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4 Addition of the Energy Correction Terms to the Fine Structure Energy Levels.The á priorí relationship for the fine structure energy levels is given by [9], Eq.(3.39) repeated here for convenience
Adding the orbital energy correction terms, (3.6) and (3.9), to (4.1) provides a new complete relationship for the electron fine structure orbital energy as
This relationship is of course, as previously stated, despite the presence of Z, only applicable in fine detail to hydrogen. It is also subject to the Selection Rules as defined in [8], and to the parametric quantisation of Ge as determined in Table 3.1. Subject to these restrictions it is used to produce the energy levels and emission spectra for hydrogen, (vacuum and for l > 2000Å, air), as presented in Appendix A and which, with all other pertinent results, are discussed in detail below. 5 Discussion of Results.With the development and addition of the energy correction terms, incorporating the Lamb Shift, in this paper, the resurrection of the Bohr/Sommerfeld theory of atomic structure has reached a state whereby a direct comparison of the achieved results may be made with data published in the literature, in particular, [5]. This reference was, in part, compiled from [12] and in doing so the results have been truncated down to four and three decimal places. Nevertheless, for the purpose of this comparison it is assumed that [5] are the most accurate published data for the electron orbital energy levels and emission spectra for hydrogen. As stated previously, the development here has, in the interests of brevity and clarity been restricted to hydrogen and it is on the results for that element that the ensuing discussion entirely concentrates. The results presented in Appendix A should be reviewed in depth to fully appreciate this discussion. 5.1 The Spectral Signature.The primary results here are those for the emission spectra resulting from transitions from any permitted orbital to the ground state. These spectra, as quoted in [5], are for transitions observed in vacuum and as can be seen from Appendix A, for those orbitals shown, the results are in complete agreement with those in [5]. That this is so not only provides credence to the concept with which the energy correction terms incorporating the Lamb Shift have been developed in this paper but, also to the prior parts of the theory presented in the earlier papers [6] to [9]. When reviewing all other spectra, it is noted that in [5] these are quoted as pertaining to spectra observed in air. Consequently cognisance of the refractive index of air, Pair, has to be taken. To ensure the comparison with [5] was specific to the different theoretical aspects only, Pair as used here was calculated from [12], and is shown in Table A.2. From the results of this comparison it can be seen that there is excellent agreement between the two sources of data, Appendix A and [5]. Differences range from 0.0001Å, {4d(+) ® 2p(+)}, at 4861Å, (2.06E-6)%, to 0.2356 Å, {8k(+) ® 7i(+)}, at 190567Å, (1.24E-4)%. This very small level of difference would be due to the very small variances in the values of the constants used in this series of papers, and those in [5] and [12]. Also, in this paper, only a single value of Pair has been calculated for each set of transitions between individual orbit shells. This will introduce a small variation across the orbitals in each shell. Finally, there are very small orbital energy differences which may also contribute. These are discussed below. 5.2 The Energy Levels.Comparison of orbital energy levels between those in Appendix A and those in [5], show there is a very small difference in all orbit shells ranging from ~ 0.0320cm-1, (3.89E-5)%, in shell 2 to ~ 0.0496cm-1, (4.50E-5)%, in shell 8. Within each shell there is a small variation about these values according to orbital. These differences will in part be due to the slightly different values of the constants used, (see Table A.1). However, the variances are all negative indicating that there is still a small systemic difference between the two theories. This is not considered a very significant point however, because the most important of the atomic structure attributes is the spectra, which can be experimentally measured and thereby compared directly with theory. Nevertheless, it is possible that the small difference in orbital energy above can be further reduced as discussed below in Section 5.6. 5.3 The Lamb Shift.The Lamb Shift is the s(+) ® p(-) energy interval for those orbit shells in which this effect is apparent. The short table below shows this interval for the first eight shells as a comparison between [5] and the results in Appendix A.
The result for the second shell is significant because the value of Ge for the 2s(+) orbital was determined in conjunction with the 2p(-) orbital to give exactly the experimental difference value for these orbitals, (0.0352834087 cm-1 or 1057.77Mc/s). The other applicable intervals for shells three to eight in the above table are then predetermined by the need to meet both the spin angular momentum quantum criteria and the conditions determining transition. These are further discussed in Section 5.4 below. As can be seen from the above table for the applicable orbital energy intervals in shells three to eight there are some very small differences between [5] and the results in Appendix A. These will be due to the same factors discussed above for the orbital energy differences. Improvement may also be effected here via the discussion in Section 5.6. 5.4 The Electron Spin Matter Wave Radius and the Mechanics of Orbital Transition.The discussion here will be aided by reference to Table 3.1 and Fig. 5.1. The latter shows two plots. The first, (1), is a plot of ewsp vs Ge for when their product is equal to c, the velocity of light, i.e. this is a plot of ewsp vs Ge at the point of orbital transition. The second plot, (2), is a plot of ewsp vs Ge at the point when the electron spin angular momentum quantum criteria is met. The important point about these plots is that they cross, at Ge » 1.3409E-11cm, ewsp » 2.2358E+21rads/sec., i.e. at point A. It is this that determines which orbitals are "unstable" and result in a transition, and those which are stable, 1s(+), or "meta-stable", 2s(+). The manner in which this is effected is examined in detail below.
