International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F, ch. 2.1, p. 53
Section 2.1.4.2. Scattering by a system of two electrons^{a}Laboratory of Biophysical Chemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands |
This can be derived along classical lines by calculating the phase difference between the X-ray beams scattered by each of the two electrons. A derivation based on quantum mechanics leads exactly to the same result by calculating the transition probability for the scattering of a primary quantum , given a secondary quantum (Heitler, 1966, p. 193). For simplification we shall give only the classical derivation here. In Fig. 2.1.4.2, a system of two electrons is drawn with the origin at electron 1 and electron 2 at position r. They scatter the incident beam in a direction given by the vector s. The direction of the incident beam is along the vector . The length of the vectors can be chosen arbitrarily, but for convenience they are given a length . The two electrons scatter completely independently of each other.
Therefore, the amplitudes of the scattered beams 1 and 2 are equal, but they have a phase difference resulting from the path difference between the beam passing through electron 2 and the beam passing through electron 1. The path difference is . Beam 2 lags behind in phase compared with beam 1, and with respect to wave 1 its phase angle is where .
From Fig. 2.1.4.3, it is clear that the direction of S is perpendicular to an imaginary plane reflecting the incident beam at an angle θ and that the length of S is given by The total scattering from the two-electron system is if the resultant amplitude of the waves from electrons 1 and 2 is set to 1. In an Argand diagram, the waves are represented by vectors in a two-dimensional plane, as in Fig. 2.1.4.4(a).^{1} Thus far, the origin of the system was chosen at electron 1. Moving the origin to another position simply means an equal change of phase angle for all waves. Neither the amplitudes nor the intensities of the reflected beams change (Fig. 2.1.4.4b).
References
Heitler, W. G. (1966). The quantum theory of radiation, 3rd ed. Oxford University Press.