It is a long-held tenet of nuclear physics, from the early work of Rutherford and Soddy up to present times that the disintegration of each species of radioactive nuclide occurs randomly at a constant rate unaffected by interactions with the external environment. During the past 15 years or so, reports have been published of some 10 or more unstable nuclides with non-exponential, periodic decay rates claimed to be of geophysical, astrophysical, or cosmological origin. Deviations from standard exponential decay are weak, and the claims are controversial. This paper examines the effects of a periodic decay rate on the statistical distributions of 1) nuclear activity measurements and 2) nuclear lifetime measurements. It is demonstrated that the modifications to these distributions are approximately 100 times more sensitive to non-standard radioactive decay than measurements of the decay curve, power spectrum, or autocorrelation function for corresponding system parameters.
Radioactivity refers to the spontaneous transformation of one kind of atomic nucleus (designated a “nuclide” in the terminology of nuclear physics) into a different kind of atomic nucleus, ordinarily with the emission of a helium-4 nucleus (alpha particle), fast electron or positron (beta particle), high-energy electromagnetic radiation (gamma photon) or, more rarely, some other particle or cluster [
with decay rate parameter λ.
The salient feature of standard radioactive decay, irrespective of the particular process by which the transmutation of nuclear identity occurs, was described by Rutherford and Soddy in 1903 [
“The radioactive constant λ has been investigated under very widely varied conditions of temperature, and under the influence of the most powerful chemical and physical agencies, and no alteration of its value has been observed. The law forms in fact the mathematical expression of a general principle…”
The Law of Radioactive Change and its underlying statistical foundations have been the basis for practical nuclear metrology for more than a century from the discovery of radioactivity up to present times. For example, taking account of the enormous developments in nuclear physics in the years following Rutherford and Soddy’s early discovery, one still finds in influential nuclear science textbooks a confirmation of the same general principle that
“No change in the decay rates of particle emission has been observed over extreme variations of conditions such as temperature, pressure, chemical state, or physical environment.” [
Actually, there are a few known physical processes such as electron-capture decay [
With regard to theory, it has in fact been known since the early development of quantum electrodynamics that the exponential decay law is an approximate result that follows from neglect of the energy-dependence of certain terms in the Green’s functions from which the associated decay amplitudes are calculated [
However, more recent model-dependent calculations with a focus on the decay of nuclear states have predicted non-exponential behavior at intermediate times as well, including the possibility of oscillatory behavior throughout the entire decay transient [
Claims of radioactivity exhibiting periodic decay rates and extra-nuclear environmental correlations are highly controversial, and refutations have been published in specific cases, e.g. [
Experimentally, claims of non-standard radioactive decay have been drawn primarily from perceived deviations from the exponential decay law (1), which follows from a Poisson distribution of decay events as expressed by the probability function
for x decays within a counting interval (bin width) Δt, given a mean count
The reported deviations were very weak, typically a few tenths of a per cent.
A much more sensitive method by which to search for a time-dependent nuclear decay rate was recently proposed by Silverman [
In order to search for non-standard radioactive decay based on the statistical distribution of decay events, it is first necessary to understand better the statistics of standard radioactive decay (i.e. at constant decay rate). In Section 2 of this paper the statistics of a time series of radioactive decays are investigated in greater detail first for standard radioactive decay with constant decay rate λ (mean lifetime
with constant decay parameter λ, amplitude
In Section 3 the effect of a variable nuclear decay rate on a different statistical distribution―the distribution of mean lifetime measurements―is examined and shown to provide another sensitive statistical method by which to search for non-standard radioactive decay. This novel method of determining nuclear half-lives was first discovered empirically [
For a time-dependent decay rate, however, the distribution is displaced, widened, and no longer of Cauchy form.
In Section 4 the effects of a time-varying decay rate on the power spectrum and autocorrelation function of a time series of nuclear activities are discussed.
Conclusions are summarized in Section 5.
