8 October 2003
Language consultation: J.B., Ontario - Canada.
radial kinetics - angular symmetry - collective motion

Hydrogen atom.

Helium atom.
Since the time of J.J. Thomson we do know that a negatively charged electron is a universal component of matter. Since the time of E. Rutherford, we also do know that almost the entire mass of the atom is situated in a positively charged point-like nucleus. Unbelievable, but one century later, we still cannot say much about the behavior of electrons bound in the Coulomb field of a heavy nucleus. This ignorance is a result of the false step that was taken at the beginning of the twentieth century. What has happened that instead of a precise picture of the atom we have today a Y function cloud, which has nothing in common with a physical reality? I did try to answer this question at the beginning of my www.course, see lectures, 1,2 - those, who know Polish can learn a little bit more on this point from my book "Sprawa atomu". In the last two lectures, 4 and 5, I presented the way, which has lead me to the identification of electron orbits in the two first elements of the Mendellev table - how do these elements look, see two animated pictures on the left. In the actual lecture I will present a general strategy of the research aimed at the construction of the modern, that is containing complete information on dimensions and the form of atomic electron orbits, periodic table of elements. Almost everyone, who is familiar with elements of differential and integral calculus has a chance of making his own contribution to this fascinating undertaking.
Spinning electron in the atom. A discovery that electrons in the atom move radially [MG, Phys.Rev.Lett. 36A,(1965)1059] was a turning point in the research on the electronic structure of the atom. This discovery, achieved on the grounds of the concept of a localized electron, was the proof that deterministic laws of classical dynamics were abandoned too early, at least. The successful deciphering of electron orbits in the hydrogen and in the helium atom has shown that really there are three basic forces which determine the behavior of electrons in the atom. Those are:
the Coulomb force:
the Lorenz force:
the magnetic force:
F m = - grad ( m . H s ) .
The Lorenz force F L and the magnetic force F m, are inversely proportional to the third power of distance
spin magnetic field:
Since spin magnetic field decreases very rapidly with distance from the electron, with a cube of the distance, the influence of the Lorenz force FL and the magnetic force Fm on the motion of the electron is limited to radii smaller than the electron Compton wavelength, see Fig. 1.
Fig. 1. Interaction energy, electrostatic and magnetic, as a function of the distance between electrons. The magnetic interaction at a distance between electrons on the order of the Bohr radius is by four orders of magnitude smaller than the electrostatic interaction.
In view of the fact that motion of atomic electrons is dominated by electrostatic interaction, the first part of our considerations on the structure of the atom can be limited to the following set of differential equations:
where Ne is the number of electrons in the atom and Z is the charge of the nucleus. To calculate electron trajectories one must specify 2 N e constants of motion defining the atom considered in our formalism. Unfortunately, our knowledge of the constants, which are in fact responsible for the global shape of the atom, is at the moment almost non-existent.

It is a chief task of the theory to find constants of motion defining the internal order in the atom, which represent a final stage of radiative evolution of the atom.

In principle, it is possible to achieve this goal in two different ways: theoretical - extending the postulates of the theory on radiative phenomena, and semi-theoretical - investigating results of measurements with the help of the mathematical formalism developed on the basis of the known laws of classical electrodynamics (the same which are used to describe the motion of atomic electrons). Finding a solution to the problem in a purely theoretical way is rather a hopeless task. All what we have at the moment at our disposal are quantum rules formulated in the early days of atomic physics, which determine conditions of radiationless motion of the electron. Among them is, the "quantum" integral discussed in the lecture 2:
p . d l = n h      n = 1, 2, 3, ... .
Unfortunately, the search for orbits satisfying this criterion, in the case of multi-electron atoms is not an easy task. Moreover, we are not sure that this criterion works correctly in the presence of spin magnetic forces. Thus, all we can do at the moment is search for help in atomic spectroscopy, X-rays spectroscopy and various atomic collision experiments - in any experimental data there is always hidden some information about electrons moving in the atom.