(i) Consider first the spin-up and spin-down orbitals other than the circular and s orbitals in Table 3.1. When an electron makes a transition into one of these orbitals, (to the left of Lines 1 and 2 in Fig. 5.1), its spin will increase in magnitude, (+ or -), according to which quadrant of the orbital it is in. At the same time its spin matter wave radius, Ge will also increase such that their product tends towards providing conformance to the spin angular momentum quantum criteria. Before the latter is accomplished however, the circumferential velocity of the electron reaches the velocity of light, i.e. the product ewspGe reaches Line 1, (above point A), in Fig. 5.1, and a further transition occurs.
In the high shell numbers, ( > 3rd), the spectra are relatively
insensitive to the value of Ge in the applicable orbitals and
therefore the initial value of Ge determined for them is
3.50E-13 cm., just above the widely quoted value of ~ 2.818E-13 cm.,
the physical radius of the free electron. The one exception is the 3p(+)
orbital for which an initial value of 1.33E-11cm. has been determined,
giving the best fit with [5] for the applicable air spectra.
(ii) Now consider the circular orbitals other than the ground state, 1s orbital. When the electron makes a transition into a circular orbital, it is not subject to spin induction for reasons discussed in [8]. Consequently, the value of ewsp possessed by the electron will remain unchanged while it is in these orbitals. Ge will therefore simply expand to meet the spin angular momentum quantum criteria. Once again, before that criteria is satisfied the velocity of the electron circumferential surface reaches the velocity of light and a further transition occurs.
The value of Ge determined for these circular orbitals is
1.33E-11 cm for 2p(+) and 1.275E-11 for the rest, which, in conjunction with the value for all other orbitals
provides the best fit with [5] for the applicable air spectra.
(iii) Now consider the s orbitals other than 1s and 2s. These orbitals only contain spin-up electrons. When an electron makes a transition into one of these orbitals its spin angular rate and spin matter wave radius will react as for the other elliptic orbitals discussed above and a further transition takes place.
The value of Ge determined for these orbitals, 1.325E-11 cm.,
provides in conjunction with the other orbitals discussed above the best fit
for the Lamb Shift for the 3rd to 8th shell. The fact that the
spin angular rate is lower and the value of Ge higher at
transition than in most other elliptic orbitals so far discussed is believed
due to the nature of the transition path into these orbitals. This last
point also affects the last two orbitals to be discussed, 1s and 2s.
(iv) The 2s orbital. This is the so called "meta-stable" orbital. When an electron makes a transition into 2s, due to the characteristics of the insertion path, its spin, (+ve), will be low enough such that Ge can increase to a value that, together with the spin induced increased angular rate, provides conformance to the spin angular momentum quantum criteria. i.e. the product ewspGe reaches Line 2, (below point A), in Fig. 5.1. At that instant the electron is stable and can remain so in the 2s orbital until it moves into the next quadrant whereupon its spin reverses and it has made an internal shell transition to the 2p(-) orbital, (same e and nj with ensp changing from + 1/ 2 to - 1/ 2 and nf * from + 1/ 2 to +1 1/ 2 ). This phenomenon was presented and described in detail in [8]. Ge falls to its 2p(-) value as the electron passes through the dead zone and the spin reverses. The electron can then make a normal 2p(-) ® 1s ground state transition. This is proposed as the means by which electrons escape from the 2s "meta-stable" orbital. If this mechanism did not exist, in a large volume of hydrogen, there would be two ionisation energies, one for ionisation from 1s, the 13.6eV existing level and one from the 2s orbital which would be ~ 3.4eV.
The value of Ge determined for the 2s orbital is that at
meta-stability, i.e. 1.37817E-11 cm., which in conjunction with that in the
2p(-) gives an orbital energy interval equivalent to the 1057.77Mc/s Lamb
Shift in shell 2 as mentioned in Section 5.3 above.
(v) Finally, consider the 1s ground state circular orbital. When an electron makes a transition into the ground state, due to the characteristics of the insertion path its spin angular rate will be sufficiently low so that Ge can increase to a value that results in their product providing conformance to the spin angular momentum quantum criteria. As the ground state orbital is circular, there is no spin induction and the electron is therefore permanently stable in this orbital. The value determined for Ge, 1.36E-11 cm. is therefore that which gives the appropriate energy correction in this orbital such that the spectral response for transitions directly into the ground state from any permitted orbital is in agreement with [5] for the vacuum spectral lines. 5.5 Ionisation.Any volume of hydrogen, irradiated at a level in excess of 13.6eV will ionise. To simulate this using the mathematical formulation in the modern quantum mechanics version of atomic structure theory, means raising the principle quantum number, n, to infinity. This in turn means that the electron must be moved to an infinite distance from the proton before it becomes disassociated from it. In reality of course this is not true as in the laboratory ionisation takes place within the physical confines of laboratory equipment.
