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On the randomness and correlation in the trajectories of alpha particle emitted from241Am: Statistical inference based on information entropy

M. El Ghazaly,1 Elsayed K. Elmaghraby,2 A. Al-Sayed,3, 1 Amal Mohamed,4, 1 and Mahmoud S. Dawood1

1Department of Physics, Faculty of Science, Zagazig University, Zagazig, 44519, Egypt.2Experimental Nuclear Physics Department, Nuclear Research Centre,

Egyptian Atomic Energy Authority, Cairo 13759, Egypt. Corresponding author.∗3Department of Physics, College of Science and Arts, Al-methnab,

Qassim University, P. O. Box 931, Buridah 51931, Al-mithnab, KSA4Department of Physics, Faculty of Science, Umm Al-Qura University, Makkah, Saudi Arabia.

(Dated: July 6, 2022)

Most particle detectors are based on the hypothesis that particles are emitted randomly uponnuclear decay. In the present work, we tested the hypothesis of the existence of correlation in therandom trajectories of alpha particles emitted from 241Am source and the null hypothesis of randomtrajectories. The trajectories were clued through the registration of track in a solid-state nucleartrack detector. The experimental parameters were optimized to identify the possible sources ofcorrelation in the track registration and the detector conditions upon exposure and etching to avoidmisleading results. The optimization included authentication of linearity in registration efficiencywith exposure time to prevent coalescence of registered tracks. The statistical inference processeswere based upon adaptive quadrates analysis of the spatial data, and entropy and divergence analysisof the quadrate data together with the null hypothesis of Poisson distribution of random trajectories.The clustering and dispersion analysis were performed with central deviation tendency, empirical K-function, radial distribution analysis, and proximity Analysis. Results showed a pattern of gainedinformation within the registered tracks that may be attributed to the alteration in the alphaparticles’ trajectories induced by the strong electric field due to atoms in the source compound andencapsulation film.

I. INTRODUCTION

A few decades ago, numerous passive detectors, suchas solid-state nuclear track detectors (SSNTD), were uti-lized to detect heavy ions from different radiation sources,including cosmic rays, accelerators, and laser-matter in-teractions [1]. The most common SSNTD is poly allyldiglycol carbonate (PADC), commonly known as Colom-bia Rains PADC. PADC detector is characterized by ahigh registration efficiency that reaches 100% for perpen-dicularly incident alpha particles. PADC tracks rangefrom several micrometers to several nanometers for scan-ning with a high spatial resolution imaging system suchas atomic force and confocal microscopes. Meanwhile,spatial resolution amounts to several micrometers forscanning with an ordinary optical microscope [1–6]. Themajor disadvantage of SSNTD is the time resolution ofthe PADC detector; it is impossible to discern the occur-rence of the tracks in the PADC detector. Because SS-NTD is a position-sensitive tool with a small spatial res-olution, if the etching conditions were adjusted correctly,it can be used for the pattern of the particle beam. Thebasic idea of SSNTD is the damage zone along the trajec-tory of the incident particle through the detector’s mate-rial (known as latent tracks) which are of some nanome-ters. The latent tracks in the PADC detector can beenlarged to be visible under an ordinary optical micro-

∗ Institutional email: elsayed.elmaghraby@eaea.org.eg; Alt. email:e.m.k.elmaghraby@gmail.com

scope by chemical etching at a proper concentration andtemperature in NaOH or KOH aqueous solution[6].

Alpha particles from 241Am, have enough energy toovercome the air deceleration and reach the surface ofthe SSNTD, inducing the latent track. The alpha-decayenergies are associated with 241Am are 5485.56 KeV for84.8% of the emitted particle, 5442.8 KeV for 13.1% ,and 5388 KeV for the remaining 1.6% [7]. The isotropicnature of the alpha decay from 241Am is based upon thefact that one could not “stimulate” atomic nuclei to decayin specific direction; i.e. there is no preferred directionof the emitted alpha particles. It could be determinedstatistically by the number of nuclei among a large as-sembly of them that would decay in a given time inter-val. Accordingly, alpha particles should be distributedrandomly in a given area if the null hypothesis is correct;furthermore, it should follow complete spatial random-ness (CSR).

