3.3 Meteorite dating

Long-lived α- or β-decay isotopes may be sensitive probes of the variation of fundamental constants on geological times ranging typically to the age of the solar system, t ∼ (4– 5) Gyr, corresponding to a mean redshift of z ∼ 0.43. Interestingly, it can be compared with the shallow universe quasar constraints. This method was initially pointed out by Wilkinson [539Jump To The Next Citation Point] and then revived by Dyson [168Jump To The Next Citation Point]. The main idea is to extract the αEM-dependence of the decay rate and to use geological samples to bound its time variation.

The sensitivity of the decay rate of a nucleus to a change of the fine-structure constant is defined, in a similar way as for atomic clocks [Equation (23View Equation)], as

∂ ln λ sα ≡ ---------. (55 ) ∂ lnαEM
λ is a function of the decay energy Q. When Q is small, mainly due to an accidental cancellation between different contributions to the nuclear binding energy, the sensitivity s α maybe strongly enhanced. A small variation of the fundamental constants can either stabilize or destabilize certain isotopes so that one can extract bounds on the time variation of their lifetime by comparing laboratory data to geophysical and solar system probes.

Assume some meteorites containing an isotope X that decays into Y are formed at a time t∗. It follows that

N (t) = N e−λ(t− t∗), N (t) = N [1 − e− λ(t−t∗)] + N (56 ) X X∗ Y X∗ Y∗
if one assumes the decay rate constant. If it is varying then these relations have to be replaced by
∫tt∗λ(t′)dt′ NX (t) = NX ∗e
so that the value of NX today can be interpreted with Equation (56View Equation) but with an effective decay rate of
1 ∫ t0 λ¯= ------- λ(t′)dt′. (57 ) t0 − t∗ t∗
From a sample of meteorites, we can measure {N (t),N (t )} X 0 Y 0 for each meteorite. These two quantities are related by
[ ¯λ(t −t) ] NY (t0) = e 0 ∗ − 1 NX (t0) + NY ∗,
so that the data should lie on a line (since NX ∗ is a priori different for each meteorite), called an “isochron”, the slope of which determines ¯λ(t − t ) 0 ∗. It follows that meteorites data only provides an average measure of the decay rate, which complicates the interpretation of the constraints (see [219Jump To The Next Citation Point, 218Jump To The Next Citation Point] for explicit examples). To derive a bound on the variation of the constant we also need a good estimation of t0 − t∗, which can be obtained from the same analysis for an isotope with a small sensitivity s α, as well as an accurate laboratory measurement of the decay rate.

3.3.1 Long lived α-decays

The α-decay rate, λ, of a nucleus AZX of charge Z and atomic number A,

A+4 A 4 Z+2X −→ ZX + 2He, (58 )
is governed by the penetration of the Coulomb barrier that can be described by the Gamow theory. It is well approximated by
( c) λ ≃ Λ (αEM, v)exp − 4πZ αEM -- , (59 ) v
where ∘ --------- v∕c = Q ∕2mpc2 is the escape velocity of the α particle. Λ is a function that depends slowly on αEM and Q. It follows that the sensitivity to the fine-structure constant is
( ) --αEM----- 1--d-ln-Q-- sα ≃ − 4πZ ∘ ------- 1 − 2 d ln α . (60 ) Q ∕2mp EM
The decay energy is related to the nuclear binding energies B (A, Z) of the different nuclei by
Q = B (A,Z ) + Bα − B (A + 4, Z + 2)
with B = B (4,2) α. Physically, an increase of α EM induces an increase in the height of the Coulomb barrier at the nuclear surface while the depth of the nuclear potential well below the top remains the same. It follows that α-particle escapes with a greater energy but at the same energy below the top of the barrier. Since the barrier becomes thiner at a given energy below its top, the penetrability increases. This computation indeed neglects the effect of a variation of αEM on the nucleus that can be estimated to be dilated by about 1% if αEM increases by 1%.

As a first insight, when focusing on the fine-structure constant, one can estimate sα by varying only the Coulomb term of the binding energy. Its order of magnitude can be estimated from the Bethe–Weizäcker formula

Z(Z − 1) EEM = 98.25 --A1-∕3---αEM MeV. (61 )

Table 9: Summary of the main nuclei and their physical properties that have been used in α-decay studies.


