I teach physics, and I'm looking for simple, easy examples from chemistry to use in order to illustrate basic ideas about quantum mechanics for my students. A conceptually simple example is the $\text{He}^-$ ion, which has a metastable state with a mean lifetime of 0.36 ms. This is for the totally isolated ion, i.e., we're not talking about a gas or plasma in which there can be collisions or reactions. (How would chemists refer to this? "In vacuo?") This is easy to relate to in chemical terms (a noble gas needs an electron like a fish needs a bicycle) and provides a simple example of quantum mechanical concepts like exponential decay.

What would be more fun for my purposes would be an example of a metastable molecule or molecular ion that would spontaneously break up in two different ways. I would imagine that for the ions, one could measure the half-life and branching ratios by storing them in a circular accelerator. So we could have something like the decay channel

$$\text{XYZ}^+ \rightarrow \text{XY}^+ + \text{Z}$$

competing with, say,

$$\text{XYZ}^+ \rightarrow \text{X}^+ + \text{YZ}.$$

We can then discuss things like the probabilities (branching ratios) and the fact that if the ion starts out in a well-defined state, presumably its ground state, then we can only predict these probabilities, not the actual outcome.

Part of what's making it hard for me to find examples is that I don't know the terminology to google on. Is this "autodetachment," or is that only used for the loss of an electron? "Autodissociation?"

Note: I really am looking for the process described above, not something else like breakup induced by a collision, nuclear decay, electromagnetic decay of an electronic excitation, electron emission, or chemical reactions that require two reactants to come in contact. Those other examples might be fine for some educational purposes, but they're not what this question is about.

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    $\begingroup$ The fragmentation patterns seen in mass spectrometry might be partly what you are looking for: chemguide.co.uk/analysis/masspec/fragment.html. Not sure how much the rates of decay have been studied, but the patterns of relative abundance are the core of the technique. $\endgroup$ – Tyberius Apr 17 '20 at 4:05
  • $\begingroup$ Phosphorescence would obey an exponential decay curve. $\endgroup$ – MaxW Apr 17 '20 at 4:22
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    $\begingroup$ Any number of radioactive isotopes decay in multiple ways. $\endgroup$ – MaxW Apr 17 '20 at 4:35
  • $\begingroup$ x-ray excitation of atoms can decay by fluorescence or Auger electron emission. $\endgroup$ – MaxW Apr 17 '20 at 4:37
  • $\begingroup$ many chemical reactions can branch to form multiple products. Copper and nitric acid is one such example. $\endgroup$ – MaxW Apr 17 '20 at 4:46

This happens all of the time in mass spectrometry following electron-impact ionization. Following ionization, each molecular ion decomposes (or fragments, to use mass spectrometry lingo) into other radicals and ions, usually following multiple pathways. As the pressure inside a mass analyzer is usually near vacuum, each fragmentation event is independent. No collisions or reactions between particles occur. The mass analyzer collects all of the fragments and generates signal that computer software converts to a mass spectrum - a frequency bar graph of the masses of species generated. For some very simple molecules, only one fragmentation pathway may predominate, but for most organic compounds, there are multiple fragmentation pathways.

For example, let's consider the fragmentation of but-3-en-2-one radical cation with the formula of $\ce{C4H6O+}$ and the structure shown below. The mass/charge ratio (m/z) is 70 for this compounds

structure of the but-3-en-2-one radical cation

There are at least two different fragmentation pathways.

The first fragmentation pathway is a loss of a vinyl group ($-\ce{C2H3}$, -27 m/z): $$\ce{C4H6O+ -> C2H3O+ + C2H3 }$$

Fragmentation of but-3-en-2-one by vinyl loss

The second pathway is a loss of a methyl group ($-\ce{CH3}$, -15 m/z): $$\ce{C4H6O+ -> C3H3O+ + CH3}$$

Fragmentation of but-3-en-2-one by methyl loss

Each of these fragmentation pathways is independent. Here is a copy of the mass spectrum from the NIST Chemical Webbook, which is a frequency bar graph of all possible fragments. You can see in addition to the two pathways mentioned above, there are many other possible fragments.

mass spectrum of but-3-en-2-one

We can estimate the relative rates of these processes from the frequencies of the peaks in the mass spectrum. The first pathway occurs with a relative frequency of 83.19 (based on the height of the m/z = 43 peak) , and the second pathway occurs with a relative frequency of 1.000 (from the height of the m/z= 55 peak). We can use these as proxy for the relative rates of decomposition. We can also see that the parent ion survives this process with a relative frequency of 0.57779 (from the m/z = 70 peak), which indicates that the parent ion has at least some stability.