Previous studies (cf. Refs. [8–16]) had shown that thedistribution of alpha particle tracks was neither uniformnor random, whatever the precaution made to make sucha fact obvious. The reason may be any process from thedecay to the registration on the detector surface. Ac-cordingly, the present work aims to study the spatial dis-tribution and correlation between alpha particle tracksfrom the 241Am source.

2

II. MATERIAL AND METHOD

PADC detectors (Tastrak PADC; density = 1.32g/cm3, molecular composition C12H18O7) of a thickness(500±4) µm were exposed to vertically incident alphaparticle from 241Am (main alpha particle energy 5.486MeV and having activity 9 µCi in air). The vertical align-ment was essential to eliminate, even, the small effect ofgravity. Predefined average energy could be obtained be-tween 0.5 and 5.5 MeV by changing the length of theair column between the 241Am source and the PADC de-tector with the aid of a collimator. Air molecules aremoving randomly at room temperature so that the scat-tering shall be random. The 241Am source is coated bya 30 nm-thick layer of gold to avoid source corrosion andneutralize the effect of the recoil of 237Np atom and theelectron negative charge1 remaining in the source mate-rial upon alpha particle emission [18]. The length of thecollimator was 1.5 cm which is suitable to reduce theaverage of 5.5 MeV alpha particles to about 4 MeV ofenergy. The cross-section diameter of the collimator was1.5 mm (area of 1.77 mm2) to ensure that most of the al-pha particle is perpendicularly striking the detector sur-face; such a narrower collimator minimizes the energy’sspreading out due to different paths in air. The alpha-irradiated PADC detectors were etched in an aqueoussolution of 6.25N NaOH at 70±1◦C for different dura-tions. The bulk-etching rate (V B) was measured usingthe well-known weight and thickness decrement meth-ods. It amounts to (1.26±0.06) µm/h. The track densityand diameters were measured with an optical microscope(Nikon, ECLIPSE, E200) equipped with Nikon digitalcamera (DS-Fi1). All photomicrographs are captured inRBG colors with a dynamic range of 216. RBG pho-tomicrographs were analyzed using ImageJ software [19],where different color channels are separated, and Green-channel is selected since it has maximum contrast.

We apply three data reduction techniques and analysison the track pattern registered on the PADC detector:(1) Divergence analysis between the null hypothesis ofrandomness and a test hypothesis of the registered pat-tern. (2) Dispersion analysis involving the central ten-dency and (3) Proximity analysis. Fig. 1 shows the ab-straction of the techniques used.

1 The alpha decay changes the energy states of the atom so that the237Np atom reconfigures the binding levels for the 95-2 electrons.This charge reconfiguration process is complex process [17]. Thetwo extra electrons will be left free in the lattice, then that bulkwill be negatively charged.

III. RESULTS AND DISCUSSION

A. Verification of linearity in registration efficiency.

The registration efficiency depends on the energy ofthe alpha particles reaching the surface of the detector.For the source-to-detector distance of 1.5 cm, the alphaparticle reaches the surface with an average energy of 4MeV. Several factors must be optimized to adjust thecondition for linearity in registration efficiency: (1) Therange of alpha particles having an energy of 4 MeV in thePADC detector is about 20.5 µm [20, 21]. The etchingtime (te) should be much less than the etching time toreach track depth at 20.5 µm (denoted tR), which leadsto detecting all incident alpha particles on the PADC de-tector [22]. The minimal etching time will cause a minortrack diameter and lower the detectability of the trackusing an ordinary optical microscope. (2) Another re-quirement is to adjust the registration efficiency associ-ated with the coalescence of registered tracks when thefluence exceeded a specific limit, i.e., determination ofthe maximum number of tracks registered per unit areato maintain linearity between track density and exposuretime. (3) Normality test for the registered tracks uponoptimized conditions.Three sets of samples were exposed to alpha particles