Key experiments and the structure of the atom. The first crucial information on the internal structure of the atom we owed to X-rays spectroscopy. The diagram, shown in Fig.2, developed by H. G. J. Moseley [Phil.Mag. 26(1913)1024] at the beginning of the twentieth century on the basis of thousands of measured lines, has led to a conclusion that electrons in the atom are grouped forming K, L, M, N... shells and s,p,d,f... sub-shells, with a precisely defined number of electrons in each of the atomic shell (sub-shell).
Fig. 2. X-rays terms, T, as a function of the atomic number Z. The diagram elaborated by Moseley enabled to look for the first time inside the atom. Since the work of Moseley, we do know that electrons in the atom are grouped forming atomic shells (K,L,M,N,O,P) and atomic sub-shells (s,p,d,f).
Moseley, introducing a characteristic for each shell numerical factor s found the rule relating frequency of the radiation v with the atomic number Z :
~ ( Z - s ) 2 .
Not much later, A. Sommerfeld interpreted the Moseley relation in a spirit of Bohr-Rutherford atomic model and on the basis of energy conservation law ascribed to the observed terms concrete energies. In this way, a concept of atomic energy levels defining positions of electrons in the Coulomb field of nucleus was born, see Fig.3.
Fig. 3. The figure shows how electrons in the atom are grouped. The fact that electrons of the same group are energetically equivalent was the first hint that electrons in the atom move collectively.
The other important information on electronic structure of the atom we owe to C. Ramsauer, who observed scattering of slow electrons from argon [Ann. Phys. 12 (1932) 529, 837]. In his results was hidden important information on electronic structure of the argon atom. Unfortunately, only in 1970 this information was deciphered [MG, Phys. Rev. Lett. 24 (1970) 45]. Analysis of experimental data I carried out with the help of small angle scattering formalism showed that the leading term of the expansion of the electric field of the argon atom in electric multipoles is a dynamic quadrupole, see Fig.4.
Fig. 4. Cross section of Argon for small angle scattering of low energy electrons: points represent experimental data, solid lines are the results of theoretical calculations at various assumptions on the character of the asymptotic form of the electrical potential of the atom; n is the power with which electric field decreases with the distance from the atom. The observed decrease of the cross section at very small electron velocities is a characteristic feature of the oscillatory interaction between the scattered particle and the scattering center.
Analysis of experimental data for other heavy noble gases gave similar result: the leading term of the expansion of the electric field of the atom in electric multipoles is a dynamic (oscillatory) quadrupole. From atom-atom collisions it was possible to identify the next term of the expansion. Careful analysis of experimental data showed that it was a static octupole with a three-fold axis of symmetry. In this way I have found that electrons in the atom are situated regularly, that the atom has the axis of symmetry and that electrons in the atom move collectively - electric field of the atom has a periodic component [MG. J. Chem. Phys. 62 (1975)2610, 2620, 2629]. The deciphered in this way picture of the electric field of the noble atom is shown in Fig.5.
  F20 = (Q20 / r 3) 1/2 (1 - 3 cos 2q) coswt            F33 = (Q33 / r 4) sis 2q) cos3j
Fig. 5. Electric field of noble gases at large distances from the atom. There is seen a periodically varying quadrupole component of the field, on the left, and a short range static component with a three-fold axis of symmetry, octupole component of the field, on the right.
All this information on the electron structure of the atom, deduced from various atomic collision experiments and X-rays measurements, put together enabled the construction of a model of the atomic shell as shown in Fig.6.
p - subshell
d - subshell
f - subshell
Fig. 6. The distribution of electrons in various atomic shells deduced on the grounds of atomic and X-rays spectroscopy. Triplets, quintets ad septets of atomic spectroscopy have origin in the angular symmetry of atomic shells.
In this way, with the help of the experiment, a hopelessly difficult multi body problem was reduced to a single particle problem, with the following equation of motion:
With the help of the relation given above one can calculate trajectories of collectively moving electrons. To perform such calculations one must specify the value of the binding energy W and the form of the screening coefficient s, identifying the character of the collective motion of electrons inside the given shell. The form of the screening coefficient may be easily found from simple geometrical considerations, see Fig. 7.
Fig. 7. There is shown geometry of the simplest 2k electron configuration consisting of two identical subsystems, which enables one to calculate the screening coefficients for various atomic sub-shells.
In a general case, for a distribution with k-fold symmetry, the screening coefficient has the following analytical form [MG, Fizika 19 (1987) 325].

