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10 pages, 256 KiB  
Article
Discrete Z4 Symmetry in Quantum Gravity
by Grigory E. Volovik
Symmetry 2024, 16(9), 1131; https://doi.org/10.3390/sym16091131 - 1 Sep 2024
Cited by 1 | Viewed by 1148
Abstract
We consider the discrete Z4 symmetry i^, which takes place in the scenario of quantum gravity where the gravitational tetrads emerge as the order parameter—the vacuum expectation value of the bilinear combination of fermionic operators. Under this symmetry operation, [...] Read more.
We consider the discrete Z4 symmetry i^, which takes place in the scenario of quantum gravity where the gravitational tetrads emerge as the order parameter—the vacuum expectation value of the bilinear combination of fermionic operators. Under this symmetry operation, i^, the emerging tetrads are multiplied by the imaginary unit, i^eμa=ieμa. The existence of such symmetry and the spontaneous breaking of this symmetry are also supported by the consideration of the symmetry breaking scheme in the topological superfluid 3He-B. The order parameter in 3He-B is also the bilinear combination of the fermionic operators. This order parameter is the analog of the tetrad field, but it has complex values. The i^-symmetry operation changes the phase of the complex order parameter by π/2, which corresponds to the Z4 discrete symmetry in quantum gravity. We also considered the alternative scenario of the breaking of this Z4 symmetry, in which the i^-operation changes sign of the scalar curvature, i^R=R, and thus the Einstein–Hilbert action violates the i^-symmetry. In the alternative scenario of symmetry breaking, the gravitational coupling K=1/16πG plays the role of the order parameter, which changes sign under i^-transformation. Full article
(This article belongs to the Section Physics)
12 pages, 249 KiB  
Article
On the Choice of Variable for Quantization of Conformal GR
by A. B. Arbuzov and A. A. Nikitenko
Universe 2024, 10(7), 294; https://doi.org/10.3390/universe10070294 - 13 Jul 2024
Cited by 2 | Viewed by 839
Abstract
The possibility of using spin connection components as basic quantization variables of a conformal version of general relativity is studied. The considered model contains gravitational degrees of freedom and a scalar dilaton field. The standard tetrad formalism is applied. Properties of spin connections [...] Read more.
The possibility of using spin connection components as basic quantization variables of a conformal version of general relativity is studied. The considered model contains gravitational degrees of freedom and a scalar dilaton field. The standard tetrad formalism is applied. Properties of spin connections in this model are analyzed. Secondary quantization of the chosen variables is performed. The gravitational part of the model action turns out to be quadratic with respect to the spin connections. So at the quantum level, the model looks trivial, i.e., without quantum self-interactions. Meanwhile the correspondence to general relativity is preserved at the classical level. Full article
(This article belongs to the Special Issue Quantum Physics including Gravity: Highlights and Novelties)
13 pages, 13236 KiB  
Article
Preparation and Characterization of High-Strength and High-Modulus Multi-Walled Carbon Nanotube/Hydroxyapatite/Carbon Fiber/Polyetheretherketone Composites
by Lijian Liu, Yongkang Zhang, Shaobo Ma, Shouxiao Zhu, Shuxuan Wu, Bin Wei and Guang Yang
Appl. Sci. 2024, 14(5), 1723; https://doi.org/10.3390/app14051723 - 20 Feb 2024
Cited by 6 | Viewed by 1600
Abstract
The materials currently used for bone prostheses are mainly metals such as titanium alloy. Compared with human cortical bone, traditional metals exhibit high elastic moduli, which may lead to stress shielding. In contrast, the mechanical strength and elastic modulus of polyetheretherketone are slightly [...] Read more.
