Why are there two different values for the electron magnetic moment?

[drodophila]

So far I have found two completely different values of the electron magnetic moment and i need help to know in which cases these are applicable:
They both refer (i think) to values of the particle as such, without orbital contribution. Just the spin contribution to the magnetic moment

1st:
the magnetic moment of the electron is 1.73 Bohr magnetons
http://hyperphysics.phy-astr.gsu.edu/Hbase/spin.html#c6"

2nd:
the magnetic moment of the electron is 1.0011596521931 Bohr magnetons
http://www.isis.rl.ac.uk/neutronSites/index.htm?content_area=/neutronSites/constants.htm&side_nav=/neutronSites/neutronsitesSideNav.htm&"
You have to look up : (Electron magnetic moment in Bohr magnetons)


the first one has to do with the fact that according to wave mechanics the magnitude of the spin angular momentum is not a multiple of the quantum number s but sqrt[s(s+1)].
It is the value normally used to predict magnetic moments in transition-metal coordination chemistry.

The second one is the value experimentally measured and appears everywhere.

Are they both correct?
When can we use which?

thanks

 

[kdv]

So far I have found two completely different values of the electron magnetic moment and i need help to know in which cases these are applicable:
They both refer (i think) to values of the particle as such, without orbital contribution. Just the spin contribution to the magnetic moment

1st:
the magnetic moment of the electron is 1.73 Bohr magnetons
http://hyperphysics.phy-astr.gsu.edu/Hbase/spin.html#c6"

Where do you see that value on that site?!? I have never seen the electron magnetic moment given as that value.

 

[drodophila]

It is at the top of the page

 

[pam]

The mu ans s at the top of the page are vectors.
g is defined n terms of their z components, where the \sqrt{s(s+1)} does not enter.
Parts of that website are confusing.
The "S-G Experiment" part seems to imply that the experiment was performed on electrons, which wouldn't work. Other parts mention that it was done on neutral atoms, and then some atomic physics theory was used to infer the spin of the electron.
The S-G expt is not accurate enough to find the difference of mu_e from 1.

 

[drodophila]

Yes, they are vectors.
g is a scalar quantity, and is called the electron gyromagnetic ratio or spin g-factor, i.e., the ratio between the MAGNITUDES of the magnetic spin angular momentum to the spin angular momentum ans is very close to two.
http://en.wikipedia.org/wiki/G-factor

Actually the very accurate value of 1.0011596521931 Bohr magnetons is exactly half of the very accurate value given for g.

However there are two things I do not understand:

Why do they assume the magnitude of the spin angular momentum (Total) to be 1/2 hbar instead of sqrt[s(s+1)] hbar, which is the real value?. 1/2 hbar is just the z-component of the total spin angular momentum!

The magnetic moment that can be measured (using a SQUID magnetometer) in any transition-metal compound with one unpaired electron and with no orbital contribution can be calculated using the SPIN-ONLY contribution to the magnetic moment (mu = g sqrt[s(s+1)] Bohr magnetons) with very good agreement with the experiment, for instance in the case of Ti +3.
http://wwwchem.uwimona.edu.jm:1104/spectra/MagMom.html"
How can this be explained?

 

[drodophila]

For instance in this example they give different formulae for the spin contribution and for the orbital contribution to the magnetic moment of the electron.
The orbital contribution is how it should look like, i.e., mu= sqrt[l(l+1)] Bohr magnetons
But not the spin constribution, which is just s.
http://en.wikipedia.org/wiki/Electron_magnetic_dipole_moment

However here it shows the opposite:
http://en.wikipedia.org/wiki/Spin_quantum_number

 

[pam]

Those two articles are confusing, at best.
The gyromagnetic ratio is always (if correctly) defined in terms of the z component of spin.
The square root factor never enters in a correct definition of the GR.
The Lande g factor mentioned in one of the articles as to do with the vector addition of
L and 2S in an atomic state, and is not the same as the g factor for an isolated electron.

 

[drodophila]

ok! After racking my brain i got the solution to the riddle.
Experimentally measured electron magnetic moment:
In order to measure the magnetic moment of a free electron a magnetic field must be applied and only the z-coordinate of the magnetic moment is measured. The question of how we measure experimentally the total magnetic moment of the electron is pointless, because the Larmor precession frequency cancels out the x and y contributions. Then, the z-component of the electronic magnetic moment is the measured value and amounts to g/2 Bohr magnetons.

the magnetic moment of the electron is 1.73 Bohr magnetons:
However, there are other indirect techniques that allow to calculate the magnitude of the whole spin vector, that gives rise to the above-mentioned z-component when a magnetic field is applied. The Curie Law for paramagnetic materials is an example of such techniques. Then 1.73 Bohr magnetons is the total magnitude of the spin magnetic moment vector for one single electron mu = g sqrt[s(s+1)] when the magnetic field tends to 0.

Thanks pam

 

[pam]

You have to learn to start with the statement "The mm of the electron is NOT 1.73 Bohr magnetons." I have seen that number nowhere, but in your posts.
You get sqrt(3) by misinterpreting that web page.
You have to understand the difference in QM between the operator $${\vec s}$$
and the operator $$s^2$$. Because the 3 components of $${\vec s}$$
don't commute, you can't use your classical "intuition". g is defined only in terms of
$$\mu_z=g s_z$$. That should leave no confusion.

 

[Hans de Vries]

pam is right, the magnetic moment of the electron is 0.00115965218085
Bohr magnetrons. This corresponds with the z-component of the spin and
not with the total spin.

The total magnetic moment corresponding to the total spin, which axis is
rotated over ~54.735 degrees with the z-component is considered to
be averaged out over all directions with only the z-component remaining.
Interestingly, This purely QM picture can be derived from classical magneto-
statics by simply requiring that the charge and spin-densities associated
with the wave-function do not lead to infinite energies.

Abandoning the classical point-charge resolves the infinite electrostatic
energy of the classical electron. The Pauli-Weisskopff interpretation spreads
out the charge and spin continuously over the wave-function.

However, this still does not resolve the infinite magneto-static energy because
the magneto-static dipole force increases with the fourth order if the distance
decreases:

$$\vec{F}_r\ =\ \frac{3\mu_o\mu_e^2}{4\pi r^4}\left[\ \left(1 - 3\cos^2(\theta) \right)\mbox{\Large \hat{r}}\ \right]$$

Compared with the electrostatic force which goes only with $1/r^2$. This is
too fast and leads to an infinite magnetostatic energy except when the angle
theta is 54.735 degrees, where the term between brakets becomes zero and
the force is neither attractive nor repulsive.

So classical magnetostatics defines the angle $\theta$ between the spin at any
certain point of the wave function, and the local average magnetic moment
corresponding to the z-component. It does not define the angle $\phi$ which is
undefined and averaged out. This is exactly the quantum mechanical picture.Regards, Hans