Atoms in fields

[Danijel Gorupec]

I would like to understand better how atomic orbitals behave in (magnetic) fields...

After reading about Zeeman effect, I understand that electrons bound in an atom might shift their energies when placed in magnetic field. I ask if this energy shift is also associated with some change in the shape of their orbitals? (I guess, this is equivalent to asking if the probability density function changes).

I would expect that orbitals that have some angular momentum (and magnetic moment) do change their shape. I read that atoms near a magnetar star could look needle-shaped... But I don't see how would an orbital with zero angular momentum (and zero magnetic moment) change its shape?

On the other hand, if no change in orbital shape (probability density) can be associated with electron energy shift in magnetic field, how do then electrons 'shift' their energies (do they speed-up, become heavier or what)?

[studiot]

14 hours ago, Danijel Gorupec said:

I would like to understand better how atomic orbitals behave in (magnetic) fields...

After reading about Zeeman effect, I understand that electrons bound in an atom might shift their energies when placed in magnetic field. I ask if this energy shift is also associated with some change in the shape of their orbitals? (I guess, this is equivalent to asking if the probability density function changes).

I would expect that orbitals that have some angular momentum (and magnetic moment) do change their shape. I read that atoms near a magnetar star could look needle-shaped... But I don't see how would an orbital with zero angular momentum (and zero magnetic moment) change its shape?

On the other hand, if no change in orbital shape (probability density) can be associated with electron energy shift in magnetic field, how do then electrons 'shift' their energies (do they speed-up, become heavier or what)?

 

This question that shows some good thinking. +1

Since the answer can be quite complicated I will try to start with a simple version.

You have asked about orbitals, which are of quantum theory origin.
But your consideration is from classical theory. 
So let's sort that out first.

There are two types of energy repository involved with the interaction of electrons (in atoms) and magnetic fields.

The first can be answered satisfactorily and results in what is called the normal Zeeman effect.
Classically the electron (mass,m)  orbits the nucleus at a particular radius, r, with velocity v_o due to electric forces between the electron and the nucleus.

So the electron is moving under the action of a central force F_o, so by Newton's second law

[math]{F_0} = \frac{{mv_0^2}}{r} = m\omega _0^2r[/math]

where v_o is the linear tangential velocity of the electron and w_o is the angular velocity due to the circular motion.

 

If a suitable magnetic field is now applied (that is perpendicular to the plane of the orbit) two things will happen.

During the time the magnetic field is being established there will be an electric field tangent to the orbit because of the emf generated by the changing magnetic flux.
At the same time there will be an additional force on the electron perpendicular to the magnetic field and to the velocity of the electron that is radial to the orbit and in its plane.

This additional force can be inwards (towards the nucleus) or outwards, depending upon the direction the electron is travelling around its orbit.

The additional force is also small compared to the electrostatic force which maintains the orbit radius r.

So there is negligable change to the orbit radius, but the electron speeds up a little or slow down a little to adjust for the two new values of F_o,  one slightly lower and one slightly higher.

The derivation is quite simple and you can have it if you like, but the result is

[math]v = {v_0} \pm \frac{{eH}}{{4\pi m}}[/math]

So classically the energy repository is due to a change in the velocity of the electron not the shape (radius) of the orbit.

Thus the ground energy state is split into two, one slightly higher and one slightly lower.

 

The spectroscopic Zeeman effect is the result of transitions between energy states (usually the ground and another), each of which is split so the number of spectral line increases.

 

But the story is not yet over since a second type of energy repository is available that is a quantum effect.

This is due to the quantum spin of the electron and leads to an additional quantum number.

Unlike the classical value above the quantum values can only come in discrete values.where the spin is aligned parallel or antiparallel to the magnetic field.

Any questions?

 

[swansont]

I’m not aware of anyone that worries about the effects on orbital shape (but I’m only really familiar with one slice of atomic physics).

Most of the emphasis I know of is on the energy shifts and possible limitations on interactions, because that has a direct impact on how you run an experiment 

It’s possible that other fields (like physical chemistry) might care, if this affects how atoms bond in the presence of a field. But as studiot has pointed out, the effects are small — the Bohr magneton, which is the characteristic value to apply to Zeeman splitting, is 1.4 MHz/gauss. The hyperfine transition in hydrogen is a thousand times bigger, and transitions that change n (from the ground state) are generally more than a thousand times bigger than that (and often 10,000-100,000)

[Danijel Gorupec]

Thanks guys... I am glad that studiot used the term 'energy repository' as I think what bothers me is understanding how the energy is stored during these energy shifts.

So could the 'spin' be a possible way to store energy? It would surprise me as the quantum spin is fixed (it can measure only two discrete values of identical magnitude).

