Scattering four-point pattern

In oriented systems (fibres or stretched films), the scattered image often appears as a two-bar or a four-point pattern with the scattering maximum at or near the meridian (fibre axis). The one-dimensional scattered intensity along the meridian must be calculated by the projection method using the following fonnalism... 

The lamellar reflections in small-angle scattering patterns from polymer fibers are often spread onto a curve symmetrical about the fiber axis. These are usually referred to as two-or four-point patterns, the latter sometimes resembling the butterfly pattern frequently found in light scattering. We recently showed that these 2-D patterns could be best analyzed if we describe die intensity distribution in elliptical coordinates because the intensity maxima of the lamellar reflections from oriented polymers fall on an elliptical curve. We now present new analysis to support this assertion. We will also discuss die physical basis for some of the features in the SAXS pattern in terms of misorientation of the lamellar stacks, deformation of the lamellae, and possible correlation between the lamellar spacing and the orientation of the lamellae. 

Fig. 18. SAXS pattern of PBZO (a) as-spun fibers, (b) schematic of HT fibers, exhibiting four-point scattering. Reprinted from Ref. 34, by courtesy of Marcel Dekker, Inc.

Interpretation of Anisotropic 2D Scattering Pattern Polymer crystallization under external fields (e.g., shear or extensional flows, stretching of polymer solids) usually exhibits preferred orientation, which results in anisotropic scattering patterns. For example, a four-point SAXS pattern was seen during solid-state uniaxial deformation of an ethylene-propylene copolymer (both experimental data and fitted 3D plot are shown in Fig. 1.17) [108]. 

In the third interval (s = 140-510%), the two-point scattering patterns are present only in the measurements carried out under stress. Those taken in the relaxed state are distinguished by a perfect four-point shape (Figure 8,... 

The graph is obtained by plotting Y,- against Y, results for each of the ten laboratories. The axes are drawn such that the point of intersection is at the mean values for Y, and 7/. As a single method is used in the trial, the circle represents the standard deviation of the pooled Y and Y data. The plot shows the predominance of systematic error over random error. Ideally, for bias-free data (i.e. containing no systematic error) the points would be clustered around the mid-point with approximately equal numbers in each of the four quadrants formed by the axes. In practice the points lie scattered around a 45° line. This pattern has been observed with many thousands of collaborative trials. 

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