The materials currently used for bone prostheses are mainly metals such as titanium alloy. Compared with human cortical bone, traditional metals exhibit high elastic moduli, which may lead to stress shielding. In contrast, the mechanical strength and elastic modulus of polyetheretherketone are slightly lower than those of human bone. In this study, we utilize multi-component co-modification in order to improve the mechanical properties of polyetheretherketone materials. Firstly, we blended three types of materials: multi-walled carbon nanotubes, nanohydroxyapatite, and carbon fibers. Then, the blended materials were used to reinforce polyetheretherketone, and the preparation of the multi-walled carbon nanotubes/nanohydroxyapatite/carbon fibers was described in detail. After the contact angle was determined, SEM, thermogravimetric analysis, and various tests were conducted on the ternary composites. Ball milling co-mingling technology was used to prepare tetrad composite specimens for tensile and compression experiments, a finite element model of the tetrad composite tensile and compression specimens was established, and the structural stresses of the specimens were analyzed under the specified loads. The experimental results show that the surface roughness of carbon fiber increased, and the multi-walled carbon nanotube/nanohydroxyapatite/carbon fiber/polyetheretherketone quaternary composites were well optimized in terms of strength and elastic modulus, which broadens the application field of polyetheretherketone materials. Full article
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10 pages, 376 KiB  
Article
Chronology Protection in f(T) Gravity: The Case of Gott’s Pair of Moving Cosmic Strings
by Franco Fiorini
Universe 2024, 10(1), 52; https://doi.org/10.3390/universe10010052 - 22 Jan 2024
Cited by 2 | Viewed by 1434
Abstract
As a consequence of the spacetime structure, defined by the tetrad field instead of the metric tensor alone, f(T) gravity seems to harbor its own chronology protection agency. When Gott’s pair of moving cosmic strings is considered, it is shown [...] Read more.
As a consequence of the spacetime structure, defined by the tetrad field instead of the metric tensor alone, f(T) gravity seems to harbor its own chronology protection agency. When Gott’s pair of moving cosmic strings is considered, it is shown that the requirement of having a global parallelization—i.e., a global smooth field of tetrads– drastically restricts the form of the tetrads on the junction surface between the two strings. The junction conditions on the tetrad field are satisfied only if the corresponding boosts needed to put the strings in motion are null on the matching surface. This seems to throw overboard Gott’s construction from the outset without the need for analyzing the divergence of the expectation value of the energy–momentum tensor on the Cauchy horizon, evading in this way bothersome quarrels concerning the choice of vacuum. Full article
(This article belongs to the Section Gravitation)
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18 pages, 2370 KiB  
Review
G-Quadruplexes in Human Telomere: Structures, Properties, and Applications
by Yan Xu and Makoto Komiyama
Molecules 2024, 29(1), 174; https://doi.org/10.3390/molecules29010174 - 27 Dec 2023
Cited by 18 | Viewed by 5064
Abstract
G-quadruplexes, intricate four-stranded structures composed of G-tetrads formed by four guanine bases, are prevalent in both DNA and RNA. Notably, these structures play pivotal roles in human telomeres, contributing to essential cellular functions. Additionally, the existence of DNA:RNA hybrid G-quadruplexes adds a layer [...] Read more.
G-quadruplexes, intricate four-stranded structures composed of G-tetrads formed by four guanine bases, are prevalent in both DNA and RNA. Notably, these structures play pivotal roles in human telomeres, contributing to essential cellular functions. Additionally, the existence of DNA:RNA hybrid G-quadruplexes adds a layer of complexity to their structural diversity. This review provides a comprehensive overview of recent advancements in unraveling the intricacies of DNA and RNA G-quadruplexes within human telomeres. Detailed insights into their structural features are presented, encompassing the latest developments in chemical approaches designed to probe these G-quadruplex structures. Furthermore, this review explores the applications of G-quadruplex structures in targeting human telomeres. Finally, the manuscript outlines the imminent challenges in this evolving field, setting the stage for future investigations. Full article
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26 pages, 382 KiB  
Article
The Relativistic Rotation Transformation and the Observer Manifold
by Satyanad Kichenassamy
Axioms 2023, 12(12), 1066; https://doi.org/10.3390/axioms12121066 - 21 Nov 2023
Cited by 1 | Viewed by 1460
Abstract
We show that relativistic rotation transformations represent transfer maps between the laboratory system and a local observer on an observer manifold, rather than an event manifold, in the spirit of C-equivalence. Rotation is, therefore, not a parameterised motion on a background space or [...] Read more.
