_{1}

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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 [_{0} radioactive nuclei, the number surviving after a time interval t is given by the exponential survival law―designated the Law of Radioactive Change by Rutherford and Soddy [

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_{1}. The probability density function (pdf) is examined as a function of the initial mean count per bin _{1} of the periodic component of

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 [_{0}, which is related to the half-life

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 x_{t} 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_{t} from a high-activity radioactive source may be represented as a Poisson-Gauss (PG) variate

Over a time interval

with

under a transformation

to the dimensionless variate _{t} at any time t could exceed

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. _{0} = 100 and initial mean activity (A) μ_{0} = 100 or (B) μ_{0} = 1000 as the number of samples comprising the time series increases from T = 1 (plot a) to T = 450 (plot g). Plot a is the distribution that would result from drawing numerous samples all from a pure Poisson population of initial mean activity μ_{0}. However, in a time series of radioactive decay measurements, ordinarily only the first measurement is drawn from distribution a, and each subsequent measurement is drawn from a residual population of lower mean activity. As the number of measurements increases and the residual mean activity decreases, the pdf of the distribution skews markedly to the left―i.e. in the direction of decreasing μ_{t}.

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 ^{−}^{1} power law (12) before plunging to 0 at x = 0. It is interesting to note that the integrand in Equation (12) has the form (to within a normalization constant) of an Inverse Gaussian (also known as a Wald) distribution, which arises in the analysis of Brownian diffusion processes [

In comparing corresponding plots (i.e. of the same color) in _{0} = 1000 is narrower than Poisson distribution a (red)

defined by μ_{0} = 100, in contrast to what one might expect, since the variance of a Poisson distribution equals the mean μ_{0}. The explanation is that the distributed variate in the figure is not x but_{0}. Apart from the two Poisson plots a, an examination of the other corresponding plots in

_{0} = 10^{2} (red) to μ_{0} = 10^{5} (orange). The higher the value of μ_{0}, the more sharply the pdf sides drop to the horizontal baseline, as indicated in _{0}, the func-

tion

cept in the immediate vicinity of

The apparent oscillatory structure of the orange plot (μ_{0} = 10^{5}) in _{n} and

in terms of the lifetime_{0} = 10^{5} of a radioactive source with lifetime T_{0} = (A) 50, (B) 125, and (C) 200. The increasing lifetimes lead to a progression of MPG distributions with (A) completely resolved, (B) partially resolved, and (C) unresolved individual PG maxima. Application of criterion (13) to

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 k^{th} moment

Summation of (14) over the range of t and expansion of ^{th} moment of the MPG random variable

in which

Expansion of Equation (15) leads to the series

in which sequential terms decrease by powers of

From the moments M_{k} 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 T_{0}, periodicity T_{1}, and count duration T (which is equal to the number of PG samples in the time series), and for amplitude

with μ_{t} given by Equation (22), as a function of normalized activity _{0} = 10^{4}, lifetime T_{0} = 100, duration T = 100 and period T_{1} = (A) 10, (B) 50, (C) 200. In each panel the color of the individual plots denotes the value of the amplitude: _{1}.

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_{1}.

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

_{1} = 10. The horizontal solid black lines mark the numerical values of the activities z corresponding to the nine peaks in

Another important feature to note is illustrated by the blue trace in _{1} = 10 in the blue transient of

It is a well-known principle of time series analysis that one cannot measure the period T_{1} of a harmonic component if the duration T of the series is shorter than the period^{22}Na [_{0} = 10^{6}, lifetime T_{0} = 1000, measurement time T = 1000 and increasing periods T_{1} = (A) 1500, (B) 2000, (C) 4000. The plots are color-coded for amplitude:

panels that relatively high amplitudes _{1} need not be a significant experimental limitation, since it is often possible to increase the duration T of the time series.

A standard procedure for measuring the half-life _{0} (inverse decay rate)― of a radionuclide is to record the decay curve as a function of time. For a single unstable state decaying exponentially, a log plot generates a straight line from whose slope the lifetime can be determined. An alternative procedure for determining nuclear lifetimes is based on the statistical distribution of two-point lifetime estimates obtained from a time series of measured activities

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 T_{0}.

Given that the activities A_{i} 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 T_{ij} 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 T_{ij} in the population of N_{t} 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 _{t} is a PG variate of variance

PG variate

The red plots in _{0} = 10^{6}, lifetime T_{0} = 500, number of time bins T = 250, and sampling multiplicity J = (A) 1 and (B) 25. From the relation following Equation (25) the total number of two-point estimates T_{ij} is N_{t} = 31,125. The solid red traces 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 0^{th} 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

_{1} = 100. Whereas the two plots are virtually indistinguishable in

were omitted although, as in ^{−}^{3} in the figure, but higher sensitivities are achievable.

^{−}^{3}, (B) 5 × 10^{−}^{4}, (C) 0. In each panel, the plots are color-coded for increasing values of J from a minimum of 1 to maximum of 225. Comparison of plots of the same color shows that with increasing J, the differences between distributions for non-standard and standard radioactive decay becomes readily discernible even at values of the harmonic amplitude ^{−}^{4}.

Deviations from standard nuclear decay due to a periodic decay rate of amplitude

A discrete time series

with _{0}) in accordance with the Shannon sampling theorem [

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 r_{k} 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 _{0} = 10^{6}, T_{0} = 500, T_{1} = 50, T = 1000, as a function of increasing amplitude

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 [_{1}. The barest indication of oscillatory structure is seen in

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 ^{22}Na a sensitivity of one part in 1000 was demonstrated [^{22}Na has a half-life of approximately 2.6 y, or lifetime ^{6} measurements were to a large extent samples from the same population of Poisson variates. The decay factor

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 [^{60}Co [

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 [^{4}, which exceeds the corresponding sensitivities of the decay curve, power spectrum, or autocorrelation function by at least two orders of magnitude. Implementation of these statistical methods requires 1) a radioactive source of initial high activity and 2) measurement of a sufficiently long time series of decay events so that the activities comprise a population of mixed Poisson-Gauss variates. Both conditions are readily achievable in nuclear metrology labs for some of the nuclides (listed in Section 1) claimed to violate the standard radioactive decay law.

The statistical methods reported here are most sensitive and quantitatively revealing when the harmonic contribution to the decay rate has a period T_{1} 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 T_{1} 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