Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 4.5, pp. 469-470   | 1 | 2 |

Section Polycrystalline fibres

R. P. Millanea* Polycrystalline fibres

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A polycrystalline fibre is made up of crystallites that are oriented parallel to each other, but are randomly positioned and randomly rotated about their molecular axes. The recorded diffraction pattern is the intensity diffracted by a single crystallite, cylindrically averaged about the Z axis. On a fibre diffraction pattern, therefore, the Bragg reflections are cylindrically projected onto the (R, Z) plane and their positions are described by the cylindrically projected reciprocal lattice (Finkenstadt & Millane, 1998[link]).

The molecules are periodic and are therefore usually aligned with one of the unit-cell vectors. Since the z axis is defined as the fibre axis, it is usual in fibre diffraction to take the c lattice vector as the unique axis and as the lattice vector parallel to the molecular axes. It is almost always the case that the fibre is rotationally disordered about the molecular axes, i.e. about c. Consider first the case of a monoclinic unit cell [(\alpha = \beta = 90^{\circ})] so that the reciprocal lattice is cylindrically projected about [{\bf c}^{*}]. The cylindrical coordinates of the projected reciprocal-lattice points are then given by [R_{hkl}^{2} = h^{2} a^{*2} + k^{2} b^{*2} + 2hka^{*} b^{*} \cos \gamma^{*} \eqno(] and [Z_{hkl} = lc^{*}, \eqno(] so that R depends only on h and k, and Z depends only on l. Reflections with fixed h and k lie on straight row lines. Certain sets of distinct reciprocal-lattice points will have the same value of [R_{hkl}] and therefore superimpose in cylindrical projection. For example, for an orthorhombic system [(\gamma = 90^{\circ})] the reciprocal-lattice points (hkl), [(\bar{h}kl)], [(h\bar{k}l)] and [(\bar{h}\bar{k}l)] superimpose. Furthermore, the crystallites in a fibre specimen are usually oriented randomly up and down so that the reciprocal-lattice points (hkl) and [(hk\bar{l})] superimpose, so that in the orthorhombic case eight reciprocal-lattice points superimpose. Also, as described below, reciprocal-lattice points that have similar values of R can effectively superimpose.

If the unit cell is either triclinic, or is monoclinic with [\alpha \neq 90^{\circ}] or [\beta \neq 90^{\circ}], then [{\bf c}^{*}] is inclined to c and the Z axis, and the reciprocal lattice is not cylindrically projected about [{\bf c}^{*}]. Equation ([link] for [Z_{hkl}] still applies, but the cylindrical radius is given by [\eqalignno{R_{hkl}^{2} &= h^{2} a^{*2} + k^{2} b^{*2} + l^{2} [c^{*2} - (1/c^{2})] + 2hka^{*} b^{*} \cos \gamma^{*}\cr &\quad + 2hla^{*} c^{*} \cos \beta^{*} + 2klb^{*} c^{*} \cos \alpha^{*} &(}] and the row lines are curved (Finkenstadt & Millane, 1998[link]).

The most complicated situation arises if the crystallites are rotationally disordered about an axis that is inclined to c. Reciprocal space is then rotated about an axis that is inclined to the normal to the [a^{*} b^{*}] plane, [R_{hkl}] and [Z_{hkl}] are both functions of h, k and l, equation ([link] does not apply, and reciprocal-lattice points for fixed l do not lie on layer lines of constant Z. Although this situation is rather unusual, it does occur (Daubeny et al., 1954[link]), and is described in detail by Finkenstadt & Millane (1998[link]).

The observed fibre diffraction pattern consists of reflections at the projected reciprocal-lattice points whose intensities are equal to the sums of the intensities of the contributing structure factors. The observed intensity, denoted by [I_{l} (R_{hk})], at a projected reciprocal-lattice point on the lth layer line and with [R = R_{hk}] is therefore given by (assuming, for simplicity, a monoclinic system) [I_{l} (R_{hk}) = {\textstyle\sum\limits_{h', \,  k' \in {\scr S} (h, \,  k)}} |F_{h' k' l}|^{2}, \eqno(] where [{\scr S} (h, k)] denotes the set of indices [(h', k')] such that [R_{h' k'} = R_{hk}]. The number of independent reflections contributing in equation ([link] depends on the space-group symmetry of the crystallites, because of either systematic absences or structure factors whose values are related.

