2.0 Resurrection of the Bohr/Sommerfeld Theory of Atomic Structure.
2.1 Quantum Number Nomenclature.
The current nomenclature for quantum numbers in atomic structure theory is
not used in this paper. A new one, (similar to that of Sommerfeld), is used
for two reasons. Firstly, it introduces a degree of rationalisation and
secondly, the list differs slightly from those currently used. The table
below provides a comparative list.
Where relativistic effects are to be included, the azimuthal quantum number in the new nomenclature will be represented by nf*. The reason for this will be discussed during the derivation.
In addition to the above, the current nomenclature lists a number of "term"
letters for values of the orbital quantum number, l. The first five of these
are:-
This term scheme is adopted, but in relation to the azimuthal quantum number nf (and nf *), in the new nomenclature. In this paper only the first three quantum numbers, n, nf, (and nf*), and nr will be involved. The other numbers resulting from the effects due to spin will appear in subsequent papers.
2.2 Justification of the Quantization Criteria & Determination of Permitted Orbits.To begin this analysis, consider again the energy component of [1], Eq.(2.1). If this is to represent the orbital energy of a bound electron in an atom, for that electron orbit to be stable, it is necessary for the bound energy to be constant over an entire single orbit, e.g. there can be no net energy loss or gain. Thus for such a case, if the energy in [1], Eq.(2.1) is to be constant, then in [1], Eq.(2.15), fv must also be constant over an entire orbit. In turn, this means that in a simple re-arrangement of [1], Eqs.(2.24) and (2.26),
the product vlsv must be similarly constant. and where
It is noted that (2.1) appears to invoke the fictitious spatial "phase velocity" of the electron matter wave, viz [1], Eq.(2.19). However, although the quantity c2/v appears in (2.1) it is only via a re-arrangement of the relationships between real parameters and is therefore acceptable as a mathematical descriptor. In a stable elliptic orbit, the orbital velocity v cannot be constant because the distance of the orbiting electron from the central nucleus is continuously changing. Consequently for (2.1) to be constant for such an orbit, lsv while being single valued over an entire orbit , must however vary proportionately in precise inverse harmony to the variation in v within the orbit. This then ensures that the elliptical orbit is stable. In circular orbits, the distance of the electron from the central nucleus is constant and so the orbital velocity is constant. Thus in this case lsv is not only single valued over a complete orbit, but also exactly constant throughout it. The so called "pendulum orbits" are discussed at the end of this Section. Now, to justify the criterion of quantisation, the single valuedness of lsv over a complete orbit, inserting the component parts of v for a basic elliptic orbit in (2.1) gives after minor re-arrangement
where
Now, in any elliptic orbit it is well known that
Where
Substitution of (2.3) into (2.2) gives after minor reduction
Where
On the RHS of (2.4) the only variable is the angular position so that it is
clear that over a complete orbit
Eq.(2.5) states that over the orbital path length of a stable orbit the orbiting electron's matter wave wavelength is an integer number. Therefore the criterion of quantisation is now linked to the requirement that, within a stable orbit, the bound energy of the orbiting electron must be constant. Accordingly, the criterion of quantisation is thus proved to be a necessary and sufficient condition for the stability of a basic electron orbit. The variability of lsv within the orbit is also clearly visible in (2.4). If e = 0 then (2.4) becomes
and is constant throughout the complete orbit. This is the electron matter wave wavelength for circular orbits, (r º L). In the case of the pendulum orbits, in (2.4) L = 0 and therefore lsv = 0. Accordingly from [1], Eq.() fsv becomes infinite, which from [1], Eq.(2.26) requires that the orbital velocity v also becomes infinite. This contravenes the criterion of existence within the Relativistic Space-Time Domain D0, and is a sufficient proof for the exclusion of the pendulum orbits. Note that for fsv to be infinite would also necessitate infinite orbital energy. In (2.4) if e = 1, it becomes
The orbit is parabolic and f varies from -(p-d) to +(p-d). Quantisation does not apply because the orbit is not closed. The same comment applies to hyperbolic orbits, (e > 1).
2.3 Quantisation of the Bohr/Sommerfeld Atom.In view of the results of the preceding Section, it is now permissible to apply the quantisation process to relativistically modified electron orbits to resurrect the Bohr/Sommerfeld theory of atomic structure.
where h is Planck's constant. Because the wavelength is to be quantised for all orbit path lengths, both sides of (2.8) are integrated over the orbital path length to give
where l is the length of the orbit path, and m the energy mass of the electron. On the LHS if l is to be an integral number of wavelengths then (2.9) can be written
Where n is an integer, the principle quantum number of the orbit. In (2.9) and (2.10) the circled integral sign indicates integration over the complete path of the orbit. It is (2.10) which will later be used as the source equation for the sample quantisation of a number of permitted orbits. To show that the above derivation leads to the same quantisation results as in the original Bohr theory, it is sufficient to demonstrate that (2.10) leads to the original quantisation rules as propounded by Niels Bohr.
insertion of this into (2.10) and expansion of the RHS gives
Splitting the RHS of (2.12) into two terms then gives
where
Eq.(2.14) is identical to the original quantisation rules of Niels Bohr with the minor exception that the momentum terms are corrected for the relativistic mass increase of the electron. Eq.(2.14) is perhaps the most elegant way of representing the quantisation process but simpler analysis results from the use of (2.10) as in Section 3 below.
2.4 Orbital Energy Levels.In order to derive quantised orbital energy levels, an energy expression suitable for use in the process is required. To derive the form required here, use is made of the solution to the orbital equation of motion, which has been effected in [2]. The result however, requires considerable preliminary analysis before insertion into the orbital energy derivation process. For clarity, this preliminary analysis is relegated to Appendix A, Section A.1, the results of which are used in the following process.
Starting with Einstein's energy/momentum equation as stated at [1],
Eq.(2.29), the bound energy of the electron is
where
It should be noted that although the atomic number has been included in this analysis, only hydrogen, (Z = 1), will be considered in detail when calculating spectra. Via binomial expansion, retaining only second order relativistic terms,
(2.15) reduces to
where
and where (A.3) has been inserted for r. An expression for v is now required.
The angular velocity of the electron is wf r which from (A.3) and (A.14) is
So that from (2.17) and (2.18)
Solving (2.20) for v2 gives
Substitution of this into (2.16) gives for the orbital energy, after some
reduction including binomial expansion to relativistic second order
Now, substituting for L from (A.14) gives, again after some reduction
This expression for the orbital energy can now be quantised for all permitted orbits by inserting the appropriate expression for Mj*2/( 1-e2 ). Non-relativistically corrected orbits can be treated by letting c ® ¥, and circular orbits by putting e = 0. Eq.(2.24) leads directly to the expanded version of Sommerfeld's equation for relativistically mass corrected elliptic orbits as will be shown in Section 3.5.
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P.G.Bass, April 2008
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