_{1}

The wave equation for two electrons in an external Coulomb field (helium-like atoms) has been shown to be a problem in a three-dimensional half-space. The wave-equation becomes quasi-separable in inertial coordinates. This allows to work out the electron motion in the frame of principal inertia axes. We find that non-adiabatic coupling terms constitute a fictitious force and lead to a deformation of the static potential surface. Incoming and outgoing modes of electron pairs are studied in detail, and applied to the threshold ionization of hydrogen-like atoms by electrons. Our analysis confirms the classical work by Wannier. However, we go beyond Wannier and present bending and stretch vibrations of electron pairs. The bending vibration has no influence onto the total ionization cross-section. The pair formation below threshold destroys the existence of high double Rydberg resonances. Finally, we describe the propagation of an electron pair through a linear chain of Rydberg atoms.

The motion of single electrons in atomic matter is basically well understood. However, our knowledge of the dynamics of few-electron complexes in external fields is still incomplete. We mention here two typical examples.

For the ionization of a neutral atom by slow electron impact, a simple phase space consideration suggests a threshold cross-section being linear in the excess energy, i.e.

the exponent given by

Z being the nuclear charge. For hydrogen (2) predicts the numerical value

The phenomenon of superconductivity is well described by a theory developed by Bardeen, Cooper and Schriefer (hereafter shortly BCS) [

The present paper investigates three-body Coulomb systems (nucleus + 2 electrons) within the frame of quantum mechanics. We re-derive (2) for ionization and present an alternative electron-electron attraction mechanism for electron pairs in an external Coulomb field.

The paper is organized as follows. Section 2 introduces a suitable set of coordinates to treat the electron pair as a whole. Section 3 derives a fictitious force which deforms the electrostatic potentials and presents dominantly correlated wave functions for electron pairs. Section 4 presents a discussion of the quantum version of the classical Wannier ionization theory. In particular we investigate intrinsic vibrations of moving electron pairs. Finally, we describe a possible travel of an electron pair through a linear chain of Rydberg atoms. Concluding remarks are presented in Section 5.

The treatment of three bodies (two electrons + nucleus) in the centre-of-mass system needs in general six space coordinates. Single electron coordinates are certainly not suitable for correlation studies. In this paper we restrict ourselves to pure S-states. Body-fixed coordinates coincide then with lab-fixed coordinates. We fall then down to three space coordinates only. This was already remarked long ago by Sommerfeld [

Without loss of generality we put the principal x-axis into the direction of the smallest moment. A right-handed coordinate system we obtain provided we use

In the above basis of principal axes the electron positions are given by, see [

It is easily verified that the inertia tensor is diagonal, i.e.

Our coordinates R and ψ are given by its components

Note that the moments

The angles ψ and φ may be expressed in terms of the electron position vectors. To do that we first calculate the squared interparticle distances. From (4) we get for the squared interparticle separations

with the e-e-separation

Note, however, that (9) holds only on the ridge (see Section 3), i.e.

Our coordinates may be regarded as standard spherical coordinates in a three- dimensional half-space. That space is spanned by 3-body configurations rather of single electron positions. The kinetic energy of the electron pair as one was already calculated in [

and observe that the angular part coincides with the squared orbital angular momentum provided we introduce the latitude angle

The three-dimensional halfspace is spanned by the Cartesian coordinates

and consists of the elements

From the definition it is obvious that

Classical zero-energy trajectories have been calculated by Wannier [

with the charge function

The equilibrium point is located at

the coefficients given by

We simplify now the kinetic energy to its value on the equator (ψ = 0) of the hemisphere and use the potential expansion (14). We arrive thus at the wave equation to be solved

Previous calculations using hyperspherical coordinates have shown that correlation is most important at radii

The Wannier radius diverges at threshold (E = 0). We need therefore an asymptotic solution of the wave equation. Below we take into account the terms in the wave equation which scale like

In order to get an overview on solutions we consider for the moment an adiabatic approximation, i.e. we put R = constant, see Macek [

Analogous to the oscillator we get for the antioscillator the eigenfunction

This adiabatic approach is far from being exact, but gives us hints how to proceed. We go now beyond the adiabatic approximation, and expect an angular part of the wavefunction which constitutes travelling waves along the angular directions. A suitable ansatz is

With an additional amplitude A we construct the complete wavefunction in the form,

In order to retain flexibility for the angular function we have allowed for slowly varying amplitudes p(ψ) and q(φ) in equation (18); we require for (15) the following boundary conditions

and for outgoing flux along φ

whereas for incoming flux we need

Dominant correlation effects at zero total energy emerge from a deformation of the 3-body Coulomb potential surface. A pilot study for the two-electron atom based on a liquid drop model has shown that a fictitious force emerges from surface tension [

The fictitious force is here hidden in the cross-term of the second derivative with respect to R, i.e.

We show below that both driving terms of

These terms modify the curvature of the potential surface; i.e. they modify the coefficients

To this end we substitute now (18, 19) into the wave equation, use for the amplitude the ansatz

and solve the wave equation in the Coulomb zone for large values of R. I.e. we neglect terms of the order

We now determine the non-adiabatic width γ in (18) and the front function

(26) may be regarded as wave equation for an harmonic oscillator located in a moving frame caused by the evolution along R. This causes a fictitious force which manifests itself here as a modification of the potential surface curvature.

