4.0   The Effect of Electron Spin Reversals Within an Elliptic Orbital.

Now that the effects of magnetic dipole coupling have been introduced into this resurrected theory, the question as to what happens when, due to spin induction, the electron spin reverses in an elliptic orbital, can, further to the brief dissertation in Section 3.1, Note 2 to Table 3.1, be discussed in more detail here.

If there were no dead zones in an elliptic orbit, the spin reversal would be ïnstantaneous" at the quadrant boundaries. This would result in a spin-orbit et al coupled energy change exemplified by the spin quantum number change D e nsp =   ±1, and therefore a Dnj change of   ±1. This could incur an inter-shell orbital transition of some kind. However, the dead zones impose a restriction on the magnitude of the spin change such there is no energy change, although there is an intra-shell transition. The manner in which these changes occur is best explained by means of an example, as follows. Consider Fig. 4.1.


Fig. 4.1

Fig. 4.1 - Electron Orbital Spin Reversals, (Elliptic Orbital)

This shows electron spin directions, up, down or no spin, around the four quadrants and dead zones of an elliptic orbit. Consider the case of an electron in an ns orbital, n > 1. As the electron proceeds through the quadrants and dead zones, the quantum numbers change according to the spin reversals as depicted in Table 4.1 below.

Q/DQuantum NumberExplanation
Q1 ensp = + 1/ 2
nf* = 1/ 2
nj = 1		 Orbital is ns(+), (See table 3.1)
D1ensp = 0
nf* = 1
nj = 1		Change in ensp causes 1/ 2 quanta reduction in spin-orbit coupled energy.
nj is already at a minimum value and can only change by an integer. Therefore nf* must increase by 1/ 2 .
The electron undergoes an intra-shell transition to ns as in [1], Table 3.1, (same e).
Q2ensp = - 1/ 2
nf* = 1 1/ 2
nj = 1		Spin induction causes ensp to change to - 1/ 2 causing a 1/ 2 quanta reduction in spin-orbit coupled energy.
nj is already at a minimum value and can only change by an integer. Therefore nf* must increase by 1/ 2.
The electron undergoes an intra-shell transition to np(-) as in Table 3.1, (this paper).
D2ensp = 0
nf*=1
nj = 1		Change in ensp causes 1/ 2 quanta increase in spin-orbit coupled energy.
nj is already at a minimum value and can only change by an integer. Therefore nf* must reduce by 1/ 2.
The electron undergoes an intra-shell transition to ns as in [1], Table 3.1, (same e).
Q3ensp = + 1/ 2
nf* = 1/ 2
nj = 1		Change in ensp causes 1/ 2 quanta reduction in spin-orbit coupled energy.
nj is already at a minimum value and can only change by an integer. Therefore nf* must reduce by 1/ 2.
The electron undergoes an intra-shell transition to ns(+) as in Table 3.1, (this paper).
D3Same as D1As per Q1 to D1
Q4Same as Q2As per D1 to Q2
D4Same as D2As per Q2 to D2
Q1Same as Q3As per D2 to Q3

Table 4.1 - Intra-Shell Transition Characteristics for ns(+), ( n > 1)


Thus in the dead zones, D1 to D4, the orbital is that of ns in [1], Table 3.1.


In quadrants Q1 and Q3, the orbital is that of ns(+) in Table 3.1, (this paper).


In quadrants Q2 and Q4 the orbital is that of np(-), {ns(-)}, in Table 3.1, (this paper).


The same sequence of intra-shell transitions would occur for np, nd orbitals etc except where the orbital was circular within which there is no spin induction and therefore no reversals.

Note that in each change the non spin-orbit coupled orbital has the same geometrical characteristics as the spin-orbit coupled orbitals in the transition sequence and therefore there is no energy change in the transition.

The angular momentum change that affects nf* , (the spin-orbit coupled environment value), will be shown in a future paper to be instrumental in the hyperfine structure of elliptic orbitals.

An important result of this feature is that for hydrogen, there is no pseudo ground state in which an electron exists in the 2s orbital. As a consequence, as no inter-shell transition path exists for a 2s orbital, an intra-shell transition of electrons that enter this orbital always takes place by which they move from the 2s orbital to the 2p(-), {2s(-)} orbital. This subsequently allows a normal inter-shell transition to the ground state 1s orbit.

It must be noted that the above dissertation does not infer that electrons in the higher, ( > 1s), orbitals can traverse the complete orbital before making an inter-shell transition. It is most likely that after insertion into any particular quadrant, due to spin induction, it will immediately make a further transition to a lower shell. Alternatively, it could traverse just one dead zone into the next quadrant before making such a transition with reversed spin. The one exception of course is an electron in the 2s orbital where, as described above, it must traverse one dead zone before making the transition to the ground state.

