International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 6.2, pp. 596598
https://doi.org/10.1107/97809553602060000601 Chapter 6.2. Trigonometric intensity factorsA summary of the expressions for the intensity of diffraction as recorded by various techniques, including the fundamental constants as well as the trigonometric factors, is provided. Notes on the polarization factor, the angularvelocity factor, the Lorentz factor and special factors in the powder method are included. Some remarks are also made about the integrated reflection power ratio formulae for singlecrystal slabs. 
The expressions for the intensity of diffraction of Xrays contain several trigonometrical factors. The earlier series of International Tables (Kasper & Lonsdale, 1959, 1972) gave extensive tables of these functions, but such tables are now unnecessary, as the functions are easily computed. In fact, many crystallographers can ignore the trigonometric factors entirely, as they are built into `blackbox' dataprocessing programs. The formulae for singlecrystal reflections (b) and (c) of Table 6.2.1.1 in the previous edition (Lipson & Langford, 1998) list only the integrated reflection power ratio (i.e. integrated reflection) under the strong absorption case. The revised formulae given here include both the reflection power ratio and the integrated reflection power ratio for a crystal slab of finite thickness with any values of the ratio of the absorption to the diffraction cross sections and under all possible kinds of diffraction geometry.

A conspectus of the expressions for the intensity of diffraction as recorded by various techniques, including the fundamental constants as well as the trigonometric factors, is given in Table 6.2.1.1. Details of the techniques are given elsewhere in this volume (Chapters 2.1 – 2.3 ) and in textbooks, such as those of Arndt & Willis (1966) for singlecrystal diffractometry and Klug & Alexander (1974) for powder techniques. Notes on individual factors follow.
Xrays are an electromagnetic radiation, and the amplitude with which they are scattered is proportional to the sine of the angle between the direction of the electric vector of the incident radiation and the direction of scattering. Synchrotron radiation is practically planepolarized, with the electric vector in the plane of the ring, but the radiation from an ordinary Xray tube is unpolarized, and it may thus be regarded as consisting of two equal parts, half with the electric vector in the plane of scattering, and half with the electric vector perpendicular to this plane. For the latter, the relevant angle is π/2, and for the former it is . The intensity is proportional to the square of the amplitude, so that the polarization factor – really the nonpolarization factor – is If the radiation has been `monochromatized' by reflection from a crystal, it will be partially polarized, and the two parts of the beam will be of unequal intensity. The intensity of reflection then depends on the angular relations between the original, the reflected, and the scattered beams, but in the commonest arrangements all three are coplanar. The polarization factor then becomes where and is the Bragg angle of the monochromator crystal. The expression (6.2.2.2) may be substituted for (6.2.2.1) in Table 6.2.1.1 whenever appropriate.
In experiments where the crystal is rotated or oscillated, reflection of Xrays takes place as a reciprocallattice point moves through the surface of the sphere of reflection. The intensity is thus proportional to the time required for the transit of the point through the surface, and so is inversely proportional to the component of the velocity perpendicular to the surface. In most experimental arrangements – the precession camera (Buerger, 1944) is an exception – the crystals move with a constant angular velocity, and the perpendicular component of the velocity varies in an easily calculable way with the `latitude' of the reciprocallattice point referred to the axis of rotation. If the reciprocallattice point lies in the equatorial plane and the radiation is monochromatic – the most important case in practice – the angularvelocity factor is If the latitude of the reciprocallattice point is , a somewhat more complex calculation shows that the factor becomes For , the expression (6.2.3.2) reduces to (6.2.3.1). In some texts, is used for the colatitude; this and various trigonometric identities can give superficially very different appearances to (6.2.3.2).
There has been some argument over the meaning to be attached to the term Lorentz factor, probably because Lorentz did not publish his results in the ordinary way; they appear in a note added in proof to a paper on temperature effects by Debye (1914). Ordinarily, Lorentz factor is used for the trigonometric part of the angularvelocity factor, or its equivalent, if the sample is stationary. (See below).
