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Materials Sciences and Applications, 2011, 2, 526-536 doi:10.4236/msa.2011.26071 Published Online June 2011 (http://www.SciRP.org/journal/msa) Copyright © 2011 SciRes. MSA Geometrical Modeling of Crystal Structures with Use of Space of Elliptic Riemannian Geometry Stanislav Rudnev1, Boris Semukhin2, Andrey Klishin1 1Tomsk Polytechnic University, Tomsk, Russia; 2Institute of Strength Physics and Materials Science SB RAS, Tomsk, Russia. Email: stas_rudnev 2004@mail.ru, bss@ispms.tsc.ru, aklishin@yandex.ru Received February 28th, 2011; revised March 21st, 2011; accepted April 6th, 2011. ABSTRACT The space of internal geometry of a model of a real crystal is supposed to be finite, closed, and with a constant Gaus- sian curvature equal to unity, permitting the realization of lattice systems in accordance with Fedorov groups of trans- formations. For visualizing computations, the interpretation of geometrical objects on a Clifford surface (SK) in Rie- mannian geometry with the help of a 2D torus in a Euclidean space is used. The F-algorithm ensures a computation of 2D sections of models of point systems arranged perpendicularly to the symmetry axes l3, l4, and l6. The results of mod- eling can be used for calculations of geometrical sizes of crystal structures, nanostructures, parameters of the cluster organization of oxides, as well as for the development of practical applications connected with improving the structural characteristics of crystalline materials. Keywords: F-Algorithm, Crystal Lattice Systems, Microstructure, Riemannian Geometry, Space of Interpretation 1. Introduction Modern material science, building on very different models of the structure of a substance, tries to create ab- solutely new materials or materials with properties needed when exploiting machines and mechanisms under unusual conditions. Let us note that some successes are recently observed in making materials with a structure modeled at very different scaled and dimensional levels it is the so-called nanomaterials technology. However, essential breakthroughs in applications of these materials when fabricating microelectronic engineering have not yet been made. In our opinion, an essential value has the fact that all up-to-date models are based only on one no- tion of the structure of a solid, namely, considered in a Euclidean space. In the present paper we offer the other, alternative ap- proach to describing both a structure and making materi- als with special or unique properties by means of inter- preting the space of experience and processing materials in a strictly symmetrized electromagnetic field. Development of physicochemical methods of investi- gating a crystal structure and processing technologies, growing needs for production of high quality crystalline materials arouse a lot of attention to new approaches grounded on modeling crystal structures with the use of different modeling spaces. At the present time, concepts of non-Euclidean phase spaces are ever more widely used for describing general evolutionary principles of various physical systems. In this connection, a particular interest of researchers to problems of a possible realiza- tion of Fedorov groups in non-Euclidean spaces is noted. The realization of Fedorov groups of symmetry was con- sidered in a pseudo-Euclidean space, Lobachevsky space and Minkowski space [1]. A lack of similar examinations is that in the specified spaces a space of infinite extent is used for a model of lattices [2,3]. It is traditionally accepted to construct the crystal structure of an ideal crystal by the multiplication of a finite number of atoms by all transformations from some Fedorov group in a Euclidean space. But such a con- struction of an ideal crystal is not related to natural causes of the growth of a real crystal, its finite (localised) character and shape restricted in the space. Besides, the problem of the relation of the internal structure and ex- ternal faceting of a real crystal remains unresolved. An ambiguity of the geometrical interpretation of the space of the real crystal of minerals has led to the necessity of developing models by using a non-Euclidean method of describing crystal structures. When modeling under the conditions of Riemannian geometry, it will be natural to maintain the term “Fe- dorov group” for discrete groups of movements, which Geometrical Modeling of Crystal Structures with Use of Space of Elliptic Riemannian Geometry527 we shall denote by F-groups. For a model of the crystal structure in an elliptic space, as well as in the Euclidean case, the conditions of the global discreteness and ho- mogeneity are fulfilled. In the case of F-groups the fi- niteness theorem for the volume of the fundamental do- main of a finite polyhedron is rather simply proved. When modeling crystal structures, the space of Rie- mannian geometry (V4) with the constant Gaussian cur- vature К = 1, that coincides in sufficiently small regions with a Euclidean space (a locally Euclidean space), was chosen as a modeling space. When constructing crystal lattices, simulated groups of transformations, which are used for computations of point systems, are the basic tool of investigating. In this case we deal with a finite space where the distance between any two points does not ex- ceed a certain value [4-8]. 