A study on electron deep orbits by quantum relativistic methods

CHAPTER

A study on electron deep orbits by quantum relativistic methods

16

J.L. Pailleta, A. Meulenbergb a

Aix-Marseille University, Marseille, PACA, France Science for Humanity Trust, Inc., Tucker, GA, United Statesb

Introduction The introduction of relativity into the quantum mechanical description of the Bohr atom presented some problems. Both the Klein-Gordon and the Dirac equations predicted levels that were consistent with the nonrelativistic work and with experimental observations. However, they also predicted levels that had never been observed and therefore have been called anomalous. This disconnect between theory and observation has produced a controversy that has sporadically erupted over many decades. Interest in one set of these anomalous levels has been raised in the last two decades [1, 2] as a result of the advent of Cold Fusion and its search for a theoretical basis to explain the presence of one or more electrons spending more time between two close-spaced nuclei.

Interest of the electron deep orbits (EDOs) for the low-energy-nuclear reaction (LENR) It was recognized early in the search that, to fit the experimental observations for bringing together two hydrogen atomic nuclei (protons or deuterons) to fuse would involve increasing the electron density between them. A probable theory for this action involved the set of deep-electron orbits (with binding energies into the 500 keV range) predicted by the Klein-Gordon and the Dirac equations. These electron deep-orbits (EDOs) are predicted by the simple versions of the relativistic equations for the hydrogen atom to have orbital radii in the femtometer range. As such, if occupied, their bound electron(s) would form femto-atoms [3, 4] and allow pairs of protons or deuterons (assuming one of either pair to be a femto-atom) to readily form a femto-hydrogen molecule or molecular ion and/or fuse. The fusion of two hydrogen nuclei to form deuterium or 4He is the foundational observable in Cold Fusion. The femto-molecule or -molecular ion becomes the basis for the observed transmutations in Cold Fusion. In brief: A hydrogen atom with its electron in a deep orbit can greatly enhance fusion with another nucleus because: 1. It is electrically neutral, at the femto-meter level, which facilitates the crossing of the potential barrier posed by the bound-electron cloud of another atom. 2. Its size, of order a few femto-meters, also allows its proton to penetrate the nuclear Coulomb potential sufficiently to reach the range of the strong nuclear interaction. Cold Fusion. https://doi.org/10.1016/B978-0-12-815944-6.00016-6 # 2020 Elsevier Inc. All rights reserved.

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Chapter 16 A study on electron deep orbits by quantum relativistic methods

During the nuclear fusion process: 1. Rapid energy transfer from the nucleus to a substrate/lattice is possible thanks to the strong nearfield coupling between excited nuclear components and the deep-orbit electron and thence to the lattice. 2. This rapid de-excitation avoids the nuclear fragmentation, particle emission, and emission of energetic EM radiations characteristic of neutron activation. Transmutation without radiation is an essential feature of the low-energy-nuclear reaction (LENR) observed during the experiments. 3. The nuclear energy absorbed by the deep-orbit electron is rapidly transferred to the adjacent lattice atoms via near-field coupling to their bound electrons and finally by enhanced internal conversion in which the deep-orbit electron leaves with the remaining nuclear energy. 4. Moreover, as the process takes place in condensed matter, the deep electron can transfer energy throughout the lattice network, through a chain of interactions between many electrons.

Starting point of our study There are, in the literature, various and numerous theoretical methods to define a state of the hydrogen atom with EDOs. Some authors use the term hydrino for denoting the special deep-orbit hydrogen states owing to the work of R.L. Mills [5] on the hypothetical existence of H atoms with orbit levels under the Bohr ground level and where the values of orbit radii are fractional values of the Bohr radius. We do not use this term, a physical concept specifically attached to the cited work, because it is not deduced from (standard) relativistic quantum equations and does not coincide with the deep level model that we are addressing. Our starting point has been the analysis of works on EDOs obtained as singular solutions of relativistic quantum equations applied to the hydrogen atom. With the relativistic quantum equations habitually used in the literature for computing the bound states of the H atom, we note that there is, in the relativistic form, a crossroad with a choice of value or a choice of sign for a square root in a parameter. According to the path chosen, the resolution process leads either to the usual “regular” solution or to one called an “anomalous” solution, usually rejected because of its singularity at r ¼ 0. These anomalous solutions, having a mean orbit of order femto-meter, are very near the nucleus and determine electrons with high binding energy. At first sight, it seems normal that such electrons be relativistic; nevertheless, we subsequently showed [6] that Special Relativity is essential to obtain EDOs. The first thing to do was to analyze the criticism existing in the literature against the singular solutions.

Arguments against the EDO states and possible solutions Mathematical arguments against the anomalous solutions of the relativistic equations have dominated the discussion of this issue for over 50 years. However, by acceptance of the physical reality of a nonsingular central potential within a nuclear region, these objections no longer pertain. Indeed, most arguments against EDOs are mathematical in nature and are based on the singularity of the Coulomb potential at the origin. The arguments against EDO states, while assuming a singular Coulomb potential in 1/r, are exposed in detail in Ref. [7], as well as the possible solutions to these questions, and in a more developed way in Refs. [8, 9]. They concern only the radial solutions of the quantum equations:

Introduction

303

 The wave function has a singular point at the origin.  The wave function cannot be “square integrable.”  The “orthogonality criterion” cannot be satisfied. We give below some quick explanations about these arguments and their solutions. The wave function has a singular point at the origin. The spatial part of the solutions of the radial equation, in the most general form, has several factors:  One factor is a decreasing exponential exp(r) such that exp(r) ! 0 when r ! +∞.  Another one is ∝ 1/rs with s a real number, due to the form of the Coulomb potential.  And there can be a further one in polynomial form. In the case of the so-called anomalous solutions, the exponent s of the factor in 1/rs is positive, then the radial function R(r) ! ∞ when r ! 0 and the wave function ψ(r,θ,φ) does not obey a boundary condition. This problem comes from the expression of the Coulomb potential in 1/r. But, if considering the nucleus has a finite size 6¼ 0, this problem disappears automatically. The wave function is not “square integrable.” If the wave function is not integrable, it cannot be normalized in the entire space. This case results essentially from the behavior of the wave function ψ at the origin and not for r ! ∞, thanks to the decreasing exponential. According to the Jacobian, the Ð Ð spherical coordinatesÐ and the corresponding norm of ψ is defined by jj ψ(r,θ,φ)jj ¼ jψ j2sinθ r2dθ dφ dr ¼ j Y(θ, φ)j2dΩ jR(r)j2r2 dr, Ð where Ω is the solid angle. Since the spherical harmonics are normalized, one has only to verify that jR(r)j2r2dr is finite, and this depends only on the behavior of j R(r)j2r2 at the origin. Nevertheless, we can cite the work of Naudts [2], where an EDO state is found by using the KleinGordon (K-G) equation and the corresponding solution is square integrable. In his paper, the K-G equation has the following form: ðiħ∂t  V Þ2 Ψ ðr, tÞ + ħ2 c2 ΔΨ ðr, tÞ ¼ m2 c4 Ψ ðr, tÞ

€ Historically, this equation was called the relativistic Schrodinger equation. Note the author considers only the case where the angular momentum is null: while expressing the Laplacian Δ in spherical coordinates, he eliminates the term involving the angular momentum operator L, so the classical term in l(l + 1) representing the eigenvalues of L2, does not occur in his computation. By means of a suitable ansatz, the author finds a regular solution and an “anomalous” one. For this latter one, he obtains an electron total energy E  mc2α  3.73 keV, where α is the electromagnetic coupling constant (also called “fine-structure” constant), with α  1/137. So the binding energy is BE  mc2(α  1)   507.3 keV. And, he shows that the conditions satisfied by his ansatz guarantee square integrability of the wave function solution. We can see that the origin is a singular point for the wave function; but, Naudts argues against this problem by saying that the nucleus is not a point. Its charge is “smeared” over a distance of about 1 fm. Solving the equation with a smeared out Coulomb potential would produce a solution not diverging at the origin, but with certain minor changes on the deep orbit state. Finally we note two things:  If the singular point at the origin (in the “Coulomb” potential or the solutions) is suppressed, the wave function is automatically square-integrable.  The square-integrability cannot be obtained for the Dirac equation in a pure Coulomb potential.

