Calculation of x-ray wave fields in perfect crystals by applying Riemann's method to solve the Takagi-Taupin equations

American Journal of Physics

Volumn 73, Issue 9, 2005, Pages 860-870

  Palmisano, C ^a   Zosi, G ^a  

^a UNIVERSITY OF TURIN   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 25144483286     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1891172     Document Type: Article

References (36)

1. S. Takagi, "Dynamical theory of diffraction applicable to crystals with any kind of small distortion," Acta Crystallogr. 15, 1311-1312 (1962).
2. D. Taupin, "Théorie dynamique de la diffraction des rayons X par les crystaux déformés," Bull. Soc. Fr. Mineral. Cristallogr. 87, 469-511 (1964).
3. U. Bonse and M. Hart, "An X-ray interferometer," Appl. Phys. Lett. 6, 155-156 (1965).
4. R. Deslattes and A. Henins, "X-ray to visible wavelength ratios," Phys. Rev. Lett. 31, 972-975 (1973).
5. P. Becker, K. Dorenwendt, G. Ebeling, R. Lauer, W. Lucas, R. Probst, H.-J. Rademacher, P. Seyfried, and H. Siegert, "Absolute determination of the (220) lattice spacing in a Silicon crystal," Phys. Rev. Lett. 46, 1540-1543 (1981).
6. G. Basile, A. Bergamin, G. Cavagnero, G. Mana, E. Vittone, and G. Zosi, "Measurement of the Silicon (220) lattice spacing," Phys. Rev. Lett. 72, 3133-3136 (1994).
7. -8. The effort to achieve this goal is known as the Avogadro project, started in 1991 by the Bureau International des Poids et Mesures. The goal of this project is to relate the international definition of the kilogram, at present a Pt-Ir artifact, to an invariant quantity of nature such as, for instance, the Avogadro constant. In this way we could count the number of atoms in a mole of Si, for instance, and therefore a new definition of the unit of mass would be traced back to an atomic mass unit.
8. A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1964), p. 52.
9. W. Zachariasen, Theory of X-ray Diffraction in Crystals (Dover, New York, 1967).
10. A. Authier, Dynamical Theory of X-ray Diffraction, 2nd ed. (IUCR/Oxford U.P., Oxford, 2001).
11. The conditions are sometimes called initial because in the T-T equations the variable z could take the role of time.
12. P. Becker, "History and progress in the accurate determination of the Avogadro constant," Rep. Prog. Phys. 64, 1945-2005 (2001).
13. G. Mana and G. Zosi, "The Avogadro constant," Riv. Nuovo Cimento 18, 1-21 (1995).
14. The change of variables has been carried out using Mathematica, 〈http:// www.wolfram.com)〉.
15. See EPAPS Document No. E-AJPIAS-73-012505 for our Mathematica notebook. A direct link to this document may be found in the online article's HTML reference section. The document may also be reached via the EPAPS homepage (http://www.aip.org/pubservs/epaps.html) or from ftp.aip.org in the directory /epaps/. See the EPAPS homepage for more information.
16. S. Takagi, "Wave optics - A dynamical theory of diffraction for distorted crystals," International Summer School on X-ray Dynamical Theory and Topography, 18-26 August 1975, Limoges, France, unpublished notes.
17. G. Thorkildsen, "Three-beam diffraction in a finite perfect crystal," Acta Crystallogr., Sect. A: Found. Crystallogr. A43, 361-369 (1987).
18. G. Thorkildsen, H. Larsen, and E. Weckert, "Approximate solution of the Takagi-Taupin equations for a semi-infinite crystal in the three-beam Laue-Laue case," Acta Crystallogr., Sect. A: Found. Crystallogr. A57, 389-394 (2001).
19. C. Hammond, The Basics of Crystallography and Diffraction (IUCTC/ Oxford U.P., Oxford, 2004).
20. M. von Laue, Röntgenstrahleninterferenzen (Akademische Verlagsgesellschaft, Frankfurt, a.M., 1960).
21. S. Takagi, "A dynamical theory of diffraction for a distorted crystal," J. Phys. Soc. Jpn. 26, 1239-1253 (1969).
22. See Eq. (1.6.4-5) in Ref. 2.
23. This quantity can depend on the polarization state of the incident wave.
24. See Eq. (40) in Ref. 21.
25. T. Katagawa and N. Kato, "The exact dynamical wave field for a crystal with a constant strain gradient on the basis of the Takagi-Taupin equations," Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. A30, 830-836 (1974).
26. F. Chukowski, K. Gabrielyan, and P. Petrashen, "The dynamical theory of X-ray Bragg diffraction from a crystal with a uniform strain gradient. The Green-Riemann functions," Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. A34, 610-621 (1978).
27. A. Authier, C. Malgrange, and M. Tournaire, "Ètude Théorique de la Propagation des Rayons X dans un Crystal Parfait ou Legèrment Déformé," Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. A24, 126-136 (1968).
28. C. Carvalho and Y. Epelboin, "Simulation of X-ray topographs: A new method to calculate the diffracted field," Acta Crystallogr., Sect. A: Found. Crystallogr. A49, 460-467 (1993).
29. G. Mana, C. Palmisano, and G. Zosi, "Effects of analyzer deformation in scanning x-ray interferometry," Metrologia 41, 238-245 (2004).
30. G. Mana, C. Palmisano, and G. Zosi, "Lattice strain in the measurement of Si lattice parameter by Laue-case double-crystal diffractometry," J. Appl. Crystallogr. 37, 773-777 (2004).
31. J. Kevorkian, Partial Differential Equations (Wadsworth & Brooks/Cole, Pacific Grove, 1990), Sees. 4.4.3 and 1.2.1.
32. A. Authier and D. Simon, "Application de la Théorie Dynamique de S. Takagi au Contraste d'un Défaut Plan en Topographie par Rayons X. I. Faute d'empilement," Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. A24, 517-526 (1968).
33. To see in greater detail how the energy propagates into the crystal for different thickness values, see Ref. 15 for four additional figures.
34. B. Batterman and H. Cole, "Dynamical diffraction of X rays by perfect crystals," Rev. Mod. Phys. 36, 681-717 (1964).
35. The roots of the Bessel functions 70 and 7, are not generally periodic except asymptotically for large z.
36. E. G. Kessler, A. Henins, R. D. Deslattes, L. Nielsen, and M. Arif, "Precision comparison of the lattice parameters of Silicon monocrystals," J. Res. Natl. Inst. Stand. Technol. 99, 1-18 (1994).