of gravitation G of the trajectory of a quantum state compared to the equato-rial plane E. Let α be this angle. The radial action does not modify the orien-tation of the basic circular trajectory. It is thus enough to determine α for such states. Let us consider the dl displacement which brings the electron from B to C (figure 1). The displacement corresponding to the increase dϕ is rdϕ. The component of the action parallel to the plane E is the product of the projections of the displacement and the projection of the momentum. One thus has:
cos2α = pdl
rd pϕ ϕ
(7.1) We saw that the projection of the angular action on the equatorial level E is uh while the total angular action is kh. It comes :
cos2 α =
k m k
u = −1/2 (7.2)
The trajectory of a state m crosses the equatorial plane in an unspeci-fied point of the circle homologous with that of radius PB of figure 1. When there are several states occupied on a same subshell the trajectories of the various states deviate the one from the others in such a way to minimise the interactions of repulsion between electrons. On the other hand, if the interac-tions electron-electron are neglected, the angle α keeps a constant value. In particular there is a minimum value. The exploitation of these properties
makes it possible to deduce certain properties characteristic of the considered atom. In particular the directions of space with a strong or small screening constant resulting of the electrons present on the atom. In other words the knowledge of these directions gives access to dissymmetries of the atom, which generate those of the crystal. Thus we proposed an interpretation of the preferential occupation of the octahedral site of spinels by chromium in MnCr2O4 [16].
8 CONCLUSION
In this study of the quantification, the motion is supposed the result of ex-changes of matter between the electron and the proton. These exex-changes take place in all the directions of space. From this fact the orbital rotation must be regarded as the result of a mechanical action which acts in two orthogonal directions. In other words the rotation can be regarded as the sum of two orthogonal rotations. As a results the magnetic field which corresponds to an action on only one direction of planes can modify only half of the action associated with the rotation. In addition the doublet have in fact for origin the possibility for the electron to be able to turn for same angular quantification with a constant or variable mass independently of radial speed and nothing indicate that the doublets correspond to state of spin-up and spin-down [13].
These results lead in examining in which extend the hypothesis of own rota-tion is confirmed.
The hypothesis of the spin was proposed in 1925 [11] to interpret the dou-blets before the discovery of the equations of wave of quantum mechanics 1926 [10] for that of Schršdinger and 1928 [17, 18] for that of Dirac. How-ever the spin i.e. the own rotation of the electron are a property of volume and the study of the properties of a point in classical mechanics or special relativity could not reveal the properties of a volume. Moreover the introduc-tion in quantum mechanics of the differential operators presumably giving again, while acting on the wave function, the properties of the centre of grav-ity of the electron, cannot more reveal the own rotation.
For lack of a clear comprehension of the quantum phenomena this remark into general is ignored. In fact the theory of Dirac have from the very start been impregnated of the model of Pauli where the wave function have two components : one being by hypothesis presumably attached to the one of the type of spin the other component being attached to the other type of spin. In theory of Dirac one can consider a similar step but the study of solutions show that it is the whole of four components of wave function which
charac-terises one quantum state. Thus the concept of spin escapes in fact from the theory of Dirac just like from the theory of Sommerfeld.
To discuss more completely the assumption of the spin, there remains the existence of the half integers angular momentum which belong according to this study to orbital rotation as well to the solutions of the equation of Dirac.
It is with them that we calculated every magnetic moment by multiplying them by the factor g=k(k+_)-1 characteristic of the subshell. They contributed, before the equations of waves, to the hypothesis of the spin. These half inte-ger numbers belonging to the model of Dirac, they consequently consolidated the assumption of the spin. However they rise not, through the studies pro-pose to date, of a property of the own rotation, but of studies describing the properties of a point. As a result, we have not the right to present them as a proof of the existence of the spin. Thus it is possible that the orbital rotation of the electron is related to that of its own rotation but it is a question which remains open.
On the other hand this study shows that the action associated with the own rotation is the Planck's constant h and that it is this action which determines all the quantum numbers. This state of things clarifies the comprehension of the periodic table. Indeed the constant of action h does not depend on the medium, i.e. of the complexity of the atom, the molecule or of the solid, liquid or gas state of the matter in which moves the electron. Then we under-stand that the quantum numbers which characterise the quantum state remain whatever the situation of the atom. It is that which allows to understand the structure of the periodic table which synthesises so many chemical and physical properties of the matter. We also understand that the total angular momentum, reflection of the action associated with rotation, subsists what-ever the complexity of the atom to which belongs the electron and the me-dium in which is the atom. It is this assumption which enabled us with George Lochak to propose a calculation of the magnetic moments of the atoms and whose coherence with the experiment remained mysterious [13, 19]. In the field of the prospects the understanding of the crystal state is en-lighten by this approach. We already showed it [16] and hope to be able to again show it in the near future.
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Manuscript received the 29 January 1998, revised 28 May 1999.