We show that relativistic rotation transformations represent transfer maps between the laboratory system and a local observer on an observer manifold, rather than an event manifold, in the spirit of C-equivalence. Rotation is, therefore, not a parameterised motion on a background space or spacetime, but is determined by a particular sequence of tetrads related by specific special Lorentz transformations or boosts. Because such Lorentz boosts do not form a group, these tetrads represent distinct observers that cannot put together their local descriptions into a manifold in the usual sense. The choice of observer manifold depends on the dynamical situation under consideration, and is not solely determined by the kinematics. Three examples are given: Franklin’s rotation transformation for uniform plane rotation, the Thomas precession of a vector attached to an electron, and the motion of a charged particle in an electromagnetic field. In each case, at each point of its trajectory, there is a distinguished tetrad and a special Lorentz transformation that maps Minkowski space to the spacetime of the local observer on the curve. Full article
(This article belongs to the Section Hilbert’s Sixth Problem)
29 pages, 13985 KiB  
Article
Integration of Whole-Rock Geochemistry and Mineral Chemistry Data for the Petrogenesis of A-Type Ring Complex from Gebel El Bakriyah Area, Egypt
by Ahmed A. Abd El-Fatah, Adel A. Surour, Mokhles K. Azer and Ahmed A. Madani
Minerals 2023, 13(10), 1273; https://doi.org/10.3390/min13101273 - 29 Sep 2023
Cited by 3 | Viewed by 2322
Abstract
El Bakriyah Ring Complex (BRC) is a prominent Neoproterozoic post-collisional granite suite in the southern part of the Central Eastern Desert of Egypt. The BRC bears critical materials (F, B, Nb, and Ta) in appreciable amounts either in the form of rare-metals dissemination [...] Read more.
El Bakriyah Ring Complex (BRC) is a prominent Neoproterozoic post-collisional granite suite in the southern part of the Central Eastern Desert of Egypt. The BRC bears critical materials (F, B, Nb, and Ta) in appreciable amounts either in the form of rare-metals dissemination or in the form of fluorite and barite vein mineralization. The complex consists of inner syenogranite and outer alkali feldspar granite that have been emplaced in a Pan-African assemblage made up of granitic country rocks (granodiorite and monzogranite), in addition to post-collisional fresh gabbro as a part of the Arabian-Nubian Shield (ANS) in northeast Africa. Granites of the BRC are characterized by enrichment in silica, alkalis, Rb, Y, Ga, Nb, Ta, Th, and U and depletion in Sr, Ba, and Ti. Geochemical characterization of the BRC indicates that the magma is a crustal melt, which originated from the partial melting of metasedimentary sources. Concentrations of rare-earth elements (REEs) differ in magnitude from the ring complex and its granitic country rocks but they have similar patterns, which are sub-parallel and show LREEs enrichment compared to HREEs. The presence of a negative Eu anomaly in these rocks is related to plagioclase fractionation. The abundance of fluorine (F) in the different granite varieties plays an important role in the existence of a tetrad influence on the behavior of REEs (TE1, 3 = up to 1.15). Geochemical parameters suggest the crystallization of the BRC granite varieties by fractional crystallization and limited assimilation. Mn-columbite and Mn-tantalite are the most abundant rare-metals dissemination in the BRC granite varieties. We present combined field, mineralogical and geochemical data that are in favor of magma originating from a metasedimentary source for the BRC with typical characteristics of A-type granites. Our geodynamic model suggests that the Gebel El Bakriyah area witnessed the Neoproterozoic post-collisional stage of the ANS during its late phase of formation. This stage was characterized by the emplacement of fresh gabbros followed by the syenogranite and alkali-feldspar granite of the BRC into an arc-related assemblage (granodiorite and monzogranite). It is believed that the mantle-derived magma was interplated and then moved upward in the extensional environment to a shallower level in the crust owing to events of lithospheric delamination. This presumably accelerated the processes of partial melting and differentiation of the metasedimentary dominated source (Tonian-Cryogenian) to produce the A-type granites building up the BRC (Ediacaran). Full article
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10 pages, 283 KiB  
Article
Exploring Substance Abuse and the Dark Tetrad in Health Sciences and Non-Health Sciences Students
by Marina Carvalho de Lima Moraes, Giulia Cunha Russo, Julia da Silva Prado, Ariela Raissa Lima-Costa, Bruno Bonfá-Araujo and Julie Aitken Schermer
Behav. Sci. 2023, 13(9), 778; https://doi.org/10.3390/bs13090778 - 18 Sep 2023
Viewed by 2245
Abstract
Substance abuse can be used as a coping strategy to manage stress related to academic activities and is a risk-taking behavior that is also associated with people with higher levels of the Dark Tetrad personality traits. Our study aimed to investigate the association [...] Read more.
