A Quantum Dynamics of Heisenberg Model of the Neutron Associated with Beta Decay

In this work we re-examine a model of the nucleons that involve the weak interaction which was once considered by Heisenberg; that is a neutron may have the structure of a dwarf hydrogen-like atom. We formulate a quantum dynamics for the Heisenberg model of the neutron associated with interaction that involves the beta decay in terms of a mixed Coulomb-Yukawa potential and the More General Exponential Screened Coulomb Potential (MGESCP), which has been studied and applied to various fields of physics. We show that all the components that form the MGESCP potential can be derived from a general system of linear first order partial differential equations similar to Dirac relativistic equation in quantum mechanics. There are many interesting features that emerge from the MGESCP potential, such as the MGESCP potential can be reduced to the potential that has been proposed to describe the interaction between the quarks for strong force in particle physics, and the energy spectrum of the bound states of the dwarf hydrogen-like atom is continuous with respect to distance. This result leads to an unexpected implication that a proton and an electron may also interact strongly at short distances. We also show that the Yukawa potential when restrained can generate and determine the mathematical structures of fundamental particles associated with the strong and weak fields.

( ) e , r Q V r r r β α − = − + (1) where , Q α and β are physical parameters that will need to be determined [5] [6]. Furthermore, in order to account for possible bound states of a dwarf hydrogen-like atom which can be identified with a neutron we will need to use a more general form of Yukawa potential, which has been studied and applied to various fields of physics, the More General Exponential Screened Coulomb Potential (MGESCP) given as ( ) where 0 V is the potential depth and the parameter α is the strength coupling constant [7] [8]. Remarkably, we will show that the MGESCP potential can be reduced to the potential that has been proposed for the interactions between the quarks for strong force in particle physics and this result leads to an unexpected implication that a proton and an electron may also interact strongly at short distances. There are also prominent features that emerge from using the MGESCP potential to describe a neutron as a dwarf hydrogen-like atom, such as the energy spectrum of the bound states is continuous with respect to distance, and, as discussed in Section 3, the Yukawa potential can be restrained to generate and determine mathematical structures of physical objects that may be identified with the quantum mediators associated with the weak and strong interactions.
With this regard, it is reasonable to suggest that functional potentials in physics may have physical mechanisms to generate mediators of associated physical fields, and these mechanisms can be formulated in terms of differentiable manifolds and their corresponding direct sums of prime manifolds as discussed in our works on the possibility to formulate physics in terms of differential geometry and topology [9].

Formulating Potentials from a System of Equations Similar to Dirac Equation
The system of equations given in Equation (3) can be rewritten in a matrix form as where ( ) A B and σ are matrices representing the quantities , , r r r ij ij j a b c , which are assumed to be constant in this work, and kJ is a matrix that represents the quantity r d , where k is a dimensional constant. As normal convention, the letter i in front of the last two terms in Equation (4) is the imaginary number i. While the quantities q, m and kJ represent physical entities related directly to the physical properties of the particle under consideration, the quantities i V represent the potentials of an external field, such as an electromagnetic field or the matter field of a quantum particle. In order to formulate a physical theory from the system of equations given in Equation (4), it is necessary to determine the unknown quantities , i i A B and σ , as well as the mathematical conditions that they must satisfy, such as commutation relations between them. The commutation relations between the matrices can be determined if we apply the op- Since the quantities , , , In order to describe the dynamics of a particular physical system, undetermined parameters given in Equation (4) must be specified accordingly. To obtain a system of partial differential equations similar to Dirac where the matrices i A and i B have been identified with Dirac matrices i γ as follows It is seen from Equation (4) that the quantity kJ represents an internal source which is associated with a dynamical process of a quantum particle. For example, a quantum particle is undergoing some form of physical evolution that changes its physical structure, such as an accumulation of mass during its formation. In fact we will show that this physical process can be formulated using the MGESCP potential in which the energy spectrum depends continuously on distance. In terms of the gamma matrices i γ then Equation (7) can be rewritten in a covariant form similar to Dirac equation for an arbitrary field with an internal source kJ as

V V V V V A A A =
then Equation (12) can be used to determine the dynamics of the components of the wavefunction (  )   T  1  2  3  3 , , , ψ ψ ψ ψ ψ = , where the term are the components of the electric field E and the magnetic field B . Now we will discuss how free quantum particles can create their own physical fields in which wavefunctions can be identified as potentials. Therefore we set  (14) In the following we will consider two cases depending on the conditions that are applied to the internal source kJ in which either 0 kJ = or 0 ≠ . For the case 0 kJ = , Equation (13) reduces to a form similar to Dirac equation for a free particle i . m µ µ γ ψ ψ ∂ = − (15) For massive particles in which 0 m ≠ , all components of the wavefunction µ ψ satisfy the Klein-Gordon equation And, in particular, if the wavefunctions are time-independent then we obtain where α and β are undetermined dimensional constants [10]. It should be mentioned here that, unlike Dirac wavefunctions that represent spinor fields, the wavefunctions µ ψ in this work are simply the components of vector fields that are solutions of a system of linear first order partial differential equations. The identification of the wavefunctions µ ψ can be viewed either as static Yukawa potential or Coulomb potential is similar to the identification that we discussed in our work on the fluid state of Dirac quantum particles in which Dirac-like wavefunctions are identified either with a velocity potential or a stream function [11]. According to Yukawa work, the wavefunctions given in Equation (18) can be associated with the strong interaction between nuclear particles.
For massless time-independent particles, the Klein-Gordon equation given in Equation (17) (20) In this case the wavefunctions µ ψ can be viewed as static Coulomb potential, where Q is an undetermined dimensional constant, which is associated with the electromagnetic interaction between elementary particles.
As mentioned in the introduction, we will discuss possible bound states of a dwarf hydrogen-like atom which can be identified with a neutron therefore we will need to use a more general form of Yukawa potential, which is the MGESCP potential given as in Equation (2 i . In particular, if the wavefunctions are time-independent then we obtain ( ) It can be verified that a solution of the form e mr a µ ψ − = , where a and m are constants, satisfies the following equation x r γ − ∂ + = ∂ (24) A differential equation for the quantity kJ can be determined by using the matrices i γ given in Equation (8), which is written in an explicit form as From Equation (25) we obtain the following equations for the quantity kJ Now we apply the results that have been obtained into the MGESCP potential given in Equation (2), one of whose components has the form With the Yukawa potential that is restrained by the condition given in Equation . This type of potential describes interactions between two quarks that can be represented in the following picture shown in Figure 1 and Figure 2.
For small values of the distance r the potential manifests as a Coulomb potential V A r = , however, for large values of distance the potential acts as a linear potential with respect to the distance V Br = . The linear potential shown in Figure 2 is a flux tube of energy in which the quantity B has the dimension of a cross-sectional energy therefore by comparison we may also interpret the quantity kJ in Dirac equation given in Equation (7) also as a cross-sectional energy.
The reduced form of the MGESCP potential also indicates that a proton and an electron can attract strongly at very short distances so that they can bind and form a dwarf hydrogen-like atom.

Topological Structures of Elementary Particles Generated by Yukawa Potential
In this section we discuss further the restraint to the Yukawa potential given in Equation (28) which has been shown to reduce the MGESCP potential to the potential that is proposed for the interaction between the quarks for strong force in particle physics. We now show that in fact the restrained Yukawa potential actually generates and determines mathematical structures of physical objects that may be identified with quantum mediators of the weak and strong interactions. Instead of giving a mathematical analysis of the restrained Yukawa potential given in Equation (28), as an illustration, we simply plot possible shapes that can be generated and determined by a restrained Yukawa potential from the relation given in Equation (28)

A Quantum Dynamics of the Weak and Strong Interactions
In this section we will discuss whether a neutron can be modelled as a dwarf hydrogen-like atom with the two different mixed potentials given in Equations (1) and (2) so that it can be used to explain the physical processes associated with the beta minus decay. As shown in Section 2, these potentials can be formed from the three forms of potentials that have been derived from the Dirac equations. First we consider the mixed potential that is formed from the Coulomb potential and the Yukawa potential as given in Equation (1). As shown below, this form of potential can be used to explain how an electron can be repelled from a dwarf hydrogen-like atom composed of a proton and an electron. By dif- From Equation (30), the corresponding force of interaction The mixed potential given in Equation (1) now takes the form And the corresponding force of interaction In order to investigate further we need to know the nature of the stationary point at r R = . From Equation (30), the second derivative of ( ) Since we are considering the case for which the mixed potential given in Equation (33) is applied to the bound system of two charges of opposite signs, namely a proton and an electron, therefore the quantity Q can be written as q is the charge of the proton, 1 q q = , and 2 q is the charge of the electron, 2 q q = − , then we have , the force is attractive for r R < and repulsive for r R > . This situation is seen similar to the case of weak interaction of beta minus decay in which a neutron turns into a proton and emits an electron and an anti-neutrino. First the neutron turns into a dwarf hydrogen-like atom, whose bound states will be described below using the MGESCP potential, then the electron moves in an orbit of zero force and then it is repelled from the dwarf hydrogen atom by a repulsive force. The force given in Equation (34) The force given in Equation (36) is assumed to be a weak force. Since R is the radius of the stationary orbit therefore we can assume that the electron is ejected from the stationary orbit because the equilibrium of this system at r R = is unstable.
It can be seen that the whole process of beta decay is a complicated physical process that actually undergoes through many different physical configurations V. B. Ho of the system, therefore it can only be described approximately by many different dynamics, only if we can formulate the whole physical process under a mathematical formulation that can give rise to each physical configuration by some form of limit associated with mathematical parameters that are used to describe the whole system. With this in mind in the following we will discuss in terms of Schrödinger wave mechanics a more complete structure of a neutron as a dwarf hydrogen-like atom using a more complete MGESCP potential given in Equation (2). Consider the time-independent Schrödinger equation in which ( ) V r is the More General Exponential Screened Coulomb Potential (MGESCP) given in Equation (2). Since the MGESCP potential is spherically symmetric, Equation (37) can be written in the spherical polar coordinates as where the orbital angular momentum operator 2 L is given by  [8]. Although this energy spectrum is discrete with respect to the quantum numbers n and l, it depends continuously on the radial distance r. In order to interpret the energy spectrum given in Equation (44) Now we may interpret this continuous spectrum of energy with respect to distance as the energy spectrum of massive mediators associated with strong force described by the potential given in Equation (29). When a physical particle is created it is being created continuously until it reaches the size that is required for the system. This process happens in a very short time therefore it seems like an instantaneous creation. In particle physics, the parameter α of the exponential term is expressed in terms of the mass m of a force carrier as 2 mc α =  . Therefore when the mass of the force carrier is being continuously created the parameter α is being getting larger, at the same time the radius r is also getting bigger, therefore the term 2 e 0 r α − → and also the term 2 e 0 r α α − → . The mass that is accumulated by the force carrier must be supplied by the neutron. When the force carrier with required mass hits the electron, the latter will move further from the proton. On the other hand the MGESCP potential is reduced to the mixed Coulomb-Yukawa potential when the process of creation of the force carrier is complete. This form of potential provides a repulsive force to move the electron away.
As a further remark on the forms of potentials given in Equations (1) and (2), it is seen that the potential given in Equation (2) cannot be reduced to the potential given in Equation (1). And as a consequence the solutions given in Equation (43) cannot be reduced to solutions to a Schrödinger wave equation that uses a potential of the form given in Equation (1). In order to reduce to the potential given in Equation (1)

