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 (23)], as
Assume some meteorites containing an isotope that decays into
are formed at a time
. It
follows that
The -decay rate,
, of a nucleus
of charge
and atomic number
,
As a first insight, when focusing on the fine-structure constant, one can estimate by varying only
the Coulomb term of the binding energy. Its order of magnitude can be estimated from the
Bethe–Weizäcker formula
Element |
Z | A | Lifetime (yr) | Q (MeV) | ![]() |
Sm |
62 | 147 | 1.06 × 1011 | 2.310 | 774 |
Gd |
64 | 152 | 1.08 × 1014 | 2.204 | 890 |
Dy |
66 | 154 | 3 × 106 | 2.947 | 575 |
Pt |
78 | 190 | 6.5 × 1011 | 3.249 | 659 |
Th |
90 | 232 | 1.41 × 1010 | 4.082 | 571 |
U |
92 | 235 | 7.04 × 108 | 4.678 | 466 |
U |
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 [399]. They are in good agreement with the ones derived from
Equation (61
) (e.g., for 238U, one would obtain
instead of
).
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 .
Wilkinson [539] considered the most favorable case, that is the decay of for which
(see Table 9). By comparing the geological dating of the Earth by different methods, he concluded that the
decay constant
of 238U, 235U and 232Th have not changed by more than a factor 3 or 4 during the last
years from which it follows
As for the Oklo phenomena, the effect of other constants has not been investigated in depth. It is clear
that at lowest order both and
scales as
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 (53
), it was argued that
, which leads to
. In a grand unify framework, that could lead to a constraint of the order of
.
Dicke [150] stressed that the comparison of the rubidium-strontium and potassium-argon dating methods to
uranium and thorium rates constrains the variation of .
As long as long-lived -decay isotopes are concerned for which the decay energy
is small, we can
use a non-relativistic approximation for the decay rate
We refer to Section III.A.4 of FVC [500] for earlier constraints derived from rubidium-strontium,
potassium-argon and we focus on the rhenium-osmium case,
The modelization and the computation of were improved in [399
], following the same lines as for
-decay.
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 . Thus, we could
adopt [399
]
As pointed out in [219, 218], these constraints really represents a bound on the average decay rate
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
. It
was shown that well-designed time dependence for
can obviate this limit, due to the time
average.
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 [500]) but this process is less well understood and less sensitive than the
- and
- 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 , 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,
,
,
…,
require a nuclear model and remain very model-dependent.
http://www.livingreviews.org/lrr-2011-2 |
Living Rev. Relativity 14, (2011), 2
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