Wavelength dispersion function

It is worth noting that unlike the instrumental and wavelength dispersion functions, the broadening effects introduced by the physical state of the specimen may be of interest in materials characterization. Thus, effects of the average crystallite size (x) and microstrain (s) on Bragg peak broadening (P, in radians) can be described in the first approximation as follows ... 

In general, three different approaches to the description of peak shapes can be used. The first employs empirical peak shape functions, which fit the profile without attempting to associate their parameters with physical quantities. The second is a semi-empirical approach that describes instrumental and wavelength dispersion functions using empirical functions, while specimen properties are modeled using realistic physical parameters. In the third, the so-called fundamental parameters approach, all three components of the peak shape function (Eq. 2.45) are modeled using rational physical quantities. 

Before the development of semiconductor detectors opened the field of energy-dispersive X-ray spectroscopy in the late nineteen-sixties crystal-spectrometer arrangements were widely used to measure the intensity of emitted X-rays as a function of their wavelength. Such wavelength-dispersive X-ray spectrometers (WDXS) use the reflections of X-rays from a known crystal, which can be described by Bragg s law (see also Sect. 4.3.1.3)... 

If we consider an absorption band showing a normal (Gaussian) distribution [Fig. 17.13(a)], we find [Figs. (b) and (d)] that the first- and third-derivative plots are disperse functions that are unlike the original curve, but they can be used to fix accurately the wavelength of maximum absorption, Amax (point M in the diagram). 

Reference wavelengths for calibration lines are corrected from Bearden s values [19] for the recent CODATA determination of lattice spacings and X-ray wavelengths [25]. The dispersion function is fitted to the 10 calibration wavelengths. The dispersion function relates the wavelength of a spectral feature located at the detector centre, to the angle of diffraction measured by clinometers. 

One example of a systematic shift is that caused by the calibration source not being in the same location as the EBIT source. Our theoretical modelling determines the shifts of < 1 arcsecond associated with this mis-location. The dispersion function is not a simple relationship between angle and wavelength but a complex (but smooth) function of reference wavelengths, clinometer values, detector scale and systematic shifts. 

Summing all errors in quadrature results is a 27 ppm-40 ppm uncertainty. The main sources of uncertainty are therefore statistical, reference wavelengths and dispersion function determination. All major error sources are soft and may be reduced further. Methods of reducing statistical uncertainty by improving spectrometer efficiency are being investigated and improved flux from the EBIT has been achieved in other studies [26],... 

The observed peak shapes are best described by the so-called peak shape function (PSF), which is a convolution of three different functions instrumental broadening, Q, wavelength dispersion. A, and specimen function, E. Thus, PSF can be represented as follows ... 

If the incident beam corresponds to several radiations with different mean wavelengths, the beam s wavelength dispersion is then described as the sum of all the Lorentzian functions associated with each radiation. 

Another point worth mentioning is the use of the metal long-wavelength dielectric function. No dispersion was allowed in the model. The wave-vector dependence of the dielectric constant is important at close proximity to the surface as was already remarked in the context of the image calculations. This approximation may be reasonable away from the surface, distances for which the LFE is most appropriate. 

Figure 2.22 The contribution of components to the total background in a wavelength-dispersive spectrometer, as a function of wavelength A = first order, B = second order, C = third order, and D = fourth order.

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