2. Geometrical Approach to Modeling Crystal Structures 2.1. Space of Interpretation RE Historically developed priority of Euclidean geometry (despite the discoveries of non-Euclidean geometries) has led to two fundamental consequences: 1) Up till now, the space of our experience is supposed to be Euclidean, 2) All the laws of physics and chemistry are supposed to be realizing in a Euclidean space. And though mathematicians and, partly, physicists do not forget to repeat that with N. Lobachevsky, J. Bolyai and B. Riemann’s discoveries Euclidean geometry has lost its unique position as the singular geometry of the space of our experience, the problem of the choice of an adequate geometrical space is still open. Nevertheless, a reliable experimental confirmation of the fulfillment of Coulomb’s law in non-Euclidean spaces has not yet been obtained, and so far it is not clear how it can be realized. However, after A. Poincaré constructed the interpreta- tion (a model) of a non-Euclidean geometry (realizing Lobachevsky’s plane), the situation changed radically. It turned out that Euclidean geometry itself and, accord- ingly, a Euclidean space are no more than one of a vari- ety of geometrical interpretations. There arises the unique possibility of considering the interpretation (rep- resentation) of one geometry by means of geometrical images of other geometry, of course, according to strictly defined rules. A considerable quantity of modern investigations which can be found in [1] is devoted to interpretation problems. For constructing geometrical models of crystal structures the approach applied in [9] has been used. A general scheme for the interpretation of elliptic Rie- mannian geometry in a Euclidean space ( R E I ) is simple enough. A certain geometrical image A of the Riemannian space 4 A V , possessing necessary properties, is chosen as an object of the interpretation in a Euclidean space, with the help of some geometrical image B of the Euclidean space, , R E I AB where The chosen geometrical image B of the Euclidean space is endowed with properties of the geometrical image A, and becomes, thus, а carrier of properties of the geometrical image of the Riemannian space in the Euclidean space. 3.BR Operations with the geometrical image B are executed according to the rules of the Riemannian space with re- gard to certain conditions stated below. Consequently, in the Euclidean space there arises a certain region RE which is called the space of interpretation. R E E I MR, (1) where M is the Riemannian space being subject to the interpretation. Properties of the space RE are sufficiently specific and taken into consideration in each specific case. A Clifford surface SK—a direct circular cylinder of the elliptic space [10] is chosen as the basic geometrical element of the elliptic Riemannian space for constructing the space of interpretation RE. The basis for such a choice is the fact that Euclidean geometry (R2) takes place on SK. D. Hilbert wrote about it that the greatness of W. Clif- ford’s discovery is that the Euclidean plane “in the small” is present in a closed and restricted curvilinear space. SK is isomeric to a Euclidean rectangle or rhomb with identified opposite sides, what leads us to an ordi- nary Euclidean torus. 2 R EK I ST, (2) where T2 is a 2D torus belonging to the Euclidean space. But with that principal difference, that a torus loses its geometrical independence in the space of interpretation and becomes a carrier of the SK properties. That is, in the space of interpretation a torus is a very surface on which Euclidean geometry is fulfilled and well-known groups of movements of a Euclidean torus are more not applica- ble to it. To operate with a torus in the space of interpre- tation one should follow other rules. In RE the motions of a torus are rotations. At the same time, using the fact that RE is “arranged” in a Euclidean space, it is possible to carry out sections of RE by a Euclidean plane and, in such a way, to study properties of structures in the space of interpretation. It is extremely important that the Clifford