*Register * or Login: Password: Quick Search: within *4 of 34* *Physica B: Condensed Matter* Volume 336, Issues 1-2 , August 2003, Pages 27-38 Proceedings of the Seventh International Conference on Surface X-ray and Neutron Scattering This Document *SummaryPlus* Full Text + Links PDF (251 K) Actions Cited By Save as Citation Alert E-mail Article Export Citation doi:10.1016/S0921-4526(03)00266-7 Copyright © 2003 Elsevier Science B.V. All rights reserved. Phase sensitive reflectometry and the unambiguous determination of scattering length density profiles * * *C. F. Majkrzak^ <#m4.cor*>^, ^ and N. F. Berk * NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899-8562, USA Received 6 February 2003; accepted 13 February 2003. ; Available online 23 May 2003. Abstract Exact methods for determining the complex neutron reflection amplitude for a thin film, which make use of multiple measurements of the specularly reflected intensities of composite systems, composed of the film adjacent to a reference layer and/or surrounding media, have been developed over the past several years. These techniques are valid even where the Born or distorted wave Born approximations break down. Thus, given both the modulus and phase of the specular reflection, a first-principles inversion can be performed which yields the scattering length density (SLD) depth profile of the film directly. Ideally, if the reflection amplitude is known for all wave vector transfers /Q/, the associated SLD profile is unique. Applying the aforementioned methods to a purely real SLD profile, which, effectively, is almost always that encountered in neutron reflection, at least two distinct reflectivity curves, corresponding to two different composite film systems, are required to determine the phase by direct algebraic computation, independently at each value of /Q/. Each of the composite systems consists of the common unknown part of the film plus a different reference layer segment and/or surrounding medium (e.g., the backing). Recently, investigations of certain classes of SLD profiles have been reported in the literature which examine whether a single X-ray reflectivity curve, given certain a priori knowledge about the system, i.e., about known parts of the film SLD and/or substrate, suffices to reconstruct the phase. Employing the exact formulation of phase sensitive reflectometry, we consider several illustrative and realistic cases in which a minimum of two reflectivity curves are required to distinguish the true SLD profile. *Author Keywords: *Neutrons; X-ray; Reflectivity; Thin film; Phase *PACS classification codes:* 61.12.−q; 68 Article Outline 1. Introduction 2. Theory 3. Illustrative examples 3.1. Free-standing film 3.2. Unknown film on known backing (substrate) 3.3. Unknown film adjacent to reference layer on a substrate 4. Conclusions References (6K) Fig. 1. General composite system SLD profile with "unkown" and reference film segments on semi-infinite backing or substrate. Dotted arrows are labeled by the normalized amplitudes of the incident (l) and reflected (r) beams. (6K) Fig. 2. Identical (neutron) reflectivity curves corresponding to mirror image SLD profiles shown in the inset. (After Fig. 1 <#fig1> of [14 <#bib14>].) (7K) Fig. 3. /Q/^2 /Re r/(/Q/) for the two SLD profiles in the inset of Fig. 2 <#fig2>. (5K) Fig. 4. Symmetry-related SLD film profiles (reflection about vertical axis through /z/=20 Å, followed by reflection about horizontal axis through -scale=3) on fixed known backing, as discussed in the text. (4K) Fig. 5. Reflectivity (neutron) curves corresponding to the SLD profiles of Fig. 4 <#fig4>. (8K) Fig. 6. /Q/^2 /Re r/(/Q/) for the SLD profiles of Fig. 4 <#fig4> without backing. (6K) Fig. 7. Composite system neutron reflectivities |/r/(/Q/)|^2 for the SLD profile of Fig. 4a <#fig4> on backings of SLD equal to 6.0×10^−6 Å^−2 (approximately sapphire value) and 2.0×10^−6 Å^−2 (roughly that for silicon). (5K) Fig. 8. /Q/^2 /Re r/(/Q/), as determined algebraically by the exact methods described in the text, from the two reflectivity model "data" sets of Fig. 7 <#fig7> compared to that computed directly for the reversed SLD profile of Fig. 4a <#fig4> (free film only, i.e., no backing). (6K) Fig. 9. Model SLD (neutron) profiles, (a) and (b), similar to two of the profiles considered by Zimmermann et al. in Fig. 3 <#fig3> of [6 <#bib6>] for X-ray reflection. Note that not only are the backings the same, but both profiles share a common "reference" or known segment between /z/=20 and 60 Å. (6K) Fig. 10. Neutron reflectivities for the two composite film systems (including backing) of Figs. 9a <#fig9> and b. Convolution for instrumental resolution has not been applied. The two curves are practically indistinguishable from one another; slight differences primarily occur in the vicinity of some of the minima. (9K) Fig. 11. /Q/^2 /Re r/(/Q/) for the (reversed) film structures of Fig. 9 <#fig9> (not including the backing but incorporating the known or reference sections of the films). These /Re r/(/Q/) correspond to what would be retrieved, for example, by phase-sensitive reflectivity experiments (for each of the two SLD profiles) in which the backing SLD density was varied according to the methods discussed in the text. In contrast to the situation illustrated in Fig. 10 <#fig10>, these curves are markedly different over a wide range of /Q/. References 1. K. Chadan and P.C. Sabattier. /Inverse Problems in Quantum Scattering Theory/, Springer, New York (1989). 2. T. Aktosun and P.E. Sacks. /Inverse Problems/ *16* (2000), p. 821. Abstract-INSPEC | $Order Document | *Full Text* via CrossRef 3. W. Leslauer and J.K. Blasie. /Acta Crystallogr. A/ *27* (1971), p. 456. 4. M.K. Sanyal, S.K. Sinha, A. Gibaud, K.G. Huang, B.L. Carvalho, M. Rafailovich, J. Sokolov, X. Zhao and W. Zhao. /Europhysics Lett./ *21* (1993), p. 691. Abstract-INSPEC | $Order Document 5. W.L. Clinton. /Phys. Rev. B/ *48* (1993), p. 1. Abstract-INSPEC | $Order Document | *Full Text* via CrossRef 6. 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B/ *63* (2001), p. 45414. ^ <#m4.bcor*>Corresponding author This Document *SummaryPlus* Full Text + Links PDF (251 K) Actions Cited By Save as Citation Alert E-mail Article Export Citation *Physica B: Condensed Matter * Volume 336, Issues 1-2 , August 2003, Pages 27-38 Proceedings of the Seventh International Conference on Surface X-ray and Neutron Scattering *4 of 34* Feedback | Terms & Conditions | Privacy Policy Copyright © 2005 Elsevier B.V. All rights reserved. ScienceDirect® is a registered trademark of Elsevier B.V.