International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 2.9, pp. 126127

Consider the glancing (smallangle) reflection of a neutron plane wave characterized by a wave vector from a perfectly flat and smooth surface of infinite lateral extent, as depicted schematically in Fig. 2.9.2.1 . Although the density of the material can, in general, vary as a function of depth [along the direction (z) of the surface normal], it is assumed that there are no inplane variations of the density. If the scattering is also elastic, so that the neutron neither gains nor loses energy (i.e.  k_{i}  =  k_{f}  = k = 2π/λ, where the subscripts i and f signify initial and final values, respectively, and λ is the neutron wavelength), then the component of the neutron wave vector parallel to the surface must be conserved. In this case, the magnitude of the wavevector transfer is Q = Q =  k_{f} − k_{i}  = 2ksin() = 2k_{z}, where the angles of incidence and reflection, , are equal, and the scattering is said to be specular. Since at low values of Q the neutrons are strongly scattered from the surface (i.e. the magnitude of the reflectivity approaches 1), the neutron wave function is significantly distorted from its freespace planewave form. The first Born approximation normally applied in the description of highQ crystal diffraction is therefore not valid for the analysis of lowQ reflectivity measurements, and a more accurate, dynamical treatment is required.
Because the inplane component of the neutron wave vector is a constant of the motion in the specular elastic reflection process described above, the appropriate equation of motion is the onedimensional, timeindependent, Schrödinger equation (see, for example, Merzbacher, 1970) where ψ is the neutron wave function [which in free space is proportional to where is the magnitude of the z component of the neutron wave vector in vacuum]. If the infinite planar boundary from which the neutron wave reflects separates vacuum from a medium in which the neutron potential energy is , conservation of the neutron's total energy requires that where is Planck's constant divided by 2π and m is the neutron mass. If 2π/Q has a magnitude much greater than interatomic distances in the medium, then the medium can be treated as if it were a continuum. In this limit, the potential energy can be expressed as (see, for example, Sears, 1989)where , i represents the ith atomic species in the material, is the number density of that species and is the coherent neutron scattering length for the ith atom (which is in general complex if absorption or an effective absorption such as isotopic incoherent scattering exists; magnetic contributions are not accounted for here but will be considered below). The quantity is the effective scattering density. Substituting the expression for given in equation (2.9.2.3) into (2.9.2.2) yields In order to calculate the reflectivity, continuity of the wave function and its first derivative (with respect to z) are imposed. These boundary conditions are a consequence of restrictions on current densities required by particle and momentum conservation. In general, given a sample with layers of varying potentials where the boundaries of the jth layer are at and and the potential, is constant over that layer, it can be shown that (see, for example, Yamada, Ebisawa, Achiwa, Akiyoshi & Okamoto, 1978) where and is the magnitude of the neutron wave vector in the jth layer [equation (2.9.2.4)].
The first equality in (2.9.2.5) relates the wave function at one boundary within the jth layer to the next boundary within the jth layer, whereas the second equality represents the continuity of the wave function and its derivative across the boundary between the jth and (j + 1)th layers. When a neutron plane wave is incident on a multilayer sample, we can take the incident amplitude as unity, set up the coordinate system to have z = 0 at the air/sample interface, and write the wave function in air as the sum of the incident and reflected waves, and the wave function in the substrate as a purely transmitted wave, where is the magnitude of the z component of the neutron wave vector in the substrate. By combining equations (2.9.2.5) through (2.9.2.8), we obtain a working equation for calculating the reflectivity: where total film thickness. The experimentally measured reflection and transmission coefficients R ^{2} and T ^{2} can be computed from (2.9.2.9). The procedure outlined above can be applied in piecewise continuous fashion to arbitrary, smooth potentials, ρ(z), which are approximated to any desired degree of accuracy by an appropriate number of consecutive rectangular slabs, each having its own uniform scattering density, , and thickness, , as depicted in Fig. 2.9.2.2 .
If , then becomes imaginary in the substrate, and total external reflection occurs. In addition, for a single layer deposited on the substrate, the reflectivity will oscillate with a periodicity characteristic of the layer thickness. Fig. 2.9.2.3 compares the ideal Fresnel reflectivity corresponding to an infinite silicon substrate and that of a 1000 Å nickel film deposited on silicon. For a barrier of finite thickness, tunnelling phenomena can also be observed (see, for instance, Merzbacher, 1970; Buttiker, 1983; Nuñez, Majkrzak & Berk, 1993; Steinhauser, Steryl, Scheckenhofer & Malik, 1980).

Neutron reflectivities calculated for an infinite Si substrate (dashed line) and 1000 Å Ni film on an Si substrate (solid line). 
With the matrix method described above, the reflectivity of any model scatteringdensity profile can be calculated with quantitative accuracy over many orders of magnitude. Unfortunately, the inverse computation of an unknown scattering density profile corresponding to a given reflectivity curve can be exceedingly difficult, in part due to the the lack of phase information on R(Q), which forces one to use highly nonlinear relations between R(Q)^{2} and ρ(z). Often, parameterized model scatteringdensity profiles are fit to the experimental data (Felcher & Russell, 1991). Recently, several authors have described modelindependent methods for obtaining ρ(z) from measured reflectivity curves (Zhou & Chen, 1993; Pedersen & Hamley, 1994; Berk & Majkrzak, 1995).
References
Berk, N. F. & Majkrzak, C. F. (1995). Using parametric Bsplines to fit specular reflectivities. Phys. Rev. B, 51, 11296–11309.Buttiker, M. (1983). Larmor precession and the traversal time for tunneling. Phys. Rev. B,27, 6178–6188.
Felcher, G. P. & Russell, T. P. (1991). Editors. Physica (Utrecht), B173, 1–210.
Merzbacher, E. (1970). Quantum mechanics, 2nd ed. New York: John Wiley.
Nuñez, V., Majkrzak, C. F. & Berk, N. F. (1993). Dynamical scattering of polarized neutrons by thin magnetic films. MRS Symp. Proc. 313, 431–436.
Pedersen, J. S. & Hamley, I. W. (1994). Analysis of neutron and Xray reflectivity data by constrained leastsquares methods. Physica (Utrecht), B198, 16–23.
Sears, V. F. (1989). Neutron optics. Oxford University Press.
Steinhauser, K. A., Steryl, A., Scheckenhofer, H. & Malik, S. S. (1980). Observation of quasibound states of the neutron in matter. Phys. Rev. Lett. 44, 1306–1309.
Yamada, S., Ebisawa, T., Achiwa, N., Akiyoshi, T. & Okamoto, S. (1978). Neutronoptical properties of a multilayer system. Annu. Rep. Res. React. Inst. Kyoto Univ. 11, 8–27.
Zhou, X.L. & Chen, S.H. (1993). Modelindependent method for reconstruction of scatteringlengthdensity profiles using neutron or Xray reflectivity data. Phys. Rev. E, 47, 3174–3190.