Oklo is the name of a town in the Gabon republic (West Africa) where an open-pit uranium mine is situated. About 1.8 × 109 yr ago (corresponding to a redshift of 0.14 with the cosmological concordance model), in one of the rich vein of uranium ore, a natural nuclear reactor went critical, consumed a portion of its fuel and then shut a few million years later (see, e.g.,  for more details). This phenomenon was discovered by the French Commissariat à l’Énergie Atomique in 1972 while monitoring for uranium ores . Sixteen natural uranium reactors have been identified. Well studied reactors include the zone RZ2 (about 60 bore-holes, 1800 kg of 235U fissioned during 8.5 × 105 yr) and zone RZ10 (about 13 bore-holes, 650 kg of 235U fissioned during 1.6 × 105 yr).
The existence of such a natural reactor was predicted by P. Kuroda  who showed that under favorable conditions, a spontaneous chain reaction could take place in rich uranium deposits. Indeed, two billion years ago, uranium was naturally enriched (due to the difference of decay rate between 235U and 238U) and 235U represented about 3.68% of the total uranium (compared with 0.72% today and to the 3 – 5% enrichment used in most commercial reactors). Besides, in Oklo the conditions were favorable: (1) the concentration of neutron absorbers, which prevent the neutrons from being available for the chain fission, was low; (2) water played the role of moderator (the zones RZ2 and RZ10 operated at a depth of several thousand meters, so that the water pressure and temperature was close to the pressurized water reactors of 20 Mpa and 300℃) and slowed down fast neutrons so that they can interact with other 235U and (3) the reactor was large enough so that the neutrons did not escape faster than they were produced. It is estimated that the Oklo reactor powered 10 to 50 kW. This explanation is backed up by the substantial depletion of 235U as well as a correlated peculiar distribution of some rare-earth isotopes. These rare-earth isotopes are abundantly produced during the fission of uranium and, in particular, the strong neutron absorbers like , , and are found in very small quantities in the reactor.
From the isotopic abundances of the yields, one can extract information about the nuclear reactions at the time the reactor was operational and reconstruct the reaction rates at that time. One of the key quantity measured is the ratio of two light isotopes of samarium, which are not fission products. This ratio of order of 0.9 in normal samarium, is about 0.02 in Oklo ores. This low value is interpreted  by the depletion of by thermal neutrons produced by the fission process and to which it was exposed while the reactor was active. The capture cross section of thermal neutron by
Shlyakhter  pointed out that this phenomenon can be used to set a constraint on the time variation of fundamental constants. His argument can be summarized as follows.
In conclusion, we have different steps, which all involve assumptions:
We shall now detail the assumptions used in the various analyses that have been performed since the pioneering work of .
By comparing the solution of this system with the measured isotopic composition, one can deduce the effective cross section. At this step, the different analyses [465, 415, 123, 220, 305, 416, 234] differ from the choice of the data. The measured values of can be found in these articles. They are given for a given zone (RZ2, RZ10 mainly) with a number that correspond to the number of the bore-hole and the depth (e.g., in Table 2 of , SC39-1383 means that we are dealing with the bore-hole number 39 at a depth of 13.83 m). Recently, another approach [416, 234] was proposed in order to take into account of the geometry and details of the reactor. It relies on a full-scale Monte-Carlo simulation and a computer model of the reactor zone RZ2  and both RZ2 and RZ10  and allows to take into account the spatial distribution of the neutron flux.