Z A Lifetime (yr) Q (MeV) sα


62 147 1.06 × 1011 2.310 774


64 152 1.08 × 1014 2.204 890


66 154 3 × 106 2.947 575


78 190 6.5 × 1011 3.249 659


90 232 1.41 × 1010 4.082 571


92 235 7.04 × 108 4.678 466


92 238 4.47 × 109 4.270 548

Table 9 summarizes the most sensitive isotopes, with the sensitivities derived from a semi-empirical analysis for a spherical nucleus [399Jump To The Next Citation Point]. They are in good agreement with the ones derived from Equation (61View Equation) (e.g., for 238U, one would obtain sα = 540 instead of sα = 548).

The sensitivities of all the nuclei of Table 9 are similar, so that the best constraint on the time variation of the fine-structure constant will be given by the nuclei with the smaller Δ λ∕λ.

Wilkinson [539] considered the most favorable case, that is the decay of 23928U for which sα = 548 (see Table 9). By comparing the geological dating of the Earth by different methods, he concluded that the decay constant λ of 238U, 235U and 232Th have not changed by more than a factor 3 or 4 during the last 9 3 –4 × 10 years from which it follows

|Δ αEM ∕ αEM | < 8 × 10− 3. (62 )
This constraint was revised by Dyson [168] who claimed that the decay rate has not changed by more than 20%, during the past 2 × 109 years, which implies
|Δ α ∕ α | < 4 × 10− 4. (63 ) EM EM
Uranium has a short lifetime so that it cannot be used to set constraints on longer time scales. It is also used to calibrate the age of the meteorites. Therefore, it was suggested [399Jump To The Next Citation Point] to consider 147Sm. Assuming that Δ λ147∕λ147 is smaller than the fractional uncertainty of 7.5 × 10− 3 of its half-life
|ΔαEM ∕ αEM | ≲ ×10 − 5. (64 )

As for the Oklo phenomena, the effect of other constants has not been investigated in depth. It is clear that at lowest order both Q and mp scales as ΛQCD so that one needs to go beyond such a simple description to determine the dependence in the quark masses. Taking into account the contribution of the quark masses, in the same way as for Equation (53View Equation), it was argued that 300– 2000 λ ∝ Xq, which leads to −5 |Δ lnXq | ≲ 10. In a grand unify framework, that could lead to a constraint of the order of |Δ lnαEM | ≲ 2 × 10− 7.

3.3.2 Long lived β-decays

Dicke [150] stressed that the comparison of the rubidium-strontium and potassium-argon dating methods to uranium and thorium rates constrains the variation of αEM.

As long as long-lived β-decay isotopes are concerned for which the decay energy Q is small, we can use a non-relativistic approximation for the decay rate

λ = Λ±Qp ± (65 )
respectively for − β-decay and electron capture. Λ ± are functions that depend smoothly on αEM and which can thus be considered constant, p+ = ℓ + 3 and p− = 2ℓ + 2 are the degrees of forbiddenness of the transition. For high-Z nuclei with small decay energy Q, the exponent p becomes ∘ ----------- p = 2 + 1 − α2 Z2 EM and is independent of ℓ. It follows that the sensitivity to a variation of the fine-structure constant is
d ln Q sα = p--------. (66 ) d ln αEM
The second factor can be estimated exactly as for α-decay. We note that Λ ± depends on the Fermi constant and on the mass of the electron as Λ± ∝ G2Fm5eQp. This dependence is the same for any β-decay so that it will disappear in the comparison of two dating methods relying on two different β-decay isotopes, in which case only the dependence on the other constants appear again through the nuclear binding energy. Note, however, that comparing a α- to a β-decay may lead to interesting constraints.

