# What is physical significance of g factor?

1. Jan 28, 2009

### xfshi2000

In quantum mechanics, magnetic moment is equal to g factor times (gyromagnetic ratio x angular momentum). For orbital angular momentum, g=1, for electron spin angular momentum, g=2, for proton, it is 3.56. Is there any physical significance for this factor? why scientist introduce this factor? Why don't they just absorb this factor into the gyromagnetic ratio? thanks

2. Jan 28, 2009

### AEM

There is a very nice discussion of a naive model of electron spin, and the gyromagnetic ratio in Peebles Quantum Mechanics , pages 198-201. The g-factor was introduced by Goudsmit and Uhlenbeck to account for the behavior of an electron in a magnetic field because naive models of the electron do not give the proper magnetic dipole moment. g is measured experimentally via the precession of the spin vector. As you probably know, Dirac's theory predicts g = 2. However, it actually is slightly larger than 2, these corrections are explained by quantum field theory of electromagnetism. Incidentally, my sources indicate that g for the proton is 5.59 rather than 3.56.

3. Jan 29, 2009

### AEM

I must apologize. When I wrote my previous post it slipped my mind that there is a beautiful little book by Sin-Itiro Tomonga (as you recall ,winner of the Nobel Prize for QED along with Feynman and Schwinger) called The Story of Spin , published by U of Chicago Press. The first chapter is a discussion of the history of spin and tells how the g-factor was introduced by Lande (not Goudsmit and Uhlenbeck as Peebles said). As you can imagine since the whole book is about spin, if you have a chance to read it, you'll have you answer in more detail than you might have wished.

4. Jan 29, 2009

### clem

The gyromagnetic ratio has complicated units and depends on the particle's mass.
The g factor is dimensionless and is useful in comparing the magnetric moments of different particles.

5. Jan 29, 2009

### xfshi2000

thank you all.

6. Jan 29, 2009

### AEM

You're welcome. However, I had a few free minutes on my hands and you question interested me so I did a little reading. This is a case where the history of physics can be helpful in sorting things out. Lande introduced his g-factor when attempting to account for the Zeeman effect and according to Tomonaga, Lande's g is just the ratio of the magnetic moment to the angular momentum. Now, when one wants to develop a Hamiltonian for the motion of an electron circulating about a nucleus (remember this work was done in the mid 1920's slightly before the invention quantum mechanics) it is helpful to work in analogy with classical physics. In classical case, when one considers a current loop in a magnetic field one finds the relationship between the magnetic moment of the loop and the angular momentum of the charge circulating in the loop to be

$$\bf \mu = ( \frac{q}{2mc}) \bf \l$$

One can show ( see Shankar Principles of Quantum Mechanics ) that the same relation holds when one constructs the Hamiltonian for the circulating electron. So what happened a few years later when the idea of spin operators and electron spin had been introduced and one wanted to model electron spin? Reading between the lines of the text I just mentioned, a first try probably was to take $$\mu$$ to be

$$\bf \mu = g (\frac{-e}{2mc}) S$$

where g is a constant to be determined by experiment and S is the spin operator. I believe that this g is the g factor you were inquiring about. It would appear that it was originally introduced for convenience and only later was found to have a theoretical basis. So in the early days it was a fudge factor. The fact that it was measured to be equal to 2 and Dirac's theory predicted it to have the value 2 was a positive for the acceptance of Dirac's theory. In quantum electrodynamics corrections to the value of g can be calculated in a power series in $$\alpha$$, the fine structure constant. The close agreement between the calculated value of g and the value found by experiment gives one confidence that QED is correct.

A reasonable conclusion is that g gives one an indication of the difference between the classical approach and the quantum mechanical approach.

Last edited: Jan 29, 2009
7. Jan 30, 2009

### granpa

https://www.physicsforums.com/latex_images/20/2054609-0.png [Broken]

this formula assumes that the spinning charge and mass have the same radius (and distribution within that radius). 'g' suggests that they do not.

Last edited by a moderator: May 3, 2017
8. May 21, 2009

### granpa

or it (gyromagnetic ratio) suggests that the effective mass of the spinning electron is half its normal mass.

9. May 22, 2009

### clem

No. That's the wrong interpretation. g=2 is a relativistic effect that comes directly out of the Dirac equation.

10. May 22, 2009

### xepma

Hmm, I'm having to trouble to accept it as a relativistic effect. To me it seems that the fact that it comes out of the Dirac equation only proves a consistency of the g-factor with special relativity. The g-factor is, in the end, a reflection of the fact that the electron carries a non-trivial spin. Spin is also a concept consistent with relativity (i.e. irreps of the Poincare Group are labeled by their spin), but that doesn't mean that the effect (spin) is relativistic. Non-relativistic spin is perfectly fine.

In other words, by taking $$c\rightarrow\infty$$ we eliminate the relativistic effects, but the concept of spin and g-factors still remain.

11. May 22, 2009

### granpa

me too. in fact now that I think about it I dont really understand it at all.

gl=1
and gs=2
but jz=ml + ms
not jz=ml + 2ms

12. May 22, 2009

### TFT

Good point. Since SO(3) has spin representation, it is perfectly fine to talk about spin in the content of non-relativistic QM. While to get the specific value of the g factor, you do need QED, which has relativity intrinsically. I think this is a reflection of the fact that we do need relativity to describe the real world, since the idea of the g factor came from experiments.