Here is what I think is the essence of my troubles: A simple atom, with only single electron in a state that only has spin magnetic moment (zero orbital magnetic moment) is immersed into magnetic field. Electron's energy shifts... But there is nothing measurable that changes about this atom. I expect that its orbital shape remains unchanged, and its spin (if measured) remains unchanged. Except, of course, its mass does change (from Einstein) - but is this it, is this mass change all that I should expect to happen with this atom?

[swansont]

1 hour ago, Danijel Gorupec said:

Here is what I think is the essence of my troubles: A simple atom, with only single electron in a state that only has spin magnetic moment (zero orbital magnetic moment) is immersed into magnetic field. Electron's energy shifts... But there is nothing measurable that changes about this atom. I expect that its orbital shape remains unchanged, and its spin (if measured) remains unchanged. Except, of course, its mass does change (from Einstein) - but is this it, is this mass change all that I should expect to happen with this atom?

A charged particle with spin has a magnetic moment. An external field means it can be attracted or repelled (depending on the spin projection) and the energy level shifts accordingly.

[Danijel Gorupec]

21 hours ago, swansont said:

A charged particle with spin has a magnetic moment. An external field means it can be attracted or repelled (depending on the spin projection) and the energy level shifts accordingly.

I believe you introduced the concept of potential energy. Specifically, the potential energy of magnetic dipole in magnetic field.

But isn't the idea of potential energy just a shorthand that we use? Isn't it always more precise to deal with the energy stored in the field? That is, I assume that the potential energy change (due to electromagnetic field) can always be represented as the electromagnetic field energy change.

The trouble is that I continuously fail to relate the potential energy of our elementary-charged-particle-with-spin to the field energy! If the particle really changes its potential energy, and this is not mirrored in the field energy, should I then accept the the potential energy is something really fundamental (not just a shorthand)? I just cannot imagine the potential energy as a thing on its own.

Oh, and yes... you might think that the energy shift of the particle is stored is in the magnetic field, but I don't think so (the energy stored in magnetic field seems to shift in a wrong way - it decreases when the electron shifts its energy up.)

[studiot]

On 7/12/2020 at 9:51 PM, Danijel Gorupec said:

Thanks guys... I am glad that studiot used the term 'energy repository' as I think what bothers me is understanding how the energy is stored during these energy shifts.

So could the 'spin' be a possible way to store energy?

I am sorry I did not ( and still don't) have time for diagrams in my last post.

  Repository was used deliberately as available energy because the energy is distributed or partitioned between different degrees of freedom.

The potential energy of a magnetic dipole in a field is the (angular) work done to align it with the field.

Some diagrams will help and I  hope to post them tomorrow.

 

Meanwhile I am not sure what your interest in this is.

The Zeeman effect is spectroscopic and only occurs for energy transitions.

The splitting of energy states due to immersion in a magnetic field happens anyway and does not require and energy transition.

It is just shown up by the Zeeman effect.

 

All will become clear with some diagrams.

 

[swansont]

3 hours ago, Danijel Gorupec said:

I believe you introduced the concept of potential energy. Specifically, the potential energy of magnetic dipole in magnetic field.

But isn't the idea of potential energy just a shorthand that we use? Isn't it always more precise to deal with the energy stored in the field? That is, I assume that the potential energy change (due to electromagnetic field) can always be represented as the electromagnetic field energy change.

The trouble is that I continuously fail to relate the potential energy of our elementary-charged-particle-with-spin to the field energy! If the particle really changes its potential energy, and this is not mirrored in the field energy, should I then accept the the potential energy is something really fundamental (not just a shorthand)? I just cannot imagine the potential energy as a thing on its own.

Oh, and yes... you might think that the energy shift of the particle is stored is in the magnetic field, but I don't think so (the energy stored in magnetic field seems to shift in a wrong way - it decreases when the electron shifts its energy up.)

The magnetic dipole interacts with the field. That’s the energy that matters, not the energy of the field.

[Danijel Gorupec]

On 7/14/2020 at 1:48 AM, swansont said:

The magnetic dipole interacts with the field. That’s the energy that matters, not the energy of the field.

Hmm... What do I do now - I don't find this answer that much revealing because I feel that terms 'interaction energy' or 'potential energy' are intentionally obscuring (they seem like aggregate terms used when it would be too complex to look into full details). Should I open another thread to clarify the difference between potential/interaction energy and field energy?

[swansont]

2 hours ago, Danijel Gorupec said:

Hmm... What do I do now - I don't find this answer that much revealing because I feel that terms 'interaction energy' or 'potential energy' are intentionally obscuring (they seem like aggregate terms used when it would be too complex to look into full details). Should I open another thread to clarify the difference between potential/interaction energy and field energy?

Sure. I’d like to know what you think field energy is, or why potential energy is mysterious.

Work is involved in rotating a magnetic dipole in a magnetic field, and energy is conserved, so personally I don’t see what is being obscured.That depends on the field where the dipole is, not the energy of the field.