The objective of this paperwork is the development of a crystal lattice parameter identification algorithm, which allows obtaining a more accurate solution compared to the Bravais unit cell estimation algorithm. To achieve the objective, we suggest solving the parameter identification problem using the steepest descent gradient method. The study of the parameter identification accuracy was conducted on a large number of modeled crystal lattices using the edges and angles similarity measures for Bravais unit cells.
Nowadays, much attention has been concentrated on reconstruction of
threedimensional objects [
The most well-known crystal lattice model was offered by Auguste Bravais. The
Bravais model is based on unit cell representation: the entire lattice can be constructed by
translation of a single cell. All unit cells are divided into seven lattice systems
according to edge lengths and angle values (Fig. 1) [
Initial data for the identification method based on estimation of the Bravais unit cell parameters are a finite set of radius-vectors of crystal lattice nodes.
The identification method involves a search of three non-coplanar vectors in the
original set of nodes: the first vector has a minimum norm; the second vector does not lie
on a straight line with a directing vector equal to the first vector; the third vector does
not lie on a plane made by two found vectors.
The method is based on calculation of six key parameters of the Bravais unit cell (Fig.
2), i.e. three edge lengths and three included angles [
The identification method involves a search of three non-coplanar vectors in the original set of nodes: the first vector has a minimum norm; the second vector does not lie on a straight line with a directing vector equal to the first vector; the third vector does not lie on a plane made by two found vectors.
In order to find volumes of the Wigner-Seitz cells (Fig. 3), we must construct planes
limiting the cell and estimate its volume using the Monte-Carlo method – a random
scattering of a large number of points into a lattice area and calculation of a number
of points fell into the limited area of the Wigner-Seitz cell [
= { ̅} =1, ̅ = ( ̅1, ̅2, ̅3)Т ∈ ℝ3.
The identification method involves calculation of Wigner-Seitz cell volumes by constructing the limiting planes: pl1x pl2 y pl3 z pl 2 0,1 l L 1 ; generating L - values of three-dimensional random vectors which are uniformly distributed in the whole lattice volume and counting the number of vectors that hit in the region limited by planes. Calculation of the cell volume is based on the fact that the probability of hit in the Wigner-Seitz cell region is proportional to its measure (volume). In the presented work, we use only information about limiting planes to reconstruct the whole lattice. It is necessary to compare all developed parameter identification methods uniformly.
In the context of this paperwork, we have developed a crystal lattice parameter identification algorithm based on the steepest descent gradient method. The fundamental element of this algorithm is the Bravais lattice model described by three translation vectors a1 , a
2 and a3 [
In this case, both algorithms shall require initial approximation as an additional input parameter. Particularly, the result vectors of the lattice identification method based on estimation of Bravais unit cell parameters can be used as the initial approximation. The objective function of optimization is as follows:
L l1 i, j,k 2 E a1, a2 , a3 min xl ia1 ja2 ka3 , where L is the number of nodes in the lattice.
Let us introduce the following notation: A = ( ̅1 nl il ̅2 jl Nl nl nlТ ;
̅3) ∈ ℝ3×3; kl Т ; wls il a1s jl a2s kl a3s xl , (1) where s is the step number in the descent.
In this case a gradient (2) and a descent factor (3) are as follows: Expressions (2) and (3) are recorded in compact form. The desired solution is the matrix (triple translation vectors). Figure 4 shows the convergence of translation vectors to solution for a two-dimensional lattice.
L L E A 2 A l1 Nl l1 xl nlT ;
L wls , E As nl s l1 L 2 2 E As nl l1 . (2) (3)
The aim of the crystal lattice parameter identification algorithm based on the steepest descent gradient method is finding the vectors a1 , a2 and a3 , minimizing the objective function (1). In addition, the objective function (1) determines the error of the obtained solution in relation to the source set of nodes.
Using computational experiments, we studied the dependence between the reduction of the objective function (1) and the lattice systems. For the experiments, 20 lattices were generated for each crystal system. All modeled lattice consisted of 125 nodes (5 translations in each direction). Then the lattices was distorted through a random offset of each node from its ideal position in a random direction at a distance no more than 0.5 Å. The solution error (1) between the original lattice and the lattice reconstructed by estimated parameters determined the accuracy of parameter identification methods. Figure 5 shows the obtained results as values of the averaged objective function.
45 35 25 15 5 -5 aP mP oP tP cP hR hP
According to experiment results, the Bravais unit cell parameter estimation algorithm is highly sensitive to the distortion of lattice structure. It is obvious, since the method analyze only 4 nearest lattice nodes but not the whole lattice structure. On the contrary, a Wigner-Seitz cell of three-dimensional lattice are described by 27 lattice nodes. As a result, the Wigner-Seitz cell volume estimation algorithm demonstrated significantly better results than the previous one. The error decreased on average by 47 %. However, the Wigner-Seitz cell volume estimation algorithm has a crucial disadvantage – an extremely high computational complexity.
The best result was obtained using the developed parameter identification algorithm based on the gradient steepest descent method. It greatly improved the results obtained by the Bravais unit cell parameter estimation algorithm and reduced almost sixfold the solution error.
In the presented work, we have developed algorithm of parameter identification based on the gradient steepest descent method. The algorithm uses the result vectors of the lattice identification method based on estimation of Bravais unit cell parameters is used as the initial approximation.
The experiments proved that the parameter identification algorithm based on the gradient steepest descent method is stable to lattice distortion and allows to reduce sixfold the solution error compared to the Bravais unit cell parameter estimation algorithm and threefold compared to the Wigner-Seitz cell volumes estimation algorithm.
This work was partially supported by the Ministry of education and science of the Russian Federation in the framework of the implementation of the Program of increasing the competitiveness of SSAU among the world’s leading scientific and educational centers for 2013-2020 years; by the Russian Foundation for Basic Research grants (# 14-07- 97040, # 15-29- 03823, # 15-29- 07077, # 16-57- 48006); by the ONIT RAS program # 6 “Bioinformatics, modern information technologies and mathematical methods in medicine” 2016.