The original incentive of surface structure analysis was the determination of ordered structures of adsobates. Such structures were originally predicted by Langmuir and observed with LEED (low-energy electron diffraction), a technique which dates back to the famous Davisson and Germer experiment, however, became the standard surface science tool only in the early sixties.

In a LEED pattern you see extra spots, in addition to the spots at the substrate positions. These new spots are due to a so-called superstructure, i.e. a lattice with a larger periodicity than the substrate lattice. Low-energy electrons scatter strongly from a surface, so considerable effort had to put into development of powerful scattering calculations so that LEED intensities can be used to determine surface structure. With the advent of synchrotrons as powerful x-ray sources, the weakly interacting x-rays developed rapidly into an alternative method for surface structure determination, which was appealing becouse of the simple kinematic scattering theory suitable for structure analysis.
 

The scattering of from a 2D lattice can be calculated straightforwardly within the kinematic approximation: The Fourier-Transform of the electron density yields a 2D lattice of d-functions, parallel to the surface plane, and diffuse streaks perpendicular to the plane. The in-plane scattering intensities are modulated according to the arrangement of the atoms within the surface unit cell.

The standard tool for determining surface structure is the collection of an in-plane data set of superstructure reflections. If we stay away from the q positions where the bulk lattice scatters, and collect only the intensities of the so-called superstructure spots we obtain information of how the atoms within the monolayer are arranged. If we Fourier backtransform the measured spot intensities, we obtain a 2D Patterson map which is the autocorrelation function of the surface electron density. Maxima in the Patterson map indicate interatomic vectors from which some first models of the surface structure can be constructed. From these models, scattering intensities can be calculated and compared to the measured data, from which the most likely model can be obtained and further refined in a fitting procedure. A pecularity of the 2D Patterson are negative maxima due to the missing intensities that overlap with substrate scattering. This makes the interpretation of 2D Patterson functions more difficult compared to the strictly positive 3D Patterson functions. 

The scattering from a monolayer perpendicular to the plane - by convention in q_z direction - is diffuse. For pointlike objects these scattering rods would have constant intensity. However, for real atoms the x-ray scattering intensity falls off according to the atomic form factor f(q). Furthermore we have to account for thermal motion of the atoms giving rise to a Debye-Waller factor. The thermal motion of atoms perpendicular to the surface plane can be considerably larger than the in-plane thermal motion, and an anisotric Debye-Waller factor may have to be introduced.

If there are several atoms within the superstructure unit cell, with various heights above the substrate termination plane, there will be interference between the scattering from the surface atoms which can give rise to more complicated modulations of the scattering rod. Hence the scattering rods also contain information on the surface structure in such cases.

Note that it is not trivial to define where the surface starts and the substrate ends: The adsobate can lead to a reconstruction of the upper plane of the substrate (see [2] for a very nice example). Almost always the so-called relaxation, i.e. the variation of the distance between the first two layers of the substrate, will be affected. The relaxation is directly related to the variation of the electronic charge density due to the presence of the interface. In fact, even a clean surface can be considered as a 1x1 structure, because of the relaxation of the top surface layer.
 

If the substrate is single-crystalline, there will also be diffuse scattering from the surface, the so-called crystal truncation rods [1]. An elegant way of deriving a decription is to consider an infinite 3D lattice. A surface can be introduced by multiplying the periodic electron density with a Heavyside function H(z) which is 0 for positive z and 1 for negative z.
Applying the Convolution Theoreme, the Fourier transform of the truncated lattice is nothing but the convolution of the Fourier transforms of each object. The Fourier-Transform of the electron density yields the reciprocal lattice of d functions whereas the Fourier transform of H(z) is i/q_z. The meaning of the convolution is that each point of the reciprocal lattice (i.e. the diffraction pattern) gets smeared out with 1/q_z tails in the direction of the surface normal. The overlap of all q_z tails yield for the scattering amplitude

S(q) = f(q) 1/sin(q_z c)

OUTLOOK: The CTRs can be identified with the 1/q_z tails of the diffraction peak profile in dynamic diffraction theory.

relaxation

The CTR form an important contribution to a crystallographic data set. At the substrate positions and inbetween substrate Bragg reflections, the scattering from the substrate and the surface overlap coherently. In particular, the in-plane intensity of the CTR contains the information, how the surface layer is positioned with repect to the substrate lattice.
 

Thin films, which is usually anything from 2 to 100 layers of atoms or molecules, is a case inbetween the scattering rod from a monolayer and the CTR. The scattering amplitude S(q) of a system of N layers can be evaluted to be

S(q) = f(q) sin(N q_z c/2)/sin(q_z c/2)

OUTLOOK : There are various effects that can complicate this simple picture. First of all there can be strain in such films, in particular in pseudomorphic films, i.e. if there is a misfit between the in-plane lattice constants of the substrate and the bulk constants of the film, however, the substrate adsorbate interaction is sufficiently strong to force the film to grow with the substrate lattice constants. The strain can result in a variation of the lattice spacing perpendicular to the surface.