Methods of cumulative quantum mechanics for describing the interaction of electromagnetic fields with matter in Vysikaylo large standing nanoscale excitons: Hydrogen-like atoms, molecules, surfaces, and twinkling crystals

Vysikaylo, Philipp I.

doi:10.1063/5.0256431

Using cumulative quantum mechanics (CQM) and the method of generalized mathematical transfer (MGMT), we analytically study quantum nanometer cumulative-dissipative structures (CDS) and the forces arising within them, which focus nanostructures into regular, fractalized systems—cumulative-dissipative standing hydrogen-like excitons (atoms, molecules, lines, surfaces) and flickering crystals we discovered for the first time. (1) We demonstrate the formation of Vysikaylo standing excitons on permittivity [ε(r)] inhomogeneities in diamond in the nanoscale regions of foreign atoms. (2) For the first time, we solve the problem of measuring ε(r) profiles in inhomogeneous nanoscale structures using Raman spectra (RS) [with an accuracy of up to 99.9% for ε(r) and a step of up to 0.3 nm depending on the distance from the impurity atom (boron)]. (3) Using our theory of Vysikaylo standing excitons, we explain the experimental observation of the degeneracy of electron spectra in standing excitons with respect to the principal quantum number n and n-1/2. By comparing the theory and experimental observations of RS in diamonds doped with boron, we solve the problem (that we formulated previously) between the de Broglie hypothesis and the classical new quantum mechanics of Dirac (which limits the ψ-functions, or prohibits symmetric de Broglie half-waves in spherically and cylindrically symmetric quantum resonators) in favor of the de Broglie hypothesis. Based on the works of Wannier and Mott, we refine the definition of the permittivity of nanocrystals as a coefficient in electric potentials [U(r) → ε(r)U(r)] rather than electric fields [D(r) = ε(r)E(r)]. We construct the most complete theory of the chemical doping of crystals (using the example of group IV crystals doped with group III and V atoms). For the first time, we raise the question of the quantum cleaning of crystals or the accumulation of dopant atoms.

ARTICLE HIGHLIGHTS

CQM is useful for describing CDS with sizes of 10^-15–10²⁶ m. In CDS, the cumulation and dissipation of masses, energies, momenta, and fields occur in accordance with fairly general laws (the virial theorem works).⁴
According to CQM, waves with wavelengths equal to one-half of the de Broglie length of electrons are realized in hollow quantum resonators. This corresponds to an increase in the resonant kinetic energy of a quantum particle by a factor of (1/2)² = 4.⁴
During chemical alloying, local disturbances of the relative permittivity ε(r) occur in the region of the introduction of a foreign atom. Nanoprofiles ε(r) can be determined from Raman spectra.

I. INTRODUCTION

The cumulation (self-focusing) of charged particles in inhomogeneous media with electric fields is a universal property of a number of cumulative-dissipative structures (CDS) with typical dimensions ranging from 10^-15 to 10²⁶ m.^1–13 Vysikaylo CDS include neutral and charged structures such as neutrons, atomic nuclei, atoms, molecules, lightning, tornadoes, stars, galaxies, intergalactic lightning, states, ethnic groups, and living organisms. An analogy with processes in CDS allows us to apply the method of generalized mathematical transfer (MGMT) of the most complete mathematical models to describe similar phenomena from well-studied areas of science in less thoroughly studied areas of the natural sciences. When transferring mathematical models in this way, we must consider the specifics of the phenomena being described.

In this paper, we use the method of generalized mathematical transfer (MGMT) to describe the phenomena of the cumulation and dissipation of electromagnetic radiation near foreign atoms chemically doping diamond crystals. This method consists of transferring mathematical models and their solutions from well-studied areas of the natural sciences to less well-studied areas of the natural sciences.

A. Definition of cumulation

The most intelligible definition of cumulation is given in the preface to Ref. 14 by Zeldovich: “cumulation is the concentration of force, energy or another physical quantity in a small volume.” We will rely on this definition. It implies that we are discussing the cumulation of a certain parameter (dynamic or static order). The main results of the study of the phenomenon of unlimited cumulation are presented in Ref. 14 as separate problems.

The difference between Vysikaylo CDS and dissipative structures (as described by Kolmogorov, Turing, Prigogine, and their followers) is the consideration of convective processes that focus energy-mass-pulse flows (EMPF) into points, lines, or surfaces of cumulation.^1–13 The difference between the CDS we previously discovered¹ and dissipative Kolmogorov–Turing–Prigogine structures (in particular, those from Busselator) is that in CDS, there is a cumulation of convective flows and the generation of new degrees of freedom (NDF, including rotation, pulsations, violation of electroneutrality, and the generation of electric and magnetic fields). Like Prigogine dissipative structures (DS), Vysikaylo CDS have several common properties that differ from those of DS. We will focus on some common properties of positively charged (+) CDS (e.g., cathode spots, atomic nuclei, stars, galaxies) and particularly on the heliosphere and the Sun.

B. Method of generalized mathematical transfer

The analogy between a number of processes in the CDS allows the use of MGMT to describe similar phenomena in various environments. However, when transferring mathematical models in this way, it is necessary to take into account the specifics of the phenomena being described. MGMT has long been used and is still used to accelerate the development of philosophy and the natural sciences. This method was used by:

Newton, to describe and generalize gravitational forces on Earth and in space.
Louis de Broglie, to describe quantum phenomena using his hypothesis.
Einstein, to describe the photoelectric effect. He applied the opposite idea, that electromagnetic waves behave like particles.
Niels Bohr, who successfully applied the planetary model of Copernicus and his followers to describe electric potentials and electron orbits and thus explain the spectra of the hydrogen atom.
Schrödinger with Dirac, who created a new quantum (wave) theory that considers the statistical wave properties of an electron in a hydrogen atom and its wave passage through two slits. As a result, in 1933, Schrödinger and Dirac received the Nobel Prize “for the discovery of new productive forms of atomic theory” (and thus for the transfer of Bohr’s theory to the status of a pseudoscientific theory).
Us, for the discovery and description of:
- Shock waves of an electric field and plasma nozzles (analogs of Laval nozzles) in a plasma with current (1985–2024). For this, we constructed the most complete theory of perturbations in a plasma with current and thus obtained a modified Navier–Stokes equation,^2,6–9 which made it possible to analytically and numerically describe all the phenomena observed in our experiments in an inhomogeneous plasma with current.
- Five points of cumulation and libration for electrons in a system of two rotating positively charged Coulomb attractors (two atoms in a molecule). Here, we took into account the generality of the laws of cumulation, which scale as ∼1/r in gravitational and Coulomb potentials. We related the number of electron accumulation points in a molecule to atomic valence.⁵
- Points, lines, and surfaces (charged strata—running and standing shock waves of the electric field) of libration and cumulation for free electrons between positively charged structures of plasma with current. We proved that Coulomb (electric) potentials in 4D spacetime function similarly to gravitational potentials.⁵
- Endoelectrons in fullerenes and, accordingly, all spectra of negatively charged fullerenes. We demonstrated (based on a number of experiments, see Ref. 4) the formation of symmetric de Broglie waves of electrons in hollow fullerenes and the corresponding degeneration of spectra in hollow fullerenes with respect to the principal quantum number n and n-1/2.⁴ Similar cos waves are formed (when drops fall into cylindrical glasses with liquid, e.g., when studying sonoluminescence phenomena) in ordinary hydrodynamics in hollow spherically and cylindrically symmetric resonators.
- Standing Vysikaylo excitons formed by the chemical doping of diamonds with boron atoms. Here, we modified the Wannier–Mott model (with a uniform value of the permittivity ε = const. throughout the crystal) into a model that considers the polarization of crystals [ε = ε(r)] when a foreign atom is introduced into their crystal lattice. We thus proposed a new method for determining the profile of the relative permittivity ε(r) from the distance to the center of a standing exciton using Raman spectra.^10,11
- The mechanism of Coulomb fractalization of meteoroids and small asteroids by a plasma tail.¹³ This mechanism was developed based on a modification of the Vysikaylo model of lightning impulse propagation (see Ref. 12) observed in experiments by Shenland in 1934–1938. Shenland experimentally proved using a Boyce chamber that electrons focused by lightning run away from it in a forward direction (in the form of runaway electron jets) with an energy of up to 5 MeV.

The MGMT goes back to Eratosthenes’ ideas, which he applied (June 19, 240 BC) when calculating the length of the Earth’s meridian. These ideas included that geometric structures are similar and that various theorems can be applied to them. Eratosthenes’ method has long been successfully used on Earth and has aided the discovery and quantitative description of “mysterious” phenomena, including CDS.

In this work, we use this general method (MGMT) to expand (transfer) the achievements of Vysikaylo cumulative quantum mechanics^4,11 (obtained by analytically describing the phenomena of the accumulation of de Broglie waves of electrons in hollow quantum resonators, using the example of hollow fullerenes—C_60,70) to describe Vysikaylo standing hydrogen-like excitons discovered in diamonds (group IV crystals) during their chemical alloying with foreign atoms with a valence other than that of carbon (group III and V atoms).

In physics, there are two ways of alloying materials:^4,11 (1) physical alloying with atomic or molecular structures with a high affinity for free electrons; and (2) chemical alloying, in which a foreign atom is embedded (with the replacement of the support lattice atom) into the crystal lattice of the support crystal.