for durations between 30 s and 300 s; the first set wasetched chemically in 6.25N NaOH at 70 ◦C for 2 h. Thesecond and the third set were etched in the same chemicalconditions but for 4 h, and 6 h, respectively, see Table I.Images were recorded through the collimated area up to1583×1583 µm2. Since there was a need to optimize thedetectible track diameter, the track size distribution wasdetermined from the two pre-samples after being chem-ically etched in 6.25N NaOH at 70 ◦C for 4 hours, seeFig. 2. Despite both photomicrographs being producedunder the same conditions, namely exposure time, etch-ing time, alpha particle energy, and PADC detector, onecan recognize the randomness in the registered patternsand the extension of the registration area to cover all the1583×1583 µm2 area. This assures that the alpha parti-cle source is isotropically distributed behind this area.The track diameter distribution histogram in Fig. 3

shows that the alpha particle track diameter followsGaussian distribution centered at (8.77±0.33 µm). Thesmall value of standard deviation points to the indepen-dence of the track diameter on the difference between al-pha particle energies originating from the 241Am sourceon one side and the good efficiency of the etching processon the other one.The histogram in Fig. 4 illustrates the variation of

track density with time; these photomicrographs are forthe samples chemically etched for 4 h. The area of eachpicture is 1583×1583 µm2. The observed circular trackdiameters ranged from 8.4 to 9.1 µm. Several coalescenttracks are apparent in Figs. 4 (d-f) due to the expansionof the tracks to nearby ones. For low exposure time, thetrack densities are low, and no significant alpha tracks co-

3

FIG. 1. Flowchart of the spatial data analysis.

TABLE I. List of investigated samples and exposure and etch-ing conditions. Letters will be used for the figure’s label. λ isthe registered track density in cm−2. All samples were etchedin 6.25N NaOH at 70 ◦C.

Etch timeExposure 2 h 4 h 6 h

time Label λ Label λ Label λ

30 s a2 7.58×104 a4 7.31×104 a6 6.03×104

60 s b2 1.39×105 b4/1 1.21×105 b6 1.07×105

60 s - b4/2 1.31×105 -60 s - b4/3 1.17×105 -120 s c2 2.58×105 c4 2.61×105 c6 1.44×105

180 s d2 3.68×105 d4/1 2.81×105 d6 1.27×105

180 s d4/2 2.82×105

240 s e2 4.89×105 e4 4.27×105 e6 1.39×105

300 s f2 5.16×105 f4 5.02×105 f6 1.10×105

alescence was observed; for instance, for irradiation timeof 30 seconds, the alpha track density is (7.3±0.7)×104

tracks.cm−2. For alpha particle irradiation time of 1 min,the alpha track density is (12.5±0.8)×104 tracks cm−2;for a maximum exposure time of 5 minutes, the alpha

track density amounts to (502±1)×104 tracks cm−2.

A comparison of the response of the PADC detectorsto exposed alpha particles for different durations and dif-ferent etching times was undertaken by measurements ofthe track density in each of the samples listed in TableI. These samples of the PADC detectors are chemicallyetched for 2 h, 4 h, and 6 h in 6.25 N NaOH at 70±1◦C, in which the response of the detector depends on thedetectible track after etching.

As shown in Fig. 5 for 4 h etching time, the tracksdensity and exposure time plot offered a non-linearityin registration efficiency. As exposure time increases, thediameters of alpha particle tracks are growing, and there-fore coalescence and registered as one alpha reducing thetrack density, especially for longer exposure time. Simi-lar non-linearity, nearly independent between the tracksdensity and exposure time, was evident for a larger etch-ing time of 6 h.

For etching time of 2 h, the alpha particle track diam-eters were 4.9±0.2 µm, the alpha particle track densitieswere linearly correlated with irradiation time to alphaparticle up to 240 s of exposure as depicted in Fig. 5.On the other hand, the maximum linearity for the sam-

4

FIG. 2. Comparison between the distribution of alpha parti-cles tracks free areas in two photomicrographs (a) for sampleb4/1 and (b) for sample b4/2 of the PADC detector irradi-ated with 4 MeV alpha particle for 1 minutes with a timeinterval of ten minutes, samples were chemically etched in6.25N NaOH at 70 ◦C for 4 hours. The area of each pictureis 1583×1583 µm2.

ples etched for 4 h is 120 s of exposure time. In suchcircumstances, the linear registration of all spatially inci-dent alpha particles grants minimum loss of information.Hence, patterns in samples f2, f4, and all samples etchedfor 6 h will give a biased conclusion on the extractedinformation.