Substance abuse can be used as a coping strategy to manage stress related to academic activities and is a risk-taking behavior that is also associated with people with higher levels of the Dark Tetrad personality traits. Our study aimed to investigate the association between substance abuse and the Dark Tetrad in students in health and non-health sciences fields. Our sample was composed of 174 college students between 18 and 58 years old (M = 25.60; SD = 9.14). Students completed self-report psychopathy, narcissism, Machiavellianism, sadism, and substance use scales. Results suggest that men consumed more substances and scored higher on the Dark Tetrad than women. Also, when comparing fields, men from health sciences tended to score higher on dark personality traits. These results emphasize the potential risk factors associated with dark personality traits and the consumption of licit and illicit substances by college students, highlighting the need for further studies with this population and the impact of these behaviors and characteristics on future professional practice. Full article
18 pages, 4494 KiB  
Article
A Comparison of Cell-Cell Interaction Prediction Tools Based on scRNA-seq Data
by Zihong Xie, Xuri Li and Antonio Mora
Biomolecules 2023, 13(8), 1211; https://doi.org/10.3390/biom13081211 - 2 Aug 2023
Cited by 8 | Viewed by 6552
Abstract
Computational prediction of cell-cell interactions (CCIs) is becoming increasingly important for understanding disease development and progression. We present a benchmark study of available CCI prediction tools based on single-cell RNA sequencing (scRNA-seq) data. By comparing prediction outputs with a manually curated gold standard [...] Read more.
Computational prediction of cell-cell interactions (CCIs) is becoming increasingly important for understanding disease development and progression. We present a benchmark study of available CCI prediction tools based on single-cell RNA sequencing (scRNA-seq) data. By comparing prediction outputs with a manually curated gold standard for idiopathic pulmonary fibrosis (IPF), we evaluated prediction performance and processing time of several CCI prediction tools, including CCInx, CellChat, CellPhoneDB, iTALK, NATMI, scMLnet, SingleCellSignalR, and an ensemble of tools. According to our results, CellPhoneDB and NATMI are the best performer CCI prediction tools, among the ones analyzed, when we define a CCI as a source-target-ligand-receptor tetrad. In addition, we recommend specific tools according to different types of research projects and discuss the possible future paths in the field. Full article
(This article belongs to the Section Bioinformatics and Systems Biology)
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10 pages, 288 KiB  
Article
Planck Constants in the Symmetry Breaking Quantum Gravity
by Grigory E. Volovik
Symmetry 2023, 15(5), 991; https://doi.org/10.3390/sym15050991 - 27 Apr 2023
Cited by 7 | Viewed by 1841
Abstract
We consider the theory of quantum gravity in which gravity emerges as a result of the symmetry-breaking transition in the quantum vacuum. The gravitational tetrads, which play the role of the order parameter in this transition, are represented by the bilinear combinations of [...] Read more.
We consider the theory of quantum gravity in which gravity emerges as a result of the symmetry-breaking transition in the quantum vacuum. The gravitational tetrads, which play the role of the order parameter in this transition, are represented by the bilinear combinations of the fermionic fields. In this quantum gravity scenario the interval ds in the emergent general relativity is dimensionless. Several other approaches to quantum gravity, including the model of superplastic vacuum and BF theories of gravity support this suggestion. The important consequence of such metric dimension is that all the diffeomorphism invariant quantities are dimensionless for any dimension of spacetime. These include the action S, cosmological constant Λ, scalar curvature R, scalar field Φ, wave function ψ, etc. The composite fermion approach to quantum gravity suggests that the Planck constant can be the parameter of the Minkowski metric. Here, we extend this suggestion by introducing two Planck constants, bar and slash /h, which are the parameters of the correspondingly time component and space component of the Minkowski metric, gMinkμν=diag(2,/h2,/h2,/h2). The parameters bar and slash /h are invariant only under SO(3) transformations, and, thus, they are not diffeomorphism invariant. As a result they have non-zero dimensions—the dimension of time for and dimension of length for /h. Then, according to the Weinberg criterion, these parameters are not fundamental and may vary. In particular, they may depend on the Hubble parameter in the expanding Universe. They also change sign at the topological domain walls resulting from the symmetry breaking. Full article
(This article belongs to the Special Issue Symmetry and Asymmetry in Gravity Research)
19 pages, 3264 KiB  
Article
Identification of Candidate Genes Involved in the Determinism of Pollen Grain Aperture Morphology by Comparative Transcriptome Analysis in Papaveraceae
by Ismael Mazuecos-Aguilera and Víctor N. Suárez-Santiago
Plants 2023, 12(7), 1570; https://doi.org/10.3390/plants12071570 - 6 Apr 2023
Cited by 2 | Viewed by 2089
Abstract
In the last decade, certain genes involved in pollen aperture formation have been discovered. However, those involved in pollen aperture shape remain largely unknown. In Arabidopsis, the interaction during the tetrad development stage of one member of the ELMOD protein family, ELMOD_E, [...] Read more.
