Fig. 2. Diagrammatic representation of lithium atom.

External electron is rather wide apart from a nucleus and from two electrons of internal shell – i.e. from the atomic core. The dimensions of this atomic core as it was found earlier are rather small - about 0,364R_B (table 1). Therefore it is possible to consider, that valent electron is in the field of a total positive unit charge of atomic core. In this respect such atoms are hydrogen-like.

It would seem that this external electron should completely replicate the behavior of electron in atom of hydrogen and satisfy formulas (3-6).

But it is not exactly so.

If radius of the external electron shell of lithium atom is as one of hydrogen (n=1) so its ionization potential would be equal to the ionization potential of hydrogen (13,6 eV). But lithium ionization potential is only 5,39 eV and specifies that the radius of external electron shell of atom of lithium is essentially (in 2,523 times) more, than of hydrogen. It follows from formula (5), which shows interdependence of electron energy in atom of hydrogen (numerically equal to potential of ionization) and radius of its electron orbit:

(13)

It means that external shell of lithium atom should contain more than one spatial period, and number n should be more then 1.

But assuming n=2, we obtain too big distance R equal to four Bohr radiuses. The ionization potential with such distance should be equal 3,4 eV, that mismatches actual value 5,39 eV.

Nevertheless, this situation has the logical settlement.

Let's determine wave length of electron proceeding from its kinetic energy which, as we saw, numerically corresponds to ionization potential. Hence

(14)

So we receive wave length of electron and corresponding radius

(15)

Substituting in (15) ionization potential of lithium 5,39 eV, we get:

Here we used new notation R _λ for conditional radius (spatial phase factor), parameter characterizing rate of spatial wave change. Actually this parameter was used already in formula (2).

As we expected, the external electron shell of lithium atom is characterized by number (n=1,593) which, unlike hydrogen (n=1), is not integer and is in an interval between 1 and 2!

However actually the external electron shell of lithium atom also is resonant and is characterized by the integral number of wave-lengths. It is achieved due to excitation of higher harmonic.

That is, some unknown to us the integer number of wave-lengths N₁, describing a shell resonance, should be result of multiplication of number 1,593 by other number N₂ - number of harmonic. Therefore we should make some simple calculations and select suitable numbers.

For example, having divided number N₁=8 by n=1,593, we receive N₂=5,02, that is, the fifth harmonic enables to obtain eight wave-lengths resonance of electron shell of lithium atom. Moreover the formula (4) for real shell radius appears valid when n is not an integer number

(16)

This value really meets actual magnitude of external electron shell radius of lithium atom, starting from formula (13)

(17)

There are many atoms similar to lithium with one electron on the external electron shell, so we have an opportunity to make analogous calculations in these cases too. Results of all calculations are shown in table 2.

We attach such importance to these calculations to be convinced of natural character of results. Data represented in table 2 let us make a conclusion on validity of hypothesis.

The nature, really, uses resonances generated on multiple harmonics, therefore atoms become more compact and in electron shells meanwhile can be "placed" greater number of electrons!

Table 2

Atom	φ1, eV	n	n^2	R/RÁ	n≈ N1/N2
3 (Li)	5,39	1,593	2,54	2,52	8/5,02
11 (Na)	5,138	1,632	2,66	2,65	5/3,06
19 (K)	4,339	1,775	3,15	3,13	7/3,94
55 (Cs)	3,893	1,875	3,51	3,49	15/8,00
37 (Rb)	4,176	1,81	3,27	3,26	9/4,97
47 (Ag)	7,574	1,342	1,8	1,79	4/2,98
78 (Pt)	8,96	1,236	1,53	1,52	5/4,04
81 (Tl)	6,106	1,498	2,24	2,23	3/2,00

Here it is necessary to stipulate, that figures in the right column of table 2 were obtained by trial-and-error method and should be confirmed by further researches (for example, by studying spatial symmetry of chemical compounds with these atoms). In particular, numbers N₁ and N₂, obviously, can have values multiple of the numbers specified in the table.

However it is not so important, whether there will be numbers multiple or commensurable as in any case these findings are extremely important for atom structure understanding.

Similar laws became apparent in characteristics of the excited states of atoms, and, hence, they can be used for the analysis and systematization of atoms spectra.

As an example let us examine spectrum of lithium atom. On fig. 3 there is Grotrian diagram of lithium atom [4] which represents excited states (energy levels) and transitions between them.

It is necessary to make some additional calculations to ease analysis of the diagram.

First, energy levels in this case are more suitable for measuring not from the lowest level corresponding ionization energy (as it is on the diagram), but from "absolute" zero (that is, from energy level at infinite separation from atom).

Secondly, we will need to calculate all the same electron characteristics at excited levels, as it was done in table 2.

Final table 3 includes also the principal quantum number n (the first column) which is attributed to each excited level on the diagram (fig. 3). It was made to have opportunity of comparison with the results of calculating n (relation N₁/N₂).

Displayed in table 3 data are rather eloquent. Calculated values of the principal quantum number (n calc.) in most cases with good accuracy coincide with integer figures in the first column by which the corresponding excited levels on Grotrian diagram are marked.

Fig. 3. Grotrian diagram of lithium atom [4].

However in six cases (almost each third excited level) authors of the diagram, probably, could not be sure with the principal quantum number because it should be fractional (N₁/N₂), instead of integer as it is specified on the diagram. Thus numbers N₁ and N₂ are detected with good accuracy by means of simple calculations.

Impressive also is the remoteness (R/R_B) of external electron from atom nucleus at high excited levels (large n).

　

Table 3

n diagr.	W, eV diagr.	W, eV absol.	n calc.	n^2 calc.	R/RB	n≈N1/N2
2	1,848	3,544	1,963578	3,85564	3,837472	2
3	3,373	2,019	2,601519	6,7679	6,736008	13/4,997
3	3,834	1,558	2,961498	8,770468	8,72914	3
3	3,879	1,513	3,005216	9,031321	8,988764	3
4	4,341	1,051	3,605735	13,00132	12,94006	18/4,998
4	4,522	0,87	3,963104	15,70619	15,63218	4
4	4,541	0,851	4,007101	16,05686	15,9812	4
4	4,542	0,85	4,009458	16,07575	16	4
5	4,749	0,643	4,60988	21,25099	21,15086	23/4,99
5	4,837	0,555	4,961907	24,62052	24,5045	5
5	4,847	0,545	5,007222	25,07227	24,95413	5
5	4,848	0,544	5,011822	25,11836	25	5
6	4,958	0,434	5,611129	31,48477	31,33641	28/4,99
6	5,008	0,384	5,965262	35,58435	35,41667	6
6	5,014	0,378	6,012419	36,14918	35,97884	6
7	5,079	0,313	6,607284	43,6562	43,45048	33/4,99
7	5,11	0,282	6,960983	48,45528	48,22695	7
7	5,114	0,278	7,010883	49,15248	48,92086	7
8	5,156	0,236	7,609202	57,89995	57,62712	38/4,99

In more complex atoms spectra thousands excited levels are observed, and to analyze them is much more difficult task. Besides fractional principal quantum numbers there are also other "complicating" factors. In particular, energy of the excited level is influenced by disturbing factor in the form of various mutual spin states of atomic core and external electron.

Naturally, greater work to detail parameters of all electron shells of atoms is needed. These extremely interesting and important questions, undoubtedly, will attract attention of many researchers.