FIG. 3. Histogram of the alpha tracks diameter in the PADCdetector for samples d4/1 and d4/2.

B. Randomness analysis

Analyzing patterns is a well-established branch of com-putational science and information technology [23]. Themost concerning abstraction is the point pattern analysis(PPA) [24] which involves the analysis of the spatial loca-tion of points in the multi-dimensional array (mostly two-dimensional). These analyses reveal deep laying informa-tion in such patterns. The divergence analysis is achievedby adopting the procedure of quadrat sampling to test itsprobability distributions and a statistical model to givepredicted probabilities that may be compared with eachof the individual probabilities in the observed frequencydistribution. The theoretical probability distribution isobtained by making the sensible assumption of the ran-domness of the registration governing the evolution ofthe features in the pattern. From those assumptions, wededuced the probability distribution that will give thecorrect prediction of the frequency distribution of thequadrats. Finally, a comparison between the predictedprobability distribution with the observed probabilitydistribution obtained by sampling the pattern was madeusing Kullback-Leibler divergence based on the Shannonentropy hypothesis. There are no particular restrictionsfor the shape and size of a quadrat if the size is reasonablecompared to the area under investigation. The selectionof quadrat size is always an arbitrary procedure but mayinfluence the subsequent interpretation of results. One ofthe most used treatments of quadrat size is the approachtaken by Greig-Smith [25]. On the basis that random-ness at a variety of scales within a square quadrat censuswhere the number of cells on each axis is some power of 2,based on the binary property of the Poisson distributionthat it mean λ equals its variance. However, in searchfor evidence of clustering at that scale, Greig-Smith hassuggested that the size of quadrat at that scale will be

5

FIG. 4. Photomicrographs of alpha particle tracks in PADCdetector exposed for alpha particles of 4 MeV for differentdurations (a) 0.5 min, (b) 1 min., (c) 2 min., (d) 3 min., (e)4 min, and (f) 5 min., PADC detector is chemically etched in6.25 N NaOH at 70±1 ◦C for 4 h. The area of the picture is1583×1583 µm2.

related to the mean area of the pattern in which the testdescribed here does not measure tendencies towards uni-formity in the pattern. In the present work, we forcedthe quadrant area to follow the relation

AQS =√2A

Nt

(1)

Where A is the studied area and Nt the total number offeatures in the whole pattern.According to Poisson distribution, the null hypothesis

of alpha tracks in SSNTD is the equal probability to hitany location in the exposed area, which implies that thenumber of hits is proportional to the detector area A ac-cording to Poisson probability distribution. However, ifthere were clustering and dispersion in the pattern regis-tered, the distribution would be different.Poisson probability distribution (q = qi = q(xi)) of the

FIG. 5. Alpha particle track density dependence on the ex-posure time at different etching times.

number of features that will occur in a quadrat is

q(xi) =λxi

Γ(xi + 1)e−λ, (2)

which gives the random probability a number of xi eventsoccur while being hit, λ is the intensity function describesboth the mean expected value and the variance of thedistribution given from the relation;

λ =NP

NPQ

(3)

where NP is the total number of features in the registeredpattern within the investigated clip and NPQ is the num-ber of quadrats to which the study area is divided. Thisanalysis was undertaken for the samples a2, b2, c3, d2,e2, f2, a4, b4/1, b4/2, b4/3, c4, d4/1, e4, and f4, aslabeled in Table I.The comparison in Fig. 6 shows a heatmap for the

number of features in each quadrate, xi = N , countedin all investigated areas; The total number of quadratesdepends on the condition in Eq. 1. Detailed informationis given in Table II. Generally, the statistics rely on thevalue of λ. So Eq. 1 grantee the closeness of the resultsupon comparison.The histograms for the probability of a number of fea-

tures xi within each quadrate deduced from the frequencystatistics of the number of quadrats having a count xi di-vided by the total number of countsNP are shown in Fig.7. For comparison, the Poisson distribution given in Eq.2 was calculated assuming the exact value of the meanλ.The photomicrographs imaged for the samples a4,

b4/1, b4/2, b4/3, c4, d4/1, e4, and f4 were analyzedusing the same method. The results are illustrated inFigs. 8-10.