In the last decade, certain genes involved in pollen aperture formation have been discovered. However, those involved in pollen aperture shape remain largely unknown. In Arabidopsis, the interaction during the tetrad development stage of one member of the ELMOD protein family, ELMOD_E, with two others, MCR/ELMOD_B and ELMOD_A, can change the morphology of apertures from colpus (elongated) to pore (round). Here, comparative transcriptome analysis is used to identify candidate genes involved in the determination of pollen aperture morphology in Papaveraceae (order Ranunculales). Furthermore, the role of ELMOD genes in the genetic determinism of aperture shape was tested by comparative analysis of their expression levels using RNA-seq data and RT-qPCR. Two pairs of species belonging to two different subfamilies were used. Within each pair, one species has colpate pollen and the other porate (Fumarioideae—Dactylicapnos torulosa, 6-colpate, and Fumaria bracteosa, pantoporate; Papaveroideae—Eschsholzia californica, 5–7 colpate, and Roemeria refracta, 6-porate). The transcriptomes were obtained at the tetrad stage of pollen development. A total of 531 DEGs were found between the colpate and porate pollen species groups. The results from RNA-seq and RT-qPCR indicate that pollen aperture shape is not determined by the relative expression levels of ELMOD family genes in Papaveraceae. However, genes related to callose wall formation or cytoskeleton organisation were found, these processes being involved in pollen aperture formation. In addition, transcriptomes from anthers with pollen during the tetrad stage of three species (D. torulosa, R. refracta, and F. bracteosa) were obtained for the first time. These data will be available for further studies in the field of floral evolution and development. Full article
(This article belongs to the Section Plant Genetics, Genomics and Biotechnology)
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10 pages, 267 KiB  
Article
Kerr–Schild Tetrads and the Nijenhuis Tensor
by José Wadih Maluf, Fernando Lessa Carneiro, Sérgio Ulhoa and José Francisco Da Rocha-Neto
Universe 2023, 9(3), 127; https://doi.org/10.3390/universe9030127 - 28 Feb 2023
Viewed by 1531
Abstract
We write the Kerr–Schild tetrads in terms of the flat space–time tetrads and of a (1, 1) tensor Sμλ. This tensor can be considered as a projection operator, since it transforms (i) flat space–time tetrads into non-flat tetrads, and vice-versa, [...] Read more.
We write the Kerr–Schild tetrads in terms of the flat space–time tetrads and of a (1, 1) tensor Sμλ. This tensor can be considered as a projection operator, since it transforms (i) flat space–time tetrads into non-flat tetrads, and vice-versa, and (ii) the Minkowski space–time metric tensor into a non-flat metric tensor, and vice-versa. The Sμλ tensor and its inverse are constructed in terms of the standard null vector field lμ that defines the Kerr–Schild form of the metric tensor in general relativity, and that yields black holes and non-linear gravitational waves as solutions of the vacuum Einstein’s field equations. We demonstrate that the condition for the vanishing of the Ricci tensor obtained by Kerr and Schild, in empty space–time, is also a condition for the vanishing of the Nijenhuis tensor constructed out of Sμλ. Thus, a theory based on the Nijenhuis tensor yields an important class of solutions of the Einstein’s field equations, namely, black holes and non-linear gravitational waves. We also demonstrate that the present mathematical framework can easily admit modifications of the Newtonian potential that may explain the long range gravitational effects related to galaxy rotation curves. Full article
(This article belongs to the Section Gravitation)
16 pages, 512 KiB  
Article
Fermionic Greybody Factors in Schwarzschild Acoustic Black Holes
by Sara Kanzi and İzzet Sakallı
Universe 2023, 9(2), 108; https://doi.org/10.3390/universe9020108 - 19 Feb 2023
Cited by 6 | Viewed by 1715
Abstract
In Schwarzschild acoustic black hole (SABH) spacetime, we investigate the wave dynamics for the fermions. To this end, we first take into account the Dirac equation in the SABH by employing a null tetrad in the Newman–Penrose (NP) formalism. Then, we consider the [...] Read more.
