A few applications of x-ray crystallography methods
The experimental methods described above make it possible to measure:
Bragg angles in diffracted beams (photographic methods)
intensities of diffracted beams (counter method)
All metallurgy applications use one or other of these parameters as summarized in the following table.
Position of diffraction peaks \((\theta)\) | nature of a metal, phase, inter-metallic compound, composition of a solid solution, order-disorder transformations |
|---|---|
Variation in the position of a diffraction peak \((\Delta\theta)\) | residual stresses |
Diffraction peak intensities \((I)\) | Phase identification |
Variation in the intensity of a diffraction peak \((\Delta I)\) | Crystal preferential orientation, texture |
Diffraction peak shape | crystallite size micro-deformations (strain hardening rate) |
Phase analysis
For a polyphase material, a Debye-Sherrer radiogram contains a recording of all phases present. The method can therefore be used for the identification of these phases from a qualitative perspective. To obtain quantitative information and therefore perform true phase dosing, it is preferable to use a counter diffractometer.
The principle of this method is based on the fact that the integrated intensities of diffraction peaks are proportional to the volume fractions of phases responsible for the diffraction.
For example, for a quenched steel composed of martensite (\(\ce{M}\)) and austenite (\(\ce{A}\)) we have:
\(I_M = K_M.V_M\) and \(I_A = K_A.V_A\)
with \(K_M\) (respectively \(K_A\)) proportionality constants, \(I_M\) (respectively \(I_A\)) integrated intensity of martensite diffraction peaks (respectively austenite), \(V_M\) (respectively \(V_A\)) martensite volume fraction (respectively austenite) and \(V_M + V_A = 1\)
\(V_A / V_M = I_A.K_M /I_M.K_A = y\) and therefore \(V_A = y / (y+1)\)
Measurement of residual stresses
Residual stresses present in the crystals and that are caused, for example, by heat treatments, welding, machining, etc., can be estimated using x-ray diffraction. The basic principle consists of using the natural length gauge that are lattice parameters (or any inter-planar spacing distance): the value depends on the residual stress field and varies in relation to the characteristic value in the stress free material. Note that only the elastic part of the deformation field is taken into account because plastic deformation does not impact lattice parameters.
For a family of diffracting planes \((hkl)\), if we differentiate, at constant \(\lambda\), the Bragg relationship, we obtain:
\(\frac{\Delta d_{hkl}}{d_{hkl}} + \frac{\cos\left(\theta\right)}{\sin \left( \theta \right)} \Delta \theta = 0\)
\(\Delta \theta = - \frac{\Delta d_{hkl}}{d_{hkl}} \tan\left(\theta\right)\)
Measuring displacement of diffraction peak \(\Delta \theta=\theta_{\textrm{strained}} - \theta_{\textrm{unstrained}}\) makes it possible to identify the deformation normal to planes \((hkl)\)
\(\epsilon = \Delta d_{hkl}/d_{hkl} = - \textrm{cotan} \left( \theta \right) . \Delta \theta\).
Based on this value, it is simple to relate the stresses to the deformations by a calculation using continuum mechanics equations.
This method is non-destructive, it can only be used with crystalline materials and only on the surface layers of the material (a few micrometres to a few dozen micrometres depending on the material studied and the radiation used). This explains why this technique is primarily used for surface characterization. However, the distribution of stresses at depth can be assessed after successive chemical or electrolytic etching (thus introducing no stress), but in this case the method becomes destructive.
Study of crystallographic textures
Where grains of a polycrystal are oriented randomly, Debye rings present uniform intensities. In cases where the grains are grouped according to preferred orientations (for example after rolling), the probability of finding grains in a diffraction position for the family of planes \((hkl)\) is no longer uniform in all directions. Debye rings then present intensity enhancements. Qualitatively, this is a mean to verify the presence of these preferred orientations. Quantitatively, textures can be described by tracing, on a stereographic projection and using measured diffraction intensities, iso-density lines for a given crystalline pole. For diffraction by planes of family \((hkl)\), we obtain the \((hkl)\) poles figure. It is important to note that certain physical and mechanical properties are heavily dependent on the direction of the crystals. Consequently, in polycrystals these properties depend on texture.