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3.0 Quantisation of Permitted Orbits.
3.1 Preamble.The orbital energy quantisations to be derived here will cover the following four cases.
All of these have been derived before and are therefore well documented in the literature, [3], [4] et al. However, they have been included here for a number of reasons as follows. First, they provide confirmation that the quantisation process presented in this paper, i.e. (2.10), is valid. Secondly, some of the derivation so produced is needed later in this and subsequent papers. Finally, it provides the means by which the values for the quantum numbers used here can be discussed, which realises several factors concerning orbit characteristics which may be new.
It should be noted that in the following derivations the mass referred to as electron mass is the effective mass of the electron as given in [3], i.e.
where
This substitution accounts for the finite mass of the nucleus as it and the electron orbit around a common orbital focal point.
3.2 A Simple Circular Orbit.In this case (2.10) becomes simply, (by letting c ® ¥)
(3.2) becomes
Thus from (2.24) the orbital energy, (e = 0, and omitting the relativistic term)
is Rydberg's constant (for hydrogen). Eq. (3.6) is well known as the Balmer energy term. Note that in this case
3.3 A Simple Elliptic Orbit.In this case (2.10) reduces to
Eq.(3.11) becomes
Using (2.19) this reduces to
Introducing (2.3) yields
Integrating the first term in (3.15) by parts gives
Now multiplying (3.16) by 2 and (3.15) by -1 and adding gives
From the first part of (2.3) this can be written
This integral is well known as twice the area of an elliptic orbit so that
it reduces to
where p and q are the semi-major and semi-minor axes of the elliptic orbit. By virtue of the standard equations for an ellipse, (3.20) finally becomes
So that from (2.24) the orbital energy is, (omitting the relativistic term)
identical to the circular case. Note that the process here, from (3.14) onwards is essentially the same as presented in [3]. It should be noted that in the above derivation both the azimuthal, nf and the radial, nr quantum numbers have been suppressed because the orbital energy is completely specified by the primary quantum number n. However, both nf and nr are of extreme importance because they dictate the shape of the orbit. The azimuthal quantum number nf is from (3.9) given by
So that from (3.21) and (3.23)
the radial quantum number is determined. The primary question is whether nf is an integer for this orbit. In the quantisation of the circular orbit just covered, nf and n are identical so nf is at all times an integer. In the case analysed here the only change to the orbit is that a radial component has been added. Consequently there is no change to the azimuthal component and with Mf still being constant by the law of conservation, it is therefore determined that nf must still be integer. Note that this was effectively invoked in the use of (3.9) to obtain (3.23). Note also from (3.25) that with n and nf being integer, so must nr be. As a result it is possible to determine exactly the characteristics of permitted simple orbits. The term schemes for the first four orbital shells are thus shown in the following table.
From this table a number of orbit features are apparent. First, in the first shell, (n = 1), only one orbit is permitted and this is clearly circular. This must be so in the atoms of all the elements. In the second shell, two orbits are allowed, one elliptical and one circular and this sequence of permitted orbits continues up through the orbit shells with an extra elliptical orbit being added per shell. The orbits correspond to the term letters shown in Section 2.1. Note that in each higher shell the eccentricity of the first orbit is higher than in the previous shell. Thus as n ® ¥, e ® 1 the orbit becomes parabolic and the atom is ionised. However, in a future paper, as additional relativistic correction terms are introduced it will be shown that ionisation takes place long before n ® ¥. Also, it can clearly be seen from the Table that if nf = 0 then nr = n, (e = 1), and the so called pendulum orbits would result. However, these orbits were proved to be excluded in Section 2. Finally, it can also be seen from the Table that although all the orbitals have, via (3.7) and (3.22), exactly the same orbital energy, the shell/orbital configuration is exactly that which will result in the spectral fine structure when relativistic mass correction is introduced. This is shown in the next two Sections.
P2 Version 2.0.1
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P.G.Bass, April 2008
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