The effect of a finite crystallite size is to smear what would otherwise be infinitely sharp reflections into broadened reflections of a finite size. If the average crystallite dimensions normal and parallel to the z axis are [l_{\rm lat}] (i.e. in the `lateral' direction) and [l_{\rm axial}] (i.e. in the `axial' direction), respectively, the profile of the reflection centred at [(R_{hk}, Z = l/c)] can be written as (Fraser et al., 1984[link]; Millane & Arnott, 1986[link]; Millane, 1989c[link]) [I (R, Z) = I_{l} (R_{hk}) S (R - R_{hk}, Z - l/c), \eqno(] where the profile function [S(R, Z)] can be approximated by [S (R, Z) = \exp [-\pi (l_{\rm lat}^{2} R^{2} + l_{\rm axial}^{2} Z^{2})]. \eqno(]

The effect of crystallite disorientation is to smear the reflections given by equation ([link] about the origin of the projected reciprocal space. If the effects of disorientation dominate over those of crystallite size, then the profile of a reflection can be approximated by (Fraser et al., 1984[link]; Millane & Arnott, 1986[link]; Millane, 1989c[link]) [\eqalignno{I (\rho, \sigma) &\simeq {I_{l} (R_{hk}) \over 2\pi \alpha_{0} l_{\rm lat} l_{\rm axial} \rho} &\cr&\quad\times\exp \left(- \left[{(\rho - \rho_{hkl})^{2} \over 2\beta_{\rho}^{2}} + {(\sigma - \sigma_{hkl})^{2} \over 2\beta_{\sigma}^{2}}\right]\right), &(\cr}] where [(\rho_{hkl}, \sigma_{hkl})] are the polar coordinates of the reflection, [\beta_{\rho}^{2} = {l_{\rm lat}^{2} l_{\rm axial}^{2} \over 2\pi (l_{\rm lat}^{2} \sin^{2} \sigma_{hkl} + l_{\rm axial}^{2} \cos^{2} \sigma_{hkl})} \eqno(] and [\beta_{\sigma}^{2} = \alpha_{0} + {l_{\rm lat}^{2} l_{\rm axial}^{2} \over 2\pi \rho_{hkl}^{2} (l_{\rm axial}^{2} \sin^{2} \sigma_{hkl} + l_{\rm lat}^{2} \cos^{2} \sigma_{hkl})}. \eqno(]

Reflections that have similar enough [(R, Z)] coordinates overlap severely with each other and are also included in the sum in equation ([link]. This is quite common in practice because a number of sets of reflections may have similar values of [R_{hk}].


First citationDaubeny, R. de P., Bunn, C. W. & Brown, C. J. (1954). The crystal structure of polyethylene terephthalate. Proc. R. Soc. London Ser. A, 226, 531–542.Google Scholar
First citationFinkenstadt, V. L. & Millane, R. P. (1998). Fiber diffraction patterns for general unit cells: the cylindrically projected reciprocal lattice. Acta Cryst. A54, 240–248.Google Scholar
First citationFraser, R. D. B., Suzuki, E. & MacRae, T. P. (1984). Computer analysis of X-ray diffraction patterns. In Structure of crystalline polymers, edited by I. H. Hall, pp. 1–37. New York: Elsevier.Google Scholar
First citationMillane, R. P. (1989c). Relating reflection boundaries in X-ray fibre diffraction patterns to specimen morphology and their use for intensity measurement. J. Macromol. Sci. Phys. B28, 149–166.Google Scholar
First citationMillane, R. P. & Arnott, S. (1986). Digital processing of X-ray diffraction patterns from oriented fibres. J. Macromol. Sci. Phys. B24, 193–227.Google Scholar

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