C. Classification of excitons

The concept of an exciton is similar to the concept of a solitary wave, soliton, or CDS. When electromagnetic energy (waves) is accumulated (absorbed and localized) by a crystal, electrons and holes (positive ions) are formed in space. At any distance, they experience a Coulomb interaction, which is weakened by the polarization of the medium localized within the characteristic size of the exciton (the distance between the electron and the ion). This interaction means that electrons and holes should be considered in 3D space as a bound electron-hole pair—an exciton. In this case, an exciton is a quasiparticle (a quantum pulsar in which an electron pulsates in the region of a positively charged hole in accordance with its total energy), arising during globally currentless excitations in semiconductors. Coulomb potentials act as lenses focusing electrons that have received additional energy. By focusing electrons that have received energy from electromagnetic waves, these potentials accumulate (localize) the energy of these electromagnetic waves in an exciton. The accumulation of electromagnetic waves by excitons leads to an increase in the total energy of an electron in an exciton, a decrease in its kinetic energy in the potential well, and a corresponding increase in the radius of its pulsation in the hole region. Conversely, the release of electron energy in an exciton (in the form of electromagnetic waves) leads to a decrease in the effective radius of electron pulsation in an exciton and a corresponding increase in its kinetic energy. This is the general scheme of pulsation of a quantum Coulomb pulsar and the formation of electron levels of excitons.¹¹ This general scheme requires a certain amount of detail in the case of excitons in crystals doped with foreign atoms.

Depending on the nature of the bond, two types of excitons localized in potential wells are noted in the literature (Fig. 1).^15,16 The first are small-radius excitons (Frenkel’s excitons) associated with a specific atom; their sizes do not exceed the interatomic distance in the crystal.^15,16 Frenkel’s excitons can move along the supporting crystal (if they were formed based on one of the atoms of the crystal lattice) or be localized if the energy spectra of the excited atoms do not coincide with the energy spectra of the surrounding atoms.¹¹ The second type are free hydrogen-like spherically symmetric excitons with large Wannier–Mott radii (WM excitons), the characteristic sizes of which reach 20 nm (tens and hundreds of interatomic distances). This type of exciton is formed from atoms or molecules of a homogeneous reference crystal [with constant relative permittivity, i.e., -ε(R) = const.]. Therefore, they can move freely along the crystal.

FIG. 1.

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The de Broglie–Bohr–Schrodinger problem for hydrogen-like nanostructures. A scheme of a stable (closed) quantum dot with an infinite potential Coulomb barrier surrounding it for particles with total energy less than zero. To describe the energy spectrum of a quantum dot, a model of a hydrogen-like atom with an “orbit radius” is used. r_n and r_n–1/2 are the characteristic radii of a quantum resonator with a Coulomb potential for quantum particles with a principal quantum number n and energies E_n (with asymmetric ψ_n) and E_n-1/2 (with symmetric ψ_n-1/2), respectively. ΔE_n-1/2,n is the Vysikaylo energy splitting of the higher levels with the main quantum number n.^10,11

We will dwell on the discovery and description of the third new type of standing (localized) excitons of large size and the formation of large-radius molecules, lines, surfaces, and flickering crystals (superlattices) from them. These Vysikaylo standing excitons (hydrogen-like, molecular-like, and linear-, surface-, and crystal-like structures from excitons) are formed by the chemical doping of crystal lattices with impurity atoms with a valence different from the valence of the atoms of the reference crystal. We prove that chemical doping leads to local changes in the relative permittivity in crystals. Let us consider the chemical doping scheme.

D. Detailed schemes of Vysikaylo excitons and the chemical doping of support crystals with acceptor impurities

Acceptor doping is the process of introducing an impurity into a material that locally lowers the Fermi level. This is achieved by doping the reference crystal with atoms of lower valence. Since the valence of the doping atom is lower, one of the valence electrons in the doped crystal is unpaired and, receiving additional energy, forms an exciton in the region of the doping atom. We will consider in detail the scheme of doping diamond (a group IV element) with boron atoms (a group III element).

Figure 2 shows the general scheme of the chemical doping of diamond with an acceptor impurity—a boron atom. Figure 2(a) shows the flat 2D scheme of an unperturbed diamond crystal. In diamond, each carbon atom forms four paired strong sp³ bonds. The distance between the centers of the carbon atoms in diamond is 0.154 nm. In this case, the value of the relative permittivity of a homogeneous diamond is ε = 5.7.

FIG. 2.

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A geometric 2D schematic of the formation of a quantum dot from carbon atoms and one boron atom. (a) The ideal crystal lattice of a group IV element. (b) A geometric 2D schematic of the modification of an ideal crystal lattice via the introduction of a boron atom into the crystal lattice (triangle). The impurity atom replaces the atom in the original (ideal) crystal lattice. e is the electron shared by the volume of the quantum dot. + is a hole moving around the boron atom at a speed corresponding to the kinetic energy of the valence electrons of the carbon atoms (the positive ion moves quickly). The gray square is the node of the ideal lattice at which the shared electron is focused. (c) A spherically symmetric quantum dot in diamond with a boron atom replacing a carbon atom: (1) An internal cavity in the region of a node of an ideal crystal lattice, caused by a boron atom bonding with only three carbon atoms. (2) The sphere of motion of a positively charged hollow hole with a radius of d ≈ 1.59 Å. (3) A boundary within which an electron is localized in a stable spherically symmetric quantum dot—a standing Vysikaylo exciton (or color center) with a radius of a_n.

When diamond is doped with a boron atom (whose valence is 3), one electron of carbon (whose valence is 4) in the region of the boron atom is unpaired and, receiving energy, can form a Vysikaylo standing exciton in this region. Figure 2(b) shows a rough 2D diagram of diamond doping with a boron atom. Numerical first-principles density functional theory (DFT) calculations¹⁷ show that the distance between the boron and carbon atoms is significantly greater than the distance between the carbon atoms in pure diamond and reaches a value of -0.159 nm. If this is so, then doping with a boron atom leads to a perturbation of the diamond crystal lattice in the region of the foreign atom (i.e., the boron atom pushes the carbon atoms apart) and thus leads to a significant local increase in the relative permittivity ε and, consequently, a change in the entire spectrum of the free WM exciton, making such an exciton a Vysikaylo standing exciton.^10,11 The fast (relative to the velocity of an electron in an exciton) movement of a hole (in the region of the doping atom) forms an exciton with a positive ion smeared in time and a hollow center [Fig. 2(c)]. Here we observe an analogy with the polarization capture of a free electron by a fullerene cavity.^4,11 Therefore, following MGMT, we can transfer the technique used for analytical calculations of the eigenenergy spectra in hollow spherically symmetric quantum resonators to the calculation of the energy spectra of Vysikaylo standing excitons in boron-doped diamonds.^10,11 According to experimental and analytical studies of the resonant capture of electrons by hollow fullerene molecules, cos-wave spectra (with symmetric ψ-functions) appear in the eigenenergy spectrum of quantum resonators, i.e., spectra with half-integer principal quantum numbers n-1/2. This quantum degeneracy in the principal quantum number was discovered by the author using cumulative quantum mechanics and experimental data; see Refs. 4, 10, and 11.

E. On the identification of the electronic spectra of Vysikaylo excitons observed in experiments

We propose that a number of long-observed Raman spectra in boron-doped diamonds belong to the electron spectra of Vysikaylo standing excitons.^10,11 However, for a number of reasons, these spectra have not yet been properly identified, due to a lack of understanding of the basics of cumulative quantum mechanics (CQM)⁴ among experimenters (see Refs. 4 and 11). When comparing with experimental observations of the energy spectra of Vysikaylo standing excitons, we will mainly focus on the spectra established in the works of Collins et al. These spectra are recognized as reliable by most researchers of boron-doped diamonds. In 1994, Collins claimed that there are no adequate theoretical descriptions of any of the processes observed in experiments, including the temperature dependence of the scattering mechanisms, the contribution of the split-off valence band, and the population of excited states in the conductivity of doped diamond crystals.¹⁸ In this case, for both polycrystalline and single-crystal homoepitaxial CVD diamond, measurements of electrical properties can be completely nullified due to the presence of a surface layer of non-diamond carbon. The task of describing all possible emission and resonant absorption spectra in doped diamonds is rather complex and ambiguous. We will apply MGMT to describe the electron spectra of Vysikaylo hydrogen-like standing excitons. This will prove that the description of Vysikaylo hydrogen-like standing exciton atoms and molecules (using MGMT) can be carried out similarly to the description of the spectra of hydrogen atoms and molecules.

Before we proceed to the description of the electron spectra of Vysikaylo standing hollow excitons [Fig. 2(c)], we will briefly dwell on the achievements of cumulative quantum mechanics based on the analytical^4,11 and experimental^19–21 study of the polarization resonance capture of free electrons by the internal cavities of various types of fullerenes, C_60,70. This will give us grounds to take into account a number of additional spectra involving the principal quantum numbers n-1/2. Thus, we will apply the basics of Vysikaylo CQM in our study and to interpreting Raman spectra in diamonds doped with boron.

II. ACHIEVEMENTS OF VYSIKAYLO CUMULATIVE QUANTUM MECHANICS

The discovery of new materials is accompanied by the development of new technologies, and attempts are being made to create mathematical models capable of describing phenomena in hollow quantum resonators—quantum dots, lines, and other cumulative-dissipative nanometer-scale three-dimensional structures. Such goals can be achieved by testing and modernizing the foundations of Dirac’s new quantum wave mechanics (NQWM) and supplementing it with the foundations of Vysikaylo cumulative quantum mechanics (CQM), which describes real cumulative and dissipative phenomena in nanometer-scale structures within the framework of de Broglie quantum mechanics. This modification of the NQWM, confirmed by experiments in the theoretical and experimental nanophysics worlds (see Refs. 4 and 11 for studies of the polarization capture of electrons by the internal cavities of fullerenes) turns out to be very useful in describing the “mysterious” cumulative-dissipative phenomena in structures with sizes ranging from 10^-15 to 10²⁶ m.^1–13 The results of our study and the experiments available in the literature (see Refs. 4 and 11) can be summarized as follows:

It is necessary to limit everywhere only the probability (or the probability density W(r) dr = χ² |ψ|²dr, taking into account the regularizing geometric coefficient – χ² = 2^kπ^kr^k, where k = 1 for cylindrical symmetry of a hollow resonator, k = 2 for spherical symmetry, and χ² = 1 for plane symmetry) of finding a particle in the volume of a resonator with different types of symmetry.^4,11 It is not necessary to limit the ψ-function everywhere, as Dirac²² and others^23,24 do. We limit everywhere only the probability density of finding a particle, W(r) = χ²|ψ|.²
Modifying Dirac’s requirement, taking into account the geometric coefficient χ, and normalizing the ψ_n-1/2-functions [ψ_n-1/2 ∼ cos(kr)/r in spherical and ψ_n-1/2 ∼ cos(kr)/r^1/2 cylindrical hollow resonators] transforms the forbidden eigenenergy spectra with energy E_n-1/2 and unlimited cumulation at the centers of hollow resonators (with ψ_n-1/2-functions) into allowed ones. In this case, all spectra previously allowed only for resonators with planar symmetry²⁵ become allowed for spherically and cylindrically symmetric hollow quantum resonators, such as:
$\begin{align} E_{n - 1 / 2} = E_{n \cos} = \pm (ℏ^{2} / 2 m) π^{2} {(n - 1 / 2)}^{\pm 2} / {(R + r_{ind})}^{2}, \\ n = 1, 2, 3, \dots H E_{n - 1 / 2} ⇄ E_{i}, \dots E_{n - 1 / 2} ⇄ E_{i - 1 / 2} \dots \end{align}$
(1)
Here “−” represents quantum wells, and “+” represents quantum resonators surrounded by potential barriers (such as polarization traps).^4,11 R is the characteristic size of the resonator, r_ind is the polarization length by which the resonator size increases, and m is the classical electron mass.
The eigenenergy spectra (1) for hollow quantum resonators with spherical and cylindrical symmetries will be called Vysikaylo energy spectra, and the degeneracy of the electron spectra with respect to the principal quantum number n and n-1/2 will be called the Vysikaylo degeneracy. These spectra, obtained analytically in Refs. 4 and 11, were in excellent agreement with the experimentally measured eigenenergy spectra of fullerenes C₆₀ and C₇₀ during the study of the resonant capture of electrons by the internal cavities of these molecules.^19–21 Thus, negatively charged endoions e_k@C_n with endoelectrons inside them are formed from hollow fullerenes, C_n. Here k = 1, …, 6 is the number of captured electrons, and n is the number of carbon atoms in the fullerene.^4,11
The complete eigenenergy spectra (taking into account the Vysikaylo spectra) of hollow quantum resonators do not depend on the symmetry of the quantum resonator.^4,11
Our analysis of the experimentally obtained resonance spectra of electron capture (with resonance energy) by fullerenes^19–21 and a comparison with our analytical calculations of these spectra within the framework of CQM^4,11 allows us to assert that the ψ-function of a particle appears only when boundary conditions are specified for the Schrödinger equation. This means that the particle always remains a particle with its mass, charge, and other parameters, and the wave properties of the particle correspond to its statistical behavior in a quantum resonator and are determined by the boundary conditions, i.e., external factors.

Thus, in the case of electron two-slit experiments, the interference pattern appears only as a statistical result. The representation of a free particle as a moving plane de Broglie wave is incorrect! The particle itself does not transform into a shell or a plane wave. Statistically, the particle behaves like a wave in a certain resonator, and this behavior creates the effect of its transformation into a wave or shell. When the characteristic dimensions of quantum resonators change, the ψ-function of a quantum particle trapped in the resonator can be changed, e.g., when an electron collapses into a proton in an atomic nucleus. In this case, according to the virial theorem proved by Fock for quantum phenomena,²⁶ half of the potential energy goes into increasing the internal energy of the proton turning into a neutron, and the other half goes into emitting a neutrino with an energy of about 0.85 MeV.^4,11 In this regard, the phenomena described within the framework of wave cumulative quantum mechanics have analogues in ordinary hydrodynamics.^4,11

III. BASIC PROVISIONS FOR THE DESCRIPTION OF VYSIKAYLO STANDING EXCITONS

In our papers, we consistently solve the problems arising (as we believe, in describing the eigenenergy spectra of quantum nanoresonators of various natures) between the de Broglie hypothesis and the classical Dirac NQWM, which limits the values of ψ-functions everywhere. As we have proved (in our analysis of the available experimental data and our theoretical studies), it is a mistake to limit ψ-functions everywhere without taking into account the normalizing geometric coefficients χ². In Refs. 4 and 11, it was shown how these problems and paradoxes (discrepancies between experimental observations and incomplete theories) caused by phenomena arising from the violation of electrical neutrality (in particular, polarization) in nanostructured composite materials are solved using CQM and taking into account the regularization of unbounded ψ_n-1/2-functions through the normalizing geometric coefficient χ², which accounts for the symmetry of the quantum resonator. CQM made it possible to discover approximately 35 quantum dimensional effects and to describe a number of previously “mysterious” phenomena.^4,11 The discovery of Vysikaylo degeneracy by the main quantum number n and n-1/2 in hollow quantum resonators significantly changes the parameters of quantum stars (white dwarfs, neutron stars, and black holes).^10,11

In this paper, we will consider similar problems in describing nanostructures that arise during the chemical doping of group IV crystals. We will do this using the example of nanostructures, i.e., large hydrogen-like atoms, hydrogen-like molecules, lines, surfaces, and flickering crystals of Vysikaylo standing excitons in inhomogeneous diamond doped with boron. For this purpose, we will modify the Wannier–Mott theory, constructed for homogeneous crystals, to describe standing excitons in inhomogeneous crystals.^10,11

A. Large running (free) Wannier–Mott excitons

Wannier–Mott^27,28 excitons^15,16 exist in semiconductors due to the high permittivity of the latter. This high permittivity, ε > 5, leads to a weakening of the electrostatic attraction between the electron and positively charged hole, which leads to a large radius of the exciton (a large hydrogen-like atom). The characteristic dimensions of Wannier–Mott (WM) excitons increase by a factor of ε, and the energy spectra decrease by a factor of ε² in relation to the parameters of the hydrogen atom:

E_{n} = - (m_{r} e^{4} / {(8 ε_{k} ε_{0} h)}^{2}) / n^{2},

(2)

where m_r is the reduced mass of the system (electron and hole) in a cubic crystal; ε_k is the constant permittivity of the crystal under consideration; n is the principal quantum number of the exciton; ε₀ is the electric constant (dielectric constant of vacuum); and h is the effective radius of the WM exciton. Wannier and Mott applied and detailed the method of Eratosthenes (MGMT) for predicting and describing large-radius excitons using a quantum model for the hydrogen atom.

The hydrogen-like electron spectrum of WM excitons was first observed in the absorption spectrum of Cu₂O in 1952 by Gross and Karyev and independently by Hayasi and Katsuki, but the excitonic interpretation was absent from the work of the latter authors. In all these studies, the permittivity ε was considered constant. Since such large-radius excitons are formed by the ionization of atoms or molecules of the supporting crystal, they move freely along the crystal and are considered running (free) WM excitons.

B. Determination of the relative permittivity for nanostructures

In 1937–1939, Wannier and Mott^27,28 proposed a model in which they predicted and described hydrogen-like structures of free excitons of large radius (up to 20 nm) in homogeneous crystals. To achieve this, in the Bohr model of the hydrogen atom, they considered the relative permittivity (2) (Fig. 3). In the case of the WM model, the relative permittivity ε(r) = ε_k was considered constant throughout a pure crystal.^15,16 Under these assumptions, Wannier and Mott^15,16,27,28 were the first to show that the hydrogen atom model, taking into account the relative dielectric polarization, which weakens the Coulomb potential by a factor of ε_k = const. > 1, could be successfully applied to describe free hydrogen-like excitons of large radius in homogeneous dielectric crystals.^15,16

FIG. 3.

A schematic of the transition of an electron from an energy level with potential energy φ(r2) to a level with φ(r1), accompanied by the emission of an electromagnetic wave (EMW) carrying, according to the law of conservation of energy, information about the internal structure of the doped dielectric through ε(r1) and ε(r2).

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A schematic of the transition of an electron from an energy level with potential energy φ(r₂) to a level with φ(r₁), accompanied by the emission of an electromagnetic wave (EMW) carrying, according to the law of conservation of energy, information about the internal structure of the doped dielectric through ε(r₁) and ε(r₂).

1. Our modification of the Wannier–Mott model

A number of researchers mistakenly associate the definition of the relative permittivity ε_k with the wavelengths of electromagnetic radiation. They forget that electromagnetic waves accumulate on electrons and thus increase (when they are absorbed by electrons) or reduce (when they are emitted by electrons) the size of the region of electron pulsation near the hole. (Within the framework of the old Bohr theory, the radius of the electron orbit in the exciton changes.) The wavelength of electromagnetic radiation changes from zero to infinity. Therefore, even within the framework of the Wannier–Mott model, the relative permittivity should be associated with the nanostructure corresponding to the characteristic orbit of the electron, and not with the wavelength of electromagnetic radiation. In the case of inhomogeneous crystals, such an understanding of the relative permittivity allows us to propose a method for determining the relative permittivity for nanosized structures.^10,11

In physics, there are two definitions of the relative permittivity ε: that related to the weakening coefficient of the electric field strength E (D= εE) (with all the assumptions; see Ref. 29, p. 58 for more information), and that related to the weakening of the Coulomb potential φ(r) → φ(r)/ε (see Ref. 29, p. 60). Such a definition for local potentials φ(r) allows us to introduce a local concept for the relative permittivity in nanocrystals ε(r) and φ(r) → φ(r)/ε(r).^10,11 These two different definitions coincide only in the case of a constant value of ε(r), i.e., in the case of a spatially homogeneous dielectric with constant ε throughout the volume of the crystal. In the case of doping a crystal with a foreign atom, the crystal, as we have seen, becomes inhomogeneous in the doping region, and its relative permittivity can locally change significantly [Fig. 2(b)]. This can significantly change the resonance electron spectrum of Wannier–Mott excitons in a crystal doped with a foreign atom.^10,11 Therefore, to solve problems in the nanoscale world, where the Schrödinger equation includes the Coulomb potential profile, we will use the definition of the relative permittivity as a coefficient weakening the Coulomb potential of a charge in a crystal at a distance r from this charge by a factor of ε(r).^10,11 This definition, according to Ref. 11, allows us to apply a modified Wannier–Mott model to explain the Raman spectra obtained in a number of works.^30–36 This allows us to calculate the local values of ε(r_n), where r_n is the radius of the Bohr orbit of the hydrogen-like standing Vysikaylo exciton with principal quantum number n, and to obtain the profile of ε(r_n) at n = 1, 2, … in the region of the doped atom.^10,11

In large-radius excitons, as well as in a hydrogen atom or a Frenkel exciton, the absorption of electromagnetic waves by atoms or excitons is possible. Electromagnetic wave generation is also possible, as is the dissipation of excitation energy into the surrounding space. Therefore, such structures can be classified as cumulative-dissipative structures (CDS).^1–13 When studying the flows of dissipative energy from such spherically symmetric quantum structures, according to the principles of quantum (wave) mechanics, it is possible to construct an internal discrete quantum structure of the CDS and its modification, both during energy absorption and during its dissipation (in particular, during the emission of electromagnetic waves) (Fig. 3). In this case, as noted above, we will take into account the weakening of the Coulomb potential in the dielectric using the spherically symmetric coefficient ε(r). According to Ref. 29, p. 60 and the principle of superposition of Coulomb potentials, this coefficient is determined by internal charges and their location in a sphere of radius r, and not by parts of the crystal external to this sphere (Fig. 3). Here, we prove the absence of the manifestation of Mach’s principle for the properties of standing Vysikaylo excitons (on the influence of all atoms surrounding the exciton on its properties).