surface is a carrier of Euclidean geometry, then in RE on a torus (or on systems of tori) all the physical laws show themselves in the same form and with the same content, Copyright © 2011 SciRes. MSA Geometrical Modeling of Crystal Structures with Use of Space of Elliptic Riemannian Geometry 528 as on the Euclidean plane and in the space. The differ- ence is that an arc lying on a torus is used instead of a segment of a Euclidean straight line. For example, in Coulomb’s law L—an arc on a torus is used in RE instead of r. The existence of systems of parallel straight lines of the Riemannian space on the Clifford surface (SK) leads to the presence of groups of parallel transports, both on the Clifford surface and generally in the Riemannian space (the so-called Clifford’s parallels), and, so, of groups of translations. It is important to remember that these groups are realized only on SK, but not on planes of the Riemannian space. In RE these groups show them- selves rather originally and will be described below. Im- ages of translations on the Clifford surface are expressed in RЕ as a rotation of the torus about own axes, and are defined by the group where 12 , ll TTT HHH ,1 l T H is a subgroup of the rotation about the axis l1, 2 l T H is a sub- group of the rotation about the axis l2, that is, as a slip on itself. , R E ST I HH (3) where S H is a translational group formed from sub- groups of the paratactic displacements of SK. Translations of the Clifford surface itself in the Rie- mannian space are expressed in RЕ as a rotation of the torus about own generatrices. 2. R ESK T I HS HT (4) Different crystallographic axes of symmetry (from l2 up to l6) are considered on different Clifford surfaces, what is expressed in RE as an application of tori with dif- ferent ratios of the interior and exterior radiuses. When unfolding tori, it shows itself in the form of rectangles with different lengths of sides and with different angles between their diagonals (for the axis l3, for example, an angle between diagonals is taken to be equal to 60 or 30 degrees, for the axis l4 this angle is equal to 45 degrees, with corresponding side lengths of the rectangle of an unfolding). 2, ii ml R ET KTii IH SHTrR, (5) where ri and Ri are the interior and exterior addresses of a torus, respectively; i l T H are subgroups of the rotation of T2; i m S H are subgroups of the rotation of SK; i is an or- der of symmetry. , R E ST I OO (6) where is a subgroup of the 2, O TT kk OHTrR rotation of a torus, the ratio k r k m Rn the radiuses. ace of interpretation RE (Riemannian in Eu being correct for By the sp clidean) we shall call a totality of all points A being equivalent to a point O through which there passes a to- rus— 2, O kk TrR with the ratio of the internal and exte- rior ra equal to m : n, on all possible transla- tions of this torus, interpreted as it was pointed out above. diuses being ,, ETT RHO (7) where T H f th is a translational subgroup; graphy and ma- te interpretation there exist three families tic field of a charge is restricted in sizes, res of crystal structures in RE is pos- si el of a T O ce is a sub- group oe rotations that form a subspaof RE. We shall call the point О by the center of the space of inter- pretation. It is easy to show that the space of interpreta- tion is restricted, closed, and continuous. A select of the value of the ratio of the radiuses of a torus will be de- fined for a given model by a type of symmetry of a crys- tal structure viewed. It is convenient to study the geo- metrical and structural features of objects in RE by means of sections of the space of interpretation by Euclidean planes. However, the most important property of the space of interpretation is that on a torus in RE we have the right to consider physical laws in their normal Euclidean interpretation, except for a replacement of Euclidean segments by arcs of a torus. For problems of mineralogy, crystallo rial science Coulomb’s law is of heightened inter- est—thanks to its simplicity and the fact that when inter- preting it the features of RE show themselves most brightly, namely: In the space of of the shortest lines (rounds with the ratio of diame- ters being equal 1:2:3) each of which becomes a fam- ily of lines of force of an electrostatic charge Q. The electrostatic lines of force do not start from the charge Q, and do not end at infinity, but touch a surface of the charge, and close up at some distance from it. This distance is calculated using usual trigonometry formulas; An electrosta closed, and continuous. At some distance from a charge Q its electrostatic field is already absent. This distance is called a radius of the field of a charge Q and denoted by Rp. It should be noted