Rulands Method

The method suggested by Ruland34 addresses two problems associated with the use of Equation (3.62) to evaluate the degree of crystallinity. The first is that lattice imperfections in the crystalline phase cause some of the diffracted intensities to be diverted from the Bragg peaks to the diffuse background. The second is that the intensity is measured experimentally only up to some finite upper limit in s and not to infinity as demanded by (3.62). 

Suppose that the scattered intensity I(s) is separated into its crystalline and amorphous components in such a way that ICT(s) obtained includes only the scattered intensity in the Bragg peaks and not the one diverted to the background. The invariant Qcr based on ICT(s) can then be written as 

The fact that the effect of thermal vibrations and other imperfections of the first kind can be represented by an expression of the form (3.64) has been explained in Section 3.4.4. Ruland34 showed that for the imperfections of the second kind the distortion effect can also be expressed in the form of (3.64) at least approximately. 

by writing the distortion factor D(s) as in (3.64), the constant B is understood to incorporate the effect of imperfections of these different kinds together. 

To deal with the second problem concerning the upper limit of integration in (3.65), we note that in any material there is always a shortest distance n below which two atoms cannot approach each other, and thus the intensity function I(s) for s beyond Smax 1/ min merely reflects the electron density distribution within individual atoms only, so that the equality 

---

WAXS the scattering is completely described by the interactions of neighboring atoms along a single chain, the so-called single-chain structure factor. Cf. descriptions of the Ruland method [14] in textbooks [7,22],... 

The following crystallinity values were obtained for poly(propylene-sfflf-ethylene) by X-ray diffraction using the Ruland method and by IR spectroscopy... 

This property is readily established from the definition of Fourier transform and convolution. In scattering theory this theorem is the basis of methods for the separation of (particle) size from distortions (Stokes [27], Warren-Averbach [28,29] lattice distortion, Ruland [30-34] misorientation of anisotropic structural entities) of the scattering pattern. 

In WAXS of soft condensed matter, studies of the intensity in absolute units are not common, unless the method for the exact determination of X-ray crystallinity according to Ruland is applied (cf. Sect. 8.2.4). 

Precision determinations of wc by means of WAXS measurements are carried out by the Ruland [14] method. The method is sufficiently described in textbooks [7, 22], A modified version adapted to automatic processing by a computer has been introduced by Vonk [108],... 

Figure 8.48 visualizes the selectivity of this Ruland-Smarsly method for... 

The influence of finite size and imperfect orientation of the entities on the shape of the reflections. Separation of unimodal orientation distributions by means of Ruland s streak method, and assessment of the analytical shape of the orientation distribution (Sect. 9.7). 

Motivation and Principle. Broadened reflections are characteristic for soft matter. The reason for such broadening is predominantly both the short range of order among the particles in the structural entities, and imperfect orientation of the entities themselves. A powerful method for the separation of these two contributions is Ruland s streak method [30-34], Short range of order makes that the reflection is considerably extended in the radial direction of reciprocal space - often it develops the shape of a streak. This makes it practically possible to measure reflection breadths separately on several11 nested shells in reciprocal space. As a function of shell diameter one of the contributions is constant, whereas the other is changing12. If the measurement is performed on spheres (azimuthal), the orientation component is constant. 

History. Wilke [129] considers the case that different orders of a reflection are observed and that the orientation distribution can be analytically described by a Gaussian on the orientation sphere. He shows how the apparent increase of the integral breadth with the order of the reflection can be used to separate misorientation effects from size effects. Ruland [30-34] generalizes this concept. He considers various analytical orientation distribution functions [9,84,124] and deduces that the method can be used if only a single reflection is sufficiently extended in radial direction, as is frequently the case with the streak-shaped reflections of the anisotropic... 

Figure 9.7. Separation of misorientation (Bg) and extension of the structural entities (1/ (L)) for known breadth of the primary beam (Bp) according to Ruland s streak method. The perfect linearization of the observed azimuthal integral breadth measured as a function of arc radius, s, shows that the orientation distribution is approximated by a Lorentzian with an azimuthal breadth Bs... 

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. 

Ruland, W. Smarsly, B. 2005. SAXS of self-assembled nanocomposite films with oriented two-dimensional cylinder arrays An advanced method of evaluation. J. Appl. Cryst. 38 78-86. 

Methods for estimating lattice distortion generally require two or more orders of a particular reflection to be present, and most polymers have only one order available. A method for estimating both crystallinity and lattice disorder, which does not need higher orders of a reflection, and indeed takes into account the whole of the diffraction trace, is that due to Buland (27). This method has been applied to many different fibres by Sotton and his colleagues, who have discussed their results both here (28) and elsewhere (12). The major problem with Ruland s method is that an arbitrary separation of the crystalline scatter from the non-crystalline scatter must be made other restrictions are that the method cannot be used to measure crystallite size and cannot give any indication of the presence of paracrystalline or intermediate-phase material. 

Nevertheless, when we carry out x-ray crystallinity measurements on textile fibers, we must consider distortions that always affect crystalline material. Even in a completely crystalline material, the scattered x-ray intensity is not located exclusively in the diffraction peaks. That is because the atoms move away from their ideal positions, owing to thermal motion and distortions. Therefore, some of scattered x-rays are distributed over reciprocal space. Because of this distribution, determinations of crystallinity that separate crystalline peaks and background lead to an underestimation of the crystalline fraction of the polymer. In this paper, we attempt to calculate the real crystallinity for textile fibers from apparent values measured on the x-ray pattern. This is done by taking into account the factor of disorder following Ruland s method (3). 

Corrections of the apparent crystallinity values of fibers materials have been carried out by taking into account a disorder parameter k, following Ruland s method. Peculiar care was taken about samples preparation (cutting and pelleting of fibers), data collection and reduction, which will be briefly described. Crystallinity and disorder parameter measurements have been performed on main textile fibers (polyester, polyamide, aramid, polypropylene, cellulosic fibers) and the results will be discussed comparatively, with those got by more conventional x-ray crystallinity determinations. The complementarities of these different approaches will be illustrated with several examples. For instance,... 

Electron diffraction crystallinity X-Ray crystallinity (RULAND s Method)... 

This process consists of obtaining the pure profde f, towards which tend the functions f when n increases, by convoluting the measured function h with the instrumental function. One of the difficult aspects of this type of method is achieving the convergence of the iterative process toward the solution. This problem was studied by Ruland [RUL 71], who defined an efficient convergence criterion. 

---