LECTURE 
15 April 2003 

Language consultation: J.B., Ontario  Canada. 

HELIUM ATOM.
TWO SPINNING ELECTRONS IN THE COULOMB FIELD OF THE NUCLEUS. 


Since the time of J.J.Thomson and E. Rutherford we do know that the Helium atom, the second element in the Mendeleev periodic table of elements, has a positively charged nucleus and two electrons. We do know, moreover, that 24.59 eV is needed to remove the first electron from the atom, while 54.42 eV is needed to remove the second electron. These two quantities measured directly in the experiment, called the first and the second ionization potential, respectively, play the role of the identification card of the atom. It is the aim of the theory to reconstruct the internal order existing inside the atom on the basis of this scrap of information and with the help of the mathematical formalism developed within a definite system of laws


A false start. A lot of research, aimed at the construction of a dynamic model of the helium atom was undertaken just immediately after Rutherford's discovery. Unfortunately, the first attempts to identify trajectories of two pointlike electrons moving in the Coulomb field of the pointlike nucleus were not crowned with a success. N. Bohr proposed, by analogy to his erroneous, as we already do know, circular model of the hydrogen atom a circular model of the helium atom, see Fig.1a. In a quite short time it became clear that the atomic model with two electrons circulating around the nucleus may have nothing in common with a physical reality, as the model had, among others, evidently wrong magnetic properties. In reality, measurements in SternGerlach experiment did not leave room for doubts that the dipole magnetic moment of the helium atom is equal to zero, while the magnetic moment calculated for the Bohr circular model was as high as 1.6 Bohr magnetons. Langmuire, to be in agreement with measurements, suggested the other form of collective motion of two electrons, see Fig.1b. In this case, however, electrostatic interaction energy of two electrons appeared to be much too high. Moreover, the helium atom is an exceptionally stable element, while the collective motion of two electrons in the Langmuir model is highly unstable. A lack of success in solving the problem was presented by advocates of quantum mechanics as an argument supporting the thesis that the atom cannot be described on the basis of deterministic laws of classical dynamics. As a result, research on the construction of the dynamic model of the atom within the concept of a localized electron moving along a welldefined orbit, was stopped for a long time. Only a few decades later, in light of great successes of the binary encounter atomic collision theory which I developed on the basis of Newtonian dynamics and the Coulomb interaction law, the problem of the model of the atom became current again. In the atomic collision theory, which I was talking about two lectures ago, the atomic model plays the role of initial conditions for a collision problem and forms the inherent part of the calculation formalism.
Atomic model and the collision theory. It is quite obvious that results of a collision more or less depend on structure of colliding atoms. In the classical atomic collision theory, theoretical results depend directly on the atomic model used in calculations. One can, therefore, get some information about the atom, by comparing results of calculations carried out for different atomic models, with experimental data. Such procedure led me in 1965 [Phys. Rev. Lett. 14 (1965) 1059] to quite a fundamental discovery. Confronting the calculation with the experiment did not leave room for doubts that 

motion of atomic electrons is dominated by radial kinetics.


In this way, I have arrived at a freefall atomic model concept with electrons symmetrically localized in space and moving collectively towards the nucleus, see Fig.1c.


Bohr (1913) 
Langmuir (1921) 
MG (1965) 

m = 1.6 m_{ 0} s = 0.25 
m = 0 s = 0.40 
m = 0 s = 0.25 

EXPERIMENTAL VALUES: m = 0, s = 0.296 

Fig. 1. Three various models of the helium atom  a history of the research on electronic structure of the helium atom.