Conclusion
In this work we have discussed a quantum dynamics of Heisenberg's model of the neutron associated with the beta minus decay through the weak and strong interactions in which a neutron may have the structure of a dwarf hydrogen-like V. B. Ho Journal of Modern Physics atom. It has been shown that the whole process of beta decay is a complicated physical process that actually undergoes many different physical states of configuration of the system. Therefore, the physical process of beta decay should be described by many different dynamics rather than a single one, only if we can formulate the whole physical process under a mathematical formulation that can give rise to each state by some form of limit associated with mathematical parameters that are used to describe the whole system. For example, a more complete mathematical formulation would require a more general form of potential as given in Equation (46) in Section 4. However, in this work we have still been able to discuss in terms of Schrödinger wave mechanics a quantum dynamics of the neutron as a dwarf hydrogen-like atom using a more complete potential which has been studied and applied to various fields of physics, the so-called the More General Exponential Screened Coulomb Potential (MGESCP). We have shown that the MGESCP potential can be derived from a Dirac-like system of equations which can be reduced from a general system of linear first-order partial differential equations. There is a particular advantage for our approach to deriving a Dirac-like system of equations from a general system of differential equations, that are Dirac-like wavefunctions which can be interpreted according to the purpose of the mathematical investigation of a physical system. They can be used to represent different type of physical objects rather than exclusive mathematical wavefunctions that are used to calculate the probability of the outcome of an experimental result. From the MGESCP potential we have also been able to derive a form of potential that is used to describe interaction between quarks in strong interaction and, in particular, the energy spectrum of the bound state of the dwarf hydrogen-like atom that is continuous with respect to space. This may be a manifestation of a continuous creation of an elementary particle in space. We have also shown that the Yukawa potential can generate and determine the physical shapes of fundamental particles associated with the strong and weak fields. As a consequence, it seems reasonable to suggest that the functional potential in physics may have physical mechanisms to generate mediators of associated physical fields, and these mechanisms can be formulated in terms of differentiable manifolds and their corresponding direct sums of prime manifolds as discussed in our works on the possibility to formulate physics in terms of differential geometry and topology.

Introduction and Background
The SHOES ( [9]. The interested reader should start with a recent FSC summary paper [10], which shows how its predictive capacity and falsifiability make it preferable in many ways to the ΛCDM concordance model.

Results
The relevant FSC equations useful for these calculations include: It is trivial to show how Equations (1) and (2) The following equation is useful in this regard: where the t value is simply the integer, or fractional number, of billions of years of cosmic age, and H 0 = 66.893 km·s −1 ·Mpc −1 .
One can then construct the following The results shown in Table 1 can be used to present a graph ( Figure 1) of log 10 (H t ) as a function of redshift z.

Discussion
The FSC publication entitled, "Temperature Scaling in Flat Space Cosmology in Comparison to Standard Cosmology" [11] makes note of the fact that the temperature curves of FSC and ΛCDM cosmology are somewhat different, particularly in the first billion years or so of cosmic evolution. This is a function of the embedded FSC scaling thermodynamic equation. A graph ( Figure 2) from the temperature scaling paper is reproduced herein.
The blue line is the radiation temperature (T R ) curve expected in the ΛCDM concordance model and the green line is the radiation temperature curve expected in FSC. The dashed red line represents the measured and projected spin temperature (T S ) and the solid red line represents the baryonic gas temperature   (T G ). In Figure 2, the concordance model temperature curve, as well as the concordance model conception of the time scale at which "cosmic dawn" occurred is presented as seen in Bowman's publication [12] on the recent redshifted 21-cm observations. One should note that, in the FSC model, the same radiation temperatures and redshift z values correlating to "cosmic dawn" occurred at about 20 -60 million years after the Planck epoch, as opposed to approximately 110 -250 million years in the concordance model. As discussed in recent FSC papers (see references [10] and [11]), the FSC model allows for earlier galaxy formation by several hundred million years in comparison to the concordance model. Thus, the FSC model, by its significantly colder temperature curve in the first billion years, offers a solution to the "The Impossibly Early Galaxy Problem" described by Steinhardt [13].
Of most importance with respect to the stated purpose of this paper, one can now see that the FSC model makes very specific predictions for the Hubble parameter values correlating with the range of currently observable galactic redshifts. The following major points need to be stressed: 1) As is true for all R h = ct models, the FSC Hubble parameter is scalar over the great span of cosmic time. R h = ct models require that the Hubble parameter value be defined according to H t = c/R t , where c is the speed of light and R t is the Hubble radius at any time t. Thus, while at any point in human time spans the Hubble parameter may appear to be a constant, any cosmic model which incorporates c/R t within its Hubble parameter equation stipulates that the Hubble term is not a true constant over the great span of cosmic time.
2) The SHOES study of the "local" Hubble constant measured galactic separation velocities per megaparsec at earlier cosmic times corresponding to a range of redshift z values of less than 0.15. It is interesting to note that their averaged "outlier" Hubble constant value corresponds roughly with the FSC Hubble parameter value correlated to a redshift z value of 0.06 (see Table 1). Moreover, when the Milky Way was used as a sole anchor for these observations, the SHOES study H 0 value of 76.18 ± 2.37 km·s −1 ·Mpc −1 can be seen to fit the middle of the 0 < z < 0.15 range of our Figure 1 curve. The log 10 (76.18) is 1.88. This may be more readily apparent if one compares the first five entries in Table 1. Similar adopted zero point measurements of Hubble parameters corresponding to higher z value ranges are fully expected by these authors to correlate with higher sections of Figure 1 curve. If so, then modern cosmologists will soon accept the idea of a time-variable, scalar, Hubble parameter.
3) Table 1 and Figure 1 show very specific predictions of the FSC model awaiting confirmation or refutation within the next few years of observations. If these observations prove that the Hubble parameter has a radically different temporal history than predicted herein, the FSC model and any similar R h = ct models would then be falsified. On the other hand, if observations in the next few years fit with Table 1 and Figure 1, the temporal behavior of dark energy would be better understood using the FSC model. In such case, the FSC model

Summary and Conclusions
The heuristic FSC R h = ct cosmology model is utilized to evaluate the tension between the SHOES local Hubble constant value and the Planck Survey and Dark Energy Survey global values. The SHOES report looked at a specific redshift z range of 0 < z < 0.15. Thus, the first approximately 11 billion years of the cosmic expansion history was not subjected to their analysis.
In the FSC model, the Hubble parameter is defined according to H t = c/R t . Therefore, the FSC Hubble parameter is time-variable and scalar according to Table 1 and The authors predict that, in the next few years, similar measurements corresponding to higher z value ranges will correlate with higher sections of Figure 1 curve. If this proves to be the case, then modern cosmologists will have to accept that what is currently called a Hubble constant is actually a time-variable, scalar, Hubble parameter. Furthermore, such a result would likely establish FSC as the new concordance model.

Abstract
In this study, we demonstrate the correctness of our 2010 hypothesis regarding the need to complete Coulomb's F C law with the term lnr, resulting in the completed F CC force. For this purpose, we consider the electrical interactions between charged microparticles (MPs), which develop as fundamental vibrations (FVs) in ether, producing the vibrational strains ε and γ and the resulting stresses σ and τ, as percussions of ether cells (ECs) upon the MP surface. The stresses σ and τ produce a resultant force F P , due to the percussions which constitute the real electric force F CC . The spatial effect of ether on F P is demonstrated by an analytical method, considering the electrical interaction between MPs through various equidistant spatial paths l i of FVs, modelled on the basis of the Huygens principle for waves. For this issue, we utilized a numerical calculation, which could be generalized. But this spatial effect of the ether leads at a very slow decreasing of the F P forces ratio r F when doubling the distance l, in contrast to Coulomb's F C forces whose ratio r F decreases accentuate with doubling l. Accordingly, the necessity of including the term ln r in the F CC force, which is limited to 1.0 for doubling l, at long distances, was justified.

Introduction
The HM16 ether model, originally proposed by the present authors in 2016 [1], will be the starting point for the present analysis, which will provide new developments in the model in terms of the composition, and behaviour and effect of ether in nature and in physics.
We consider in our HM16 model that in nature, all microparticles (MPs) have either a positive or a negative electric charge, denoted MP + and MP − . However, in the general case in which the MP sign (+ or −) is not important, we will utilize the simple notation MP in the text, for both MP + and MP − . In addition, it is accepted in physics that at the macro scale, MP + and MP − are in parity, or even in pairs, and matter is neutral in nature.
To date, in physics, the nature of positive +q and negative −q electric charges has been neither defined nor explained. Furthermore, it has not been justified physically and materially.
In the following sections, we will initially present the modelling of the physical nature of the electric charge by highlighting new properties of the HM16 model of ether and of MPs, related to electric charges.
The paper brings in some novelties, such as the mechanical percussion nature of corrected electric forces F CC , or the many simultaneous paths in ether of electric force F CC transmission from close to closer distances, in contrast to the classical Coulomb force F C transmitted at a distance.
The main new contribution of the paper is the analytical demonstration that the F CC forces decrease very slow with the distance l, thus justifying the new term lnr, in contrast to the classical Coulomb force F C which decreases with distance.
But this spatial effect of the ether leads at a limit to 1.0 in the decrease of the F P forces ratio when doubling the distance l, in contrast to Coulomb's F C forces whose ratio decreases to 0.25 with doubling l.
Accordingly, the necessity of including the term lnr in the F CC force, which too is capped to 1.0 for doubling l, at long distances, was demonstrated.
In all the following graphical representations of space filled with ether, we will also indicate the indispensable reference frame (RF), usually as a Cartesian Oxyz frame, attached to a certain MP immovable in ether or to a certain cell of the ether (EC). Hence any Oxyz frame will be an absolute RF, referring to the ether.

