3.2 The Oklo phenomenon

3.2.1 A natural nuclear reactor

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 [382Jump To The Next Citation Point]. 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 16429Sm, 16513 Eu, 15645Gd and 15645Gd 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 14629Sm ∕16427Sm 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 [465Jump To The Next Citation Point] by the depletion of 149Sm 62 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 149 62 Sm

n + 149Sm − → 150Sm + γ (38 ) 62 62
is dominated by a capture resonance of a neutron of energy of about 0.1 eV (Er = 97.3 meV today). The existence of this resonance is a consequence of an almost cancellation between the electromagnetic repulsive force and the strong interaction.

Shlyakhter [465Jump To The Next Citation Point] 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 [465Jump To The Next Citation Point].

3.2.2 Constraining the shift of the resonance energy

Cross sections.
The cross section of the neutron capture (38View Equation) strongly depends on the energy of a resonance at Er = 97.3 meV and is well described by the Breit–Wigner formula

2 σ(n,γ)(E ) = g0π--ℏ---------Γ nΓ-γ------ (39 ) 2 mnE (E − Er )2 + Γ 2∕4
where −1 −1 g0 ≡ (2J + 1)(2s + 1) (2I + 1) is a statistical factor, which depends on the spin of the incident neutron s = 1∕2, of the target nucleus I, and of the compound nucleus J. For the reaction (38View Equation), we have g = 9∕16 0. The total width Γ ≡ Γ + Γ n γ is the sum of the neutron partial width Γ n = 0.533 meV (at Er = 97.3 meV and it scales as √ -- E in the center of mass) and of the radiative partial width Γ γ = 60.5 meV. 15645Gd has a resonance at Er = 26.8 meV with Γ n = 0.104 meV, Γ γ = 108 meV and g = 5∕8 while 16547Gd has a resonance at Er = 31.4 meV with Γ n = 0.470 meV, Γ γ = 106 meV and g = 5∕8. As explained in the previous Section 3.2.1, this cross section cannot be measured from the Oklo data, which allow only to measure its value averaged on the neutron flux n (v, T), T being the temperature of the moderator. It is conventionally defined as
∫ -1-- ˆσ = nv σ (n,γ)n (v,T)vdv, (40 ) 0
where the velocity − 1 v0 = 2200 m ⋅ s corresponds to an energy E0 = 25.3 meV and ∘ ------- v = 2E ∕mn, instead of
∫ --σ(n,γ)n(v,T-)vdv- ¯σ = ∫ n (v, T)vdv .
When the cross section behaves as σ = σ0v0∕v, which is the case for nuclei known as “1∕v-absorbers”, ˆσ = σ0 and does not depend on the temperature, whatever the distribution n(v). In a similar way, the effective neutron flux defined
∫ ˆϕ = v0 n(v,T )dv, (41 )
which differs from the true flux
∫ ϕ = n(v,T )vdv.
However, since ¯σ ϕ = ˆσϕˆ, the reaction rates are not affected by these definitions.

Extracting the effective cross section from the data.
To “measure” the value of ˆσ from the Oklo data, we need to solve the nuclear reaction network that controls the isotopic composition during the fission. The samples of the Oklo reactors were exposed [382] to an integrated effective fluence ∫ ϕˆdt of about 21 −2 −1 10 neutron ⋅ cm = 1 kb. It implies that any process with a cross section smaller than 1 kb can safely be neglected in the computation of the abundances. This includes neutron capture by 16442 Sm and 16428Sm, as well as by 15645Gd and 16547Gd. On the other hand, the fission of 29325U, the capture of neutron by 143Nd 60 and by 149Sm 62 with respective cross sections σ5 ≃ 0.6 kb, σ143 ∼ 0.3 kb and σ149 ≥ 70 kb are the dominant processes. It follows that the equations of evolution for the number densities N147, N148, N149 and N235 of 147 62 Sm, 148 62 Sm, 149 62 Sm and 235 92 U takes the form

dN147 ------= − ˆσ147N147 + ˆσf235y147N235 (42 ) ˆϕdt dN148- ˆ = σˆ147N147 (43 ) ϕdt dN149-= − ˆσ N + ˆσ y N (44 ) ˆϕdt 149 149 f235 149 235 dN235-= − σ5N235, (45 ) ˆϕdt
where yi denotes the yield of the corresponding element in the fission of 235 92 U and ˆσ5 is the fission cross section. This system can be integrated under the assumption that the cross sections and the neutron flux are constant and the result compared with the natural abundances of the samarium to extract the value of ˆσ149 at the time of the reaction. Here, the system has been closed by introducing a modified absorption cross section [123Jump To The Next Citation Point] σ ∗ 5 to take into account both the fission, capture but also the formation from the α-decay of 239 94 Pu. One can instead extend the system by considering 239 94 Pu, and 235 92 U (see [234Jump To The Next Citation Point]). While most studies focus on the samarium, [220Jump To The Next Citation Point] also includes the gadolinium even though it is not clear whether it can reliably be measured [123Jump To The Next Citation Point]. They give similar results.