The above intra-shell transition sequences also lend significant credence to the renamed Term Scheme, {the "Term, (New)" column, in Table 3.1}.

5.0   Comparison with the Atomic Structure Theory of Modern Quantum Mechanics.

The subject matter of this comparison is, in the quantum mechanics theory, the three relativistic terms that give rise to the fine structure and the splitting thereof into doublets, as presented in [4]. (The quantum mechanics quantum number l, referred to below, is the azimuth quantum number omitting spin-orbit coupling).

5.1   The Relativistic Mass Increase Effects.

5.1.1  In Quantum Mechanics theory for l = 0, the Darwin Term.

Adopting for this Section the energy designator of [4], this term is given in [4] as


DE/// = hRhy Z 2
n 2
 k2Z 2
n
(5.1)

which in conjunction with the "relativistic mass increase effects" from [4] gives for the orbital energy in s orbitals


E = - hRhy Z 2
n 2
 ì
í
î 		1+ k2Z 2
n 2
 æ
è 		n - 3
4
 ö
ø 		ü
ý
þ
(5.2)

In the resurrected theory proposed here, the orbital energy is given by, (spin-orbit coupling omitted), [1], Eq.(3.46), repeated here for convenience


E = - hRhy Z 2
n 2
 ì
í
î 		1+ k2Z 2
n 2
 æ
è 		n
nf*
- 3
4
 ö
ø 		ü
ý
þ
(5.3)

and for s orbitals nf* = 1, (the non-spin-orbit coupled value), so that (5.3) reduces to (5.2). Thus the resurrected Bohr/Sommerfeld theory is equivalent to the quantum mechanics theory at this stage.

5.1.2  In Quantum Mechanics theory for l ¹ 0.

Omitting spin-orbit coupling, the quantum mechanics theory gives for the mass increase effect, [4]


E = - hRhy Z 2
n 2
 ì
í
î 		1+ k2Z 2
n 2
 æ
è 		n
l + 1/2
 - 3
4
 ö
ø 		ü
ý
þ
(5.4)

Reference to (5.3) shows that at this stage in the comparison


nf* = l + 1/2
(5.5)

and consequently, in concert with Section 5.1.1 above, when


l = 0,    nf* = l +1
l ¹ 0,    nf* = l + 1/2
(5.6)

The difference of nf* to l for the two conditions of l in (5.6) is due solely to the fact that in the quantum mechanics theory the Darwin term is no longer present for l ¹ 0 and spin orbit coupling is not yet included.

5.2   Spin-Orbit Coupling.

The incorporation of spin-orbit coupling in quantum mechanics theory, (l ¹ 0), gives, [4]


E = - hRhy Z 2
n 2
 ì
í
î 		1+ k2Z 2
n 2
 æ
è 		n
j + 1/2
 - 3
4
 ö
ø 		ü
ý
þ
(5.7)

where j is the inner quantum number given by j=l ± 1/ 2 , so that (5.7) becomes


E = - hRhy Z 2
n 2
 ì
í
î 		1+ k2Z 2
n 2
 æ
è 		n
l + 1/2 ± 1/2
 - 3
4
 ö
ø 		ü
ý
þ
(5.8)

In the resurrected Bohr/Sommerfeld theory under this condition the orbital energy is


E = - hRhy Z 2
n 2
 ì
í
î 		1+ k2Z 2
n 2
 æ
è 		n
nj
- 3
4
 ö
ø 		ü
ý
þ
(5.9)

Where


nj = nf* + e nsp
(5.10)

Inserting (5.5) and [3], Eq.(3.4) gives


nj = l + 1/2 ± 1/2
(5.11)

thus equating (5.9) with (5.8).

Also note that this means that for all values of l including zero


nj = j + 1/2
(5.12)

so that nj is always integer as j is always half integer. As nj is the quantum number associated with the total angular momentum of the electron, (as is j), this value of nj is more in keeping with de Broglie's quantum hypothesis.

If desired this last comparison can be term specific wherein the isolated spin-orbit coupling term can be shown to be equivalent in both theories. However, this only applies when electron spin-orbit coupling is assumed to affect both the orbits of the electron and the proton nucleus, which has already been shown to be incorrect.

It is also clear that the above relationship comparison means that effectively, nf* is in part equivalent to the 1/ 2 quantum number zero point energy term in modern quantum mechanics theory.