Origin of the Interaction Force between Microparticles with Electric Charges MP + and MP − in the Case of the HM16 Ether Model
Since we admit that the electric charge is intimately linked to MPs, the electric charge can be explained by the composition and functioning of MPs in the HM16 ether model.
In the following analysis, we will utilize the same abbreviations used in [1], which will be redefined in this section as follows.
HM16 is our ether model initiated in 2016, MP stands for microparticle, SMP  The Type A HM16 ether model is the main focus of the present analysis.
Of course, the detailed mechanisms developed between ether components and the forces acting between MPs, which will be presented and utilized in the next part of this paper, are only hypothesized, because until now it has not been possible to observe or directly measure them at their very small scale.
However, our HM16 ether model and its functioning is based on, and is in agreement with, actual existing experimental data from physics concerning the behaviour of matter and radiation at an observable scale. Any such hypothesis will be confirmed in physics, when the results correspond to the real behaviour of matter and to the structure/composition of radiation at an observable scale.
Furthermore, as the purpose of this study, we will show that the results yielded by our HM16B model will be in accordance with the results of physical experiments. Accordingly, our initial hypothesis concerning the HM16 ether model will be confirmed.
It should be noted, as in [1], that the first-order components of HM16 ether, namely the ether particles themselves, are of two types: EPα and EPβ. These are the α and β etherons, which are grouped into the elemental cells (ECs) of ether, with at least four etherons (tetrahedrons) or as is more justified by volume symmetry, with eight etherons (cubes).
Ether is considered to be in its natural, undisturbed state, thus presenting no deformations/displacements but containing the fundamental vibrations (FVs) induced by all existing MPs, which are considered immobile/fixed in space at the starting moment of any analysis.
We acknowledge that the MPs in ether constitute physical support for the electric charges +q and −q and we refer to them as MP + and MP − respectively. We should state that, similarly to the claims made in section III of [1], the FVs of the HM16 ether comprise the linear specific (unitary) deformations ε and the angular specific (unitary) distortions γ, of the crystalline spatial net of ECs. Both ε and γ are periodic or pulsatory deformations, with frequency ν corresponding to the pulsation ω of the MPs' rotations or oscillations. These specific deformations ε and γ of a body, are known in mechanics as strains.
In the present analysis, we will discuss only the plane problem of deformations or of the strains ε and γ. Although in the general analysis of the HM16 ether model we are dealing with a spatial problem, in which the plane deformations ε and γ are replaced by the three-dimensional spatial tensors of deformations ε and γ, this does not diminish the utility of studying the plane problem, which can then be generalized.
These vibrations in ether FV + and FV − will create the effect and mode of interaction between MPs, as the electrical forces, which are specific to MP + and dif-  This specific behaviour of MP + or MP − is illustrated in Figure 3, which presents MP + and MP − disposed in opposition on a principal diagonal D1, i.e. in the two opposing quadrants of a Cartesian Oxyz reference system.
Then, MPs in contact with surrounding ECs will transmit and create FVs, via their own PVs.
The FVs in the ether are also of two types: FV + representing strain couples with the same sign (+ε and +γ or −ε and −γ) and FV − representing strain pairs with opposite sign (+ε and −γ or −ε and +γ).
Accordingly, in the HM16 ether model, the specific properties of electric charges +q and −q are intrinsically attached to MP + and to MP − by their specific mode of vibration PV + or PV − manifested mechanically.
The detailed mechanics of PVs (as oscillations, vortexes or other kinds of movements) is not yet known, but it is not strictly necessary here.
Concerning the mechanism of the transmission of vibrations of the electrical force F CC between electrically charged MPs via the ether, we advance here a hypothesis regarding the manifestation and continuous action of percussion forces p i between the involved physical objects considered to be in direct contact (MPs, ECs), in which force transmissions are produced from close to closer distances (and not at a distance), as will be presented in the next section of this paper. If an MP1 is present at a point in the ether and an MP2 is found in the area ( Figure 2), the MP2 will be subjected over its entire external surface to a contact of FVs through the ECs, caused by MP1.

I. Has et al.
This contact action appears because the vibrating ECs of ether, around MP2, have direct contact with the MP2 surface and as such will create normal stresses σ (two−way stresses, i.e. −σ compression and +σ stretching) and tangential stresses τ (with two senses +τ and −τ) over the entire surface of the MP2 ( Figure  2), with a resultant force +f or −f upon a unit area. Obviously, the phenomenon is similar for MP1 when the cause is an MP2.
In HM16, FVs caused by an MP + will create vectors (tensors in 3D) of the stresses σ and τ (or of the resultant force f) on the unit surface of an MP, disposed in the opposite directions to the σ and τ stresses (or force f) caused on the same MP unit surface by an MP − .
Adding the two types of stress vectors (tensors) σ and τ (or the resultant f) upon the whole surface of an MP will result in a +F force of attraction or a −F force of repulsion ( Figure 2, Figure 3) between the two types of MPs, depending on whether they have identical or opposite electric charges +q or −q.
Accordingly, in HM16, in the presence of MP + (Figure 2, Figure 3), the vectors of the strains ε and γ produced in the ether, as well as the resultant vectors of the stresses σ and τ (or resultant f) acting on the MP − surface, will have opposite senses, resulting in an attraction force +F on this MP − , while the same +F force will be simultaneously applied to the MP + .
Regarding the resultant force caused by vibrational waves starting from two MPs of the same sign (MP + and MP + or MP − and MP − ), the vectors of the strains ε and γ and the vectors of the stresses σ and τ (or resultant f) will have the same senses ( Figure 2, Figure 3). In turn, there will be a repulsive force −F on the Journal of Modern Physics second MP (MP + /MP − ), while the same repulsive force −F will appear on the first MP at the same time.
Consequently, the difference in the behaviour of interactions between MP − and MP + is manifested according to the above presentation.
However, it is possible that between MPs of the same sign, attraction forces arise (and between the MPs of opposite signs, repulsion forces appear) according to the completed law of F CC force [2]. For these forces, there will be different formulas, that depend on the distance l i between the MPs or between the material bodies (MBs).
However, these interaction forces, having the opposite sense to those usually currently considered, will not change the physical behaviour of the MPs, which possess a property equivalent to the actual sign of the electric charge.
The above presentation can be considered to be the mechanism of the ap- Today, in physics, the Coulomb force F C is considered to be a manifestation of interaction at distance between the so-called electric charges +q and −q. However, these charges do not exist as physical entities, only as manifestations of specific vibrations of MPs and of the HM16 ether model ( Figure 3).
Note that the sense of the F DC force between two electric dipoles will always be attraction [2], resulting in a gravitational pull exerted by the F DC via the corrected F CC force given in Euation (21) in [3].

The Occurrence Mode of the Interaction Force between Charged Microparticles in the Case of Type A HM16 Ether
To the best of our knowledge, in no classical analysis of electricity [4] [5] [6] [7] [8] is there any discussion of the possibility that the Coulomb forces F C could, in fact, be the result of the unitary forces σ or τ, which might act on the surface of MPs due to ε or γ deformations of the ether.
We note that in the classical approach, considering the Coulomb electric interaction between two MBs or MPs endowed with electric charges ±q, only the existence of concentrated F C forces of the Coulomb type are acknowledged. It is also classically understood that F C forces appear and act at a distance instantly and mutually between MB1 and MB2, with the sense of the action provided by the electric charge signs ±q. The F C force is considered to be a linear, one-directional vector, which is transmitted directly on a straight line between MB1 and MB2.
We observe that, similar to the case of the electric force, in the case of Newtonian gravitational interaction, the gravitational force F N is classically considered to act between two MBs. It is acknowledged that the force F N is also a focused force, the size of which is a function of the masses m of the bodies and of the distance 1/r 2 [7] [8].
However, in our analysis of the presence of HM16 ether, we consider that the cause of F C forces is the appearance of the unitary stresses σ and τ acting on the entire surface of an MP2 ( Figure 2). Furthermore, the σ and τ stresses do not occur instantly at a distance but are the result of the specific ε and γ deformations of the ECs around the MP2. In addition, ε, γ strains are produced by FVs from ether, caused by the PVs vibrating permanently inside the corresponding MP1, whose strains ε and γ are permanently transmitted by the MP1 directly to the ether as FVs.
In Moreover, the classical transmission of F C as a linear vector from MP1 directly to the destination (MP2) cannot occur, because it is highly likely that the F C straight line will encounter another MP, or even several MPs, existing in any MBs. Furthermore, such a classical type straightforward F C force in the form of a straight line (linear vector) cannot bypass an obstacle, or even an MP. As a result of bypassing an MB, the F C force can no longer be considered to be a linear vector or a straight line.
Due to the above situation, the spatial transmission of F CC in the form of spatial FVs, in terms of ether volume, prompts an additional argument.
In fact, the ε and γ deformations of the ether acting on MP2 are caused by the corresponding MP1 itself. Furthermore, MP1 is in an area of the ether where a quantity of energy (ΔE) has accumulated, which is manifested in the production of its own PVs as vortexes/oscillations in the EC group in the volume of the MP1 (Figures 1-3).
These PV vibrations in the EC group constituting the MP1 may in fact be periodic deformations (or even rotations/whirlpools) inside these ECs from MP1, of type ε p and γ p and with frequency ω (Figures 1-3). They are similar to the two types of specific deformations in compact MBs, known in mechanics as ε m and γ m . However, ε p and γ p do not transmit energy like ε m and γ m , owing to the ether's special (ideal) properties.
Meanwhile, MP1, by being in intimate contact with ether, precisely conveys its vibrations to it as ε p and γ p deformations, which will create periodic ε and γ deformations in the ECs of the adjacent ether ( Figure 2, Figure 3). The ε and γ deformations of ether will have an amplitude that may be different from that of I. Has et al. ε p and γ p from the MP, but they will have their own preserved frequency ω across the entire network of the surrounding ECs.
It follows that these ε and γ deformations in the whole EC network are continuously present over time, as long as there exists an MP1 at their origin and no energy consumption/transmission is involved. In addition, these ε and γ deformations in ether can only be transmitted from near to near in the whole three-dimensional ether network, with a fundamental speed c f given by ether's physical properties. These properties are currently unknown, but we only can suppose they are the properties of an ideal body having an extremely high modulus E and an extremely low density ρ, consequently resulting in a wave speed c f that is extremely high, in our estimate. The P2 terminus point of the FVs' vibration paths can only be located at the point of contact of FV1 with an MP2 surface. For the MP2, the effect of the ε and γ(ω) deformations in the surrounding ether will be manifested by the MP2's own EC deformations, i.e., ε p and γ p . These deformations imply a modification of the stress distribution σ, τ on the surface of the MP2, resulting in the F P type of interaction force described in Section 4, as a result of these stresses. Note that this F P force is the effect of FV1 through σ and τ, appearing on the entire outer surface of the MP2.
However, if the point P3 is located anywhere in the ether network in an area in which there is no MP (Figure 2), the path of FV1 will not have P3 as an end point, and so the FV1 waves will continue beyond P3. Beyond P3, there will be a continuation of the FV1 vibration transmission and of the ε and γ(ω) ether deformation. Accordingly, at the P3 point, there is no longer an end-point effect of the deformation, as in the case of P2.

