Natural nuclear reactor at Oklo and variation of fundamental constants: Computation of neutronics of a fresh core

Using modern methods of reactor physics, we performed full-scale calculations of the Oklo natural reactor. For reliability, we used recent versions of two Monte Carlo codes: the Russian code MCU-REA and the well-known international code MCNP. Both codes produced similar results. We constructed a computer model of the Oklo reactor zone $\mathit{RZ}2$ which takes into account all details of design and composition. The calculations were performed for three fresh cores with different uranium contents. Multiplication factors, reactivities, and neutron fluxes were calculated. We also estimated the temperature and void effects for the fresh core. As would be expected, we found for the fresh core a significant difference between reactor and Maxwell spectra, which had been used before for averaging cross sections in the Oklo reactor. The averaged cross section of ${}_{62}^{149}\mathrm{Sm}$ and its dependence on the shift of a resonance position $E_r$ (due to variation of fundamental constants) are significantly different from previous results. Contrary to the results of previous papers, we found no evidence of a change of the samarium cross section: a possible shift of the resonance energy is given by the limits $-73⩽\Delta E_r⩽62$ meV. Following tradition, we have used formulas of Damour and Dyson to estimate the rate of change of the fine structure constant α. We obtain new, more accurate limits of $-4\times {10}^{-17}⩽\stackrel{·}{\alpha }/\alpha ⩽3\times {10}^{-17}\hspace{1em}{\mathrm{yr}}^{-1}$. Further improvement of the accuracy of the limits can be achieved by taking account of the core burn-up. These calculations are in progress.

Figure 1

Present-day abundance of uranium (in volume percent of dry ore) in dependence on the dry ore density ${\gamma }_U$ in the Oklo reactor [43]. Three initial variants of the $Y_{U,i}(T_0)=35,45,55\%$ U in dry ore are shown.

Figure 2

Dependence of water density ${\gamma }_{H_{2}O}$ on temperature $T$ for different pressures $P$.

Figure 3

Power effect [44, 45]. Dependence of reactivity ρ (in %) of the fresh core of the Oklo reactor on temperature $T_C$ at a pressure of $P_C=100$ MPa for three different initial compositions of the active core and different proportions of water: 1: 49.4 vol.% U in ore, ${\omega }_{H_{2}O}^0=0.455$; 2: 49.4 vol.% U in ore, ${\omega }_{H_{2}O}^0=0.405$; 3: 38.4 vol.% U in ore, ${\omega }_{H_{2}O}^0=0.355$. Calculations were done using code MCU-REA.

Figure 4

$T_C$ dependence of $K_{\infty }(T_C)$ and $M^2(T_C)$ for a core containing 49.4 vol.% of uranium in the present-day dry ore and a water content of ${\omega }_{H_{2}O}^0=0.355$ at $T_C=300$ K and $P_C=0.1$ MPa. Calculations using code MCU-REA [44, 45].

Figure 5

Neutron spectrum in the bare fresh core RZ2 at different initial uranium concentrations (water content ${\omega }_{H_{2}O}^0=0.355$). Calculations using code MCNP4C [44, 45].

Figure 6

Strong absorber as a sensitive detector of a variation of $E_r$ [55]. Left: energy level density of the compound nucleus $n{+}^{A}Z\to {}^{A+1}{Z}^{*}$. Right: resonances in the cross section of the $n\gamma$ reaction. The capture cross section behaves like ${\sigma }_{\gamma }\simeq ({\Gamma }_{\gamma }/E_r){}^2$, where $E_r$ is the distance from the resonance and ${\Gamma }_{\gamma }$ is its width. Neutron capture is strongly affected by a shift of $\Delta E_r$.

Figure 7

Comparison of the calculated (crosses) and measured (circles) concentrations of fission fragments $N_A$ relative to the ${}^{143}$Nd content for one Oklo sample [10, 56].

Figure 8

Dependence of the cross section ${\stackrel{\wedge }{\sigma }}_{\gamma ,\mathrm{Sm}}$, averaged over a Maxwell neutron spectrum, on the resonance shift $\Delta E_r$ and on the temperature $T=300\text{−}800$ K, Eq. (29). The curves are for fixed temperatures with intervals of 100 K. The highest curve is for $T=300$ K.

Figure 9

Dependence of thermal neutron capture cross section of ${}_{62}^{149}\mathrm{Sm}$ on core temperature $T_C$ and on resonance shift $\Delta E_r:{\stackrel{\wedge }{\sigma }}_{\gamma ,\mathrm{Sm}}(T,\Delta E_r)$ [44, 45]. The curves are for cross sections, averaged over the reactor spectrum of the fresh core at three different initial states: 1: $Y_{U2}(0)=38.4$ vol.% U, ${\omega }_{H_{2}O}=0.355,T_C=670$ K; 2: $Y_{U2}(0)=49.4$ vol.% U, ${\omega }_{H_{2}O}=0.405,T_C=725$ K; 3: $Y_{U2}(0)=49.4$ vol.% U, ${\omega }_{H_{2}O}=0.455,T_C=780$ K, $P_C=100$ MPa. For comparison we show the cross section averaged over the Maxwell spectrum (curve 4) for initial composition $Y_{U2}(0)=38.4$ vol.% U, ${\omega }_{H_{2}O}=0.405,T_C=725$ K. Shown is the error corridor of measured values ${\stackrel{\wedge }{\sigma }}_{\gamma ,\mathrm{Sm}}^{\mathrm{Exp}}$ [11].

Figure 10

Change of the thermal ($T=25$ meV$=273$ K) capture cross section of ${}_{62}^{149}\mathrm{Sm}$ for a uniform shift of all resonances by $\Delta E_r$ [8].

Figure 11

Dependence of the thermal neutron capture cross section on the resonance shift $\Delta E_r$ for the two temperatures of the Maxwell neutron spectrum (Damour and Dyson [11]). Energies of the crossing of the lower limit of data are $\Delta E_{r1}^M=-120$ meV and $\Delta E_{r2}^M=90$ meV.

Figure 12

Dependence ${\stackrel{\wedge }{\sigma }}_{\gamma ,\mathrm{Sm}}(T_C,\Delta E_r)$ for the Maxwell distribution at $T_C=450,570,670$ K (curves 1–3) (Fujii et al. [12, 64]). Data correspond to Table ; ${\stackrel{\wedge }{\sigma }}_{\gamma ,\mathrm{Sm}}^{\mathrm{Exp}}=90.7\pm 8.2$ kb, $\Delta E_{r2}^M=20$ meV. For comparison, we show curve 4 for the reactor spectrum at $T_C=670$ K; $Y_{U1}(0)=38.4$ vol.% U; ${\omega }_{H_{2}O}^0=0.355$; $P_C=100$ MPa.

Figure 13

Data of Srianand et al. (filled circles) are plotted against the redshift [69]. Each point is the best fitted value obtained for individual systems using ${\chi }^2$ minimization. The open circle is measurement from the Oklo reactor. The weighted average and $1\sigma$ range measured by Murphy et al. [70] are shown with horizontal dashed lines. Most of Srianand et al. measurements are inconsistent with this range. The shaded region represents the weighted average and its $3\sigma$ error: $\langle \delta \alpha /\alpha \rangle {}_W=(-60\pm 60)\times {10}^{-8}$. Within $3\sigma$ there is no variation of fundamental constants within the limits: $-25\times {10}^{-17}⩽(\Delta \alpha /\alpha \Delta t)⩽12\times {10}^{-17}$ yr${}^{-1}$.