Rulands Theory of Affine Deformation

In a fundamental paper [265] Ruland develops an advanced method for the analysis of scattering patterns showing moderate anisotropy. The deduction is based on a 3D model and the concept of highly oriented lattices. The addition of distortion terms makes sure that the theory is applicable to distorted structures and their scattering. 

According to his deduction the common finding of ellipsoidal deformation of the reflections is indicative for affine deformation. Moreover, he arrives at an equation that permits to determine with high accuracy the microscopical draw ratio, A, of the stmctural entities from the ellipticity of the deformed Debye sphere. This value can be compared to the macroscopical draw ratio. Even the intensity distribution along the ellipsoidal ridge is predicted for a bcc-lattice of spheres, and deviations of experimental data are discussed. 

Let (yi2, S3) be the coordinates of reflection maxima determined on radial rays in the scattering pattern. Then a linearizing plot of the ellipsoidal shape is 

The clear linearity of the data demonstrates the affine character of the deformation. From the intercept, tP-, the semiminor axis of the rotation ellipsoid is determined. After transformation to units of reciprocal-space, the meridional long period follows from Eq. (10.3). 

Modeling the initial structure by spherical domains in a bcc-lattice the theoretical intensity along the ellipsoidal ridge as a function of the angle y/ between fiber axis and the direction of the radial beam is 

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