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. 141-142

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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Fig. 2.5.24[link] illustrates two diffraction cones for backward diffraction. The regular diffraction cone (dashed lines) is from the powder sample with no stress, so the 2θ angles are constant at all γ angles. The diffraction ring shown as a solid line is the cross section of a diffraction cone that is distorted as a result of stresses. For a stressed sample, 2θ becomes a function of γ and the sample orientation (ω, ψ, ϕ), i.e. [2\theta = 2\theta (\gamma, \omega, \psi, \varphi)]. This function is uniquely determined by the stress tensor. The strain measured by the 2θ shift at a point on the diffraction ring is [{{\varepsilon }}_{{{(\gamma, \omega, \psi, \varphi)}}}^{{{\{ hkl\} }}}], based on the true strain definition[\varepsilon _{(\gamma, \omega, \psi, \varphi)}^{\{ hkl\} } = \ln {d \over {{d_o}}} = \ln {{\sin {\theta _o}} \over {\sin \theta }} = \ln {\lambda \over {2{d_o}\sin \theta }},\eqno(2.5.71)]where do and θo are the stress-free values and d and θ are measured values from a point on the diffraction ring corresponding to [(\gamma, \omega, \psi, \varphi)]. The direction of [\varepsilon _{(\gamma, \omega, \psi, \varphi)}^{\{ {{hkl}}\} }] in the sample coordinates S1, S2, S3 can be given by the unit-vector components h1, h2 and h3. As a second-order tensor, the relationship between the measured strain and the strain-tensor components is then given by[\varepsilon _{(\gamma, \omega, \psi, \varphi)}^{\{ hkl\} } = {\varepsilon _{ij}} \cdot {h_i} \cdot {h_j}.\eqno(2.5.72)]The scalar product of the strain tensor with the unit vector in the above equation is the sum of all components in the tensor multiplied by the components in the unit vector corresponding to the first and the second indices. The expansion of this equation for i and j values of 1, 2 and 3 results in[\varepsilon _{(\gamma, \omega, \psi, \varphi)}^{\{ hkl\} } = h_1^2{\varepsilon _{11}} + 2{h_1}{h_2}{\varepsilon _{12}} + h_2^2{\varepsilon _{22}} + 2{h_1}{h_3}{\varepsilon _{13}} + 2{h_2}{h_3}{\varepsilon _{23}} + h_3^2{\varepsilon _{33}}.\eqno(2.5.73)]Or, taking the true strain definition,[\eqalignno{&h_1^2{\varepsilon _{11}} + 2{h_1}{h_2}{\varepsilon _{12}} + h_2^2{\varepsilon _{22}} + 2{h_1}{h_3}{\varepsilon _{13}} + 2{h_2}{h_3}{\varepsilon _{23}} + h_3^2{\varepsilon _{33}} &\cr&\quad= \ln \left({{{\sin {\theta _0}} \over {\sin \theta }}} \right),&(2.5.74)}]where θo corresponds to the stress-free d-spacing and θ are measured values from a point on the diffraction ring. Both θ and {h1, h2, h3} are functions of [(\gamma, \omega, \psi, \varphi)]. By taking γ values from 0 to 360°, equation (2.5.74)[link] establishes the relationship between the diffraction-cone distortion and the strain tensor. Therefore, equation (2.5.74)[link] is the fundamental equation for strain measurement with two-dimensional X-ray diffraction.

[Figure 2.5.24]

Figure 2.5.24 | top | pdf |

Diffraction-cone distortion due to stresses.