In the Wannier–Mott model itself, the potential energy is weakened by the value of the relative permittivity ε_k. This method requires a more precise definition of what the relative permittivity is for hydrogen-like nanometer structures and depends on whether it is possible to take into account the inhomogeneity of doped crystals and thus modify the Wannier–Mott model by replacing ε_k = const. over the entire crystal with ε(r), which depends only on the internal part of the crystal, located in the Vysikaylo standing exciton. This is very important for studying the propagation and localization of large-radius excitons in inhomogeneous crystals.

Therefore, naturally, problems arise for nanostructure excitons that are similar to problems in atomic physics): (1) the absorption of the energy of electromagnetic waves (the wavelengths of which can vary from thousands of nm to 100 nm) into the excitation energy of an exciton (the size of which can be less than 1 angstrom for Frenkel excitons and about 20 nm for large Wannier–Mott excitons), (2) the localization and self-focusing of excitons, and (3) the transformation of the exciton energy, such as the transfer and accumulation of this energy in the battery.

Within the framework of the new quantum theory, to describe hydrogen-like excitons of large radius with a charge Z, the solution of the Schrödinger equation with a Coulomb potential weakened by a factor of ε_k is used (2) (Ref. 15, p. 352). In (2), in the Wannier–Mott model, there is a discrepancy between the local value of the potential U(r) and the excessive requirement of constancy for the relative permittivity, ε(r) = ε_k = const., throughout the crystal with any dimensions. This question was first posed in the author’s previous works.^10,11

Such rough accounting for polarization in crystals (accounting for the relative permittivity in the Wannier–Mott model) led to a corresponding decrease in all proper energy spectra of the hydrogen atom E_n by a factor of ε_k² (Ref. 15, pp. 352–353) and a corresponding increase in the characteristic sizes of hydrogen-like structures r_n by a factor of ε_k in relation to the size of the hydrogen atom. In this case, the sizes of excited excitons, as in the hydrogen atom, increase with the growth of the principal quantum number proportionally to n² [Ref. 16, formula (9.41)]:

r_{n} = ε_{k} n^{2} a_{0}

(3)

So, the question: “Is it possible to take into account the local heterogeneity of the relative permittivity or the weakening of the Coulomb potential by a factor of ε(r), at least in spherically symmetric problems, and thus determine this coefficient in experiments at the nanometer level?,” is posed in Refs. 10 and 11. There, this problem is solved using the principle of superposition of electric fields, according to which the weakening of the Coulomb potential by ε(r) is caused not by the entire crystal, but only by the polarization of charges inside a sphere of radius r (Fig. 3).

In Ref. 11, according to , , and the law of conservation of energy, transitions between the intrinsic energy electron levels E_n and E_m, accompanied by electron spectra of the EMW hν_n,_m, for Vysikaylo standing hydrogen-like excitons of large radius are written in the following general form (Fig. 3):

\begin{align} h v_{n, m} & = E_{n, m} = E_{n} - E_{m} = U_{n} (r) - U_{m} (r) \\ = - Z e^{2} / (ε (r_{n}) ε_{0} r_{n}) + Z e^{2} / (ε (r_{m}) ε_{0} r_{m}) \end{align}

(4)

According to (4), within the framework of the Bohr model of the hydrogen atom, for hydrogen-like excitons, it is possible to determine the local heterogeneity of crystals in the region of spherically symmetric perturbations using their electron spectra.¹¹

The coincidence of the results of the old Bohr and new quantum theories in determining the eigenenergies of a quantum system—the hydrogen atom—lies in Bohr’s successful choice of the condition for quantizing the angular momentum of an electron on an allowed circular orbit mv_nr_n = nℏ or the product of the orbit length and the particle momentum 2πmv_nr_n = λ_np_n = nh. Here h = 2πℏ is the Planck constant. It should be noted that the first radius of the Bohr orbit in the semiclassical Bohr theory of the hydrogen atom coincides with the value of the radius r₁ = Λ/α, at which the probability of finding an electron (2²πr² |ψ|²) reaches its maximum value. Here, Λ is the Compton wavelength of the electron, and α is the fine structure constant. A similar coincidence is observed for all subsequent radii r_n corresponding to a principal quantum number n greater than 1. Only this coincidence provides a certain basis for retaining the term Bohr radius in the new quantum theory. In this case, within the framework of the new quantum theory for hydrogen-like structures in any media, the Bohr radius of the orbit is:^10,11

r_{n} = a_{0} ε (r) n^{2}

(5)

Here, a₀ is the first-level brow radius, and ε(r) is the local spherically symmetric attenuation coefficient of the Coulomb potential or the local spherically symmetric relative permittivity of the medium (a small part of the crystal) in which a hydrogen-like structure with characteristic dimensions determined by is formed. For vacuum, ε(r) = const. = 1, i.e., the limiting transition from excitons in inhomogeneous crystals to a hydrogen atom in vacuum within the framework of this model, which generalizes the description of hydrogen-like structures in inhomogeneous crystals, occurs without significant problems.

If the relative permittivity of the crystal (or the local attenuation coefficient of the Coulomb potential) ε(r) changes with the distance to the dopant atom introduced into the crystal lattice, then the resonant exchange of electron energies cannot occur. This localizes the exciton generated or focused by the Coulomb potential (Figs. 2 and 3) in the region of the dopant, where ε(r) has different values than in places farther from the dopant atom. Thus, in Refs. 10 and 11, the possibility of the appearance of standing Vysikaylo excitons is substantiated.

Taking into account the Vysikaylo degeneracy of the principal quantum number in hollow quantum resonators determines the possibility of the appearance of symmetric de Broglie waves (cos waves). Moreover, within the framework of Vysikaylo cumulative quantum mechanics for hollow hydrogen-like structures in any media, the Bohr radius of the orbit for such waves is:^10,11

r_{n - 1 / 2} = a_{0} ε (r) {(n - 1 / 2)}^{2}

(6)

IV. PONDEROMOTIVE FORCES IN INHOMOGENEOUS CRYSTALS

In inhomogeneous crystals (with inhomogeneous permittivities) in electric fields, ponderomotive electric forces qU(r)∇ε(r) arise, acting on charged particles due to the gradient of the relative permittivity:

F = q \nabla (ε (r) U (r)) = q ε (r) \nabla U (r) + q U (r) \nabla ε (r)

A detailed analysis of these forces remains to be carried out by us in the future.

V. ELECTRONIC SPECTRA OF VYSIKAYLO STANDING EXCITONS OF LARGE RADIUS

Vysikaylo large-radius standing excitons arise in the region of introduction of a foreign atom (with a valence different from the valence of the atoms of the reference crystal) into the crystal lattice of the reference crystal. The electron and the hole are localized in the region of the foreign atom.^10,11 The introduction of a foreign atom generally leads to local quantum oscillations, including of the permittivity ε(r).^10,11 This also leads to the energy localization of standing Vysikaylo excitons in the region of introduction of a foreign atom into the crystal lattice of the reference crystal.^10,11 Such phenomena occur, for example, during the chemical doping of diamond (or other group IV crystals) with boron atoms (or other group III atoms) or nitrogen (or other group V elements). In this case, as we have shown, the permittivity in boron-doped diamond changes locally by up to 7%.^10,11

A. Spectra of Wannier–Mott excitons in diamond with ε = const

Under these assumptions (ε = 5.7), according to Wannier–Mott theory, when introducing a foreign atom into a diamond crystal lattice, taking into account the cos waves of the Vysikaylo spectra , we have the following spectrum for free (running) excitons of large Wannier–Mott radius with constant permittivity:

\begin{align} E_{n - 1 / 2} = - (1.674 (1- 1 / 2); 0.186 (2- 1 / 2); 0, 067 (3- 1 / 2); 0.034 (4- 1 / 2); \\ 0.021 (5- 1 / 2); 0.014 (6- 1 / 2); 0.01 (7- 1 / 2); \dots [eV]) \\ E_{n} = - (0.4185 (1); 0.104 (2); 0.0465 (3); 0.026 (4); 0.0167 (5); \\ 0.0116 (6); 0.008 54 (7); 0.0065 (8); 0.0052 (9) \dots [eV] \end{align}

(7)

In our calculations, we use the following values for the hydrogen atom: a radius of a₀ = 0.529 Å and an ionization potential of I = 13.598 eV. Possible transitions from (1) (E_n-1/2 ⇆ E_m; E_n-1/2 ⇆ E_m-1/2) are not considered here.

