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Electron-electron bound state

in light of magnetic dipole-dipole interaction

2020

Arayik Emilievich Danghyan

E-mail: araik_d@hotmail.com

Introduction

There are many publications out there dealing with the problem of magnetic interaction between

elementary particles with intrinsic dipole moments. Basically, the magnetic interaction becomes

significant at sufficiently small distances; therefore, the problem is complicated by the need to take

into account relativistic effects. The derived equations with the composite potential of the Coulomb

and magnetic dipole-dipole interactions generally do not have a clear and simple analytical solution.

In this paper, we propose an approach to study a particular case of electron-electron interaction by

numerically solving the M2 equation [7] [9].

Keywords: electron-electron pair, magnetic dipole, quantum mechanics, relativistic equation.

Composite electromagnetic potential

The potential energy of magnetic dipole-dipole interaction between two electrons can determined

using formula (1).

( )( )

2

1 2 1 2

05

3

4

e e e e

r r r

Wr

−

=

(1) where

12e e B

==

(2) represents

an intrinsic magnetic dipole moment, which, as we know, is equal to the Bohr magneton

2

Bem

=

(3),

r

– the distance between electrons,

0

– the magnetic constant.

Let us rewrite formula (1), taking into account scalar products and formula (2).

2

012 1 2

3

cos 3cos cos

4B

Wr

−

=

(4) where

12

represents the angle between vectors

1e

and

2e

,

1

and

2

– the angles between the magnetic moment vectors and the line joining particles.

It is clear that the potential energy (W) depends on the mutual arrangement of dipoles.

The following mutual arrangements of dipoles [1] are of interest to us (Fig. 1)

r

r

1e

2e

1e

2e

1e

2e

1e

2e

a) b) c) d)

Fig. 1 Mutual arrangements of the intrinsic magnetic moments of electrons.

Cases a) and c) represent electron attraction; hence; the energy is negative. Cases b) and d), on the

other hand, represent electron repulsion; therefore, the energy is positive.

It is clear that electron pairing requires attraction. Therefore, we can determine the energy (W) for

cases a) and c) using formula (4), in which case we obtain:

a)

0

12 180

=

,

12

cos 1

=−

,

0

12

90

==

12

cos cos 0

==

,

2

03

4B

a

Wr

=−

c)

0

12 0

=

,

12

cos 1

=

,

0

12

0

==

,

12

cos cos 1

==

,

2

03

2

4B

c

Wr

=−

Therefore, the attractive potential energy for case c) is twice the value of the energy for case a). Note

that case a) corresponds to oppositely directed electron spins. Therefore, the total spin is equal to 0.

The electron spin direction coincides for case c); therefore, the total spin is equal to 1.

The Coulomb potential energy of interaction between two electrons is represented as follows:

2

0

4

Coulomb e

Wr

=

The total potential energy of electromagnetic interaction for case c) can be calculated

as follows:

22

030

2

44

B

c Coulomb e

U W W rr

= + = − +

(5)

Let us rewrite formula (5) to make it more convenient. To do this, we will use the well-known relation

00 2

1

с

=

, which can be rewritten as follows:

02

0

1

с

=

(6). Let us substitute values

0

(Formula 6)

and

B

(Formula 3) into the potential energy formula (5). We obtain:

2 2 2

2 3 2

00

2

() 4 4 4

ee

Ur с r m r

= − +

(7). Let us rewrite formula (7) using Hartree atomic units.

In the Hartree atomic unit system, the following values of physical constants are accepted:

00

1, 1, 1, 1, 137.03599971,4 1a m e c

= = = = = =

Finally, we obtain the following equation for the

composite potential energy of electromagnetic interaction:

23

11

() 2

Ur с r r

= − +

(8). Let us graph the

resulting equation (Fig. 2)

Fig. 2 Graph of the composite potential energy of electron-electron interaction.

As we can see, the obtained potential consists of a Coulomb barrier and a deep potential well and has

a characteristic maximum.

Let us determine the coordinates of the maximum potential energy.

0.008937

Max

r=

in Hartree atomic

units. If we multiply this by the Bohr radius, we will obtain

0.008937*52.9 0.472788

Max

r==

pm.

( ) 74.592950

Max

Ur =

in Hartree atomic units. If we multiply this value by the Hartree energy, we will

obtain:

( ) 74.592950*27.2 2028.928244

Max

Ur ==

eV.

M2 radial equation for the electron-electron pair

Let us designate the electron-electron pair by the “ee” symbol. The ee pair is a hydrogen-like entity.

And since the masses of these two electrons are equal, it is necessary to replace the mass (m) with the

reduced mass (

2

m

=

) in the equation for the hydrogen atom [8]. Let us write the M2 radial equation

for the ee pair:

( )

( )

2 4 6 22

2

2 2 2

1

21 0

()

ll

d R dR c

R c R

dr r dr r E U r

+

+ − − − =

−

(9)

It is difficult to solve the resulting equation using the analytical methods due to the complex

dependence of the composite potential. Therefore, we will apply a numerical solution method using

FlexPDE software package http://www.pdesolutions.com/.

