Expressions for intensity of diffraction
Trigonometric intensity factorsIntensity factorstrigonometricIntensity factorsin singlecrystal methodsThe expressions for the intensity of diffractionDiffractionintensities 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 programsComputer programsdata processingData processingprogram for intensity factorsIntensity factorsdataprocessing programs for. 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 Classification of experimental techniquesJ. R. Helliwell2.1
Singlecrystal Xray techniquesJ. R. Helliwell–
Powder and related techniques: Xray techniquesW. ParrishJ. I. Langford2.3
) and in textbooks, such as those of Arndt & Willis (1966) for singlecrystal diffractometryIntensity factorsin singlecrystal methods and Klug & Alexander (1974) for powder techniques. Notes on individual factors follow.
(a) Crystal element 
(b) Reflection from a crystal slab of thickness t 
1. Symmetrical Bragg geometry 
2. Asymmetrical Bragg geometry, when the reflecting planes are inclined at an angle to the crystal surface, and the surface normal is in the plane of the incident and reflected beams.angle of incidence and angle of emergence to the crystal surfaceDefine and : 
(c) Transmission from a crystal slab of thickness t 
1. Symmetrical Laue geometry 
2. Asymmetrical Laue geometry, when the reflecting planes are at to the crystal surface, with the normal in the plane of the incident and reflected beams.angle of incidence and angle of emergence to the normal to the crystal surfaceDefine and 
(d) Powder halo: no absorption correction includedwhere P is the diffracted power. 
(e) Debye–Scherrer lines on cylindrical film: no absorption correction includedwhere l is the length of line measured and r is the radius of the camera. P_{l} is the power reflected into length l. 
(f) Reflection from a thick block of powdered crystal of negligible transmission 
(g) Transmission through block of powdered crystal of thickness twhere δ′, δ are the densities of the block of powder and of the crystal in bulk, respectively. 
(h) Rotation photograph of small crystal, volume V 
1. Beam normal to axis 
2. Equiinclination Weissenberg photograph 
Symbols 
Q 
Integrated reflection from a crystal of unit volume 

Volume of crystal element 
e, m 
Electronic charge and mass 
c 
Speed of light 
λ 
Wavelength of radiation 
μ 
Linear absorption coefficient for Xrays or total attenuation coefficient for neutrons 
2θ 
Angle between incident and diffracted beams 

In (b) and (c), as defined; in (h), latitude of reciprocallattice point relative to axis of rotation 
V 
Volume of crystal, or of irradiated part of powder sample 
N 
Number of unit cells per unit volume 
ξ 
In (b) and (c), as defined; in (h), radial coordinate xi used in interpreting Weissenberg photographs 
I_{0} 
Energy of radiation falling normally on unit area per second 
hkl 
Indices of reflection 
F 
Structure factor of hkl reflection 

Distribution function of the mosaic blocks at angular deviation from the average reflecting plane 
σ 
Diffraction cross section per unit volume 
σ_{0} 
Diffraction cross section per unit volume at 
b 
Asymmetry parameter 
τ 
Reduced thickness of the crystal slab 
P_{H}/P_{0} 
Reflection power ratio, i.e. the ratio of the diffracted power to the incident power 
ρ 
Integrated reflection power ratio from a crystal element 
ρ′ 
Integrated reflection power ratio, angular integration of reflection power ratio 
p′ 
Multiplicity factor for singlecrystal methods 
p′′ 
Multiplicity factor for powder methods 

Summary of formulae for integrated powers of reflectionIntegrated intensityformulae for
Summary of formulae for integrated powers of reflectionIntegrated intensityformulae for
The polarization factorPolarization factorIntensity factors
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 radiationSynchrotron 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.
The Lorentz factorLorentz factorIntensity factors
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).
Special factors in the powder methodPowder diffractionspecial factors in
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 factorMultiplicity factor (the number of permutations of hkl leading to the same value of ). For the flatplate frontreflection arrangement, the variation becomes Combining the polarizationIntensity factorspolarization, angularvelocityAngularvelocity factorsIntensity factorsangular velocity, 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.
Some remarks about the integrated reflection power ratio formulae for singlecrystal slabsIntensity factorsin singlecrystal methods
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 Xray absorptionE. N. Maslen6.3.3.1
, 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).
Other factors
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 Intensity of diffracted intensitiesP. J. BrownA. G. FoxE. N. MaslenM. A. O'KeefeB. T. M. Willis6.1
.
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 (1940Lorentz–polarization factor, errors). For the retigraph, see Mackay (1960).