Is the Fine-Structure Constant Actually Constant? Research Notes on the Oklo Reactor and Quasar Spectra — Epoche C1
Research note. Purpose: to state what "do the constants of nature change?" can actually mean, and to weigh the strongest probes of the fine-structure constant across three epochs. Framework and conventions follow Uzan's 2003 review. 1. The problem, stated so that it can be tested The standard objection holds that the question is empty. If every constant drifted, our rulers, clocks and detectors — built from atoms governed by those same constants — would drift in proportion, and nothing would show. The objection is half right, and locating the right half does most of the work. It is right for constants carrying units. The speed of light $c$ converts metres into seconds, and since 1983 the SI has fixed $c = 299\,792\,458\ \mathrm{m\,s^{-1}}$ by definition, so a "drift in $c$" alone is a statement about the metre rather than about nature. The same holds for the reduced Planck constant $\hbar$ taken on its own. Only a dimensionless number — one whose value is identical in every system of units — can change in a way that is not a change of bookkeeping. Uzan's review sets out the argument in full and its consequence for what may be measured. The central dimensionless example is the fine-structure constant, the pure number fixing the strength of the electromagnetic interaction: $$\alpha=\frac{e^{2}}{4\pi\varepsilon_{0}\hbar c}\approx\frac{1}{137.036},$$ with $e$ the elementary charge and $\varepsilon_{0}$ the vacuum permittivity. (The quantity meant throughout is the low-energy value; in quantum electrodynamics $\alpha$ also runs with the energy scale of the process, reaching about $1/128$ at the mass of the Z boson, which is a different phenomenon and not at issue here.) The escape from the objection is that different physical systems depend on $\alpha$ in different, non-proportional ways, so a real drift cannot cancel out of every comparison at once. A convincing test therefore needs a quantity that depends steeply on $\alpha$, was recorded by nature long ago, and can be read out today — and it needs at least two probes with independent physics at different epochs, so that a systematic error in one cannot masquerade as cosmology. Three such probes now exist. 2. Probe one: a natural reactor, two billion years old In 1972 isotope analysis of ore from the Oklo uranium deposit in Gabon revealed that a natural fission reactor had run there about $2\times10^{9}$ years ago. The reason such a thing was possible then and is not now is a matter of two decay constants. Uranium-235, with a half-life of $7.0\times10^{8}$ years, decays about six times faster than uranium-238, with $4.5\times10^{9}$ years. Running the abundances backwards from today's $0.72$ per cent $^{235}\mathrm{U}$ by the factors $e^{\ln 2 \times 2/0.70} \approx 7.2$ and $e^{\ln 2 \times 2/4.5} \approx 1.36$ respectively gives $$\frac{0.0072 \times 7.2}{0.0072 \times 7.2 + 0.9928 \times 1.36} \approx 0.037,$$ that is, about 3.7 per cent — close to the enrichment of a modern power reactor. Groundwater percolating through the ore body served as the moderator, slowing neutrons enough to sustain a chain reaction. Shlyakhter pointed out in 1976 that the ore therefore preserves a set of nuclear measurements made two billion years ago, and that comparing them with the same measurements today bounds any change in the constants that govern them. One entry in the reactor's ash is decisive. The samarium isotope $^{149}\mathrm{Sm}$ was strongly depleted by neutron capture, and its capture is dominated by a single resonance — a narrow peak in the capture probability — at a neutron energy of $E_{r} = 97.3\ \mathrm{meV}$. Near such a resonance the cross-section (the effective target area for capture) follows the Breit–Wigner form, falling off as $\left[(E-E_{r})^{2} + \Gamma^{2}/4\right]^{-1}$ with $\Gamma$ the resonance width, here of order tens of millielectronvolts. The neutrons available to be captured had a thermal distribution of energies with mean of order $k_\mathrm{B}T$, which at the several-hundred-kelvin temperatures of the reactor zones is a few tens of millielectronvolts — the same order as $E_{r}$ itself. The capture rate was therefore acutely sensitive to where the resonance sat. 3. Why the resonance amplifies a change in $\alpha$ The resonance energy is small not because the nuclear energies involved are small but because it is a near-cancellation between two large ones: the nuclear binding energy of the compound state and its Coulomb energy, the electrostatic repulsion among the protons. Only the Coulomb part scales with $\alpha$, so scaling $\alpha$ shifts the resonance by the corresponding fraction of a quantity of order an MeV rather than of order the resonance energy. Damour and Dyson estimated the relevant Coulomb energy difference at about $1.1\ \mathrm{MeV}$, which fixes the sensitivity: $$|\Delta E_{r}| \approx (1.1\ \mathrm{MeV})\left|\frac{\Delta\alpha}{\alpha}\right| .$$ Set the two scales side by side: $1.1\times10^{6}\ \mathrm{eV}$ against a resonance energy of order $10^{-1}\ \mathrm{eV}$, seven orders of magnitude apart. A fractional change in $\alpha$ of one part in $10^{7}$ would move the resonance by about $0.1\ \mathrm{eV}$ — past its own energy, and clean out of the thermal neutron range — and the ancient capture rate would have been visibly different from today's. The measured isotopic ratios show that the effective capture cross-section operating then agreed with the modern one, which is what pins down the resonance position. Damour and Dyson obtained $|\Delta E_{r}| \lesssim 0.1\ \mathrm{eV}$, hence $|\Delta\alpha/\alpha| \lesssim 0.1/(1.1\times10^{6}) \approx 10^{-7}$ over two billion years. Fujii and collaborators, working with better-preserved samples and a more careful treatment of the neutron spectrum, narrowed the allowed shift to about $0.02\ \mathrm{eV}$ and hence $|\Delta\alpha/\alpha| \lesssim 2\times10^{-8}$. Read as a steady drift, that is $(2\times10^{-8})/(2\times10^{9}\ \mathrm{yr}) = 10^{-17}$ pe