that an electro- static field of a charge Q does not occupy the entire space of interpretation completely. Thus two different charges, Q1 and Q2, can interact or not, depending on a distance between them, as well as on their own lin- ear dimensions. Studying the featu ble in sections by its Euclidean plane in two ways: Case 1: Symmetry of a structure is known. A mod crystal structure is considered in the form of point sys- tems where points simulate a distribution of centers of Copyright © 2011 SciRes. MSA Geometrical Modeling of Crystal Structures with Use of Space of Elliptic Riemannian Geometry529 structure is unknown, or a str l lattice system in a Eu here is the othe si tures in In mescribing the to mod- el n el- lip odeling of lattice structures of a given structures of ng of microstructures of crystals in nt for ob e chemical composition of an explored st ture of electrostatic components of pa structing sublattices for ea atoms in lattice points—the so-called R-systems which are computer-simulated using a special program. A sec- tion by the Euclidean plane is carried out either perpen- dicularly to the axis of symmetry of the structure studied, or at an angle interesting us. Case 2: Symmetry of a ucture has been badly studied from the point of view of symmetry. Under these circumstances the task be- comes complicated and is solved in a few stages, what will be considered below (Item 1.2). The principal difference of an idea clidean space and in RE consists in the following: in a Euclidean space an ideal crystal has to be either infinite large-sized or infinitesimal. An ideal Euclidean crystal has no center of symmetry, zonality, and sectoriality, and is not faceted. The question about the faceting of an ideal crystal is solved with the help of introducing different sets of the so-called “boundary conditions”. Simply speaking, the ideal Euclidean model of a crystal does not possess those structural features that are characteristic for real crystals, except for a fragmentary coincidence to restricted fragments of a plane lattice. In the history of mineralogy, crystallography, and geology generally, no real crystal coinciding on its own structural and symme- try characteristics with its ideal Euclidean models has been met. The fact is well-known, but somehow slips attention of researchers all the time. In the space of interpretation RE tr crystal in accordance with its Fedorov group of sym- metry, realized in the elliptic Riemannian space with the involvement of visualizing the model construc- tions. Building on data obtained when modeling point systems with a given Fedorov group, it is possible to form judgments about the morphology of a structure, clustering, and a type of zonality, and to make suppo- sitions of different types of anisotropy; Theoretical modeling of a family of tuation. An ideal crystal in the space of interpretation is restricted in sizes, has a certain shape and symmetry, is zonal and sectorial, and also possesses the center—that is, it possesses practically a complete set of the structural and symmetry characteristics which the real crystal of a mineral has. If one takes into account that both the radius of an electrostatic field Rp and linear dimensions of the space of interpretation RE are calculated in accordance with values of the ionic radius Ri and atomic radius Ra of a given substance, we can always estimate both real sizes of microcrystalline blocks and distances at which there are electrostatic interactions between real ions. 2.2. Principles of Modeling Crystal Struc Elliptic Riemannian Geometry odern structural examinations, when d organization and processes of the growth of a crystal structure, information about the type of a space in which an explored process being watched is not enough used [11-14]. The examinations of different non-Euclidean methods of describing elements of a crystal lattice [4] are the foundation of the suggested theoretical approach to modeling crystal structures under the conditions of Rie- mannian geometry. In order to achieve results the inter- pretation of the geometrical objects (SK, F-groups, and symmetries) in Riemannian geometry is used. In a three-dimensional Euclidean space a 2D torus 2 T (see formula 2), on which the basic geometrical transrma- tions (lattices, elements of Fedorov groups) are consid- ered, corresponds to the Clifford surface (SK). The basic difference from existing approaches fo ing consists in a statement according to which the or- ganization of a lattice structure happens in accordance with a certain F-group (() Ф R E IF) operating in the Riemannian space V4. An a in a model of the lattice system is considered as a point site, and point sys- tems are studied in the initial stage