B. Spectra of Vysikaylo excitons in diamond at ε = ε(r)

We believe that the complexity of interpreting the absorption and emission spectra of Vysikaylo large-radius standing excitons in experiments is determined to a greater extent by the profiles of the relative permittivity ε(r) of the reference crystal perturbed by the dopant atom (diamond in our case) than by the spectra of the dopant atom. For this reason, the most accurate values of the electron spectra of Vysikaylo large standing excitons can be obtained at cryogenic temperatures of crystals, when the temperature and concentration effects are small, and the probe radiation is strictly collimated, including in energy.^10,11 Such experiments were carried out, for example, in Ref. 35 (see Table I). Unfortunately, Ref. 35 presents the electron spectrum of Vysikaylo excitons with energies ranging from 2 to 43 meV. The most informative sources for the next proof of the validity of Vysikaylo cumulative quantum mechanics are the electron spectra of Vysikaylo standing excitons in the energy range from 400 to 50 meV (see Table I).

TABLE I.

The dependence of the relative permittivity ε on r, ε(r), was calculated according to Vysikaylo theory [in a standing ε(r) wave]. The electron energy levels E_n–1/2 = −13.56 ((n–½)ε(r))^–2 eV, E_n=−13.56 (nε(r))^–2 eV and the characteristic radii r_n of a hydrogen-like hollow quantum dot in diamond doped with boron (the electron spectra of Vysikaylo standing excitons). Note: ? indicates an absence of observation of these levels in experiments known to the author. These data require additional experimental studies.

	E, meV	E, meV	ε, relative units
n	WM theory	experiment	Vysikaylo theory	r_n, Angstrom
1-1/2	−1674	?		0.8
1	−418.5	369^32,33	6.07	3.2
		Lyman–Vysikaylo series	6.7
2-1/2	−186	From 161 to 165^30,31,34	6.14-6.05	7.15
2-1/2	−186	Vysikaylo series	6.14-6.05	7.15
2	−104.6	∼−92.54?	6.06	12.82
		-From 140 to 142³⁴
3-1/2	−67	∼−60.47?	6.0	19.84
3	−46.5	−42.9³⁵	5.93	28.25
3	−46.5	Pashen–Vysikaylo series	5.93	28.25
		−37.2³⁵
		−35.08³⁵
4-1/2	−34.1	−30.99³⁵	5.98	38.77
4-1/2	−34.1	Vysikaylo series	5.98	38.77
See Z = 2		−28.02³⁵
4	−26.15	−25.54³⁵	5.77	48.81
4	−26.15	Brackett–Vysikaylo series	5.77	48.81
		−22.56³⁵
5-1/2	−20.67	−17.36³⁵	5.92	63.42
5	−16.74	−15.36³⁵	6.0	79.29
6-1/2	−13.83	−13.51³⁵	5.77	92.3
6	−11.62	−11.41³⁵	5.75	109.57
7-1/2	−9.9	−9.3³⁵	5.88	131.48
7	−8.54	−7.93³⁵	5.92	153.3
8-1/2	−7.44	⋯		174.67
8	−6.54	⋯		195.0
9-1/2	−5.79	−5.58³⁵	5.8	222.0
10	−4.18	−3.97³⁵	5.85	309.6
11	−3.46
12	−2.9
13	−2.48
14	−2.135
15-1/2	−1.99	−1.98	5.71	635.
15	−1.86
5-1/2, Z=2	−41.3	−37.2³⁵	6.2	32.2
5, Z=2	−33.48	−28.02³⁵	6.2	41.2
6-1/2, Z=2	−27.67	⋯
6, Z=2	−23.25	−22.56³⁵	5.78	55.1

	E, meV	E, meV	ε, relative units
n	WM theory	experiment	Vysikaylo theory	r_n, Angstrom
1-1/2	−1674	?		0.8
1	−418.5	369^32,33	6.07	3.2
		Lyman–Vysikaylo series	6.7
2-1/2	−186	From 161 to 165^30,31,34	6.14-6.05	7.15
2-1/2	−186	Vysikaylo series	6.14-6.05	7.15
2	−104.6	∼−92.54?	6.06	12.82
		-From 140 to 142³⁴
3-1/2	−67	∼−60.47?	6.0	19.84
3	−46.5	−42.9³⁵	5.93	28.25
3	−46.5	Pashen–Vysikaylo series	5.93	28.25
		−37.2³⁵
		−35.08³⁵
4-1/2	−34.1	−30.99³⁵	5.98	38.77
4-1/2	−34.1	Vysikaylo series	5.98	38.77
See Z = 2		−28.02³⁵
4	−26.15	−25.54³⁵	5.77	48.81
4	−26.15	Brackett–Vysikaylo series	5.77	48.81
		−22.56³⁵
5-1/2	−20.67	−17.36³⁵	5.92	63.42
5	−16.74	−15.36³⁵	6.0	79.29
6-1/2	−13.83	−13.51³⁵	5.77	92.3
6	−11.62	−11.41³⁵	5.75	109.57
7-1/2	−9.9	−9.3³⁵	5.88	131.48
7	−8.54	−7.93³⁵	5.92	153.3
8-1/2	−7.44	⋯		174.67
8	−6.54	⋯		195.0
9-1/2	−5.79	−5.58³⁵	5.8	222.0
10	−4.18	−3.97³⁵	5.85	309.6
11	−3.46
12	−2.9
13	−2.48
14	−2.135
15-1/2	−1.99	−1.98	5.71	635.
15	−1.86
5-1/2, Z=2	−41.3	−37.2³⁵	6.2	32.2
5, Z=2	−33.48	−28.02³⁵	6.2	41.2
6-1/2, Z=2	−27.67	⋯
6, Z=2	−23.25	−22.56³⁵	5.78	55.1

The spectra of Vysikaylo standing excitons observed in experiments in diamonds doped with boron were identified by us as follows:^{4,10,11,18,30–36}

\begin{align} E_{n - 1 / 2} = ? (1 - 1 / 2); - 0.160 \div - 0.165 {(2 - 1 / 2)}^{30, 31}; ? (3 - 1 / 2); \\ 0.03099 {(4 - 1 / 2)}^{35}; 0.01351 {(6 - 1 / 2)}^{35}; 0.0097 {(7 - 1 / 2)}^{35}; [eV] \\ E_{n} = - 0.369 {(1)}^{18, 30}; - ? (2); 0.0429 {(3)}^{35}; 0.02554 {(4)}^{35}; \\ 0.01513 {(5)}^{35}; 0.01139 {(6)}^{35}; 0.00836 {(7)}^{35}; . . [eV] \end{align}

(8)

Previously, it was believed that an impurity atom perturbs the crystal at the level of internode dimensions (via a bound Frenkel exciton). Now we see (Fig. 2, Table I) that there are standing (bound) excitons of large radii caused by quantum effects [wave profiles ε(r)] caused by the introduction of a foreign atom into the crystal lattice of the doped crystal. This occurs in full accordance with the hypothesis of de Broglie. Thus, in this work, using QCM, a completely new method for determining the relative permittivity profiles in nanosized structures in doped crystals by Raman scattering is proposed for the first time [Fig. 2(b)]. According to the author’s model (1), (5), (6), (8), and (9), based on QCM, the step in determining the ε(r) profile is ε(r) – Δr ≈ 0,529ε(r)(n–1/4)/Z, which is more than two times smaller than the step that follows from the classical Wannier–Mott model. The Z-charge is localized in the nanostructure, and n is the principal quantum number of the standing exciton of large radius formed on the ε(r) structure.

C. Classification of the electronic spectra of Vysikaylo standing excitons of large radius

The electronic spectra of Vysikaylo standing excitons can be similarly divided, like the spectra of the hydrogen atom, into Vysikaylo–Lyman, Vysikaylo–Balmer, Vysikaylo–Paschen, and Vysikaylo–Brackett lines, etc. (Fig. 4). Unlike for Wannier–Mott spectra, the following formulas should be used to calculate Vysikaylo spectra, taking into account the changes in the relative permittivity ε(r_i) of diamond in the regions of the doping atom:⁴

v = 1 / λ = R (1 / (ε {(r_{i} i)}^{2} - 1 / {(ε (r_{m}) m)}^{2}),

(9)

where R is the Rydberg constant (equal to 109737.31568539 cm⁻¹).

FIG. 4.

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Electron levels during the absorption and emission of electromagnetic waves by a Vysikalo standing exciton with principal quantum numbers n and n-1/2.

Based on a number of experiments (in accordance with the works of Bohr, Schrödinger, Born, Wannier, Mott, Vysikaylo, and others), we have proven that in addition to electronic spectra with integer principal numbers (5), in a number of experiments,^30,31,34 a line is clearly observed that arises from an electronic level with the principal quantum number i =2-1/2.^4,11 Therefore, to calculate all possible spectra of standing Vysikaylo excitons according to (5), it is necessary to take into account the Vysikaylo degeneracy (in principal quantum numbers), both with integer i and m and half-integer principal quantum numbers (i = 1, 2-1/2, 2, 3-1/2,…; m = 1, 2-1/2, 2, 3-1/2,…), and all possible spectra between transitions with integer and half-integer principal quantum numbers (Fig. 4). Spectra with half-integer principal quantum numbers (cos waves with symmetric ψ-functions) were first resolved for hollow quantum resonators with spherical and cylindrical symmetries within the framework of Vysikaylo cumulative quantum mechanics,^4,11 and this theory was confirmed by numerous experiments with resonant electron capture by fullerenes.^19–21

VI. OSCILLATIONS OF THE RELATIVE PERMITTIVITY IN VYSIKAYLO EXCITONS

From a comparison of the analytical spectra and the identified experimental spectra , it is possible to construct the oscillations of the permittivity ε(r) (Fig. 5) depending on the distance r_i = a₀ε(r_i)i² to the center of the Vysikaylo standing exciton (the dopant atom, Fig. 2). Here i =1/2, 1,3/2, 2, 5/2,…:

\begin{align} ε (r) & = 6.07 (1; 3.21 Å); 6.14 \div 6.05 (2 - 1 / 2; 7.15 Å); \\ 5.93 (3; 28.25 Å); 5.97 (4 - 1 / 2; 38.48 Å); \\ 5.75 (4; 48.86 Å); 6.0 (5; 79 Å); 5.8 (6 - 1 / 2; 93 Å); \\ 5.75 (6; 109.5 Å); 5.787 (7 - 1 / 2; 129.4 Å); 5.768 (7; 149.5 Å) \end{align}

(10)

Identification of the electron level spectra of standing (frozen into the crystal) hydrogen-like Vysikaylo excitons of large radius is a complex arithmetic problem if we take into account all possible electron spectra analogous to the spectra of atomic hydrogen: Lyman, Balmer, Paschen, and clearly observable Vysikaylo spectra with principal quantum numbers i = n-½, as well as the oscillations of the relative permittivity ε_i(r_i) (Fig. 5). The fundamental role in substantiating the basics of the CQM for Vysikaylo standing excitons is played by the long-observed experimental electron levels with principal quantum numbers n=1; 2-½. These spectra have been observed since 1968. Of interest are the spectra obtained in Ref. 35 with low energy (significant excitation of standing excitons with large principal quantum numbers due to laser excitation). Of particular interest is the search for spectra theoretically predicted by us in the region of 0.2–0.06 eV (i = 2, 3-½) and transitions between electron levels with i = 1; 1.5; 2; 2.5; 3.

