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Benford's Law and β-Decay Half-Lives

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2009 Commun. Theor. Phys. 51 713

(http://iopscience.iop.org/0253-6102/51/4/25)

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Commun. Theor. Phys. (Beijing, China) 51 (2009) pp. 713–716c© Chinese Physical Society and IOP Publishing Ltd Vol. 51, No. 4, April 15, 2009

Benford’s Law and β-Decay Half-Lives∗

NI Dong-Dong,1,† WEI Lai,1 and REN Zhong-Zhou1,2,‡

1Department of Physics, Nanjing University, Nanjing 210093, China

2Center of Theoretical Nuclear Physics, National Laboratory of Heavy-Ion Accelerator, Lanzhou 730000, China

(Received June 5, 2008)

Abstract The experimental values of 2059 β-decay half-lives are systematically analyzed and investigated. We have

found that they are in satisfactory agreement with Benford’s law, which states that the frequency of occurrence of

each figure, 1-9, as the first significant digit in a surprisingly large number of different data sets follows a logarithmic

distribution favoring the smaller ones. Benford’s logarithmic distribution of β-decay half-lives can be explained in terms

of Newcomb’s justification of Benford’s law and empirical exponential law of β-decay half-lives. Moreover, we test the

calculated values of 6721 β-decay half-lives with the aid of Benford’s law. This indicates that Benford’s law is useful for

theoretical physicists to test their methods for calculating β-decay half-lives.

PACS numbers: 23.40.-s, 21.10.TgKey words: Benford’s law, β-decay, half-lives

1 Introduction

In 1881, Newcomb[1] noticed that the beginning pagesof the logarithmic table books were dirtier than the endpages, which indicated that the tables of logarithms werenot used in a uniform way. This observation led to theconclusion that 1 as the first significant digit occurs morefrequently than any other figures, and the frequency di-minishes up to 9. A half century later, in 1935, FrankBenford[2] rediscovered that the first significant digits ofa statistically large number of data in decimal form arenot equiprobably distributed but follow a logarithmic dis-tribution favoring the smaller ones. And he provided aprobability distribution function for the first significantdigits, which can be written as

P (d) = log10

(

1 +1

d

)

, (1)

where P is the probability and d is the first digit in ques-tion. Nowadays the law is called Benford’s law, althoughit was previously discovered by Newcomb.

Newcomb not only firstly discovered the first digit law,but also made the simplest explanation on the law.[1] Hisbasic argument is that any positive number can be writ-ten as ic+s (i being 10 in the common system), where c isan integer and s is a decimal fraction on the interval [0,1). Because changing the integer only changes the decimalpoint without affecting the first significant digit of ic+s,we just need to pay attention to the decimal fraction s.Newcomb thought that the fractional part was uniformlydistributed on the interval [0, 1), so Benford’s law cameinto being.

Moreover, this law is “scale invariant”[3] and “baseinvariant”.[4−6] That is to say, Benford’s law holds for any

rescaling of the data, since it does not depend on anyparticular choice of units. And if we change the base ofa number system, for example, from the decimal base-10system to the octal base-8 system, this law is equally valid.

In 1992, Buck et al. found both measured and cal-culated values of the half-lives of 477 unhindered α de-cays satisfy Benford’s law very well.[7] However, we do notknow whether β-decay half-lives follow Benford’s law as α-decay half-lives do, because there are some differences be-tween α-decay and β-decay. α-decay is governed by stronginteractions and electromagnetic interactions whereas β-decay is governed by weak interactions. In addition, α-decay mostly proceeds in heavy and super-heavy nucleiwhereas β-decay proceeds in all nuclei including light,medium, heavy and super-heavy nuclei. In this paper,we firstly investigate the experimental values of the half-lives of β−-decay, β+-decay, orbital electron capture, andβ-decay. Then, we explain why they are in good agree-ment with the predictions of Benford’s law. Finally, wetest the calculated values of β-decay half-lives in terms ofBenford’s law.

2 Benford’s Law and β-Decay Half-Lives

Since Fermi proposed the theory of β-decay in the1930s, much attention has been paid to the investiga-tions of β-decay. However, it has never been examinedwhether the frequencies of occurrence of the first digits ofthe values of β-decay half-lives fulfil Benford’s law. Mostempirical studies focused on the distribution of the val-ues of β-decay half-lives (see, for instance, Ref. [8] or [9]).Along with the accumulation of more and more experi-mental data of nuclei, it is interesting to see whether β-decay half-lives follow Benford’s law. To this end, we now

∗This project supported by the National Natural Science Foundation of China under Grant Nos. 10675090, 10535010, and 10775068,

the National Fund for Forstering Talents of Basic Science under Grant No. J0630316, the 973 State Key Basic Research and Development

Program of China under Grant No. 2007CB815004, the CAS Knowledge Innovation Project under Grant No. KJCX2-SW-N02, and the

Research Fund of Doctoral Points under Grant No. 20070284016†E-mail: dongdongnick@gmail.com‡E-mail: zren@nju.edu.cn

714 NI Dong-Dong, WEI Lai, and REN Zhong-Zhou Vol. 51

systematically investigate the frequencies of occurrence ofthe first digits of the experimental values of β-decay half-lives, and compare them with the predictions of Benford’slaw. The experimental values of β-decay half-lives used inthis article are taken from the Nubase table of nuclear and

decay properties created by Audi et al.[10] Our results are

listed in Table 1 and the first digit distribution of the ex-

perimental values of β-decay half-lives is shown in Fig. 1,

compared with Benford’s logarithmic law.