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Chapter 16 A study on electron deep orbits by quantum relativistic methods

This point is quickly proven by Naudts on a simplified form of radial Dirac equations. The case of the Dirac equation, much more interesting and fruitful, is treated further. The “orthogonality criterion” cannot be satisfied. This argument is more subtle. The Hamiltonian, representing the total energy, has to be a Hermitian operator in order for its eigenvalues to be real. This implies the following condition: eigenfunctions corresponding to distinct values have to be orthogonal. In Ref. [10], the author indicates the following conditions to obtain orthogonality:  For the K-G equation: (uk∗(duk1/dr)  uk1 (duk∗/dr)) ! 0 when r ! 0, where uk(r) ¼ r R(r) ∗ ), when r ! 0  For the Dirac equation, a condition on the components f,g: (f k∗ gk1  fk gk1 Note that the components f, g appear as the Dirac equation is transformed into a system of two first order differential equations for radial solutions. From some results indicated in this paper, one can deduce that for the light atoms, the orthogonality criterion is satisfied for l ¼ 0, and thus for the EDO solution found in Ref. [2]. On the other hand, for the Dirac case, the author indicates that only the regular solutions for the component functions f, g can satisfy the orthogonality. We can also note several works [11, 12] on the self-adjoint extension of operators for potentials with singularity. In particular, the first cited work explicitly shows that, for the Klein-Gordon equation in the case of the pure Coulomb potential, the “singular” solutions satisfy the orthogonality condition and also satisfy the boundary condition: when r ! 0, lim u(r) ¼ u(0) ¼ 0. We can also observe that the orthonormal set of eigenfunctions solutions of any quantum equation has to satisfy the completeness relation, often represented by a closure relation, see, e.g., Ref. [13]. The question of the orthogonality for the Dirac equation is solved later in our section “The question of orthogonality of the solutions, and the boundary conditions.” We end here the analysis of the most common arguments against EDOs and the possibilities to solve these questions with a criticism that is of a physical nature rather than mathematical and is not related to the singularity of Coulomb potential. It concerns the behavior of the binding energy while the coupling strength decreases. Indeed, it can be very instructive to make the following “thought experiment” to imagine variations of the coupling constant α and observe consequences on eigenfunctions of the quantum equation. Doing this in Ref. [14], Dombey points to a very strange phenomenon concerning the EDO solutions of the Klein-Gordon as well as those of a two-dimensional Dirac equation: while considering the expression of the electron total energy E  mc2α for the “anomalous” solution giving an EDO (as already seen above, for the solution found in Ref. [2]), he observes that E decreases and tends to 0, when one makes α decrease and tend to 0. This implies that the absolute value of binding energy j BE j ¼ mc2  E increases and tends to mc2. In fact, we think this result is obtained in the context of an ill-defined system, uniquely on a purely mathematical basis. From a physical point of view, we can see the coupling constant α is actually entangled with several fundamental constants, in particular, the Planck constant, the velocity of the light, and the elementary electric charge. So, modifying α without caution can certainly lead to paradoxical physical results (e.g., letting α go to zero means that the charge does also; thus, there are no bound states and no BE). There is another well-known example of changing a physical constant to obtain results: the nonrelativistic limit of a relativistic theory can be obtained if one lets c tend to infinity, and thus the relativistic coefficient γ becomes 1 fοr any speed. However, as noted in Ref. [15], if doing this on the Dirac operator in an electromagnetic field, one has to proceed carefully because of terms

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such as mc2, which would tend to infinity, and as the term (e/c)A that would turn off the vector potential A if c tends to infinity. Note that the author is led to develop specific techniques and to define some concepts needed to account for the nature of the so-called c-dependence of the Dirac operators. A second physical argument against the EDOs is based on the HUR. According to the HUR, confinement of an electron to the nuclear region would require it to have kinetic energy on the order of 100 MeV. This would appear to be impossible since the Coulomb potential binding electrons to the nucleus are only on the order of 1 MeV at the nucleus. A surprise answer to this argument comes from relativity and is detailed in Section “Relativistic confinements and the question of the Heisenberg uncertainty relation (HUR).”

The works of Maly and Va’vra on “DDL’s” The anomalous solutions of the Dirac equation We analyzed first a specific work of Maly and Va’vra in Ref. [1] on deep orbits as solutions of the Dirac equation. These orbits were named by those authors Deep Dirac Levels (DDLs) because they present the most complete solution and development available and include an infinite family of EDO solutions. The Dirac equation for an electron in the central “external” Coulomb field of a nucleus, can take the following form:   iħ∂t + iħc α:r  β mc2  V Ψ ðt, xÞ ¼ 0

where α and β represent the Dirac matrices. α is in fact a 3-vector of 4  4 matrices built from the wellknown Pauli matrices, and V is the Coulomb potential, defined by  e2/r. Maly and Va’vra refer to and use the method developed in Ref. [16], by starting with the system of radial equations obtained after separating the variables in spherical coordinates. In fact, the authors consider the general case of “hydrogen-like” elements, by taking V ¼  Ze2/r (like in the cited reference) with Z ¼ the atomic number. But here, we only consider the hydrogen atom (H or possibly an isotope) by letting Z ¼ 1. During the solution process with an ansatz, the following condition must be satisfied by a parameter occurring in the ansatz: s ¼  (k2  α2)1/2. The scalar α represents the coupling constant (not to be confused with the vector of Dirac matrices α occurring in the Dirac equations above). If taking the positive sign for s, one has the usual “regular” solutions for energy levels, whereas with the negative sign, one has the so-called anomalous solutions. The general expression obtained for the energy levels is the following: " E ¼ mc 1 + 2

α2

# 1 2

ðn0 + sÞ2

So, while considering the “anomalous” solution, with negative s, the expression of E reads 3 1

2 6 E ¼ mc2 41 + 

α 7 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 5 0 2 2 n  ðk  α Þ 2

2

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Chapter 16 A study on electron deep orbits by quantum relativistic methods

where n0 is the radial quantum number, while k is the specific Dirac angular quantum number, which can take any integer value 6¼ 0. It is important to note the following fact that all solutions expressed by E do not correspond to deep orbits, but only the ones satisfying the relation n0 ¼ j kj, i.e., equality between the radial quantum number and the Dirac angular quantum number in absolute value (because k can be <0). Indeed, we can see that if j kj ¼ n0 , the subexpression D occurring at the denominator of the expression E, D ¼ n0  (k2  α2)1/2 becomes D ¼ n0  (n0 2  α2)1/2, which is very small since D  α2/2n0 , and so E  mc2α/2n0 . Then jBE j  mc2 (1  α/2n0 ) and j BE j is close to the rest mass energy of the electron, 511 keV. Note that, since k cannot equal 0, then neither can n0 . Here, we summarize some features of the “anomalous” solutions of the Dirac equation. • •

•

if j kj > n0 , the solutions correspond to negative-energy states. if j kj ¼ n0 , these special solutions correspond to positive-energy states, and they are the only ones providing EDOs. Moreover, we can observe, in the energy tables in Ref. [1], the following property: the binding energy in absolute value, j BE j, increases when n0 increases. This is a behavior opposite to that of the “regular” states. Note that one can also directly deduce this property from the algebraic expression of j BE j  mc2 (1  α/2n0 ) if jkj < n0 , each solution corresponds to a positive energy state, but E is very close to the energy of a regular level corresponding to a value of the principal quantum number N taken equal to n0  j kj.

We named these solutions “pseudo-regular.” We will come back to the question of the sign of solutions. Note that in Ref. [1], the authors also give EDOs as “anomalous” solutions of the relativistic Schr€ odinger equation (i.e., Klein-Gordon equation), but contrary to Ref. [2], they impose no restriction on the angular quantum number l. For doing this, they use the method developed in Ref. [16]. In a similar way as for the Dirac equation, there is a condition to be satisfied by a parameter noted as s occurring in the ansatz mentioned above, but now: s must satisfy the equation s(s + 1)  l(l + 1)  α2 ¼ 0. This quadratic equation classically has two roots: s ¼ ½  [(l + ½)2  α2]½. When choosing the sign “+,” one obtains the “regular” solutions, whereas the sign “” gives the “anomalous” solutions. For the latter, the expression of energy levels is the following: 3 1

2 6 6 E ¼ mc2 61 +  4

2

α  2 1 1 n0 +  l + α2 2 2 2

7 7

2 7 5 1 2

where n0 is the radial quantum number and l the angular number. As for the Dirac equation, all solutions expressed by E do not correspond to deep orbits, but only the ones satisfying the relation n0 ¼ l, i.e., equality between the radial and the angular number. One can show this equality drastically reduces the expression in the#denominator of the fraction occurring in E. " " # 2 2 1 1 1 1  α2 2 and l +  α2 2  l + 12  α2/(2l + 1) Let λ2 be this expression, with j λj ¼ n0 + 12  l + 2 2 since α ≪ 1, so j λ j  n0  l + α2/(2l + 1); then n0 ¼ l implies jλ j α2/(2l + 1) and again, as α ≪ 1, one

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has E  mc2α/(2l + 1), thus j BEj  mc2 (1  α/(2l + 1)), i.e., jBEj is close to the rest mass energy of the electron. Moreover, one has similar inequality relations, as in the case of the Dirac solutions, between the angular and the radial number, which determine solutions corresponding to negative energy states or to positive energy states, where the values are close to the energy of regular levels.