FIG. 5.

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Vysikaylo quantum oscillations ε(r) in acceptor-doped diamond in the region of the introduction of a boron atom into the lattice.^10,11

The phenomenon of oscillation of the relative permittivity is associated with oscillations of the electron density in the region of the implantation of a foreign atom. In metals with defects, a similar phenomenon is called Friedel oscillations. In this case, the electrons screening the impurity charge form a halo around the defect center with alternating regions of condensation and rarefaction of their density.

In a gas discharge, the analog of this phenomenon is pulsating or standing striations in space, which were first noted by Faraday. In photographs, the running waves become standing waves (Fig. 6).

FIG. 6.

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Waves on water, according to de Broglie’s hypothesis, are similar to ε(r) waves in doped diamond in the region of the introduction of a boron atom into the lattice (see Table I and Fig. 5).

VII. LIMITS OF APPLICABILITY OF THE METHOD FOR DETERMINING THE ELECTRONIC SPECTRA OF VYSIKAYLO EXCITONS

The radius of a quantum dot (QD), including a Vysikaylo standing exciton a_n (or a_n–1/2) in a doped crystal, depends on the profile of the relative permittivity of the crystal ε(r), which arises when an impurity is introduced into the crystal lattice depending on the value of the principal quantum number n (or n-1/2) of the excited level of a hydrogen-like QD (the total energy of an electron in the QD).

The condition for the applicability of formulas (5) and (6), according to Ref. 15, consists of the requirement of “a sufficiently large value of the orbit radius of the Wannier–Mott exciton,” a_n∼ℏ²εn²/me² ≫ a₀ [Fig. 2(b)]. This condition is obviously fulfilled for large n, but in crystals with large ε, it can also be fulfilled for n ∼ 1 (Ref. 15, p. 353).

Similarly, we substantiate the condition of applicability (1)–(10) for Vysikaylo standing excitons of large radius by the requirement: a_n–1/2 ≫ a₀ (or a_n ≫ a₀) [Fig. 2(b)].^10,11

However, in the case of formation of a standing exciton of large radius, this condition is modified to a_n–1/2 > d or a_n > d. Here, d is the distance between the nearest atoms in the crystal lattice of the supporting crystal (diamond, silicon, germanium, etc.). This condition is associated with the impossibility of forming lower energy electron levels in the structure of a standing exciton with profile ε(r) and characteristic sizes a_n–1/2 < d or a_n < d [Fig. 2(c)]. Inside this region, there is no positive charge for the electron [Figs. 2(b) and 2(c)], and therefore, a hydrogen-like exciton is not formed.¹¹

For diamond, d ≈ 1.54 Å. Therefore, the lower energy state for a standing exciton of large radius with ψ_1–1/2, energy E_1–1/2≈−1.476 eV, and a_1–1/2 ≈ 0.80 Ǻ < d =1.54 Ǻ, as a standing exciton of large radius, is not realized in diamond (Table I). For a similar reason, the states of standing excitons of large radii with Z = 2, 3, a_n–1/2 < 1.54 Å, and a_n < 1.54 Å are not realized.

Standing excitons with Z = 2, 3 arise in the region of the doped atom during the ionization of carbon atoms surrounding the boron atom. In this case, positively charged holes can intensively move around the boron atom along the nearest carbon atoms due to energy resonance.

VIII. CALCULATIONS OF VYSIKAYLO EXCITON MOLECULES AND FLICKERING CRYSTALS

By analogy with hydrogen molecules, flickering molecules and crystals from large-radius standing excitons can be formed through the overlapping of their excited electron shells. Using MGMT, it is possible to calculate the concentration of the acceptor-dopant impurity during the formation of flickering molecules and crystals from Vysikaylo large-radius standing hydrogen-like excitons. The difference between standing exciton molecules and hydrogen molecules is as follows. Since the exciton atoms in exciton molecules are strictly fixed in crystals, they have no rotational degrees of freedom. Oscillations are possible due to the movement of a hole around the doping atom. In this case, the electron spectra of standing exciton molecules are significantly richer than those of hydrogen molecules. This is due to the electron spectra with half-integer principal quantum numbers (cos waves with symmetric wave ψ-functions). Standing flickering molecules and crystals of standing excitons of large radius can be formed via the overlapping of excited electron shells of two or more nearby standing excitons and can be observed at concentrations of dopant (for example, boron) of:

N_{B, n} = 1 / {(2 a_{n})}^{3} = 1 0^{24} / {(2.0, 529 \cdot ε (r) \cdot n^{2})}^{3} c m^{- 3}

(11)

or for symmetric states of de Broglie waves of an electron at:

N_{B, n - 1 / 2} = 1 / {(2 a_{n - 1 / 2})}^{3} = 1 0^{24} / {(2.0, 529 \cdot ε (r) \cdot {(n - 1 / 2)}^{2})}^{3} c m^{- 3}

(12)

For diamond doped with boron, the formation of flickering crystals (superlattices), according to (11), is observed at n = 5 and concentrations of doped boron of N_B,5 ≈ 2.5 · 10¹⁷ cm^–3, at n = 4, N_B,4 = 9 ⋅ 10¹⁷ cm^–3, and at n = 1 N_B,1 = 4.7 ⋅ 10²¹ cm^–3.

It is possible to form hydrogen-like molecules with overlapping orbits of excitons and different values of the principal numbers n and m-1/2.

The main difference between hydrogen-like exciton molecules and hydrogen molecules is the absence of rotational and vibrational degrees of freedom in their spectra (at least they are significantly stabilized in the crystal). This allows one to study quantum phenomena without considering rotational degrees of freedom and is an additional method for verifying the electronic spectra of ordinary hydrogen molecules and other hydrogen-like molecules with large dimensions in various doped crystals.

Based on MGMT (the method of Wannier and Mott modified by us to describe standing excitons of large radius) and Raman spectra, it is possible to calculate all the electronic spectra of standing flickering molecules with excited inside them flickering standing super-atoms (standing excitons of large radius) of two types: those with symmetric and asymmetric wave ψ-functions. Such multiphase solid-state excited systems (Fig. 7), with the division of levels with the principal quantum number n into two sublevels, may prove very promising in the conversion of thermal energy into electromagnetic energy and in other practical applications.

MGMT goes back to the works of Eratosthenes (and was used by Wanier,²⁷ Mott,²⁸ Mandelstam, and others). Mandelstam said in one of his lectures: “You all know such systems as a pendulum and an oscillatory circuit, and you know that from an oscillatory point of view, this is the same thing. Now all this is trivial, but the remarkable thing is that it is trivial.”

We similarly used the generalization of the solutions of the Poisson equation (first obtained by Euler in 1767 and verified by Lagrange and Roche using celestial mechanics) to describe the interaction of electrons with positively charged Coulomb attractors—plasmoids or holes in crystals. In plasma, studies of such phenomena have already begun.⁵ These Euler solutions, describing the role of potential long-range forces or their fields in the contraction of gravitating small particles to libration points (cumulation) in celestial mechanics, have not yet been used for a similar solution of a number of paradoxes in crystals with standing or running excitons.

The role of analogs in physics lies in prompting and verifying models and their solutions. Indeed, the mathematical generality of problems in any long-range potential fields (Δφ = ρ) provides grounds to use not only all discoveries in celestial mechanics to discover new 3D phenomena that have not attracted the attention of researchers of inhomogeneous plasma (with violation of electroneutrality), but also in many other similar phenomena.

The molecular spectra of standing and running excitons, analogs of hydrogen molecules H₂ or positive ions H₂⁺, have not yet been identified and require experimental studies. However, it is already possible to note the special role of nanometer (large) electron cumulation points (L₁ and the features of the electron flow through these points in flickering crystals) in forming the properties of flickering crystals from standing excitons. Similarly, based on MGMT, one can expect hyperconductivity effects of flickering systems upon excitation of standing excitons of large radii.

Using MGMT, it is possible to predict and calculate the forces of attraction of dopant atoms to each other in the reference crystal during the excitation of excitons.

Vysikaylo cumulative points L₁ are formed between any spatially separated attractors with long-range potentials. The presence of cumulation points in the plasma leads to the formation of a Faraday dark space between two positively charged Coulomb attractors.⁵

The obtained profiles of relative permittivity pulsations [oscillations ε(r)] (Fig. 5) allow us to calculate with good accuracy the concentrations of the dopant for the formation of large-radius exciton molecules, lines of standing excitons (Fig. 7), and the parameters of flickering 2D surfaces and 3D crystals.

FIG. 7.

