=Paper=
{{Paper
|id=Vol-1490/paper19
|storemode=property
|title=Modeling and identification of centered crystal lattices in three-dimensional space
|pdfUrl=https://ceur-ws.org/Vol-1490/paper19.pdf
|volume=Vol-1490
}}
==Modeling and identification of centered crystal lattices in three-dimensional space==
https://ceur-ws.org/Vol-1490/paper19.pdf
Computer Optics and Nanophotonics
Modeling and identification of centered crystal lattices
in three-dimensional space
Kirsh D.V.,
Samara State Aerospace University
Kupriyanov A.V.
Image Processing Systems Institute, Russian Academy of Sciences
Abstract. The paper offers a method that allows to model crystal lattices in
three dimensional space. The method is based on a description of crystal lattices
in the six-dimensional Euclidean space G6. Transformations of three basic
translation vectors from centered types to primitive types have been derived.
Using these transformations, the proposed method provides a uniform way to
model both primitive and centered Bravais lattices. On a set of simulated
lattices, the possibility of structural identification of primitive and centered
crystal lattices has been investigated. The previously developed parametric
identification method based on estimation of Wigner-Seitz cell volume has
demonstrated the best result of separation of centered lattices from primitive
ones. In addition, this method allowed to identify the type of centering for five
out of seven centered Bravais lattices.
Keywords: crystal lattice, primitive lattice, centered lattice, Bravais unit cell,
Wigner-Seitz cell, structural identification, similarity measure
Citation: Kirsh D.V., Kupriyanov A.V. Modeling and identification of сentered
crystal lattices in three-dimensional space. Proceedings of Information
Technology and Nanotechnology (ITNT-2015), CEUR Workshop Proceedings,
2015; 1490: 162-170. DOI: 10.18287/1613-0073-2015-1490-162-170
1. Introduction
A three-dimensional crystal lattice is a mathematical model which does not only
allow to describe a structure of any crystalline material, but also determines its basic
physical and chemical properties.
The problem of structural identification of crystal lattices still remains one of the
main problems related to X-ray diffraction analysis. In his paper, Kupriyanov [1] has
proposed an efficient approach to its solution which consist of estimating a number of
parameters in the lattices under investigation and their subsequent comparison with
parameters of reference lattices. The lattices either previously investigated or derived
by modeling can be used as samples. Thus, the accurate structural identification of
crystal lattices requires a large database of predetermined reference parameters.
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The analysis of existing methods to solve the problem of structural identification
has shown that the methods are mainly studied for lattices of seven primitive types,
while the centered lattices are completely ignored.
In this paper, we propose a modeling method for centered crystal lattices and
investigate the possibility of structural identification of primitive and centered lattices
by using the existing and developed parametric identification methods.
2. Modeling Method for Centered Crystal Lattices
A crystal lattice model used in previous researches was based on three translation
vectors [2]. Its basic parameters are:
three edge lengths l1 , l2 , l3 and three angle values 1 , 2 , 3 of a unit cell (Fig. 1);
a number of lattice points by each of the translation vectors N1 , N2 , N3 .
Fig. 1. – Unit cell model constructed on three basic translation vectors
This model has one significant disadvantage: it can only describe lattices of seven
lattice systems – primitive Bravais lattices (index P). However, there are seven more
centered Bravais lattices, which, in turn, are divided into the following three types [3]:
body-centered (index I): one additional point at the center of the unit cell;
base-centered (indices A, B, C): additional points at the centers of two parallel
sides of the unit cell;
face-centered (index F): additional points at the centers of each face of the unit
cell.
Therefore, all crystal lattices can be assigned to one of the 14 Bravais lattice types
(Fig. 2).
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Fig. 2. – Primitive and centered types of Bravais lattices
To describe centered lattices, the used model has to be expanded by adding up to
three translation vectors (Fig. 3). Doubling the number of parameters makes the
model overly complicated.
Fig. 3. – Choice of extra translation vectors within a centered lattice
An analogue of this model was suggested by Andrews and Bernstein [4]. It
describes the lattice in the form of a six-dimensional vector g in the space G6. In
addition, Andrews and Bernstein proved that each centered lattice gT might be
transformed to a primitive representation g P by the following transformation:
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g P A gT . (1)
Each centered type has its own general form of the transformation matrix A. By
solving the system of linear equations (1) with respect to basic translation vectors
a1 , a2 , a3 , we deduced the following solutions:
a1P a1I a1P a1C a1P 0.5a1F 0.5a2 F
a2 P a2 I ; a2 P 0.5a1C 0.5a2C ; a2 P 0.5a1F 0.5a3 F . (2)
a 0.5a 0.5a 0.5a a a a 0.5a 0.5a
3P 1I 2I 3I 3P 3C 3P 2F 3F
In other words, the transformations (2) make a choice of three new translation
vectors. These vectors also allow to simulate all lattice points, however their
parameters correspond to a primitive lattice that can be allocated within a centered
lattice (Fig. 4).
Fig. 4. – Selection of translation vectors of a primitive lattice within a centered lattice
This result gives a completely different look at the problem of modeling crystal
lattices. On the one hand, if it is necessary to model a large set of random lattices, this
set can be simulated by changing each parameter of a unit cell (lengths of the edges
and values of the angles) independently in a specified range with a given partition.
The resulting set of lattices will cover all primitive as well as centered lattices.
On the other hand, if the experiment requires a certain number of lattices for each
type of centering, it would be possible to obtain a necessary number of lattices using
the above transformations (2).
In the framework of new ideas, three basic translation vectors are to be sufficient to
describe any crystal lattice in three-dimensional space. It is only needed to
supplement them with the transformations (2) allowing to simulate a lattice of a
particular centered type.