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 [465Jump To The Next Citation Point, 415Jump To The Next Citation Point, 123Jump To The Next Citation Point, 220Jump To The Next Citation Point, 305Jump To The Next Citation Point, 416Jump To The Next Citation Point, 234Jump To The Next Citation Point] differ from the choice of the data. The measured values of ˆσ149 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 [123Jump To The Next Citation Point], SC39-1383 means that we are dealing with the bore-hole number 39 at a depth of 13.83 m). Recently, another approach [416Jump To The Next Citation Point, 234Jump To The Next Citation Point] 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 [416Jump To The Next Citation Point] and both RZ2 and RZ10 [234Jump To The Next Citation Point] and allows to take into account the spatial distribution of the neutron flux.

Table 8: Summary of the analysis of the Oklo data. The principal assumptions to infer the value of the resonance energy E r are the form of the neutron spectrum and its temperature.
Ore neutron spectrum Temperature (∘ C) ˆσ149 (kb) ΔEr (meV) Ref.
? Maxwell 20 55 ± 8 0 ± 20 [465Jump To The Next Citation Point]
RZ2 (15) Maxwell 180 – 700 75 ± 18 –1.5 ± 10.5 [123Jump To The Next Citation Point]
RZ10 Maxwell 200 – 400 91 ± 6 4 ± 16 [220Jump To The Next Citation Point]
RZ10 –97 ± 8 [220Jump To The Next Citation Point]
Maxwell + epithermal 327 91 ± 6 +7 − 45−15 [305Jump To The Next Citation Point]
RZ2 Maxwell + epithermal 73.2 ± 9.4 –5.5 ± 67.5 [416Jump To The Next Citation Point]
RZ2 Maxwell + epithermal 200 – 300 71.5 ± 10.0 [234Jump To The Next Citation Point]
RZ10 Maxwell + epithermal 200 – 300 85.0 ± 6.8 [234Jump To The Next Citation Point]
RZ2+RZ10       7.2 ± 18.8 [234Jump To The Next Citation Point]
RZ2+RZ10       90.75 ± 11.15 [234Jump To The Next Citation Point]

Determination of Er.
To convert the constraint on the effective cross section, one needs to specify the neutron spectrum. In the earlier studies [465Jump To The Next Citation Point, 415], a Maxwell distribution,

( mn )3∕2 −-mv2 nth(v,T ) = ----- e 2kBT , 2 πT
was assumed for the neutron with a temperature of 20∘C, which is probably too small. Then v0 is the mean velocity at a temperature T0 = mnv20∕2kB = 20.4 ∘C. [123Jump To The Next Citation Point, 220Jump To The Next Citation Point] also assume a Maxwell distribution but let the moderator temperature vary so that they deduce an effective cross section ˆσ(Rr, T). They respectively restricted the temperature range to ∘ ∘ 180 C < T < 700 C and ∘ ∘ 200 C < T < 400 C, based on geochemical analysis. The advantage of the Maxwell distribution assumption is that it avoids to rely on a particular model of the Oklo reactor since the spectrum is determined solely by the temperature.

It was then noted [305, 416Jump To The Next Citation Point] that above an energy of several eV, the neutron spectrum shifted to a 1∕E tail because of the absorption of neutrons in uranium resonances. Thus, the distribution was adjusted to include an epithermal distribution

n(v) = (1 − f)nth(v,T ) + f nepi(v),
with nepi = v2∕v2 c for v > vc and vanishing otherwise. vc is a cut-off velocity that also needs to be specified. The effective cross section can then be parameterized [234Jump To The Next Citation Point] as
ˆσ = g (T )σ0 + r0I, (46 )
where g(T) is a measure of the departure of σ from the 1∕v behavior, I is related to the resonance integral of the cross section and r0 is the Oklo reactor spectral index. It characterizes the contribution of the epithermal neutrons to the cross section. Among the unknown parameters, the most uncertain is probably the amount of water present at the time of the reaction. [234Jump To The Next Citation Point] chooses to adjust it so that r 0 matches the experimental values.

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 [220Jump To The Next Citation Point, 416Jump To The Next Citation Point, 234Jump To The Next Citation Point] find two branches for ΔEr = Er − Er0, with one (the left branch) indicating a variation of Er. Note that these two branches disappear when the temperature is higher since ˆσ (Er, T ) is more peaked when T 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 [416Jump To The Next Citation Point] indicates that the curves ˆσ(T,Er ) lie appreciably lower than for a Maxwell distribution and that [220Jump To The Next Citation Point] argues that the left branch is hardly compatible with the gadolinium data.

