Tables for
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 2.5, pp. 136-138

Section Fundamental equations

B. B. Hea*

aBruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA
Correspondence e-mail: Fundamental equations

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The α and β angles are functions of γ, ω, ψ, ϕ and 2θ. As shown in Fig. 2.5.19[link](a), a pole has three components h1, h2 and h3, parallel to the three sample coordinates S1, S2 and S3, respectively. The pole-figure angles (α, β) can be calculated from the unit-vector components by the following pole-mapping equations:[\eqalignno{\alpha &= {\sin ^{ - 1}}\left| {{h_3}} \right| = {\cos ^{ - 1}}({h_1^2 + h_2^2})^{1/2},&(2.5.53)\cr \beta &= \pm {\cos ^{ - 1}}{{{h_1}} \over {({h_1^2 + h_2^2})^{1/2} }}\matrix{ {} & {} & {\left\{ {\matrix{ {\beta \ge 0^\circ } \ {\rm if} \ {{h_2} \ge 0} \cr {\beta\, \lt \,0^\circ } \ {\rm if} \ {{h_2} \,\lt\,0} \cr } } \right.}, \cr }&\cr&&(2.5.54)} ]where α takes a value between 0 and 90° ([0^\circ \le \alpha \le 90^\circ ]) and β takes values in two ranges ([0^\circ \le \beta \le 180^\circ ] when [{h}_2 \,\gt\, 0] and [ - 180^\circ \le \beta \,\lt \,0^\circ ] when [{h}_2\, \lt \,0]). The condition for reflection-mode diffraction is [h_3\, \gt \,0]. For transmission diffraction it is possible that [h_3\, \lt \,0]. In this case, the pole with mirror symmetry about the S1S2 plane to the diffraction vector is used for the pole-figure mapping. The absolute value of h3 is then used in the equation for the α angle. When [h_2 = 0] in the above equation, β takes one of two values depending on the value of h1 ([\beta = 0^\circ ] when [h_1 \ge 0] and [\beta = 180^\circ ] when [h_1\, \lt \,0]). For Eulerian geometry, the unit-vector components [\{h_1,h_2,h_3\}] are given by equation (2.5.11[link]).

The 2θ integrated intensity along the diffraction ring is then converted to the pole-density distribution along a curve on the pole figure. The α and β angles at each point of this curve are calculated from ω, ψ, ϕ, γ and 2θ. The sample orientation (ω, ψ, ϕ) and 2θ for a particular diffraction ring are constants; only γ takes a range of values depending on the detector size and distance.

For a textured sample, the 2θ-integrated intensity of a diffraction ring from a family of (hkl) planes is a function of γ and the sample orientation (ω, ψ, ϕ), i.e. [I_{hkl} = I_{hkl}(\omega, \psi, \varphi, \gamma, \theta)]. From the pole-figure angle-mapping equations, we can obtain the integrated intensity in terms of pole-figure angles as[{I_{hkl}}(\alpha, \beta) = {I_{hkl}}(\omega, \psi, \varphi, \gamma, \theta).\eqno(2.5.55)]The pole density at the pole-figure angles (α, β) is proportional to the integrated intensity at the same angles:[{P_{hkl}}(\alpha, \beta) = {K_{hkl}}(\alpha, \beta) {I_{hkl}}(\alpha, \beta),\eqno(2.5.56)]where [I_{hkl}(\alpha, \beta)] is the 2θ-integrated intensity of the (hkl) peak corresponding to the pole direction [(\alpha, \beta)], [K_{hkl}(\alpha, \beta)] is the scaling factor covering the absorption, polarization, background corrections and various instrument factors if these factors are included in the integrated intensities, and [P_{hkl}(\alpha, \beta)] is the pole-density distribution function. Background correction can be done during the 2θ integration and will be discussed in Section[link]. The pole figure is obtained by plotting the pole-density function based on the stereographic projection.

The pole-density function can be normalized such that it represents a fraction of the total diffracted intensity integrated over the pole sphere. The normalized pole-density distribution function is given by[g_{hkl}(\alpha, \beta) = {2\pi P_{hkl}(\alpha, \beta) \over \int_0^{2\pi } \int_0^{\pi /2} P_{hkl}(\alpha, \beta)\cos \alpha \, {\rm d}\alpha \, {\rm d}\beta }.\eqno(2.5.57)]The pole-density distribution function is a constant for a sample with a random orientation distribution. Assuming that the sample and instrument conditions are the same except for the pole-density distribution, we can obtain the normalized pole-density function by[{g_{hkl}}(\alpha, \beta) = {{{I_{hkl}}(\alpha, \beta)} \over {I_{hkl}^{\rm random}(\alpha, \beta)}}.\eqno(2.5.58)]

The integrated intensity from the textured sample without any correction can be plotted according to the stereographic projection as an `uncorrected' pole figure. The same can be done for the sample with a random orientation distribution to form a `correction' pole figure that contains only the factors to be corrected. The normalized pole figure is then obtained by dividing the `uncorrected' pole figure by the `correction' pole figure. This experimental approach is feasible only if a similar sample with a random orientation distribution is available.

If the texture has a rotational symmetry with respect to an axis of the sample, the texture is referred to as a fibre texture and the axis is referred to as the fibre axis. The sample orientation containing the symmetry axis is referred to as the fibre axis. The fibre texture is mostly observed in two types of materials: metal wires or rods formed by drawing or extrusion, and thin films formed by physical or chemical deposition. The fibre axis is the wire axis for a wire and normal to the sample surface for thin films. Fibre texture can also be artificially formed by rotating a sample about its normal. If the fibre axis is aligned to the S3 direction, the pole-density distribution function becomes independent of the azimuthal angle β. For samples with fibre texture, or artificially formed fibre texture by rotating, the pole-density function is conveniently expressed as a function of a single variable, [{g_{hkl}}(\chi)]. Here, χ is the angle between the sample normal and pole direction.[\chi = 90^ \circ - \alpha \ {\rm or}\ \chi = {\cos ^{ - 1}}\left| {{h_3}} \right|.\eqno(2.5.59)]

The pole-density function for fibre texture can be expressed as a fibre plot. The fibre plot [{g_{hkl}}(\chi)] can be calculated from the relative intensity of several peaks (He, 1992[link]; He et al., 1994[link]) and artificial fibre texture can be achieved by sample spinning during data collection.


He, B. (1992). X-ray Diffraction from Point-Like Imperfection. PhD dissertation, Virginia Tech, USA.Google Scholar
He, B., Rao, S. & Houska, C. R. (1994). A simplified procedure for obtaining relative x-ray intensities when a texture and atomic displacements are present. J. Appl. Phys. 75, 4456–4464.Google Scholar

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