Electron g-factor

T   
he electron g-factor involves the fact that in the ground state of the hydrogen atom the electron and proton revolve about a common center of mass rather than with the proton at the exact center of the electron's orbit.

This: Not this:

For practical considerations one may assume a proton at the orbital center since the effect is quite small (the proton is 1836 times more massive). However, for purposes of calculation (to 1 part in 50,000,000) it must be accounted for.

The velocity of the electron in the ground state must be modified by recalculating the radius of centripetal force, i.e. the electron is at one end of a dumbell balanced at a point between the electron and proton.

Namely, 1836.1527... / 1837.1527... of radius=h/aB .

Since h=1 in our previous calculation, the new radius is

(1836.../1837...) / (aB) = .99945568 x 251619.002 = 251482.0407

The velocity of the electron must therefore be somewhat slower because it traverses a shorter circumference in the same time.

Then,

ar = a / (1 - [ a/(1+B) ]^2 )^1/2

From Eq.#2 sec.19 ... ar = a / (1 - a^2)^1/2

Replacing the right side of Eq.#2 with the right side of the equation given here above ... into the final equation in sec.19

ar = [ 3/2^1/3 {Br ^4 / (2/3^2/3)}^1/2 ]^1/3


as well as Br = B(1-[a/(1+B)]^2)
Eq.#1..........Brevised = B (1-a^2)

yields a new general equation relating the e/P mass ratio and FSC with due consideration for the electron g-factor.

I did not accually make a new general equation because of algebraic difficulties. Rather, I simply made a constant of /(1+B) and tried it out to see if the correction was adequate. Since it did not do what I wanted, I didn't bother to make another "bad" equation.

Unfortunately, this "fix" only affects the eighth digit (1836.1517...) yielding a final value of 1836.15195...

Whereas we are looking for 1836.1527... a "shortfall" of .0008 , i.e. 1836.1519 + .0008 = 1836.1527.

I am here accepting that the seventh digit, 2, is "secure" meaning that the past history of Codata type evaluations indicates that this digit is very unlikely to change with further refining experimentation.

I consider this shortfall to be caused by some gross error in my approach rather than something trivial.


Addendum: 3-28-2000
New Codata estimates (1998) favor me slightly. If evaluated with the new numbers my "shortfall" would be about .0005 instead of .0008. I don't consider this enough to get me on the right track though. I wouldn't want to be off by more than .0002 at this juncture.

Addendum: 12-14-2004
New Codata estimates (2002) Now, evaluated with the new numbers my "shortfall" would still be about .0005 instead of .0008. So, nothing new is indicated.

codata 2002
 7.297 352 568(24) x 10-3  =  FSC
 1836.152 672 61(85)       =  e/P mass ratio

Evaluated to ~12 digits (in my shorthand for eq. parts):
1:54 AM 12/13/2004

a....6.23373009138 x10-4 =alpha^1.5
b....2/3 ^1/3 = .873580464736
c....[1-alpha^2]  = .9999467486455 .... ^1.75 = .9999068119905

axb= .00054456648302665
axb/c = .000544617234822705
1/[axb/c] = 1836.151954
and figure another .0002 for g-factor correction 
still gives a .0005 shortfall .... 1836.1521

Note:
Since the velocity of the electron in orbit is the fine structure constant (~1/137 of light velocity in this system of measure), the magnetic field produced by the electron is dependent on a square root as in [1-v2]^1/2.
(The differences considered here for orbital radii (and subsequently orbital velocity) determine the Bohr magneton and then the electron g-factor.)
We can then see why the g-factor is stable to so many more significant digits than the FSC or e/P mass ratio. Observe:

[ 1-1/137.03598952 ]^1/2 = .99997337397
[ 1-1/137.01932172 ]^1/2 = .99997336749

A large shift in velocity (in the 5th significant digit) produced a change in the Bohr magneton generating output in only the 8th digit (and a smallish change at that). The robustness of g-factor calculations (about which scientists sometimes seem overly proud) are byproducts of the forms of the equations rather than from experimental "technique".
In fact, scientific experimental technique has never surpassed the level of back-calculated equations.



Next Page


Ebtx Home Page