6

FIG. 6. Heatmap for the investigated local track densities ofsamples a2, b2, c3, d2, e2, and f2 after being etched for 2h. The grayscale represents the number of features in eachquadrate, xi, while the number of quadrates increases pro-gressively to fulfill Eq. 1. The dashed squares (red color on-line) show clipped areas containing quadrats used for Poissondistribution analysis; see text.

TABLE II. Detailed analysis of samples under study.

Poisson Analysis Entropy & divergence

Sample NP NPQ λ D(p||q)KLD σX σY QD

a2 677 121 5.595 0.102 10.69 10.20 0.2470b2 1662 289 5.751 0.034 7.524 7.563 0.0195c2 2565 441 5.816 0.012 5.519 5.642 0.0615d2 3677 625 5.883 0.043 4.647 4.657 0.0050e2 5651 961 5.880 0.053 4.036 3.992 0.0220f2 5327 900 5.919 0.041 3.905 3.904 0.0005a4 667 121 5.512 0.118 10.38 10.85 0.2365

b4/1 1396 225 6.084 0.051 7.988 8.242 0.1270b4/2 1486 256 5.804 0.064 8.337 8.317 0.0100b4/3 1274 225 5.662 0.050 7.859 7.885 0.0130c4 2589 441 5.871 0.013 5.487 5.615 0.0640

d4/1 3004 529 5.678 0.029 5.321 5.271 0.0250e4 4717 784 6.017 0.056 4.302 4.255 0.0235f4 5158 900 5.725 0.120 3.927 3.991 0.0320

FIG. 7. Probability histograms for the test hypothesis p forsamples a2, b2, c3, d2, e2, and f2 versus the null hypothesis qversus the number of features xi within each quadrate. Thenull hypothesis is randomness based on Poisson distribution,in Eq. 2 assuming the same value of λ as given in Table II.

C. Entropy and divergence

The amount of information concerning the variabilityof a random variable (uncertainty in randomness) of astatistical system of events directly indicates the system’sShannon entropy [26].

H(p) = − 1

logNpQ

∑

i

pi log pi (4)

where p = {pi} is the probability of the event in the i ∈ χquadrat that belongs to the same probability space,χ, ofthe observables. The most crucial concept in Shannon’sEntropy is its ability to measure the extent to which thedata are spread out over its possible values; lower en-tropy values refer to high information content and aremost likely to develop a strong rule or correlation. Forrandom data, the Shannon entropy value is equal to 1.In other terms, increased observability must lead to de-creased uncertainty and entropy.The difference between true random process and signal

and more deterministic processes can be obtained using

7

FIG. 8. Constructed as in Fig. 6 for the investigated localtrack densities of samples a4, b4/1, c4, d4/1, e4, and f4 afterbeing etched for 4 h.

the Kullback-Leibler divergence (KLD) [27] divergencetheoretical models [28]. KLD is a measure of dissimilar-ity between two probability distributions p and q = {qi}usually represents the probability distribution of data,the observations, and the probability distribution of itsrepresenting random model optimized for p. For the dis-crete case of data [29], KLD comprises

DKLD(p ‖ q) =∑

i

pi log

(

pi

qi

)

(5)

The positive value of DKLD(p||q) represents the informa-tion gain achieved from p instead of the random model q.Based on Bayesian inference, DKLD(p ‖ q) is the informa-tion gained upon measurement having posterior proba-bility distribution p compared to the priori known proba-bility distribution q or vice versa, the lost inference whenforced random distribution q is used instead of measuredp [30]. The value of DKLD(p ‖ q) goes to zero as the twoprobability distributions become the same.The DKLD(p ‖ q)results are given in Table II and em-

FIG. 9. Probability histogram for the test hypothesis p forsamples a4, b4/1, c4, d4/1, e4, and f4 versus the null hypoth-esis q versus the number of features xi within each quadrate.They were constructed as in Fig. 7.

bedded within Figs. 7, 9, and 10. The last-mentioned fig-ures are calculated based on the prior probabilities of ran-dom events based on Poisson distribution. The greaterthe prior uncertainty of such an occurrence, the greaterthe information gained if such a non-random event oc-curs. Criteria for defining an information statistic sug-gest that the measure would vary from zero to infinityand that the measure would be additive between inde-pendent events. The result showed embedded informationwithin the track pattern. Information could be extractedfrom patterns in samples a2, b2, c2, d2, e2, a4, b4/1,c4, d4/1, and e4. However, this information could bemisleading due to other effects, as discussed below.

