GISAXS Diffraction Indexing

Indexing of diffraction peaks from a film, where crystallites are oriented along only Z direction and thus are 2D powders. This is also called the fiber symmetry.

In GISAXS experiment for spin-casted organic films, the films often have the fiber symmetry.

This figure shows the Ewald sphere and a reciprocal lattice that is oriented so that the real space lattice's <001> vector is parallel to Z axis in other word its (001) plane is parallel to the film.

If the film contains lots of 2D powders, then there will be the same number of the reciprocal lattices that are rotated around Z axis by as much as each 2D powder is rotated around the Z axis.

The trace of each diffraction peak for the 2D powder is then a circle. The figure, for instance, shows two circles; one for 100 and the other for 101. If the lattice parameter a and b are the same, then the circle for 100 includes 010, -100, and 0-10.

Each circle will meet the Ewald sphere at typically 2 times and minimum 0 depending on the orientation of the fiber axis and peak of interest. In the above example, the fiber axis is Z axis, which is not necessarily true for other systems.

Additionally even in the figure, the trace of 001 peak that is a spot instead of a circle will never hit the Ewald sphere unless the sample is rotated around the h vector.

Followings are the 2D powder diffraction patterns from various lattice types that are common in BCP.

1. BCC. Lattice parameter a = 1.

2. FCC

Space group Fm-3m, Lattice parameter 1

3. Double Gyroid.

Space group Ia-3d, Lattice parameter a = 1;

GISAXS indexing

In GISAXS indexing, one has to consider the reflection and refraction effects. The former produces at least one additional set of diffraction peaks, and the latter shifts positions of all the peaks toward +Z direction (normal to the film surface).

This reflection and refraction effects are both highly incident angle dependent. Thus I calculated patterns with some realistic numbers.

X-ray energy : 8keV

Pixel size in the image: 0.172mm (Pilatus Detector)

Sample to detector distance : 2000m

Incident angle: 0.17 degree

Electron density of a film : 340e/nm3.

On the image, direct beam position: [0, 0]

Lattice parameter: 350nm

Red circles: diffraction peaks by the transmitted beam

Blue squares: diffraction peaks by the reflected beam

1. BCC

<110> // Z is very common orientation in BCP because the particles on the first layer would like to possess 2D hexagonal packing.

2 FCC

<111>//Z orientation is the most common because the particles on the first layer would like to possess 2D hexagonal packing.

3. Gyroid