APPENDIX A.
A.1 Preliminary Analysis Involving the Solution to the
Orbital Equation of Motion.
The relativistic orbit of the electron about the proton nucleus can be
pictorially represented as in Fig. A.1
Fig. A.1 - Relativistic Electron Orbit.
The solution of the orbital equation of motion was, for a general case,
effected in [2], Eq.(5.40) and is repeated here for convenience
|
m = |
é
ë
|
{ ( 1 - h2m02 /c2 )1/2 + F0 m0 /m0 c2 }
( m0 h2/F0)( 1 + F02 /m02 c2h2 )
|
ù
û
|
|
é
ë
|
1 + |
{ m0 h2m0 /F0 - ( 1 - h2m02 /c2 )1/2 }cosj
{ F0 m0 /m0 c2 + ( 1 - h2m02 /c2 )1/2 }
|
ù
û
|
|
| (A.1) |
where
|
j = f |
æ
è
|
1- |
F02
m02 c2h2
|
ö
ø
|
1/2
|
|
| (A.2) |
Here, for conformity with the nomenclature used in this paper, F in
[2],Eq.(5.40) has been replaced with j.
Eq.(A.1) is of the form
Where
L is the relativistically corrected orbit semi-latus rectum.
e is the relativistically corrected orbit eccentricity.
In order to obtain the required terms in the nomenclature of atomic
structure as used in this series of papers, it is necessary to obtain
suitable expressions for m0, h and L. Note that h in the above equation
(A.1), is not Planck's constant but the swept area constant of the orbit as
defined in [2]. Also note that in all of the ensuing analysis, all terms
involving the velocity of light, c, will, where necessary, be binomially
expanded to retain only those of relativistic second order.
First for m0, this can be determined from the eccentricity terms
in (A.1) and (A.3), thus
|
e = |
|
|
ì
í
î
|
m0 h2m0
F0
|
- |
æ
è
|
1+ |
h2m02
c2
|
ö
ø
|
1/2
|
ü
ý
þ
|
|
|
ì
í
î
|
æ
è
|
1+ |
h2m02
c2
|
ö
ø
|
1/2
|
- |
F0 m0
m0 c2
|
ü
ý
þ
|
|
|
| (A.4) |
Solving this for m0 gives
|
m0 = |
|
|
ì
í
î
|
æ
è
|
1- |
F02
m02 c2h2
|
ö
ø
|
|
æ
è
|
1- |
F02 e2
m02 c2h2
|
ö
ø
|
ü
ý
þ
|
1/2
|
|
|
|
| (A.5) |
Now, substitution of this into the semi-latus rectum half of (A.1), then
gives after reduction
|
L= |
m0 h2
F0
|
|
ì
í
î
|
1- |
F02
2m02 c2h2
|
( 1+e2 ) |
ü
ý
þ
|
|
| (A.6) |
To determine a suitable expression for h, from [2], Eq.(5.22)
and from (A.2) this becomes
|
h = |
wj r 2
|
|
æ
è
|
1- |
F02
m02 c2h2
|
ö
ø
|
1/2
|
|
æ
è
|
1- |
v2
c2
|
ö
ø
|
1/2
|
|
|
|
| (A.8) |
Which can be expressed as
|
h = |
Mj*
|
m0 |
æ
è
|
1- |
F02
m02 c2h2
|
ö
ø
|
1/2
|
|
|
| (A.9) |
Solving (A.9) for h gives
|
h = |
Mj*
m0
|
|
æ
è
|
1+ |
F02
c2Mj *2
|
ö
ø
|
1/2
|
|
| (A.10) |
Substitution of this into (A.6) then gives for L,
|
L = |
Mj *2
m0 F0
|
|
ì
í
î
|
1+ |
F02
2c2Mj *2
|
( 1-e2 ) |
ü
ý
þ
|
|
| (A.11) |
Also from (A.8) and (A.10)
|
wj r 2 = |
Mj*
m0
|
|
æ
è
|
1- |
v2
c2
|
ö
ø
|
1/2
|
|
| (A.12) |
and also from (A.2) and (A.10)
|
wf = wj |
æ
è
|
1+ |
F02
c2Mj *2
|
ö
ø
|
1/2
|
|
| (A.13) |
Eqs.(A.11), (A.12) and (A.13) are the required subsidiary equations to determine orbital energy and can be converted to atomic structure
nomenclature by putting F0 = Ze2, thus
|
L = |
Mj *2
Ze2m0
|
|
ì
í
î
|
1+ |
Z2e4
2c2Mj *2
|
( 1-e2 ) |
ü
ý
þ
|
|
| |
|
wf = wj |
æ
è
|
1+ |
Z2e4
c2Mj *2
|
ö
ø
|
1/2
|
|
| (A.14) |
|
wj r 2 = |
Mj*
m0
|
|
æ
è
|
1- |
v2
c2
|
ö
ø
|
1/2
|
|
| |
These results are used in the derivation of electron orbital energy in the
main text, Section 2.4, and in the quantisation of sample orbits thereafter.
A.2 Derivation of Sommerfeld's Full Relativistically Mass Corrected Elliptic Orbit Energy Equation.
In Section 3.5, the expanded version of Sommerfeld's equation for a
relativistically mass corrected electron orbit energy level was derived. The
full version is derived here for interest.
Multiplying out (2.24), adding the rest mass energy and binomially
contracting gives
|
Eor = m0 c2 |
é
ë
|
1+ |
Z2e4
c2Mj 2
|
( 1-e2 ) |
ù
û
|
-1/2
|
- m0 c2 |
| (A.15) |
Inserting (3.34) and (3.44) then gives
|
Eor = m0 c2 |
é
ê
ë
|
1+ |
k2Z2
|
ù
ú
û
|
-1/2
|
- m0 c2 |
| (A.16) |
This finally reduces to
|
Eor + E0 = E0 |
é
ë
|
1+ |
k2Z2
{nr +( nf*2 - k2Z2 )1/2 }2
|
ù
û
|
-1/2
|
|
| (A.17) |
which is Sommerfeld's equation, and where E0 has been written for
m0 c2.
P2 Version 2.0.1
Ó
P.G.Bass, April 2008
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