The numerical solution of the radial equation in the spherical coordinate system

Fig. 3 The radial wave function of the electron-electron pair

Fig. 4 The radial probability density of the electron-electron pair

As a result of the numerical solution of equation (9), the following parameters of the ee pair were

obtained: the orbital radius is equal to 0.134876 pm. Note that this is the distance between the

electrons, between the maxima of the probability density. The binding energy is equal to 265363.7

eV. As we can see, the convergence of the equation is very high, which means that there is a high

probability that this bound state of electrons may actually exist.

The numerical solution of the M2 equation in the cylindrical coordinate system

As we can see, the radial equation (9) includes the angular momentum quantum number

l

. But, as

we know, it doesn’t include the magnetic quantum number m. Nor does it give us a complete picture

of the spatial configuration of electron shells and the magnitude of the magnetic quantum number m.

This data can be obtained by solving the equation in the cylindrical coordinate system.

The numerical solution of the equation in the cylindrical coordinate system showed us that the

convergence is possible only if the value of the magnetic quantum number is

1m=

. We also note that

further research is needed in order to obtain a solution for the orientation of magnetic moments a).

Most importantly, the possibility of existence of a deep bound state of the electron-electron pair has

been theoretically proven.

If this theory is experimentally confirmed, it will be further developed.

Fig. 5 Wave function of the ee pair in the cylindrical coordinate system

Fig. 6 Probability density in the cylindrical coordinate system

Fig. 7 Projection of probability density in the cylindrical coordinate system

Therefore, as a result of the numerical solution of the equation in the cylindrical coordinate system,

the following results have been obtained: the orbital radius is equal to 0.134793 pm. Again, note that

this is the distance between the maxima of the probability density, which is the diameter. The binding

energy is equal to 265363.7 eV. And, as we said, the magnetic quantum number is

1m=

. In addition,

we see that the spatial distribution of the electron density is represented by a toroid. However, if we

take a look at the solutions of the radial equation (9) in the spherical coordinate system, we will see

that it resembles a spherical distribution with an empty region inside.

Results and discussions

Simultaneous consideration of the Coulomb electrostatic repulsion and magnetic dipole-dipole

attraction between two electrons gives us reason to believe that a stable bound state may exist when

the electrons approach each other close enough. This is also mentioned by many other authors in their

publications. However, it has not yet been possible to obtain an accurate calculation of all the

parameters for such a state.

In this paper, the possibility of existence of a deep bound state of the electron-electron pair is proved

based on the numerical solution of the M2 relativistic equation. The exact values of the ground state

parameters have been obtained. The orbital radius is equal to 0.1348 picometers. In fact, this is the

diameter, since the equation is set up taking into account the reduced mass. Therefore, the variable

r

represents the distance between the electrons. The binding energy: 265363.7 electron volts. The total

spin:

1s=

. The magnetic quantum number:

1m=

. The wave functions and probability density

distributions have been obtained in the spherical and cylindrical coordinate systems. It has been

shown that this distribution is represented by a toroid. This characteristic form of the probability

density gives us reason to believe that the ee pair in its ground state is a neutral entity.

In addition to the fact that the above calculations and results are of theoretical interest, they may also

be of great practical value. As shown, the formation of the ee pair leads to the release of energy at

265363.7 eV. This is of great practical value, which means that environmentally friendly power plants

could be built based on this phenomenon.

We think there has been provided enough evidence to conduct an experimental search for an elusive

particle.

List of publications:

1. TO THE POSSIBILITY OF BOUND STATES BETWEEN TWO ELECTRONS Alexander

A. Mikhailichenko, Cornell University, LEPP, Ithaca, NY 14853, USA

2. Electromagnetic Theory of the Nuclear Interaction

Bernard Schaeffer

3. On certain features of electron-proton interaction

Popenko V.I. Scientific and Production Corporation "Kiev Institute of Automation"

4. F. Mayer, J. Reitz, “Electromagnetic Composites at Compton Scale”, arXiv:1110.1134v1

[physics.genph] 10 Sep 2011

5. Solution of the Dirac equation with Coulomb and magnetic moment interactions

Barut, A. O. Kraus, J.

Journal of Mathematical Physics, Volume 17, Issue 4, pp. 506-508 (1976).

April 1976

6. Two-body problems with magnetic interactions Hesham Mansour, Ahmed Gamal

Journal of Nuclear and Particle Physics 2019; 9(2): 51-58

7. Danghyan A.E. “New equation of relativistic quantum mechanics”

8. Dangyan A.E. “Hydrogen Atom. Exotic states. Part one"

9. Dangyan A.E. “Hydrogen Atom. Exotic states. Part two”