Transition from microscopic to macrosopic description:
Molecular monolayers and polymer thin films

The problem of of molecule sitting on a surface is rather involved, due to the large number of degrees of freedom. If the structure of the molecule is known, then it is possible to use the rigid molecule approximation. The molecule is moved around its Eulerian angles f, c, and q and the three translations x, y and z rather than trying to handle all 3N atom coordinates separately.

Sometimes the atomic description can be inconvenient altogether, for instance, let's think of a monolayer of molecules at the air-water interface or on a solid surface. Amphiphilic molecules consist of a hydrophilic head group (such as -COOH or -OH) and a hydrophobic tail, often a long alkyl chain of a dozen to two dozen CH2 groups. Such molecules form ordered phases where the molecules are free to rotate at their lattice position. If we do the rotational averaging of the atomic electron density before we build our structure model, we obtain an averaged electron density:
rav(r) = òrat( R(f)r) Prob(f) df
where the rotational probability density Prob(f) is simply 1/2p in this case and R(f) is the associated rotation matrix. Obviously, this approach can be easily generalized to other applications. The form factor of such an object is the Fourier transform of rav(r).

If we are considering the form factor of a rod-like molecule including rotational averaging, we get a disk in reciprocal space. Als-Nielsen et al. have shown, how this form factor can be used to determine the tilt angle of rod-like molecules that crystallize at the air-water interface: The 2D lattice of the molecules gives rise to scattering rods as we have seen before. The scattering intensity along the rods depends as usual on the product of form factor and structure factor. When the molecules are tilted, the form factor disk tilts along with them and scattering maxima appear on certain rods. By analyzing the maxima on all low-index rods not only the tilt, but also the direction of the tilt (nearest neighbor, next nearest neighbor, inbetween) can be determined. This is a remarkable result, since the molecules only form a 2D powder at the air-water interface and the scattering rods are rotationally averaged into cylinders of scattering.
 

Polymer thin films

We can go one step further and consider films of diblock copolymers. A diblock copolymer consists of two inmiscible polymer chains that are chemically bound to each other. In order to minimize their surface energy, diblock copolymers form a variety of ordered structures, the simplest being lamellae (symmetric blocks, i.e. same chain length), or regularly spaced spheres or cylinders. In the bulk such systems form 3D powders, however, in a thin film the objects can be aligned. for instance parallel and perpendicular lamellae or cylinders have been found. Polymer thin films of typically 500 to 2000 A thickness are too subtle objects to study with conventional SAXS. However, in grazing-incidence geometry the scattering can be observed. The so-called GISAXS technique combines features from GID and XR. Presently there is an active research program at CHESS and G-line, to exploit GISAXS for the characterization of polymer thin films  (Smilgies, Gruner, Ober, Wiesner).