without regard to the chemical features of atoms. By using the approach under consideration, modeling the cluster organization of mi- crostructures of oxides is carried out, as well as practical applications aimed at perfecting crystalline materials and improving their physical properties are developed. General principles of modeling crystal structures i tom or ion tic Riemannian geometry may be reduced to the fol- lowing: Theoretical m electrostatic fields of ions of elements having partici- pated in the formation of lattices of a given crystal; constructing the potential surfaces answering to minimum energy; Theoretical modeli the model sections perpendicular to axes of symmetry, with regard to: a composition, symmetry, a structure of electrostatic fields of ions and their systems, the principle of closest packing; as well as crystallo- graphic analysis and interpreting data obtained. Thus a general procedure of computing experime taining a model of a crystal structure of a necessary chemical composition and structure adds up to the fol- lowing stages. Stage 1: Th ructure is detected. Stage 2: The struc rameters of ions and atoms constituting the structure of a given substance is computed. Stage 3: Regularities of con ch of elements composing a given structure are re- vealed, according to the series of crystallochemical activ- ity [15]. Copyright © 2011 SciRes. MSA Geometrical Modeling of Crystal Structures with Use of Space of Elliptic Riemannian Geometry 530 Stage 4: Upon the Clifford surface’s unfolding a frag- m appropriate to- ru ent of the sublattice is constructed using the computa- tions of the parameters of electrostatic components of a model for each ion, and then the incorporation of sublat- tices of the rest of atoms and ions is conducted in accor- dance with the computed parameters, according to the series of crystallochemical activity [15]. Stage 5: By the unfolding obtained an s is folded, and all the necessary transformations (par- allel translations, rotations about axes of symmetry, etc.) are applied to it, according to the scheme for the inter- pretation R E I . Stage 6: For visualizing the constructions a necessary se rithm in the examination of properties and ction of the space RE by a Euclidean plane is con- structed, where the features of a studied structure are considered. 2.3. F-Algo The important place features of the organization of a crystal structure in the space RE is occupied with constructions of point systems with the use of special computational algorithms for solving of tasks of the visualization and interpretation of constructed models. For modeling lattice systems in the space RE an algorithm has been developed, with the help of which constructions of lattices are produced with regard to the fact that for each Fedorov group Ф of the Euclid- ean space R3 in the Riemannian space there is an appro- priate F-group satisfying to the condition () R E IF . For the proof it is enough to take the pacerojective sp RP and 3 and to use the homeomorphism theorem for the groups SO(3) and SU(2) [16]. From an epimorphism π: SU(2) SO(3) it is possible to obtain F-groups, as subgroups of the group SU(2). The so-called binary groups arise: *1 π , nn DD *1 π, nn *1 π, nn TT *1 π, nn QQ I I a binary group of a dihedral, a binary group of a tetrahe- lgorithm suggested in this work defines rules for co dron, a binary group of an octahedron, and a binary group of an icosahedron. The binary groups, as well as orthogonal representations, in whole, arise naturally in describing a physical system with a spin and in calculat- ing characteristics of gravitational fields in Riemannian spaces. The a nstructing point systems for a given F-group and there- fore will be denoted by F-algorithm. The given F-algorithm, realizing the translational subgroup S H in the Riemannian space when interpreting S K S R E IH 2T H T, will describe a representation b point on a Euclidean plane: y the fongllowi rule for a 32π exp , s li rl 1 1 1, , s s n n (8) 1. F rr G (9) As parameters of a model the ac following variables were cepted: GF is a matrix of a subgroup of the rotations T O induced in a section by a given F-group, where RF E I ; rs is the radius of the surface SK; t r is the appropriate torus 2 T, where R radius of an E st I rr ; n is an amount of the points lyingn the sece sur- face SK. A set of points computed by the F-algorithm (8) or- ga otion of th nizes a system of the points on a plane, which we shall denote by K-system. The parameters defining a K-system are: a Fedorov group Ф, F-groups, and a basic axis of symmetry li. A K-system can be determined with the help of its constituent sets i n T and represented in the follow- ing form: 123 ,, nn n K TT T, (10) where are sequentially generat i n T ca ed sets in Figure 1 (for these 6n ), satisfying to the conditions given below: 222 22221 , 32222 , ,,1,, ,,,,1,, ,,,,1,. iiit O njj jjtn xy nkk kktn xy yxyrin Txy xyrxyTin Txy xyrxyTkn (11) A set of points computed by the F-algorithm (9) or- a 1 ni Tx g nizes a system of the points on a plane, which we shall denote by R-system. This implies that the generation of an R-system is carried out by performing a group of the transformations 2T H T over a K-system. 