Volume: Page/Article: ------------------------------------------------------------------------ Article Collection: View Collection Help (Click on the Check Box to add an article.) ------------------------------------------------------------------------ Phys. Rev. B 51, 5297?5305 (1995) [Issue 8 ? 15 February 1995 ] [ Previous article | Next article | Issue 8 contents ] Add to article collection View Page Images or PDF (1528 kB) ------------------------------------------------------------------------ X-ray reflection and transmission by rough surfaces D. K. G. de Boer Philips Research Laboratories, Professor Holstlaan 4, 5656 AA Eindhoven, The Netherlands Received 5 July 1994 Expressions are given for the coherent (specular) and incoherent (diffuse) reflection and transmission of x rays by rough surfaces. In particular, the results from the distorted-wave Born approximation are critically compared with those obtained by us using the Rayleigh method. It is shown that the validity of the various expressions depends on the values of the perpendicular wave vector, the root-mean-square surface roughness, and the lateral correlation length of the roughness. The conservation of intensity in the various approximations is considered. ©1995 The American Physical Society URL: http://link.aps.org/abstract/PRB/v51/p5297 DOI: 10.1103/PhysRevB.51.5297 PACS: 61.10.Dp, 68.35.Bs ------------------------------------------------------------------------ Add to article collection View Page Images or PDF (1528 kB) [ Previous article | Next article | Issue 8 contents ] ------------------------------------------------------------------------ References (Reference links marked with dot may require a separate subscription.) 1. Surface X-Ray and Neutron Scatterings, edited by H. Zabel and I. K. Robinson (Springer-Verlag, Berlin, 1992); Proceedings of International Conference on Surface X-Ray and Neutron Scattering SXNS-3, edited by H. J. Lauter and V. V. Pasyuk [Phys. B 198 (1994)]; Proceedings of Nanometer-Scale Methods in X-ray Technology, edited by D. K. G. de Boer and J. P. Chauvineau [J. Phys. (Paris) III 4, 1503 (1994)]. 2. E. L. Church and P. Z. Takacs, Proc. SPIE 645, 107 (1986) [dot INSPEC ]. 3. S. K. Sinha, E. B. Sirota, S. Garoff and H. B. Stanley, Phys. Rev. B 38, 2297 (1988) . 4. D. K. G. de Boer, Phys. Rev. B 44, 498 (1991) ; W. W. van den Hoogenhof and D. K. G. de Boer, Spectrochim. Acta 48B, 277 (1993); D. K. G. de Boer, A. J. G. Leenaers, and W. W. van den Hoogenhof, X-Ray Spectrom. (to be published). 5. L. Névot and P. Croce, Rev. Phys. Appl. 15, 761 (1980) [dot INSPEC ]; and, 11, 113 (1976). 6. D. K. G. de Boer, Phys. Rev. B 49, 5817 (1994) . 7. H. A. Lorentz, Versl. Gewone Vergad. Wis Natuurkd. Afd. K. Akad. Wet. Amsterdam 14, 345 (1905). 8. If radiation is diffracted from a grating with period xi, diffraction can occur at wave vectors p vec with parallel components | p vecpara| = | k vecpara| +- 2 pi / xi. No diffraction is possible if | p vecpara| > | k vec |, which is found to be equivalent to xi k20/ | k vec | < 4 pi. Since in a correlation function like Eq. (A1) larger periods than xi occur, we will suppose that if xi k20/ | k vec | << 1 no incoherent scattering is present. In that case, a self-consistent calculation yields Eq. (1) for the coherent reflection. 9. B. Vidal and P. Vincent, Appl. Opt. 23, 1794 (1984) [dot SPIN ][dot INSPEC ]. 10. R. Pynn, Phys. Rev. B 45, 602 (1992) . 11. A. Caticha, in Physics of X-Ray Multilayer Structures, 1994, Technical Digest Series Vol. 6 (Optical Society of America, Washington, D.C. 1994), p. 56. 