Table 1 The frequency of occurrence of the figure d as the first significant digit, obtained from thevalues of the experimentally measured half-lives (in seconds) of β decays. The predictions of Benford’slaw, together with their expected root-mean-square errors, are shown in brackets.

dβ−-decay (986) β+-decay (1049) Capture (129) β-decay (2059)

Nobs (NBen) Nobs (NBen) Nobs (NBen) Nobs (NBen)

1 304(296.8 ± 14.4) 314(315.8 ± 14.8) 36(38.8 ± 5.2) 631(619.8 ± 20.8)

2 159(173.6 ± 12.0) 173(184.7 ± 12.3) 28(22.7 ± 4.3) 336(362.6 ± 17.3)

3 123(123.2 ± 10.4) 119(131.1 ± 10.7) 18(16.1 ± 3.8) 241(257.2 ± 15.0)

4 97(95.6 ± 9.3) 98(101.6 ± 9.6) 12(12.5 ± 3.4) 200(199.5 ± 13.4)

5 89(78.1 ± 8.5) 100(83.1 ± 8.7) 9(10.2 ± 3.1) 192(163.0 ± 12.2)

6 73(66.0 ± 7.8) 62(70.2 ± 8.1) 4(8.6 ± 2.8) 133(137.8 ± 11.3)

7 37(57.2 ± 7.3) 64(60.8 ± 7.6) 9(7.5 ± 2.6) 103(119.4 ± 10.6)

8 54(50.4 ± 6.9) 58(53.6 ± 7.1) 11(6.6 ± 2.5) 116(105.3 ± 10.0)

9 50(45.1 ± 6.6) 61(48.0 ± 6.8) 2(5.9 ± 2.4) 107(94.2 ± 9.5)

χ2 12.60 11.35 11.01 14.95

Fig. 1 The first digit distribution of data for the experi-mentally measured half-lives of 2059 β decays, comparedwith Benford’s logarithmic law.

In Table 1, we compare the frequency of occurrence ofthe first digit, Nobs(d), which is obtained from the half-lives of 2059 β-decays, with the frequency predicted byBenford’s law, NBen(d). We also present the expectedroot-mean-square error, ∆[N(d)], which is defined as

∆[N(d)] =√

NP (d)[ 1 − P (d) ] . (2)

Here, N is the total number of decays considered and P (d)is the prediction of Benford’s law. To measure the good-ness of agreement between Nobs(d) and NBen(d), we defineχ2 as

χ2 =9

∑

d=1

{ Nobs(d) − NBen(d)

∆[N(d)]

}2

. (3)

If the observed first digit distribution is in acceptableagreement with Benford’s logarithmic law, the value ofthis quantity should be close to 9. And the smaller it is,the better the agreement is.

In Table 1, we have also divided up the β-decay half-lives according to the kinds of β-decay in order to discernwhether the first digit distribution of β-decay half-lives de-pends on this factor. Within the expected error limits, wecan observe no significant differences among the first digitdistributions obtained respectively from the half-lives ofβ−-decay, β+-decay and orbital electron capture. All ofthem can be found in good agreement with Benford’s law.

After Becquerel discovered nuclear radioactivity in1896, a number of studies on β-decay following differentempirical and theoretical approaches were carried out, anda new interaction called the weak interaction was found.In 1931, Pauli hypothesized a new particle named neu-trino emitted in the process of β decay, which successfullysolved the energy conservation problem of β decay. Basedon this, Fermi proposed a basic formulation of the β de-cay theory in 1934.[11,12] In 1956 and 1957, Lee and Yangmade an important contribution to the theory of β de-cay. They suggested that parity might not be conservedin a weak process such as β decay,[13] which was imme-diately confirmed in an experiment completed by Wu et

al.[14] Since then, the theory of weak interactions has been

developed rapidly. However, due to the inherent complex-ity of both the weak interaction and nuclear many bodyproblem, the calculations of β-decay half-lives are rathercomplicated and model dependent.[15−23] Fortunately, inrecent years, empirical studies on β-decay half-lives haveachieved a lot. New exponential laws of β−-decay half-lives and of β+-decay half-lives were proposed.[8,9] Thenew exponential law states that the relation between the