The quantitative detection of radioactivity is ordinarily made by counting emitted particles in discrete time windows or bins. (Sometimes the detected signal is an ionization current which, when necessary, can be converted to a particle count per unit time.) In nuclear terminology “activity”
where the constant c denotes the instrumental detection efficiency. The fundamental SI unit of activity is the Becquerel (1 Bq = 1 decay/s). As used in this paper―the primary objective of which is theoretical, i.e. to elucidate the statistics of nuclear decay―the bin width Δt is taken to be 1 time unit (e.g. second, hour, day, etc.), c is taken to be 100%, and the activity xt at discrete time t is therefore a pure number (no units or dimensions) equal to the number of counts in Δt. The temporal index t is an integer denoting the number of unit intervals Δt. Similarly, the total duration TΔt of counting is simply the integer T.
From a statistical perspective the counting of particles emitted from a radioactive source that decays at a constant rate is tantamount to sampling a population of independent Poisson variates of some mean value
with
Over a time interval
with
under a transformation
to the dimensionless variate
In marked contrast to the pictorial representations of Poisson distributions frequently seen in nuclear science textbooks as well as in the research literature, the true pdf (8) or (9) of a long time series of radioactive decays bears no resemblance to a Poisson distribution.
Although there is no closed form for the mixed Poisson-Gauss pdfs (8) or (9), a very accurate expression can be derived for
(since the sum extends over all integer values of t from 0 to
which reduces to a x−1 power-law in the long-time limit. The exact calculations (solid curves) of pdf (9) in
In comparing corresponding plots (i.e. of the same color) in
defined by μ0 = 100, in contrast to what one might expect, since the variance of a Poisson distribution equals the mean μ0. The explanation is that the distributed variate in the figure is not x but
tion
cept in the immediate vicinity of
The apparent oscillatory structure of the orange plot (μ0 = 105) in
in terms of the lifetime
We conclude this section by examining the statistical moments of a mixed Poisson-Gauss random variable with probability density (8), which can be obtained by summation of the moments of the independent PG variates. The kth moment
Summation of (14) over the range of t and expansion of
in which
Expansion of Equation (15) leads to the series
in which sequential terms decrease by powers of
From the moments Mk given by (15), one can calculate the variance
the skewness Sk
and kurtosis K
which are the statistics most commonly used to characterize a probability distribution in atomic and nuclear physics. Skewness describes the asymmetry about the mean, and kurtosis is a measure of the concentration of probability around the shoulders (i.e. at about ±1σ from the mean) and tails. A distribution with high kurtosis would be sharply peaked with fat tails, i.e. with higher than normal probability of outliers (such as produced by a Cauchy distribution). Thus, the shapes of the pdfs plotted in
Explicit expressions for relations (18)-(20) are complicated and will not be given here. It is to be noted, however, that from the form
reached a maximum at around
A characteristic of non-standard radioactive decay predicted or reported in publications cited in Section 1 is the harmonic variation of the decay rate. This feature leads to a time-dependent mean activity of the form
in the simplest case of a single harmonic component. The statistical consequences of relation (22) are examined in detail in this section for various relative values of the lifetime T0, periodicity T1, and count duration T (which is equal to the number of PG samples in the time series), and for amplitude
with μt given by Equation (22), as a function of normalized activity
The explanation of the second property is reasonably self-evident from the form of expression (22). In the limiting case of
The manifestation of the first property may likewise seem unsurprising, but there is a subtlety to the question why oscillations occur in the first place. It is important to keep in mind that the function
(23)―or the transformed equivalent
The occurrence and number of periodic maxima and minima in a plot of pdf (23) as a function of activity can be accounted for by an explanation similar (but not identical) to the explanation of oscillatory structure in the orange plot of
Thus a histogram governed by pdf (23) with
Another important feature to note is illustrated by the blue trace in
It is a well-known principle of time series analysis that one cannot measure the period T1 of a harmonic component if the duration T of the series is shorter than the period
panels that relatively high amplitudes
A standard procedure for measuring the half-life
For each pair of activities
・ calculate the lifetime from the two-point relation
・ make a histogram of the
・ locate the center of the resulting distribution.
Under the conditions that (1) the number of decays per sampling interval Δt is sufficiently high, (2) the number of sequential activity measurements is sufficiently large, and (3) the lifetime is sufficiently long compared to time intervals between pairs of samples, the probability density of two-point estimates is virtually indistinguishable from a Cauchy distribution centered on the true lifetime T0.
