II. DIFFRACTION VECTOR CONTAINING γ-VARIABLE

In XRD² data analysis, the diffraction intensity distribution in both 2θ and γ-directions should be considered. In order to include the diffraction intensity variation in the γ-direction, the diffraction vector (also referred to as scattering vector) should include not only the 2θ variable, but also the γ-variable. Figure 3 illustrates the diffraction cone and diffraction vector cone in the laboratory coordinates X_LY_LZ_L. The unit vector of the incident beam is s_o and the unit vector representing a direction given by 2θ and γ on the diffraction cone is s. The diffraction vector H and its unit vector h_L in the laboratory coordinates are given by

(1)$$\eqalign{{\bi H} & =\displaystyle{{\bi s} - {\bi s}_0 \over \lambda} = \displaystyle{1 \over \lambda} \left[\matrix{\cos 2\theta - 1 \cr - \sin 2\theta \sin \gamma \cr - \sin 2\theta \cos \gamma} \right]\quad {\rm and} \cr & \quad {\bi h}_{\bi L} = \displaystyle{{\bi H} \over \left\vert {\bi H} \right\vert} = \left[\matrix{{\bi h}_x \cr {\bi h}_y \cr {\bi h}_z} \right]=\left[\matrix{ - \sin \theta \cr - \cos \theta \sin \gamma \cr - \cos \theta \cos \gamma} \right]}$$

Figure 3.

Diffraction cone and diffraction vector cone in laboratory coordinates.

For the Eulerian geometry with three sample rotation angles (ω, ψ, ϕ), the three components of the unit diffraction vector are given by:

(2)$$\eqalign{{\bi h}_1 & =\sin \theta \lpar \sin \varphi \sin \psi \sin \omega+\cos \varphi \cos \omega \rpar \cr & \quad +\cos \theta \cos \gamma \sin \varphi \cos \psi \cr & \quad - \cos \theta \sin \gamma \lpar \sin \varphi \sin \psi \cos \omega - \cos \varphi \sin \omega \rpar \cr {\bi h}_2 & =- \sin \theta \lpar \cos \varphi \sin \psi \sin \omega - \sin \varphi \cos \omega \rpar \cr & \quad - \cos \theta \cos \gamma \cos \varphi \cos \psi \cr & \quad +\cos \theta \sin \gamma \lpar \cos \varphi \sin \psi \cos \omega+\sin \varphi \sin \omega \rpar \cr {\bi h}_3& =\sin \theta \cos \psi \sin \omega - \cos \theta \sin \gamma \cos \psi \cos \omega \cr & \quad - \cos \theta \cos \gamma \sin \psi }$$

It will be shown in the following section that, many important equations can be derived from the unit diffraction vector.

III. TEXTURE

For a sample with texture, the diffraction intensity becomes a function of the sample orientation and γ value. Figure 4 shows the pole density distribution in a pole figure corresponding to the intensity variation along γ from a single 2D pattern. The pole density at (α,β) is proportional to the integrated intensity at the same angles (Bunge and Klein, Reference Bunge and Klein1996):

(3)$$P_{hkl} \lpar \alpha\comma \; \beta \rpar =K_{hkl} \lpar \alpha\comma \; \beta \rpar \cdot I_{hkl} \lpar \alpha\comma \; \beta \rpar$$

where I _hkl (α, β) is the 2θ-integrated intensity of the (hkl) peak corresponding to the pole direction(α, β), K _hkl (α, β) 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 (α, β) is the pole density distribution function. The pole figure angles (α, β) can be calculated from the unit diffraction vector components by the following pole mapping equations:

(4)$$\eqalign{\alpha & =\sin ^{ - 1} \left\vert h_3 \right\vert =\cos^{ - 1} \sqrt{h_1^2+h_2^2} \quad {\rm and} \cr & \quad \beta = \pm \cos^{ - 1} \displaystyle{h_1 \over \sqrt{h_1^2+h_2^2}} \quad \left\{\matrix{\beta \geq 0^{\degr}\comma \; &{\rm if} & h_2 \geq 0 \cr \beta \lt 0^{\degr} \comma \; &{\rm if} &h_2 \lt 0} \right.}$$

Figure 4.