I. Has et al. Journal of Modern Physics
Meanwhile, at the P3 point, which is not a terminus point, the continuation of the waveform path of FVs and the ε and γ(ω) deformations of ether can be pursued according to the Huygens principle, which also applies to ether, as shown in Section 5. This is because the ether constitutes a special, albeit material, structure, present in all space, so that anywhere there is a P3 free point, where an FV1 arrives, the Huygens principle of waves can be applied [7] [8]. According to the Huygens principle, the point P3 becomes a source of a secondary vibration FV1' of the ether, which has the same nature as FV1 but moves only in the forward direction.
However, if at the point P3 a second vibration FV2 arrives concomitantly with FV1, the travel direction of which makes an angle α with the direction of FV1 ( Figure 2), it is logical that in P3, a certain effect or phenomenon of special interference of the Huygens type appears, in which both FV1 and FV2 will have an effect upon the ether, as will be shown in Section 4.

Mechanism of the Occurrence and Calculation of the FP Force between the Considered Electrically Charged Microparticles, by a Real Interaction Force as a Result of Ether Percussions
The FVs produced by all other MPs in the ether of the universe, including those of MP1 (Figures 1-3), act upon MP2.
Any of these FVs must be considered to act on the point P2 on the surface S of MP2, via the strains ε(ω) and γ(ω), which create the stresses σ and τ ( Figure 2, Figure 3). However, it must be stated that the principle of the superposition of effects is valid, meaning that we can separately analyse the effects of MP1 (or any MP i ) upon MP2.
We can consider the plane problem further, even though the stress phenomenon occurs in space ( Figure 4).
In fact, on the entire outer surface S of MP2, there will be variations in stresses σ and τ, due to the differences in strains ε(ω) and γ(ω), which are precisely created on the surface S of MP2 by the action of the FVs of the ether produced by MP1.
We can consider the action of any FV waves on PM2, by exerting small percussions p i with the FV frequency(pulsation) ν(ω) (Figure 4). The percussions p i are the results of the variations in the stresses σ and τ produced by any vibrating ECs, due to the FVs of the ether, consisting of the deformations ε(ω) and γ(ω) with frequency/pulsation ν/ω of the FVs.
However, we consider that the small percussions p i of an EC, act with the frequency ν of the FVs (Figure 4). If the frequency of an FV is ν, the EC and the produced p i will act ν times per second, and so the percussions p i , which are discrete in time, will result in a continuous equivalent force f i of an EC, given by the    Figure 4). This is similar to the vibration energy of a harmonic oscillator. The magnitude of p i will also depend on the distance l between MP1 and MP2, with a physically justifiable variation, along with the inverse of the square of the distance l 2 , as in Coulomb's F C force Equation (1) in In Equation (1), k e is an elastic coefficient of the type of Hooke's modulus E from the field of mechanics: showing the proportionality between p i and the amplitude A m . It follows that the measurement unit (in SI) of k e will be It follows that the measurement unit (in SI) of E from Equation (2) will be: The unitary force f i given by an EC, which is considered to be a continuous force in time resulting from the percussions p i from an EC upon a MP, will be given by introducing (1) into (0), resulting in: We will make an approximate assessment of the force f i by adopting a possible  (1). We do this by utilizing its similarity to the value of the elastic coefficient E in Hooke's law for a solid crystalline body (closest to the behaviour of the Type A ether in [1]).
For steel, we have the following value from mechanics.
Now, it follows that in Equation (2c) the expression with the same measure- From Equations (2a) and (3) we find the equivalence of terms.
We obtain a value of the ether modulus E′ of about three orders of magnitude greater than that for steel from Equation (2d): 1). The case of astronomical distances. Now, we substitute in (3a) an astronomical distance l = 10 7 m, and a frequency for the FVs approximately the same as for γ rays, i.e. ν = 10 22 Hz, obtaining: ( ) Let us calculate approximately the number n of ECs, for a diameter d 1 , acting from the direction of MP1 on MP2 with diameter d 2 ( Figure 4). We consider that the relevant area for the number n of ECs in contact with MP2, is that of the spheres of MP2 and that of the large circle of the EC. This can be calculated as the ratio of the spherical surface S of MP2 in contact with ECs and the area s of the large circle around the EC sphere.
The ECs are located with distances between them of approximately 2d 1 . We consider that the diameter d 1 of an EC is d 1 = b e = 10 −27 m, according to [1]. We consider that the diameter of an electron/proton MP is d 2 = 10 −15 m. Entering these values into (4) results in the following number (pcs) of ECs in contact.
We will consider that the FV amplitude A m is of the order of 10 −3 d 1 , similar to the working strain ε in the mechanics of solids.
The effects of percussions p i (and of f i ) acting at 360˚ will be given by their projections on the Ox axis f ix , which gives a mean projection angle of 45˚, resulting in However, f ix acts in two senses on the Ox axis, giving a distribution of approximately +50% and −50% in the two senses. The forces +f ix are given by the distance l 0 while the forces −f ix are given by the distance (l 0 + d 1 ). We can obtain from Equations (4b) and (2c): i  i The total force of interaction F P upon an MP, given by the net percussions Δf i of all the ECs around it, can be calculated from Equations (4a) and (5b).
It is noted that in Equations (6) and (6a), the two forces F P and F C were obtained with values that were not equal but close, with F P < F C .
This can also be seen by considering that the value of F P obtained in (6) is the force resulting from only one electric interaction path l i out of multiple interaction paths and multiple corresponding forces F Pi , as will be seen in Section 5.
The difference is also due to the approximations of the physical parameters of ether and of the simplified forces f i , made in the above numerical calculation.
2). The case of a laboratory distance. Now, we substitute in Equation (3a) a laboratory distance l = 10 −1 m and the same frequency for the FVs, and we obtain ( ) From Equations (4c) and (7) we obtain: ( ) i i It is noted that, in Equations (10) and (10a), the two F P and F C forces obtained were not equal, but were in the same large domain, with P C F F  , which is also due to the approximations of the physical parameters of ether and the simplified forces f i , used in the above numerical calculation and which is more suitable for astronomical distances.
Comparing this result with the result from i), we conclude that the method utilized above and also in [3], is more suitable for the astronomical domain, while for the laboratory and atomic domains, it must be improved.
This can also be seen by considering that the value of F P obtained in (10) is the force resulting only from only one electric interaction path l i from among many interaction paths and multiple corresponding forces F Pi , as shown in Section 5.

Presentation of the Mechanism of the Physical Interaction between Microparticles with Electric Charges
We will now analyse the physical mechanism for producing an electrical interaction between the MPs, which we consider as being the only carriers of the positive and negative electric charges. This hypothesis implies the existence of only charged MP + and MP − microparticles, but in this paper the signs + and − will not be indicated but only supposed. Even particles which are considered neutral, such as neutrons, are composed of MP + and MP − pairs, giving zero total charge.
As a result of the electrical interactions between MPs in the ether, there will be a force that will act on each electrically charged MP. Classically, this force is considered today to be Coulomb's F C electric force.
However, this force was recently identified and named by the present authors [3] the Coulomb completed force or F CC . The F CC will be created as an effect of ε and γ ether deformations developed on the surface of the MP2, produced by the FVs originating from MP1. We can assume that the FVs are transmitted through ether, by rays, roads or paths, starting from MP1 and inclined at different angles α i (Figure 4). Next, we will denote by l i the lengths of the complete paths between MP1 and MP2, starting at radius r i . The effect of ε and γ ether deformations is also seen by changes in the stresses σ and τ acting on the surface of MP2 (Figure 2, Figure 3), which create F CC ac-Journal of Modern Physics cording to the mechanism presented in Section 2.
The FVs travel at a speed c f , with the first FVs reaching MP2 being those that travel the shortest direct path l 0 , arriving after time t 0 (Figure 4, Figure 5).
The first FVs will leave MP1 at time t i . For simplicity, we adopt t i = 0, without thereby affecting the generality of the analysis. These FVs create ε and γ (both with pulsation ω) deformations of the ether cell network, on the surface of MP2 ( Figure 2, Figure 4, Figure 5).
Similarly, the FVs travelling along path l 1 > l 0 reach MP2 after time t 1 . , , , , n r r r r  and moving along paths 1 2 3 , , , , n l l l l  , will be able to reach MP2 ( Figure 5, Figure 6). However, they left MP1 in the moments preceding t 0 and their paths must observe the condition of the total accumulated intervals Δl ti , given that we also have: , and hence (11d) However, each of these FV i vibrations creates a percussion force F P on MP2, according to Equation (6) in Section 3.
All these F P forces, when summed up over all FV i vibrations on the entire surface S of MP2, create the Coulomb completed force F CC , according to our hypothesis in Equation (21) from [3].
We can depict the elementary force of the F P percussion type in a simple and natural general form similar to Equation (6), as a variation proportional to the inverse of the square root of the ray r travelled by the FVs in ether free of any other EPs.
At time t 0 , all the F Pi forces created by FVs on MP2, leaving MP1 at times t i from Equation (11c), preceding t 0 and corresponding to the lengths l i of the travelled paths, simultaneously act directly on MP2. The different forces F Pi given by (12) oriented according to the rays r i inclined at angles α i , are as follows ( Figure 5, Figure 6).   The total force tot P F is obtained by summing all the F Pi forces from Equations (13), as they are simultaneous actions with simultaneous percussions p i in time.
However, the sum must be vectorial.
This real tot P F total force will only be obtained correctly if each force F Pi in the series of forces in (13) is of the F Pi type, i.e., is the result of a single full vibration FV i , which will produce a single set of percussions p i . This result can only be obtained if, when starting from MP1, the FV i waves are separated in time by a time period T i . This fact will correspond, on arrival at MP2, with the distance between two successive wavelengths of FV i .
On MP2b, the F 2i forces have the same formulas as in Equation (13), but with new distances l i adapted according to Figure 6.  Obviously, according to Figure 7, the deformations ε and γ(ω) in the ether, at a point P i on the surface of MP2, located at distances l a , l b , l c from MP1a, MP1b and MP1c, decrease inversely relative to the square of these distances, when taking into account the formula for F C obtained in Equation (1) from [1] and the formula of the F P force obtained in Equation (13). This behaviour of FVs in the ether, considered as a special mechanical matter, is also justified by its similarity to a common physical phenomenon depending on 1/r 2 in normal matter (solid, liquid or gas).
Therefore, the unitary forces σ and τ, exerted upon MP2 by the ECs due to the strains ε and γ(ω) created by MP1a, MP1b, …, will have to be inversely proportional to the square of these distances In Figure 7, only the effect of the direct rays r a , having the slope α 0 = 0˚ with path length l a , is represented as a grid of ECs near MP2 and as percussions p i .
However, it is necessary to analyse the effects upon MP2 produced by other possible indirect paths starting from MP1, on various radii r 1i drawn in any direction with inclination angles α i (Figures 5-7).
As the rays starting from MP1 are inclined at angle α i , they are able to reach MP2 on an indirect path in the form of an isosceles triangle ( Figure 6). This isosceles path will be the shortest path starting from MP1 at any angle α i .
Such a path can be accomplished physically by applying the Huygens principle to the midpoint C i of the radius r 1i ( Figure 5). Here, the circles (in the plane problem, but real spheres in the spatial problem) of the Huygens secondary waves with a centre at C i will allow the construction of the symmetrical radius 1i r′ of the radius r 1i .
In fact, at the point C i (Figure 5), the waves which will travel along the ray r 1i , will arrive at the same time as those arriving from the r 2i radius that started from MP2 simultaneously with r 1i . As a result, the interaction between MP1 and MP2 must be symmetrical in space and in time ( Figure 5, Figure 6).
Therefore, it can also be considered that, at point C i (Figure 5), a special interference of FVs appears, with no terminus point between the waves in the rays r 1i and r 2i . At this point, these rays mutually interchange their path, moving in the forward direction beyond point C i . This phenomenon is symmetrical due to the symmetry of MP1 and MP2, including the ether homogeneity and hence the The rays i i r r′ + , which are inclined at the angle α i , will travel the corresponding distances l i ( Figure 5, Figure 6) to MP2 in a longer time t i .
However, at a certain moment (t 0 ), various waves with rays r i , which left MP1 at various moments preceding t 0 (the shortest time), will arrive at MP2 with forces F Pi according to Equation (13), giving a total force tot P F . A question arises as to whether the overlapping of rays with various angles α i > 0 from the other MPs existing in the range close to MP2 (Figure 6), influence the final result of the interaction force F P between MP1 and MP2. The situation occurs in compact MBs (solids, liquids) when the distance d 0 between neighbouring MPs is less than 1 0 at a certain density.
However, as we discuss and analyse for two different physical situations for MP2, isolated or not isolated, the result for the F P force must be different. Consequently, it is necessary to introduce a correction reducing F P based on the MP density. Such a correction is possible using various mathematical procedures, including correcting the power of 2 in the distance l 2 .
However, in our current stage of analysis, we evaluate the interaction between any MP1 and MP2 as isolated, considering the normal contribution of all the rays with angles α ≤ 45˚ (a cone, in space).