13. May 24, 2009

### granpa

I think the above formula gives the fine structure splitting (due to spin-orbit coupling) minus the lamb shift and with zero external magnetic field. (the energy levels are a function of j only). someone please correct me if I'm wrong.

mj ranges from −j to +j in steps of one

http://en.wikipedia.org/wiki/Zeeman_effect#Weak_field_.28Zeeman_effect.29

apparently gl and gs are replaced by gj when calculating the splitting of individual fine structure lines in the presence of a magnetic field . (the splitting is proportional to the total magnetic moment?)

the quantity in square brackets is the Lande g-factor gJ of the atom (gl = 1 and gs = 2) and mj is the z-component of the total angular momentum. For a single electron above filled shells s = 1 / 2.
(VM) is the total shift in energy.

the above formulas dont work for hyperfine structure/splitting though. that apparently uses different variables.

Last edited by a moderator: Apr 15, 2017
14. May 26, 2009

### granpa

this image gives the fine structure + lamb shift + hyperfine doubling.
the fine structure is a function of j and n.

http://www.pha.jhu.edu/~rt19/hydro/img194.gif [Broken]

hyperfine structure uses:
https://www.physicsforums.com/latex_images/22/2211684-1.png [Broken]

Last edited by a moderator: May 4, 2017
15. Oct 9, 2009

### Narayanan.S

When an electron is bound in an atom it is subjected to spin - orbit coupling.The g-factor is the ratio of electron magnetic moment [eCre)xSqrt2/Bohr Magneton) + Bohr magneton] and this sum is again divided by Bohr magneton.C-spin velocity at "C" and re-classical radius of electron.
Stumbled upon this formula for electron anomaly from google search
May be it means something more.
Cheers

16. Oct 10, 2009

### ytuab

Uhlenbeck tried to explain spin g-factor 2 using the spinning electron sphere model.
So, he thought that total charge e is distributed only over the surface of an electron.
(If the charge is uniformly distributed, the spin g-factor becomes 1, which is the same as the electron orbital g-factor).

But if the electron has the classical radius size, (this means that an electron is as big as an proton or an neutron.) the spinning sphere speed leads to more than 100 times the speed of light. Of course if an electron is smaller than an proton, the spinning speed becomes much faster than that.
So he gave up the idea that the electron is a real thing (as Pauli did.)
Then the spinor matrices were introduced.

It remains a mystery what the electron spin g-factor 2 means if the electron is a real thing.

17. Oct 14, 2009

### granpa

g relates the angular momentum of an electron to its magnetic moment. it should be 1. instead its 2. I think this may have to do with 'effective mass'.

18. Oct 15, 2009

### Narayanan.S

Re: what is physical significance of g factor?

--------------------------------------------------------------------------------

I find it hard to accept that electrons are close to point particle (<10^-18m) and thought of considering classical radius for the following reasons
1) It is well known that the magnetic moments of electron are greater than those of nucleons and therefore I thought even modern physics should fundamentally expect a larger radius for the electrons(re-classical).Though experiments may show smaller and smaller radius as more and more energy is imposed on it.Is it that the interpretation of experiment is not correct?
2) Does it not mean violation of "C" will not reflect on mass energy relation?
3) At the instant of a neutron beta decay into a proton and since electrons are close to a point like particles the distance between them is about (0.5x10^-15- the accepted emperical value for nucleon radius) is well within range of nuclear forces
of (1.7x10^-15m).The binding energy between them should be (>95Mev) and the beta decay is not possible?
4) The electron size is not precisely known yet, then how to accept a precise value for the magnetic moment of an electron?.
5) Could not find an answer for the very high electric field intensity (point like charged electron) seems not addressed?

Therefore did an exercise with classical radius of electron and found that it agrees extremely well with g-factor correction.And this value for electron magnetic moment is considerably less than Bohr magneton and could have turned out to be anomalous Zeeman effect.
Request can anyone help me in clarifying on the above "5" points
Thanks

19. Oct 15, 2009

### Narayanan.S

By the introduction of gyromagnetic ratio tha angular momentum gets cancelled.
What remains is magnetic moment which is the only measurable quantity and for me I had to work back from this magnetic moment by dividing this by a suitable gyromagnetic ratio.
This is the answer I got

When an electron is bound in an atom it is subjected to spin - orbit coupling.The g-factor is the ratio of electron magnetic moment [eCre)xSqrt2/Bohr Magneton) + Bohr magneton] and this sum is again divided by Bohr magneton.C-spin velocity at "C" and re-classical radius of electron.
The g then gets related to (1/sqrt2)=Cos45 degrees
Classical magnetic moment "Mu"=IxArea
g=2 for spin may have been arrived at from projected area at Bohr orbit (2pixrB^2) and for electron it is (2pixre^2) for hemisperical projected area,hence g=1 but the directional quantum number
(ms=+-)1/2) is now (+-)sqrt2)
Stumbled upon this formula for electron anomaly from google search
May be it means something more.

20. Oct 15, 2009

### ytuab

Why do you think of the classical electron radius (2.8 x 10^-15 m) ?

Foundations of Quantum Physics by Charles E.Burkhardt
In page 264

------------------------------
They imagined that the electron is a spherical shell having total charge e uniformly smeared over its surface, reminiscent of the model used to derive the classical radius of the electron in Section 1.2.5.
This spinning sphere creates a magnetic moment identical with that of a bar magnet.
Is this model consistent with the classical radius of the electron? No-- as can be seen by
equating the angular momentum of the spinning sphere to 1/2 hbar. Solving for the speed of a point on the sphere leads to a speed that is roughly 100 times the speed of light.
(For a spinning shell with radius equal to the classical radius of the electron(Re), equate the angular momentum Iw to 1/2 hbar to show the spinning speed would be 100 times the speed of light.)
--------------------------------

So If the spinning speed does not lead to the speed of light, the electron must be much bigger than the classical radius size, an proton (10^-15 m) or an neutron.

For example, such a big electron can be captured by the nucleus?

Last edited: Oct 15, 2009