No. 4 Benford’s Law and β-Decay Half-Lives 715

logarithm of half-lives of β−-decay (β+-decay) and theparent nuclei far from the β-stable line is described as

log10 T1/2 = (c1Z + c2)N + c3Z + c4 , (4)

where T1/2 is the half-lives (in seconds) of β−-decay (β+-decay), Z and N are respectively the proton number andthe neutron number of the parent nuclei. The values ofc1,c2,c3 and c4 are determined according to the availableexperimental data of the half-lives. The relation betweenthe β+-decay half-lives and the parent nuclei is under thecondition that the forbiddenness of β+-decay is the same,whereas the relation of β−-decay half-lives is valid for allnuclei far from the stability-line. However, this differencebetween β+-decay half-lives and β−-decay half-lives doesnot matter for our investigation, because we only care thedistribution of the exponential [(c1Z + c2)N + c3Z + c4].

According to Newcomb’s justification of Benford’s law, ifthe fractional part of the exponential is uniformly dis-tribution, it is certain that β−-decay half-lives and β+-decay half-lives have first digits, which obey Benford’slaw. Since the exponential depends on the proton num-ber and the neutron number of the parent nuclei, it seemsthat the fractional part of this quantity should be equallydistributed. More precisely, the exponential of β+-decayhalf-lives depends not only on the parent nuclei but alsoon the forbiddenness of β+-decay, that is to say, the dis-tribution of its fractional part approaches to uniform onemore than that of β−-decay half-lives. Therefore, β+-decay half-lives obey Benford’s law better than β−-decayhalf-lives, which we can see from Table 1 and Figs. 2 and3.

Fig. 2 The first digit distribution of data for the experi-mentally measured half-lives of 986 β− decays, comparedwith Benford’s logarithmic law.

Fig. 3 The first digit distribution of data for the ex-perimentally measured half-lives of 1049 β+ decays, com-pared with Benford’s logarithmic law.

Table 2 The frequency of occurrence of the figure d asthe first significant digit, obtained from the calculatedvalues of β-decay half-lives. The predictions of Benford’slaw, together with their expected root mean square er-rors, are shown in brackets.

d β-decay(6721)

Nobs (NBen ± ∆[N(d)])

1 2139 (2023.0 ± 37.6)

2 1142 (1182.9 ± 31.2)

3 836 (840.1 ± 27.1)

4 612 (651.3 ± 24.3)

5 489 (532.3 ± 22.1)

6 405 (449.6 ± 20.5)

7 366 (389.8 ± 19.2)

8 404 (344.1 ± 18.1)

9 328 (307.1 ± 17.1)

χ2 36.43

Looking at the intensities of three kinds of β-decay inNUBASE2003, we can find (i) that β−-decay and β+-decay

take an important place in nuclear β-decay, and (ii) that

β-decay of one nuclide is in most cases either β−-decay or

β+-decay, that is, it is very infrequent that one β-decay

contains β−-decay and β+-decay. Since β−-decay half-

lives and β+-decay half-lives follow Benford’s law well, it

is perhaps not so surprising that β-decay half-life data also

satisfy Benford’s law.

Since the experimental values of β-decay half-lives sat-

isfy Benford’s law, we believe that the same phenomenon

should exist for the calculated values of β-decay half-lives.

Therefore, we now investigate the frequencies of occur-

rence of the first digits of the calculated values of β-decay

half-lives, and compare them with the predictions of Ben-

ford’s law. The calculated values of β-decay half-lives

used in this article are taken from the table of calcu-

lated nuclear ground-state properties by Moller et al.[15]

Table 2 summarizes our findings and our results for the

first digit distribution of the calculated values are com-

pared graphically with Benford’s logarithmic law in Fig. 4.

716 NI Dong-Dong, WEI Lai, and REN Zhong-Zhou Vol. 51

Fig. 4 The first digit distribution of data for the cal-culated half-lives of 6721 β decays, compared with Ben-ford’s logarithmic law. We can see that the calculatedvalues have some tendency to follow Benford’s law, whichindicates that they are in a way reasonable.

We can see, though the agreement is not prefect, thatthe calculated values of β-decay half-lives show some ten-dency to follow the logarithmic law predicted by Benford’slaw. This indicates that they are in a way acceptable.

3 Summary

In summary, we systematically investigate experimen-tal data for the β-decay half-lives. The results show thatthey are in good agreement with Benford’s law. Moreover,we explain Benford’s law’s applicability to the statistics ofβ-decay half-lives in terms of Newcomb’s justification ofBenford’s law and empirical exponential laws of half-livesof β−-decay and of β+-decay. Finally, we test the calcu-lated values of β-decay half-lives with the aid of Benford’slaw. The results show that the calculated values are rea-sonable in a way. We hope that the first digit distributionof β-decay half-lives might give us information about thephysics process of β-decay. It is worth deeper and fullerinvestigation in the near future.

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