Section 2.5.1.3. Resolution

The momentum resolution in energy-dispersive diffraction is limited by the angular divergence of the incident and diffracted X-ray beams and by the energy resolution of the detector system. The observed profile is a convolution of the profile due to the angular divergence and the profile due to the detector response. For resolution calculations, it is usually assumed that the profiles are Gaussian, although the real profiles might exhibit geometrical and physical aberrations (Subsection 2.5.1.5). The relative full width at half-maximum (FWHM) of a diffraction peak in terms of energy is then given by where is the electronic noise contribution, F the Fano factor, the energy required for creating an electron–hole pair (cf. Subsection 7.1.5.1 ), and the overall angular divergence of the X-ray beam, resulting from a convolution of the incident- and the diffracted-beam profiles. For synchrotron radiation, can usually be replaced by the divergence of the diffracted beam because of the small divergence of the incident beam.

Fig. 2.5.1.3 shows as a function of Bragg angle . The curves have been calculated from equations (2.5.1.1) and (2.5.1.2) for two values of the lattice-plane spacing and two values of , typical for Bremsstrahlung and synchrotron radiation, respectively. It is seen that in all cases decreases with decreasing angle (i.e. increasing energy) to a certain minimum and then increases rapidly. It is also seen that the minimum point of the curve is lower for the small d value and shifts towards smaller values for decreasing . Calculations of this kind are valuable for optimizing the Bragg angle for a given sample and other experimental conditions (cf. Fukamachi, Hosoya & Terasaki, 1973; Buras, Niimura & Olsen, 1978).

Figure 2.5.1.3| top | pdf | Relative resolution, , as function of Bragg angle, , for two values of the lattice plane spacing: (a) 1 Å and (b) 0.5 Å. The full curves have been calculated for , the broken curves for .

The relative peak width at half-height is typically less than 1% for energies above 30 keV. When the observed peaks can be fitted with Gaussian functions, one can determine the centroids of the profiles by a factor of 10–100 better than the value of equation (2.5.1.2) would indicate. Thus, it should be possible to achieve a relative resolution of about 10⁻⁴ for high energies. A resolution of this order is required for example in residual-stress measurements.

The detector broadening can be eliminated using a technique where the diffraction data are obtained by means of a scanning crystal monochromator and an energy-sensitive detector (Bourdillon, Glazer, Hidaka & Bordas, 1978; Parrish & Hart, 1987). A low-resolution detector is sufficient because its function (besides recording) is just to discriminate the monochromator harmonics. The Bragg reflections are not measured simultaneously as in standard XED. The monochromator-scan method can be useful when both a fixed scattering angle (e.g. for samples in special environments) and a high resolution are required.