In the theory presented here, in order to simulate ionisation using (4.2),
it is only necessary to raise n to the values shown in the following table.
The reason is that all terms in (4.2) decrease in magnitude as n increases, but one term, that for the energy correction term resulting from the dynamic distance of the proton from the central point of orbital rotation, decreases slower than all other terms, (proportional to 1/n). This term is positive and becomes the dominant term at the above values of n, i.e. the orbital energy has become positive which means that the electron has become disassociated from the proton and the atom has ionised. The above values of n at ionisation follows the orbital eccentricity pattern as is seen by comparing Table 5.3 with [9] Table 3.1. The eccentricities of the above orbitals are however very much closer to unity than in the reference. For instance, the apparent eccentricity in the first entry in Table 5.3 is (1 - 2.199406639E-14)1/2. The apparent eccentricity of the orbital in the last entry is (1 - 2.199406965E-14)1/2. The calculation of these virtually identical values, [9], Eq.(3.18), does not however, take into account the very small additional energy terms determined in this paper which provide the difference to make e exceed unity.
It is appropriate to designate the above ionisation quantum numbers as
ni . It is a matter of interest that the value of ni for any orbital can be calculated from the following simple formulae.
Where,
k is the orbital identifier number in Table 5.3. Finally, it should be obvious that ionisation cannot take place at any orbital with an eccentricity less than those shown above. The ionisation energy from the ground state in the resurrected Bohr/Sommerfeld theory presented here is 13.59843288eV.
5.6 Summary.This summary is to draw together, in simple form, the salient points reviewed above.(i) The vacuum spectra determined here are in perfect agreement with those in [5], for shells 1 to 8 for all permitted transitions, (in Ångstroms to four decimal places).
Although not reported in Appendix A, spot checks of np(+) to 1s(+) spectral
emissions were also compared with [5] for shells 10, 20, 30, 40 and 50 again
with perfect agreement.
(ii) The air spectra determined here are in agreement to within (1.24E - 4)% worst case with those in [5] for shells 2 to 8 for all permitted transitions, (in Ångstroms to three decimal places).
Although not reported in Appendix A, spot checks of air spectral emissions
for a number of random transitions involving very high shell numbers were
also compared with [5] with equal or better accuracy to that quoted above.
(iii) The orbital energy levels determined here are in agreement to within
typically (4.0E - 5)% with those in [5] for shells 1 to 8, (in cm-1
to four decimal places).
(iv) The orbital energy level intervals determined here are in agreement
from 0cm-1 to 0.0009cm-1 for the s to p(-) intervals, shells 2 to 7.
(v) In the dissertation on the mechanics of electron transitions, a means by
which electrons ëscape" from the meta-stable 2s(+) orbital was identified.
(vi) A central factor in the determination of an orbital energy level
correction term due to the distributed nature of electron charge, is the
variability of the electron spin matter wave radius, a necessity to achieve
conformance with the spin angular momentum quantum criteria. This factor is
also central to the manner in which orbital transitions are initiated.
(vii) The orbital energy equation has been shown to incorporate the effect of ionisation of the atom and permit the determination of the principle quantum number at ionisation for all orbitals. 5.7 Possible Further Additions/Improvements.It was stated above that some of the differences between the results achieved here and those in [5] could possibly be further reduced. The means by which that could be effected is threefold as follows.
(i) The Introduction of Magneto-Gyric Effects. - In [9] the
derivation of orbital and spin magnetic dipoles assumed that both the
electron and the proton nucleus were point source charges. This tacitly
assumed that that the so called Magneto-Gyric factors associated with these
parameters were unity, (the "2" associated with the spin magnetic dipole was
shown to be due to a coupling feature). If cognisance is taken of the fact
that the charges on these two particles are uniformly distributed on their
surfaces, this will result in their Magneto-Gyric factors being very
slightly different from unity. This will in turn make a small adjustment to
the orbital energy of the electron which will then result in a similar
adjustment to the spectral signature.
(ii) Optimisation of Pair, the Refractive
Index of Air. - Although derived with a very high degree of precision, only
one value of Pair was used to calculate the air spectra in Table
A.3 in the transitions of one orbit shell to another. Account was not taken
of the individual transitions from the separate orbitals within each shell.
To do so would result in an improvement in the calculated air spectra in
Appendix A to those in [5]. To effect this would mean deriving a very
precise algorithm that related Pair to spectral wavelength. To
ensure sufficient precision, this derivation would have to be based upon
Table A.2 and interpolation around each point effected manually.
(iii) Optimisation of Ge, the Electron
Spin Matter Wave Radius. - The values of Ge in (4.2) were only
effectively optimised for the first two orbit shells, n = 1 and 2, and 3p(+).
Precise optimisation for all other orbit shells was not fully explored,
partially because it should only be attempted after all systemic features
have been incorporated, i.e. as in (i) and (ii) above. Subsequently, it may
be possible to further reduce differences in orbital energy and spectral
signature between Appendix A and [5] via this parameter.
The exploration of the three above potential improvements to the theory is planned.
P5 Version 1.1.1
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P.G.Bass, April 2008
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