2TTK. RH (12) The initial forms of the organiz ar E ation of real crystals e R-systems. The real structure of a crystal is supposed to be consisting of rows of particles, integrated into a system, and to have concrete sizes of the space of inter- pretation. With the help of the apparatus of elliptic Rie- mannian geometry, a distribution of atoms and a filling of the crystalline space can be interpreted as a compact locally Euclidean set. The misorientation and displace- ment of R-systems are accompanied by the appearance, in the integrated space of a crystal, of blocks and bound- ary spaces separating them [15]. The process of the organization of a crystal structure in R consists in sequential introducing new particles into consideration, arranged at the shortest distances (put on a torus) from fixed ones in accordance with a given Fedorov Copyright © 2011 SciRes. MSA Geometrical Modeling of Crystal Structures with Use of Space of Elliptic Riemannian Geometry531 (a) (b) (c) Figure 1. Stages of constructing a K-system. A sequence of the formation of the sets: (a)b; (c) hoizon of a lat- ce in RE can be presented as a multistage process in n number of the la of a crystalline polyhe- dr along geodetic lines of the elliptic space, spreading the 1 T; ( 6)2 6 T3 6 T. group, up to the faceting. Tus the rganati ti which bonding of chemically responsive particles is un- der way, beginning from the zero point. On a K-system, nodes are connected by arcs of circles and arranged from each other at a certai ttice periods (E). An ideal crystal is usually constructed by the multiplication of a finite number of atoms by all the transformations of some Fedorov group, but such a construction of the ideal crystal is not related to natural causes of the crystallization [4]. The organization of a lattice system in the space of in- terpretation RE, up to the faceting on, consists of the structurally-forming (Figure 2(a)) and structurally-filling (Figure 2(b)) stages, each of newly fixed particles behaving like a new center of the organization of the lattice. The realization of bonds passes (a) (b) Figure 2. (a) Structurally-forming (), and structur- ally-filling stages of the organizationtal stcture; (b) The sets (have bonstr. apec Riemannian ace. degrees of order of atoms inside a crystal and at its res in the space developed, ons of crys- 1 12 T of a crys ucted ru 2 12 T,3 12 T) een c forces of long-range interaction to the whole closed re- gion, what is scific feature of an ellipti sp The noted features of a distribution of lattice points on a R-system are in good conformity with really observed different near-surface layers, different reticular densities of particles on different areas of the same face. The con- structed model possesses all the attributes of a real crys- tal: a center, an exterior faceting, symmetries of a certain Fedorov group, finite sizes, and zonality. 3. Visualization of Lattice Systems For computer simulation of crystal structu RE an rСrystal software package has been allowing to conduct the visualization of secti tal structures and to construct models of the electrostatic components of parameters of ions. There is the possibil- ity of inputting various types of crystal structures with the use of a special technique of coding. The software package for modeling point systems uses the F-algorithm Copyright © 2011 SciRes. MSA Geometrical Modeling of Crystal Structures with Use of Space of Elliptic Riemannian Geometry 532 for visualizing constructions and the scheme for the in- terpretation R E I of geometrical objects on the Clifford surface in the Riemannian space. Examples of the computations of point systems with the use of th-algorithm have been given for modeling the symmetry axes: l3 (Figure 4), e F l4 (Figure 5), and l6 (F d sections perpendicularly to the axis l6, igure 3). It is obvious from Figure 3(a) and Figure 3(b) that the character of the distributions of the point systems, on the constructe essentially differs in a central region. If the point system in Figure 3(a) has a homogeneous distribution, then for the point system in Figure 3(b) a block structure is (a) Figure 4. The organization of a lattice system in the section plane, perpendicularly to the axes