  • $\begingroup$ I was hoping to see a mass spec answer. It seems to fit with what the original post is looking for, with multiple fragmentation pathways and fairly well characterized ratios between those paths based on stability of given types of fragments. $\endgroup$ – Tyberius Apr 21 '20 at 14:51
  • $\begingroup$ This is nice, thanks. It would be cool to have an example in which there was a known state in the parent ion, and a measurable lifetime. $\endgroup$ – Ben Crowell Apr 21 '20 at 21:10

The key operation is to make a metastable particle that has at least two decomposition pathways and a short lifetime. Fragmentation is probably the key word you were looking for. Pyrolytic fragmentation of complicated molecules (1-methoxycyclopropylamines) is one way https://www.sciencedirect.com/science/article/pii/S0040403900741402. Photolytic fragmentation of acetaldehyde, a relatively simple molecule, has been investigated https://aip.scitation.org/doi/10.1063/1.4878668 , and many others have been investigated with lasers. Another well known area is mass spectrometry fragmentation https://en.wikipedia.org/wiki/Fragmentation_(mass_spectrometry). For example, toluene is bombarded with electrons, loses one to form a

enter image description here

cation, then decomposes to form a benzyl cation or a phenyl cation.

An advantage of using mass spectrometric data is that there is a lot out there, and the idea can also be hypothetical, in that a molecule can be selected on the basis of its simplicity and likely decomposition products to allow theoretical calculations, e.g., for educational purposes, even in the face of insufficient actual data.

  • $\begingroup$ This looks interesting, thanks. Is the parent ion in a defined state before it breaks up? Is there a measurable lifetime and measurable branching ratios? $\endgroup$ – Ben Crowell Apr 21 '20 at 21:08
  • $\begingroup$ Original state for volatile compounds is just the ground state; branching ratios will be measured as output; lifetimes will be more difficult, not really sure how that's done now. I did mass spec work as a grad student, many moons ago. $\endgroup$ – James Gaidis Apr 22 '20 at 12:49

There are many compounds that will break down into different products when involved in collisions with inert molecules. For example, $\ce{F3-}$ can separate into $\ce{F2 + F-}$ or into $\ce{F. + F2-}$. But atoms or molecules that will breakdown without being involved in a collision are rarer (simply because their instability means they are harder to observe).

One example, though, is the radioactive decay of the bismuth-212 isotope. It decays ~64% of the time by $\beta-$ decay, producing polonium-212: $\ce{^{212}_{83}Bi -> ^{212}_{84}Po + e- + \nu_e}$. The other ~36% of the time it decays by $\alpha$ decay to thallium-208: $\ce{^{212}_{83}Bi -> ^{208}_{81}Tl + ^4_2He}$.

More info can be found here: http://nucleardata.nuclear.lu.se/toi/nuclide.asp?iZA=830212

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    $\begingroup$ Thanks, but this doesn't answer the question. The question is about spontaneous breakup, and it's a molecular physics question, not a nuclear physics question. $\endgroup$ – Ben Crowell Apr 20 '20 at 22:38

This is not really a complete self-answer, but I thought it was close enough that it was worth posting. I would be happy if others could come up with better examples.

The ions $\text{N}_2^{2+}$ and $\text{CO}^{2+}$ have been studied experimentally and theoretically in Pandey, Bapat, and Shamasundar, J. Chem. Phys. 140, 034319 (2014). Pandey's thesis also has a helpful table on p. 122. Their experimental setup is only sensitive to a breakup in which both fragments are charged. They do a total kinematic reconstruction of the two fragments and infer the kinetic energy release (KER). Their resolution is not quite good enough to resolve contributions from individual vibrational states in their KER spectrum, but you can see structures in them which they explain as arising from those vibrational states. They didn't store the fragments, just let them fly through a spectrometer, so they could only see decays that happened within a few milliseconds. With these two dications, you don't get tunneling on these time-scales when the initial state is the vibrational and electronic ground state. They do observe it from the vibrational ground state coupled to an electronic excitation and vice versa. They don't actually seem to have measured any lifetimes, and they don't give references to anyone else who has. There are no branching ratios to measure, because their molecules were binary, and their detector needed to see two charged fragments.

Eland et al., https://arxiv.org/abs/1908.11441.pdf (2019), have studied the breakup of $\text{HNCO}^{2+}$ and $\text{HNCO}^{3+}$. There are cases where different breakup channels compete, and there seem to be some lifetimes on the order of microseconds. But the experimental setup basically seems pretty messy, and it doesn't seem like they got a lot of detailed data.

Relevant keywords seem to be "dissociative ionization" and "vibrational autoionization."

These are both pretty recent, and I was kind of surprised that the state of the art was so crude. To me as a physicist, these seem like obvious things for a molecular physicist or quantum chemist to want to study. No doubt my ignorance of the experimental techniques is my reason for not understanding why this kind of experiment is so hard. I wonder if there are any systems that happen to be particularly easy to study and that have been better characterized than these.

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    $\begingroup$ It's not clear to me how one knows that the cations are all in the ground state before dissociation. Couldn't the breakup channel be dependent on the initial state produced by the ionization collision (eg angle, energy, etc)? I think that's the challenge of finding a system of the sort you are looking for. One needs to have molecules that are unstable enough that they fall apart, but stable enough that they have time to relax to a common ground state as the starting state before dissociation. $\endgroup$ – Andrew Apr 21 '20 at 13:50
  • $\begingroup$ @Andrew: In the Pandey experiment, none are in the ground state. See Pandey's table on p. 122, and the text of my answer starting at "With these two dications..." $\endgroup$ – Ben Crowell Apr 21 '20 at 21:06

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