The Bohr/Sommerfeld theory of atomic structure as re-stated in [1], [2] and [3], is herein further developed to incorporate the effects of magnetic dipole coupling.

The splitting of the fine structure spectral emissions of single electron atoms and ions, is, in classical terms, attributed to a coupling between the magnetic flux density generated by the orbiting electron, and the magnetic moment produced by its spin. Even in the modern quantum mechanics theory, which considers the electron as a probabilistic wavefunction, this phenomenon is, in elementary treatises, initially described in such terms, [4],[5].

In the resurrected Bohr/Sommerfeld theory presented in this series of papers, in which the electron is treated as a real physical particle orbiting the nucleus, the above description is an exact one, and it is therefore necessary that its mathematical representation is developed accordingly. Consequently, its derivation in this paper is pursued along two lines. Firstly by a re-evaluation of the total electron orbital angular momentum via the principle quantum number criteria stated in [1]. This leads directly to a re-statement of the electron orbital energy equation with spin-orbit magnetic coupling fine structure splitting incorporated. From this relationship, the simple subtraction of the orbital energy without spin-orbit coupling, yields the spin-orbit energy term itself in isolation.

The second derivation is then completed in which this same energy term is obtained from first principles via consideration of the basic electrostatic, magnetic and mechanical concepts involved. Also, in this third Section, all other magnetic coupling terms, i.e. those involving the nucleus are also developed for inclusion into the extended electron orbital energy equation, and for use in future papers. In order to develop these last derivations from first principles, it is necessary to determine the manner in which the magnetic effects are generated. This is the subject of Section 2 of the paper, and also includes consideration of the effects due to the nucleus.

In the fourth Section the effects of electron spin reversal within an orbital are examined via presentation and discussion of a detailed example.

The development of this resurrected theory has, with the incorporation of the effects derived herein, now reached the point where it can, in detail, be compared with the equivalent theoretical relationships of the modern quantum mechanics theory of atomic structure. This comparison is effected in the fifth Section.

In the third and final Appendix, calculated emission spectra for the first 7 to 3 orbital shells of hydrogen are presented, that can, in part, be compared with the listings in [6]. Calculated orbital energy levels are also included.

Finally, because specific magnetic attributes of the nucleus are included in this paper, all analyses will be restricted to the hydrogen atom, although all equations so derived will continue to incorporate the Atomic Number Z.

2.0  The Electric and Magnetic Characteristics of the Electron and the Nucleus, Pertinent to Magnetic Dipole Coupling.

2.1  Preamble.

Prior to attempting any mathematical derivation, it is necessary to discuss the contribution of the proton nucleus to this effect. In all treatises dealing with fine structure splitting, it has been attributed to the coupling between the magnetic dipoles generated by the electron's orbit and its spin. Much less mention has been made of the contributory effects of the proton nucleus. The fact that the proton orbits a common centre of rotation with the electron has, to some degree, been incorporated in previous papers in the series et al, by the use of the reduced electron mass m. However, the proton is also a charged particle and therefore due to its orbital rotation generates a magnetic field. In addition, it was stated in [3] that due to the nature of the proposed cause of electron spin, the proton would also be subject to spin induction. This spin of the proton would also induce the generation of a magnetic dipole. A fact which is generally accepted to be a constituent cause of the hyperfine structure. Consequently, both the orbital and spin magnetic dipoles of the proton could potentially couple with each other and with those of the orbiting electron. All of these potential reactions must be considered when analysing the magnetic contributions to atomic spectra and are therefore so included in this paper.

2.2   The Magnetic Coupling Characteristics of the Electron.

The orbital and spin magnetic dipoles of the electron are derived in Appendix B and are caused by the circular motion of its electrostatic charge. It is well known that the dipole due to the spin is purported to be ëxactly" twice that due to its orbital motion. Because this is so for one and the same particle it infers that the effective charge in the spinning motion is twice that in the orbital. This can obviously not be so and therefore an alternative cause must be found. To address this point it is proposed that this effect is really a result of the way in which the two dipoles couple to produce a precessional force. To consider this, refer to Fig. 2.2 below. This figure shows the magnetic field lines generated by a spinning, orbiting electron.

Fig. 2.1 - Electron Magnetic Field Coupling Geometry.

Assume the electron orbital motion is out of the page which represents a current flow into the page. The magnetic field so generated will then be as shown, (clockwise). If the electron is also spinning clockwise, (spin - up), the charge on its RHS is also moving out of the page so that the current flow this motion represents is also into the page. At point A the spin field and the orbit field add to cause a force F_R to be imparted to the electron. Because the spin rate of the electron is much greater than its orbit rate, at all points on the orbit the charge rotating past point A will be the full charge on the electron. The force F_R will therefore be proportional to this full charge.

On the opposite LHS of the electron the charge spin motion is into the page so that the representative current is out of the page, resulting in the spin field line shown in Fig. 2.1. This field subtracts from the orbit field line thus causing force F_L on the electron. For the same reason as above F_L will also be proportional to the full charge e. Consequently, the total force on the electron, F_R+F_L must be proportional to twice the electron charge. This can be mathematically represented in magnetic coupling equations by doubling the value of the spin magnetic dipole.

Thus (B.15) may be re-stated as

e Ysp = eh e nsp 2pme c

(2.1)

The orbit dipole remains at that derived in Appendix B, (B.7) as

e Yor = Zehnj 4pme c

(2.2)

Where

_eY_sp   is the spin magnetic moment of the electron.

_eY_or   is the orbital magnetic moment of the electron.

e   is the electron charge.

h   is Planck's constant.

_en_sp   is the spin quantum number of the electron.

m_e   is electron mass, (rest).

c   is the velocity of light

n_j   is the orbital quantum number of the electron

Z   is the atomic number

2.3   The Magnetic Coupling Characteristics of the Nucleus.

The magnetic dipoles of the proton nucleus are not currently theoretically derivable but the orbital dipole is known, as a result of experimental measurements, to be a function of the nuclear magneton thus

p Yor = - Zehnj gp 4pmp c

(2.3)

Where

_pY_or   is the orbital magnetic moment of the proton.

m_p   is the rest mass of the proton.

g_p   is a constant of proportionality determined from experimental measurements to be 2.79275, [4].

The constant of proportionality in (2.3), g_p, is thought to be due to the internal structure of the proton affecting the manner in which the motion of the charge generates a magnetic field, and consequently, for the same reason, the proton's spin magnetic dipole will be similarly affected. Also, for the same reason as above concerning the magnitude of the electron spin magnetic dipole, that of the proton nucleus would also be expected to be twice that of its orbit dipole. Consequently, it is proposed that this dipole can be represented by the following relationship

p Ysp = - Zeh p nsp gp 2pmp c

(2.4)

Where

_pY_sp   is the spin magnetic moment of the proton nucleus.

_pn_sp   is the spin quantum number of the proton nucleus.

The proton dipoles are of opposite sign to those of the electron because of the opposite polarity of their respective electrostatic charge and because the proton is orbiting and spinning in the same direction as the electron which, as implicit in [3], is the norm.

Eqs.(2.1), (2.2), (2.3) and (2.4) can now be used to develop isolated magnetic dipole coupled energy terms. This is the subject of Section 3.3.

P4 Version 1.3.0
Ó P.G.Bass, April 2008