# Complex Assessment of X-ray Diffraction in Crystals with Face-Centered Silicon Carbide Lattice

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{8}

^{9}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results and Discussion

_{1}, x

_{2}, … x

_{n}laid from some arbitrary beginning, was used. f

_{1}, f

_{2}, … f

_{n}are the scattering factors corresponding to different N atoms. The amplitude of the wave scattered by the n atom is f

_{n}times the amplitude of the wave scattered by the isolated electron, which we took to be 1. It was assumed that the object was small enough and that the absorption in it could be neglected; i.e., radiation of the same amplitude falls on each atom.

_{0}and S are unit vectors parallel to the incident beam and the direction of observation, respectively. The amplitude of the wave resulting from the addition of N elementary waves is:

#### 3.1. Ewald Structure

_{0}, and its absolute value is as follows:

_{0}.

_{0}has the direction s, and this diffraction angle is associated with the wavelength in Equation (2).

_{0}with a wavelength of λ falls on this stationary body, then a diffraction pattern can be obtained using the Ewald construction [32] (Figure 1). Then, we need to cross out in reverse space the Ewald sphere of radius $\frac{1}{\lambda}$ centered at point O so that $OM=\frac{{S}_{0}}{\lambda}$. Point M is the origin of the inverse space.

_{1}, x

_{2}, … x

_{n}are introduced into it. Since atoms are considered impermeable, each point where ρ(x) is not zero is in the electronic cloud of one of the N atoms. The value ρ(x) is the sum of the functions ρ

_{n}(x − x

_{n}) expressing the density at the points of the electron cloud of the nth atom, the center of which is x

_{n},

_{n}, thus resulting in Equation (3).

#### 3.2. Scattering Capacity of the Object under Study (Electron Scattering Intensity)

_{N}(s) is the ratio between the emission intensities scattered by the object and the number of free electrons. According to another definition, this is the effective number of free electrons that, scattering independently of each other, produce the same effect under test conditions as the object. The scattering ability refers to an atom if an object consists of N atoms and, more generally, to an elementary motif (grouping) if the object consists of N identical motifs (groupings, such as molecules or crystal cells). This single (per unit) scattering capacity is denoted as I(s):

#### 3.3. Diffraction Pattern and the Nature of the Location of Stacking Faults

^{s−}

^{1}(s is the correlation range required to describe the defective structure). However, in many cases, the correlation range can be lowered by moving from correlations at the locations of the close-packed layers to correlations at the locations of the stacking faults themselves. For example, sections of the 18R phase with Zhdanov symbols in the 3C phase can be represented either as interlayers with the application of close-packed layers according to the ABCABABCABCBCABCAC law inside the main ABC sequences or as portions of the original structure with single-layer (subtraction-type) stacking faults in each sixth layer. The description of the transition 3C→18R first requires consideration of the correlations in the arrangement of at least six layers (s = 6), while a description using the second method only requires consideration of the correlations in the arrangement of the nearest stacking faults, assuming that the faults are predominantly formed five layers apart.

#### 3.4. Basic Diffraction Equation

_{k}is the number of the layer in which the k stacking fault is located, ξ = 2πh

_{3}/3, h

^{3}is the variable along the reverse lattice axis coinciding with the direction of node diffusion, φ

_{0}= 2π(H − K)/3, and H and K are the reflex indices on hexagonal axes.

_{k}–m

_{k+}

_{p}may take different values. By averaging over k and considering that the crystal, although large, has a finite size L and, therefore, N stacking faults, we obtain:

_{p}

^{t}is the probability that, between the first and (p + 1) single-layer stacking faults in an arbitrary sequence (p + 1) of single-layer stacking faults, there is a distance of t layers; C’ = Csin

^{2}φ

_{o}/sin

^{2}1/2 (ξ + φ

_{o}); cs is the complex conjugate; and 〈…〉 is the averaging sign.

#### 3.5. Application of Correlations in Arrangements of Single-Layer Stacking Faults

_{p}

^{t}requires considering the correlations between the locations of single-layer stacking faults. The presence of a correlation between single-layer defects means that the probability of single-layer defects occurring in a given layer depends on the relative position of the defect considered from among the previous single-layer defects. Therefore, we can introduce the probability ${P}_{{i}_{s}{i}_{s-1}}{\dots}_{\hspace{0.33em}{i}_{2}\hspace{0.33em}{i}_{1}}$ that a single-layer stacking fault will appear in a given plane, provided that the first adjacent defect is at a distance of i

_{1}layers, the second is at a distance i

_{s}from the first, and so on, until, finally, the s neighbor is at a distance i

_{s}from the x − 1 neighbor. The considered events turn out to be connected in a complex Markov chain. It can be reduced to a simple chain if the probability ${P}_{{i}_{s}{i}_{s-1}}{\dots}_{\hspace{0.33em}{i}_{2}\hspace{0.33em}{i}_{1}}$ is presented as the probability of transition from the configuration of single-layer stacking faults described by indices (i

_{s}i

_{s−1}…i

_{2}) to the configuration (i

_{s−1}i

_{ε−2}…i

_{2}i

_{1}):

_{s−1}i

_{s−2}…i

_{1}), j = (j

_{s−1}j

_{s−2}… j

_{1}), and (s−1) indicates member configurations.

_{ij}≠ 0 only for i

_{s}

_{−2}= i

_{s}

_{−1}, i

_{s}

_{-3}= i

_{s}

_{-2}, …i

_{1}= i

_{2}.