3. Methods of Crystal Lattice Parametric Identification
The basic methods among the existing crystal lattice parametric identification
methods are the following:
parametric identification method based on estimation of atomic packing factor [5];
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parametric identification method based on estimation of distances between
isosurfaces [6].
In the parametric identification method based on estimation of atomic packing
factor, each unit cell is represented as close packing of spheres (Fig. 5). The atomic
packing factor can be calculated by dividing volume of atoms in unit cell by volume
of unit cell.
Fig. 5. – Close packing of spheres
In the parametric identification method based on estimation of distances between
isosurfaces, a set of isosurfaces are constructed for each unit cell (Fig. 6). Then the
root mean square distance and the Hausdorff distance between the isosurfaces are
calculated.
Fig. 6. – Types of isosurfaces constructed for a cubic lattice
The investigation of above two methods has shown that they have a number of
disadvantages: a strong dependence of the crystal lattice identification accuracy from
the type of the crystal system and a high sensitivity to distortions of coordinates of
crystal lattice points. In order to eliminate these drawbacks, we have developed two
new parametric identification methods.
The first of the developed method is based on estimation of Bravais unit cell
parameters (lengths of three edges and values of three angles between the edges). By
using the rotation of the analyzed lattice about coordinate axes, the proposed
algorithm chooses three lattice points and calculates lengths of their radius vectors
and angles between them. The radius vectors of the chosen points are non-coplanar
and have minimum lengths [7].
The second of the developed method is based on estimation of Wigner-Seitz cell
volume. The proposed algorithm uses Monte Carlo method to calculate the volume. It
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constructs planes limiting Wigner-Seitz cell and scatters randomly a great number of
points in the lattice area. The amount of points that fall into the limited area
determines the Wigner-Seitz cell volume [8].
All the methods listed above use normalized similarity measures to compare
estimated parameters with reference parameters. The similarity measures take the
greatest value of unit when parameters completely coincide.
4. Identification of Centered Crystal Lattices
We conducted a series of computational experiments in order to investigate the
possibility of separation centered lattices from primitive ones, as well as the
possibility of delimiting types of centering.
Using the developed modeling method, crystal lattices of 11 Bravais types were
simulated: monoclinic (mP, mC), orthorhombic (oP, oC, oI, oF), tetragonal (tP, tI)
and cubic (cP, cI, cF). Triclinic, rhombohedral and hexagonal crystal systems were
excluded from our consideration since all lattices of these systems are primitive. As
simulation parameters we used the parameters of crystal lattices of natural minerals
which have the unit cells similar in volume.
Then, the identification methods were used to estimate parameters of each crystal
lattice and to calculate similarity measures between parameters of primitive and
centered lattices. The results of computational experiments in the form of diagrams
are presented below in Fig. 7 and Fig. 8.
Fig. 7. – Similarity between centered and primitive Bravais lattices.
Comparing atomic packing factors and distances between isosurfaces
Fig. 7 shows that RMS and Hausdorff distances are weakly dependent on the type
of centering. Moreover, for two centered lattices, the value of their similarity with
primitive lattices was about 0.95. Thus, the identification method based on estimation
of distances between isosurfaces is not applicable to separate centered lattices from
primitive ones. The obtained result can be explained by the fact that this method
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estimates the maximum and the average distances between lattice points, whereas the
centering has the greatest influence on the minimum distance. A similar situation can
be observed in Fig. 8 for the identification method based on estimation of Bravais unit
cell parameters (edges and angles).
Fig. 8. – Similarity between centered and primitive Bravais lattices.
Comparing edges and angles of unit cell and Wigner-Seitz cell volumes
The identification methods based on estimation of atomic packing factor and
Wigner-Seitz cell volume have demonstrated the best result among other
identification methods. The maximum value of their similarity with the primitive
types was 0.66 and the average value of difference between the centered types was
0.2. Therefore, these methods allow to identify the type of centering with high
accuracy. However, for orthorhombic system, it proved to be impossible to separate
body-centered lattices from base-centered lattices.
It should be noted that the parametric identification method based on estimation of
Wigner-Seitz cell volume does not require any information about basic translation
vectors of a unit cell. As a consequence, this method is more universal than the
parametric identification method based on estimation of atomic packing factor. At the
same time, the parametric identification method based on estimation of Wigner-Seitz
cell volume has the highest computational complexity because of the Monte Carlo
method taken as its basis.
5. Conclusion
The developed crystal lattice modeling method allows to simulate any primitive or
centered lattice in three-dimensional space. Besides, by changing each parameter of a
unit cell (lengths of the edges and values of the angles) independently, a large set of
lattices can be simulated to cover all 14 types of Bravais lattices. This set is a
necessary condition for correct structure identification of crystal lattices.
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The performed investigation of possible identification of centered crystal lattices
has demonstrated that the parametric identification method based on estimation of
Wigner-Seitz cell volume allows to separate with the high accuracy the centered
lattices from the primitive lattices. In addition, it can identify the type of centering for
five out of seven centered Bravais lattices (mC, oF, tI, cI, cF).
As it has been shown in previous papers, the parametric identification method
based on estimation of Bravais unit cell parameters provides the high accuracy of
lattice system identification. However, the Bravais type is determined not only by the
lattice system but also by the type of centering. Therefore, it is essential to use both
developed methods for the most accurate structural identification of crystal lattices.
Acknowledgements
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 (# 14-01-00369-a, # 14-07-97040-p_ povolzh'e_a, # 15-29-03823, #
15-29-07077); by the ONIT RAS program # 6 “Bioinformatics, modern information
technologies and mathematical methods in medicine” 2015.
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