The theory and analysis of NEUTRON SPECULAR REFLECTIVITY is a major
research interest in the NIST Center for Neutron Research.

Experimental Demonstration of Phase Determination in Neutron Reflectometry by
Variation of the Surrounding Media
C.F. Majkrzak, N.F. Berk, C.W. Meuse, and V. Silin
Proc. of Sixth Surface X Ray and Neutron Scattering Conference (6sxns), Nordwijkerhout,
Sep., 1999. To appear in Physcia B.
The method of phase determination using surround
variation was applied to several thin films and the resulting real parts of the
reflection coefficients, Re r(Q), were inverted to retrieve the veridical scattering length
density (SLD) profiles within the spatial resolution of the measurements (about 20 A).
In one experiment two backings, air and heavy water, were used. In another, Si and sapphire
served as two frontings. The SLD profiles thus obtained were in excellent agreement with
expectations. A new method of diagnosing the quality of measured films using the imaginary part
of a measured reflection coefficient is also given. It is proved here that Im r(Q) for
a perfect but arbitrary film necessarily has a regular sequence of zeros and that this zero set
is sensitive to imperfections, such as largescale lateral inhomogenieties. A simulated example
is given.
This work was a cooperative effort
among scientists in the NCNR (Majkrzak and Berk) and the Biotechnology
Division (Meuse and Silin) at NIST.
 Using Polarized Neutrons to
Determine the Phase of Thin Film Structures
A.
Schreyer,C.F. Majkrzak, N.F. Berk, H. Gruell, and C.C. Han.
Proc. of Frontiers in Neutron Scattering, Tokyo, Nov., 1998. To
appear in Journ. of the Phys. and Chem. of Solids, (1999).
The method of phase determination using polarized
neutrons was applied to a 400 A thick polymer blend film (dPB/PI)
on a reference consisting of sputtered Si on ferromagnetic Fe. A new
phase fitting procedure using parametric B
splines was developed to unambiguously determine the imaginary part
of the reflection amplitude from noisy data and to extract the
scattering length density profile of the polymer film from two
polarized neutron reflectivity spectra. The results confirm a
preferential adsorption of the hydrogenous component of the dPB/PI
blend at both film edges at room temperature.
This continuing study is a collaboration of
the NCNR (Schreyer, Majkrzak, and Berk) and the NIST Polymers Division
(Gruell and Han).

Inverting Neutron Reflectometry from Layered Film Structures Using Polarized
Neutron Beams
C.F. Majkrzak and N.F. Berk.
Physica B, (1999) in press.
Experimental applications of the reference layer
phase determination method and reflectivity inversion are shown,
using a 2measurement variation of the original 3measurment
procedure. Systems studied include a Au layer over a buried
ferromagnetic Fe reference layer, and an organic film on Au over a
buried ferromagnetic Fe layer. The single buried ferromagnetic layer
effects two references when polarized neutron beams are used.

Exact Determination of the Phase in Neutron Reflectometry by Variation of the
Surrounding Media
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 thinfilm systems involving
either gasliquid or solidliquid 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'fandLevitanMarchenko 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
current instrumentation.
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
Supported Potentials
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'fandLevitanMarchenko integral
equation. The method is general, to the extent that symmetric
potentials can be formed by abutting two identical specimens of a film
of interest.
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
XRay and Neutron Scattering Conference in Oxford (July, 1997), and at
the International Conference on Neutron Scattering in Toronto (August,
1997).

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'fandLevitanMarchenko equation is discussed in the paper on
symmetric potentials.
 Using
Parametric B Splines to Fit Specular Reflectvities
N. F. Berk and C. F. Majkrzak, Phys. Rev. B51, 11296
(1995).
Parametric Bspline 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 densitydepth plane which provide graded local influence
on profile shape. These profiles exist in vector spaces defined by
Bspline order and parameter knot set, which can be systematically
densified during analysis using the Oslo spline refinement algorithm.
An interactive fitting strategy using the NelderMead simplex method is
described.
The lack of uniqueness inherent in profile determination is
discussed.
References to related methods are given.