Although the simple freefall atomic model enabled to solve various problems of atomic physics, in particular, to describe quite accurately atomic diamagnetism and Van der Waals forces and to disclose the physical nature of the Ramsauer effect [J. Chem. Phys. 62 (1965) 2610, 2620, 2629], it was clear from the very beginning that a simple radial motion may be considered only as a starting point for a description of a more sophisticated physical reality.
The important hint showing the direction of further research towards deciphering the internal structure of the atom was delivered by measurements of small angle scattering of slow electrons from atoms  in the particular case of the helium atom it was a small hump in scattering cross section at energy of few electronvolts. Theoretical analysis of this hump has shown that it has its origin in dipole oscillations of the electric field of the helium atom. It was quite a strong argument that the collective motion of two electrons cannot be exactly radial. Two spinless electrons in a collective motion in the Coulomb field of the nucleus. To make a step further beyond a simple free fall atomic model, collective motion of two electrons in the Coulomb field of the nucleus, was subjected to a detailed analysis [Fizika, 19 (1987) 325]. Results of this research may by summarized in the following way. 

Each orbit is determined by the value of the binding energy W which is directly related to the measured in the experiment ionization potentials, U_{ i} and U_{ ii} in a following way


W = 1/2 (U_{ i} + U_{ ii}) .


The screening parameter s, which represents interaction energy between electrons, which for each of the closed orbits may be calculated theoretically, is given by the relation


( Z  s )^{ 2} U_{i}^{H} = W ,


where U_{ i}^{H} is the ionization potential of the hydrogen atom. The analysis shows that there exists a whole infinite discrete spectrum of closed orbits, some of them are shown in Fig.2.


Fig. 2. The three simplest closed orbits obtained by a numerical integration of Newton equations of motion for two electrons moving collectively in the Coulomb field of the nucleus.


Comparing the value of the screening parameter calculated for the given orbit with that obtained from the relation given above, one could try to identify the orbit which could represent the motion of electrons in the real helium atom. Unfortunately, on the orbit which could represent the helium atom, electrons come to each other to a distance at which spinspin magnetic interaction can no longer be neglected, see Fig.3.


SCREANING COEFFICIENT s 

SPINSPIN INTERACTION: 
STRONG 
WEAK 

Fig. 3. The screening parameter s for various closed orbits as a function of energy of transversal oscillations  E_{ q}.
A gray horisontal line on the figure marks the experimental value of the screnning parameter: 0.296. 

In view of the above it was necessary to include into theory spin properties of the electron.
Electron is a spinning body. An extension of our theoretical considerations on magnetic properties of the electron implies inclusion into the theory Euler equations of motion describing behavior of rotating bodies. Fortunately, slow changes of orientation of electron spin axis that result from noncentral magnetic interactions can be neglected and the problem can be investigated, analogously as it has been done in the case of the hydrogen atom, in so called rigid top approximation. In such a case, motion of electrons is determined by three various types of interaction. To write these terms explicitly let us introduce a following notation: let us denote a distance of the first electron to the nucleus by r_{ 1}, the distance of the second electron by r_{ 2} and the distance between the electrons by r_{ 12} 

r_{ 12} = r_{ 1}  r_{ 2} =  r_{ 21} ,


v_{ 12} = v_{ 1}  v_{ 2} =  v_{ 21} .


1. Electrostatic interaction of the electrons with the nucleus and the mutual interaction of the two electrons:


(1) 

2. Spinspin magnetic interaction between the electrons:


(2) 

3. Electromagnetic interaction between the electrons and between each of the electrons and the nucleus:


(3) 

where A_{ s} is the vector potential describing spin magnetic field of the electron, m is the magnetic moment of the electron and s_{ 1} and s_{ 2} are unit vectors describing orientation of the considered electrons in space.
In view of the above, the Lagrangean describing the behavior of the considered electrons in the Coulomb field of the nucleus will have the form: 

(4) 

where L represents angular momentum of the given electron (1 or 2) with respect to the nucleus


(5) 

Two spinning electrons in a collective motion. Detailed analysis of the Lagrangean given above has shown that there are in principle two possible forms of a collective motion. The first, with parallel oriented spins and the vector of angular momentum parallel to spin axes of the electrons and the other with antiparallel oriented spins and angular momentum equal zero:


and


and

L = 0 .