5.3   Discussion.

In both theories, the primary quantum number, n, is a measure of the single valuedness of the electron matter wave around its orbit. In this respect the two theories are identical encompassing the same tenet. To some extent the same comment applies to the spin quantum numbers, s in quantum mechanics, (QM) theory; # nsp in the Bohr/Sommerfeld, (B/S) theory. Both have the same numerical value but, in view of the different treatment of the electron, there the similarity ends. In B/S theory ensp results from the angular momentum of a real physical particle. This then leads directly to a properly defined quantised spin magnetic moment. In QM theory the spin quantum number, s, is still purported to be due to electron ßpin", but because of the nature of the electron in this theory, electron spin cannot be adequately defined. Accordingly, in QM theory the ßpin" magnetic moment of the electron is subject to the same difficulties.

The main difference between the two theories however, concerns the azimuth quantum number, (l in QM, nf* in B/S). This comparison needs to be conducted in several parts because of the peculiarity in the interpretation of l.


(i) s Orbitals :- l = 0 : nf* = 1, (B/S Theory Spin-Orbit Coupling Omitted). In these orbitals in QM theory, azimuthal angular momentum is zero, so that accordingly spin-orbit coupling is also zero. Therefore, l = 0, and j = 1/ 2, and the orbital energy is given by, (i) the ubiquitous "zero point energy" of quantum mechanics, the " 1/ 2 " in the denominator of (5.4), and, (ii) the appearance of the Darwin term which only applies to s orbitals. The fact that there is no azimuthal angular momentum in QM theory is considered somewhat of an anomaly. Even with the electron treated as an orbiting probabilistic matter wave, this matter wave, as shown in [2], is a representation of kinetic energy, and accordingly will possess an equivalent mass in line with Einstein's renowned energy/mass relationship. The electron should still therefore generate an effective angular momentum. Consequently, spin-orbit magnetic coupling should also appear in QM theory in s orbitals.


In the resurrected B/S theory, in s orbitals, (with spin-orbit coupling omitted), nf* is effectively equivalent to the zero point energy plus the Darwin terms of QM theory. It cannot be actually so equated because it represents a different concept in that it is directly related to the angular momentum of a real physical electron particle. As a result spin-orbit coupling is present as is evident in Table 3.1. This leads directly to the next Sub-Section.


(ii) s Orbitals :- l = 0 : nj = 1, (B/S Theory Spin-Orbit Coupling Incorporated). In this case the comments in (i) concerning the azimuth quantum number l of QM theory are still fully applicable. In discussing the resurrected B/S theory however, it is now nj that is equivalent to the zero point energy and Darwin terms of QM theory, and as is clear from Table 3.1, nj is composed of the azimuth quantum number nf* , ( 1/ 2 in s orbitals), plus the spin-orbit magnetic coupled quantum number, ensp, (+ 1/ 2 in s orbitals).


(iii) All Other Orbitals :- l ¹ 0, j = l ± 1/ 2 : nf* > 1, nj = nf* + ensp. In these orbitals both theories contain exactly the same components to make up the orbital energy and the differences are related to the meaning attributed to the azimuth quantum number l. Here l is the azimuth quantum number related to the angular momentum of the electron, (in wavefunction form), thus enabling spin-orbit coupling, so resulting in the inner quantum number j. The zero point energy term is still a separate addition. On the other hand nf* is now equivalent to l, this equivalence including the zero point energy term. The comparison between j and nj is consequently identical to that between l and nf* .


(iv) Other Differences. Apart from the quantum number nomenclature and meanings, and the presence of spin-orbit coupling in s orbitals in the B/S theory, as discussed above, there are three other differences to the modern QM theory. Although at this juncture they have insignificant impact on the emission spectra, they will do so when the Lamb Shift and hyperfine structure are incorporated. These differences are exemplified in the derivations in Section 3.3 and discussed there. They are:-


(a) The coupling energy between electron spin and electron orbit is only coupled into the electron orbit, not that of the proton nucleus. Hence the appearance of the term me/m0 in (3.34). This correcting term is absent in the QM theory equivalent expression as represented by (5.7).


(b) The spin-orbit energy coupled into the electron orbit also, in the B/S theory, incorporates a term to reflect that effectively resulting from the proton dipole. Hence the appearance of the term
(1-gp me /mp ) in (3.34). This correcting term is absent in the QM equivalent expression as represented by (5.7).


(c) The magnetic dipole energy coupled into the electron orbit, in the B/S theory, also incorporates a term to reflect that resulting from the proton spin dipole, as represented by (3.38). No such term exists in the QM theory in the representative expression, (5.7), although this couple does receive detailed attention in the literature, [4], in discussion of the hyperfine structure for l > 0.

All of these other differences have very little impact on the orbital energy levels of the electron at this stage, but should be included because as the development continues to incorporate appropriate terms for the Lamb Shift and the hyperfine structure, they become to some extent the dominant features.



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

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