_{p}

^{t}is expressed through the introduced transient probabilities as follows:

_{1}+ l

_{1}+ … + n

_{1}+ j

_{1}= t), where f

_{i}is the relative fraction of the i configuration, and k

_{1}, l

_{1}, … n

_{1}, j

_{1}is the distance between the last two single-layer stacking faults in the corresponding (s−1) membered configurations.

_{ij}= f

_{j}; (Q)

_{ij}= P

_{ij}exp [i(ξ − φ

_{0})j

_{1}]. The matrix F consists of similar rows. In the case of correlation with one single-layer stacking fault, the matrix Q has the same form, since the probability that the two nearest and adjacent single-layer stacking faults are located at a distance j will not depend on the distance i from the first neighbor to the next single-layer stacking fault P

_{ij}= P

_{j}. In general, a correction in s single-layer stacking faults may limit the number of preferred (or ordered) configurations of three or more single-layer stacking faults and, therefore, limit the number of rows in the Q matrix other than those for all other configurations.

_{1}is the matrix with only one row other than zero and Q

_{1}is the matrix with R + 1 rows other than zero (R is the number of ordered configurations of three or more single-layer stacking faults). The trace of the matrix does not change even when rearranging rows and then rearranging the corresponding columns. Therefore, it is possible to place all the zero rows of matrices F

_{1}and Q

_{1}as the top rows. When multiplying such matrices, only the first R + 1 elements of the rows are important. Therefore, matrices F

_{1}and Q

_{1}can be considered square matrices of the order R + 1. In the future, any matrix after the specified transformations will be denoted by a letter with index (1).

_{m}is the coefficient of decomposition of a characteristic polynomial |E

_{λ}− Q

_{1}|.

_{1}P

_{1}= F

_{1}.

#### 3.6. Analysis of Diffraction Effects during Transitions

_{ij}= j

_{1}δ

_{ij}, (PT)

_{ij}= P

_{ij}j

_{1}.

_{1}—it is possible to determine their theoretical distributions for the intensity of ray reflection in reverse space.

_{1}| = 1 − A. Then:

#### 3.7. FCC Crystal Studies

_{n+1}and m

_{n′+1}are the coordinates of the beginning of the (n + 1) and (n′ + 1) groups, and m

_{r}and m

_{r′}are the coordinates of the r single-layer stacking fault in the (n + 1) group and the r′ single-layer stacking fault in the (n′ + 1) group.

#### 3.8. Diffraction Effects for Models of Defective Structures

_{1}is the probability of there being multi-layer stacking faults with h = b, provided that that the closest multi-layer stacking fault lies at a distance of a layers and has a thickness of h = d. β

_{2}is the probability of there being multi-layer stacking faults with h = d, provided that the closest multi-layer stacking fault lies at a distance of c layers and has a thickness of h = b. Then, α

_{1}and α

_{2}are the probabilities of the independent appearance of a multi-layer stacking fault with h = b and h = d, respectively (in the absence of the ordering β

_{1}= α

_{1}and β

_{2}= α

_{2}).

_{p}

^{t}(h

_{1}, h

_{2}, …, h

_{p+1}) that, in the sequence p + 1 of multi-layer stacking faults, the first has a thickness h

_{1}, the second h

_{2}, etc. (h

_{i}= b, d), while, between the beginning of the first and the p + 1 multi-layer stacking faults, there is a distance of t layers (m

_{p+1}− m

_{1}= t). Using this and considering that, in this case, g(i) = h

_{i}and m

_{r}= r, we can rewrite Equation (20) as follows:

_{j}at a distance of x

_{i}layers after the multi-layer stacking faults with thickness h

_{i}; this probability is easily expressed with the introduced probabilities (β

_{1}, β

_{2}, α

_{1}, α

_{2}) and ${f}_{{h}_{i}}$, the relative fraction of multi-layer stacking faults in the layers. Then:

_{m}are coefficients of the characteristic polynomial of matrix A; (*) is the complexly conjugate value; and ${A}_{{h}_{i}{h}_{j}}$ are functions of the introduced probabilities α

_{1}, α

_{2}, β

_{1}, and β

_{2}. The calculation of these values using Equation (14) does not cause any complications. The relative fractions of multi-layer stacking faults f

_{b}and f

_{d}can be determined using an obvious system of equations:

#### 3.9. Diffraction Effects

#### 3.10. Calculation of the FCC Twinned SiC Crystal

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Bosikov, I.I.; Martyushev, N.V.; Klyuev, R.V.; Tynchenko, V.S.; Kukartsev, V.A.; Eremeeva, S.V.; Karlina, A.I.
Complex Assessment of X-ray Diffraction in Crystals with Face-Centered Silicon Carbide Lattice. *Crystals* **2023**, *13*, 528.
https://doi.org/10.3390/cryst13030528

**AMA Style**

Bosikov II, Martyushev NV, Klyuev RV, Tynchenko VS, Kukartsev VA, Eremeeva SV, Karlina AI.
Complex Assessment of X-ray Diffraction in Crystals with Face-Centered Silicon Carbide Lattice. *Crystals*. 2023; 13(3):528.
https://doi.org/10.3390/cryst13030528

**Chicago/Turabian Style**

Bosikov, Igor I., Nikita V. Martyushev, Roman V. Klyuev, Vadim S. Tynchenko, Viktor A. Kukartsev, Svetlana V. Eremeeva, and Antonina I. Karlina.
2023. "Complex Assessment of X-ray Diffraction in Crystals with Face-Centered Silicon Carbide Lattice" *Crystals* 13, no. 3: 528.
https://doi.org/10.3390/cryst13030528