The General Case of the Electric Interaction Problem
A calculus demonstrating the capping trend of decreasing of the F P force with distance l (according to the term lnr), is presented in the following analysis. Here, we analyse the variation of electric force F P with distance, starting from the percussions of the ether (Sections 4 and 5). The next calculation of F P is based on multiple paths l i of FVs between two MPs.
We consider the situation of two MPs, MP1 and MP2, each with an elementary electric charge ±q, located at a direct distance l 0 (Figure 8).
The electrical interaction between the two MPs occurs via all FVs, which can occur outside the direct path l 0 and travel on other deviating paths, with the general form of concave curves ( Figure 8). As a simplified particular case, we adopt here simple paths in the form of isosceles triangles, as discussed in Section 5.
In the general case, we accept that we have discretized the area of influence between MP1 and MP2 in a number n + 1 of curvilinear paths l i (Figure 8).
These paths are arranged on one half of the interaction surface, which is bounded at the bottom by the Ox axis and at the top by the line inclined at 45˚, in the case of the plane problem. Figure 8 presents the case with 6 paths (red), l 0 , l 1 , l 2 , l 3 l 4 , and l 5 , which applies to the plane problem of the interaction between MP1 and MP2.  Further, in the general case with n + 1 paths 0 1 2 3 , , , , , n l l l l l  , for the total force from the percussions tot P F , noting the differences in the paths For simplicity, we consider the length differences of paths Δl to be constant and identical between any two paths l i and l i-1 . This does not significantly affect the final result. These paths differences Δl must be at least equal to the wavelength λ, according to Section 5, but may be multiples of it, as in the general case.
In turn, the lengths of successive paths l i can be written as: In this case, we can also write: We will define the relative length difference ε as follows: We will directly calculate the magnitude of the total corrected force tot P F , adding, term by term, the F Pi forces, which arise from the various FV waves that have passed along the paths l i (Figure 6, Figure 8 It is noted that, in Equation (17), after the first term, there appears a series with n terms with the variable ratio q, where ( ) The sum of this series is too laborious to compute analytically.
Consequently, we will apply the numerical calculation method to the series sum, whose results can then be generalized as follows.

Situation with the Maximum Angle α = 45˚
a.1) Situation Involving the Initial Distance l 0 , Considering the Upper

Half of the Plan Domain, Comprising 6 Paths
In this section, we use an approximate numerical empirical calculation by considering some numerical cases with a small number of paths (n + 1). The result obtained via this procedure can be then generalized and will be useful at the current stage of the HM16 model.
In the first situation a.1) (in the parentheses in (17)), we consider the initial case ( Figure 8) in which n = 5 and there are 6 paths which are distributed initially on only half of the domain above the Ox axis, in the plane problem. For simplicity, we replace the curved paths in Figure 8 with isosceles triangular paths, according to Section 5, as shown in Figure 6 and Figure 9. In this case, for the angle α max = 45˚ from Equation (16a), we obtain: By introducing (21) into Equation (17), we have:   By introducing (23a) into Equation (17), we obtain:  In the second case in Figure 9, between MP1 and MP2 we have a distance 2l 0 and n = 5 with 6 paths, so we replace l 0 with 2l 0 in Equations (16), (17) and (21), resulting in: By introducing (26b) into Equation (17), we obtain:  Now, we double the number of paths by thickening the 6 paths considered in b.1) according to Figure 9, including the dashed paths, resulting in 11 paths for which n = 10. For simplicity, we also replace the curved paths from Figure 7 with isosceles triangles from Figure 9, according to Section 5. In this case, for the angle α max = 45˚, we obtain from (26a):     Let us now consider the complete plane situation shown in Figure 9, including the symmetrical paths relative to the horizontal centre axis Ox. This is a plane problem, doubling the terms in cases a) and b) above, which results in the following cases.  We introduce (37a) into Equation (17), resulting in: We now consider the case in which α = 60˚. We add nine additional paths giving 10 + 9 = 19 paths from 6.2.1. The interval number is n = 19 − 1 = 18.
We introduce (39a) into Equation (17) The increase in the coefficient will be 2.4674/1.962 = 1.25 = 125%. Consequently, the favourable effect of increasing the angle α from 45˚ to 60˚ is confirmed above (+25%). But the paths extension beyond 45˚ is not justified physically because it do not match with ether symmetry and homogeneity, and will be abandoned.

Case of the Spatial Problem a) General Situation
In order to be able to generalize to the spatial problem from any of the plane problems in Section 6.2, the number of paths which it is possible to consider in the spatial case will be determined.
For this, we note that the central path l 0 considers the ether as a cylinder of maximum radius r 0 in the middle part of the path (Figure 8, left). Further, the second row of cylindrical paths, each having a maximum radius r 0 , is arranged in a circular manner with the cylinders centred on the paths in the form of a circle with radius r 1 = 2r 0 . The length of this circle is ( ) The circles in row 2, which are evenly distributed, will have centres located on the L 1 circle to which a length reduction of approximately 5% is applied, to take account of the curvature of the route. Under these conditions, the reduced cylinder diameter will be The number of uniformly distributed circles will be: Similarly, the paths in the fourth row with radius r 0 can be distributed with the centres of the cylinders on the circle having a radius r 3 = 6r 0 . The length of this circle is: The number of adjacent circles, evenly distributed, will be: We now recalculate the magnitude of the force of interaction F P between MP1 and MP2, considering all spatial paths for the initial distance l 0 , starting from the We now recalculate the magnitude of the force of interaction F P between MP1 and MP2, by considering all spatial paths for the initial distance l 0 , starting from the six paths in Section 6.2. The number of these paths will be in accordance with the rule above: We now recalculate the magnitude of the force of interaction F P between MP1 and MP2, by considering all spatial paths for the doubled initial distance 2l 0 , starting from the 6 paths in Section 6.2. The number of these paths will be in accordance with the rule above: We now recalculate the magnitude of the force of interaction F P between MP1 and MP2, by considering all spatial paths for the doubled initial distance 21 0 , starting from the 11 paths in Section 6.2. The number of these paths will be in accordance with the rule above: We now recalculate the magnitude of the force of interaction F P between MP1 and MP2, by considering all spatial paths for the doubled initial distance 21 0 , starting from the 21 paths in Section 6.2. The number of these paths will be in accordance with the rule above:

Comparison of the Results from the Previous Situations from Sections 6.2.1 and 6.3, with Conclusions
In Sections 6.2.1 and 6.3, based on the analysis of the variation with distance l of the force of electrical interaction F P between the two MPs (See Section 6.1.), a series of numerical calculations was performed for some particular cases involving the distance l 0 between MP1 and MP2, the limit of angle α, the spatial distribution of paths and the path deformation ε. The initial conditions and results of calculations for the forces F P are presented in Table 1.
The purpose of this calculation was to establish a new correlation between the magnitude of the force F P and the change in the distance l between MP1 and MP2. For the distance between MP1 and MP2, multiple integer values of an initial distance l 0 , i.e. 0 0 0 , 2 ,3 , l l l  were considered.
However, it was sufficient for generalization purposes to use only the first two multiples of the distance l 0 , i.e., l 0 and 2l 0 . For n paths l i between MP1 and MP2, 6, 11 and 21 paths were considered, which proved to be sufficiently precise for generalization.  It was intended to obtain information on the F P force variation when changing both the above parameters: the distance l and the number of paths n + 1, where n represents the side paths, but not the central path.
However, in order to have comparable values for the two values of the F P force, when both of the above parameters (number n of paths and distance l 0 ) change, we firstly compare the situation for the distance 2l 0 with that for the distance l 0 (Figure 7, Figure 9). A). For the plane situation, to highlight the effect of doubling the distance l 0 →21 0 on F P , the following 4 combinations of cases a.) to f.) from Table 1 are considered and compared, analysing the forces ratio r F = F P2 /F P1 for 2l 0 and for l 0 at corresponding paths number n + 1, of 6, 11 and 21.
i). The plane situation with upward paths (u) with 11/6 paths, Case b.2) with F P2 , versus Case a.1) with F P1 , give the next values for the forces F P ratio r F :. 3.045 6.432 0.473 One remark here that increasing the paths number n + 1, from 6 to 9, gives only a little influence on r F , so the number of 6 paths utilised above is correct and sufficient.
ii). The plane situation with upward paths (u) with 21/11 paths, Case b.3) with F P2 , versus Case a.2) with F P1 , give the next values for the forces F P .
The plane situation for the paths results in a decrease in F P2 of approximately 54.4% (mean ratio r F = 0.456) of F P1 , by doubling the distance l between MPs.
However, the decrease in F C force ratio r F = according to Coulomb's law for doubling the distance l, is: This indicates a decrease in F P force with only 54.7% compared with 75.0% decrease for F C force. Even in this case, with only incomplete plane paths, result of Coulomb's law differs markedly from Equations (57)-(60).  Table 1 are considered and compared. i). The spatial situation, with 11/6 paths, Case f.2) with F P2 , versus Case e.1) with F P1 , give the next values for the forces F P ratio r F . 2 1 52.52 58.29 0.90 ii). The spatial situation, with 21/11 paths, Case f.3) with F P2 , versus Case e.2) with F P1 , give the next values for the forces F P . 2 1 199.0 210.1 0.947 One remark here that the spatial situation from B), compared with the plane situation from A) gives an important favourable influence on r F ratio, which has increased strongly, so only the spatial situation must be considered in future, which is also physically justified.
The spatial situation of paths from ii). gives an r F ratio of F P forces from relation (63) which indicate a diminution of F P forces with about 5.3% (ratio 0.947), by doubling the distance between MPs.
This also must be compared with the r F ratio of F C forces according to Coulomb's law, where doubling the distance l 0 from (61), results in: Also, in this spatial case, Coulomb's law in Equation (61) is more clearly not confirmed by result from (63).
This indicates a decrease in F P force with only 5.3% compared with the decrease of 75.0% (ratio 0.25) for F C force by doubling distance l 0 .
It appears probable, that by increasing the number n of paths beyond n = 21, one could obtain a ratio r F = F P2 /F P1 greater than 0.947 or equal to 1.0, or even greater than 1.0.
However, the result from (63) provides strong support for justifying the introduction of the term lnr in the F CC formula from [3].
This justification appears because, in this case of F CC , the effect of doubling the distance from l 0 to 2l 0 is the next: We observe in (65) that for long distances l 0 , the first term becomes zero and the ratio r F of F CC forces tends to 1.0.
Consequently, the result from (63) is almost the same as that from (65) for long distances, and hence this result justifies the presence of the term lnr in the formula for the F CC force from [1].
We remark that in any of the pairs of comparable cases above from A) and B), the condition of the equality of maximum intervals Δd 2 = Δd 1 was considered, between two successive paths, at the midpoint of the distance between the two MPs ( Figure 8). This is the physical justification of the equivalence and comparability of any pair of paths as above from A) and B). We observe that the result obtained in (63) for the electrical interaction force F P between MP2 and MP1, indicates a spatial effect of the ether which leads at a limit of r F to 1.0 in the decrease of the F P forces ratio when doubling the distance l, in contrast to Coulomb's F C forces whose ratio r F decreases to 0.25 with doubling l. But this result for F P is in concordance with the presence of term lnr in formula of F CC force from [1].
We must remark that results for all the Cases above analysed, when we doubled the distances from l 0 to 2l 0 , can be generalised for new doubled distances: 0 0 0 0 0 2 ,4 ,8 ,16 ,32 , l l l l l  , when the first forces F ratio r F , must be put at the correspondent power: In case of F C force, the first ratio r F = 0.25 put at rising powers from (66) will tend quickly to zero.
But in case of F P force, the first ratio 0.95, ,1.0 F r =  , put at rising powers from (66) will tend to approx. 1.0, as tends also the r F ratio for term lnr, (in 65), so being justified its presence.
The physical explanation for this result from (63) lies in the presence around MP1 and MP2 of ether cells (ECs), which transmit the percussion forces F P , not by a single direct path but by multiple paths between MP1 and MP2, spatially distributed, but some of them simultaneously arriving at MP2, so that their effect is superposed.
These multiple forces F P continue to arrive at MP2 at any moment in time, and hence they accumulate. The number n of paths l i increases as the distance is doubled, compensating for the diminution of the percussion forces F P by 1/l 2 term.

Conclusions and Consequences
In this paper, we analysed the role of the ether and the effect of its presence in nature, in the form of our HM16 model, with reference to the interaction between two MPs situated at the distance l. The interaction consists of forces F P , produced by mechanical percussions between MPs through the ether vibrations FVs, and depending on 1/l 2 . But the forces F P , constitute in fact the so called electric Coulomb's forces F C , between so called electric charges, which charges in fact do not exist as a physical quantity.
We have shown that the force F P , regarded as the basic elementary electric force, can be considered as the resultant force of multiple percussions p i on an Was calculated that individual F P force given by the ether percussions at astronomical distances is practically equal with F C , but at laboratory distance discrepancies appear, indicating the theory applicability at long distances. An initial assessment of the all F P forces upon an MP was made in the paper, using a simplified numerical calculation, aimed to determine the effect of all FVs travelling along various rays/paths l i through the ether, between two MPs considered today as having electric charges.
However, a series of such paths departing in a given time interval from MP1, at different angles from the direct path to MP2, but inside an angle of 45˚, will arrive at the same moment t 0 at MP2, with the result that their individual forces F Pi cumulates their effects, and so compensating the decrease of F P due to the term in 1/l 2 .
As a result of these multiple paths and of associated forces F Pi , a limited to 1.0 decrease in the final F P force was obtained in the calculations, along with an increase in the distance l, in contrast to Coulomb's law, where F C decreases to zero with increasing l, due to the term in 1/l 2 .
The effects of these multiple paths, of associated forces F Pi , and of the spatial presence of the ether, were evaluated by numerical calculus which furnished a limit to 1.0 in the decrease of the F P forces ratio, when doubling the distance l. This result is in contrast to Coulomb's F C forces, whose ratio decreases to 0.25 with doubling l, and to zero after multiples doublings. Accordingly, the possibility of including the term lnr in the F CC forces was demonstrated, because the ratio of decreasing of lnr is capped also to 1.0 for doubling l at long distances.
These results justify the possibility and necessity of introducing the term lnr into the F CC force in Equations (21) in [3], which will be maintained in our future analyses.
As a consequence of the above results validating the F CC force formula, the electrical nature of gravity based on the electrical dipole constitution of matter, is confirmed, since numerical results given by F DC force as gravity force, are very close to those gave by Newton's law force F N [3]. Our "new gravity" theory, starting from the F CC electric force in [3], involves a more complex force law of gravity, considering for this the F DC force between the electrical dipoles of all matter. The formula of F DC force is constituted by a series of powers of r x , which is different from Newton's simple gravitational force F N .
However, the new form of the gravitational force F DC will explain special phenomena observed elsewhere.

Introduction
The study of recombination processes in irradiated semiconductors as a nonequilibrium state, mainly in silicon crystals, is of significance because it determines the behavior of the electro-physical parameters for these materials: conductivity, concentration and mobility of charge carriers, and a characteristic property, which is the lifetime of the minority charge carriers. The main factors that prevent the achievement of an equilibrium state under conditions of irradiation are the energetic parameters of impurity redistributions and vacancy-interstitials, both as initially present, and those which are created during ir- [5]. These radiation defects are often responsible for the degradation of devices that contain silicon crystals, and even in the absence of degradation, the performance of a device can be impeded by the presence of defects. Those defects which have energy levels within the bandgap can act as recombination centers, and affect the carrier collection within a device. These effects can sometimes be used to advantage, since the frequency characteristics of devices may be increased by the presence of a high concentration of defects, and thus, the optical generation of THz pulses is possible, in semiconductors with sufficiently large defect densities that the carrier lifetimes are as short as a few picoseconds [4]. The radiation-induced changes in properties of the silicon used in the tracking detectors, that are employed in different areas of science and technology, mainly occur as a result of the formation and interaction of radiation defects with impurities, and these fundamentally influence the electrical parameters of the detectors, their efficiency [6] [7]. The formation of a non-equilibrium state in irradiated host crystals is well known [7] [8] [9]; however only a few initial investigations have been made using radiation applied at very short pulse durations [10] [11] [12], whereas such investigations are highly demanded both in theory and practice, particularly in space applications. The concept of a non-equilibrium state provides a basis for reinterpretation of the following radiation effects: formation of stable state of the crystal properties; differences in the radiation, thermal and time stability of materials with different impurity contents; injection of impurity atoms into the positions where intrinsic interstitial atoms are trapped; and thermal processes that occur in irradiated crystals: the latter is not feasible in the current study, because the thermal processes are not activated on the short timescale of pico-second pulse irradiation [11].
The influence of high-density pulse irradiation in current study-which results in an intensive ionization of semiconductors, with the generation of additional non-equilibrium charge carriers-is significant, since it may give rise to large non-stationary currents and electrostatic forces in electronic devices, and hence, damage them. Journal of Modern Physics In the present paper we report the results from some experiments in which silicon crystals were irradiated using pico-second duration pulse electrons, to create a non-equilibrium state. This state is characterized by recombination processes, which take place in semiconductor materials and act as trapping centers for charge carriers, and are determined by the carrier lifetimes (often those of minority carriers) at the centers. In silicon, which is an indirect-bandgap semiconductor, the recombination centers are efficient when the radiation defect level is close to the middle of the bandgap, since it has a large capture cross-section for holes and electrons [13] [14]. Hence, it can be concluded that the lifetime of the minority charge carriers (τ) is the most useful parameter for the characterization of non-equilibrium processes in irradiated semiconductors, in particular, crystalline silicon. Indeed, τ is very sensitive to the effect of irradiation, and changes by an order of magnitude at low radiation doses, although the specific resistivity shows practically no change. Below the results of measurements and discussion are given.