Pole density mapping from γ-profile to pole-figure

IV. STRESS

X-ray diffraction evaluates stress (either residual stress or applied stress) by measuring the lattice distortion associated to the strain. The sin²ψ method is mostly used for stress measurement with the conventional X-ray diffractometer, which uses either a point detector or line detector to measure the XRD profiles at different ψ-tilt angles (Noyan and Cohen, Reference Noyan and Cohen1987). The fundamental equation for stress measurement with XRD² can be given as:

(5)$$\eqalign{& {{\bi S}_1}\lpar {\sigma _{11}} + {\sigma _{22}} + {\sigma _{33}}\rpar + \textstyle{1 \over 2}{{\bi S}_2}\lpar {\sigma _{11}}{\bi h}_1^2 + {\sigma _{22}}{\bi h}_2^2 \cr & + {\sigma _{33}}{\bi h}_3^2 + 2{\sigma _{12}}{{\bi h}_1}{{\bi h}_2} + 2{\sigma _{13}}{{\bi h}_1}{{\bi h}_3} + 2{\sigma _{23}}{{\bi h}_2}{{\bi h}_3}\rpar \cr & = \ln \left(\displaystyle{{\sin {\theta _0}} \over {\sin \theta }} \right)}$$

where S ₁ and 1/2 S₂ are the macroscopic elastic constants, and h₁, h₂, and h₃ are the three components of the unit diffraction vector. By taking various γ values, it defines the 2θ value variation measured at different γ angles for a particular stress tensor and sample orientation. Figure 5 shows the simulated 2θ shift as a function γ for carbon steel with equibiaxial stress σ₁₁ = σ₂₂ = 1000 MPa. The straight solid line represents the stress free 2θ = 156.1°. The goniometer angles set at ω = 78° and ϕ = 0°. The four plots from the left to the right are for ψ = 0°, 15°, 30°, and 45°, respectively. The solid curves represents 2θ shift based on isotropic elasticity and the dotted curves are based on anisotropic factor A _RX = 1.49 (He and Smith, Reference He and Smith1998).

Figure 5.

2θ vs. γ plots for steel with σ₁₁ = σ₂₂ = 1000 MPa at various ψ angles.

V. CRYSTAL SIZE

The size of the crystallites in a polycrystalline material has significant effects on many of its properties, such as thermal, mechanical, electrical, magnetic, and chemical properties. For instance, the mechanical strength of polycrystalline metals and alloys are strongly dependent on the crystallite size. Another example is the particle size of pharmaceutical substances, which may determine their stability and solubility. Crystal size can be measured by many techniques, such as optical microscopy, electron microscopy, or atomic force microscopy. X-ray diffraction has been used for crystallites size measurement for over 90 years based on the diffraction line broadening and line profile analysis of the convention 2θ-profile. However, the conventional 2θ-profile analysis is suitable for crystallite sizes below 100 nm (1000 Å). The γ-profile analysis is suitable for crystallite sizes from as large as tens of μm down to 100 nm with a small X-ray beam size. Figure 6 shows the atomic force microscope (AFM) images and XRD² pattern of a sample crystallized from an organic glass with large and small crystal size (Yu, Reference Yu2011). The XRD² pattern from the sample with fine crystal size shows a smooth diffraction ring, whereas the XRD² pattern from the sample with larger crystals shows spotty diffraction rings. The crystal size can be measured by the spottiness of the diffraction ring by γ-profile analysis.

Figure 6.

(Color online) AFM images and XRD² patterns of samples crystallized from organic glass with large and small particle sizes.

