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Phys. Rev. B 45, 602–612 (1992)

[Issue 2 – 1 January 1992 ]

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Neutron scattering by rough surfaces at grazing incidence

Roger Pynn

Manuel Lujan, Jr. Neutron Scattering Center, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

There are a number of calculations in the literature of the effect of statistical surface roughness on the specular and diffuse scattering of neutrons and x rays which impinge on surfaces at grazing incidence. These calculations do not agree with one another and in some cases lack internal consistency. In this paper I show that these discrepancies can be resolved within the distorted-wave Born approximation. The result I obtain for the specular reflectivity is the widely used version of Nevot and Croce, while that for the diffuse scattering is the same as a recent calculation by Sinha, Sirota, Garoff, and Stanley. The approximations made by Nevot and Croce are clearly revealed by the distorted-wave Born approximation. A simple numerical algorithm is proposed for the calculation of diffuse scattering from a surface whose roughness is self-affine, and the form of the scattering is calculated for a range of parameters within this model. The calculation of diffusion scattering is extended in this paper to the case of films with rough surfaces. Correlation between roughness at the two film surfaces is considered and shown to be important for the description of neutron reflectivity data obtained with a thin film of titanium deposited on a sapphire substrate.

©1992 The American Physical Society

URL: http://link.aps.org/abstract/PRB/v45/p602
DOI: 10.1103/PhysRevB.45.602
PACS: 61.12.Ex, 68.90.+g, 68.35.Bs

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References

1. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, Norwood, MA, 1987).
2. L. Nevot and P. Croce, Rev. Phys. Appl., 15, 761 (1980) [ INSPEC].
3. P. Croce and L. Prod'homme, Nouvo. Rev. Opt. 7, 121 (1976).
4. P. Croce, Nouvo. Rev. Opt. 8, 127 (1977).
5. Except for values of q_z close to the critical value, the difference between q_z and q^t_z is very small for both x rays and neutrons, because the refractive index only differs from unity by an amount of order 10^-6.
6. S. K. Sinha, E. B. Sirota, S. Garoff and H. B. Stanley, Phys. Rev. B 38, 2297 (1988).
7. A. Steyerl, Z. Phys. 254, 169 (1972).
8. A. Messiah, Mecanique Quantique, Tome II (Dunod, Paris, 1960).
9. Y. Yoneda, Phys. Rev. 131, 2010 (1963); O. J. Guentert, J. Appl. Phys. 30, 1361 (1965); A. N. Nigam, Phys. Rev. 4, 1189 (1965).
10. J. Lekner, Theory of the Reflection of Electromagnetic and Particle Waves (Martinez Nijof, Dordrecht, Holland).
11. W. A. Hamilton and R. Pynn, Physica B 173, 71 (1991) [ INSPEC].
12. M. O. Robbins, D. Andelman and J. -F. Joanny, Phys. Rev. A 43, 4344 (1991).
13. R. F. Voss, in Scaling Phenomena in Disordered Systems, edited by R. Pynn and A. T. Skjeltorp (Plenum, New York, 1985).
14. H. E. Hurst, Trans. Am. Soc. Civ. Eng. 116, 770 (1951).
15. J. Feder, Fractals (Plenum, New York, 1988).
16. Mathematica is a program published by Wolfram Research Inc. See S. Wolfram, Mathematica: A System for Doing Mathematics by Computer (Addison-Wesley, Reading, MA, 1988).
17. P. Thompson, D. E. Cox and J. B. Hastings, J. Appl. Cryst. 20, 79 (1987) [ INSPEC].
18. W. A. Hamilton, G. S. Smith, and R. Pynn (unpublished).
19. Sir Isaac Newton, Opticks (originally published in London, 1704; new edition by Dover, New York, 1952), p. 289 ff.
20. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980).
21. S. Garoff, E. B. Sirota, S. K. Sinha and H. B. Stanley, J. Chem. Phys. 90, 7505 (1989) [ SPIN][ INSPEC].
22. The fact that Fig. 6 shows the measured specular and diffuse scattering to be in phase for a ``titanium'' film on sapphire confirms the expectation that such a thin film oxydizes almost completely when exposed to air.
23. I. M. Tidswell, T. A. Rabedeau, P. S. Pershan and S. D. Kosowsky, Phys. Rev. Lett. 66, 2108 (1991).

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