If the binding energy W is specified one can perform integration and calculate the trajectory.

Free-fall configuration as a model of the electron shell. Among various solutions of the problem there is one of special interest for our considerations. This is the case, when angular components of the interaction between electrons cancel each other and all electrons move along exactly radial free-fall trajectories - a particular free fall configuration is shown in the animated drawing on the left. In such a case, the screening coefficient is a constant quantity, which only depends on the geometry of electrons situated regularly on the surface of the sphere. Thus,
and, therefore,
Z eff = Z - s ff .
To relate the purely dynamical considerations to the atom one must take into account that at specific conditions only an electron does not emit radiation. The radiation-less states of the electron can be identified by the path integral (6). Performing elementary calculations we obtain the following relation describing a discrete spectrum of energy levels:
W ff = W 0 ( Z - s ff ) 2 / n 2 ,
W 0 = 1/2 m c 2 a 2 = 13.6 eV .
Since in the case of a Kepler motion the binding energy W and the main axis of the ellipse, 2 a ', are related in a following way:
2 a ' = Z e 2 / W ,
therefore, replacing Z by Z eff we can write the relations describing the radius of the considered free-fall configuration:
r ff = 2 a 0 n 2 / ( Z - s ff ) ,
a 0 = h 2 / 4 p 2 m e = 0.503 A .
Analogically, taking into account that the period of motion T and the binding energy W are in the Kepler motion related in the following way:
we obtain for the considered free-fall configuration the following spectrum of frequencies
w ff = w 0 ( Z - s ff ) 2 / n 3
w 0 = 4 p 2 m e 4 / h 3 = 4.13 . 10 16 Hz .
Now, we will present some examples showing how closesly a simple free-fall model of the atomic shell may represent a physical reality.

Free-fall theoretical model and the experiment. One can be surprised but the above presented, trivial in fact, free-fall model of the atomic shell appeared to be a very effective tool in theoretical analysis of various atomic phenomena. Now, a few examples how this model works.