The deep orbits, as solutions of the Dirac equation with a corrected potential for a nucleus of finite size In a second work [17], Maly and Va’vra determine again the wave functions of EDOs for hydrogen-like atom solutions of the Dirac equation. But this time, they consider the nucleus not to be point-like, and thus the potential inside the nucleus is finite at the origin r ¼ 0. We have seen, in the previous section on criticisms that this allows eliminating the problems related to the singularity of the classical Coulomb potential in 1/r. To solve the Dirac equation, the authors start with a method (Fluegge [18]) different from that of Schiff [16] used in the first paper. The process includes again separate angular and radial variables and leads again to a system of coupled first-order differential equations on both radial functions f(r) and g(r). The exact form of this equation system is dependent on some choices made in the variable separation process. In the method of solution of Fluegge (p. 195), the equation system is transformed into a second-order differential equation, a Kummer’s equation. The general solutions of this equation take the following form, with confluent hyper-geometrical series requiring suitable convergence conditions:    1 r s + p r g ¼ Crs1 er=a 1 F1 s + p, 2s + 1; 2  F s + p + 1, 2s + 1; 2 2 a k + q1 1 a    s+p  i r r s1 r=a f ¼  Cr e F s + p, 2s + 1; 2 F s + p + 1, 2s + 1; 2 + 1 1 2μ a k + q1 1 a

The parameters a and μ are parameters including the energy E; p and q are defined by means of μ. It is unnecessary to write out all their expression because our discussion will focus on the solutions “inside” the nucleus. The parameter s is again s ¼  (k2  α2)1/2, as seen above, and s + p ¼  n0 (i.e., minus the radial number). To solve the equation with a nucleus of finite size 6¼ 0, the authors carry out the following steps: – To choose a radius R0, the so-called matching radius, delimiting two spatial domains: an “outside” one, where the potential is correctly expressed by the usual Coulomb potential, and an “inside” one, where the potential cannot be expressed by the 1/r Coulomb potential. Of course, this choice may seem arbitrary, but it takes physical meaning if one chooses a value R0 close to the “charge radius” Rc of the nucleus. For example, for a hydrogen H atom, the nucleus is reduced to one proton and this one has Rc  0.875 F from CODATA [19]. So one can reasonably choose 1F < R0 < 1.3 F. – To choose a “suitable” expression for the “inside potential.” It is again an arbitrary point, but we observed (see further, results of computations) this choice has weak influence on the numerical results that interest us, especially the value of mean radius as a function of the radial number n0 . – To satisfy continuity conditions at the matching radius R0 for connecting the inside and outside potentials. The potential chosen by the authors is derived from the Smith-Johnson

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Chapter 16 A study on electron deep orbits by quantum relativistic methods

potential, corresponding to a uniformly distributed spherical charge, whose expression is the following: V ðr Þ ¼ 

  2 3 1 r2 Ze + β0  R0 2 2 R20

where the continuity condition for the potentials in R0 implies β0 ¼ 0. – To solve the system of radial equations for the “outside potential,” i.e., Coulomb potential that gives the outside solution composed of two components: functions fo and go. Here, the “outside” functions fo and go are respectively, the functions f and g expressed above while choosing s < 0, i.e., s ¼ (k2  α2)1/2, to have “anomalous” solutions and by putting j kj ¼ n0 to discriminate the special solutions corresponding to EDOs. – To solve the system for the chosen inside potential, that gives the “inside solution” composed of two components: functions fi and gi. – To satisfy continuity conditions for connecting the inside and outside solutions.

Ansatz used for finding the “inside” solutions and continuity conditions The choice of ansatz is a very important element for finding the solutions fi and gi of the system of radial equations. Moreover, its expression is determinant to satisfy the continuity condition. A complete analysis of this question and is given in Ref. [9]. Here, we give only indications of the solution. In their paper [17], the authors put the ansatz in the following form: gi ¼ ArSi 1 G2 ðr Þ fi ¼ iBrSi 1 F2 ðr Þ

where G2(r) and F2(r) are power series, i.e., G2 ðrÞ ¼ a1 r + a2 r 2 + a3 r3 + ⋯ and F2 ðr Þ ¼ b1 r + b2 r 2 + b3 r 3 + ⋯,

However, one may consider approximations of these series by polynomials, by taking into account the following facts:  fi and gi must be defined for r < R0  for r < R0, very small, the higher-power terms vanish as the degree increases. The authors use polynomials of degree 5. The classical method used, after inserting the ansatz into the equations, allows one to determine the exponent si and the polynomial coefficients in order to obtain the solutions. For doing this, the authors deduce a couple of interdependent recurrent formulas for computing the coefficients of both power series G2(r) and F2(r). This leads to “inside solutions” fi and gi, but it seems the information given in the paper is incomplete, or more precisely, the chosen ansatz is not complete and it does not contain enough free parameters to satisfy the continuity condition for both couples of functions (fi, gi) and (fo, go) in R0. In fact, useful information was included in another paper by the same authors, referenced as “to be published” but never published.

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309

To resolve this problem, we looked for a more complex ansatz including an additional free real parameter λ, necessary to connect in a suitable manner the inside and outside functions, where the series/polynomials have the following form: G2 ðrÞ ¼ a1 ðλr Þ + a2 ðλr Þ2 + a3 ðλr Þ3 + ⋯ and F2 ðr Þ ¼ b1 ðλrÞ + b2 ðλr Þ2 + b3 ðλr Þ3 + ⋯

The continuity conditions {gi(R0) ¼ go(R0), fi(R0) ¼ fo(R0)} lead to a system of two algebraic equations. We showed in Ref. [9] that, for any degree k of the polynomials, the maximal power of λ in this system of equation remains constant and equals 2, and so this system provides suitable solutions.

The question of orthogonality of the solutions, and the boundary conditions If we consider a couple of “inside” functions (gi, fi) and because of the initial conditions of the recurrence relations, the term of a minimal degree of the polynomial gi is jk j and for fi it is j kj  1. We saw that in Ref. [10] an orthogonality criterion (Section “Arguments against the EDO states and possible solutions”oncriticism)is expressed by means of the condition P ¼ (f k∗gk1  fk1 gk∗) ! 0 when r ! 0, for any k and k1. One can see that each term of the subtraction is a polynomial whose term of minimal degree d ¼ j kj + j k1 j  1. As k and k1 are 6¼ 0, one has d  1, and the polynomial P cannot contain any constant term. We can deduce the corresponding global solution (including the internal and the external ones) satisfies the orthogonality condition. Next, if we look at the boundary conditions, expressed by gi ! 0 and fi ! 0 when r ! 0, we can see this property is verified for any j kj  1. Now, we have to consider a further question arising from the existence of both usual and peculiar eigenstates (corresponding to the EDOs) in the same system, i.e., the total set of solutions of the Dirac equation (for a hydrogen atom): the usual ones and the peculiar (corresponding to the EDOs). This leads to mathematical problems about the total Hilbert space including both sets of states: the two sets of states are not orthogonal to each other, and the system is “over-complete.” To resolve this problem, one can adopt the solution used in a similar situation (that of the states of positronium) by Crater [20]. He introduces an observable quantum number, noted ζ and called “peculiarity,” corresponding to new operator ^ζ which allows him to specify the usual eigenstates with ζ ¼ 0 and the peculiar states with ζ ¼ 1. The details, to express and expand a general wave function by this means, can be found in the cited reference.

Results obtained by computations of the DDL wave functions for modified potentials, further developments, and discussion The value of the mean radius is an essential parameter for the LENR, since the range of the strong nuclear force is on the order of femto-meters (fm or F) and quickly decreases at distances >3 to 5 F. The mean radius of its electron orbitals determines the “size” of the (femto-)atom and the value of its repulsive radius. This radius can be estimated to be approximately [17] the value where the electron probability density drops to 1/10 of its peak value.

Computation process for orbital mean radii

Summarily, the computation process for mean orbit radius for a given value of n0 ¼ j kj includes the following steps:  To determine both couples (fo, go) and (fi, gi) of respective outside and inside solutions. At this step, the four functions fo, go, fi, and gi include parameters still to be determined

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Chapter 16 A study on electron deep orbits by quantum relativistic methods

 To connect them in a suitable manner, e.g., by satisfying the continuity conditions, in order to obtain a couple of “global” wave function solutions (F,G). During this step, the unknown parameters included in the initial functions fo, go, fi, and gi are fixed. The functions, thereby completely defined, can be denoted by Fo, Go, Fi, and Gi  To compute the normalization constant N by using the following formula: R ð0

1=N ¼

+∞ ð

ElDi dr +

ElDo dr R0

0

where ElDi represents the electron probability density corresponding to the couple of inside functions (Fi, Gi):   ElDi ¼ 4π r 2 jFi j2 + jGi j2

and likewise ElDo for the outside functions:   ElDo ¼ 4π r 2 jFo j2 + jGo j2

 Finally, to compute the mean radius hri by using the following formula: hr i ¼ N

ð R 0



ð +∞ rElDi dr +

rElDo dr R0

Note that the numerical results of hri should depend on the following preliminary choices:  The choice of the matching radius R0: even if its value is “reasonably” chosen to fit physical data, such as the charge radius of the considered nucleus, it is rather fuzzy.  The choice of the inside (nuclear) potential: apart from a common condition requiring it to be finite at r ¼ 0, there are multiple possibilities, each depending on modeling and approximations for the nuclear structure. Two of the most used examples are the following:  A simple constant potential equal to the value of the Coulomb potential at the surface of the nucleus and corresponding to a uniformly charged empty spherical shell.  The potential function defined by the expression written for V(r) above (Section “The deep orbits, as solutions of the Dirac equation with a corrected potential for a nucleus of finite size”) and corresponding to a uniformly charged solid sphere.  Nevertheless, one can consider more complex potentials or intermediate forms of both previous ones.  A more subtle choice related to the precision of the inside functions and depending on the approximation degree chosen for the polynomials of the ansatz, i.e., the power degrees of these polynomials.  The choice between atom H, or hydrogen-like atoms such as Li, Na, etc.