|Ore||neutron spectrum||Temperature ()||(kb)||(meV)||Ref.|
|?||Maxwell||20||55 ± 8||0 ± 20|||
|RZ2 (15)||Maxwell||180 – 700||75 ± 18||–1.5 ± 10.5|||
|RZ10||Maxwell||200 – 400||91 ± 6||4 ± 16|||
|RZ10||–97 ± 8|||
|–||Maxwell + epithermal||327||91 ± 6|||
|RZ2||Maxwell + epithermal||73.2 ± 9.4||–5.5 ± 67.5|||
|RZ2||Maxwell + epithermal||200 – 300||71.5 ± 10.0||–|||
|RZ10||Maxwell + epithermal||200 – 300||85.0 ± 6.8||–|||
|RZ2+RZ10||7.2 ± 18.8|||
|RZ2+RZ10||90.75 ± 11.15|||
It was then noted [305, 416] that above an energy of several eV, the neutron spectrum shifted to a tail because of the absorption of neutrons in uranium resonances. Thus, the distribution was adjusted to include an epithermal distribution
These hypothesis on the neutron spectrum and on the temperature, as well as the constraint on the shift of the resonance energy, are summarized in Table 8. Many analyses [220, 416, 234] find two branches for , with one (the left branch) indicating a variation of . Note that these two branches disappear when the temperature is higher since is more peaked when decreases but remain in any analysis at low temperature. This shows the importance of a good determination of the temperature. Note that the analysis of  indicates that the curves lie appreciably lower than for a Maxwell distribution and that  argues that the left branch is hardly compatible with the gadolinium data.
The energy of the resonance depends a priori on many constants since the existence of such resonance is mainly the consequence of an almost cancellation between the electromagnetic repulsive force and the strong interaction. But, since no full analytical understanding of the energy levels of heavy nuclei is available, the role of each constant is difficult to disentangle.
In his first analysis, Shlyakhter  stated that for the neutron, the nucleus appears as a potential well with a depth . He attributed the change of the resonance energy to a modification of the strong interaction coupling constant and concluded that . Then, arguing that the Coulomb force increases the average inter-nuclear distance by about 2.5% for , he concluded that , leading to , which can be translated to
The following analysis focused on the fine-structure constant and ignored the strong interaction. Damour and Dyson  related the variation of to the fine-structure constant by taking into account that the radiative capture of the neutron by corresponds to the existence of an excited quantum state of (so that ) and by assuming that the nuclear energy is independent of . It follows that the variation of can be related to the difference of the Coulomb binding energy of these two states. The computation of this latter quantity is difficult and must be related to the mean-square radii of the protons in the isotopes of samarium. In particular this analysis  showed that the Bethe–Weizäcker formula overestimates by about a factor the 2 the -sensitivity to the resonance energy. It follows from this analysis that with gadolinium, found the favored result , which corresponds to
The more recent analysis, based on a modification of the neutron spectrum lead respectively to  .
Olive et al. , inspired by grand unification model, reconsider the analysis of  by letting all gauge and Yukawa couplings vary. Working within the Fermi gas model, the over-riding scale dependence of the terms, which determine the binding energy of the heavy nuclei was derived. Parameterizing the mass of the hadrons as , they estimate that the nuclear Hamiltonian was proportional to at lowest order, which allows to estimate that the energy of the resonance is related to the quark mass by
Similarly, [207, 467, 212] related the variation of the resonance energy to the quark mass. Their first estimate  assumes that it is related to the pion mass, , and that the main variation arises from the variation of the radius of the nuclear potential well of depth , so that
Then, in , the nuclear potential was described by a Walecka model, which keeps only the (scalar) and (vector) exchanges in the effective nuclear force. Their masses was related to the mass of the strange quark to get and . It follows that the variation of the potential well can be related to the variation of and and thus on by . The constraint (48) then implies that
In conclusion, these last results illustrate that a detailed theoretical analysis and quantitative estimates of the nuclear physics (and QCD) aspects of the resonance shift still remain to be carried out. In particular, the interface between the perturbative QCD description and the description in term of hadron is not fully understand: we do not know the exact dependence of hadronic masses and coupling constant on and quark masses. The second problem concerns modeling nuclear forces in terms of the hadronic parameters.
At present, the Oklo data, while being stringent and consistent with no variation, have to be considered carefully. While a better understanding of nuclear physics is necessary to understand the full constant-dependence, the data themselves require more insight, particularly to understand the existence of the left-branch.
Living Rev. Relativity 14, (2011), 2
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