Given that the activities Ai are (to excellent approximation) Poisson-Gauss (PG) variates, and that the inverse of the logarithm of the ratio of PG variates involves complicated transformations of the Gaussian probability density, it is perhaps highly surprising that the distribution of the variates Tij turns out to be described by a simple, symmetric Cauchy function. Actually, the exact density function is far more complicated than a Cauchy function, but reduces to the latter under the previously enumerated conditions, as derived in [
Let θ be a continuous random variable whose realizations are the samples Tij in the population of Nt samples obtained from the sequential measurement of a time series of T activities. And let J be the “multiplicity” of a measurement, whose significance will be explained shortly. Then the probability density function of two-point lifetime estimates takes the form
which will be denoted simply by
The derivation of relation (26) involves three sequential transformations of the pdfs of functions of the variables
Step 1:
Step 2:
Step 3:
in which the transformation at each step is implemented by a relation of the form of Equation (10)
where
There are three principal differences between Equation (26) and the corresponding pdf published previously in [
The third difference is the inclusion of the multiplicity J in (26), which is absent from the analysis in [
give the mean activity
PG variate
The red plots in
An important point worth noting because of its experimental consequences is that a Cauchy distribution, in contrast to Poisson and Gaussian distributions, has no finite moments (apart from the 0th moment, which equals 1 as required by the completeness relation for probability). The moment-generating function does not exist, and although the characteristic function (Fourier transform of probability density) does exist, it does not lead to finite moments [
In view of the preceding remarks, the question may arise as to why, if relation (26) reduces for all practical purposes to a Cauchy distribution in the case of standard radioactive decay, does a multiplicity
The derivation of pdf (26) does not depend on the form of the decay rate, but is valid for any non-pathological functional form for the mean activity
were omitted although, as in
Deviations from standard nuclear decay due to a periodic decay rate of amplitude
A discrete time series
with
To any time series of finite length T sampled at intervals Δt there is a fundamental frequency
and a cut-off frequency
If the series contains a periodic component at frequency
The discrete autocorrelation function rk of lag k (in units of Δt) is defined by
with series mean
It is usual procedure to detrend a series, i.e. transform to a series of zero mean and zero trend, before performing the operation (33). This also leads to
The left suite of panels in
spectrum (standard radioactive decay) for comparison. The theoretical power spectrum of the decay curve generated from the time-dependent activity (22) contains spectral lines corresponding to periods
Since the autocorrelation is calculable from the power spectrum by means of the Wiener-Khinchine relations [
To summarize, analysis indicates that a periodic contribution to the radioactive decay rate would be detectable in the power spectrum and autocorrelation of the decay curve for harmonic amplitudes
Violations of the standard radioactive decay law, such as cited in Section 1, are weak at best and controversial. It is this author’s opinion that, at the present stage of investigation, alleged correlations, if indeed they exist, between the disintegration of radioactive nuclei and external events of a geophysical, astrophysical, or cosmological nature are more likely to be attributable to unanticipated instrumental effects resulting from known physical interactions than to violations of current physical laws or to the manifestation of some new physical interaction. Nevertheless, physicists have been surprised before by unexpected violations of principles thought to have been previously well-established. The violation of parity conservation [
Because non-random nuclear decay, or nuclear decay influenced by environmental conditions external to the nucleus (apart from known processes such as electron capture), has far-reaching fundamental implications, it is important to search for such phenomena by sensitive methods that have the potential to yield reliable, unambiguous results. In this paper two such methods were investigated and found to be capable of yielding a higher sensitivity than any method yet tried: 1) the statistical distribution of nuclear activities and 2) the statistical distribution of two-point estimates of nuclear lifetime (or half-life).
Theoretical analyses and numerical simulations of non-standard radioactive decay processes undertaken for this paper and an earlier brief report [
The statistical methods reported here are most sensitive and quantitatively revealing when the harmonic contribution to the decay rate has a period T1 shorter than the duration T of the time series of measured activities. However, this ought not to be a serious constraint in effecting a search for violations of the radioactive decay law since 1) the periods T1 of primary interest are already known (i.e. they are the periods claimed to have been observed in published papers), and 2) the length of the time series is an experimentally adjustable parameter, which can be made larger by taking more data.
M. P.Silverman, (2015) Effects of a Periodic Decay Rate on the Statistics of Radioactive Decay: New Methods to Search for Violations of the Law of Radioactive Change. Journal of Modern Physics,06,1533-1553. doi: 10.4236/jmp.2015.611157