3.4 A Relativistically Mass Corrected Circular Orbit.
In this case (2.10) becomes
From (A.14) this becomes
|
nh = |
æ
è
|
1+ |
Z2e4
c2Mj *2
|
ö
ø
|
| ó
(ç)
õ
| mwj rdl |
| (3.27) |
Here
so that (3.27) becomes
|
nh = |
æ
è
|
1+ |
Z 2e4
c2Mj *2
|
ö
ø
|
1/2
|
| ó
(ç)
õ
| mwj r 2dj |
| (3.29) |
which becomes
|
nh = |
æ
è
|
1+ |
Z 2e4
c2Mj *2
|
ö
ø
|
1/2
|
Mj * |
ó
õ
|
2p
0
|
dj |
| (3.30) |
which integrates to
|
nh = 2pMj * |
æ
è
|
1+ |
Z 2e4
2c2Mj *2
|
ö
ø
|
|
| (3.31) |
Re-arranging
|
Mj *2 - |
nh
2p
|
Mj * + |
Z 2e4
2c2
|
= 0 |
| (3.32) |
Solving (3.32) for Mj * gives,
|
Mj * = |
nh
2p
|
- |
k2Z 2h
4pn
|
or |
k2Z 2h
4pn
|
|
| (3.33) |
where k is Sommerfeld's Fine Structure Constant given by
Eq.(3.33) shows that theoretically, the electron can have two such circular
orbits which satisfy the quantisation criteria. However, the second can be
discounted for the purpose of determining atomic structure because it puts
the electron ïnside" the nucleus.
Insertion of the first root for Mj * in (3.31) into (2.24) then
gives the orbital energy thus
|
Eor = - |
Z 2e4m0
|
2 |
æ
è
|
nh
2p
|
- |
k2Z 2h
4pn
|
ö
ø
|
2
|
|
|
ì
ï
í
ï
î
|
1 - |
3
4
|
|
Z 2e4
|
c2 |
æ
è
|
nh
2p
|
- |
k2Z 2h
4pn
|
ö
ø
|
2
|
|
ü
ï
ý
ï
þ
|
|
| (3.35) |
which with the insertion of (3.8) and (3.34), and because k2
<< 1, (3.35) can be reduced to
|
Eor = - |
hRhy Z 2
n2
|
|
æ
è
|
1+ |
k2Z 2
4n2
|
ö
ø
|
|
| (3.36) |
Eq.(3.36) is the orbital energy for a relativistically corrected electron
mass in a circular orbit and as expected is dependent only upon the
principle quantum number. Nevertheless it is still of interest to discuss
the quantisation value of the azimuthal quantum number nf. This
number is still given by (3.9) but in the relativistically corrected case it
is clear that it cannot be an integer. This is because Mf is the
angular momentum of the rest mass which in this case does not represent the
total angular momentum of the orbiting electron. The relativistically added
mass results in an increase in the angular momentum,
and it is this plus the angular momentum of the rest mass which is quantised
by an integer value, i.e. nf*.
3.5 A Relativistically Mass Corrected Elliptic Orbit.
In this case (2.10) reduces to
|
nh = | ó
(ç)
õ
| m |
æ
è
|
×
r
|
2
|
+wf2 r 2 |
ö
ø
|
1/2
|
dl |
| (3.37) |
which becomes with insertion of (3.12) for dl
|
nh = | ó
(ç)
õ
| m |
æ
è
|
×
r
|
2
|
+wf2 r 2 |
ö
ø
|
dt |
| (3.38) |
In this integral the radial and angular terms must be treated separately.
