3   Derivation of the Total Electron Orbital Energy and the Isolated Spin-Orbit Magnetic Dipole Coupling Energies.

3.1   Derivation of Electron Orbital Energy Incorporating Spin-Orbit Coupling Via Re-Assessment of the Principle Quantum Number Criterion.

This is the simplest means of deriving the electron orbital energy incorporating spin-orbit magnetic coupling. However, it is to be noted that it only covers the magnetic coupling between the electron spin and the electron orbit dipoles. This is because this couple is by far the largest contributor to fine structure splitting. Also, the results so obtained can thereby be compared directly with those of the comparable equations of the modern quantum theory of atomic structure. There are however, other smaller magnetic dipole coupling contributions which are derived in Section 3.3 and added to the extended electron orbital energy equation as appropriate.

The analysis is initiated from an expanded version of [1] Eq.(2.10). This approach is adopted here because it expresses more clearly the manner in which spin-orbit coupling is effected.Thus

Where now

n   is the primary quantum number

v   is the total orbital velocity of the electron due to both the central coulomb force and the spin-orbit magnetic coupling force.

m₀   is the reduced rest mass of the electron.

dl   is an element of the orbit path length

The first integral in (3.1) is identical in form to that for a basic elliptic orbit and therefore has the solution

Here

M_f   is the angular momentum of the rest mass in the original elliptic orbit resulting from the central coulomb force

M_so   is the angular momentum of the rest mass resulting from the spin-orbit magnetic coupling force.

e   is the eccentricity of the elliptic orbit.

The second integral in (3.1) is the rotation of the elliptic orbit due to the sum of the relativistic mass increase and spin-orbit precessions and is evaluated as follows

Where

_ew_f   is the angular rate due to the central coulomb force.

_ew_so   is the angular rate due to spin-orbit coupling.

r_e  is the radius vector magnitude from the electron to the focal point of rotation.

In (3.3),

Where

f   is the azimuth angle.

and (3.5) evaluates to

This gives in (3.7)

Where

n_f   is the azimuth quantum number associated with the rest mass of the electron.

The second term applies because the spin-orbit couple must be a direct function of electron spin and therefore directly dependent upon the spin quantum number. It is noted that this means that in this representation the spin-orbit angular momentum is identical to the spin angular momentum of the reduced electron mass as derived in [3].

Insertion of (3.10) into (3.9) gives

Now, (3.7) can be expressed as

Where

M_j^*   is the total orbital angular momentum of the electron due to the combined effects of the central coulomb field and spin-orbit coupling.

Accordingly, with

Where

n_j   is the quantum number associated with M_j^* and is real, i.e. is integer and where adjacent values differ by unity.

Thus from (3.17) and (3.18)

Where

n_f ^*   is the azimuth quantum number associated with the relativistic energy mass of the electron.

The bound energy of the electron with spin-orbit magnetic coupling incorporated is then obtained by simply replacing n_f ^* in [1], Eq.(3.46) by n_j thus

Where

_eE_or   is the orbital energy of the electron

_e n_sp = ± 1/ 2

n = 1 ® ¥

n_j   is given by (3.20)

R_hy   is Rydberg's constant for hydrogen

and the Selection Rules of [3] apply. It is a simple matter to prove that the above substitution of n_j for n_f ^* is valid.

It is clear that (3.21) still takes the form of Sommerfeld's reduced equation as derived in [1], Eq. (3.46). A similar derivation exists for circular orbits.

It should be noted that the effective use of m₀, the reduced electron rest mass in the above derivation assumes that some of the energy in this expression is coupled into the proton orbit. This is incorrect, but the error so introduced is extremely small. This will be further discussed in Section 3.3.

To best demonstrate how (3.21) controls the emission spectra at this point in the development, it is recollected that n, the principle quantum number, must from the results of [1], be integer and its successive values differ by unity. Also, from the results of [3], _e n_sp can only possess the values of  ± 1/ 2 . Because n_j is the quantum number associated with the total angular momentum of the electron, it must also, in accordance with the results of [1], be an integer equal to or less than n, and successive values differ by unity. Accordingly, n_f ^* the quantum number associated with the orbital angular momentum of the relativistically corrected electron mass, in order that the above criteria are met, must now take half integer values and successive values also differ by unity. These criteria can in turn best be demonstrated by rewriting [1] Table 3.1 as follows.

Table 3.1 shows how the four quantum numbers so far considered combine to produce the first four orbital shells and is largely self explanatory. There are however, two points worthy of comment and they are as follows.

Note 1: A second "Term" column has been added in Table 3.1, "Term (New)". In this column it is seen that the term for the _e n_sp = - 1/ 2 has been re-allocated. This is because it is clear from the values of n_j and e that the orbitals in question are identical to the orbitals with _e n_sp = + 1/ 2 and n_f ^* one quanta lower within the same orbital shell. This will be discussed in more detail in Section 4.0.