The detailed theoretical analysis can be found in (He, Reference He2009), but the theory can be simply explained by a “candy box”. The effective diffraction volume, also referred to as gauge volume, can be considered as a candy box. As the number of candies to fill the box depends on the size of the candy and candy box, the number of spots along the diffraction ring in a given γ range is also associated to the crystallite size (also referred as gain size or particle size in some articles). Based on the effective diffraction volume, the crystallographic nature, and crystal size of the sample, the equation for crystal size measured in reflection mode is given as:

(6)$${\bi d}=\left\{{\displaystyle{{3{\bi p}_{hkl} {\bi A}_0 \beta \arcsin \lsqb\! \cos \theta \sin \lpar \Delta \gamma /2\rpar \rsqb } \over {2\pi ^2 \mu {\bi N}_{\rm s} }}} \right\}^{{1 / 3}}$$

where d is the average diameter of the crystallite particles, p_hkl is the multiplicity of the diffracting planes, A₀ is the cross-section of the incident X-ray beam, β is the divergence of the incident X-ray beam, μ is the linear absorption coefficient, and N_s is the number of crystallites contributing to the diffraction pattern. The Δγ is the measured range of the diffraction ring. Introducing a scaling factor covering all the numeric constants, incident beam divergence, and calibration factor for the instrument, we obtained an equation for the crystallite size measurement in the reflection mode.

(7)$${\bi d}={\bi k}\left\{{\displaystyle{{{\bi p}_{hkl} {\bi b}^2 \arcsin \lsqb \cos \theta \sin \lpar \Delta \gamma /2\rpar \rsqb } \over {2\mu {\bi N}_{\rm s} }}} \right\}^{{1 / 3}}$$

where b is the size of the incident beam and k = (3β/4π)^1/3 if the instrumental broadening in the 2θ direction is given, but we may treat k as a calibration factor, which can be determined from the 2D diffraction pattern of a known standard. Since only a limited number of spots along the diffraction ring can be resolved, it can be seen from the equation that a smaller X-ray beam size and low multiplicity peak should be used if a smaller crystallite size is to be determined.

For measurement in transmission mode with a sample thickness of t, we have:

(8)$${\bi d}={\bi k}\left\{{\displaystyle{{{\bi p}_{hkli} {\bi b}^2 {\bi t}\arcsin \lsqb \cos \theta \sin \lpar \Delta \gamma /2\rpar \rsqb } \over {{\bi N}_{\rm s} }}} \right\}^{{1 / 3}}$$

The effective sampling volume reaches a maximum for transmission mode diffraction when t = 1/μ. It can be observed from the above equations that the effective sampling volume is proportional to the beam cross-section area and is lower for materials with higher linear absorption coefficients. The sample absorption has a different effect on the sampling volume in reflection and transmission modes. In reflection mode, the linear absorption coefficient determines how fast the incident beam is attenuated within the sample, but there is no clear cutoff for the penetration depth. Therefore, the effective sampling volume has to be used. In transmission mode with the incident beam perpendicular to the sample surface, the linear absorption coefficient affects the relative scattering intensity (counting statistics), but not the actual sampling volume. In other words, all the sample volume within the beam path contributes to the diffraction. Therefore, it is reasonable to ignore the absorption effect for crystallite size analysis as long as the sample is thin enough for transmission mode diffraction.

Figure 7 shows the measurement range of the crystal size analysis by the 2θ-profile collected with conventional laboratory diffractometer and the γ-profile collected with a 2D detector. The 2θ-profile analysis from peak full width at half maximum (FWHM) by the Scherrer equation, or the profile analysis by Stokes and Wilson, is suitable for crystal size below 100 nm, whereas the γ-profile analysis is suitable for crystal size from sub-micrometer to a few millimeters. The size range of γ-profile analysis can be further extended by instrumentation and data collection strategy.

Figure 7.

Crystal size measurement range of the conventional 2θ-profile and γ-profile analysis.