Strain/stress diffraction analysis

A. Segmuller and M. Murakami. X-Ray Diffraction Analysis of Strains and Stresses in Thin Films. In Analytical Techniques for Thin Films. (K.N. Tu and R. Rosenberg, eds.) Academic, San Diego, 1988, p.l43. 

For many decades the principal aim of strain/stress analysis by diffraction was the determination of the average strain and stress tensors e and s in materials. The determination was based on the supposition that the elastic strain and stress tensors e(g), s(g) in a crystallite are connected to the average tensors s, e as follows ... 

Arguments for recent developments of the spherical harmonics approach for the analysis of the macroscopic strain and stress by diffraction were presented in Section 12.2.3. Resuming, the classical models describing the intergranular strains and stresses are too rough and in many cases cannot explain the strongly nonlinear dependence of the diffraction peak shift on sin even if the texture is accounted for. A possible solution to this problem is to renounce to any physical model to describe the crystallite interactions and to find the strain/ stress orientation distribution functions SODF by inverting the measured strain pole distributions ( h(y)). The SODF fully describe the strain and stress state of the sample. 

The chosen technique was X-ray diffraction that is widely used for non-destructive surface measurement of applied and residual stresses. Stress analysis relies on the determination of the lattice strain using the interplanar spacing as a gauge by measuring the peak shifts in a fixed O-direction for different /-tilt of the sample [3]. Stresses are calculated from measured strains using diffraction elastic constants which were calculated theoretically. As the Co phase takes up a certain amount of W and transforms after cooling into a solid solution with a variable W content, the measurements were limited to the WC phase. 

X-ray and electron diffraction methods are applied in order to measure atomic distances in the crystal lattice and their changes. Hence, the diffraction methods are also basically suitable for measuring the strain/stress behaviour in thin films. However, since the film thickness and the crystallite size in thin films are small, some line broadening already arises from this. In order to determine what contribution the mechanical stresses have on the diffuse lines, careful analysis of the line profiles must be undertaken [148, 151]. This method is less suitable for routine determination of stresses in thin films. In some cases, it is possible though rarely applied to determine the stresses in the films through their influence on other, known film properties, at least approximately. Such properties are, for example, the position of an absorption edge [152], the Hall effect [153], electron spin resonance spectra [155] and in the case of superconducting films, variations in the critical transition temperature [156]. However, these effects can, unfortunately, also arise for other reasons, and thus these techniques can usually only be used as supplemental experiments. 

Several experimental procedures can be used to measure the residual stresses. The three preferred methods involve diffraction (X-ray or neutron), beam deflection, and permanent strain determination. X-ray diffraction measurements have the limitation that the penetration depth is small, such that only near-surface information is obtained. Moreover, in composites, residual stresses are redistributed near surfaces.47 Consequently, a full stress analysis is needed to relate the measured strains to either q or a. ... 

X-ray diffraction patterns and infrared Raman spectra show specific changes through this a transition (Tashiro Tadokoro, 1987 Jakeways et al, 1976 Ward Wilding, 1977). The stress and strain dependence of the molar fraction of the form, Xp, can be evaluated by quantitative analysis of the infrared spectra. The characteristic behaviour of this phase transition observed in a uniaxially oriented sample is as follows Xp increases drastically above the critical stress / Xp is almost linearly proportional to the strain the transition is reversible and the... 

Consequently, if the peak shifts for one or more peaks are measured as a function of T in the range (0, ujl) at y and y + re for three fixed values of y e.g., 0, 71/4 and nj2) the stress tensor elements 5, can be determined from the intercept and the slopes of these lines. It is presumed that the single-crystal elastic constants are known and the diffraction elastic constants in Equations (109) and (110) can be calculated following one of the models presented before. This is the conventional sin T method. Alternatively Equation (107) can be used in a least-square analysis or implemented in the Rietveld codes. If diffraction patterns measured in several points (T, y) are available the stress tensor elements 5,- can be refined together with the structural and other parameters. The implementation in GSAS is the Voigt formula Equation (90) and not Equation (107). In this case refinable parameters are the strain tensor elements e,. 

This well-known result is the foundation of the residual stress (actually residual strain) analysis by diffraction techniques, dealt with in Chapter 12. As shown schematically in Figure 13.4b, macrostrain produces a shift in PD reflections. The effect can be measured for different sample orientations and different... 

Although not all of the experiments needed to determine residual stresses have been completed, some useful data has been gathered. As temperature increased, the diffraction peaks shifted to lower two theta values, indicating an increase in d-spacing correlating to thermal expansion of the composite material.(Figure 5) Once analysis of the powder samples is complete, the inherent thermal expansion of each phase can be calculated, which will allow the strains to be determined for each phase in the composite material. The diffraction patterns from the experiments also provided evidence that the use of B to produce the parts made neutron diffraction possible. The patterns were visible within the first few minutes of data acquisition, which indicated the excellent scattering from the material. 

Applications of neutron analysis developed much later than those found in X-ray and electron diffraction. Like electron diffiaction, neutron analysis has often been used in conjunction with X-ray techniques. For example, researchers were able, in such a combined study, to elucidate the internal dynamics of protein molecules. Independently, neutron diffraction has allowed investigators to study the details of atomic movements in substances. This basic scientific knowledge has helped others to develop better products, including window glass, semiconductors, and other electronic devices. In industry, neutron diffraction has been used to study tbe stresses, strains, and textures of various building materials. Eor example, metal alloys and welds often exhibit cracks or expansion as well as shrinkage, which limits the value of the respective products. Indeed, this practice is so prevalent that it has been named engineering diffraction. ... 

Furthermore, the presence of defects in a crystal lattice may also alter the diffraction pattern Depending on the type and the concentration of the defects, systematic peak broadening, peak shifts as well as peak splitting may be observed, and stress and strain may also influence the diffractograms [20, 21]. Thus the detailed analysis of measured peak positions, their widths and intensities can be used for the identification of the defects existing in a particular sample. 

X-ray analysis methods (including diffraction and reflectometry) described in Chap. 1 are the most widely used tools for the identification of crystalline properties of materials, in addition to materials strain, texture, stress, density, and surface roughness—properties that are key parameters for various industrial applications. Chapter 2 covers a wide range of optical characterization techniques with focus on ellipsometry, Raman scattering, Fourier transform infrared spectroscopy, and spectrophotometry. Those methods, covering a wide range of photon energy and laser... 

Bulk analytical methods, including laser beam deflection and X-ray or neutron diffraction techniques, have been used to measure lattice strains and stresses. These methods, however, are limited to measuring lattice volume expansion due to li-ion intercalation on the molecular scale. They cannot detect extended defects, cracks, or microfractures, and thus cannot be directly used for mechanics analysis of particles or porous particle networks. Even with this limitation, diffraction or deflection techniques can stiU provide valuable information on lattice structural... 

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