Equation (26), finally, becomes an equation for

Inspection of (27) shows that the fictitious force mentioned above has changed the potential curvature

the shift being independent of R, see below. We eliminate all terms

This is a Riccati equation which can be solved exactly. We put

Because of the boundary condition (20) we have rejected the solution

The appendix shows that the function

This is a typical oscillator spectrum except that only even values of M occur, and the zero-point energy is one unit instead of 1/2. The first three polynomials standardized to

We come now to the function

The corresponding equation for

We eliminate all terms

Here we observe that the curvature of the potential has been changed to

with

The function

The Appendix shows that due to the travelling wave boundary condition

Electron exchange is described in our coordinates by the replacement

We conclude, therefore, that even quantum numbers

We come, finally, to the radial function A(R), see (24). That function satisfies the equation

With the ansatz (24) we derive relations for the parameters σ, and τ,

So far we have solved the stationary wave equation at zero energy

Our analysis gives insight into the process of electronic excitation. Let us consider a hydrogen-like target bombarded by a slow electron. Due to the attractive

interaction

collinear configuration (electron-nucleus-electron). In this configuration the

unstable repulsive interaction

function carries in the incoming wave mode a factor

pair switches after reflexion into the outgoing wave mode. The outgoing radial wavefunction carries now an amplification factor

We investigate now the ionization cross section of an hydrogen-like atom by electron impact near threshold. The total cross-section is then given by the ratio of outgoing flux divided by incoming flux, i.e.

with

We rewrite the second equation of (42) as pair of aggregates for outgoing and incoming flux, and subtract the incoming component from the outgoing one. We find

which yields after trivial rearrangements

(46) coincides in the vibrational ground state (N = 0) with Wannier’s formula (2). In agreement with Wannier we also confirm that the total threshold ionization cross-section is independent of the parameter

Only the groundstate N = 0 is independent of φ. We conclude that only the groundstate shows a uniform energy sharing; all triplet events and excited singlets do not.

According to Wannier the electrons escape sharply into opposite directions corresponding to an angular distribution

We conclude that excitation of a bending mode corresponding to

The observation of stretch vibrations is difficult because the cross-sections become rather small for increasing excitation

From

The immediate decay of a pair after its creation may be employed to transport an electron through a solid. We consider for simplicity a linear chain of one- electron atoms, and hit an atom at one chain end by a slow electron. According to our analysis the two electrons enter into a collinear configuration. During the further penetration of the impact electron a pair will be created. After the pair

N | μ |
---|---|

0 | 1.127… |

1 | 3.381… |

2 | 5.635… |

3 | 7.889… |

4 | 10.143… |

reflection from the turning surface the pair decays under the influence of the repulsive fictitious force between the electrons. One electron will be trapped into a Rydberg orbital whereas the other one escapes. The next neighbour atom experiences the escaping electron as an incoming one in its own frame. It will be attracted by that atom to form a new pair which creates after its decay another pair in the atom #3. At the end of the chain we obtain again one free electron. Concluding, an observer from outside may say that one electron has travelled through the chain. Actually, a macroscopic transport of electric charge has not taken place, but a wave of electron pairs has propagated.

The reader of this article might claim that the fictitious force derived here and its unusual consequences have emerged due to artefacts from the use of rather unusual coordinates. We stress, however, that this criticism is not justified. We believe that the three-body Coulomb problem is not separable in any coordinate system. The deformation of the potential surface as described above must always happen due to the non-separability of the system. Our coordinates have the great advantage that we are able to treat exactly, and surprisingly simply, the non- separability between radial and angular motion.

We remark that except for atomic hydrogen all atomic and molecular systems are non-separable. Our above treatment may therefore be regarded as roadmap to treat non-separable systems. In the present case of two electrons a NAIV consideration would have expected highly excited double Rydberg states of the form

Such unstable equilibrium configurations, however, seem always exist. We have shown that the three-electron equilibrium is an equilateral triangle, one electron in each corner and the nucleus in the center [

The prototype of a non-separable system is the simplest molecule

This worked was started long ago under the support by DFG contract SFB 276. The author acknowledges gratefully that support during 1970-80.

Klar, H. (2017) Formation of Slow Electron Pairs in an External Coulomb Field. Journal of Modern Physics, 8, 1029-1042. https://doi.org/10.4236/jmp.2017.87065

We investigate here the functions

We start with p. Its eigenvalue equation reads after multiplication with

where

For the moment it is convenient to put

We try to solve with a power series expansion, i.e.

The recurrence relation for the coefficients

The asymptotic form of

That destroys however the incoming wave in a classically forbidden region

As in the bound state situation we overcome this difficulty by a truncation of the power series. We stress, however, that a travelling wave boundary condition forces us to that step. This has nothing to do with normalization.

To this end we return to (A1) and solve it with the expansion

After trivial rearrangements this leads to the recurrence relation

The condition for truncation is

eigenvalue

follows. It is evident from (A1) that only functions even in

finite. The term

The first three polynomials standardized to

The function

and show its asymptotic behaviour

At a finite polynomial of degree

The first three polynomials standardized to

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