Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 4.5, p. 568   | 1 | 2 |

Section Fibre specimens

R. P. Millanea Fibre specimens

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A wide variety of kinds of fibre specimen exist. All exhibit preferred orientation; the variety results from variability in the degree of order (crystallinity) in the lateral plane (the plane perpendicular to the axis of preferred orientation). This leads to categorization of three kinds of fibre specimen: noncrystalline fibres, in which there is no order in the lateral plane; polycrystalline fibres, in which there is near-perfect crystallinity in the lateral plane; and disordered fibres, in which there is disorder either within the molecules or in their crystalline packing (or both). The kind of fibre specimen affects the kind of diffraction pattern obtained, the relationships between the molecular and crystal structures and the diffraction data, methods of data collection, and methods of structure determination.

Noncrystalline fibres are made up of a collection of molecules that are oriented. This means that there is a common axis in each molecule (referred to here as the molecular axis), the axes being parallel in the specimen. The direction of preferred orientation is called the fibre axis. The molecule itself is usually considered to be a rigid body. There is no other ordering within the specimen. The molecules are therefore randomly positioned in the lateral plane and are randomly rotated about their molecular axes. Furthermore, if the molecule does not have a twofold rotation axis normal to the molecular axis, then the molecular axis has a direction associated with it, and the molecular axes are oriented randomly parallel or antiparallel to each other. This is often called directional disorder, or the molecules are said to be oriented randomly up and down. The average length of the ordered molecular segments in a noncrystalline fibre is referred to as the coherence length.

Polycrystalline fibres are characterized by molecular segments packing together to form well ordered microcrystallites within the specimen. The crystallites effectively take the place of the molecules in a noncrystalline specimen as described above. The crystallites are oriented, and since the axis within each crystallite that is aligned parallel to those in other crystallites usually corresponds to the long axes of the constituent molecules, it is also referred to here as the molecular axis. The crystallites are randomly positioned in the lateral plane, randomly rotated about the molecular axis, and randomly oriented up or down. The size of the crystalline domains can be characterized by their average dimensions in the directions of the a, b and c unit-cell vectors. However, because of the rotational disorder of the crystallites, any differences between crystallite dimensions in different directions normal to the fibre axis tend to be smeared out in the diffraction pattern, and the crystallite size is usefully characterized by the average dimensions of the crystallites normal and parallel to the fibre axis.

The molecules or crystallites in a fibre specimen are not perfectly oriented, and the variation in inclinations of the molecular axes to the fibre axis is referred to as disorientation. Assuming that the orientation is axisymmetric, then it can be described by an orientation density function [\Omega (\alpha)] such that [\Omega (\alpha)\;{\rm d}\omega] is the fraction of molecules in an element of solid angle [{\rm d}\omega] inclined at an angle α to the fibre axis. The exact form of [\Omega (\alpha)] is generally not known for any particular fibre and it is often sufficient to assume a Gaussian orientation density function, so that [\Omega (\alpha) = {1 \over 2\pi \alpha_{0}^{2}} \exp \left(-{\alpha^{2} \over 2\alpha_{0}^{2}}\right), \eqno(]where [\alpha_{0}] is a measure of the degree of disorientation.

Fibre specimens often exhibit various kinds of disorder. The disorder may be within the molecules or in their packing. Disorder affects the relationship between the molecular and crystal structure and the diffracted intensities. Disorder within the molecules may result from a degree of randomness in the chemical sequence of the molecule or from variability in the interactions between the units that make up the molecule. Such molecules may (at least in principle) form noncrystalline, polycrystalline or partially crystalline (described below) fibres. Disordered packing of molecules within crystallites can result from a variety of ways in which the molecules can interact with each other. Fibre specimens made up of disordered crystallites are referred to here as partially crystalline fibres.

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