Introducing the elasticity of materials, one obtains[\eqalignno{& - {\nu \over E}({\sigma _{11}} + {\sigma _{22}} + {\sigma _{33}}) &\cr&\quad+ {{1 + \nu } \over E}({\sigma _{11}}h_1^2 + {\sigma _{22}}h_2^2 + {\sigma _{33}}h_3^2 + 2{\sigma _{12}}{h_1}{h_2} + 2{\sigma _{13}}{h_1}{h_3} + 2{\sigma _{23}}{h_2}{h_3}) &\cr&\quad\quad= \ln \left({{{\sin {\theta _0}} \over {\sin \theta }}} \right)&(2.5.75)}]or[\eqalignno{&{S_1}({\sigma _{11}} + {\sigma _{22}} + {\sigma _{33}}) &\cr&\quad+ {\textstyle{1 \over 2}}{S_2}({\sigma _{11}}h_1^2 + {\sigma _{22}}h_2^2 + {\sigma _{33}}h_3^2 + 2{\sigma _{12}}{h_1}{h_2} + 2{\sigma _{13}}{h_1}{h_3} + 2{\sigma _{23}}{h_2}{h_3})&\cr&\quad\quad = \ln \left({\sin \theta _0 \over \sin \theta } \right).&(2.5.76)}]

It is convenient to express the fundamental equation in a clear linear form:[{p_{11}}{\sigma _{11}} + {p_{12}}{\sigma _{12}} + {p_{22}}{\sigma _{22}} + {p_{13}}{\sigma _{13}} + {p_{23}}{\sigma _{23}} + {p_{33}}{\sigma _{33}} = \ln \left({{{\sin {\theta _0}} \over {\sin \theta }}} \right),\eqno(2.5.77)]where pij are stress coefficients given by[p_{ij} = \cases{(1/E)[(1 + \nu)h_i^2 - \nu] = {\textstyle{1 \over 2}}{S_2}h_i^2 + {S_1} & {\rm{if }}\ $i = j$,\hfill \cr 2(1/E)(1 + \nu){h_i}{h_j} = 2{\textstyle{1 \over 2}}{S_2}{h_i}{h_j}&{\rm{if }}\ $i \ne j$.\hfill} \eqno(2.5.78)]In the equations for the stress measurement above and hereafter, the macroscopic elastic constants ½S2 and S1 are used for simplicity, but they can always be replaced by the XECs for the specific lattice plane {hkl}, [S_1^{\{hkl\}}] and [{\textstyle{1 \over 2}}S_2^{\left\{hkl \right\}}], if the anisotropic nature of the crystallites should be considered. For instance, equation (2.5.76)[link] can be expressed with the XECs as[\eqalignno{&S_1^{\{ hkl\} }({\sigma _{11}} + {\sigma _{22}} + {\sigma _{33}}) &\cr&+ {\textstyle{1 \over 2}}S_2^{\{ hkl\} }({\sigma _{11}}h_1^2 + {\sigma _{22}}h_2^2 + {\sigma _{33}}h_3^2 + 2{\sigma _{12}}{h_1}{h_2} + 2{\sigma _{13}}{h_1}{h_3} + 2{\sigma _{23}}{h_2}{h_3}) &\cr&\quad\quad= \ln \left({{{\sin {\theta _0}} \over {\sin \theta }}} \right).&(2.5.79)}]

The fundamental equation (2.5.74)[link] may be used to derive many other equations based on the stress–strain relationship, stress state and special conditions. The fundamental equation and the derived equations are referred to as 2D equations hereafter to distinguish them from the conventional equations. These equations can be used in two ways. One is to calculate the stress or stress-tensor components from the measured strain (2θ-shift) values in various directions. The fundamental equation for stress measurement with 2D-XRD is a linear function of the stress-tensor components. The stress tensor can be obtained by solving the linear equations if six independent strains are measured or by linear least-squares regression if more than six independent measured strains are available. In order to get a reliable solution from the linear equations or least-squares analysis, the independent strain should be measured at significantly different orientations. Another function of the fundamental equation is to calculate the diffraction-ring distortion for a given stress tensor at a particular sample orientation [(\omega, \psi, \varphi)] (He & Smith, 1998[link]). The fundamental equation for stress measurement by the conventional X-ray diffraction method can also be derived from the 2D fundamental equation (He, 2009[link]).


He, B. B. (2009). Two-dimensional X-ray Diffraction. New York: John Wiley & Sons.Google Scholar
He, B. B. & Smith, K. L. (1998). Computer simulation of diffraction stress measurement with 2D detectors. Proceedings of 1998 SEM Spring Conference on Experimental and Applied Mechanics, Houston, USA.Google Scholar

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