random probability analysis of heavy-element data

*Corresponding author.E-mail address: [email protected] (N.J. Stoyer).

Nuclear Instruments and Methods in Physics Research A 455 (2000) 433}441

Random probability analysis of heavy-element data

N.J. Stoyer!,*, M.A. Stoyer!, J.F. Wild!, K.J. Moody!, R.W. Lougheed!,Yu.Ts. Oganessian", V.K. Utyonkov"

!Lawrence Livermore National Laboratory, University of California, Livermore, L-231, P.O.Box 808, Livermore, CA 94551, USA"Joint Institute for Nuclear Research, 141980 Dubna, Russian Federation

Received 12 January 2000; received in revised form 21 April 2000; accepted 22 April 2000

Abstract

In this paper, we present the Monte Carlo Random Probability (MCRP) calculational method details that weredeveloped for the determination of random correlations in a set of unrelated data. After "nding random correlations, wefurther process the correlations by applying nuclear property systematics. We compare the results of MCRP withmethods presented in other references. The MCRP method can provide a conservative estimate of the randomprobability associated with observed events that takes into account the entire background observed in the experimentand any other running conditions (noise, decay of long-lived species, etc.) which may have been sporadic or intermittent.We discuss a particular example of a set of correlated alpha decays and its interpretation as a candidate decaychain. ( 2000 Elsevier Science B.V. All rights reserved.

1. Introduction

A key question relating to the discovery of a newelement or isotope is the probability, P

%33, that the

event sequence observed is due to a random cor-relation of unrelated events. The magnitude of thisprobability allows readers and experimenters tojudge the validity of the interpretation, and is a ne-cessary argument for or against such a discovery.The method used for calculating this probabilitycan cause much debate among a collaboration orwithin the scienti"c community.

During the analysis of the data from the experi-ment described in Ref. [1], we had to determine theprobability that the decay sequence, which is at-

tributed to element 114, was, instead, due to a ran-dom correlation of unrelated events. For the eventsequence shown in Fig. 1, three di!erent methodswere used to calculate this probability, which in-clude two previously published methods as well asa new method which involved a Monte Carlo tech-nique. This paper will compare the results obtainedusing each of these methods and contain a thor-ough description of the new Monte Carlo method.

2. Description of experiment and data

The calculations described in the following sec-tions are from the data collected during theElement-114 experiment conducted at JINR inNovember and December 1998 by JINR in collab-oration with LLNL and described more completelyin Ref. [1]. In this experiment, we bombarded

0168-9002/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 8 - 9 0 0 2 ( 0 0 ) 0 0 5 0 3 - 9

Fig. 1. This is the event sequence for the 114 event as reported inRef. [1].

a 244Pu target with 236 MeV 48Ca ions at the U400heavy-ion cyclotron in Dubna. At that energy, thepeak cross section was expected to be the 3n evap-oration channel. A total of 5.2]1018 projectileswere delivered to the target.

The reaction products recoil out of the target, #ythrough the Dubna Gas-"lled Recoil Separator,where they pass through a time-of-#ight (TOF)system, and are implanted in the focal-plane de-tector. The focal-plane detector is a Si solid-statedetector that consists of 12 vertical strips with verti-cal position sensitivity. On the four sides surround-ing the Si detector array, similar detectors areplaced so that any decay products escaping out ofthe detector can also be detected. Behind this arraywas another detector for rejection of events whichwere not stopped in the focal-plane detector. Thedetector e$ciency is 87% for detection of a par-ticles, averaged over the position-sensitive array.The separator suppressed unwanted reaction prod-ucts a factor of 5105 and scattered beam by a fac-tor of 51015 but the event rate in the detector wasstill &15 Hz.

The data format for the experiment includedinformation about the particle identity, energy,position, TOF, and event-rejection-detector signal.Over the three and a half months of the experiment,196 "les containing both calibration data and ex-perimental reaction data were collected. Of these

196 "les, 91 "les contained 48 800 min of data onthe 48Ca#244Pu reaction used for the productionof element 114. The remainder were calibration "lesand a few "les that were not analyzed because someimportant part of the experimental setup was notworking properly.

3. Method descriptions and results

Each of the described methods will be applied tothe data set described in Section 2. For the newMonte Carlo Random Probability (MCRP)method, external criteria based on the Geiger}Nuttall (GN) relationship between a-particledecay energies and decay lifetimes will also beapplied.

3.1. GSI methods

There are two methods used by GSI described inRef. [2] by Karl-Heinz Schmidt. Both will be brief-ly described here, (for more details, see Ref. [2]).Provided the expectation value for the number ofevent chains due to background #uctuations ismuch less than one, a probability for the eventchain being random can be easily calculated fromEq. (3) in Ref. [2]. This equation is

P%33

+

nnmb

nm!

(1)

where nm

is the number of event chains observedand n

bis the expectation value for the complete

event chain assuming each of the events in thechain are independent of each other. The expecta-tion value for both methods is calculated usingthe counting rates for various types of events ofinterest, time limits for the correlation to occur,and the total time for the experiment. When there isone event sequence (i.e. n

m"1), then P

%33+n

b.