Results obtained from parameters near those of Maly and Va’vra Here, we give the values of hri computed for the hydrogen atom H, while following approximately the choices of Maly and Va’vra:

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 R0 ¼ 1.2 F.  A nuclear potential defined by the expression chosen by the authors and given previously: it approximates the proton by a uniformly charged solid sphere.  The polynomials of our ansatz have degree 6, whereas the ones of M&V, for a simpler ansatz, have degree 5. We have the following values of the mean orbital radius hri, for different values of the radial number n0 , with only three digits because of uncertainties in the considered method: • • • • •

n0 ¼ 1, hri  6.62 F n0 ¼ 2, hri  1.65 F n0 ¼ 3, hri  1.39 F n0 ¼ 10, hri  1.22 F n0 ¼ 20, hri  1.20 F

From these computation results, we can note the following facts:  We obtain values of the same size order as that in Ref. [17], while we used a method which is likely different: it is a good confirmation of the prior results. The authors indicate explicitly the value 5.2 F for the DDL atom H for j kj ¼ n0 ¼ 1, and they give only this case for atomic H.  The mean radius decreases when n0 increases, which is consistent with the fact that the binding energy in absolute value j BE j increases when n0 increases, as expected in Ref. [7].  After a gap between the values for n0 ¼ 1 and n0 ¼ 2, the value of the radius asymptotically tends to the value of the matching radius R0 ¼ 1.2 F.  The reduction in average value, with increasing n0 , could indicate the circularization of the orbit as the electron settles deeper into the potential well about the nucleus but limited by an increasing repulsive centrifugal core. However, with l ¼ 0, this repulsion is much weaker than that usually odinger equation. based on the l(l + 1)ℏ2 term of the Schr€ In Fig. 1, we plot the normalized electron probability density functions (NEPD) for n0 ¼ 1, 2, and 3. The peak values for NEPD correspond to R0, a minimum in the effective potential well, which, in turn, depends on the effective nuclear charge distribution and the relativistic characteristics of the EDO (to be described later).

Varying the parameters Here we only report conclusions about methods and results described in detail in Ref. [9] – With different values for R0, we obtain the same kind of progression when n0 increases, i.e., hri decreases when n0 increases and hri still asymptotically tends to R0. We also note a near-linear shift of the values of hri in the same direction as the shift of R0. This effect is most important for the first values of n0 . For example, with R0 ¼ 0.78 F (only a style exercise) we obtain hri  4.6 F for n0 ¼ 1; and with R0 ¼ 2.8 F, a reasonable choice for Li6 atom (charge radius Rc  2.59 with Z ¼ 3), we have hri  13.4 F for n0 ¼ 1. – When taking polynomials of higher degree, we have still the same progression, with slightly smaller values of hri for the smallest values of n0 ; but there is convergence. – Surprisingly, a change of the nuclear potential has almost no influence on the results. Of course, we tested “reasonable” changes, i.e., such that the potential does not increase (in absolute value) for

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Chapter 16 A study on electron deep orbits by quantum relativistic methods

3. × 1015 2.5 × 1015 2. × 1015 1.5 × 1015 1. × 1015 5. × 1015 0

0

0.5

1

1.5

1

2.5 r

3

3.5

4

4.5

5

FIG. 1 NEPD, for n0 ¼ 1 (cross), n0 ¼ 2 (asterisk), and n0 ¼ 3 (point). The radius ρ is in F.

r < R0. More precisely, we defined a parameterized potential, which can be fixed in intermediate forms between the potential previously tested and a constant potential for r < R0 In fact, this result is because the “inside” (nuclear) potential has only a very weak influence on the results: the electron probability density inside the nucleus has a weak weight. In conclusion:  The values of the mean radius hri and orbit “shape” of the EDOs are dependent on the angular number k and thus, the radial quantum number n0 .  For any considered changes of the parameters, the progression of the values when n0 increases is always the same: the values of hri decrease and tend to the value of the matching radius.  The values of hri, globally, are nearly independent of the parameters except for the value of the matching radius R0. This seems logical from a physical point of view, when recognizing the charge radius of the nucleus. Nevertheless, the mathematical method introduces an intrinsic degree of arbitrariness in the choice of R0 that cannot be eliminated.

Some criticisms of the considered method of corrected potential, and attempts to correct discrepancies The lack of dependence of the inside solutions on the nuclear charge potential, and the coherence of the values of energies We note in Ref. [21] a subtle criticism about some lack of dependence of the solutions on the nuclear potential, which we verified in our computations. In fact, the outside functions are fully dependent on the Coulomb potential. But, the inside functions depend on both the nuclear potential and the total energy E of DDLs involved in some parameters included in the recurrent relations used to determine the

The works of maly and va’vra on “DDL’s”

313

inside functions. Then, if we compute, in an approximate manner, the value of binding energies BE corresponding to the computed values of the mean radius, it seems there is some discrepancy. To correct this discrepancy, we use a method of iterative computation with convergence, which is precisely described in Ref. [9]. Summarily, for “injecting more dependence” on the nuclear potential at each computation cycle, one inserts energies computed from previously computed radii, into the equations, until they reach a fixed point. We carry out this whole process for each n0 ¼ 1, 2, 3, 10 with the following results:  for n0 ¼ 2, 3, and 10, the process quickly reaches fixed values, and we obtain the values of BE ¼  275, 301, and 320 keV, respectively, for mean radii 1.65 F, 1.39 F, and 1.22 F.  for n0 ¼ 1, the successive values approximately behave as geometrical series and there is convergence at radius value 12 F, that would give BE   56 keV. We conjecture that these values may be the actual values of the EDO’s binding energy for the corrected potential, instead of the values given in the Section “Results obtained from parameters near those of Maly and Va’vra” above, which are greater than 509 keV. Note that it is not the introduction of spin that causes the EDOs (the results of the K-G and Dirac equations are within 2 keV in the simple case), but the introduction of the rest mass energy term, mc2, in the construction of the relativistic equations. Developed explanations are given in the next section. The mean EDO radii are not sensitive to the nuclear potentials; however, they are quite sensitive to the radial quantum number. As are the atomic orbitals for low n values, they are noncircular for the low angular momentum values of small n0 . This large variation in average radius creates large variations in the kinetic and binding energies of specific EDOs. Since there are other contributions (as seen below) to the deep orbits that have not yet been included, the suggested values are only representative and comparative at this stage.

The discontinuity of the derivative of solutions A recent criticism was reported by a colleague, about the discontinuity of the derivative of the wave functions at the matching radius. Indeed, in the method exposed above, one satisfies the continuity of the corresponding inside and outside functions at the matching R0, but not the continuity of their derivatives. The discontinuity of derivative leads, on Fig. 1, to the angular point at the top of the plotted curves of electron probability functions. This seems to be a common feature observed in other works as the Dirac equation is a first-order differential equation, even if this breaches canonical rules that require a wave function to be everywhere continuously differentiable. From a discussion on the subject, this discontinuity appears to be equivalent to supposing an additional virtual potential ΔP at R0, so creating a well or a barrier according to the sign of ΔP. In fact, after some analysis and computations, the discontinuity, as well as the “ghost” potential, is seen to be a simple artifact due to technical imperfections of the method of corrected potential. Nevertheless, we wanted to look for possible ways to study/correct this imperfection, while observing in all our computations of normalization constants and hri (see the previous section) that the components of g, both the inside function gi and the outside one go, dominate. This dominance of g over f can even reach several orders of magnitude for the outside functions as n0 increases. So, to compute the mean radius hri with the only largest component yields results close to the results including both components. We carried out numerous computations in various ways [22]: e.g., to use an additive parameter β0 (as indicated in the expression of the inside nuclear potential, Section “The deep orbits, as solutions of

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Chapter 16 A study on electron deep orbits by quantum relativistic methods

the Dirac equation with a corrected potential for a nucleus of finite size”) as an additive potential to balance the virtual potential ΔP for the largest component, or to satisfy both continuity and derivative continuity for this component. These computations always give results close to the ones indicated in the previous section. As a conclusion, we think it is not useful to look for a more complex method to resolve this question.

