The sensitivity of the decay rate of a nucleus to a change of the fine-structure constant is defined, in a similar way as for atomic clocks [Equation (23)], as

is a function of the decay energy . When is small, mainly due to an accidental cancellation between different contributions to the nuclear binding energy, the sensitivity maybe strongly enhanced. A small variation of the fundamental constants can either stabilize or destabilize certain isotopes so that one can extract bounds on the time variation of their lifetime by comparing laboratory data to geophysical and solar system probes.Assume some meteorites containing an isotope that decays into are formed at a time . It follows that

if one assumes the decay rate constant. If it is varying then these relations have to be replaced by

The -decay rate, , of a nucleus of charge and atomic number ,

is governed by the penetration of the Coulomb barrier that can be described by the Gamow theory. It is well approximated by where is the escape velocity of the particle. is a function that depends slowly on and . It follows that the sensitivity to the fine-structure constant is The decay energy is related to the nuclear binding energies of the different nuclei byAs a first insight, when focusing on the fine-structure constant, one can estimate by varying only the Coulomb term of the binding energy. Its order of magnitude can be estimated from the Bethe–Weizäcker formula

Element |
Z | A | Lifetime (yr) | Q (MeV) | |

Sm |
62 | 147 | 1.06 × 10^{11} |
2.310 | 774 |

Gd |
64 | 152 | 1.08 × 10^{14} |
2.204 | 890 |

Dy |
66 | 154 | 3 × 10^{6} |
2.947 | 575 |

Pt |
78 | 190 | 6.5 × 10^{11} |
3.249 | 659 |

Th |
90 | 232 | 1.41 × 10^{10} |
4.082 | 571 |

U |
92 | 235 | 7.04 × 10^{8} |
4.678 | 466 |

U |
92 | 238 | 4.47 × 10^{9} |
4.270 | 548 |

Table 9 summarizes the most sensitive isotopes, with the sensitivities derived from a semi-empirical
analysis for a spherical nucleus [399]. They are in good agreement with the ones derived from
Equation (61) (e.g., for ^{238}U, one would obtain instead of ).

The sensitivities of all the nuclei of Table 9 are similar, so that the best constraint on the time variation of the fine-structure constant will be given by the nuclei with the smaller .

Wilkinson [539] considered the most favorable case, that is the decay of for which
(see Table 9). By comparing the geological dating of the Earth by different methods, he concluded that the
decay constant of ^{238}U, ^{235}U and ^{232}Th have not changed by more than a factor 3 or 4 during the last
years from which it follows

As for the Oklo phenomena, the effect of other constants has not been investigated in depth. It is clear that at lowest order both and scales as so that one needs to go beyond such a simple description to determine the dependence in the quark masses. Taking into account the contribution of the quark masses, in the same way as for Equation (53), it was argued that , which leads to . In a grand unify framework, that could lead to a constraint of the order of .

Dicke [150] stressed that the comparison of the rubidium-strontium and potassium-argon dating methods to uranium and thorium rates constrains the variation of .

As long as long-lived -decay isotopes are concerned for which the decay energy is small, we can use a non-relativistic approximation for the decay rate

respectively for -decay and electron capture. are functions that depend smoothly on and which can thus be considered constant, and are the degrees of forbiddenness of the transition. For high- nuclei with small decay energy , the exponent becomes and is independent of . It follows that the sensitivity to a variation of the fine-structure constant is The second factor can be estimated exactly as for -decay. We note that depends on the Fermi constant and on the mass of the electron as . This dependence is the same for any -decay so that it will disappear in the comparison of two dating methods relying on two different -decay isotopes, in which case only the dependence on the other constants appear again through the nuclear binding energy. Note, however, that comparing a - to a -decay may lead to interesting constraints.We refer to Section III.A.4 of FVC [500] for earlier constraints derived from rubidium-strontium, potassium-argon and we focus on the rhenium-osmium case,

first considered by Peebles and Dicke [406]. They noted that the very small value of its decay energy makes it a very sensitive probe of the variation of . In that case so that ; a change of of will induce a change in the decay energy of order of the keV, that is of the order of the decay energy itself. Peebles and Dicke [406] did not have reliable laboratory determination of the decay rate to put any constraint. Dyson [167] compared the isotopic analysis of molybdenite ores (), the isotopic analysis of 14 iron meteorites () and laboratory measurements of the decay rate (). Assuming that the variation of the decay energy comes entirely from the variation of , he concluded that during the past years. Note that the discrepancy between meteorite and lab data could have been interpreted as a time-variation of , but the laboratory measurement were complicated by many technical issues so that Dyson only considered a conservative upper limit.The modelization and the computation of were improved in [399], following the same lines as for -decay.

The dramatic improvement in the meteoric analysis of the Re/Os ratio [468] led to a recent re-analysis
of the constraints on the fundamental constants. The slope of the isochron was determined with a precision
of 0.5%. However, the Re/Os ratio is inferred from iron meteorites the age of which is not determined
directly. Models of formation of the solar system tend to show that iron meteorites and angrite meteorites
form within the same 5 million years. The age of the latter can be estimated from the ^{207}Pb-^{208}Pb
method, which gives 4.558 Gyr [337] so that . Thus, we could
adopt [399]

As pointed out in [219, 218], these constraints really represents a bound on the average decay rate since the formation of the meteorites. This implies in particular that the redshift at which one should consider this constraint depends on the specific functional dependence . It was shown that well-designed time dependence for can obviate this limit, due to the time average.

Meteorites data allow to set constraints on the variation of the fundamental constants, which are comparable to the ones set by the Oklo phenomenon. Similar constraints can also bet set from spontaneous fission (see Section III.A.3 of FVC [500]) but this process is less well understood and less sensitive than the - and - decay processes and.

From an experimental point of view, the main difficulty concerns the dating of the meteorites and the interpretation of the effective decay rate.

As long as we only consider , the sensitivities can be computed mainly by considering the contribution of the Coulomb energy to the decay energy, that reduces to its contribution to the nuclear energy. However, as for the Oklo phenomenon, the dependencies in the other constants, , , …, require a nuclear model and remain very model-dependent.

Living Rev. Relativity 14, (2011), 2
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