l3, for a model of the trigonal system. Figure 5. The organization of a lattice system in the section plane, perpendicularly to the axes l4, for a model of the cu- bic system. on in the space and have a centre and a zonal ructure. ures 7 and 8) and a series of its polymorphic m clearly seen. All the constructed point systems fill a re- stricted regi st With the use of the rCrystal software package the com- plex modeling of the crystal structures SiO2 (Figure 6), Al2O3 (Fig odifications α-, β-, θ-, and γ-Al2O3 [17,18] has been carried out. A method of modeling mesoatomic ensem- bles, clusters and crystal structures, permitting to explore the geometrical properties of oxide systems, has been (b) Figure 3. The homogeneous (a) and block (b) structures (an ideal system) in a section perpendicular to the axes l6. Copyright © 2011 SciRes. MSA Geometrical Modeling of Crystal Structures with Use of Space of Elliptic Riemannian Geometry533 po- ne tiocharge distribution on a surface developed. On the basis of the model computations a minimum structural element Al4O6 has been selected, parameters of which were computed in the space RE. Us- ing the above mentioned approach a model of the crys- talline structure Al2O3 has been computed. The geomet- rical configuration of a micropolyhedron Al4O6 and a set of the external physicochemical conditions (a structure of the electromagnetic field, thermodynamic parameters, crystallophysical parameters) define methods of packing of the micropolyhedrons in different motifs in accor- dance with the principle of closest packing (Figures 7(а) and (b)). The structure of an electrostatic field modeled in the space RE, being formed by the structural element Al4O6, defines possible methods of integrating into a macrostructure in accordance with the most probable joining mechanism. 4. Computation of Pair Interaction Potentials When Modeling Structures in the Space RE In the space RE the models of the electrostatic com nts of parameters of ions in the case of pair interac- ns with regard to a have been constructed for the following ions presented in Table 1. When computing, a model of the charged spheres with different charge distributions on a surface was used. The suggested approach uses the scheme , R E I what allows to study the features and character of the interaction of elements in a modeled crystal structure, and to explore regularities of its formation. When computing the geometrical parameters of poten- tials of the interatomic interaction, it was supposed that the interaction occurs under the conditions of the elliptic O Si Figure 6. The model of the crystal structure SiO2. The sec- tion arranged perpendicularly to the axis l3 has been shown. closed Riemannian space V4. The structural features of an electrostatic field in the Riemannian space, considered for the case of interacting the unlike charges of ions, im- pose the special conditions which lead to the formation of coherent states and to the formation of local closed (a) (b) Figure 7. The motif I1 of the model of the crystal structure Al2O3, satisfying to the principle of closest packing (a), the corundum motif I2 (b); the models has been presented in a projection on the plane [0001]. Copyright © 2011 SciRes. MSA Geometrical Modeling of Crystal Structures with Use of Space of Elliptic Riemannian Geometry 534 regions of the attraction and repulsion, what speaks of the reasonability of the applied approach [4,15]. In the last column of Table 1 the computed values of radiuses of the electrostatic fields in the space RE have been given. The atomic and ionic radiuses used when modeling were taken from N. Belov’s work [19]. When interpreting the total electrostatic field in mod- els of ionic crystals, it becomes possible to take into ac- count the multiparticle and long-range parts of the inter- actions that occur under the conditions of elliptic Rie- mannian geometry, and also there appears the possibility for investigating their role in the stabilization of probable polymorphic structures. The energy of the atomic bunching under the condi- tions of a given atomic volume in ionic crystals is char- acterized by the interatomic interaction potential V(r The suggested model potential becomes strongly repul- on the distance between adjacent ions in a ntial curves for a series of the systems underated byf the su teavn According to the principles of modeling, s 1.3e comion of the eters of coents of the electrostatents (Fres 9-1aving par in the tion of thystallinblattices ostals have out. I the space RE a model of the electrostatic field of a charge is chrized by: tricted size, the closure, ae cony. The stre of the moelectro- static field has a distinct zonal character. The shape of a diution nergy