Question of the sign of the EDOs solutions As a consequence of our previous publications, a reviewer has stated that EDOs represent negative energy states. This question comes from known problems encountered in using the Dirac equation. For example, the well-known Brown-Ravenhall disease [23], also referred to as the “continuum dissolution” problem, happens while applying the Dirac equation for at least two atomic electrons. On double excitation of a pair of correlated electrons, one electron can end in the negative energy continuum (positrons), while the other lands in the positive energy continuum, the total system energy is retained. Of course, as noted by Rusakova [24], the “Brown-Ravenhall disease” appears only in many-body cases. Nevertheless, it is quite legitimate to ask the question of the sign of the EDO energies, as this sign cannot be deduced from the relation E2 ¼ hexpressioni obtained at the end of the Dirac equation solution. A clear solution to this issue has been given in Ref. [25]. It is based on the fact that, during the process of solution, E appears in some parameters without the square. We can refer to the process used by Fluegge [18], already cited previously about the works of Maly and Va’vra. In this solution method, the final step to obtain the expression of the energy E is carried out by solving the following algebraic relation: 1 α 2

(rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) mc2 + E mc2  E  ¼ ðn0 + sÞ 2 mc  E mc2 + E

where n0 is the radial number, s has to satisfy s ¼  (k2  α2)1/2, and α is the coupling constant. It is easy to see that the left side has the same sign as E, and thus E has the sign of (n0 + s). The EDOs are obtained when j kj ¼ n0 and s < 0, i.e., s ¼  (k2  α2)1/2. As α < 1, one has n0 + s ¼ | k |  (k2  α2)1/2 and j k j is always greater than the real (k2  α2)1/2 so we can affirm the EDO energy E is positive Furthermore, as already mentioned above in Section “The anomalous solutions of the Dirac equation”:  If jk j > n0 , the energy E is <0, so it does not concern physical solutions.  If jk j < n0 , the energy E is >0. We have named the corresponding solutions “pseudo-regular” [8]. We can conclude that the case j kj ¼ n0 separates the positive- and negative-energy solutions and it falls on the positive-energy side.

Involvement of special relativity in the EDOs In early works, e.g., Ref. [22], we observed that nonrelativistic equations give singular solutions but these do not correspond to orbits with high binding energy (in absolute value). A physical

Involvement of special relativity in the EDOs

315

reason is that an electron in a deep orbit is necessarily relativistic. We also observed a purely mathematical way to sort the EDO among the singular solutions of a relativistic equation, when we have the algebraic expression of energy levels: e.g., to make j k j ¼ n0 in the case of the Dirac equation, or to make j l j ¼ n0 in the Klein-Gordon equation. Furthermore, we found a deeper reason by comparing the relativistic version of the Schr€odinger equation with its classical nonrelativistic version. It is seen in the relativistic correction to the Coulomb potential [26, 27], which is not taken into account for the usual atomic orbits, being too weak at these levels. We developed a complete analysis of this question in Ref. [6]. We report here only some essential elements and conclusion.

Comparing the relativistic and the nonrelativistic versions of the Schr€ odinger equation We give here both versions for the hydrogen atom, as extracted from Ref. [16], where R represents the radial wave function and l is the angular quantum number:

  1 d λ 1 lðl + 1Þ  α2 2 dR R¼0 ρ +   ρ2 ρ2 dρ dρ ρ 4

  0 1 d λ 1 lðl + 1Þ 2 dR  R¼0 ρ +  ρ 4 ρ2 dρ dρ ρ2

The former is the relativistic equation, whereas the latter is nonrelativistic. We can see that there is only one parameter making the difference between the two versions. The coupling constant α occurs, squared, only in the relativistic version (note: in Ref. [16], the symbol γ is used instead of α). Of course, λ 6¼ λ0 and the transformation of the initial radius r into the dimensionless variable ρ is not the same for both equations. This term α2 is the source of EDO solutions. We saw, in Section “The works of Maly and Va’vra on “DDL’s,” that the energy levels of the relativistic Schr€odinger equation are given by the expression of the following form:  ½ E ¼ mc2 1 + α2 =λ2

And we noted the EDO solutions are given by letting s ¼ ½  [(l + ½)2  α2]½ in the ansatz used for solving the equation. It provides a total energy of E  mc2α/(2l + 1), when n0 ¼ l, and so a very high binding energy j BE j ¼ mc2[1  α/(2l + 1)]. Again, the term α2 occurring in the expression of E above directly comes from the one occurring in the relativistic equation. On the other hand, it is not possible to set n0 ¼ l in the case of the nonrelativistic equation, where we recall that singular solutions (usually dismissed) are obtained when taking s ¼ (l + 1). Indeed, λ0 has to satisfy the condition λ0 ¼ n0 + s + 1, for all types of solutions, which gives λ0 ¼ n0  l for the singular ones. As the nonrelativistic energy levels are given [16] by E ¼  mc2α2/2λ0 2, making n0 ¼ l leads to λ0 ¼ 0, which is not possible because it produces an undefined value of E.

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Chapter 16 A study on electron deep orbits by quantum relativistic methods

Meaning of the term α2 appearing in the equation The key is the relativistic equation being built from the relativistic expression of total energy in free space E2 ¼ c2p2 + m2c4, where p is the momentum vector. Next one introduces an exterior electromagnetic field in covariant form into E, and as the nuclear Coulomb potential has spherical symmetry, this leads to (E  V)2 ¼ c2p2 + m2c4, where V is the usual Coulomb potential. The expression (E  V)2  m2c4, developed into E2  m2c4  2VE + V2, gives rise to several “energy factors” during the building of the Schr€ odinger equation. However, the term α2/ρ2, distinctive in the relativistic equation, is particularly interesting: it comes from and is proportional to V2. Moreover, in the relativistic equation, this term is added to the term λ/ρ, positive by construction, which comes from 2VE and is proportional to j V j. So, the real meaning of the occurrence of α2 in the relativistic equation is a cause of the existence of EDOs. It corresponds to a dynamic relativistic correction to the Coulomb potential energy V in the form of a term proportional to V2, which strengthens the static potential energy V. The general form of this correction (see Refs. [26, 27]) leads to the following effective “dynamic” potential:   Veff ¼ V E=mc2  V 2 =2mc2

In case of a relativistic electron, whose relativistic coefficient γ is known, the expression of Veff can be transformed into the following one: Veff ¼ γV + V 2 =2mc2

Note that, while looking at both expressions of Veff given above, it is not clearly apparent that one always has j Veff j > j V j, i.e., strengthening of the “normal” Coulomb potential. In Ref. [6, 22], we showed, in the case of quasi-circular orbit approximation, the following result:  one always has j Veff j > j V j and Veff is attractive (a negative value) and  j Veff j quickly increases as a function of jV j, with a parabolic behavior in j V j2 when j V j ! +∞. Nevertheless, while making more recent computations (see Section “Relativistic confinements and the question of the Heisenberg uncertainty relation (HUR)”) based on the Heisenberg relation, we obtain this strengthening of the Coulomb potential for EDO orbitals of any form. Moreover, as we indicate further (Section “Relativistic confinements and the question of the Heisenberg uncertainty relation (HUR)”), before using Heisenberg’s relationship and computing the relativistic confinement energies in Refs. [25, 28], we strongly underestimated the relativistic coefficient γ of the EDOs solutions of the Dirac equation: this information cannot be extracted directly from these solutions, be it the wave functions, the (total) energy levels, or the binding energies.

Study of the magnetic interactions near the nucleus At atomic levels, the magnetic interactions such as spin-orbit interactions or spin-spin interaction have little effect on the energy levels of bound electrons, leading only to very small energy shifts. Their essential interest is to break the degeneracy of energy levels and split the spectral lines. Nevertheless,

Study of the magnetic interactions near the nucleus

317

at deeper levels, they can become very strong and might even be dominant over the Coulomb interaction, as they have a behavior in 1/r3. In these conditions, it is important to study the effects of magnetic interactions on deep orbitals, and in particular to analyze the existing work in the field of magnetic interactions. In Ref. [25], we did a thorough study developed on this subject. We report here the most important elements and our conclusions, perhaps provisional.

Summary of the magnetic interactions near the nucleus We only consider the atomic H or D (when we explicitly indicate it).

Interactions involving only the electron spin It is essentially the spin-orbit interaction, automatically included in the Dirac equation: one considers the relative orbital movement of the nucleus in the electron reference frame, where a magnetic field is “seen” by the electron, i.e., acting on the magnetic moment associated with the electron spin. We recall that the general quantum expression of the spin-orbit interaction reads HSO ¼

1 2 m 2 c2

  1 ∂V LS r ∂r

where L and S, respectively, denote the electron angular momentum and spin operators. As V is the Coulomb potential energy, by replacing V by its expression as a function of r, one obtains HSO ¼

μ 0 e2 1 L  S ¼ ξ ðr Þ L  S 8 πm2 r 3

By using the rules on the composition of angular momenta, one shows L ∙ S can be expressed by means of the eigenvalues l, s (fixed ¼ ½), and j, of L and S and the total angular momentum J ¼ L + S, respectively. If l 6¼ 0, one can write LS¼

ħ2 ħ2 ½jðj + 1Þ  lðl + 1Þ  sðs + 1Þ ¼ ½jðj + 1Þ  lðl + 1Þ  3=4 2 2

Here j ¼ l  ½ and – If l ¼ 0, there is no SO interaction. – For any value of l > 0, one has two cases:

j ¼ l  1/2 ¼ > LS ¼ (ℏ2/2) (l + 1), that gives an attractive potential, since ξ(r) is >0

j ¼ l + 1/2 ¼ > LS ¼ +(ℏ2/2) l, giving a repulsive potential To obtain numerical values for the SO potential energy as function of hri, one replaces the physical constants occurring in ξ(r) by their values in a chosen unit system, e.g., the SI standard. Then the average energy hESOi for l ¼ 1 reads: 

 

 hESO i  1:71  1053 = r 3 J  1:07  1034 = r 3 eV

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Chapter 16 A study on electron deep orbits by quantum relativistic methods

Note this generalization is made, by supposing the rules for angular momenta can be extended for any radius. However, when high energies are concerned, we have to take into account other spin and relativistic corrections, e.g., Ref. [28].

