Oklo is the name of a town in the Gabon republic (West Africa) where an open-pit uranium mine is
situated. About 1.8 × 10^{9} 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., [509] for more details).
This phenomenon was discovered by the French Commissariat à l’Énergie Atomique in 1972
while monitoring for uranium ores [382]. Sixteen natural uranium reactors have been identified.
Well studied reactors include the zone RZ2 (about 60 bore-holes, 1800 kg of ^{235}U fissioned
during 8.5 × 10^{5} yr) and zone RZ10 (about 13 bore-holes, 650 kg of ^{235}U fissioned during
1.6 × 10^{5} yr).

The existence of such a natural reactor was predicted by P. Kuroda [303] 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 ^{235}U and
^{238}U) and ^{235}U 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 ^{235}U 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 ^{235}U 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 [465] 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

is dominated by a capture resonance of a neutron of energy of about 0.1 eV ( today). The existence of this resonance is a consequence of an almost cancellation between the electromagnetic repulsive force and the strong interaction.Shlyakhter [465] 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.

- First, the cross section strongly depends on the energy of a resonance at .
- Geochemical data allow to determine the isotopic composition of various element, such as uranium, neodynium, gadolinium and samarium. Gadolinium and neodium allow to determine the fluence (integrated flux over time) of the neutron while both gadolinium and samarium are strong neutron absorbers.
- From these data, one deduces the value of the averaged value of the cross section on the neutron flux, . This value depends on hypothesis on the geometry of the reactor zone.
- The range of allowed value of was translated into a constraint on . This step involves an assumption on the form and temperature of the neutron spectrum.
- was related to some fundamental constant, which involve a model of the nucleus.

In conclusion, we have different steps, which all involve assumptions:

- Isotopic compositions and geophysical parameters are measured in a given set of bore-hold in each zone. A choice has to be made on the sample to use, in order, e.g., to ensure that they are not contaminated.
- With hypothesis on the geometry of the reactor, on the spectrum and temperature of the neutron flux, one can deduce the effective value of the cross sections of neutron absorbers (such as samarium and gadolinium). This requires one to solve a network of nuclear reactions describing the fission.
- One can then infer the value of the resonance energy , which again depends on the assumptions on the neutron spectrum.
- needs to be related to fundamental constant, which involves a model of the nucleus and high energy physics hypothesis.

We shall now detail the assumptions used in the various analyses that have been performed since the pioneering work of [465].

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 [123], 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 [416] and both RZ2 and RZ10 [234] 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 | [465] |

RZ2 (15) | Maxwell | 180 – 700 | 75 ± 18 | –1.5 ± 10.5 | [123] |

RZ10 | Maxwell | 200 – 400 | 91 ± 6 | 4 ± 16 | [220] |

RZ10 | –97 ± 8 | [220] | |||

– | Maxwell + epithermal | 327 | 91 ± 6 | [305] | |

RZ2 | Maxwell + epithermal | 73.2 ± 9.4 | –5.5 ± 67.5 | [416] | |

RZ2 | Maxwell + epithermal | 200 – 300 | 71.5 ± 10.0 | – | [234] |

RZ10 | Maxwell + epithermal | 200 – 300 | 85.0 ± 6.8 | – | [234] |

RZ2+RZ10 | 7.2 ± 18.8 | [234] | |||

RZ2+RZ10 | 90.75 ± 11.15 | [234] | |||

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 [416] indicates that the curves lie appreciably lower than for a Maxwell distribution and that [220] 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 [465] 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 [123] 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 [123] 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

which, once combined with the constraint on , implies at level, corresponding to the range if is assumed constant. This tight constraint arises from the large amplification between the resonance energy () and the sensitivity (). The re-analysis of these data and also including the data of [220] with gadolinium, found the favored result , which corresponds to and the other branch (indicating a variation; see Table 8) leads to . This non-zero result cannot be eliminated.The more recent analysis, based on a modification of the neutron spectrum lead respectively to [416]

and [234] at a 95% confidence level, both using the formalism of [123].Olive et al. [399], inspired by grand unification model, reconsider the analysis of [123] 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

Using the constraint (48), they first deduced thatSimilarly, [207, 467, 212] related the variation of the resonance energy to the quark mass. Their first estimate [207] 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 [467], 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
http://www.livingreviews.org/lrr-2011-2 |
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