b, a group ofmetries ation of su rystal on s the interpretation of geomet- ric (a) ). sive at a distance R, less than some critical value 2R0 (Figure 8). Let us note that the potential V(r) directly influences crystal. It can be expected that the distance between the nearest neighbors will be near Rmin (a minimum point). The parameters1 of pote der consi d aph, h ion, comput e beiven i means o Table 2. tated in item ggesproacen g , thputatparammpon ic fields of 1) h ions for different ticipated elem forma igu e cre suf crybeen carried n aractea res nd thtinuitucturdel stribof the eands sym (b) Figure 8. The model curves of the radial section of the elec- trostatic potential of the ions: Н+ (a) and O2– (b), built in the space RE. Table 1. The computation of radiuses of electrostatic fields of the ions in the space RE. ion atomic radius (a.u.) ionic radius (a.u.) radius of electrostatic field (a.u.) and the energy parameters determine a type of the con- figuration of ions participating in the organiz blattices. 5. Conclusions The possibilities for constructing a model of a real c other principles immediately connected with the in- volvement of a non-Euclidean method of describing have been considered. The elliptic closed space of Riemannian geometry V4 with the constant Gaussian curvature К = 1, that coincides in sufficiently small regions with a Euclid- ean space, was chosen as a modeling space. For visualiz- ing the model construction H+ 0.25 1.36 23.19 O2– 0.66 1.36 11.04 Fe2+ 1.18 0.78 6.26 Fe3+ 1.18 0.65 5.79 Mg2+ 1.45 0.72 6.87 Al3+ 1.26 0.53 5.70 al objects on a Clifford surface (SK) in Riemannian 1The Notes: R0is the radius of the hard component of the electrostatic field, where the repulsive forces have a dominating character; Rmin is the coordinate of the minimum of the potential V(r); ε is the value o f the potential well V(r). Copyright © 2011 SciRes. MSA Geometrical Modeling of Crystal Structures with Use of Space of Elliptic Riemannian Geometry535 Table 2. The parameters of potential curves of the intera- tomic interaction in the space RE. bond type R0, Rmin, depth radius of electrostatic (a.u.) (a.u.) (ε) field (a.u.) [H+] – O2– 17.96 20.81 –6.14 23.19 [O2–] – Fe2+ 7.88 8.76 –8.51 11.04 [O2–] – Fe3+ 7.88 8.92 –8.81 11.04 [O2–] – Al3+ 7.98 8.97 –8.90 11.04 [Fe2+] – O2– 5.07 5.68 –5.81 6.26 [Fe3+] – O2– 4.71 5.51 –16.22 5.79 [Mg2+] – O2– 5.60 6.24 –6.32 6.87 [Al3+] – O2– 4.66 5.44 –17.41 5.70 Figure 9. The equatorial section of 3+ the model of the electro- static potential of the ion Al. The computation has been carr the RE. ied out inspace Figure 11. The equipotential picture of the electrostatic field of the ion O2– relatively the ion H+. The frontal section. geometry with the help of a 2D torus in a Euclidean space E3 is used. The process of the organization of a crystal structure in the space of interpretation RE is pre- sented as a multistage process consisting in sequential introducing the point systems into consideration, each point of whic accordance with a given Fedorov group, beginning from a zero point and up to the faceting of a crystal, each of newly fixed point behaving like a new center of the or- ganization of the lattice. Properties of the internal space of a real crystal, such as zonality, boundedness in sizes (finiteness), sectoriality and the occurrence of a center, are naturally deduced from properties of the modeling space. The main characteristic of improvements in the suggested model constructions is that the local changes The developed F-algorithm for constructing crystalline lattice systems ensures a computation of 2D sections of models of point systems, arranged perpendicularly to the symmetry axes l3, l4, and l6. As an example of the reali- zation of the developed principles of modeling, the computations of components of the electrostatic fields of the ions Al3+, O2–, Fe2+, and Mg2+, having participated in the formation of the crystalline sublattices of α-, β-, θ-, and γ-Al2O3, have been presented. The linear parameters of crystal lattices, obtained from the model computations, are connected with the character of their constituent at- oms and can be used to determine a type of chemical bonding between them. The suggested approach can be used for computing the as for the development of practical 2T RHTK h is fixed on a torus in of structural features, as well as curved regions (defects), can be more completely taken into account. geometrical parameters of the cluster organization of nanostructures of oxides and many other nonequilibrium materials, as well Figure 10. The equipotential picture of the electrostatic field of the ion H+ relatively the ion O2–. The frontal section. 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