Since the bulk of the present work deals with neutron reflectometry, the technique will be developed below in greater detail than any of the other analysis techniques. Neutron reflectometry is a technique in which neutrons at thermal energies are used to probe the near surface structure of material systems. In neutron reflectometry, neutrons are incident on a surface at a grazing angle of less than 3. At these small angles, the potential for scattering can be approximated by a continuous value, as will be shown later, and this value is called the scattering length density (SLD) usually denoted in equations as . It is a product of the scattering length of each isotope and that isotope's number density.
For neutrons of thermal energies, the very small angles of incidence can be sufficient to cause the neutrons to be totally reflected from the surface. The position of this transition to total reflection yields information about the average SLD of the material. The calculation of this total reflection (critical) angle will be discussed below. The reflectivity away from the total reflection angle holds information about the change in scattering length density with depth. It can be analyzed to determine a film's total thickness, material composition, periodicity, and even roughness.
The geometry of a typical reflectometry experiment is shown in Fig. 1. The data are measured as
(1) | |||
(2) |
When , , and the scattering is called specular. This is the condition that is used to determine the structure of the material in the direction, perpendicular to the surface. When dealing with specular reflectivities the data are usually presented as a function of .
The geometry that was assumed in the discussion above and shown in Fig. 1 is the forward scattering geometry. In this geometry, the direction perpendicular to the plane of reflection, the -direction, is neglected. The slits which collimate the beam of neutrons are very wide in this direction (> 1 cm), so any measurements are assumed to be integrated over the out-of-plane direction. For this reason, all of the equations presented here will have the and dependence eliminated or integrated over.
Most reflectivity curves have three basic features.
The first is the critical wave-vector transfer . Neutrons are
totally reflected for values of below .
For a uniform film, is given by
A second feature of reflectivity profiles is the decrease in intensity (or reflectivity) with which, for a smooth sample, becomes proportional to as increases. A thin film can also show oscillations, the third feature, around this continuously decreasing reflectivity; the result of an interference effect between the air/film and film/substrate interfaces. The amplitude of these thickness oscillations is proportional to the difference in SLD between the thin film and the substrate, often referred to as the SLD contrast. An estimate of the film's thickness is given by , where is the oscillation period.
(5) |
(6) |
The 3-D steady state Schrödinger equation in three dimensions is given
by
(8) |
Since in specular
reflection we are only concerned with the structure of the medium in the
direction, perpendicular to the surface, and since off-specular scattering
requires a special treatment, we will assume that
is a function of
only. This assumption is appropriate to the presentation of the basic
development of the different equations, but it is not completely appropriate
when calculations are made for real surfaces which usually have some kind of
roughness. The only assumption that must be made about the SLD in the
direction is that it has the limiting values
(9) |
(10) |
where
satisfies the equation
Since
as
, the above equation
is just the equation for a wave in free space above the sample
where is the reflectance of the wave from the surface. If the wave
equation is determined above the surface of the sample it will have the form
(13) |
The square modulus of the coefficient , the complex amplitude of the reflected wave, is called the reflectivity, and it is this quantity that is measured. For specular reflectivity, since and are conserved, the momentum transfer vector has only one component: .
In order to begin a calculation of the reflectance, , we first begin by
setting the scattering length density equal to
If we define an operator such that
(15) |
This Green's function is a wave traveling from the point ,
and Eq. 16 can be rewritten as an integral equation using this
Green's function and the previous definitions
The solution to Eq. 17 has the asymptotic form
When Eq. 20 and Eq. 21 are substituted into
Eq. 19 is can be seen just by inspection that the equation has
the form of Eq. 12 and that the reflection coefficient, ,
can be expressed as
Also called the first Born approximation or just the Born approximation,
this approximation assumes that the entire scattering event is just a
perturbation of the incoming beam so we set[1]
(23) |
(25) |
When Eq. 24 and this Green's function are substituted into
Eq. 22 the reflected amplitude becomes
No assumption has been made about the form of the incoming wave yet in
Eq. 26, so
the above equation is still technically an exact result.
For the plane wave Born
approximation, we assume
and the reflected
amplitude is
(27) |
The DWBA assumes that the unperturbed wave is a wave reflected
from a completely smooth surface so we set[1]
(28) |
(29) |
We know that the solution of Eq. 17 has the form of a
wave that is partially reflected and partially refracted at :
(30) |
(31) |
(32) |
(33) | |||
(34) |
The final method of calculating reflectivities presented here is an iteratively exact method called Parratt's recursion relation[2]. The basic principle for this relation is very simple. The model scattering length density profile is broken up into a series of layers as shown in Fig. 2. The layer has a thickness and the component of the wave number , where is the incident wave number in air in the
(35) |
Each of these waves are originating from a depth in
the medium. The first derivative of the wave in each layer is given by
(36) |
(37) | |||
(38) |
(41) |
(42) |
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