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Thomas-Ehrman effect

The Thomas-Ehrman effect, or Thomas-Ehrman shift, is an effect observed between pairs of mirror nuclei, which causes a shift in a the energy of a level (also known as a spin-eigenstate of the nucleus) when the level is unstable against proton emission.

Mirror nuclei are nuclei which are the $$t_{z}=-t$$ and $$t_{z} = t$$ members of an isospin multiplet; that is, they are nuclei $$A_{1} = N_{1}+Z_{1}$$ and $$A_{2}= N_{2}+Z_{2}$$ where $$N_{1}=Z_{2}$$ and $$Z_{1}=N_{2}$$ i.e. they "mirror" each other in proton-number and neutron-number.

The internucleon interaction has isospin symmetry (it only contains terms of the form $$\tau_{1}\cdot\tau_{2}$$), except for the Coulomb interaction term (which can be expressed in terms of $$\tau_{z}$$ only, and as such does not have a form $$\tau_{1}\cdot\tau_{2}$$, and so makes the internucleon interaction dependent on orientation of isospin ($$\tau_{z}$$ and breaks isospin-symmetry). Thus, the energy of a nuclear state should be invariant with regards to orientation of isospin.

All nuclei in an isomultiplet thus should exhibit the same eigenstates at the same energy when the effects of the Coulomb interaction are subtracted out, since they differ only in orientation of isospin and no other term in the internucleon interaction will give them different energies based on the orientation of isospin. A great example of this is the nuclei $$^{20}$$Ne, $$^{20}$$F and $$^{20}$$Na, which all exhibit the same spin states at very close energies.

However, it is observed that when one of the members of a mirror pair has a proton excess, and one of those protons is in an s-wave state, that the Coulomb shift of energy levels is somewhat reduced. This is the Thomas-Ehrman shift: "an effect of the Coulomb force on a loosely-bound or unbound proton occupying an s-orbit" Ogawa. The effect thus comes into play when there is a level (or spin eigenstate) of one of the nuclei of a mirror pair, based on a resonance between a proton and the rest of the nucleus (the core).

Intuitively, the concept is easy to understand: When one of the protons is in an s-wave, it can tunnel far out of the nucleus (because it does not see a centrifugal barrier). This means that it will spend much of its time outside the nuclear core. This reduces the total positive charge in the core and thus reduces the Coulomb shift of the energy levels.

There have recently been studies Mich which extend this idea of Thomas-Ehrman effect to explore the effects of resonances of any type on the energies of mirror-pair nuclei. The argument is that proximity to a resonance is associated with structural changes in the nucleus (e.g. clustering), and since each nucleus in a mirror pair will have different clustering thresholds, each nucleus will exhibit different structures for a particular spin-eigenstate and so the energies of these states will be rather different. These differences cannot be argued away purely with Coulomb considerations.