We refer to Section III.A.4 of FVC [500Jump To The Next Citation Point] for earlier constraints derived from rubidium-strontium, potassium-argon and we focus on the rhenium-osmium case,

187 187 − 75 Re − → 76 Os + ν¯e + e (67 )
first considered by Peebles and Dicke [406Jump To The Next Citation Point]. They noted that the very small value of its decay energy Q = 2.6 keV makes it a very sensitive probe of the variation of αEM. In that case p ≃ 2.8 so that s ≃ − 18000 α; a change of 10−2% of α EM will induce a change in the decay energy of order of the keV, that is of the order of the decay energy itself. Peebles and Dicke [406] did not have reliable laboratory determination of the decay rate to put any constraint. Dyson [167] compared the isotopic analysis of molybdenite ores (λ187 = (1.6 ± 0.2) × 10−11 yr−1), the isotopic analysis of 14 iron meteorites (λ187 = (1.4 ± 0.3) × 10 −11 yr− 1) and laboratory measurements of the decay rate (−11 −1 λ187 = (1.1 ± 0.1) × 10 yr). Assuming that the variation of the decay energy comes entirely from the variation of αEM, he concluded that −4 |Δ αEM ∕αEM | < 9 × 10 during the past 3 × 109 years. Note that the discrepancy between meteorite and lab data could have been interpreted as a time-variation of αEM, but the laboratory measurement were complicated by many technical issues so that Dyson only considered a conservative upper limit.

The modelization and the computation of sα were improved in [399Jump To The Next Citation Point], following the same lines as for α-decay.

( ) Δ-λ187 = pΔQ-- ≃ p 20-MeV-- Δ-αEM- ∼ − 2.2 × 104Δ-αEM- λ187 Q Q αEM αEM
if one considers only the variation of the Coulomb energy in Q. A similar analysis [147Jump To The Next Citation Point] leads to −2.2 Δ ln λ187 ≃ 104Δ ln[α EM X −q 1.9(Xd − Xu )0.23X −e0.058].

The dramatic improvement in the meteoric analysis of the Re/Os ratio [468] led to a recent re-analysis of the constraints on the fundamental constants. The slope of the isochron was determined with a precision of 0.5%. However, the Re/Os ratio is inferred from iron meteorites the age of which is not determined directly. Models of formation of the solar system tend to show that iron meteorites and angrite meteorites form within the same 5 million years. The age of the latter can be estimated from the 207Pb-208Pb method, which gives 4.558 Gyr [337] so that λ187 = (1.666 ± 0.009 ) × 10 −11 yr− 1. Thus, we could adopt [399Jump To The Next Citation Point]

|| || |Δ-λ187|< 5 × 10 −3. | λ187 |
However, the meteoritic ages are determined mainly by 238U dating so that effectively we have a constraint on the variation of λ187∕λ238. Fortunately, since the sensitivity of 238U is much smaller than the one of the rhenium, it is safe to neglect its effect. Using the recent laboratory measurement [333] (λ187 = (− 1.639 ± 0.025) × 10−11 yr−1), the variation of the decay rate is not given by the dispersion of the meteoritic measurement, but by comparing to its value today, so that
| | ||Δ-λ187||= − 0.016 ± 0.016. (68 ) | λ187 |
The analysis of Ref. [400], following the assumption of [399], deduced that
−7 Δ αEM ∕αEM = (− 8 ± 16) × 10 , (69 )
at a 95% confidence level, on a typical time scale of 5 Gyr (or equivalently a redshift of order z ∼ 0.2).

As pointed out in [219, 218], these constraints really represents a bound on the average decay rate ¯ λ since the formation of the meteorites. This implies in particular that the redshift at which one should consider this constraint depends on the specific functional dependence λ(t). It was shown that well-designed time dependence for λ can obviate this limit, due to the time average.

3.3.3 Conclusions

Meteorites data allow to set constraints on the variation of the fundamental constants, which are comparable to the ones set by the Oklo phenomenon. Similar constraints can also bet set from spontaneous fission (see Section III.A.3 of FVC [500Jump To The Next Citation Point]) but this process is less well understood and less sensitive than the α- and β- decay processes and.

From an experimental point of view, the main difficulty concerns the dating of the meteorites and the interpretation of the effective decay rate.

As long as we only consider αEM, the sensitivities can be computed mainly by considering the contribution of the Coulomb energy to the decay energy, that reduces to its contribution to the nuclear energy. However, as for the Oklo phenomenon, the dependencies in the other constants, Xq, GF, μ…, require a nuclear model and remain very model-dependent.

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