Reciprocal-space

Reciprocal-space is a conceptual three-dimensional space which contains the full 3D scattering pattern of a given sample. It is the 3D Fourier transform of the sample's realspace electron-density distribution.

The distribution of intensity in reciprocal-space can be arbitrarily complex. For a sample with a distinct and non-trivial internal structure, the Fourier transform will be highly complex and difficult to interpret. However, samples of scientific interest generally have certain regularities, which give rise to a reciprocal-space with recognizable features, making the data easier to understand and analyze. In particular, a well-defined repeating structure in realspace gives rise to a sharp peak in reciprocal-space. The position of the peak encodes the repeat-spacing (the peak width encodes the correlation length, etc.).

Detector Image

The concept of reciprocal-space is extremely useful, because it allows one to connect between experiment geometry and the scattering equations. In particular, the scattering observed on a 2D detector plane during an experiment arises is actually a particular 'slice' through the 3D reciprocal-space. Thus, one can probe different regions of reciprocal-space by reorienting the sample with respect to the beam.

The 'slice' through reciprocal-space is not strictly a plane: it is actually the surface of a sphere, known as the Ewald sphere.

Caveats

Although it is extremely useful to imagine that the detector image is a slice through reciprocal-space, one must keep in mind other effects can influence what one observes on the detector. For instance, anything which affects the scattered beams as they travel to the detector will influence the image one sees. For example, ambient air will scatter the beam, giving rise to a background signal (which will look like diffuse scattering). As another example, portions of the instrument (e.g. sample windows) can attenuate or obstruct the x-ray beam.

The sample itself can also affect the scattered rays, again making the detector image not strictly a map of reciprocal-space. The scattering may be attenuated as it transits through the sample; this attenuation may be non-uniform. In GISAXS, the observed image is distorted due to refraction of both the incident beam and the scattered rays. This refraction distortion is non-trivial. Dynamic scattering also gives rise to multiple scattering peaks (some due to the direct beam, and some due to the reflected beam), as well as more complex effects such as the Yoneda streak and waveguide streaks.

Thus, especially in GISAXS, one must be careful about naively interpreting the detector image as being a direct replica of reciprocal-space. On the other hand, all of the complications noted above can be accounted for; thus GISAXS data can be quantitatively mapped into the 'true' reciprocal-space.

An example of a detector image affected by the experimental setup: the dark regions on the right side of this GIWAXS image are actually the shadow of a screw (sitting on top of washers) that was at the corner of the sample-holding bar. It blocked the scattering from the sample. Although silly, this image is a good reminder that the experimental setup (sample chamber windows, air scattering, etc.) affects the x-ray scattering image.