D. Clustering and dispersion analysis

Dispersion, skewness, and other major parameters canbe clued from the central tendency analysis. While Clus-tering requires density analysis (including entropy andconvergence) and distance analysis using pair correla-tion function (radial distribution function) and Ripley’s

8

FIG. 10. Comparison between heatmap for the investigatedlocal track densities of samples (a) b4/1, (c) b4/2, and (e)b4/3 and its associated probability histograms (b), (d), and(f), respectively. Constructed as figures 7 and 8.

K function [31] involved in the spatial analysis method.The basic descriptive centric technique for a real dataanalysis is the featured center (Xc,Yc) in which

Xc =1

N∑N

i=1 wi

N∑

i=1

wiXi, (6)

Yc =1

N∑N

i=1 wi

N∑

i=1

wiYi (7)

wi is the weighting factor for the feature, which may beconsidered a reciprocal uncertainty of the existence ofthat point (Xi,Yi) within the center area of the feature.For definitely shaped points, wi=1, the variance of thedistribution of the data may be different in the directionsX and Y,

σ2Y =

1

(N − 1)∑N

i=1 wi

N∑

i=1

wi (Yi − Yc)2

(8)

σ2X =

1

(N − 1)∑N

i=1 wi

N∑

i=1

wi (Xi −Xc)2. (9)

The distribution deviation is determined by the relation

σ2D =

σ2X + σ2

Y

2, (10)

while the quality of the distribution is determined by therelation

QD =

∣

∣

∣

∣

σ2X − σ2

Y

2

∣

∣

∣

∣

. (11)

The standard deviation in two dimensions is defined by

σ2 =

∑N

i=1 wi

(

(Xi −Xc)2+ (Yi − Yc)

2)

(N − 2)∑N

i=1 wi

. (12)

The (N -2) provides an unbiased estimate of standard dis-tance since there are two constants related to a real de-viation. Note that σ2 6= σ2

D if a circular clip of patternwas taken whether or not QD 6= 0.As shown in Table II, the photomicrograph patterns of

the alpha particle tracks do not offer a uniform spatialdistribution around the center of the data. The QDvaluesspan a range from 0.005 to 0.247. A large value of QD atlow exposure time is attributed to a limited number ofregistered tracks in the detectors to attain random data.Hence, patterns a2, b2, a4, and b4/1 contain remnantinformation of randomness despite a2, and a4 have largevalues of entropy divergence. Similarly, at a large ex-posure time of 300 s, tracks coalescence may disturb thegained information. At the intermediate track densities,the value of QD begins to reach 0, the nominal value ofrandom track registration. Conversely, its value may in-crease due to the accumulation of clustering informationwithin the alpha particle tracks.

E. The empirical K-function

The empirical distribution function is the pairwise dis-tances used to search for anomalies in the feature pat-terns. The second moment of this distribution functionis the differential Radial Distribution Function (RDF) asa function of distance r. Our focus is on the distanceor spacing between features in the registered pattern.Each ordered pair of points had a measured distancedi,j = ||ri−rj || which may contain the information aboutthe alpha particles’ spatial pattern.There are two different definitions used in the present

work of the RDF as a function of distance r, first,

H1(ri) =1

λ

∑

i=1

~1(ri−1 < di,c < ri) (13)

di,c is the distance between the features labeled i and the

center of the data, ~1(condition) is the indicator function

for the satisfaction of the condition and λ is the aver-age number of features per unit area. In this case, the

9

FIG. 11. Plot of the Radial Distribution Function H1(ri)andH2(ri)for samples a2, b2, c2, d2, e2, and f2 after being etchedfor 2 h.

cutoff radial distance is the radius of the clipped pattern(denoted dc). Also, as a function of pair distances

H2(ri) = N

∑

i

∑

j 6=i~1(di,j < ri)

λ(No. of interdistances)(14)

