The relationship between Bragg reflections and disorder in crystalline polymers*

Theory is developed for describing diffraction from polycrystalline fibers of helical molecules, in which the constituent crystallites contain correlated lattice disorder and uncorrelated substitution disorder. The theory utilizes a recently reported description of cylindrically averaged diffraction by distorted lattices that is based in real space and uses an imposed correlation field to describe correlated disorder [Stroud & Millane (1996). Proc. R. Soc. London Ser. A, 452,. The theory developed here is implemented as an efficient numerical algorithm for calculating diffraction patterns from fibers containing correlated disorder. Simulations are used to explore the effects of this kind of disorder and to characterize the disorder in a polynucleotide fiber whose diffraction pattern indicates its presence. This leads to a significant improvement in the agreement between the calculated and measured diffraction patterns over that from a model of only uncorrelated lattice disorder.

Section: Introductionsupporting

confidence: 68%

Diffraction by Polycrystalline Fibers with Correlated Disorder

Stroud¹,

Millane²

1996

“…It has been understood for a considerable time from X-ray diffraction (Clark & Muus, 1962b) and NMR (Hyndman & Origlio, 1960) that individual chains in polytetrafluoroethylene crystals undergo torsional motions after the 293 K transition and that these motions appear to be of even greater magnitude after the 303 K transition. A previous treatment (Clark & Muus, 1962a)of the effect of this disorder dealt only with the crystal Bragg reflections (that is, the sampled intensities on layer lines). We believe that our treatment extends this to explain the appearance of the unsampled continuous layer-line intensities arising from the disorder.…”

Section: The Fourier Transform Of Actinmentioning

confidence: 99%

Structural studies onA-cation-deficient perovskite-related phases. II. Microdomain formation in ThNb₄O₁₂

Labeauu¹,

Grey²,

Joubert³

et al. 1982

Electron diffraction/microscopy and X-ray diffraction techniques have been used to study the thorium/ vacancy ordering and microdomain structures of quenched and slow-cooled samples of the A-cationdeficient perovskite-related phase ThNb4Olz. In both types of samples, there is primary ordering of thorium atoms into the cuboctahedral sites in alternate (001)p layers. The quenched and slow-cooled samples have different secondary orderings of thorium atoms and vacancies in the occupied (001)p layers. In the quenched samples, the thorium atoms and vacancies are ordered in alternate rows parallel to [100]p and [010]p. Short segments (20-50A) of the two orientation variants are statistically distributed in a type of tweed pattern, separated by boundaries that are aligned predominantly parallel to (ll0)p and (ll0)p. In the slow-cooled samples, the ordering of columns of thorium atoms and vacancies parallel to [ll0]p or [1.10]p occurs in microdomains, with domain boundaries parallel to (100)p and (010)p and with average separations of 6ap, 6bp =-24 A. The domains corresponding to the two orientations of thorium columns form a checkerboard pattern of two interpenetrating sets of corner-shared squares. In either set, the ordering of columns is propagated along diagonal rows of corner-shared domains, but there is no correlation between adjacent rows. The NbO 6 octahedra are tilted about the [ll0]p and [ll0]p axes, parallel to the thorium-column orientations, and the domain boundaries act as mirror-twin planes for the octahedral tilt systems. This periodic change in the tilt-axis orientation * Permanent address: CSIRO Division of Mineral Chemistry, PO Box 124, Port Melbourne, Victoria 3207, Australia. 0567-7394/82/060753-09501.00gives rise to characteristic clusters of split superlattice spots in the diffraction patterns for ThNb40~2. Optical transform methods were used to check the validity of microdomain models for both the quenched and the slow-cooled samples. IntroductionIn part I (Alario-Franco, Grey, Joubert, Vincent & Labeau, 1982), we presented electron and X-ray diffraction results for samples of the A-cation-deficient perovskite-related phase Th0.25NbO 3 (i.e. ThNb40~2), which were slowly cooled from the melt. The diffraction patterns showed both strong, sharp reflections and groups of very weak, diffuse satellite reflections. The sharp reflections, which indexed with a tetragonal perovskite supercell (a t = ap, c t = 2Cp), resulted from long-range ordering of thorium atoms into alternate (001)p layers of cuboctahedral A-cation sites, as originally reported by Trunov & Kovba (1966). The diffuse satellite reflections were interpreted as resulting from short-range ordering of thorium atoms and vacancies within the (001)p layers. Splitting of certain groups of satellite reflections was observed, characteristic of a periodic two-dimensional array of domains, with domain boundaries oriented parallel to (100)p and (010)p. However, in order to simplify the interpretation of the diffraction patterns, we consid...

“…The formulas for axial and for translational disorder in the thermal mode are of interest, however. The formula for thermal translational disorder (Clark & Muus, 1962) is similar: it has the same Gaussian transform as we have obtained, that is, exp (-2~V).…”

Section: C'(~ ~)=Exp[-t(a+ G)] Xexp[-t(a-r) Cos2(%+me)]mentioning

confidence: 71%

“…The disorder due to axial, translational and screw displacements along the helical axis of chain molecules in a crystalline array has been previously studied and equations have been derived for these effects (Clark & Muus, 1962). Our treatment refers to disorder within the heli~ itself and is, therefore, not directly comparable.…”

Section: C'(~ ~)=Exp[-t(a+ G)] Xexp[-t(a-r) Cos2(%+me)]mentioning

confidence: 99%

See 1 more Smart Citation

Helical diffraction. I. The paracrystalline helix and disorder analysis

Worthington¹,

Elliott²

1989

In a new approach to helical diffraction a helix generating function is defined, and thence an expression for the autocorrelation function (a.c.f.) for a helix is obtained. The Fourier transform of this a.c.f, gives a new expression for the diffracted intensity, which is shown to be equivalent formally to the classical expression of Cochran, Crick & Vand [Acta Cryst. (1952), 5, 581-586] and A. R. Stokes (unpublished). The new expression allows straightforward examination of the effects of helical disorders on the diffracted intensity. The thermal and paracrystalline effects of disorders with cylindrical symmetry are shown, and examples are given from the diffraction of a model of the actin helix. The general case, disorder with no symmetry, is derived and the effects of axial and radial disorder, separately and together, are computed, again for the model actin helix. Translational disorder is also included, and its effects are explained. The new results are compared with existing accounts of the effects of helical disorders on fibre diffraction.