BINDING ENERGIES. The most important parameter describing the atom are binding energies of electrons in various electron shells of the atom. Those were for the first time identified on the grounds of X-spectroscopy and later confirmed on the grounds of atomic collision physics. Some values obtained in this way for some closed electron shells are given in the table below. For these shells values calculated using theoretical formula (10) with the screening constant calculated according to the relation (8) are also given. The obtained results are given in the table below. The agreement with the experiment is astonishing!
Tabl. 1. Atomic shell energies.
type of the shell
1s shell
39.5 eV
99.0 eV
1279 eV
4273 eV
s 2 = 0.25
41.6 eV
1293 eV
4285 eV
(2s+2p) shell
119 eV
659 eV
s 8 = 2.47
104 eV
623 eV
(3s+3p) shell
72.2 eV
s 8 = 2.47
46.6 eV
ELECTRON SHELL RADII. There are some physical phenomena which depend on the electron shell radii. One of them is the absolute value of the leading multipoles describing electric field of the atom, which can be measured in low energy experiments. Results of calculations carried out with the help of the formula (13), for some of the few simplest atoms are given below in table 2.
Tabl. 2. Atomic shell radii ( in 2 a 0 )as calculated using formula 13.
1s shell
(2s+2p) shell
(3s+3p) shell
Among various physical properties, which are directly related to the dimensions of electron shells of the atom, is atomic diamagnetism. This property can be easily calculated with the help of free-fall model of the atomic shell and the following formula, for the coefficient of magnetic susceptibility, derived on the grounds of classical electrodynamics:
where the factor b depends on the orientation of the atom with respect to the external electric field (for isotropic oriented atoms it is equal to 2/3). Thus all that we need to calculate atomic diamagnetism is to calculate dimensions of all atomic shells with the help of equation (13). Comparison of simple arithmetic with measurements is again surprising, see Fig.6.
Fig. 8. Coefficient of magnetic susceptibility for various atoms and molecules measured in the experiment and calculated on the grounds of simple free-fall atomic model [MG,  Journ.of Magn. and Magn. Mat. 71 (1987)53].
FREQUENCIES. With the help of formula (16) we can calculate the other important atomic parameter: the oscillation frequency of the atomic field, which can be quite accurately determined from small angle scattering experiments I was talking about in lecture 3. Comparison of calculations with the experiment is presented in the table below.
Tabl. 3. Oscillations frequencies, as calculated from free fall atomic model and derived from electron atom scattering experiments.
shell frequency (in w 0)
theoretical estimation
Eight decades old enigma solved. At the beginning of the twentieth century various rules were established describing reach x-rays spectra. One of the most exciting was a discovery that outer shell electrons can screen inner shell electrons, what is clearly seen on a diagram worked out by A. Sommerfeld, see Fig.8.
Fig. 9. Screening constants as deduced from x-rays spectroscopy by A. Sommerfeld (circles) and constants calculated on the basis of free-fall atomic model. 
With the formulation of free-fall atomic model concept, the old puzzle: how can outer shell electrons screen inner shell electrons - a simple explanation was found - just radially moving outer shell electrons spend some time closer to the nucleus than inner shell electrons [MG, Procc. of the III-rd Siberian Conf. on mathematical problems of the space - time (STP 2000) Nowosibirsk, 22-24 July 2000, Ed. Institute of Mathematics - 2001, page 135]. The phenomenon is visualized on Fig. 10.
Fig. 10. Illustration of how external shell electrons screen the internal shell electrons.
With the free-fall atomic model it was possible not only to explain qualitatively the astonishing phenomenon, but to give quite a good quantitative estimation for the value of the outer shell screening factor. Differences in the theoretical values of the screening coefficients of s,p,d - subshells
D s s p = s p - s s
D s p d = s d - s p
quite well reproduce the observed differences. Taking into account that motion in the whole atom is perfectly synchronic and assuming that periods of motion of subsequent electron layers: K,L,M,N,O, are related in a following way:
T k / T l = 2 k
we immediately obtain the following relation for the value of the outer shell screening coefficient:
s out = S N out 2 - ( n - l ) / ( n - 1 )
where n l is number of electrons in the given outer layer (each layer contains s,p,d - subshells) and the sum is taken over all outer layers l = 1,2,3,4,5 (K,L,M,N,O). The values of the screening constant for the L shell obtained in this way are shown in a diagram developed by A. Sommerfeld, see Fig.9.

Using a simple free-fall atomic model it was possible to explain a lot of phenomena, like the one shown in the figure below.
Fig. 11. Electron structure of L-electron shell and X-ray transitions on the vacancy in the K-shell.
What further? One cannot have doubts that the simple free fall model of the atomic shell, although very useful, may be considered as the first step on the way to a precise description of the structure of the atom. Further progress towards a construction of a realistic model of the multi electron atom needs, however, like in the case of H and He atoms, spin properties of the electron included into the theory. But this exciting stage of research is still before us
Fig. 12. Electrostatic energy as a function of the polar angle q for two particular cases of planar motion in a six electron group. At some value of the polar angle q ( q = q ff ) the polar components of the force disappear and electrons can move radially to the nucleus.
Fig. 13. A possible configuration of spinning electrons in a p-subshell.