View large Download slide

The formation scheme of a flickering crystal from standing excitons of large radius at different irradiation doses: (a) A doped crystal at a weak energy source; (b) the overlap of electron shells at an increased irradiation dose and (1) the formation of linear flickering crystals with high conductivity; (2) The area of maximum radiation; and (3) the formation of Vysikaylo cumulation points⁵ L₁ (for electron flows) between positive ions of standing excitons of large radius.¹¹

Solid-state flickering crystals (crystals of standing excitons), described by the Vysikaylo model in Refs. 10 and 11, contained inside the base crystal, can be very convenient and useful both practically and in scientific research. Examples of the latter include investigations of the general properties of metallic hydrogen-like crystals and the spectra of exciton molecules. These molecules of Vysikaylo standing excitons do not have rotational and vibrational degrees of freedom (they are suppressed by interatomic bonds in the crystal lattice of the supporting crystal). This property can be used to verify the electronic spectra of hydrogen molecules.

IX. DISCUSSION

A specialist in doped crystals may assume that we propose two completely new theoretical foundations: cumulative quantum mechanics (CQM) and the method of generalized mathematical transfer (MGMT) for describing Vysikaylo standing excitons. Indeed, here we have applied these two methods¹¹ for the first time to describe the quantum-mechanical interaction between external electromagnetic fields and matter in nanostructured systems (doped crystals).

MGMT was used to calculate the radius of the Earth by Eratosthenes and was used by Newton, Einstein, and now us to describe similar electrical and electromagnetic phenomena based on gravitational interactions previously well studied by Euler. The use of MGMT has led to a number of Nobel Prizes, and for this reason, does not require further validation for describing nanosystems.

In the case of Vysikaylo CQM foundations, the following should be noted: “everything new is well hidden old.” In any classical textbook on atomic physics, in the general case for the plane symmetry of hollow resonators, the fundamentals of cumulative quantum mechanics (taking into account cos waves or waves with ½ wavelength) are well obscured. In these textbooks, all solutions for plane-symmetric problems with hollow quantum resonators are shifted by π/2, and thus the cos wave turns into a ½ sin wave. For example, in Ref. 25, cos waves are explicitly taken into account for hollow nanoscale resonators with plane symmetry. In Ref. 4, the author first solved the problem of how an electron determines the symmetry of a quantum resonator. As a result, the author weakened Dirac’s requirement on the boundedness of the ψ-function of de Broglie waves everywhere in quantum resonators to the requirement that only the probability density of finding a particle is bounded everywhere.

The Wannier–Mott formalism is also based on an application of the idea of Eratosthenes (MGMT) in the field of quantum mechanics, i.e., the transfer of the classical wave model of the hydrogen atom (from the angstrom-scale world of the hydrogen atom) to describe running (free) hydrogen-like excitons of large radius (on the nanometer scale) under the assumption of constancy of the relative permittivity throughout the crystal. Following Eratosthenes, we have already modified the Wannier–Mott method and applied the formalism to describe Vysikaylo standing excitons. For this, we assumed that during chemical doping, the relative permittivity ε(r) in the region of introduction of a foreign atom into the crystal lattice changes! The fact that the characteristic distance between carbon and boron atoms increases by 0.005 nm compared to the distance between carbon atoms was established using numerical density functional theory (DFT) calculations.¹⁷ Our method may be useful for determining the charge density profile at a sharp interface due to electrostatic polarization. To do this, we need to learn how to use the experimentally obtained Raman spectra together and apply DFT (i.e., solve the inverse problem). We have not found any specialists in Russia who have already done this.

A number of experiments have shown that doping diamond with boron and introducing boron into the diamond lattice leads to the formation of “gaps” consisting of several layers with an interplanar distance increased from 0.206 to 0.25 nm. This means that the presence of boron atoms in these areas weakens the diamond lattice. This leads to a partial rupture of sp³ bonds in diamond and thus to a local increase in the relative permittivity in the area of the doping atom. The introduction of foreign atoms into the crystal lattice, such as Ge (or other group IV atoms), can lead to a local increase in the relative permittivity in the area of the doping atom. When doping group IV crystals with group V atoms (e.g., when the length of the covalent bond C–N is 0.147 nm), the permittivity may decrease (locally, in the region of the doping atom). With such doping, the spectra determined by the principal quantum numbers n-½ may not be observed due to the localization of the positive ion in the doping atom.

The phenomenon of increasing the permittivity of diamond under (forced and not subject to elimination) doping with boron atoms is often compensated for by experimenters using special doping with nitrogen.

However, there are still no numerical calculations of how ε(r) changes with the distance to the doping atom using the DFT method. These calculations could validate the spherically symmetric Vysikaylo model for standing excitons of large radius.

Nevertheless, we have already proposed for the first time a completely unique method (using Raman spectra) for measuring the relative permittivity in nanoscale regions near a doping atom with a step of up to 0.3 nm. Based on the fundamentals of wave quantum mechanics (including Vysikaylo CQM), we assert that during the chemical doping of crystals with both donor and acceptor impurities (to obtain semiconductors of the “n” and “p” types) in the region of the introduction of a foreign atom, local perturbations of the relative permittivity ε(r) in the crystal lattice of the supporting crystal arise. These perturbations can be studied using Raman spectra. In this case, for semiconductors of the “n” type, ε(r) locally increases by 7% and the disturbance attenuates at distances of the order of 10 nm (see Fig. 5 and Table I). In the case of semiconductors of the “p” type, one should expect a local decrease ε(r) when doping diamond with nitrogen. We will explore this phenomenon in a future paper.

In this paper, the phenomenon of the self-focusing (cumulation) of charged particles in nanoscale inhomogeneous media has been investigated, and this mechanism has been integrated into both the CQM and CDS models. The CQM is based on the Schrödinger equation, which we previously used at the nanoscale level to explain resonant electron spectra during the polarization capture of electrons by fullerenes C₆₀ and C₇₀.⁴ It showed excellent results.

Using Vysikaylo CCM (for all types of fullerenes), we analytically determined all eigenenergy spectra for the polarization capture of electrons. For the available experiments,^19–21 the analytical cross-sections and eigenenergies coincided with an accuracy of 2%.⁴

In Ref. 4, more than 33 quantum-size effects discovered by the author and observed in experiments were considered, explained, and quantitatively characterized for the first time. In particular, we solved the problems of the polarization capture of electrons with resonant energy by hollow fullerenes, the Coulomb levitation of nanocrystals colored with fullerenes and the impossibility of their recrystallization, manifestations of various catalytic properties of electric fields in the region of charged structures in thermoelectrics and phosphors, and the formation of fullerenes on an electron with resonant energy. As a result of the application of CQM, previously “mysterious” phenomena received an explanation and numerical estimates, including for the nanoscale and femtoscale worlds of atomic nuclei.⁴

The standing excitons of large radius we discovered are a special case of the implementation of Vysikaylo CDS. The CQM is useful for describing CDS with sizes of 10^-15–10²⁶ m. This is the area of operation of Coulomb’s law. If, in the region from 10^-15 to the mesoscale, this law has been studied quite well, then for the macroscopic world of stars, galaxies, and intergalactic lightning, such studies are just beginning, as is the study of CDS from lightning, ordinary, and quantum stars to intergalactic lightning in the M 87 region (https://science.nasa.gov/wp-content/uploads/2023/04/m87-jet-jpg.webp). In CDS, the cumulation and dissipation of masses, energies, moments, and fields occur in accordance with fairly general laws (the virial theorem works). According to CQM, waves with half the de Broglie length of electrons are realized in hollow quantum resonators. This corresponds to an increase in the resonant kinetic energy of a quantum particle⁴ by a factor of (½)² = 4.

In traditional models (based on the Dirac restriction of the ψ-function everywhere), for hollow quantum resonators of spherical and cylindrical symmetries, theorists reject electronic spectra involving levels with a principal quantum number of n-½ (de Broglie cos waves). Within the framework of CQM (modifying Dirac’s requirement to a more lenient one—the limited probability of finding a quantum particle everywhere), these spectra are taken into account in hollow quantum resonators, which makes it possible to explain the polarization phenomena of resonant electron capture by the inner cavities of fullerenes⁴ and the spectra of the Vysikaylo standing excitons discussed in this paper. All problems with the singularity (the ψ-function of de Broglie waves indefinitely accumulating at the center of a sphere or cylinder) are solved by correctly considering the regularizing geometric coefficient χ² = 2^kπ^kr^k, where k = 1 for cylindrical symmetry of a hollow resonator, k = 2 for spherical symmetry, and χ² = 1 for plane symmetry. Our method (which converts a number of spectra forbidden by Dirac’s requirement into allowed ones) can undoubtedly be applied to calculations of functional nanocomposites and multiphase heterostructures, as was proved by calculating the electronic levels of fullerenes with high polarization affinity for electrons.⁴

To describe quantum effects in all our previous works, we used the classical electron mass and the classical Schrödinger equation, with reasonable profiles for the classical electric (polarization) potentials (formed by positive ions) and the corresponding potential barriers that form channels for electron flows. In all these studies, we did not need to introduce vacuum polarization effects, spatially localized nonequilibrium configurations that are inexplicable in their essence, or quantum interference. The real profiles of pulsating or stationary Coulomb (polarization) potentials were sufficient.⁴

The author did not uncover any restrictions on the use of MGMT and CQM when describing physical effects in Vysikaylo CDS (in any extreme conditions such as ultra-high pressure or ultra-low temperatures) with characteristic sizes from 10^-15 to 10²⁶ m.^1–13

Some researchers believe that the value of the relative permittivity of a material is associated only with the wavelength of electromagnetic radiation. This is a mistake. According to our model of hydrogen-like standing excitons, the relative permittivity is a local primary characteristic of a crystal [ε(r) is a coefficient weakening the Coulomb potential of an exciton hole in a radially inhomogeneous crystal on a sphere of radius r in the doping region] and can change at the nanometer level, reflecting by what factor [ε(r)] the local Coulomb potential of an external charge located at a distance r from the positively charged center of an exciton changes. New nanostructured materials with nanosized crystals and unique properties are currently being developed. Of particular interest among these are materials formed from nanocrystallites with significantly different relative permittivities. Thus, to increase the efficiency of nanostructured luminophores, crystallites with a relative permittivity of ε ∼ 1500 are sintered with nanographite powder or fullerenes of nanometer dimensions. Models with a constant value of the relative permittivity over the entire mesostructure are, in principle, inapplicable to describe excitons in such structures. We propose such a theory for nanostructures with large ε.