3.2.3 From the resonance energy to fundamental constants

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 V ≃ 50 MeV 0. He attributed the change of the resonance energy to a modification of the strong interaction coupling constant and concluded that ΔgS ∕gS ∼ ΔEr ∕V0. Then, arguing that the Coulomb force increases the average inter-nuclear distance by about 2.5% for A ∼ 150, he concluded that Δ αEM ∕αEM ∼ 20 ΔgS ∕gS, leading to |α˙EM ∕αEM | < 10−17 yr−1, which can be translated to

− 8 |ΔαEM ∕αEM | < 1.8 × 10 . (47 )

The following analysis focused on the fine-structure constant and ignored the strong interaction. Damour and Dyson [123Jump To The Next Citation Point] related the variation of Er to the fine-structure constant by taking into account that the radiative capture of the neutron by 14629Sm corresponds to the existence of an excited quantum state of 150Sm 62 (so that Er = E ∗ − E149 − mn 150) and by assuming that the nuclear energy is independent of α EM. It follows that the variation of α EM 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 [123Jump To The Next Citation Point] showed that the Bethe–Weizäcker formula overestimates by about a factor the 2 the αEM-sensitivity to the resonance energy. It follows from this analysis that

ΔEr αEM ------ ≃ − 1.1 MeV, (48 ) Δ αEM
which, once combined with the constraint on ΔEr, implies
− 0.9 × 10−7 < Δ αEM ∕αEM < 1.2 × 10− 7 (49 )
at 2σ level, corresponding to the range −17 − 1 −17 −1 − 6.7 × 10 yr < α˙EM ∕αEM < 5.0 × 10 yr if αE˙M is assumed constant. This tight constraint arises from the large amplification between the resonance energy (∼ 0.1 eV) and the sensitivity (∼ 1 MeV). The re-analysis of these data and also including the data of [220] with gadolinium, found the favored result −17 − 1 αE˙M ∕αEM = (− 0.2 ± 0.8 ) × 10 yr, which corresponds to
Δ αEM ∕αEM = (− 0.36 ± 1.44) × 10 −8 (50 )
and the other branch (indicating a variation; see Table 8) leads to − 17 −1 αE˙M ∕αEM = (4.9 ± 0.4 ) × 10 yr. This non-zero result cannot be eliminated.

The more recent analysis, based on a modification of the neutron spectrum lead respectively to [416Jump To The Next Citation Point]

Δ αEM ∕αEM = (3.85 ± 5.65) × 10−8 (51 )
and [234]
−8 Δ αEM ∕αEM = (− 0.65 ± 1.75) × 10 , (52 )
at a 95% confidence level, both using the formalism of [123Jump To The Next Citation Point].

Olive et al. [399Jump To The Next Citation Point], 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 mi ∝ ΛQCD (1 + κimq ∕ΛQCD + ...), they estimate that the nuclear Hamiltonian was proportional to mq ∕ΛQCD at lowest order, which allows to estimate that the energy of the resonance is related to the quark mass by

( ) ΔEr-- 17 -mq--- Er ∼ (2.5 − 10) × 10 Δ ln ΛQCD . (53 )
Using the constraint (48View Equation), they first deduced that
| ( ) | | mq | −8 ||Δ ln Λ----- || < (1 − 4) × 10 . QCD
Then, assuming that αEM ∝ m50q on the basis of grand unification (see Section 6.3 for details), they concluded that
|Δ α ∕α | < (2 − 8) × 10−10. (54 ) EM EM

Similarly, [207Jump To The Next Citation Point, 467Jump To The Next Citation Point, 212] related the variation of the resonance energy to the quark mass. Their first estimate [207Jump To The Next Citation Point] assumes that it is related to the pion mass, m π, and that the main variation arises from the variation of the radius R ∼ 5fm + 1∕m π of the nuclear potential well of depth V0, so that

δR δm δEr ∼ − 2V0--- ∼ 3 × 108---π, R m π
assuming that R ≃ 1.2A1 ∕3r 0, r 0 being the inter-nucleon distance.

Then, in [467], the nuclear potential was described by a Walecka model, which keeps only the σ (scalar) and ω (vector) exchanges in the effective nuclear force. Their masses was related to the mass ms of the strange quark to get m σ ∝ m0.s54 and m ω ∝ m0.s15. It follows that the variation of the potential well can be related to the variation of m σ and m ω and thus on mq by V ∝ m −3.5 q. The constraint (48View Equation) then implies that

| ( )| | ms | −10 ||Δ ln ------ || < 1.2 × 10 . ΛQCD
By extrapolating from light nuclei where the N-body calculations can be performed more accurately, it was concluded [208Jump To The Next Citation Point] that the resonance energy scales as ΔEr ≃ 10(Δ lnXq − 0.1Δ lnαEM ), so that the the constraints from [416] would imply that 0.1 − 9 Δ ln(Xq ∕α EM) < 7 × 10.

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 ΛQCD 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.

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