Experimental Procedure and Results
Although many energy levels are created in the forbidden gap of crystalline silicon when it is irradiated, the behavior of minority charge carriers' lifetime (τ) is determined by just one or two such levels. For example, the A-centre (Oxygen atom with a complex vacancy) is the dominant radiation defect formed in silicon crystals that have been grown using the Czochralski method, which is located in the upper half (E c -0.17 eV) of the bandgap and acts as an acceptor level in n-Si.
The E-centre (Phosphorus atom with a complex vacancy) is the most dominant recombination centre in silicon that has been grown using the floating-zone method (doping with Phosphorus), which is located at the E c -0.41 eV level of the bandgap and also acts as an acceptor level in n-Si [1].
In previous studies, we have shown that the application of pico-second duration pulse irradiation mainly generates A-centre type defects (in the form of clusters) in n-Si crystals [12], and in the following, we describe measurements of this center specifically. In the current paper a determination is made of the conductivity, concentration of charge carriers, their mobility, the lifetime of the minority charge carriers, in irradiated samples, and a detailed analysis of the results was made in order to elucidate the nature of the recombination centers. A standard Hall effect method [15] and a lifetime of the minority charge carriers' recombination method, based on HF (high-frequency) light-absorption [16] are used. All measurements were carried out both before and after irradiation at room temperatures.
The irradiation was performed using the linear electron accelerator AREAL at the CANDLE organization [11]. Time dependent measurements of the resistivity of the silicon crystals were made, in-situ, during irradiation, and the occurrence of recombination was observed on oscilloscope. The use of a stabilized current source and a high-speed oscilloscope, fitted with a preamplifier, enabled the ob- co-Coulomb and the frequency was 12 Hz, which corresponds to an intensity of 10 9 el/cm 2 •s. The n-type silicon single crystal samples were of dimension, 10 mm × 4 mm × 1 mm, and had a dual-cross form and a specific resistance of 120 Ω•сm. The irradiation scheme is presented in Figure 1. Following the irradiation, the main parameters of n-Si were again measured.
The lifetime of the minority charge carriers (τ) in Si was determined nondestructive by measuring the decay time of the signal of the recombination process of nonequilibrium charge carriers in a semiconductor. Non-equilibrium charge carriers are generated by illumination of a sample with a radiation wavelength of 1.05 µm, the decay process of which is detected by microwave absorption at a frequency of 10 GHz, which is recorded on an oscilloscope. The accuracy of the measurement is ±10% [16]. A significant aspect of these experiments is that all parameters were measured on the same sample, for both in-situ and ex-situ regimes.
Experimental results are given in Figures 2-4     for one month (stage duration) it increased significantly above the initial value. As a result of further irradiation and storage the peak decreases, and finally almost disappears as the degree of irradiation increases. The dose dependencies of the electro-physical parameters of samples measured at room temperatures both before and after irradiation are presented in

Discussion
The transformations of defect complexes may occur by direct interaction with vacancies, interstitials or impurities. Since the impurity concentrations often are higher than the saturation limit, a redistribution of the concentrations of the impurities may occur in sites and interstitial positions, and interact with point crystal defects to form defect complexes such as A-or E-centers, or other defects that involve the participation of the impurity atoms. Hence, the displacement of impurity atoms from the lattice sites in interstitial positions will occur in any crystal with impurity concentrations higher than at the non-equilibrium state and this process will become more important for higher concentrations of impurities and will be dependent on the conditions (e.g. dose) of the irradiation. It has been mentioned in the literature [17] [18] [19] that a divacancy type radiation defect is the main trap for free charge carriers, and its formation in irradiated Si crystals leads to an increase in the resistivity of the material which is an amphoteric center and has three energy levels in the Si bandgap. However, in our experiments, despite the fact that the resistivity increases during irradiation, three energy levels were not observed and we consider that A-centers are the main recombination centers for minority charge carriers. In any case, it is more likely that divacancy type radiation defects will be created in silicon, by irradiation with very high energy electrons (higher than 10 MeV), neutrons or protons [18] [19] [20].

Conclusions
Thus, the in-situ observation of the behavior of the electro-conductivity for silicon crystals under conditions of pico-second duration pulse electron irradiation with an energy of 3.5 MeV, suggests that the transition of the crystal from a non-equilibrium state to a "steady state", takes place by a process involving the recombination of charge carriers, and has some particular characteristic which is a consequence of their interaction times.
1) The value of the "characteristic time", , observed under in-situ condition is 10 times higher than the lifetime of the minority charge carriers ( ) D τ , measured at ex-situ, despite the fact that both decrease with increasing irradiation dose.
2) Meta-stable defects were observed, which can be annealed at room temper- 3) It was shown that at low doses of pico-second pulse irradiation, the specific resistivity change of silicon is determined by the mobility of the charge carriers, rather than by their concentration.

Introduction
Numerical approach of many-electron systems is extremely difficult computation. Density functional theory (DFT) [1] [2] is a computational quantum mechanical modeling method used to investigate many-electron systems, in particular atoms, molecules, and the condensed phases system is. The DFT proves accurate and computationally much less expensive than usual ab-initio wave function methods, such as a Hartree Fock method. However, the majority of the applications are restricted to the ground states. The exchange-correlation energy functional, which is a functional of the total electron density, is not known exactly, and thus approximate exchange-correlation energy functional must be used. Moreover, while the highest-occupied orbital energy corresponds to the negative of the ionization potential for well-behaved potentials vanishing at infinity, energies of other occupied and unoccupied orbitals have no rigorous correspondence to the excitation energies. Nevertheless, it has been shown recently [5] that the unoccupied true Kohn-Sham eigenvalues can also provide good excitation energies which the commonly used approximate density functionals usually do not satisfy. Nevertheless, numerous attempts [3] [6] have been made in this direction over the years; the ensemble density functional method [7] [8] [9] [10] [11], the method based on partitioning of the wave function, the time-dependent density functional theory (TDDFT) approach [12] [13] [14] [15]. The DFT based upon the Hohenberg-Kohn (HK) energy functional [3] [4] focuses on the solution of exchange-correlation energy and it had been used in many calculations of ground state properties an atomic system. An efficient and accurate grid method for solving the time-dependent Schrödinger equation for an atomic system interacting with an intense laser pulse has been shown by Liang-Y.Peng and A.F. Starace [16]. Instead of the usual finite difference (FD) method, the radial coordinate is discretized using the discrete variable representation (DVR) constructed from Coulomb wave functions. The DVR method has its origin in the transformation method devised by Harris et al. [17], where it was further developed by Dickinson and Certain [18]. Light et al. [19] first explicitly used the DVR method as a basis representation for quantum problems, where after different types of DVR methods have found wide applications in different fields of physical and chemical problems. The DVR method gives an idea; associated basis functions are localized about discrete values of the coordinate under consideration. The DVR simplifies the evaluation of Hamiltonian matrix elements. The matrix elements of kinetic energy can also be calculated very simply and analytically in most cases [20]. Since the CWDVR method is able to treat the Coulomb singularity naturally, it is suitable for atomic systems [16]. The Kohn-Sham equation is shown to be solved by the Coulomb wave function discrete variable representation method.
To the best of our knowledge, no one reported the solution of the Kohn-Sham equation on the ground state problem for the many-electronic atoms by the CWDVR method. Furthermore, we accurately present the ground state energies of the many-electronic atoms by the CWDVR method.
This paper consists of methodology and results obtained within the CWDVR method. We show that ground state energy values calculated by the present method are in good agreement with other precise theoretical calculations. Finally, we present conclusions. Here, atomic units (a.u.) are used throughout this paper unless otherwise specified.

CWDVR Method
In this section, we first give a brief introduction to the DVR constructed from orthogonal polynomials and Coulomb wave functions, which will be used to solve the Kohn-Sham equation for many-electron atomic systems. The DVR approach basis functions can be constructed from any complete set of orthogonal polynomials, defined in the domain with the corresponding weight function [21].
It is known that a Gaussian quadrature can also be constructed using nonclassical polynomials. The DVR derived from the Legendre polynomials has been shown by Machtoub and Zhang [22] to provide very precise results for the metastable states of the exotic helium atom. An appropriate quadrature rule for the Coulomb wave function was given by Dunseath et al. [23] with explicit expressions for the weights.
It is known that a Gaussian quadrature can also be constructed using nonclassical polynomials. The DVR derived from the Legendre polynomials has been shown by Machtoub and Zhang [22] to provide very precise results for the metastable states of the exotic helium atom. An appropriate quadrature rule for the Coulomb wave function was given by Dunseath et al. [23] with explicit expressions for the weights.  1 The simple local parameterized Wigner-type correlation energy functional [25] used for ground states:  (9) One of the main features of the DVR is that a function ( ) , R t r can be approximated by interpolation through the given grid points: The DVR greatly simplifies the evaluation of Hamiltonian matrix elements.
The potential matrix elements involve merely the evaluation of the interaction potential at the DVR grid points, where no integration is needed.
The DVR basis function ( ) j f r is constructed from the cardinal function j g as follows here the weight j ω is given by [23]: The second derivative of the cardinal function j g′′ is given, where k a is given by Equation (18) and k c . Here kinetic energy matrix elements ij D calculated using: In the Equation (15), to expand ( ) j R r in the eigenvectors of the Hamiltonian 0 H , we first solve the eigenvalue problem for 0 H after discretization of coordinate, the differential equation for this problem can be written as: here ij D denotes the symmetrized second derivative of the cardinal function that is given as, 1 , Equation (2) is then numerically solved to achieve a self-consistent set of orbitals, using the DVR method. These orbitals are used to construct various Slater determinants arising out of that particular electronic configuration and its energies computed in the usual manner.
A key step in the time propagation of Equation (9)  Thus we choose the Slater wave function as our initial state at 0 t = . Note that, the differential equation for time propagation is normalized at each time step.
Here the 152 grid points are used for the DVR discretization of the radial coordinates and 0.001 a.u t ∆ = , with 500 iterations is used in the time propagation to achieve convergence.

Results and Discussions
In this section we present results from nonrelativistic electronic structure calcula-  (5) is well accurate, compared to those of Amlan J Roy [26], which show a closer agreement with Hartree-Fock exchange energies. The "exact" correlation energies are considered for the Li, Be, B, C, N and O atoms in Table 1 due to the comparison with other results. The Wigner-type correlation energy functional likely seems to be sufficiently enough for the systems considered. Compared with Hartree-Fock results for He, Be, C and O atoms, it is nearly exact, otherwise underestimated by about 0.24% -4.94%; the Li, B and N atoms are being the worst case. The calculated correlation energy for He atom is nearly exact to Amlan J Roy, but worse results for other atoms.
We note that the primary purpose of this work is to explore the feasibility of extending the CWDVR to the solution of the Kohn-Sham type differential equation with imaginary time propagation. LDA-type xc energy functional can be easily adopted in the present CWDVR approach. Table 1 shows that the virial theorem is nearly satisfied for all atoms. The calculated kinetic energy term for the O atom is reasonably exact to Hartree-Fock, while for rest atoms there is an underestimation by 1.03% -3.44%.
The results for total energies are given in the same table separately. The results from total energies calculations of the present calculations show a good agreement with the Hartree-Fock results [3]. For the Be, and O atoms, the calculated exchange energy is nearly exact, while for rest atoms there is an underestimation by 1.02% -2.25%. This indicates that the total energy functional is well accurate, compared to those of Amlan J Roy [26], which show a closer agreement with Hartree-Fock total energies.
Results from the calculations of radial densities of atoms were created images of correspondent plots. Examples of radial density plots are shown in Figure 1 and Figure 2. In Figure 1, we report the radial density plots for beryllium. The inset (a) reports the result from the present calculation; the inset (b) shows the N. Davgiikhorol et al.  Hartree-Fock plot for comparison. Here, the radial density plot shape from our calculation is in good agreement with Hartree-Fock plot [3]. Figure 2 shows the radial density plot for carbon. We note that the radial density of carbon calculated maintain the expected shell structure and closely resemble the Hartree-Fock density, where Hartree-Fock plot is not shown.

