Tables for
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 6.4, p. 609

Section 6.4.1. Introduction

T. M. Sabinea

aANSTO, Private Mail Bag 1, Menai, NSW 2234, Australia

6.4.1. Introduction

| top | pdf |

In a diffraction experiment, a beam of radiation passes into the interior of a crystal via an entrance surface. When the crystal is set for Bragg reflection, a diffracted beam passes out of the crystal through the exit surface. The scattering vector is allowed to sweep over a small region around the exact Bragg position, and the total intensity scattered into a detector is recorded. This record (called the integrated intensity) contains all the information available to the crystallographer, and includes the starting point for crystal structure analysis, which is the relative magnitude of the structure factor for the reflection under examination. In order to determine the relationship between the structure factor and the observed integrated intensity, it is necessary to take into account all processes that remove intensity from the incident and diffracted beams during their passage through the crystal.

The standard theory, which is called the kinematic theory, assumes that there is no attenuation of either the incident beam or the diffracted beam during the diffraction process. The fact that this is not so in a real crystal is expressed in the following way. [I^{\rm obs}=EI^{\rm kin}.]E is equal to unity for a crystal, called the ideally imperfect crystal, which scatters in accordance with the kinematical approximation. In kinematic theory, the integrated intensity is proportional to the square of the magnitude of the structure factor, [\left | F\right | ^2]. The factor E has a value very much less than unity for a perfect crystal, to which the dynamical theory of diffraction applies. In the dynamical theory, the integrated intensity is proportional to the first power of the structure-factor magnitude, [|F|]. All crystals in nature lie between these limits, either because of the microstructure of the crystal, or because of the removal of photons or neutrons by electronic or nuclear processes. A real crystal may behave as a perfect crystal for some reflections and as an imperfect crystal for others. It is the purpose of this section to give formulae by which the value of [\left | F\right |] can be extracted from the measured intensity of a reflection from a real crystal without resorting to mechanical methods of changing its state of perfection. Those methods, such as quenching or irradiation with fast neutrons, usually introduce problems that are more difficult than those that they were designed to solve.

The formulae are constructed so that they apply at all angles of scattering, and are either analytic or rapidly convergent for ease of use in least-squares methods.

It should be noted that whenever the symbol F subsequently appears it should be interpreted as the modulus of the structure factor, which always includes the Debye–Waller factor.

In common with all published theories of extinction, the theory presented here is phenomenological, in that the assumption is made that the wavevector within the crystal does not differ from the wavevector in vacuum whatever the strength of the interaction between the incident radiation and the crystal. The results will be compared with a solution in the symmetric Bragg case in which the dynamic refractive index of the crystal has been taken into account (Sabine & Blair, 1992[link]).


Sabine, T. M. & Blair, D. G. (1992). The Ewald and Darwin limits obtained from the Hamilton–Darwin energy transfer equations. Acta Cryst. A48, 98–103.

to end of page
to top of page