Using MGMT, we have analyzed many years of experimental observations of Raman spectra (laser radiation Raman scattering spectra) of diamonds doped with boron atoms, which began in the last century (before 1968) and continue to the present day.^37–40 Before our work, there was no worthy model of quantum transitions in such crystals.¹⁸ This was due to an assumption about the constancy of the permittivity ε in the region of the introduction of a foreign atom into the crystal lattice and Dirac’s erroneous prohibition of electron spectra with principal quantum number n-½ (cos waves). As the author proves based on theoretical foundations of quantum mechanics and numerous experimental observations, these provisions of Dirac²² and his followers^23,24 should be substantially modified. In this paper, it has been proved that the Dirac requirement modified by the author and the modified Wannier–Mott–Bohr model,^10,11 taking into account oscillations (wave inhomogeneities, Fig. 5) of the relative permittivity at the nanometer level (11), allow for a satisfactory description of the Raman spectra (8) obtained in experiments in the study of diamonds doped with boron (see Table I of the experimental data in Refs. 30–36).

X. CONCLUSIONS

In this theoretical study:

We have identified and analytically described (based on Vysikaylo cumulative quantum mechanics) practically all electron spectra (observed in experiments^30–36) of Vysikaylo standing excitons in diamond doped with boron atoms;
Our modified Wannier–Mott theory explains all electron spectra observed during the irradiation of diamonds doped with boron atoms;^10,11
Dirac’s requirement on the limitedness of the ψ-functions of de Broglie waves in excitons is excessive. It is sufficient to limit the probability of finding a particle in the volume of a standing Vysikaylo exciton and take into account the normalization geometric coefficient (also considering the symmetry of the quantum resonator χ² = 2^kπr^k: k = 2 for spherical symmetry of the quantum resonator, k = 1 for cylindrical symmetry, and χ² = 1 for plane symmetry);
The Vysikaylo degeneracy of the principal quantum number n (asymmetric sin-waves) and n-1/2 (symmetric cos-waves) is observed in experiments not only during the capture of electrons by the internal cavities of fullerenes,^4,11,19–21 but also during the formation of standing Vysikaylo excitons;^{10,11,30–35}
The relative permittivity ε(r) changes significantly (by up to 7%) in the nanoregion (up to 10 nm; see Fig. 5) of the doped atom embedded in the crystal lattice of the reference crystal (e.g., diamond);
In accordance with the change in the relative permittivity ε(r) in the region of the dopant atom (at nanometer distances), a corresponding change in the Raman spectra (the spectra of combination radiation observed in the experiments^30–36) occurs;
Using Raman spectra,^30–35 it is possible to determine the nanoprofiles of the relative permittivity ε(r) of the reference crystal in the region of the introduction of a foreign atom into the crystal lattice (Fig. 5);^10,11
Experimental observations of Raman spectra^30–36 of Vysikaylo standing excitons prove the basic provisions of Vysikaylo’s cumulative quantum mechanics, as formulated in Refs. 4 and 11;
The theory we have developed (the modified Wannier–Mott theory) of the formation of inhomogeneous permittivities in inhomogeneous crystals can applied at the level of nanostructures.^10,11

The knowledge we have obtained can be applied to the study of nanocomposite materials formed via the chemical doping of dielectrics (diamond, silicon, germanium, and other group IV crystals) with foreign atoms from groups III and V of the periodic table. The unique technique we have developed, based on the Wanier–Mott model we modified, allows one, based on the experimentally established Raman spectra, to calculate the profiles of the relative permittivity at the nanometer level (from 0.3 to 20 nm) in chemically doped diamond. The results obtained indicate the erroneousness of the theoretical approach used in Ref. 35, which does not consider the wave (quantum) oscillations of the relative permittivity in the region of the doping center. The author believes that there is no point in advertising spin-orbital inventions, as in Ref. 35. It is better to modify the models of Wannier, Mott, and Dirac, relying on the classical works of Euler, Frenkel, de Broglie, Bohr, Vysikaylo, and others,^4,10,11 as verified by comparing experiments with full-fledged theories.

The obvious presence in experiments^19–21 of lines with the principal quantum number n-1/2 (E_n-1/2) in the resonance capture of free electrons by fullerenes and in the Raman spectra of boron-doped diamonds^30–35 once again proves the validity of the foundations of Vysikaylo cumulative quantum mechanics^4,11 and thus proves that Dirac’s requirement that ψ-functions are bounded everywhere is excessive and can be replaced by the requirement of the boundedness of the probability of finding a particle in a quantum resonator.^4,11

We have proved that quantum mechanics is a complete statistical theory capable of prediction if we do not attribute wave properties to particles. Particles behave as waves in the statistical approach, but they remain particles if their behavior is considered particle-like.^4,11

According to the work of Einstein, Podolsky, and Rosen and Gödel’s incompleteness theorem, standard quantum mechanics is incomplete. This is true if the particle is assigned wave properties. However, if wave properties are assigned to the statistical behavior of the particle in the resonator, another point of view is formed, according to which quantum mechanics is a complete theory. This is substantiated on the basis of the law of conservation of energy and recognition of the probabilistic nature of the statistical behavior of a particle in the resonator.

In our work, we have used classical parameters of electrons (mass and charge), and on this basis, we have explained a number of previously “mysterious” statistical phenomena, including experiments in chemically doped crystals. Studies of standing molecular excitons, flickering lines (nanolightning), planes (nanostrata), and Vysikaylo crystals (in chemically doped crystals) are of great scientific and practical interest for the further development of quantum mechanics. In such exciton molecules, lines, planes, and crystals, there are no external influences that lead to the excitation of vibrational and rotational degrees of freedom (which are significantly weakened in crystals by interatomic bonds).

The results obtained in this work prove the predictive power of Vysikaylo statistical cumulative quantum mechanics and the method of generalized mathematical transfer (MGMT) for studying and describing the structure of nanoscale cumulative-dissipative systems.

AUTHOR DECLARATIONS

Conflict of Interest

The author has no conflicts to disclose.

Author Contributions

Philipp I. Vysikaylo: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available within the article.

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Methods of cumulative quantum mechanics for describing the interaction of electromagnetic fields with matter in Vysikaylo large standing nanoscale excitons: Hydrogen-like atoms, molecules, surfaces, and twinkling crystals

ARTICLE HIGHLIGHTS

I. INTRODUCTION

A. Definition of cumulation

B. Method of generalized mathematical transfer

C. Classification of excitons

D. Detailed schemes of Vysikaylo excitons and the chemical doping of support crystals with acceptor impurities

E. On the identification of the electronic spectra of Vysikaylo excitons observed in experiments

II. ACHIEVEMENTS OF VYSIKAYLO CUMULATIVE QUANTUM MECHANICS

III. BASIC PROVISIONS FOR THE DESCRIPTION OF VYSIKAYLO STANDING EXCITONS

A. Large running (free) Wannier–Mott excitons

B. Determination of the relative permittivity for nanostructures

1. Our modification of the Wannier–Mott model

IV. PONDEROMOTIVE FORCES IN INHOMOGENEOUS CRYSTALS

V. ELECTRONIC SPECTRA OF VYSIKAYLO STANDING EXCITONS OF LARGE RADIUS

A. Spectra of Wannier–Mott excitons in diamond with ε = const

B. Spectra of Vysikaylo excitons in diamond at ε = ε(r)

C. Classification of the electronic spectra of Vysikaylo standing excitons of large radius

VI. OSCILLATIONS OF THE RELATIVE PERMITTIVITY IN VYSIKAYLO EXCITONS

VII. LIMITS OF APPLICABILITY OF THE METHOD FOR DETERMINING THE ELECTRONIC SPECTRA OF VYSIKAYLO EXCITONS

VIII. CALCULATIONS OF VYSIKAYLO EXCITON MOLECULES AND FLICKERING CRYSTALS

IX. DISCUSSION

X. CONCLUSIONS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Methods of cumulative quantum mechanics for describing the interaction of electromagnetic fields with matter in Vysikaylo large standing nanoscale excitons: Hydrogen-like atoms, molecules, surfaces, and twinkling crystals Open Access

ARTICLE HIGHLIGHTS

I. INTRODUCTION

A. Definition of cumulation

B. Method of generalized mathematical transfer

C. Classification of excitons

D. Detailed schemes of Vysikaylo excitons and the chemical doping of support crystals with acceptor impurities

E. On the identification of the electronic spectra of Vysikaylo excitons observed in experiments

II. ACHIEVEMENTS OF VYSIKAYLO CUMULATIVE QUANTUM MECHANICS

III. BASIC PROVISIONS FOR THE DESCRIPTION OF VYSIKAYLO STANDING EXCITONS

A. Large running (free) Wannier–Mott excitons

B. Determination of the relative permittivity for nanostructures

1. Our modification of the Wannier–Mott model

IV. PONDEROMOTIVE FORCES IN INHOMOGENEOUS CRYSTALS

V. ELECTRONIC SPECTRA OF VYSIKAYLO STANDING EXCITONS OF LARGE RADIUS

A. Spectra of Wannier–Mott excitons in diamond with ε = const

B. Spectra of Vysikaylo excitons in diamond at ε = ε(r)

C. Classification of the electronic spectra of Vysikaylo standing excitons of large radius

VI. OSCILLATIONS OF THE RELATIVE PERMITTIVITY IN VYSIKAYLO EXCITONS

VII. LIMITS OF APPLICABILITY OF THE METHOD FOR DETERMINING THE ELECTRONIC SPECTRA OF VYSIKAYLO EXCITONS

VIII. CALCULATIONS OF VYSIKAYLO EXCITON MOLECULES AND FLICKERING CRYSTALS

IX. DISCUSSION

X. CONCLUSIONS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Submit your article

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Methods of cumulative quantum mechanics for describing the interaction of electromagnetic fields with matter in Vysikaylo large standing nanoscale excitons: Hydrogen-like atoms, molecules, surfaces, and twinkling crystals