3 Quantisation of Permitted Orbits.
3.1 Preamble.
The orbital energy quantisations to be derived here will cover the following
four cases.
(i) A simple circular orbit.
(ii) A simple elliptic orbit.
(iii) A relativistically mass corrected circular orbit
(iv) A relativistically mass corrected elliptic orbit.
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
me is the mass of the electron.
mN is the mass of the nucleus.
m# is defined as rest mass, (# = 0), or energy mass, (# absent)
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 ® ¥)
and with
(3.2) becomes
so that
Thus from (2.24) the orbital energy, (e = 0, and omitting the relativistic term)
|
Eor = - |
2p 2Z 2e 4m0
n 2h 2
|
|
| (3.6) |
which finally becomes
where
is Rydberg's constant (for hydrogen). Eq. (3.6) is well known as the Balmer
energy term. Note that in this case
so that, in this case only
3.3 A Simple Elliptic Orbit.
In this case (2.10) reduces to
|
nh =m0 | ó
(ç)
õ
| |
æ
è
|
×
r
|
2
|
+ w 2r 2 |
ö
ø
|
1/2
|
dl |
| (3.11) |
and with
|
dl = |
æ
è
|
×
r
|
2
|
+w 2r 2 |
ö
ø
|
1/2
|
dt |
| (3.12) |
Eq.(3.11) becomes
|
nh =m0 | ó
(ç)
õ
| |
æ
è
|
×
r
|
2
|
+w 2r 2 |
ö
ø
|
dt |
| (3.13) |
Using (2.19) this reduces to
|
nh =Mf |
ó
õ
|
2p
0
|
|
ì
í
î
|
1
r 2
|
|
æ
è
|
dr
df
|
ö
ø
|
2
|
+1 |
ü
ý
þ
|
df |
| (3.14) |
Introducing (2.3) yields
|
nh =Mf |
ó
õ
|
2p
0
|
|
ì
í
î
|
e 2sin 2f
( 1+ecosf ) 2
|
+1 |
ü
ý
þ
|
df |
| (3.15) |
Integrating the first term in (3.15) by parts gives
|
nh =Mf |
ó
õ
|
2p
0
|
|
ì
í
î
|
1- |
ecosf
( 1+ecosf )
|
ü
ý
þ
|
df |
| (3.16) |
Now multiplying (3.16) by 2 and (3.15) by -1 and adding gives
|
nh =-Mf |
é
ë
|
ó
õ
|
2p
0
|
|
ì
í
î
|
e 2sin 2f
( 1+ecosf ) 2
|
+1 |
ü
ý
þ
|
df -2 |
ó
õ
|
2p
0
|
|
ì
í
î
|
1- |
ecosf
( 1+ecosf )
|
ü
ý
þ
|
df |
ù
û
|
|
| (3.17) |
which reduces to
|
nh =Mf |
ó
õ
|
2p
0
|
|
1-e 2
( 1+ecosf ) 2
|
df |
| (3.18) |
From the first part of (2.3) this can be written
|
nh =Mf |
( 1-e 2 )
L 2
|
|
ó
õ
|
2p
0
|
r 2df |
| (3.19) |
This integral is well known as twice the area of an elliptic orbit so that
it reduces to
|
nh =2pMf |
( 1-e 2 )
L 2
|
pq |
| (3.20) |
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)
and with, by definition
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.
| n | nf | nr | e | Current
Term |
|
| 1 | 1 | 0 | 0 | s |
| 2 | 1 | 1 | 0.87 | s |
| 2 | 0 | 0 | p |
| 3 | 1 | 2 | 0.94 | s |
| 2 | 1 | 0.75 | p |
| 3 | 0 | 0 | d |
| 4 | 1 | 3 | 0.97 | s |
| 2 | 2 | 0.87 | p |
| 3 | 1 | 0.66 | d |
| 4 | 0 | 0 | f |
Table 3.1 - Basic Orbit Characteristics for the First Four Shells.
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
Ó
P.G.Bass, April 2008
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