Volume: Page/Article: ------------------------------------------------------------------------ *Your access to PROLA is provided through the subscription of Central Research Institute* ------------------------------------------------------------------------ *MyArticles:* View Collection Help (Click on the to add an article.) ------------------------------------------------------------------------ Phys. Rev. B 51, 11296?11309 (1995) [Issue 17 ? 1 May 1995 ] *[ *Previous article* | *Next article* | *Issue 17 contents* ]* View Page Images or PDF (2427 kB) ------------------------------------------------------------------------ Using parametric /B/ splines to fit specular reflectivities N. F. Berk and C. F. Majkrzak /Materials Science and Engineering Laboratory, National Insitute of Standards and Technology, Gaithersburg, Maryland 20899-0001/ Received 5 January 1995 Parametric /B/-spline curves offer a flexible and appropriate mathematical description of scattering length density profiles in specular reflectivity analysis. Profiles combining smooth and sharp features can be defined in low dimensional representations using control points in the density-depth plane which provide graded local influence on profile shape. These profiles exist in vector spaces defined by /B/-spline order and parameter knot set, which can be systematically densified during analysis. Such profiles can easily be rendered as adaptive histograms for reflectivity computation. /B/-spline order can be chosen to accommodate the asymptotic (large-/Q/) behavior indicated by reflectivity data. We describe an interactive fitting strategy in which the Nelder and Mead simplex method is used in the /B/-spline control point space to guide the discovery of profiles that can produce given reflectivity data. Examples using actual and simulated spectra are discussed. ©1995 The American Physical Society *URL:* http://link.aps.org/abstract/PRB/v51/p11296 *DOI:* 10.1103/PhysRevB.51.11296 *PACS:* 61.10.-i, 61.12.-q ------------------------------------------------------------------------ View Page Images or PDF (2427 kB) *[ *Previous article* | *Next article* | *Issue 17 contents* ]* ------------------------------------------------------------------------ References (Reference links marked with may require a separate subscription.) 1. Proceedings of the Workshop on ``Methods of Analysis and Interpretation of Neutron Reflectivity Data," edited by G. P. Felcher and T. P. Russell, [Physica B *173*, 1 (1991)]. 2. T. M. Roberts, Physica B *173*, 157 (1991) [ INSPEC ]. 3. S. K. Sinha, M. K. Sanyal, K. G. Huang, A. Gibaud, M. Rafailovich, J. Sokolov, X. Zhao, and W. Zhao, in /Surface X Ray and Neutron Scattering/, edited by H. Zabel and I. K. Robinson (Springer Verlag, Berlin, 1992), p. 85. 4. D. S. Sivia, W. A. Hamilton, G. S. Smith, T. P. Rieker and R. Pynn, J. Appl. Phys. *70*, 732 (1991) [ADS ][CAS ][ SPIN ][ INSPEC ]. 5. D. S. Sivia, W. A. Hamilton and G. S. Smith, Physica B *173*, 121 (1991) [ INSPEC ]. 6. J. Penfold and R. K. Thomas, J. Phys. Condens. Matter *2*, 1369 (1990) [CAS ][ INSPEC ]. 7. J. S. Pedersen, J. Appl. Crystallogr. *25*, 129 (1992) [ INSPEC ]. 8. N. Singh, M. Tirrel and F. S. Bates, J. Appl. Crystallogr. *26*, 650 (1993) [CAS ][ INSPEC ]. 9. X. L. Zhou and S. H. Chen, Phys. Rev. E *47*, 3174 (1993) . 10. V. O. de Haan and G. G. Drijkoningen, Physica B *198*, 24 (1994) [ INSPEC ]. 11. K. Kunz, J. Reiter, A. Götzelmann and M. Stamm, Macromolecules *26*, 4316 (1993) [CAS ]. 12. J. S. Pedersen and I. W. Hamley, Physica B *198*, 16 (1994) [ INSPEC ]. 13. I. W. Hamley and J. S. Pedersen, J. Appl. Crystallogr. *27*, 29 (1994) [CAS ][ INSPEC ]. 14. J. S. Pedersen and I. W. Hamley, J. Appl. Crystallogr. *27*, 36 (1994) [ INSPEC ]. 15. The rendering step is unnecessary in the Born approximation if the Fourier transforms of the bases functions are known exactly, as in Refs. onlinecitePedersen0,Singh,Pedersen1,Pedersen2,Pedersen3. Use of the Born approximation, however, entails an assumption that is not always correct and needs to be tested for each spectrum where low /Q/ data are available. 16. O. Glatter, J. Appl. Crystallogr. *10*, 415 (1977) [ INSPEC ]. 17. A version of this work first was presented at /New Horizons: A Workshop on the State of the Art in Neutron Reflectivity/, National Institute of Standards and Technology, Gaithersburg, MD, December 9 10, 1993 (unpublished). 18. C. de Boor, /A Practical Guide to Splines/ (Springer Verlag, New York, 1978). 19. L. L. Shumaker, /Spline Functions: Basic Theory/ (Wiley, New York, 1981). 