12. R. Pynn and S. Baker, Physica B 198, 1 (1994), derived another expression for the reflection coefficient of a sample having facets with a Gaussian height distribution: r-tildek= rkexp [ - (k0+ k1)2 sigma 2/ 2 ]. They calculate the T matrix assuming that each facet has its own final state. Subsequently the amplitudes for all facets are added. In the case of small xi the latter is correct, but it seems to be in contradiction with the assumption that the final state does not depend on the adjacent facets. We think that this is an inconsistency which may be the reason why this approach does not yield Eq. (1) [dot INSPEC ]. 13. F. Stanglmeier, B. Lengeler, W. Weber, H. Göbel and M. Schuster, Acta Crystallogr. Sec. A 48, 626 (1992), introduce an averaging method leading directly to Eqs. (1) and (2). However, they only give the a posteriori justification that the results obtained in this way agree well with experiment. 14. B. Pardo, T. Megademini and J. M. André, Rev. Phys. Appl. 23, 1579 (1988), pointed out that a calculation for a transition layer with an average refractive index and a thickness 2 sigma yields the results Eqs. (1) and (2) up to O(k20 sigma 2). This procedure can be expanded by using a transition layer with the real refractive index profile, which for a Gaussian random surface is an error function. In that case, the reflectivity and transmissivity can be calculated numerically. We found that the results obtained in this way agree well with Eqs. (1) and (2), at least if k0 sigma ]. 15. P. Croce, L. Névot and B. Pardo, C. R. Acad. Sci. Paris 274, 803 (1972); and,, 855 (1972). 16. A. Steyerl, Z. Phys. 254, 169 (1972). 17. W. Weber and B. Lengeler, Phys. Rev. B 46, 7953 (1992) . 18. Pynn (Ref. 10) gives P(k0-> p0) = |kc|4| tk|2|t-tildep|2/(16 pi 2), but admits that his expression does not fulfill the reciprocity principle (Refs. 7 and 20). We think that this is because Pynn implicitly takes separate averages of the factors leading to P(k0-> p0) and S ( p vecpara- k vecpara, k1+ p1), instead of that of the product of the two. Weber and Lengeler (Ref. 17) used P(k0-> p0) = |kc|4|t-tildek|2|t-tildep|2/ (16 pi 2), with the argument that it gives a better agreement with experiment. However, the use of Eq. (4) with even more realistic material parameters can give a comparably good fit of their data [M. Tolan and D. Bahr (private communication)]. Holý (private communication) recently approached the problem by using a DWBA-type calculation with as a starting point the (numerical) solutions for an error-function-shaped refractive index profile. He found that Eq. (4) gives a better description than the formula proposed in Ref. 17. 19. The incoherent scattering in the second-order DWBA remains correctly given by Eqs. (5) and (7) up to O(k20 sigma 2). 20. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963). 21. J. W. Strutt and Baron Rayleigh, The Theory of Sound, 2nd ed. (Macmillan, London, 1896), Sec. 272a. 22. A. Caticha (unpublished). 23. Note that if the incident flux is constant, which is often the case in AD-XRF, Eq. (14) has to be multiplied by Ak0/ | k vec |. In AD-XRF the absorption of the detected x-ray fluorescence radiation can also be taken into account (Ref. 4). 24. P. Croce, L. Névot and B. Pardo, Nouv. Rev. Opt. Appl. 3, 37 (1972) [dot INSPEC ]. 25. It can be shown that the AD-XRF of impurity atoms at a rough interface is proportional to the transmission factor (Refs. 4 and 27). 26. G. Palasantzas and J. Krim, Phys. Rev. B 48, 2873 (1993) . 27. D. K. G. de Boer, A. J. G. Leenaers and W. W. van den Hoogenhof, J. Phys. III 4, 1559 (1994) [dot INSPEC ]. 28. P. Meakin, Prog. Solid State Chem. 20, 135 (1990) [dot INSPEC ]. ------------------------------------------------------------------------ Add to article collection View Page Images or PDF (1528 kB) [Show Articles Citing This One] Requires Subscription [ Previous article | Next article | Issue 8 contents ] ------------------------------------------------------------------------ [ APS | APS Journals | PROLA Homepage | Browse | Search ]