In Schwarzschild acoustic black hole (SABH) spacetime, we investigate the wave dynamics for the fermions. To this end, we first take into account the Dirac equation in the SABH by employing a null tetrad in the Newman–Penrose (NP) formalism. Then, we consider the Dirac and Rarita–Schwinger equations, respectively. The field equations are reduced to sets of radial and angular equations. By using the analytical solution of the angular equation set, we decouple the radial wave equations and obtain the one-dimensional Schrödinger-like wave equations with their effective potentials. The obtained effective potentials are graphically depicted and analyzed. Finally, we investigate the fermionic greybody factors (GFs) radiated by the SABH spacetime. A thorough investigation is conducted into how the acoustic tuning parameter affects the GFs of the SABH spacetime. Both the semi-analytic WKB method and bounds for the GFs are used to produce the results, which are shown graphically and discussed. Full article
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9 pages, 272 KiB  
Article
Lorentz Gauge and Coulomb Gauge for Tetrad Field of Gravity
by Tao Zhou
Universe 2022, 8(12), 659; https://doi.org/10.3390/universe8120659 - 15 Dec 2022
Cited by 1 | Viewed by 1862
Abstract
In general relativity, an inertial frame can only be established in a small region of spacetime, and local inertial frames are mathematically represented by a tetrad field in gravity. The tetrad field is not unique due to the freedom to perform Lorentz transformations [...] Read more.
In general relativity, an inertial frame can only be established in a small region of spacetime, and local inertial frames are mathematically represented by a tetrad field in gravity. The tetrad field is not unique due to the freedom to perform Lorentz transformations in local inertial frames, and there exists freedom to choose the local inertial frame at each spacetime point. The local Lorentz transformations are known as non-Abelian gauge transformations for the tetrad field, and to fix the gauge freedom corresponding to the Lorentz gauge μAμ=0 and the Coulomb gauge iAi=0 in electrodynamics, the Lorentz gauge and Coulomb gauge for tetrad fields are proposed in the present work. Moreover, properties of the Lorentz gauge and the Coulomb gauge for tetrad fields are discussed to show their similarities to those in electromagnetic fields. Full article
(This article belongs to the Section Field Theory)
14 pages, 1566 KiB  
Article
Measurements of Energetic States Resulting from Ion Exchanges in the Isomorphic Crystals of Apatites and Bioapatites
by Andrzej Kuczumow, Tomasz Blicharski, Mieczysław Gorzelak, Jakub Kosiński, Agnieszka Lasota, Jacek Gągała, Jakub Nowak, Maciej Jarzębski and Mirosław Jabłoński
Molecules 2022, 27(24), 8913; https://doi.org/10.3390/molecules27248913 - 15 Dec 2022
Cited by 7 | Viewed by 1676
Abstract
Developments in the field of nanostructures open new ways for designing and manufacturing innovative materials. Here, we focused on new original ways of calculating energy changes during the substitution of foreign ions into the structure of apatites and bioapatites. Using these tools, the [...] Read more.
Developments in the field of nanostructures open new ways for designing and manufacturing innovative materials. Here, we focused on new original ways of calculating energy changes during the substitution of foreign ions into the structure of apatites and bioapatites. Using these tools, the energetic costs of ion exchanges were calculated for the exemplary cases known from the literature. It was established that the most costly were ion exchanges of some cations inside apatites and of anions, and the least costly exchanges in tetrad channel positions. Real energy expenses for bioapatites are much smaller in comparison to mineral apatites due to the limited involvement of magnesium and carbonates in the structure of hard tissues. They are of the order of several electron volts per ion. The rigorous dependences of the energy changes and crystallographic cell volumes on the ionic radii of introduced cations were proved. The differentiation of the positioning of foreign ions in locations of Ca(I) and Ca(II) could be calculated in the case of a Ca-Pb reaction in hydroxyapatite. The energetic effects of tooth aging were indicated. The ability of energy change calculation during the ion exchange for isomorphic substances widens the advantages resulting from X-ray diffraction measurements. Full article
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