Exact Determination of the Phase in Neutron Reflectometry by Variation of the
C.F. Majkrzak and N.F. Berk.
Phys. Rev. B 58, 15416 (1998).
We give an extension of the phase determination
method which utilizes controlled variations of the scattering
length density of the incident and/or substrate medium instead of
reference layers of finite thickness. The procedure algebraically
yields the real part of the reflection amplitude uniquely from two
reflectivity measurements, which is sufficient for inversion. This
technique is of practical importance for thin-film systems involving
either gas-liquid or solid-liquid interfaces in which the scattering
length density of the liquid can be varied in a known way, as in
deuterated aqueous media.
Experiments are underway at the NCNR to test
the surround variation method.
Phase Determination and Inversion in Specular Neutron Reflectometry
C.F. Majkrzak, N.F. Berk, J.A. Dura, S.K. Satija, A. Karim,
J. Pedulla, and R.D. Deslattes. Physica B
248, 338 (1998).
Experiments have been performed to test recent phase determination
methods developed at NIST for scattering length density profiles of
general shape and for the special case of
symmetric fims. Using layers of Cu, Ni, and
Mo as references, the real and imaginary parts of the complex
reflection amplitude were measured from neutron reflectivities for an
asymmetric composite film consisting of deuterated polystyrene and Si.
The reflection amplitude was also measured from neutron reflectivity
without references for a symmetric deuterated polystyrene film. These
amplitudes were inverted using the Gel'fand-Levitan-Marchenko equation
to produce scattering length density profiles for the films studied.
The inverted profiles compared reasonably well to the expected
potentials, and we conclude that such methods are practical with
This work was a cooperative effort
among scientists in the NCNR, the Polymer Division, and the Physics
Division at NIST.
These experiments are also discussed in the
Proceedings of the International Conference on Neutron Scattering,
Toronto (August, 1997), Physica B in press.
Inverting Specular Neutron Reflectivity from Symmetric, Compactly
N. F. Berk and C. F. Majkrzak,
Proc. Int. Symposium on
Neutron Optics and Related Research Facilities, Kumatori, 1996.
J. Phys. Soc. Jpn., 65, Suppl. A, 107 (1996).
A method is described for inverting specular neutron reflectivities
from real symmetric, compactly supported potentials of known thickness.
For such potentials, the phase of the complex reflection coefficient is
equal to the phase of the transmission coefficient plus a known phase
shift and thus can be retrieved from a single measurement of
reflectivity using a logarithmic dispersion relation for the
transmission. The resulting reflection coeffieicent can be inverted to
find the potential by solving the Gel'fand-Levitan-Marchenko integral
equation. The method is general, to the extent that symmetric
potentials can be formed by abutting two identical specimens of a film
Experiments are in progress to test this
method and the one described below. Preliminary reports of these were
given at the 1997 March Meeting of the APS and at the Fifth Surface
X-Ray and Neutron Scattering Conference in Oxford (July, 1997), and at
the International Conference on Neutron Scattering in Toronto (August,
Exact Determination of the Phase in Neutron Reflectometry
C. F. Majkrzak and N. F. Berk, Phys. Rev. B 52, 10827 (1995).
By using a known reference layer having three tunable values of
scattering density, an exact determination of the complex amplitude
R=Re R+iIm R for neutron specular reflection can be made for
any unknown real potential (i.e., no absorption). This straightforward
yet remarkable general result is valid even in the dynamical regime
(where the Born approximation fails) and makes it feasible
to consider direct inversion methods for obtaining the scattering
length density profile normal to the reflecting surface.
An equivalent method was found
independently and published in tandem by V.O. deHaan, A.A. van Well,
S. Adenwalla, and G.P. Felcher, Ibid, p. 10830.
The inverse scattering problem using the
Gel'fand-Levitan-Marchenko equation is discussed in the paper on
Parametric B Splines to Fit Specular Reflectvities
N. F. Berk and C. F. Majkrzak, Phys. Rev. B51, 11296
Parametric B-spline curves offer a flexible mathematical
description of scattering length density profiles in specular
reflectivity analysis. Profiles mixing smooth and sharp features
can be defined in low dimensional representations using spline 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 using the Oslo spline refinement algorithm.
An interactive fitting strategy using the Nelder-Mead simplex method is
The lack of uniqueness inherent in profile determination is
References to related methods are given.