Conclusion
In this paper, we present that nonrelativistic ground-state properties of He, Li,  [26]. The approach likely opens a road to the solution of ionization and excitation states of many electron atoms.

Introduction and Background
Before 1998, the energy density of the universe was believed to be wholly comprised of approximately 5% visible matter (i.e. the visible stars, warm molecular gas clouds and cosmic dust), an unknown quantity of dark matter within and haloed around the visible galaxies, and a comparatively negligible amount of radiation energy. Observations prior to 1998 consistent with current spatial flatness provided support for the idea that our universe is at, or nearly at, critical density. And yet there didn't seem, at the time, to be sufficient galactic and peri-galactic (i.e. virial) dark matter to account for this. However, since the 1998 discovery of dark energy, the balance of the previously missing critical density has been wholly ascribed to a mysterious energy within the intergalactic vacuum. As such, cosmological general relativity equations now incorporate a "cosmo-Journal of Modern Physics logical constant" to represent this non-matter dark vacuum energy. Furthermore, observations of galactic rotation [1], the Milky Way [2], and the cosmic microwave background (CMB) anisotropy appear to confirm that the galactic and peri-galactic dark matter mass is approximately 5 to 1 5 2 times the visible galactic matter mass. For instance, the Gaia-Hubble Collaboration reported in March 2019 that the Milky Way has a total virial mass of approximately 1.5 trillion Mʘ and a visible matter mass of 250 billion Mʘ, in support of a 5:1 dark matter to visible matter ratio. A recent consensus [3] is that the visible baryonic matter of the universe comprises 4.95%, the galactic and peri-galactic dark matter comprises 26.55%, and the dark vacuum energy comprises 68.5% of the critical density (ρ c ), now estimated to be 8.533 × 10 −27 kg·m −3 (using the 2018 Planck Collaboration H 0 consensus value, 67.4 ± 0.5 km·s −1 ·Mpc −1 ).
One of the puzzling things about the current cosmological energy density partition of approximately one-third total matter mass-energy and approximately two-thirds dark vacuum energy is the fact that, at present, these energy densities are of the same order of magnitude! This appears to be an extraordinary coincidence, because the ΛCDM concordance model stipulates that the total matter mass-energy density must have been many orders of magnitude greater than the vacuum energy density in the early universe and will be many orders of magnitude smaller than the vacuum energy density in the future of the universal expansion. How then is it that we happen to be living at just the right time in the history of the universe that these energy densities are nearly equal? Or, are we wrong in assuming radically different energy density partitions in the remote past and the distant future? These and other questions arise concerning this unexpected coincidence, which is known among cosmologists as the "coincidence problem". As an aside, there is an interesting link between the coincidence problem and the cosmological constant problem, but that is beyond the scope of the current paper. In the history of progress in our understanding of the universe, unexplained coincidences have often been signposts leading to a deeper understanding.
One of the questions one might ask concerning the coincidence problem is whether there could be some systematic error in our quantitation of dark matter that, once corrected, could point to a 50%/50% energy density partition (i.e. 50% total matter mass-energy and 50% dark energy). This could be achieved, for instance, if approximately one-fifth of the critical energy density currently being classified as a component of the vacuum energy is actually hidden dark matter within the deep intergalactic vacuum. This has long been one of the predictions of the Flat Space Cosmology (FSC) R h = ct model [4] [5]. In fact, this model stipulates a perpetual 50%/50% energy density partition as a falsifiable prediction. This is in stark contrast to the approximately 31.5%/68.5% energy density partition now widely accepted, within tight constraints, and claimed by proponents of the ΛCDM concordance model (see [3]). Over the last 40 years, standard inflationary cosmology has been repeatedly subject to ad hoc adjustments from E. T. Tatum Journal of Modern Physics new observations, and has made relatively few falsifiable predictions in comparison to the new FSC model [6].
Once-tight constraints, however, can sometimes be broken when new discoveries and/or theories arise. Prior to the discovery of dark energy, for instance, the very idea of a cosmological constant was thought to be highly improbable. A new case in point, as emphasized in this paper, has been the assumption that the dark matter within and haloed around the galaxies is all, or nearly all, of the discoverable dark matter within the universe.
There is now a new and very plausible theory of dark matter which should loosen the ΛCDM density partition constraints considerably. It is the purpose of Atomic Hydrogen) theory of dark matter [7] can be used to identify dark matter within the deep intergalactic vacuum which should be removed from the dark energy ledger column and placed under the total matter mass-energy ledger column.

Results
It has long been known that the intergalactic vacuum is exceedingly sparse with respect to matter. However, it is not entirely empty! Observational 21-cm studies of the cold ground state neutral atomic hydrogen within the intergalactic vacuum, of a similar nature to those made of the interstellar vacuum of the Milky Way, have determined an average density of approximately one atom per cubic meter [8]. Thus, the average cold ground state neutral atomic hydrogen density of the intergalactic vacuum is approximately 1.67 × 10 −27 kg·m −3 . This is one million times less dense than that within the Milky Way galactic disc (1.67 × 10 −21 kg·m −3 ). Nevertheless, one can readily see that the following ratio equation is relevant to the purpose of this paper: This is approximately one-fifth of the observed critical density!

Discussion
The ratio equation given in the Results section shows that the currently-observed average density of intergalactic cold ground state neutral atomic hydrogen is 19.5% of the critical density determined from the 2018 Planck Collaboration report (see [3]). This is a component of the cosmic total matter mass-energy and should not be considered as a component of the vacuum dark energy. By this calculation, one can readily see that 19.5% of the current critical density has been mi- With correct re-apportionment of this additional dark matter to the total matter mass-energy, the cosmic critical energy density partition now becomes approximately 51% total matter mass-energy and 49% dark energy. The current negligible radiation energy contribution to the critical density does not affect the above approximations. Thus, the results of this new C.G.S.I.S.A.H. calculation may be considered to be well within the acceptable margin of observational error, and in support of the FSC prediction of 50%/50%. It should be remembered that there is no cosmological coincidence problem in the FSC model, which stipulates perpetual equality of the cosmological matter mass-energy and non-matter vacuum energy.

Summary and Conclusions
The Furthermore, this discovery of a universe comprised of 50% matter mass-energy and 50% dark energy suggests that the recent resurrection of the Dirac sea idea [9], as it may pertain to positive and negative energy eigenstates (rather than positive and negative mass eigenstates) within the FSC model, may be a guide to understanding matter and dark energy as the perpetual yin and yang of universal expansion. This is to be expected in a finite perpetual matter-generating cosmological model which begins from a zero-energy state and follows conservation of energy [10].

Dedications and Acknowledgements
This paper is dedicated to Dr. Stephen Hawking and Dr. Roger Penrose for their groundbreaking work on black holes and their possible application to cosmology. Dr. Tatum also thanks U.V.S. Seshavatharam for his co-authorship of the seminal papers of FSC, and Dr. Rudolph Schild of the Harvard-Smithsonian Center for Astrophysics for his past support and encouragement.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.

Introduction
Quantum field theory has been applied in the research on critical phenomena, for example, by means of renormalization group theory, perturbation theory, and Feynman graphs to compute critical points and critical exponents [1] [2] [3].
The renormalization group itself is a symmetry group, in which transformation we found coexistence of ordered blocks and disordered blocks. On this basis, the concept of isomorph spin is set up, and then some hidden symmetries have been revealed [4]. The two types of blocks make us recall the coexistence of protons and neutrons, and further, we think that there must be a gauge field to maintain n + -blocks are ordered blocks, with which coexisting the n − -blocks are disordered. Each lattice point is similar to a fermion, and they can form an n + -block spin behaving as a fermion. We call the n + -block as fermi-block. An n + -block spin has Z (coordinate number) nearest neighbor n + -blocks, and these Z coupled block spins have different energy states due to Pauli exclusion principle, leading to that this block spin has Z different spin states at least. This is the reason why we yet introduce hyperspin Y f for the blocks, besides the isomorph spin I f [4]. Furthermore, a block is a set of lattices; it is these lattices that give the block a peculiar ability to possess more states a particle not having. For example, two states of the non-spin interaction, which will be illustrated in Section 2.
The spin interaction is essentially electromagnetic interaction in which spins take part. Since each block contains charged lattice spins, the block always has nonzero charge although it may have zero-spin. The so-called non-spin interaction is different from neither weak interaction nor strong interaction; it is just an electromagnetic interaction in which no spin involve. Particle physics tells us that the electromagnetic field is, in nature, an Abelian gauge field introduced to maintain the invariance U(1) of electromagnetic interaction. Non-Abelian gauge field is available for those interactions to keep the invariance of SU(2), etc. [5].
Studying continuous phase transition, however, we see that there is localized SU(2) symmetry describing non-spin electromagnetic interaction, indicating that non-Abelian gauge field is also suitable for electromagnetic interaction. For this reason, we neglect mass to explore the gauge field in this paper.
In Section 2, at first, we introduce a gauge field on a finite hierarchy of the self-similar transformation at temperature T, and get W + -blocks, W − -blocks, and Z 0 -blocks carrying the non-spin interaction like bosons. We call them as bose-blocks. Then, we discover Fermi-Bose duality for blocks, we've revised the explanation of the excitation states and the intermediate states of reference [4].
In Section 3, we discuss gauge field relating to the block type inversion when temperature charges and system entropy evolution, finding constant k B ln2 and minimization entropy reduction. We explain why the scaling law only occurs nearby the first critical temperature T cf and there is degeneracy of critical temperatures for experimental specimens; point out that the thermodynamic dark parameters will play important roles in the further research on phase transition. Section 4 is conclusion remark.

Gauge Field and Three Conservation Laws
On the r-th hierarchy, 0 r < < +∞ , at temperature T m , the state of an ordered its conjugate state is ψ . The Lagrangian density of the interaction in which a gauge field participates is expressed by [6] ( ) 1 The kinetic energy term for the non-Abelian field is ( ) Journal of Modern Physics etc. thermodynamic dark parameters, which affect the system's property as much as the conventional parameters do.

Conclusion Remark
It is first found that the non-Abelian gauge field is suitable for the non-spin electromagnetic interaction in the continue phase transition, in which the blocks illustrate Fermi-Bose duality. The gauge field maintains the self-similar transformation and makes system take the ordered path to minimum entropy reduction. The degeneracy of critical temperatures interprets the reason why the scaling law shows only nearby the first critical temperature for the real experimental specimens. The concept of thermodynamic dark parameters associated with the non-spin interaction is proposed firstly.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.