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., [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 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 [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 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 [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
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.
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 [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 (![]() |
![]() |
![]() |
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
The more recent analysis, based on a modification of the neutron spectrum lead respectively to [416]
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
Similarly, [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.
http://www.livingreviews.org/lrr-2011-2 |
Living Rev. Relativity 14, (2011), 2
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