Double Stern-Gerlach with an Intermediate Electron Cloud

[Roberto Pavani]

TL;DR

If spin-polarized silver atoms are allowed to capture electrons between two Stern-Gerlach analyzers, does one expect the beam to remain in the selected branch, collapse predominantly into a non-magnetic component, or split into additional magnetic states? Has this been tested experimentally?

I am trying to understand what standard atomic physics predicts for the following thought experiment, and whether anything similar has ever been performed experimentally.

Silver was used in the original Stern-Gerlach experiment because neutral Ag has a single unpaired 5s valence electron

[Kr] 4d10 5s1

and therefore a well-defined magnetic moment.

Consider the following setup:

SG1 --> select UP branch only
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v
electron cloud
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v
SG2 (same orientation)
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v
transverse electric field
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v
detector

The first Stern-Gerlach analyzer (SG1) selects only one branch of the Ag beam (call it "UP").

The selected atoms then pass through a region where electron attachment is assumed to occur with non-negligible probability (regardless of the practical difficulty of achieving this experimentally):

Ag + e− -> Ag−

The beam then enters a second Stern-Gerlach analyzer (SG2) with the same orientation as SG1.

Finally, a transverse electric field is applied before detection so that charged and neutral particles can be distinguished.

My questions are:

1. What would be the dominant electronic state of Ag− formed in such a process?

2. Since the Stern-Gerlach splitting of neutral Ag is dominated by its single unpaired 5s electron, would formation of Ag− be expected to strongly reduce the magnetic moment by filling the 5s shell?

3. What pattern would SG2 be expected to produce?
- only the original UP branch?
- an UP branch plus a central (non-deflected) component?
- additional components corresponding to excited magnetic states?

4. Are there metastable negative-ion states of silver with non-zero magnetic moments that could survive long enough to reach SG2 before radiative decay?

5. After the transverse electric field:
- would a central component be identifiable as Ag− ions?
- would the surviving UP component correspond mainly to neutral Ag atoms that never captured an electron?

If SG2 (and the electric field analyzer) were rotated by 90° relative to the first one, would the surviving neutral Ag atoms split 50/50 while any non-magnetic Ag− component remained undeflected? Would this provide a clean way to distinguish electron attachment from survival of the original spin-polarized beam?

My naive expectation is based on the idea that electron attachment would preferentially produce a closed-shell 5s² configuration, eliminating the single unpaired 5s electron responsible for the magnetic moment of neutral Ag.

If electron attachment occurs with appreciable probability, many atoms might therefore be driven toward a low-energy Ag− configuration with a strongly reduced or vanishing magnetic moment. In that case SG2 might show a dominant central component together with a weaker residual UP component from neutral Ag atoms that never captured an electron.

However, I do not know whether standard atomic physics would actually predict this outcome, or whether there are important effects (electron capture cross sections, selection rules, metastable states, decay lifetimes, etc.) that would lead to a different beam structure.

Even if such an experiment is impractical, I would be interested in the theoretical prediction.
I am mainly interested in the expected beam composition and magnetic signatures, rather than in the practical feasibility of the setup itself.

Has anything similar ever been attempted experimentally, either with silver or another atomic beam?

Any references or qualitative estimates would be greatly appreciated.

 

[PeterDonis]

Ag + e− -> Ag−

This means the silver atom is now charged when it passes through the second S-G apparatus--which completely breaks the apparatus, since it's designed to work with neutral atoms, not charged ones. Indeed, that's the reason neutral silver atoms were used in the first place--because you couldn't just use single charged electrons, which would have been easily produced by a cathode ray tube; the S-G apparatus wouldn't work with them.

So the theoretical prediction for your thought experiment is simple: it won't work.

 

[PeterDonis]

electron attachment

It seems like the reason you are looking at this is to see if reducing or eliminating the magnetic moment changes what happens in an S-G apparatus. The theoretical prediction is simple: of course it does, since the apparatus acts on the magnetic moment.

My question is, why do you care?

I think you need to take a step back and think about what underlying issue you are really trying to get at--and then just ask about that issue, directly.

 

[Roberto Pavani]

This means the silver atom is now charged when it passes through the second S-G apparatus--which completely breaks the apparatus, since it's designed to work with neutral atoms, not charged ones. Indeed, that's the reason neutral silver atoms were used in the first place--because you couldn't just use single charged electrons, which would have been easily produced by a cathode ray tube; the S-G apparatus wouldn't work with them.

So the theoretical prediction for your thought experiment is simple: it won't work.

You're right. Once electron attachment occurs, the resulting Ag- ions are no longer neutral, so a conventional second Stern-Gerlach stage would need to be modified.

My idea was that the final transverse electric field could separate charged and neutral components, allowing them to be identified independently.
The purpose of the final electric-field stage is precisely to distinguish neutral Ag from Ag- ions

The reason for using Ag rather than free electrons is that the Ag beam can first be spin-selected by the initial Stern-Gerlach stage, while still retaining the possibility of electron attachment afterward.