Magnetic interactions involving the nuclear spin The analysis of these magnetic interactions, some being involved in the so-called hyperfine structure, is complex; but, we only need to deal with the energy. Also, we can make some simplifications while evaluating orders of magnitude of the coupling energies. So, the spin-orbit interaction SpO concerning the nuclear spin and the direct orbital motion of the electron represented by L, SpO can be neglected relative to SO potential energy indicated just above, because the magnetic moment of the electron is about 670 times greater than that of the proton. For the same reason, the “spin-spin” interaction, which is much weaker than the SO interaction, will be neglected when we take into account SO. But if L ¼ 0 one has neither SO nor SpO coupling. Thus, one has to take into account the interactions between the nuclear and electron spins. This interaction can be expressed by the following general formula (see, e.g., Refs. [29, 30]): HSS ¼ 



  8π μ0 1   ^ ^ 3 M  r ð M  r Þ  M  M  M δ ð r Þ M + p e p e p e 4π r 3 3

where Mp and Me are the respective magnetic moments of the proton and the electron, and r^ denotes a radial unit vector. The magnetic moments are related to the respective electron and nuclear (i.e., proton, here) spins Se and Sp by the following formulas:   Me ¼ ðe=2me Þ Se and Mp ¼ 2:79j ej =mp Sp

By introducing the total spin S ¼ Se + Sp, one can write the following relation:   2   2  Se  Sp ¼ ð1=2Þ S2  ðSe Þ2  Sp ¼ ℏ =2 ½sðs + 1Þ  3=2

The only possible values of s are s ¼ 1 (“triplet” state) and s ¼ 0 (“singlet” state), which gives two cases: • •

s ¼ 0 ¼ > Se  Sp ¼ (3/4) ℏ2, i.e., “attractive case” s ¼ 1 ¼ > Se Sp ¼ +(1/4) ℏ2, i.e., “repulsive case”

One knows, e.g., the energy ESS of the spin-spin interaction in the attractive case: hESSi ¼(¾) A(r) where, for hri ¼ a0, the Bohr radius, A(a0) is proportional to 1/a30 and has a value 9.39  1025 J  5.87  106 eV. We extrapolate this result to a general expression of the spin-spin interaction energy ESS for any radius hri: 

 

 hESS i  1  1055 = r 3 J  0:64  1036 = r 3 eV

Note that for a particle in a relativistic regime, the spin tends to lean in the direction of the motion of the particle [31], which could imply a weakening of the energy ESS.

Study of the magnetic interactions near the nucleus

319

Diamagnetic terms Such a term comes from expressions of form (P  eA)2 associated with the minimal coupling between one charged particle and an “exterior” EM field, as e.g., in a Pauli equation. P is the kinetic momentum of the particle, e its electric charge, m its mass, and A the vector potential of the EM field. The complete energy term associated with A2 has the form e2A2/2 m and is considered [29, 32] to be expressing diamagnetic energy with a behavior in 1/r4. This interaction is not very well-known, not involved by the Dirac equation, and generally neglected at atomic levels because it is very weak; but, it can have considerable importance near the nucleus. In fact, there are actually two similar diamagnetic terms to consider, as we can see when considering the Hamiltonian of a two-body system electron + proton, as in Refs. [33, 34]: one is caused by the interaction of the electric charge of the electron with the intrinsic magnetic moment of the proton spin, and another is caused by the symmetric interaction, between the charge of the proton and the magnetic moment of the electron. But the latter interaction energy is equal to 240 times the former one and “absorbs” it completely. So, the kept term has the form C/r4, where the coefficient C is computed by the following expression:      C  ðμ0 =4π Þ e4 ħ2 = 4 me 2 mp  1:3  1071 in SI units here : J  m4

The Vigier-Barut model The Vigier-Barut (V-B) model and the works related to this model derived from works of Barut, e.g., Refs. [35, 36], which were carried in a relativistic context, with a more complete Dirac equation. However, the above-named V-B model and the related works were made in a nonrelativistic framework.

Works of Barut, as a source of the V-B model Barut is the author of many works, but here we indicate only the works that inspired the V-B model. In these articles, the author looks for an analytic solution of the Dirac equation for a charged lepton with anomalous magnetic momentum (AMM) in a Coulomb potential. For doing this, in the former cited reference, the author uses a Dirac equation where the vector potential is completed by a term introducing the electron AMM and is expressed by means of the EM tensor Fμν and Dirac matrices γ. After numerous and complex transformations, he obtains a second-order differential radial equation having a form similar to a radial Schr€ odinger equation. Nevertheless, this equation includes a specific dynamic potential V expressed as a sum of inverse powers of the radius r. More precisely, the differential  equation reads:  d2  V ð r Þ + E2 dr 2

ψ ¼ 0, where the potential V has the following form: V ðr Þ ¼ A=r + B=r 2 + C=r 3 + D=r 4

We note the occurrence of E2 in the equation that is the “signature” of Relativity. A, B, C, and D are built from physical constants: the mass or the reduced mass, the energy E, and the initial parameters of the equation, written in natural units, i.e., with ℏ ¼ c ¼ 1. In the latter reference, the equation has a similar form. It is this form of potential, expressed as a sum of inverse powers of the radius r, which is at the origin of the V-B model. In fact, Barut does not apply

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Chapter 16 A study on electron deep orbits by quantum relativistic methods

r,a.u.

0

5

10

50

100

r

Electric zone

–0.005 Magnetic zone –0.010

–0.015

–0.020

V,a.u

FIG. 2 Potential with two wells.

the equation to the atomic H, but to a coupled system (e +, e) called “positronium.” Doing this, he finds two potential wells as pictured in Fig. 2, and he looks for resonance states of positive energy. Note V reaches a maximum at the huge energy Vmax of order 35 GeV when the radius becomes very small and orders-of-magnitude smaller than the region corresponding to the regular (atomic) positronium bound state. He fails to comment on where 35 GeV comes from when starting in a system with only 1.022 MeV mass energy plus limited kinetic energy. In another paper [37], Barut uses a very different method to treat tight molecules and finds interatomic binding energy of 50 keV. On the other hand, in Ref. [38] he extends his model, by including the short-ranged, strong, magnetic forces, to the high-energy physics, including strong and weak interactions.

Vigier-Barut model, and related works We can cite the research of Vigier [39], Samsonenko et al. [33], and Dragic et al. [40], on possible tight orbits under the Bohr level for atomic H (or D), and even for tight molecules. Of course, this list does not exhaust the concerned subject. All these works are attempts to explain LENR results by screening effects in “tight” orbits, in particular, the initial experiment of Pons and Fleishmann [41], which was the latest at the time. These works are made in a nonrelativistic context. Moreover, in the expression of the dynamical potential used by Barut and indicated above, the expressions of the coefficients B, C, and D are complex, sometimes energy dependent, and difficult to interpret physically. To the contrary, in the works of the V-B model, the meaning of the terms of the potential is the following: – The term in 1/r corresponds to the Coulomb potential energy, attractive; its coefficient A is <0. – The term in 1/r2 has a clear meaning of the “centrifugal barrier” energy, repulsive; B > 0.

Study of the magnetic interactions near the nucleus

321

– The term in 1/r3 represents magnetic interactions, such as those described in Section “Summary on the magnetic interactions near the nucleus”: (i) Either spin-orbit, available only if the angular quantum l is not null. It can be attractive or repulsive, depending on the total angular number j. (ii) Or spin-spin, involving the nuclear spin, which can be involved in the Pauli equation. It is attractive (case of “singlet” state) or repulsive (case of “triplet” state). So C can be >0 or <0. – The term in 1/r4 is the diamagnetic term (as indicated in Section “Diamagnetic terms”), repulsive, so D > 0. If C < 0, we have the following succession of potentials, written in the order of the decreasing powers of r: D/r4 [repulsive], C/r3 [attractive], B/r2 [repulsive], and A/r [attractive]. As a visual example, for some combinations of values of the coefficients where V(r) has three “zeros,” there are three different values r1, r2, and r3 such that V(r1) ¼ V(r2) ¼ V(r3) ¼ 0. Then V(r) includes two wells.  The first well, for small values of the radius r, corresponds approximately to a zone where the magnetic interactions are dominant: one can call it the “magnetic region” as in Ref. [42].  The second well, occurring when r increases toward the Bohr radius, corresponds to a zone where the electric Coulomb potential is dominant: the “electric region” [42]. In Fig. 2, we represent a pure abstract mathematical exercise to simply show that some combinations of coefficients can give rise to two local minima for the plotted curve of a potential like V. This curve does not correspond to an actual physical case. Here, as well as in Ref. [39] and in these conditions, the values on the axes have arbitrary units. In the works of the V-B model, the analytic methods meet numerous difficulties, which lead to solving the equation numerically. Under these circumstances, as written in Ref. [33]: “there is a set of contradictory estimates of energy values and quantum-orbit radii based on the starting Hamiltonian.” It is not a criticism of these works, because they are important and difficult studies in a field that has been barely explored, if at all: the domain of the magnetic interactions near to the nucleus. Concerning numerical results, we can see  In Ref. [33], the authors indicate a tight state of energy 40 keV. We recall that, as it is a study in nonrelativistic context, the electron total energy TE is usually noted as negative, and it is equal to the binding energy.  In Ref. [40], the authors give numerous results for the spectra of the hydrogen atom, at different energy levels and for different values of quantum numbers l, s, and j. Nevertheless, the indicated energies are similar to classical atomic values, i.e., a few eV. In fact, they did not find tight orbits with high energies of order keV. Note also, that Vigier [39] develops a theoretical study for computation of new tight orbits based on the causal mathematical formalism of L. de Broglie [43] and Bohm [44], but he does not give numerical results for these tight orbits. On the other hand, his work includes a long study about a three-body problem, by considering possible new “tight” molecules: two nuclei aligned with an electron between them. For these problems, the author indicates numerical results of, 28.1 keV for H2 + and 56.2 keV for D2 + . In a similar work [37], Barut indicates a similar-size order of energies, e.g., 50 keV for D2 + .

