Phys. Rev. Lett. Phys. Rev. A Phys. Rev. B Phys. Rev. C Phys. Rev. D Phys. Rev. E Phys. Rev. ST AB Phys. Rev. ST AB Rev. Mod. Phys. Phys. Rev. (Series I) Phys. Rev. Volume: Page/Article:

Phys. Rev. B 57, 2740–2758 (1998)

[Issue 5 – 1 February 1998 ]

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Effects of partial coherence on the scattering of x rays by matter

S. K. Sinha

Advanced Photon Source, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439

M. Tolan

Institut für Experimentalphysik der Universität Kiel, Leibnizstraße 19, D-24098 Kiel, Germany

A. Gibaud

Faculté des Sciences, Université du Maine, 72017 Le Mans Cedex, France

We discuss the scattering of x rays by matter using the Huygens-Fresnel method, i.e., in the kinematic regime. We derive expressions for how the mutual coherence function (MCF) of the scattered radiation defined across an exit aperture, arises from the MCF of the incident radiation across the entrance aperture and the electron density distribution of the scatterer, and in particular calculate the intensity measured in a detector placed behind the exit aperture as a function of the nominal wave vector transfer q. We discuss the exact relationship between this intensity function and the usual density-density correlation function of the scatterer, and discuss the relationship between coherence and instrumental resolution effects in various regimes. The Fraunhofer and Fresnel regimes are distinguished and the incoherent and coherent limits are discussed. We illustrate the results with explicit calculations for (a) Bragg reflections from crystals and (b) scattering from surfaces.

©1998 The American Physical Society

URL: http://link.aps.org/abstract/PRB/v57/p2740
DOI: 10.1103/PhysRevB.57.2740
PACS: 61.10.Dp, 42.25.Fx

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References

1. G. Grübel, J. Als-Nielsen, D. Abernathy, G. Vignaud, S. Brauer, G. B. Stephenson, S. G. J. Mochrie, M. Sutton, I. K. Robinson, R. Fleming, R. Pindak, S. Dierker, and J. F. Legrand, ESRF Newsletter 20, 14 (1994).
2. B. Lin, M. L. Schlossman, M. Meron, S. M. Williams, and P. J. Viccaro, Rev. Sci. Instrum. 66, 1 (1995) [ADS][ SPIN][ INSPEC].
3. E. Vlieg, S. A. De Vries, J. Alvarez, and S. Ferrer, J. Synchrotron Radiat. 4, 210 (1997) [ INSPEC].
4. M. Sutton, S. G. J. Mochrie, T. Greytak, S. E. Nagler, L. E. Berman, G. A. Held, and G. B. Stephenson, Nature (London) 352, 608 (1991) [ADS][ INSPEC].
5. Z. H. Cai, W. B. Lai, W. B. Yun, I. McNulty, K. G. Huang, and T. P. Russell, Phys. Rev. Lett. 73, 82 (1994) [SPIRES].
6. S. B. Dierker, R. Pindak, R. M. Fleming, I. K. Robinson, and L. Berman, Phys. Rev. Lett. 75, 449 (1995); ; T. Thurn-Albrecht, W. Steffen, A. Patkowski, G. Meier, E. W. Fischer, G. Grübel, and D. L. Abernathy, 77, 5437 (1996); S. G. J. Mochrie, A. M. Mayes, A. R. Sandy, M. Sutton, S. Brauer, G. B. Stephenson, D. L. Abernathy, and G. Grübel, 78, 1275 (1997).
7. R. A. Shore, B. J. Thompson, and R. E. Whitney, J. Opt. Soc. Am. 56, 733 (1966).
8. J. E. Trebes, K. A. Nugent, S. Mrowka, R. A. London, T. W. Barbee, M. R. Carter, J. A. Koch, B. J. MacGowan, D. L. Matthews, L. B. Da Silva, G. F. Stone, and M. D. Feit, Phys. Rev. Lett. 68, 588 (1992).
9. H. Kaiser, S. A. Werner, and E. A. George, Phys. Rev. Lett. 50, 560 (1983); ; H. Rauch, H. Wölwitsch, H. Kaiser, R. Clothier, and S. A. Werner, Phys. Rev. A 53, 902 (1996).
10. P. N. Pusey, in Photon Correlation Spectroscopy and Velocimetry, edited by H. Z. Cummins and E. R. Pike (Plenum, New York, 1977), p. 45.
11. W. H. de Jeu, J. D. Schindler, and E. A. L. Mol, J. Appl. Crystallogr. 29, 511 (1996) [ INSPEC].
12. A. Gibaud, G. Vignaud, and S. K. Sinha, Acta Crystallogr., Sect. A: Found. Crystallogr. 49, 642 (1993).
13. A. Gibaud, J. Wang, M. Tolan, G. Vignaud, and S. K. Sinha, J. Phys. I 6, 1085 (1996) [CAS][ INSPEC].
14. H. Dosch, Critical Phenomena at Surfaces and Interfaces (Evanescent X-Ray and Neutron Scattering), Vol. 126 of Springer Tracts in Modern Physics (Springer-Verlag, Berlin, 1992).
15. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1993).
16. S. M. Durbin, Acta Crystallogr., Sect. A: Found. Crystallogr. 51, 258 (1995).
17. K. A. Nugent, J. Opt. Soc. Am. A 8, 1574 (1991) [ADS][CAS][ SPIN][ INSPEC].
18. K. A. Nugent and J. E. Trebes, Rev. Sci. Instrum. 63, 2146 (1992) [ADS][CAS][ SPIN].
19. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
20. A. C. Schell, Ph.D. thesis, MIT, Cambridge, MA, 1961.
21. S. R. Andrews and R. A. Cowley, J. Phys. C 18, 6427 (1985) [CAS][ INSPEC].
22. I. K. Robinson, Phys. Rev. B 33, 3830 (1986).
23. B. E. Warren, Phys. Rev. 59, 693 (1941).
24. Here we have to mention that a Schell-model source with xi _x, xi _y=0 is not able to generate a beam (for details see Ref. 19), i.e., it would not radiate at all. However, the limit xi _x, xi _y-->0 is not problematic in our theory.
25. S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, Phys. Rev. B 38, 2297 (1988).
26. M. Tolan, G. König, L. Brügemann, W. Press, F. Brinkop, and J. P. Kotthaus, Europhys. Lett. 20, 223 (1992) [CAS][ INSPEC].
27. M. Tolan, W. Press, F. Brinkop, and J. P. Kotthaus, Phys. Rev. B 51, 2239 (1995).
28. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
29. M. Tolan, D. Bahr, J. Sü beta enbach, W. Press, F. Brinkop, and J. P. Kotthaus, Physica B 198, 55 (1994) [ INSPEC].
30. T. Salditt, H. Rhan, T. H. Metzger, J. Peisl, R. Schuster, and J. P. Kotthaus, Z. Phys. B 96, 227 (1994) [CAS].
31. J. L. Libbert, R. Pindak, S. B. Dierker, and I. K. Robinson, Phys. Rev. B56, 6454 (1997). .

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