Nuclear isomers are metastable nuclear states, typically protected from decay by having very different angular momentum from the ground state. Some are very stable; my favorite $^{180m}$Ta has a half-life greater than $10^{15}$ years.

The ground state $^{180}$Ta is however unstable with a half-life of a few hours. Had it been stable the system might have provided an interesting form of atomic information storage that could have been read using isomeric shift (and written by suitable handwavy nuclear technology).

This leads to my question: could there be some other very stable nuclear isomer that we have so far overlooked?

Or do we have good reason to think that due to studies of isomeric shifts and theoretical models of nuclei that it is unlikely any metastable isomers exist that have been mistaken for ground state isotopes?


Have we found all stable nuclear isomers?

The language of the title doesn't quite make sense. In nuclear physics, an isomer is defined as a state that is not the ground state, and therefore is not stable, but has an unusually long half-life. Only nuclear ground states can be absolutely stable, since their decay can be forbidden by conservation of mass-energy. You also don't need to say "metastable isomers," since an isomer is by definition metastable.

typically protected from decay by having very different angular momentum from the ground state

This is not really typical. It's atypical. This is a special type of isomer called an yrast trap. The yrast line consists of all the points on a plot of E versus J such that the state has the minimum E for a given J. Normally it is approximately parabolic in shape, as expected for a classical rotor. An yrast trap is a case where the yrast line has the unusual behavior of temporarily dipping rather than continuing to rise.

Most isomers are much less exotic. A common type of isomer occurs in odd and (especially) odd-odd nuclei, where the density of states near the ground state is large, so that the gamma-ray transition energies are very low.

Even in the case of a high-spin isomer, there are actually two different mechanisms that may make the half-life long. One is that, if it's an yrast trap, we simply need a gamma-ray transition with a high multipolarity. The other is that if the nucleus has a rigid prolate shape, there is an approximately conserved quantum number K, which is the projection of the angular momentum along the symmetry axis. Sometimes an yrast state has a high K, but the states it can decay to have very different values of K. The difference in spin may be small, but the transitions have small matrix elements.

So maybe what you're really asking is whether we have found all isomers that have very long half-lives due to being yrast traps and/or K selection? I think the answer is that we can't possibly have observed all such states. There are many, many isotopes that are predicted to exist (i.e., to be inside the proton and neutron drip lines) but that have never even been successfully produced and detected. We can't possibly know about isomeric states in these nuclei.

Even in nuclei that can be relatively easily produced by fusion reactions with stable beams and targets, it is very easy to miss isomers. Many gamma-ray experiments simply aren't designed to detect anything but prompt gamma decay. They may use a thin foil as a target, so that the recoiling products fly out the back, and then their decays can only be seen if they decay within about a nanosecond. Or they may use high gamma-ray multiplicity in prompt coincidence as a trigger, which isn't what you expect when decay hangs up for a while in an isomeric state.

It is possible to do specialized surveys for high-spin isomers, and they have been done. I think some of these were done in the 80's. I did my PhD research ca. 1995 on high-spin isomers, and we detected one in 176W basically by accident. I thought it would have been fun to do such a survey with more recent equipment, but it basically isn't considered a flashy area of research these days, and I don't know if anyone has devoted the necessary time to such a project. It would be a big project.

If you wanted to search for high-spin isomers, I think a good technique to use would be to let recoiling nuclei fly out of the back of the target and down a short beam pipe to a stopper foil, which would be surrounded by a castle of high-efficiency BGO detectors. You could look for events in which you got a high multiplicity of gamma rays in these detectors. Because of the high efficiency and high selectivity, you could check a reaction with only a very small amount of beam time.

You could look for regions of nuclei where such isomers are expected to occur -- basically regions where one of the Fermi levels is near a lot of high-K states. You would want to concentrate on regions that couldn't have been studied easily before, e.g., ones that can only be reached with radioactive beams.

Once you knew that a certain reaction was producing a high-spin isomer, you could go back and study it in more detail with high-resolution germanium detectors.

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