State Of Decay 2 Map Change

[Fisseha Aranda]

Quantummechanics says that if a system is in an eigenstate of the Hamiltonian, then the state ket representing the system will not evolve with time. So if the electron is in, say, the first excited state then why does it change its state and relax to the ground state (since it was in a Hamiltonian eigenstate it should not change with time)?

The problem is that what determines time evolution is the total Hamiltonian of the system, and in Nature, the total Hamiltonian includes all forms of interactions. We usually neglect most interactions to get the overall description of the system, and then add secondary effects using perturbation theory. In this sense, the atom is very accurately described by $H_0$, but it is not the end of the story: there are many more terms that contribute to the real dynamics.

There are two ways of looking at it. One way is to recognize that the atom is not isolated. It is always coupled to the electromagnetic field. Even if field itself is in the ground state, there are "zero-point" fluctuations in the field amplitude. Thus, the atom is always feeling the influence of an external field. The zero-point fluctuations have components at all frequencies, including the atomic transition frequency. So the spontaneous decay of an excited atom can be thought of as stimulated emission due to the zero-point fluctuations.

The second way of looking at it is to take the system of interest to be the atom and the electromagnetic field. In this case, the state with no excitation of the field, and the atom in an excited state is not an energy eigenstate. The total wave function amplitude will start entirely atomic, but evolve to include field excitation.

As others have said you can't get rid of EM fields in the "real world". EM fields, which are waves of electric & magnetic fields generated by charged particles are omnipresent because electric fields decay by $$E = \frac\textconstantr^2$$ disappear only at infinity. So the transition rates would always be non-zero.If transition rates are non-zero, it would de-excite even if there is no external time-dependent disturbance.

Long Answer: "One of the paradigms of nuclear science since the very early days of its study has been the generalunderstanding that the half-life, or decay constant, of a radioactive substance is independent of extranuclearconsiderations". (Emery, cited below.) Like all paradigms, this one is subject to some interpretation. Normaldecay of radioactive stuff proceeds via one of four mechanisms: alpha decay: the emission of an alpha particle (a helium-4 nucleus), which reduces the numbers of protons andneutrons present in the parent nucleus each by two;

beta decay: this encompasses several related phenomena in which a neutron in the nucleus becomes a proton, or aproton becomes a neutron, along with some other things happening that involve electrons, positrons, neutrinos, oranti-neutrinos. These other things are, as we shall see, pertinent to several questions involving perturbation of decayrates;

gamma decay: the emission of one or more gamma rays (very energetic photons) that take a nucleus from an excitedstate to some other (typically ground) state. Some of these photons may be replaced by "conversion electrons", of whichmore shortly;

spontaneous fission: here, a sufficiently heavy nucleus simply breaks apart. Most of the discussion aboutalpha particles also applies to spontaneous fission.

"Conversion electrons" are produced by the process of "internal conversion", in which the photon that would normally beemitted in gamma decay is virtual and its energy is absorbed by an atomic electron. The absorbed energy isusually sufficient to eject the electron from the atom.

Now for the tie-in to decay rates. Both the electron-capture and internal conversion phenomena require an electronsomewhere close to the decaying nucleus. In any normal atom, this requirement is satisfied in spades: the innermostelectrons are in states such that their probability of being close to the nucleus is both large and insensitive toenvironmental influences. The decay rate depends only very weakly on the electron wave functions, i.e., on how much oftheir time the inner electrons spend very near the nucleus.

But this general rule has its exceptions. The most notable is the astrophysically important isotope beryllium-7. Be-7 decays purely by electron capture (positron emission being impossible because of inadequate decay energy) with a half-lifeof somewhat over 50 days. It has been shown that differences in chemical environment result in half-life variations ofthe order of 0.2%, and high pressures produce somewhat similar changes. Also, a 2004 paper (see the references) measuresa 0.8% reduction in half-life for Be-7 atoms enclosed in carbon-60 cages. Other nuclei in which decay-rate changes areknown to occur are zirconium-89 and strontium-85, also electron capturers; technetium-99m ("m" implying an excited state),which decays by both beta and gamma emission; and various other "metastable" isotopes that decay by gamma emission withinternal conversion. With all of these other cases the magnitude of the effect is less than is typically the case withBe-7.

