Interest to the method of X-ray diffraction by precipitation related crystals is very high. Ni-based superalloys belong to strongly distorted crystals, in which the effect of hardening is caused by isomorphous Ni$_{3}$Al based precipitations.
For the alloys with isomorphous precipitations one of the main parameters is misfit which characterizes the discrepancy between matrix and precipitation lattice parameters. It's sign and value defines refractory properties of the alloy and is connected with the type of the intensity distribution in the vicinity of the reciprocal lattice points. It allows us to determine the sign and value of the misfit for the alloys after different crystal growth conditions and following treatment by analysing intensity distribution. As the researched alloys are strongly distorted they can be analysed in the framework of kinematic theory of scattering. According to this theory the influence of the precipitations parameters on X-ray scattering essentially depends on Debye-Waller factor exponent value 2M. If 2M<<1 crystals are slightly distorted; when 2M>>1 they are strongly distorted.
In such two cases the intensity distribution can be analytically analysed. The most complicated is the case when 2M>1.X-ray analysis of such materials reciprocal lattice points showed that it was rather complicated as in radial direction (parallel to the reciprocal lattice vector) as in azimuthal plane (perpendicular to it). According to kinematic theory of scattering intensity distribution in the azimuthal plane characterizes dislocation substructure of the crystal. In the direction parallel to the reciprocal lattice vector X-ray intensity distribution often consisted of two or three maximums. It is connected with the origin of the diffuse X-ray scattering by such aging alloys because for them factor 2M>1.
According to the theory in the case 2M>1 both regular I0 and diffuse I1 peak of the matrix have approximately the same intensity and are present in the intensity distribution simultaneously. So the resulting intensity distribution I consists of two items I=I0+I1. The presence of the diffuse matrix peak often is not taken into account when interpreting experimental data. This case can be analysed only by means of numerical calculation. It becomes much more complicated when the precipitations are isomorphs to matrix and their lattice parameter is close to that of matrix as in the case of Ni$_{3}$Al based precipitations in Ni-based superalloys. Such precipitations form their own intensity peaks in the vicinity of the matrix reciprocal lattice points. In that case intensity distribution consists of three peaks: regular and diffuse peaks of the matrix and precipitations peak. The picture can become still more complicated when as it often happens the dislocation structure is formed in such crystals, causing its own diffraction effects. In order to get correct information about real crystal structure we have worked out the diffraction procedure of such materials analysis taking into account all the above factors.
To analyse the experimental data we carried out theoretical calculations of X-ray scattering with the following modelling of the resulting intensity profile by means of IBM-PS-IT at different meanings of sign and value of the lattice misfit between matrix and '-phase and also it's different volume fraction. We found good agreement between theoretical and experimental results under certain meaning of parameters. It allowed us to determine real structure parameters of the analysed crystals.