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, p. 136

Section Pole density and pole figures

B. B. Hea*

aBruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA
Correspondence e-mail: Pole density and pole figures

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XRD results from an `ideal' powder in which the crystallites are randomly oriented normally serve as a basis for determining the relative intensity of each diffraction peak. The deviation of the grain orientation distribution of a polycrystalline material from that of an ideal powder is measured as texture. The pole figure for a particular crystallographic plane is normally used to represent the texture of a sample. Assuming that all grains have the same volume, each `pole' represents a grain that satisfies the Bragg condition. The number of grains satisfying the Bragg condition at a particular sample orientation can be larger or smaller than the number of grains for an ideal sample, and likewise for the integrated intensity of that peak. The measured 2D diffraction pattern contains two very important parameters at each γ angle: the partially integrated intensity I and the Bragg angle 2θ. Fig. 2.5.18[link] shows a 2D frame for a Cu thin film on an Si wafer collected with a microgap 2D detector. It contains four Cu lines and one Si spot. The diffraction intensity varies along γ because of the anisotropic pole-density distribution. For each diffraction ring, the intensity is a function of γ and the sample orientation (ω, ψ, ϕ), i.e. I = I(γ, ω,ψ, ϕ).

[Figure 2.5.18]

Figure 2.5.18 | top | pdf |

Diffraction frame collected from a Cu film on an Si substrate showing intensity variation along γ due to texture.

Plotting the intensity of each (hkl) line with respect to the sample coordinates in a stereographic projection gives a qualitative view of the orientation of the crystallites with respect to a sample direction. These stereographic projection plots are called pole figures. As is shown in Fig. 2.5.19[link](a), the sample orientation is defined by the sample coordinates S1, S2 and S3. For metals with rolling texture, the axes S1, S2 and S3 correspond to the transverse direction (TD), rolling direction (RD) and normal direction (ND), respectively. Let us consider a sphere with unit radius and the origin at O. A unit vector representing an arbitrary pole direction starts from the origin O and ends at the point P on the sphere. The pole direction is defined by the radial angle α and azimuthal angle β. The pole density at the point P projects to the point P′ on the equatorial plane through a straight line from P to the point S. The pole densities at all directions are mapped onto the equatorial plane by stereographic projection as shown in Fig. 2.5.19[link](b). This two-dimensional mapping of the pole density onto the equatorial plane is called a pole figure. The azimuthal angle β projects to the pole figure as a rotation angle about the centre of the pole figure from the sample direction S1. When plotting the pole density into a pole figure of radius R, the location of the point P′ in the pole figure should be given by β and[r = R\tan\left ({\pi \over 4} - {\alpha \over 2}\right) = R\tan {\chi \over 2}.\eqno(2.5.52)]

[Figure 2.5.19]

Figure 2.5.19 | top | pdf |

(a) Definition of pole direction angles α and β; (b) stereographic projection in a pole figure.

For easy computer plotting and easy angular readout from the pole figure, the radial angle α may be plotted on an equally spaced angular scale, similar to a two-dimensional polar coordinate system. Other pole-figure mapping styles may be used, but must be properly noted to avoid confusion (Birkholz, 2006[link]).


Birkholz, M. (2006). Thin Film Analysis by X-ray Scattering, pp. 191–195. Weinheim: Wiley-VCH.Google Scholar

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