish, W., Advances in X-Ray Diffractometry and X-Ray Spectrography, Centrex Publishing Company, Eindhoven, The Netherlands, 1962.Google Scholar
34. Parrish, W. and Lowitzsch, K., “Geometry, Alignment and Angular Calibration of X-Ray Diffractometers,” Am. Mineralogist. 44: 765, 1959.Google Scholar
35. Parrish, W., “Geiger, Proportional and Scintillation Counters,” International Tables for X-Ray Crystallography, Vol. III, The Kynoch Press, Birmingham, England, 1962, p. 144.Google Scholar
36. Leher, S., Personal communication, July, 1961.Google Scholar
37. Taylerson, C. O., “Testing Circular Division by Means of Precision Polygons,” The Machinist, (London) 71: 1821, 1947.Google Scholar
38. Haven, C. E. and Strong, A. G., “Assembled Polygon for the Calibration of Angle Blocks”, J. Research Natl. Bur. Standards 50: 45, 1953.Google Scholar
39. Pike, E. R., “Counter Diffractometer—The Effect of Vertical Divergence on the Displacement and Breadth of Powder Diffraction Lines,” J. Sci. Instr. 34: 355, 1957; Addendum, ibid. 36: 52, 1959.Google Scholar