In the first case the motion of both electrons is axially symmetric while in the second case the motion has a mirrorsymmetry. In both cases the distance to the nucleus and the speed of both electrons are identical and their spin axes are at all times perpendicular to the vector of their relative position r_{ 12}:


 r_{ 1}  =  r_{ 2}  = r ,


 v_{ 1}  =  v_{ 2}  = v ,


and


Finally, the Lagrangian describing the collective motion of two electrons in the Coulomb field of the nucleus assumes the form:


(6) 

Equations of motion for two electrons with paired spins derived from the Lagrangean given above and written in xy coordinates have the form:


(7) 

(8) 

Unfortunately, although the integral of energy is known:


(9) 

equations of motion must be integrated numerically.
Numerical integration. To perform numerical integration it is convenient to write equations of motion in a dimensionless form. Thus, introducing the following unit of length 

(10) 

and the following unit of time


(11) 

the three equations describing the problem assume the form:


(12) 

(13) 

(14) 

where underlined symbols represents respective variables in a dimensionless form, for instance r = r / l_{ 1}. Coefficients k and g are given by


(15) 

(16) 

where m_{ e} is the magnetic moment of the electron in Bohr magnetons


(17) 

In the case of helium atom Z = 2, W = (U_{ 1} + U_{ 2}) / 2 = 39.5 eV and, therefore


k_{ He} = 1.317 10^{ 4} ,


g_{ He} = 1.052 10^{ 4} .


Until the electrons are not too close to each other, that is until


g / x^{ 2} << k ,


motion of both electrons is almost the same as in a spinless case we have discussed in the
previous paragraph.
Starting the calculations we meet, however, in the case of antiparalel orientation of spins a quite fundamental difficulty. Calculations show that two electrons may approach each other at a distance at which spinspin magnetic interaction exceeds Coulomb repulsion and the problem looses its physical sense. Thus, we have evidently arrived at a point where the concept of the point electron can no longer be used.
To avoid spin annihilation catastrophe it was necessary to introduce some restrictions on the spin magnetic interaction energy of two electrons. It was evident that interaction energy cannot exceed total energy of the system, which is equal to the rest energy of two electrons. Assuming, therefore, that half of the rest energy of the electron has a magnetic nature and half has electrostatic nature we have the following estimate for spin magnetic radius of the electron:


(18) 

Fortunately, a global shape of the electron orbit appeared to be insensitive to the value of the magnetic radius of the electron and we could solve the undertaken problem.
Solution of the problem. Searching for a closed orbit which could represent the real atom was simplified greatly by the fact that the dominant part of the orbit is determined by Coulomb interaction. One could, therefore, have closed trajectories of a spinless problem as the basis for numerical integration of equations of motion for electrons with a spin. Taking into consideration the experimental fact that the helium atom does not posses permanent dipole moment it was clear that electron orbital should be symmetrical with respect to the yaxis of the system. It was convenient, therefore, to define initial conditions for numerical integration at a point 

x = x_{ 0} ,


y = 0 .


Components of the initial velocity coresponding to this point, that is:


d x /d t  _{ x = x o, y = 0} = 0 ,


d y /d t  _{ x = x o, y = 0} = v_{ y0} ,


are determined by energy conservation law. At the spinspin interaction the simplest closed orbit appeared to have the form as shown in the figure below.


Fig. 4. Electronic structure of the helium atom.


Although the nicely looking orbital, as shown in fig. 4, was a strong emotional argument that we have discovered one of the biggest wonders of the invisible microworld, it was necessary to give a rigorous proof that the found solution describes a physical reality. The fact that the calculated value of the screening parameter


s = ( 1 / 4x ) / ( 1 / r ) ,


that determines the binding energy of the atom is in perfect agreement with the experiment:


s^{ exp.} = 0.296 ,


s^{ theor.} = 0.302 ,


is a quite important argument that the picture shown in fig. 4 represents accurately motion of the electrons in the helium atom. To see some other arguments please look at the fig. 5 and click here.


Fig. 5. Helium atoms in a super conductiung state bount by Van der Waals forces form a thin flexible rope. Motion of the rope on the vall is determined by the gravitational force which is proporcinal to the difference of the lengthes of the rope on both sides of the vall.


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