The condition di,j < dc−ri was introduced to enforce thecalculation to run only to the pattern within the clippedcircle and reduce the edge effect of the data counting.Value is normalized to the new counted features. Theplot of the functions H1(ri)and H2(ri) is shown in Fig.11 and 12.The variation of the central distribution function re-

veals a sort of correlation in the pattern, which may besymmetric around the center of the registered pattern.Such central tendency is a consequence of the existenceof the collimator and the inverse-square law in which theemitted alpha particles from the source may be inclinedalong the diagonal of the collimator rather than parallelto its axis. The pair RDF cannot detect such a patterndue to its moving average nature.Here, the radial distribution function H1(ri)and

H2(ri)gave two crucial pieces of information. H1(ri)givesthe distribution around the center of the data from equa-tions 6 and 7, in which the central symmetry of the func-tion compensates for the effect of the non-randomnessof the data. The data is truncated at the end of thefield-of-view (FoV). H2(ri), on the other hand, is a pairdistribution function that is sensitive to clustering anddispersion of the pattern and the edge effect. Collima-tion of alpha particles on a determined region on thedetector causes two effects: (1) the pairs near the edge ofthe area have fewer neighbors from one side, which lesserthe value of H2(ri)near the end of the FoV. (2) remoteintradistant neighbors from the other directions gave avalue of H2(ri)at distances greater than the end of FoV.The difference between these orders, within a radius of

FIG. 12. Plot of the Radial Distribution Function H1(ri)andH2(ri)for samples a4, b4/1, c4, d4/1, e4, and f4 after beingetched for 4 h.

about 1/3 of the diameter of the data, is another clue tothe existence of clustering or dispersion in the pattern,as shown in Fig. 11. Similar behavior was observed forthese samples etched for 4 h (see Fig. 12.)The K-function, on the other hand, is a Cumulative

Radial Distribution Function (CRDF), obtained basedon H1(ri) and H2(ri), as

K1(r) =∑

i = 1ri ≤ r

wi,cH1(ri) (15)

K2(r) =∑

i = 1ri ≤ r

wi,cH2(ri) (16)

wi,c is a generally used edge-corrected estimator. Theweight function has a value between 0 and 1 that pro-vides higher weight to the points needing the center ofthe investigated area rather than the points at the edge.In the present work, we shall use wi,c = 1. i.e.

K1(r) =1

λ

∑

i

1(di,c < r) (17)

K2(r) = N

∑

i

∑

j 6=i 1(di,j < ri; di,j < dc − ri)

λ(No. of points)(18)

The plot of the functions K1(ri)and K2(ri) are shown inFigs. 13 and 14. It is obvious that the K1 or 2-functionsand H2 -function do not uniquely define the pattern.Still, they can be used to detect if there were a directinteraction between processes causing the pattern, i.e.,two different patterns may have the same K-function,see Refs. [32, 33].

10

FIG. 13. Plot of the cumulative Radial Distribution FunctionK1(r)and K2(r)for samples a2, b2, c2, d2, e2, and f2. Thevalues of πr2 were added for comparison.

FIG. 14. Plot of the cumulative Radial Distribution FunctionK1(r)and K2(r)for samples a4, b4/1, c4, d4/1, e4, and f4.The values of πr2 were added for comparison.

In this aspect, The null hypothesis of the K-function isthat the number of features lying closer than a distancer has expected value K1 or 2(r), i.e., the variation as πr2

and deviations from this expectation indicate scales ofclustering and/or dispersion [34, 35]. An inhibited pro-cess causes a lake of formation of the feature and willusually have K1 or 2(r) < πr2, while an enhanced pro-cess causes clustered feature and will have K2(r) > πr2,for appropriate values of r. While K1 is related tothe nearest-center distribution and related mainly to theanisotropy of the radial signature of the features, K2 isassociated with the nearest-neighbor distribution and isrelated to the none-stationary processes causing the fea-ture, also known as Ripley’s K [36, 37].

