Arbitrary angle, scattering

Beaucage, G., Small-angle scattering from polymeric mass fractals of arbitrary mass-fractal dimension, J. Appl. Crystallogr, 29, 134, 1996. 

Fig. 8. Small angle X-ray scattering pattern of a sample aligned in a 1 T magnetic field. B is the field direction, which is set at an arbitrary angle of =40° to the main capillary axis. The white rectangle at the center of the figure is due to the beam stop...

In general, the principal axes of a will be at some arbitrary angle relative to the laser polarization. The scattered intensity over all orientations of the polarizability tensor for the vibration contained within the scattering volume is given by... 

The ix axis has been chosen in the iz-i plane this choice is arbitrary and serves to define the axis from which the polar scattering angle e is measured. Thus we have ... 

All samples in this chapter are assumed to consist of multiple scattering and fluorescent planar layers of arbitrary thickness d. One surface of the layer is irradiated monochromatically at A with parallel light ofintensity /o (per unit area perpendicular to the direction of propagation) under the angle ao relative to the normal of the surface... 

Angular light scattering measurements are sometimes classified as either absolute or relative. In an absolute measurement 7s//(, which is directly related to the differential scattering cross section (13.2), is determined in a relative measurement the irradiance is referred to some arbitrary scattering angle, say 10°, so that (assuming azimuthal symmetry)... 

Relative measurements are considerably easier to make and are the type most commonly reported. However, absolute measurements are of importance, for example, in comparing measured scattering cross sections of nonspherical particles with calculations for equivalent spheres. Note that absolute as we are using the term here means that scattering is not normalized to some arbitrary reference angle it does not mean that absolute irradiances are measured, as with calibrated detectors. In both relative and absolute measurements, it is relative (i.e., dimensionless) irradiances that are determined. 

Therefore, the Zimm-plot gives a family of parallel lines which appear shifted along the abscissa by the arbitrary constant k. These line can be extrapolated for each concentration to zero scattering angle, and for each angle an extrapolation to zero concentration can be carried out. This Zimm-plot finally results into the two limiting lines Kc/R0 c = 0... 

Consider one arbitrary voxel of the extended object located at the position r relative to some coordinate scheme. If the voxel is small relative to the dimensions of the measuring arrangement, solid angle factors can be assumed to be constant and the total scatter signal can be represented in the following form ... 

---