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Chapter 16 A study on electron deep orbits by quantum relativistic methods

In conclusion, except maybe for the result in Ref. [33], the nonrelativistic models do not seem to provide tight orbits with a high binding energy of several tens of keV or greater. These high energies seem quite normal to us, according to our study on the essential role of relativity for the existence of EDOs [6, 45, 46].

Relativistic confinements and the question of the Heisenberg uncertainty relation (HUR) In Ref. [25], we computed the confinement energy expected for EDOs from the HUR and we noted that magnetic interaction energies could perhaps be high enough to balance the confinement energy. Nevertheless, even if it is certain that EDOs are highly relativistic, we had not a clear expectation of the relativistic coefficient γ and, mostly, the question of the HUR problem for the EDO’s was not actually solved. For this reason, in Ref. [28], we applied another resolution strategy:  To deduce the relativistic coefficient, directly from the HUR as applied to a confined electron.  To deduce the relativistic correction leading to the effective Coulomb potential energy Veff (see Section “Meaning of the term α2 appearing in the equation”). Doing this, we can see the value of Veff is strong enough to confine an electron in an EDO. Here, we give only some elements of the computations.

Computation of the coefficient γ To make such a computation using HUR, it is usual [29, 47] to consider that the dispersion (uncertainty) on the norm of the momentum j p j satisfies Δ j p j Δr  ℏ/2; to accept Δ j p j as an average estimate of the momentum and to attribute hri to Δr. So, we write p  ℏ/2r, where p stands for jp j and r for hri, in order to simplify notation. Then, we consider the relativistic expression of momentum: p ¼ γmv, where m is the mass of the electron and v its velocity. Thus, we have to satisfy the relation γ mv  ℏ/2r, i.e., γ v  ℏ/ 2mr. From this, we can deduce γ  [1 + (λc)2/4r2]1/2, where λc is the reduced Compton wavelength of the electron, equal to ℏ/mc. As λc  386 F and for the EDOs, where r is on the order of a few F, one can simplify the previous relation into γ  λc =2r

Of course, this is valid only if r ≪ λc/2  193 F. Moreover, in the previously cited references, the coefficient “1/2” is removed to give an order of size for the momentum p: p  ℏ/r. Under this condition, one can show the following relation: γ  λc =r

To give a size order of γ for EDOs: if computing γ with this formula and for r ¼ 2F, we can expect a relativistic coefficient of order 193, i.e., close to 200.

The effective potential Veff is strong enough to confine electrons in deep orbits Now we consider the effective potential energy Veff, for a relativistic electron having a high value of its relativistic coefficient γ, as previously computed.

Question about the stability of the EDOs

323

The expression used to compute Veff, in the case of a relativistic electron is (Section “Meaning of the term α2 appearing in the equation”): V eff ¼ γV + V 2 =2mc2

With the expected expression of γ as function of r indicated above, the expression of Veff reads   V eff ¼  αℏ2 =mr2 ð1  α=2Þ   αℏ2 =mr2

If we consider only the inequality γ  λc/2r, valid for r ≪ λc/2  193 F, we obtain jVeff j  αℏ2/2mr2. Under these conditions, we can show the following results about Veff: (1) Veff is always attractive (2) jVeff j  j V j. So one has a strengthening over the static Coulomb potential (3) Veff has a behavior in K/r2 when r decreases (thus j V j increases), with K  9  1041 in SI units. To have an idea of the size order of Veff near the nucleus, by computing it for r ¼ 2F, we obtain the following approximate values: Veff is of order 140 MeV, whereas the kinetic KE ¼ (γ  1) mc2  98 MeV. With such a high value, Veff can definitely confine an electron in the deep orbit region. We previously showed [6, 22], that Special Relativity is the source of the EDOs. Here we have shown that the HUR, which seemed an impediment for the concept of EDOs, provides its proper resolution thanks to Relativity.

Question about the stability of the EDOs The deep orbit electrons have the following features:  They are highly relativistic  They are subjected to several electromagnetic interactions of high intensity, some of which are not involved in the Dirac equation used until now for determining the EDOs for a single particle.  Note also that, in the “nuclear zone,” the deep-orbit electrons are certainly subject to fairly high radiative corrections. But the Coulomb electric field strengthened by relativistic effect, corresponding to Veff, seems sufficient by itself to retain an electron in the nuclear zone. In these conditions, the question of the EDO stability seems a very difficult problem to solve. Nevertheless, to have the first estimate of a possible stable resonance, we can use a well-known semiclassical approximation [48], which consists of seeking a local minimum of total energy. In fact, we combine two approaches for doing this:  To attempt to determine which interactions have the greatest roles in the generation of a resonance  To compute a total energy, while respecting the HUR For the latter point, we consider the relativistic expression of energy, in which the norm of momentum j pj is replaced by ℏ/r, that gives the following term, noted EH (“H” for Heisenberg) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ħ2 c2 EH ¼ + m2 c4 r2

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Chapter 16 A study on electron deep orbits by quantum relativistic methods

We consider the total energy E in EM interactions, defined by E ¼ EH + V, where V represents potential energy. Then, we look for a local minimum of energy (LME) E for various combinations of potentials included in the term V and we determine the radius of this local minimum. In fact, a potential is “interesting” for the resonance, i.e., to be kept for the rest of the study, not only by considering the energy levels, but also if the average radius for the local minimum is acceptable, i.e., near and preferably outside the nucleus (a proton). This study on resonance started in Ref. [25] and continued in Ref. [28], involves numerous calculations to evaluate the respective roles of the various interactions, most being indicated in the previous sections. We give here only some results on:  The potentials that seem to have a determinant (interesting) role for a resonance.  Possible values of the mean radius of a resonance. Note that some potentials have a repulsive version or an attractive one, depending on parameters such as spin orientation. The study leads us to choose one of the versions rather than the other. The “unchosen” versions may be just as real; however, they would lead to different consequences [25, 28].

Potential energy terms for expecting a resonance. Seeking local energy minimum •

•

•

Of course, V systematically includes Veff and has the form V ¼ Veff + V4 + OptPot, where V4 denotes the “diamagnetic term” (Section “Diamagnetic terms”), V4  1.3  1071/r4 in SI units. Even very weak, it cannot be neglected at small r, and OptPot is a combination of further potentialenergy terms. The computations show the spin-orbit interaction energy ESO, under both repulsive or attractive versions, cannot be kept in V, because they could have too high values in the nuclear zone, of order several GeV, which seems unreasonable. Moreover, in the attractive 1/r3 version, it “pulls” the LME almost to the center of the proton, even when combined with a repulsive centrifugal term Vc ¼ l(l + 1) ℏ2/2mr2. Under these conditions, we must assume that l ¼ 0, which gives ESO ¼ 0 and implies Vc ¼ 0. As Veff becomes very great near the nucleus, if we include the attractive spin-spin interaction energy ESS into OptPot, we again obtain a LME far inside the nucleus (at r  0.16 F and with forces possibly as strong as those of the nucleus). Taking the repulsive version of the spin-spin interaction energy (ESSR) corresponding to a triplet state, we have ESSR ¼ jESS j/3  3.4  1056/r3 SI units.

Making a trial with E ¼ EH + Veff +ESSR + V4  EH + Veff + 3.4  1056/r3 + 1.3  1071/r4 and l ¼ 0, the LME is reached at r  1.1 F, i.e., outside the proton, where E   61 MeV. We have also γ  365, KE  185 MeV, PE   250 MeV. Moreover, the potential wall due to the HUR is of order 20 MeV at r  5F. Of course, this result about a LME, with a potential well outside the nucleus, is just a coarse approximation. Nevertheless, it gives size orders and an “interesting” combination of potentials capable of obtaining a realistic resonance for EDOs, without yet using quantum equations. Note that for an electron to be inside the nucleus would not be scandalous per se since one knows the electron of an s orbital has a finite probability to be inside the nucleus. In a more extreme case, if considering a muonic atom of lead, one knows [29] the muon orbit is more inside the nucleus than outside, where the potential, according to a classical approximation, is parabolic. But, similar to this last example, the expressions for the potential energies are completely different inside the nucleus from the ones outside. So the results for computations of local minima, such as those indicated above with LME inside, are, just as the singularity for the 1/r Coulomb potential, only representative of a simplifying assumption.