Consequently, trends of K1(r)follow the πr2 trend to

FIG. 15. Plot of the L-Functions L1(r)and L2(r)for samplesa2, b2, c2, d2, e2, and f2.

the end of FoV while the trends of K2(r)follow the πr2

trend up to 1/3 of the diameter of the collimator (2/3 ofthe radius to the end of FoV).Because of the difficulty of comparison, we consider the

difference L-Functions

L1(r) =

√

k1(r)

π− r (19)

L2(r) =

√

k2(r)

π− r (20)

These functions are plotted in Figs. 15 and 16. Since theL-function has the dimension of distance, the confidenceinterval is just the confidence interval of each feature, i.e.,the 0.3 µm deduced from the analysis of track diameter.The values of L-function greater than this value representcorrelation among clusters, which occur at about around200-500 µm.From these results, the essential reasonable informa-

tion could be extracted from samples c2 and c4. How-ever, c4 has more features, as depicted in Table I. Hence,sample c4 was the most candidate pattern to extract in-formation.

F. Proximity Analysis

To find out what is near or within a certain distanceof one or more features, we use a common geographicinformation system process that includes a buffer as atool that creates a new feature class of buffer polygonsaround a specified input feature based on some factor.Our factor is the reciprocal of nearest neighbor distance(NND). The value of NND is inversely proportional tothe density of the features and tells much about whether

11

FIG. 16. Plot of the L-Functions L1(r)and L2(r)for samplesa4, b4/1, c4, d4/1, e4, and f4.

FIG. 17. Proximity analysis diagram based on a buffer of re-ciprocal NND. Semitransparent circles show some correlatedfeatures.

data points are clustered or dispersed. In this aspect,Fig. 17 shows the proximity analysis based on that bufferobtained from the analysis of radial distribution.

The data of the proximity analysis in Fig. 17 showedlong-range correlational symmetry between regions ofclusters. Semitransparent circles show some correlatedfeatures having a dimension of diameter 100 micrometers.Similar clustering in the registered tracks has been ob-served in all samples. These are obvious in the heatmapsin Figs. 6, 8, and 10. The origin of such a correlation is

unknown. The present results throw doubt on the valid-ity on the random model of estimating the trajectory ofalpha particles, especially in the case of use of isotopicsource as an initiator for the neutron emitting reactions(e.g. 241AmBe neutron source [38]). In radiation detec-tion, the phenomenology of assessing the radioactivitymay be influenced by the configuration mixing betweenparticle decay and its interaction [39] and interfere withthe possible time varying-decay rates [40–50].

IV. CONCLUSION

The phenomenology of charged particle emission uponthe nuclear decay showed a spatial correlation among tra-jectories of the particle. The statistical inference meth-ods, specifically the information entropy, track proximityanalysis, and Kullback-Leibler divergence, was used totrack the information gained/lost due to the alpha par-ticles trajectories. The alpha particle from an 241Amsource have to pass through a thin film of gold and reg-istered at the end of its trajectory on the SSNTD. Allsources of a possible influence on the results were testedand eliminated from the experiment and the analysis.This includes the authentication of linearity in registra-tion efficiency with exposure time to avoid coalescenceof registered tracks, gravitational alinement, electromag-netic interferences, etc; so that all experimental param-eters were optimized to identify the best conditions ofexposure and etching to avoid misleading results. Theadaptive quadrates analysis of the spatial data showedthat the trajectories do not follow the null hypothesisof Poisson trajectories upon hitting the detector surface.Entropy and divergence analysis of the quadrate datawith the null hypothesis of Poisson distribution. Entropyanalysis showed information gained upon registration ofalpha particles tracks on SSNTD. The clustering and dis-persion analyses were undertaken with central deviationtendency, empirical K-function, radial distribution analy-sis, and proximity analysis. Results showed the existenceof pattern information within the registered tracks thatmay be attributed to the structure of the source mate-rials or coherence among emitted alpha particles. Thesource of correlation may be any process from the cre-ation of the alpha particle inside the 241Am nucleus, theelectron density in the source material (the deposited saltof Am) or its crystal structure, the electromagnetic in-teractions with atoms in encapsulation gold film, or evenresonance with an ambient electromagnetic field. One ofthe main consequences from the presented results is thepossibility of influencing the accuracy of the particle de-tectors and other experimental techniques that are usedin high-energy physics and nuclear physics, in which thecalibration is based on the randomness hypothesis.

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