This is because the radial term must be integrated around the orbit as a
function of the angle j, whereas the angular term which includes
the relativistic rotation of the orbit, must accordingly be integrated
around the orbit as a function of the angle f. Effecting this and,
with insertion of (2.17) and (A.14), (3.38) becomes
|
nh = Mj * |
é
ë
|
ó
õ
|
2p
0
|
|
ì
í
î
|
1
r 2
|
|
æ
è
|
dr
dj
|
ö
ø
|
2
|
ü
ý
þ
|
dj + |
ó
õ
|
2p
0
|
|
æ
è
|
1+ |
Z 2e4
c2Mj *2
|
ö
ø
|
1/2
|
df |
ù
û
|
|
| (3.39) |
Eq.(3.39) can most easily be solved by rewriting thus
|
nh = Mj * |
é
ë
|
ó
õ
|
2p
0
|
|
ì
í
î
|
1
r 2
|
|
æ
è
|
dr
dj
|
ö
ø
|
2
|
+1 |
ü
ý
þ
|
dj - |
ó
õ
|
2p
0
|
dj + |
ó
õ
|
2p
0
|
|
æ
è
|
1+ |
Z 2e4
c2Mj *2
|
ö
ø
|
1/2
|
df |
ù
û
|
|
|
which, having inserted the standard equations for the semi major and minor
axes of an ellipse, integrates to
|
|
nh
2p
|
= |
Mj*
( 1-e2 )1/2
|
+ |
Z2e4
2c2Mj *2
|
|
| (3.40) |
Re-arranging (64)
|
|
Mj *2
( 1-e2 )
|
- |
nh
2p
|
|
Mj *
( 1-e2 )1/2
|
+ |
Z 2e4
2c2( 1-e2 )1/2
|
=0 |
| (3.41) |
Solving (3.41) for Mj */( 1-e2 )1/2
then gives, taking the positive root
|
|
Mj *
( 1-e2 )1/2
|
= |
nh
2p
|
- |
k2Z 2h
4pn( 1-e2 )1/2
|
|
| (3.42) |
Where (3.34) has also been inserted. Now (3.24) also applies to this case
except that the eccentricity is the relativistically modified value and so
(3.24) becomes
Inserting this into the RHS of (3.42) finally gives
|
|
Mj *
( 1-e2 )1/2
|
= |
nh
2p
|
|
æ
è
|
1- |
k2Z 2
2nnf*
|
ö
ø
|
|
| (3.44) |
insertion of (3.34) into (2.24) gives the orbital energy thus
|
Eor = - |
2p2Z 2e4m0
|
n2h2 |
æ
è
|
1- |
k2Z 2
2nnj
|
ö
ø
|
2
|
|
|
ì
ï
í
ï
î
|
1- |
3
4
|
|
4p2Z 2e4
|
n2c2h2 |
æ
è
|
1- |
k2Z 2
2nnf*
|
ö
ø
|
2
|
|
ü
ï
ý
ï
þ
|
|
| (3.45) |
With the further introduction of (3.34) together with (3.8), and because
k2 << 1 this finally reduces to
|
Eor = - |
hRhy Z 2
n2
|
|
ì
í
î
|
1+ |
k2Z 2
n2
|
|
æ
è
|
n
nf*
|
- |
3
4
|
ö
ø
|
ü
ý
þ
|
|
| (3.46) |
This is the expanded version of Sommerfeld's equation for the quantised
energy levels of relativistically mass corrected elliptic orbits. For
interest, the full version is derived in Appendix A.2.
In (3.46) it can be seen that the orbital energy is dependent upon the
azimuthal quantum number as well as the principle.
Clearly the effect of the rotating orbit is to increase the orbital angular
momentum by a very small amount according to the relativistic increase in
mass of the orbiting electron. This rotating orbit is only treated as a
unique additional factor in ensuring that the electron orbital path is an
integral number of matter wave wavelengths, e.g. in (2.10). This ensures
that the principle quantum number n and the relativistic azimuthal quantum
number nf* are both integers. The non-relativistic azimuthal
quantum number, nf, being a descriptor of the rest mass angular
momentum only, is therefore no longer an integer. The difference of
nf from an integer value is however, relativistically small.
Thus from (3.46) it can be seen that each of the orbitals in Table 3.1,
(with nf and nr replaced by relativistic counterparts
respectively), now possess a slightly different energy level by virtue of
the presence of nf* in (3.46). This results in the spectral fine
structure as electrons make transitions from these orbitals to lower energy
shells.
3.6 Selection Rules.
It is of course well known that in the spectral output of say hydrogen, only
certain emission lines appear. For instance between shells 4 and 3 in Table
3.1 of the 12 apparently possible transitions, only 5 appear. This is due to
the so called Selection Rules that govern which transitions are permitted.
These rules are in turn governed by the manner in which the electron emits a
photon during the process of an orbital transition. The Selection Rule
universally quoted at this point in development is
That is, that the electron in making the orbital transition will lose/gain
exactly one quanta of orbital momentum, as well as one or more quanta of
orbital energy. Because the mechanism resulting in (3.47) is closely related
to the concept of electron spin, (3.47) is simply accepted for the purpose
of this paper, and will be fully justified for a corpuscular theory of
atomic structure in the next paper which will deal exclusively with the
mechanical effects of electron spin.
P2 Version 2.0.1
Ó
P.G.Bass, April 2008
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