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325

Local energy minimum, with a relative weakening of near-nuclear interactions One also has to take into account effects that can weaken the Coulomb potential before even arriving at the proton. In particular, some radiative corrections derived from QED, such as the electron self-energy and the vacuum polarization, generate a polarized cloud of virtual pairs of electrons and positrons confined in a localized region around the electron. Electron self-energy decreases the binding energy, while vacuum polarization tends to increase it, but the sum of both effects gives a repulsive action. At atomic levels, this causes the well-known “Lamb shift” of the order a few 105 eV. Nevertheless, radiative corrections increase with the intensity of the electric field to which the electrons are exposed; so, one can expect these corrections to become much stronger for an electron localized near the nucleus. Observations on heavy atoms can already give an idea of the size of these corrections: e.g., for the ground state of an H-like uranium ion (U91+), one observes a Lamb shift of almost ½ keV [49]. Moreover, there is also the possibility of weakening for interactions involving spin: one can note [31] that a relativistic velocity transverse to the spin axis affects the direction of the spin; more precisely, as the velocity approaches c, the spin tends to become aligned with the helicity. Effective calculations using QED and other considerations for near-nuclear effects are still beyond the scope of this study. For the present, we make simple simulations of weakening for Veff, ESSR, and V4, when computing localization of an LME near the nucleus. Without reporting tedious details of our calculations, we can indicate the following:  we simulate a linear weakening VCbw(r) of the static Coulomb potential VCb(r), by a coefficient K when approaching the nucleus, i.e., at a radius r1 > r0, r0 is the charge radius of the nucleus. Next, we deduce the dynamical effective potential Veffw from VCbw.  We consider weakening of the magnetic potentials ESSR and V4, by putting E ¼ EH + Veffw +ESSR/C + V4/D, where C and D are constants >1. We report here the following example of computation results we carried out: For r1 ¼ 2.5 F, K ¼ 0.55, C ¼ 1.8, and D ¼ 2, we have an LME at r slightly greater than 1.6 F, where E   5 MeV. In Fig. 3, we plot the curve of E as a function of the radius taken in the interval [1.3 F, 2.5 F].

Conclusions, question, and perspectives 1. At this point, it is important to relate the results obtained for deep LME, even if only approximate, with the EDO solutions of the Dirac equation. – The values for the location of a deep LME are of the same order as the values of the mean radius of the Dirac EDOs. Indeed, while varying attenuation parameters of the EM interactions, we obtained LME locations between 1.1 and 2 F and, on the other hand, the mean radii of EDOs obtained by the Dirac equation (see Section “Results obtained by computations of the DDL wave functions for modified potentials, further developments, and discussion”) are from 1.2 to 1.6 F, assuming a matching radius at 1.2 F. Only the value obtained for n0 ¼ 1 deviates a little, as it is 6.2 F. This relative convergence of both methods is positive since both methods include several approximations. – Seeking an LME, with fixed potentials, provides only one value corresponding to energy well, whereas the Dirac equation provides infinity of EDOs levels. First, it is apparently the same when one seeks the size order of electron LME in the simple Coulomb potential of a proton, as, e.g., in

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Chapter 16 A study on electron deep orbits by quantum relativistic methods

p 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

p, F

–1 –1.5 –2 –2.5 –3 –3.5 –4 –4.5

V, MeV

FIG. 3 Plot of E (in MeV as a function of ρ in F) based on the weakening of the potentials inside the nuclear region (see text).

Ref. [48]: one finds the fundamental Bohr level at 53 pm; and on the other hand, the Schr€odinger equation provides an infinity of energy levels, including the fundamental level and excited ones. Nevertheless, for the Dirac EDOs level, the progression of energy levels is opposite that of the regular levels: the mean radius decreases when the radial number increases, and it tends to a finite limit equal to the “matching radius” R0; as a consequence and taking into account the blurring caused by the HUR, it seems realistic to think there are actually only a small number of distinct EDO levels. The exact number of levels, as the physical explanation of this unusual opposite progression, is still an open question. – These first calculations, carried out essentially to find a deep LME, do not automatically give values of binding energy comparable to those resulting from solutions of the Dirac equation for the deep electrons, or even realistic values. Nevertheless, recent calculations (not yet published) performed to adjust electron BE in a window of values consistent with EDOs, e.g., 150–510 keV, give positive results that are in agreement with possible LME values. – At the end of the section “Magnetic interactions involving the nuclear spin,” we noted a possible direction change of spin axis for a particle in relativistic movement, which could imply a weakening of the energy ESS, as this interaction involves two bodies moving relative to one another at relativistic speed. This question is not yet resolved, but we could use the Lorentz transformations for this.

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327

2. One can find in the literature (e.g., Ref. [16]) the following remark about the anomalous solutions of relativistic equations: if considering a finite size (6¼ 0) for the nucleus, the electric potential has no singular point and the solution which is finite at r ¼ 0 approaches the Coulomb point “regular” solution. Put another way, when the nucleus is not a point, the anomalous solution disappears for lack of singularity. But this remark is made in a context where the relativistic enhancement of the Coulomb potential V, leading to the effective potential Veff, is never taken into account, although being tacitly contained in the equation. During the process determining the relativistic corrections involved in the equation, an approximation (legitimate for atomic states) is made while considering jV j≪ mc2 (that implies Veff  V). Under these conditions (with no jVeff j>jVj) one can understand that an “anomalous” solution appears only when considering a point nucleus: as in Section “Varying the parameters,” the singularity of V at r ¼ 0 “pulls” the radial wave function toward the singularity. Unfortunately, this solution must be eliminated, as it diverges at the singularity (implying all mathematical problems analyzed in section “Arguments against the EDO states and possible solutions.” To the contrary, if taking into account jVeff j>jVj, this amplified potential pulls the wave function toward the proton (no need for the singularity) and the finite nucleus allows us to have a wave function that is not divergent. 3. For our calculations in Section “Question about the stability of the EDOs,” we have taken into account combinations of high energy potentials that are partially converted into actual kinetic energy for deep electrons. These energies are of order 100–200 MeV, as we have systematically eliminated potentials that give unrealistic energies (of order of GeV) due to angular momentum effects (SO interaction and “centrifugal potential”). But, one can legitimately ask “where do these high energies come from?” We can reasonably think this energy is taken from the rest mass of the proton, which is of order 1 GeV. It is now known from the experiment in the LHC [50] and from electron-proton inelastic scattering (e.g., Ref. [51]), that a proton is actually a “soup” of quarks, antiquarks, and gluons in a perpetual shuffling (creation/destruction of pairs) and in highly relativistic movement. More precisely, the soup contains two up-quarks (with “base” mass 4 MeV) and one down-quark (with “base” mass 5 MeV), called the “valence” quarks, and a great number of gluons and of virtual pairs {quark, antiquarks} resulting from the highly relativistic kinetic energy of the quarks. So, the mass of a proton comes mainly from the relativistic energy of its constituents. Of course, the process of energy transfer from the proton-reservoir to the relativistic deep electron, with its proximate intense fields, should be the subject of a detailed study. 4. Seeking an LME with fixed potentials is a preliminary study, carried out in a semiclassical way, which should then lead to further studies based entirely on relativistic quantum methods. To progress in this direction, we still have to deepen the way radiative corrections actually act in the nuclear area. It would be interesting to express this by a potential energy with one part, concerning the vacuum polarization, derived from the Uehling potential [52, 53] and another part, concerning the electron self-energy, deduced from results expressed by means of a data table, e.g., Ref. [54]. Moreover, the study of works on heavy atoms (e.g., Ref. [49]), recently started (see Section “Local minimum of energy, with a relative weakening of interactions near the nucleus”), can give us some leading paths, as one observes a considerable increase of the radiative corrections: e.g., Lamb shift [55, 56], due to the intense electric field caused by the heavy nuclei.

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So, in our most recent paper [57] submitted and accepted for publication in ICCF21 Proceedings, we made progress in the study (Section “Question about the stability of the EDOs”) about the existence of a local minimum energy for EDO, while deepening the Lamb shift effect for EDO and expressing it in a more explicit way, under the form of a quasi-potential. 5. In the introduction, Section “Interest of the electron deep orbits (EDO) for the low-energy-nuclear reaction (LENR),” we said that a deep-orbit electron certainly allows fast energy transfer from the nucleus to a substrate/lattice, thanks to strong near-field coupling [28, 58, 59] between excited nuclear components and the deep electron. So, this rapid de-excitation could avoid nuclear fragmentation and the emission of gamma rays, explaining an interesting and useful feature of the observed LENR. In the same topic, we think that EDOs can increase the probability of the known phenomenon of internal conversion. This is why we also study the possibility of interactions between ultra-relativistic deep electrons and mesons responsible for nucleon bonds.

Acknowledgment This work is supported in part by HiPi Consulting, Windsor, VA, USA and by the Science for Humanity Trust, Inc., Tucker, GA, USA.

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