20. R. H. Bartels, J. C. Beatty, and B. A. Barsky, /An Introduction to Splines for Use in Computer Graphics and Geometric Modeling/ (Morgan Kaufmann, Los Altos, CA, 1987). 21. In general the / /rho/ (z)/ are complex valued functions, in order to account both for scattering (real part) and absorption (imaginary part). Thus in the parametric /B/ spline representation, the { /rho/ _i }, and hence, / /rho/ (u)/, are complex also. We leave this implicit, but it means that the function space of concern actually is larger than the notation suggests. For neutrons, absorption is negligible in most cases, and the assumption of real valued / /rho/ (z)/ is a good approximation. For x rays, the imaginary part of / /rho/ (z)/ often is important, and the fitting procedure then must allow for this additional freedom. In fact the methods we describe are not restricted to real valued density profiles, but effective fitting strategies should be informed of the need to find complex valued profiles when they are required. 22. Reentrant ``/ /rho/ (z)/" do not present computational problems, /per se/. The various methods of calculating reflectivities from density profiles need only to associate a / /rho/ / value with every /z/ value, without necessary regard for the possibility of recurring /z/. The parametric representation accomplishes this, since every /u/ value is mapped to only one (/z/, / /rho/ /) coordinate, regardless of where the point falls. Inappropriate /z/ recurrences are algorithmically innocuous. The resulting reflectivities are ``correct" for the given density profile, even if they are not physically meaningful. 23. Reference onlineciteBBB, Sec. 20.3. 24. E. Cohen, T. Lyche and R. Riesenfeld, Comput. Graphics Image Process. *14*, 87 (1980) [ INSPEC ]. 25. H. Prautzsch, Comput. Aided Geom. Design *2*, 329 (1985). 26. References onlinecitePedersen1,Pedersen2,Pedersen3 employ an informal notion of smoothness as a least squares fitting constraint. 27. M. J. Lighthill, /An Introduction to Fourier Analysis and Generalized Functions/ (Cambridge University Press, New York, 1958). 28. S. K. Sinha, E. B. Sirota, S. Garoff and H. B. Stanley, Phys. Rev. B *38*, 2297 (1988) . 29. C. F. Majkrzak, N. F. Berk, S. K. Satija and T. P. Russel, Proc. SPIE *1738*, 282 (1992) [CAS ]. 30. B. A. Barsky, Comput. Ind. *3*, 17 (1982). 31. J. A. Nelder and R. Mead, Comput. J. *7*, 308 (1965). 32. S. L. S. Jacoby, J. S. Kowalik, and J. T. Pizzo, /Iterative Methods for Nonlinear Optimization Problems/ (Prentice Hall, Englewood Cliffs, NJ, 1972). See pp. 79 83 for a lucid definition of the Nelder Mead algorithm, including a detailed flow diagram. 33. See also W. M. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, /Numerical Recipes/ (Cambridge University Press, Cambridge, England, 1990). 34. If a_1 = b_1 and a_2 = b_2 , then /alpha/ a_1 + /beta/ a_2 = /alpha/ b_1 + /beta/ b_2 for all / /alpha/, /beta/ /. This, of course, is the generalization of the rule, ``equals added to equals are equal.". 35. If a_1 geq b_1 and a_2 geq b_2 , then /alpha/ a_1 + /beta/ a_2 geq /alpha/ b_1 + /beta/ b_2 when / /alpha/, /beta/ geq 0/; but otherwise, not always. 36. Fitting /E/ to /D/ on a logarithmic scale accomplishes a similar goal, of course, but is more expensive computationally and produces flatter objectives, which slow the movement of the simplex. 37. P. Croce and B. Pardo, Nouv. Rev. Opt. Appl. *1*, 229 (1970) [CAS ]. 38. S. Yamada, T. Ebisawa, N. Achiwa, T. Akiyoshi and S. Okamoto, Annu. Rep. Res. Reactor Inst. Kyoto Univ. *11*, 8 (1978) [CAS ]. 39. See Ref. onlineciteZhou for a different solution of this problem. 40. D. G. Wiesler and C. F. Majkrzak, Physica B *198*, 181 (1994) . 41. The scattering length density for the backing medium has a small effective imaginary part (due to isotropic incoherent scattering), Im /rho/ _b =0.015 x 10^-6 AA^-2 , which has negligible effect on the fits over the given /Q/ range. 42. J. F. Ankner and C. F. Majkrzak, Proc. SPIE *1738*, 260 (1992). 43. S. K. Satija and T. P. Russell (private communication). 44. Reference onlineciteBBB, Secs. 20.2, 20.4, 20.5. 45. Reference onlineciteBBB, Sec. 20.6. ------------------------------------------------------------------------ View Page Images or PDF (2427 kB) [Show Articles Citing This One] /Requires Subscription/ *[ *Previous article* | *Next article* | *Issue 17 contents* ]* ------------------------------------------------------------------------ *[ APS | APS Journals | PROLA Homepage | Browse | Search ]* E-mail: prola@aps.org