 

[Roberto Pavani]

My question is, why do you care?

The deeper reason I am interested in this setup is that, in the standard Stern-Gerlach sequence with orthogonal analyzers, the second analyzer produces a 50/50 split even though the incoming beam was selected in a single branch by SG1.

My question is whether, after electron attachment, the available final states are constrained by the Pauli principle in a way that leaves an observable signature. In particular, if Ag- is predominantly formed through a closed-shell 5s25s^25s2 configuration, the second electron cannot simply occupy the same orbital with the same spin.
The final electronic configuration must satisfy Pauli, and I am wondering whether that constraint can be probed through the subsequent magnetic analysis.

 

[PeterDonis]

in the standard Stern-Gerlach sequence with orthogonal analyzers, the second analyzer produces a 50/50 split even though the incoming beam was selected in a single branch by SG1.

Yes, because the orthogonal analyzer is measuring a different observable, and eigenstates of the first are equal-amplitude superpositions of eigenstates of the second.

My question is whether, after electron attachment, the available final states are constrained by the Pauli principle in a way that leaves an observable signature.

The theoretical answer to that would be "of course", but trying to measure that the way you are proposing seems like a clunky way of doing it. I believe there are methods already developed for probing the configuration of electron orbitals in an atom or ion; using those would seem like a better approach. Or, for that matter, just measuring the magnetic moment of the ion directly, the same way we do for any charged particle; it should be zero if the orbital configuration is what you expect.

 

[Roberto Pavani]

Suppose electron attachment to a Stern-Gerlach selected Ag beam occurs with non-negligible probability.

Is the capture expected to proceed almost entirely into the closed-shell $5s^2$ ground state, or is there also a significant population of excited states such as $5s^1 5p^1$ with parallel spins?

If such excited states are populated, would they survive long enough to produce additional Stern-Gerlach components before relaxing to the ground state?

For example, could a sequence such as

$5s^1\uparrow\,5p^1\uparrow \;\rightarrow\; 5s^1\uparrow\,5p^1\downarrow \;\rightarrow\; 5s^2 + \gamma$

(or any other allowed relaxation path) occur on a timescale comparable to the flight time between the two analyzers?

In other words, should one expect only the original Ag component plus a predominantly non-magnetic $Ag^{-}$ component, or are additional metastable magnetic components expected?

 

[PeterDonis]

Is the capture expected to proceed almost entirely into the closed-shell $5s^2$ ground state, or is there also a significant population of excited states such as $5s^1 5p^1$ with parallel spins?

The way to answer that would be to calculate the energy of each state. Note that by calling the $5s^2$ state the "ground" state you are implicitly assuming that it has lower energy than the $5s^2 5p^1$ state. Intuitively I would expect that to be the case, but intuition isn't always a good guide, especially with atoms that have that many shells. Also, the $5s^1 5p^1$ state wouldn't necessarily have to have parallel electron spins; again, you would want to do a calculation of the energy to see if parallel or antiparallel spins had lower energy. Or possibly some linear combination of the two might have lower energy still.

If such excited states are populated, would they survive long enough to produce additional Stern-Gerlach components before relaxing to the ground state?

Again, I don't think it's a good idea to have "Stern-Gerlach" as your experimental procedure here. That introduces complications that are irrelevant to the actual issue you say you're interested in.

are additional metastable magnetic components expected?

Once more, this is a question that would need to be answered by a calculation. I don't know if there are any such calculations in the literature. I don't think it's a question that can be answered with simple intuitive arguments.

 

[Roberto Pavani]

I agree that spectroscopy would probably be a much more direct way to determine the final electronic configuration.

My reason for keeping the Stern-Gerlach analysis is slightly different.

If attachment proceeds mainly through

$5s^1 + e^- \rightarrow 5s^2,$

the unpaired 5s electron disappears and I would expect a strongly reduced magnetic moment.

On the other hand, if a significant fraction of captures populate excited configurations such as

$5s^1 5p^1,$

then some atoms might retain (or even increase) their magnetic moment, at least temporarily.

If such states cannot decay directly to the closed-shell $5s^2$ configuration, they may first require a change in spin coupling before relaxation.

In that sense I am not using SG primarily as a spectroscopy tool, but as a way to distinguish between capture channels that lead to different magnetic moments and possibly different relaxation pathways.

 

[PeterDonis]

My reason for keeping the Stern-Gerlach analysis is slightly different.

Your reason is basically that you want to measure the magnitude of magnetic moments. But there are much simpler ways of measuring the magnitude of magnetic moments that work on charged particles. The purpose of the SG apparatus is not to measure the magnitude of magnetic moments, particularly not on charged particles (for which, as already noted, it doesn't work at all). The purpose of the SG apparatus is to measure the direction of the particle's magnetic moment (as a proxy for the direction of its spin). But that's not necessary for what you're trying to find out.