In the powder method, all rays diffracted through an angle lie on the surface of a cone, and in the absence of preferred orientation the diffracted intensity is uniformly distributed over the circumference of the cone. The amount effective in blackening film, or intercepted by the receiving slit of a diffractometer, is thus inversely proportional to the circumference of the cone, and directly proportional to the fraction of the crystallites in a position to reflect. When allowance is made for these geometrical factors, it is found that for the Debye–Scherrer and diffractometer arrangements the intensity is proportional to where p′′ is the multiplicity factor (the number of permutations of hkl leading to the same value of ). For the flatplate frontreflection arrangement, the variation becomes Combining the polarization, angularvelocity, and special factors gives a trigonometric variation of for the Debye–Scherrer and diffractometer arrangements, and for the flatplate frontreflection arrangement.
Nowadays, angledispersive experiments are normally carried out by stepping the sample and detector in small angular increments, both being stationary while the intensity at each step is recorded. The Lorentz factor for a random powder sample is then of the form . The factor arises from the fact that spherical shells of diffracted intensity in reciprocal space intersect the Ewald sphere at an angle that depends on , and the surface area of the shells increases as , which is embodied in the factor . It turns out that this factor is equivalent to a combination of the polarization factor (6.2.2.1), the angularvelocity factor (6.2.3.1) and (6.2.5.1), and the form of (6.2.5.3) is thus unchanged.
The transfer equations for intensity may be rewritten in the form of onedimensional power transfer equations (Hu & Fang, 1993). The in (b) and (c) for a mosaic crystal slab under symmetrical and unsymmetrical Bragg and Laue geometries are the general solutions of power transfer equations employing three dimensionless parameters b, ξ and τ. For a crystal slab with a rectangular mosaic distribution, considering multiple reflection, the integrated reflection power ratio, ρ′, can be obtained by substituting σ_{0} for σ in the formulae for and multiplying the result by the mosaic width. However, for crystals with other kinds of mosaic distribution, the corresponding ρ′ can be obtained only by integrating the expression for over the whole range of . Formulae (1)–(3) listed in Table 6.3.3.1 , i.e. the transmission coefficient A multiplied by Q, QA, are identical to those of (b) and (c) for the case of , which is the integrated reflection power ratio for a crystal slab based on the kinematic approximation without consideration of multiple reflection.
The secondary extinction factor for Xray or neutron diffraction in a mosaic crystal slab can be obtained as ρ′/(QA), in which the integrated reflection power ratio with consideration of multiple reflections can be obtained as described above.
Both the transmission power ratio and the absorption power ratio can also be obtained by solving the power transfer equations. For details, see Hu (1997a,b), Werner & Arrott (1965) and Werner, Arrott, King & Kendrick (1966).
The various expressions in Table 6.2.1.1 contain F^{2}, the square of the modulus of the structure factor. The relation of F to the atomic scattering factors, the atomic positional coordinates, and the temperature is treated in Chapter 6.1 .
For the factors relevant for the precession method (Buerger, 1944), see Waser (1951a,b), Burbank (1952), and GrenvilleWells & Abrahams (1952). For the de Jong–Bouman method, see Bouman & de Jong (1938) and Buerger (1940). For the retigraph, see Mackay (1960).
References
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Buerger, M. J. (1944). The photography of the reciprocal lattice. American Society for Xray and Electron Diffraction, Monograph No. 1.
Burbank, R. D. (1952). Upper level precession photography and the Lorentz–polarization correction. Part I. Rev. Sci. Instrum. 23, 321–327.
Debye, P. (1914). Interferenz von Röntgenstrahlen und Wärmebewegung. Ann. Phys. (Leipzig), 43, 49–95.
GrenvilleWells, H. J. & Abrahams, S. C. (1952). Upper level precession photography and the Lorentz–polarization correction. Part II. Rev. Sci. Instrum. 23, 328–331.
Hu, H.C. (1997a). A universal treatment of Xray and neutron diffraction in crystals. I. Theory. Acta Cryst. A53, 484–492.
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Klug, H. P. & Alexander, L. E. (1974). Xray procedures for polycrystalline and amorphous materials. New York: John Wiley.
Mackay, A. L. (1960). An axial retigraph. Acta Cryst. 13, 240–245.
Waser, J. (1951a). The Lorentz factor for the Buerger precession method. Rev. Sci. Instrum. 22, 563–566.
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Werner, S. A. & Arrott, A. (1965). Propagation of Braggreflected neutrons in large mosaic crystals and the efficiency of monochromators. Phys. Rev. 140, A675–A686.
Werner, S. A., Arrott, A., King, J. S. & Kendrick, H. (1966). Propagation of Braggreflected neutrons in bounded mosaic crystals. J. Appl. Phys. 37, 2343–2350.