APPENDIX B.

Derivation of Electron Magnetic Dipole Strengths.

B.1 The Orbital Magnetic Moment.

A magnetic moment elemental due to the radial normal component of the orbital motion of the electron is

D  æ
è 		e
-
Y
 
f 
 ö
ø 		= i
2c
 æ
è
-
r
 
  x d
-
l
 
 ö
ø
(B.1)
Where
d
-
l
 
   is a vector element of orbit path.


-
r
 
  is the radius vector of the orbit.

i  is the effective current due to the motion of the electron through the central coulomb field.

Thus

 e
-
Y
 
f 
 = 1
2c
 ó
(ç)
õ
i
-
r
 
  x d
-
l
 
(B.2)
and thus
 e
-
Y
 
f 
 = Z
2c
 ó
õ 		e

0 
-
r
 
  x 
-
v
 
 de
(B.3)
So that, evaluating the vector cross product and taking the magnitude,
 e Yf = ZMj *
2c
 ó
õ 		e

0 
 de
me*
(B.4)

= ZMj *
2cme
ó
õ 		e

0 
 æ
è 		1- v 2
c 2
 ö
ø 		1/2

 
 de
(B.5)
The relativistic term may be ignored because the spin-orbit coupling energy is already of relativistic magnitude. Eq.(B.5) therefore reduces to

 e Yf = ZeMj *
2cme
    =     Zehnf *
4pcme
(B.6)
Eq.(B.6) is expressed in terms of a non-spin-orbit coupled environment. To express it in terms of a spin-orbit coupled environment nj is substituted for nf * to give
 e Yor = Zehnj
4pcme
(B.7)

This substitution is valid because nf * has the same quantum value in a non-spin-orbit environment as nj in a coupled environment.

Of course when Z = nj = 1, (B.7) becomes the smallest unit of orbital magnetic dipole moment, the Bohr magneton.

B.2 The Spin Magnetic Moment.

In order to complete this derivation it is necessary to consider the physical construction of the electron. For it to possess the appropriate spin angular momentum, it is proposed that it exhibits the mechanical attributes of a very thin wall spherical shell, with the electrostatic charge uniformly distributed on the outside surface. The derivation of spin magnetic dipole then proceeds as follows.


Picture 3

Fig. B.1 - Electron Charge Configuration.

Referring to Fig. B1, if the charge on the elemental is

De = - e
4p
 sinq dq df
(B.8)
The effective current due to the spin motion of this elemental charge is then
i = velocity    x   charge
path
 = - e wsp esinq dq df
8p2
(B.9)

Where

ewsp   is the electron spin rate.

The dipole due to this elemental is then


e Ysp = DL i
c
 = - e wsp Ge2 esin3q dq df
8pc
(B.10)

Where

DL   is the area bounded by the path of the elemental.

c   is the velocity of light.

Integrating over the complete surface of the electron shell

e Ysp = - e wsp Ge2 e
8pc
 ó
õ 		p

0 
 ó
õ 		2p

0 
 sin3q dq df
(B.11)
These integrals are simple and evaluate to
e Ysp = e wsp Ge2 e
3c
(B.12)

From [3], Appendix A the non-relativistic angular momentum of the spinning electron shell may be stated as

e Msp = 2/3me  e wsp Ge2
(B.13)

and because magnetic dipole coupling is of relativistic magnitude, (B.13) may be, for the purpose of this derivation, approximated by

e Msp = 2/3me  e wsp Ge2 = e nsp h
2p
(B.14)
Inserting (B.14) into (B.12) yields finally
e Ysp = eh e nsp
4pme c
(B.15)
This value is half that recognised in the literature, but is further addressed in the main text, sub-Section 2.2, to eliminate the difference.


P4 Version 1.3.0
Ó P.G.Bass, April 2008

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