What makes these cases special? The answer is that one or more of the usual starting assumptions (such asinsensitivity of electrons near the nucleus to external forces, or availability of the innermost electrons forcapture/conversion) are not completely valid. Atomic beryllium has only 4 electrons to begin with, and so the "innermostelectrons" are also practically the outermost ones, and are thus much more sensitive to chemical effects than isusual. With most of the other cases, the foibles of nuclear structure produce very little energy from the decay: aslittle as a few electron volts, as compared with most radioactive decays, which release hundreds or thousandsof kiloelectron volts. In these low-energy cases, the innermost electrons can't undergo internalconversion. Remember that converting an electron requires dumping enough energy into it to expel it from the atom (moreor less); "enough energy" is typically some tens of keV, and so they don't get converted at all in these cases. Conversion therefore works only on some of the outer electrons, which again are more sensitive to the environment.

A real anomaly is the beta emitter rhenium-187. Its decay energy is only about 2.6 keV, practically nothing by nuclearstandards. "That this decay occurs at all is an example of the effects of the atomic environment on nuclear decay: thebare nucleus rhenium-187 [i.e., stripped of all orbital electrons] is stable against beta decay [but not to bound-state betadecay, in which the outgoing electron is captured by the daughter nucleus into a tightly bound orbital], and it is thedifference of 15 keV in the total electronic binding energy of osmium [to which it decays] and rhenium [...] which makes thedecay possible" (Emery).

A 1996 paper (see the references) discusses this bound-state decay of bare-nucleus rhenium-187. Whereas neutralrhenium-187 has a half-life of 42 109 years, the authors measured fully ionised rhenium-187 to have a halflife of just 33 years! They discuss the cosmological implications of the altered half life of rhenium-187 in variousdegrees of ionisation in stellar interiors, and what that implies for our knowledge of galactic ages.

Alpha decay and spontaneous fission might also be affected by changes in the electron density near the nucleus, for adifferent reason. These processes occur as a result of penetration of the "Coulomb barrier" that inhibits emission ofcharged particles from the nucleus, and their rate is very sensitive to the height of the barrier. Changes inthe electron density could, in principle, affect the barrier by some tiny amount. But calculations show that themagnitude of the effect is very small. For a few alpha emitters, the change has been estimated to be of theorder of 1 part in 107 or less (!), which is not measurable, given that the alpha emitters' half lives aren't knownto that degree of accuracy to begin with.

All told, the existence of changes in radioactive decay rates due to the environment of the decaying nuclei is on solidgrounds, both experimentally and theoretically. But the magnitude of the changes is nothing to get very excited about,except for the case of neutron bombardment. Controlled neutron bombardment is the basis for nuclear reactors (andexponentially increasing nuclear bombardment is the basis of a nuclear fission bomb).

In the country of Gabon in mid-western Africa lies a uranium deposit known as Oklo. This is the only known naturalself-sustaining nuclear fission site.&nbsp As discussed in a 1989 paper (see the references), its uranium-235 concentration is0.48% to 0.68% of the total uranium present, in contrast to the 0.72% concentration that is normally found in uranium. This depletion is consistent with the effects of nuclear fission.

Although Oklo is the only such natural reactor site known, the generation of neutrons by lightning discharge has beenreported since at least 1985; so if you leave fissionable material out in a storm, its fission rate might be temporarilyincreased by lightning-induced neutrons.

Perhaps the best review article on this subject is G. T. Emery, Perturbation of Nuclear Decay Rates, Annual Reviewof Nuclear Science 22, pg 165 (1972). Papers describing specific experiments are cited in that article, whichcontains considerable arcane maths but also gives a reasonable qualitative feel for what is involved.

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1 IPCC (Intergovernmental Panel on Climate Change). 2021. Climate change 2021: The physical science basis. Working Group I contribution to the IPCC Sixth Assessment Report. Cambridge, United Kingdom: Cambridge University Press. www.ipcc.ch/assessment-report/ar6.

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