Peak profile instrumental aberrations

Regardless of the quality of the adjustments, the geometry of the diffractometer leads to a few aberrations, resulting in peaks that are not quite symmetrical, generally wide and shifted with respect to the expected difftaction angle. We will discuss each of the aberrations caused by the device s various elements. The function obtained from the convolution of the different elementary functions associated with each aberration is called the instmmental function. 

The profile of diffraction peaks depends on two types of contributions first, the instrumental function and, second, the stractural defects that also lead to changes in the intensity distribution. This last comment is at the core of microstructural analysis, which will be the subject of the second part of this book and will not be discussed here. Peak profiles can be described by a function h(e), where the e variable corresponds in every point to the difference with respect to the theoretical diffraction angle. The function h(e) can be expressed as the convolution product of f(e), which represents the pure profile associated with the sample s specific effects, and g(e), which constitutes the instrumental function. The function h can then be expressed as  

The profile observed is actually more complex because the measurement noise 8(e), on the one hand, and the continuous background b(e), on the other hand, which have already been mentioned, must be taken into account. Hence, the expression of the peak profile  

Microstractural studies require the pure profile, meaning the function f, to be determined. Solving equation [3.2] is a complex problem, dealt with later on. In this chapter, we will simply describe the different components of the instrumental function. The expressions and the relative importance of the different functions gj(e) depend on the device. Clearly, the complete description of a given diffractometer s experimental profile requires taking into account all of its elements. We will now describe the main effects observed in virtually every case. 

Generally speaking, the profile of a beam produced by a tube approximately corresponds to a Ganssian function and therefore the function gi(e) is expressed as  

---

The diffraction lines due to the crystalline phases in the samples are modeled using the unit cell symmetry and size, in order to determine the Bragg peak positions 0q. Peak intensities (peak areas) are calculated according to the structure factors Fo (which depend on the unit cell composition, the atomic positions and the thermal factors). Peak shapes are described by some profile functions 0(2fi—2fio) (usually pseudo-Voigt and Pearson VII). Effects due to instrumental aberrations, uniform strain and preferred orientations and anisotropic broadening can be taken into account. 

There is no reason for an experimental Bragg peak to be exactly described by a simple analytical function. Bragg peaks are generally very complicated objects. In the early usual simplified formalism, still in use, the experimental function h x) describing a broadened Bragg peak profile observed on a powder diffraction pattern is due to the convolution of the instrumental aberration function g(x) with the sample function fix) ... 

As we have already mentioned, the development of high resolution X-ray diffraction devices made it possible to show that Gaussians did not fit the experimental profiles well. Generally speaking, when instrumental aberrations lead to symmetrical increases in peak width, the peaks can usually be approximated as Gaussian or Loientzian functions. The latter are expressed as follows ... 

The dispersion of the distribution, or diffraction line broadening, is measured by the full width at half the maximum intensity (FWHM) or by the integral breadth ifi) defined as the integrated intensity (J) of the diffraction profile divided by the peak height (/3 = ///q). Line broadening arises from the convolution of the spectral distribution with the functions of instrumental aberrations and sample-dependent effects (crystallite size and structure imperfections). 

---