Some questions about spin in relation to the stern-gerlach experiment

[xortdsc]

Hi,

I have a few questions regarding the experimental outcome of the stern-gerlach experiment.
Let's suppose the following setup: We have a magnetic field whose field-lines point towards the positive z axis and the intensity of that field becomes stronger towards the positive z axis, so there is a gradient in the magnetic field strength with respect to the z-axis.

(1) When shooting a electrically neutral silver atom along the x-axis through this field it will be deflected slightly along the z axis by a specific amount depending on the spin it has. And since it can only have +1/2 or -1/2 spin it follows either one of 2 possible paths.
So a single beam of silver-atoms will split (along the z axis) into 2 sharp beams (assuming it was composed equally of +1/2 and -1/2 spin atoms).
This should be the standard setup of the stern-gerlach experiment. Did I understand that correctly ?

(2) In case the magnetic field would be homogenous no such splitting is seen ?

(3) When performing the original experiment using electrons instead of silver atoms, will the same splitting along the z axis occur ? Is the only difference that the electrons trajectory will additionally bend into a circle in the xy plane because of its electric charge ?

(4) Is the splitting only seen for moving particles ? So an atom at rest within the (inhomogenous) magnetic field will NOT feel any force along the z axis ?

(5) How do the beams behave for other spins like 0, 1 and 3/2 compared to the case of 1/2 ?

(6) How do we know that neutrinos have 1/2 spin ?

I hope somebody can give me some clarity on this. :)
Cheers.

 

[Simon Bridge]

Hi,

I have a few questions regarding the experimental outcome of the stern-gerlach experiment.
Let's suppose the following setup: We have a magnetic field whose field-lines point towards the positive z axis and the intensity of that field becomes stronger towards the positive z axis, so there is a gradient in the magnetic field strength with respect to the z-axis.

(1) When shooting a electrically neutral silver atom along the x-axis through this field it will be deflected slightly along the z axis by a specific amount depending on the spin it has. And since it can only have +1/2 or -1/2 spin it follows either one of 2 possible paths.
So a single beam of silver-atoms will split (along the z axis) into 2 sharp beams (assuming it was composed equally of +1/2 and -1/2 spin atoms).
This should be the standard setup of the stern-gerlach experiment. Did I understand that correctly ?

That's good enough - I'd say that the beams are "well defined".

(2) In case the magnetic field would be homogenous no such splitting is seen ?

If the field is uniform then there is no torque on the dipole so yeah - no splitting.

(3) When performing the original experiment using electrons instead of silver atoms, will the same splitting along the z axis occur ? Is the only difference that the electrons trajectory will additionally bend into a circle in the xy plane because of its electric charge ?

The electrons in the beam will also repel each other so the beam spreads out. The effect pretty much wipes out the effect of the electron spins.

You could use very low intensity beams though:
http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1031&context=physicsgay
... suggests that it is a feasible experiment. Don't know if it has been done.

(4) Is the splitting only seen for moving particles ? So an atom at rest within the (inhomogenous) magnetic field will NOT feel any force along the z axis ?

You can check the math for that if you like - does a stationary dipole feel a force in an inhomogeneous B field?

(5) How do the beams behave for other spins like 0, 1 and 3/2 compared to the case of 1/2 ?

... the behavior would be in accordance with the available spin states.

(6) How do we know that neutrinos have 1/2 spin ?

Conservation of angular momentum in neutron experiments?

You could send He3 nuclei through an SG apparatus one at a time.
But why not just do the Stern-Gerlach experiment with neutrons?
http://frhewww.physik.uni-freiburg....genLoesungen/Uebungen_05_SS2011-ExIV-HELM.pdf

 

[xortdsc]

Thank you very much.

You can check the math for that if you like - does a stationary dipole feel a force in an inhomogeneous B field?

I'd think the dipole should accelerate along (or against, depeding on spin) the gradient of the B field, is that right ?

... the behavior would be in accordance with the available spin states.

So that means that a beam of 0-spin particles will not split and a beam of 1-spin particles will split in two just like the 1/2-spin do, but with double the magnitude ?

 

[jtbell]

does a stationary dipole feel a force in an inhomogeneous B field?

I'd think the dipole should accelerate along (or against, depeding on spin) the gradient of the B field, is that right ?

The usual gradient vector applies to a scalar field, but $\vec B$ is a vector field. There is a force, but it's not a simple "gradient of $\vec B$":

http://en.wikipedia.org/wiki/Magnet..._external_magnetic_field_on_a_magnetic_moment

 

[xortdsc]

ah okay. thank you :)

 

[Simon Bridge]

I'd think the dipole should accelerate along (or against, depeding on spin) the gradient of the B field, is that right ?
...

It sounds like you are guessing.

The answer requires doing some maths - as suggested.
You will only get suggestions like that if I think the answer is within your ability.

You can always look up the answers:
http://www.physicsinsights.org/force_on_dipole_1.html

 

[xortdsc]

It sounds like you are guessing.

The answer requires doing some maths - as suggested.
You will only get suggestions like that if I think the answer is within your ability.

I wasn't sure what math to apply, so yes, I was just guessing.
However the link provided by jtbell made it clear what the math is.

Thanks both of you :)

 

[xortdsc]

You can always look up the answers:
http://www.physicsinsights.org/force_on_dipole_1.html

That link is very nice, too.

So finally it can be said that the spin of a particle acts like there would be some current running in a infinitesimal loop and therefore creating a small magnetic dipole field with all the effects that follow from that (such as interaction with other magnetic fields) ? Or does it behave different from that in some situations ?

 

[Simon Bridge]

No worries.

Usually when a student learns about S-G we start them off by deriving the expression for the deflection angle, which means they have the math already. You can't get very far from qualitative descriptions but fortunately the extra stuff is easy to look up.
Enjoy.

The spin is so-named exactly because particles behave like they have some sort of current loop - which you'd expect from a spacially distributed charge that is rotating.

A real current would have a thickness, so the field close to it would depart from the field of an intrinsic dipole.
Also: given the charge and the magnetic moment of, say, an electron, you can work out what radius the "current loop" would have to be and compare with the measured particle size. (Similarly you could work out the smallest spatial distribution of charge that would give rise to that magnetic moment to compare.)

So there are differences - what the links are using is a mathematical model.
WE can also do the calculations from the intrinsic moment without pretending there is a current loop.