Note 2: Some orbitals for n ³ 2 are elliptical and within them the electron is therefore subject to spin induction as described in [3]. The question therefore arises as to what happens when the electron traverses into the following quadrant and the direction of spin reverses, i.e. when in say 2s, _e n_sp is changed from + 1/ 2 to - 1/ 2 . This reversal represents a net reduction of one quanta of orbital angular momentum. Consequently, by the law of conservation of angular momentum there must be a net increase of one quanta to balance this reduction. The only way this can be effected without violating other criteria associated with n or n_j is for n_f ^* to increase by one quanta. This represents a zero energy change intra-shell orbital transition from say 2s, {or 2s(+) in the New Term column}, to 2p(-), {2s(-)}, and is clearly associated with the comments under Note 1 above. The process so described is the manner in which the potential anomalies discussed briefly in [3], Section 4.0, (iv) and (vi) are avoided. Further more detailed discussion of this feature is the subject of Section 4.0 where it is tracked around a complete orbit.

3.2   Isolation of the Electron Spin-Orbit Coupling Energy Term by Subtracting the Orbital Energy Without Spin-Orbit Coupling from that With.

It is important to note that (3.22) is not the same as [1], Eq.(3.46). The former is the orbital energy incorporating spin-orbit coupling but with the spin-orbit quantum term mathematically removed while [1], Eq.(3.46) is the orbital energy in a non spin-orbit coupled environment, i.e. there is a difference in the value of n_f ^* in the two relationships as discussed at the end of sub-Section 3.1 above.

Subtracting (3.22) from (3.21) gives the isolated spin-orbit coupling energy term as

Calculation of levels using (3.24) must take account of the Selection Rules as effectively embedded in Table 3.1. Of course in (3.24) n_f ^* is the value in a spin-orbit coupled environment. Also the comment following (3.21) concerning the presence of m₀ also applies here.

3.3   Derivation of All Isolated Magnetic Coupling Energy Terms from First Principles.

In any theory in which all events are to be described by real physical parameters, it is necessary that each event be separately derivable from first principles. In this resurrected Bohr/Sommerfeld theory of atomic structure it is therefore mandatory that each magnetic dipole coupling effect be so derivable. This must include those involving the proton nucleus. Such derivations are the subject of this sub-Section.

The coupling energy between any two magnetic dipoles is given by, [5]

Where

- Y

or

is a magnetic dipole vector due to orbital motion.

- Y

sp

is a magnetic dipole vector due to spin motion.

- r

is the separation vector, magnitude r.

m₀  is the permeability of free space.

In all cases considered here the dipole vectors are parallel so that (3.25) reduces to, (m₀ = 4p in cgs units),

3.3.1  Relationship Between the Magnetic Dipoles of the Orbits of the Electron and the Proton Nucleus.

3.3.2   Coupling Energy of the Electron Spin Dipole and the Net Orbit Dipole.

Eq.(3.28) is the electron spin-orbit coupling energy resulting from the orbital motions of the proton/electron pair due to the central coulomb force. However, the extra precessional motion of these orbits due to magnetic dipole coupling itself will also contribute to the coupling energy. This can be inserted into (3.28) by its product with the appropriate quantum number ratio thus

Eq.(3.29) shows that the electron spin-orbit coupling energy is a variable inversely proportional to the cube of the distance between the dipoles. To obtain a value that can represent the mean energy throughout the complete orbit means determining an appropriate average value for r_e. To maintain continuity here, that derivation is relegated to Appendix A. The result is

Where

ár_e ñ   is the average value of r_e around the complete orbit.

L_e   is the semi-latus rectum of the electron orbit.

With

Where

M_j^*  is the total orbital angular momentum incorporating that component due to spin-orbit magnetic coupling.

From (3.19)

With reference to the comment following (3.21) and (3.24), note here the appearance of the ratio m_e/m₀. Comparison of (3.34) with (3.24) is effected in sub-Section 3.4 below.

Eq.(3.34) is the main contributor to electron spin-orbit coupled precession but there are other magnetic coupling effects which contribute either directly or indirectly. These are derived below.

3.3.3   Coupling Energy of the Spin Dipole of the Nucleus and the Net Orbit Dipole.

Although this couple does not affect the electron orbital energy in a direct sense, it does so in an indirect way. This will be demonstrated in the next paper. The coupling energy is derived as follows. Insertion of (2.4) and (3.27) into (3.26) gives

Where

r_p  is the radius vector magnitude of the proton orbit.

Via an identical process to that in Section 3.3.2 above, the energy coupled into the proton orbit by the proton spin dipole becomes

3.3.4   Coupling Energy of the Electron and Nucleus Spin Dipoles.

Substitution of (2.1) and (2.4) into (3.26) gives

Where

_e,pE_ss(f)  is the energy coupled into both the electron and proton orbits by the spin-spin magnetic dipole coupling of both particles.

Note that because this couple is of relativistic magnitude, r_e can be used instead of r to simplify the analysis. This substitution also ignores the fact that r is a dynamic function of f due to the different precession rate of the proton orbit, the effect of this variation on the magnetic couple also being of insignificant magnitude.

The parameter d _p is a constant of proportionality introduced to account for the effect of the rotation of the proton spin dipole about the common orbital point relative to the electron, also due to the different precession rate of the proton. This results in a field of magnetisation about the orbital focal point, the shape and therefore flux density of which, in a similar manner to g_p, will depend upon the shape and internal structure of the proton. The value of d _p will, in a future paper, via a semi-empirical analysis associated with the hyperfine structure, be shown to be of the order of 3.3558912.

Via analysis similar to that above in Section 3.3.2, the energy coupled into both orbits by these spin dipoles is

3.4   Total Electron Orbital Energy.

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