The diffraction patterns for all compounds of the database of crystal structures of intermetallic and inorganic compounds were processed using above algorithms. A DataBase of diffraction images (about 28,000 units of information) was formed, which is included in the program for qualitative phase analysis at the software level . (more details).

 

The equation for calculating the integrated intensity of diffraction reflections is as follows:

,   (1)

where I(hkl) is the integral intensity of a reflection hkl; К is the proportional coefficient; F(hkl) – structural amplitude of a reflection hkl; Lp– product of the Lorentz and polarization factors (Eq. (2) for polycrystals), q – the Bragg’s angle); P – multiplisity; A – absorption factor (for reflection mode use a formula A = 1-exp(-2mR/sinq) , R – sample thickness, m – linear absorption coefficient); e – extinction factor; T – texture multiplier (more details).

                                         ,    (2) 

The structural amplitude in general form (without taking into account the symmetry elements of a specific space group) can be calculated by the equation:

 

            ,    (3)

 

where f_i is the atomic scattering factor of the i-atom (taking into account thermal vibrations (4)), B is the isotropic temperature factor, l is the wavelength of X-ray radiation); x_j, y_j, z_j are the coordinates of the i-atom; n is the number of atoms in the i-position; N is the number of point positions occupied by atoms in the structure; g_i is the coefficient of filling the i-position with atoms (g_i  £  1).

 

                                                ,   (4)

Thus, with the information available on the crystal structure of the compound in the DataBase (more details) and the appropriate software, it is possible to calculate:

 interplanar distances d_hkl  according to the general quadratic formula:

 

,   (5)

 

where  ,

,      ,

,     ,

,     

 

and at a certain radiation, the integral reflection intensities I(hkl) according to Eq. (1).

Since the integral intensities of diffraction reflections significantly depend on the wavelength of the radiation (the relationship between the radiation angle and the wavelength ) due to impact of Lp factor, it obviously makes sense to store I(hkl)/Lp values in the DataBase of diffraction patterns of compounds instead of the integral intensities of reflections I(hkl)​​. In the phase analysis program (more details), the intensities of the reference diffraction pattern are reproduced from these values ​​for a certain radiation, which was applied to collect the experimental diffraction pattern. This approach allows to avoid discrepancies arising when comparing the diffraction pattern obtained, for example, on chromium radiation, with the standard, PDF-2 base, recorded on copper radiation.

The diffraction pattern of a certain compound can also be presented in the form of a table with following parameters ​​calculated for each reflection: 2q angle, d(A) interplanar distance, I intensity (1000-point scale), h, k, l plane indices (Fig. 1).

Fig. 1. Screenshot of the diffraction pattern processing for Mg₇Zn₃ compound

on chromium radiation

In addition to the analytical representation of the diffraction pattern of each compound, this diffraction pattern can also be obtained in a graphical mode (Fig. 2). The values ​​of the integral intensities of the reflections of the compound selected, as well as the angular positions of its peaks on a certain X-ray radiation, are used to calculate the envelope curve of the sum of all available peaks. In this case, the Lorentz distribution was assumed for the peak profile of a separate reflection (a₁– and  a₂ -components).

 

Fig. 2. Screenshot of a fragment of the diffraction pattern of Mg₇Zn₃ compound on chromium radiation

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