The "rst method is given by Eq. (5) in Ref. [2].This method assumes that the order of events isknown. The equation to calculate n

bis

nb"¹

<Ki/1

ji

(+Ki/1

ji)K~1

K~1<j/1

M1!e~+ Ki/1ji *tj,j`1N (2)

434 N.J. Stoyer et al. / Nuclear Instruments and Methods in Physics Research A 455 (2000) 433}441

1Throughout this paper a window will refer to a relative cut(e.g. $1.0 MeV of some energy) while a gate will refer to a "xedcut (e.g. 9.0}10.0 MeV).

where ji

is mean counting rate for event group(events of same type within a de"ned energy gate1)i, K is the number of di!erent event groups, *t istime interval between the successive events, and thevariable ¹ is the e!ective counting time (totalcounting time t times f ), which includes a positionfactor, f, the e!ective number of pixels that canrecord the events. In our experiment we estimate

f"12 strips]40 mm/strip

2.4 mm/pixel"200. (3)

With this method, a P%33

of 0.00010 is calculatedusing an 8.5}10.0 MeV gate for determining thecounting rate for the a groups and the appropriateenergy gate for the EVRs. The counting rate for SFswas calculated from all SF events observed withinan energy gate of 130}200 MeV. Because averagecounting rate is used, an assumption made is thatthe background is constant both in time and spaceover the whole detector system.

The second method used by GSI is a relatedcalculation given by Eq. (7) in Ref. [2]. This calcu-lation assumes only that the "rst event is followedby the other events in an unknown order. It wouldmake no sense to apply this equation to our eventsequence since a "ssion followed by an a is asunreasonable as an a followed by an EVR.

Event-rate method for entire detector system(Lazarev method)

This method was used in Ref. [3] and describedin a private communication [4]. This randomprobability calculation involves multiplying thenumber of EVR}a

1correlations within a given time

and position windows by the ratio of correlationsto start the events for each succeeding event, thus,for a "ve-member chain, EVR}a

1}a

2}a

3}SF, the

expectation value is

nb"C

EVR~a1]

Ca1~a2

Na1]

Ca2~a3Na2

]Ca3~SFNa3

(4)

where Nx

is number of x type events and Cx~y

isnumber of position correlations between x andy type events in a given time window. P

%33is cal-

culated from nb

as in Eq. (1). Using the describeddata set with an a energy gate of 8.5}10.0 MeV, anEVR gate as appropriate for the "le and the timeintervals given in Fig. 1 for the appropriate timewindows, and a position window of 1.2 mm for a}acorrelations and 2.0 mm for a}SF and EVR}acorrelations, we obtain a P

%33of 0.0041.

3.3. Monte Carlo method

Since other methods assume that the back-ground is constant over the duration of the experi-ment, as evidenced by using overall event rates ortotal number of events, we set out to generatea method that would take into account back-ground variations and possible hidden correlationswithin the data set. A Monte Carlo techniqueseemed to provide the right approach for thisundertaking. If we arti"cially introduce into thedata set a speci"c decay in the event sequence ofinterest that must be present based on either theor-etical or known decay properties, we can search forevent sequences correlated with that randomly in-troduced decay. The randomly correlated eventsequences could be interpreted in the same manneras the event sequence of interest. In this manner, wedetermine the probability that the event sequenceobserved is due to the random presence of thisde"ning decay.

Our data consisted of events that were recordedin list mode detailing the detector(s), magnitude ofsignals, position in the detector, TOF start andstop, and the time of the event. For each "le of data,a list was generated that gave the event type, focalplane detector number, position in the focal planedetector, energy deposited in the focal plane de-tector, energy deposited in the side detector, theevent time, and the TOF. Based on theoreticalpredictions of the decay modes for element 114 andits daughters, we assumed that any valid decaychain ends in a spontaneous "ssion. Using this listwe were able to insert into the data arti"cial "ssionsdistributed over the duration of the experiment andthe spatial extent of the focal plane detectors. Eacharti"cial "ssion was generated using a Monte Carlo

N.J. Stoyer et al. / Nuclear Instruments and Methods in Physics Research A 455 (2000) 433}441 435

Table 1Parameters used for the MCRP baseline calculation

Parameter Value

Ea1

8.5}10.0 MeVE

a28.5}10.0 MeV

Ea3

8.5}10.0 MeVEEVR

(First 30 "les) 9}18 MeVEEVR

(Last 61 "les) 6}15 MeVTotal position window 2.0 mmTotal time interval 2043.8 sNumber of RFs 105

Distribution type Flat, 3}37 mmRNG d1

technique to determine the time, detector number,and position in the detector. We used several di!er-ent spatial distributions to determine if there wasany correlation between the distribution of random"ssions (RF) and the calculated probability. It isthis random probability method that was used tocalculate the probability per "ssion of 0.006 re-ported in Ref. [1].

The search algorithm was based on the param-eters for decay chains of interest for element 114. Inorder to not de"ne the parameters too narrowly, wechose them such that predicted E

afor elements

with Z"110}114 with t1@2

between +0.1 s and+10 000 s were not excluded and that the candi-date element 114 event sequence was not excluded.We looked for three a's and an EVR preceding eachRF within a given time window, energy window,and position window. Our standard parameters aregiven in Table 1. The position correlation deter-mination was composed of three parts: (1) "ndinga's and EVR events that were within a reasonableposition window (2 mm) of the RF event, (2) check-ing that two sequential events met the 595%con"dence level for position correlation based oncalibration data, and (3) that the entire event se-quence was within the total position window (aparameter in the MCRP code).

In the search for the EVR and three a's precedinga RF, we performed two di!erent types of timewindow searches: total event time window or separ-ate time windows for each decay. A time windowfor the entire event means that from the RF onlylook at the preceding time window for a

3, a

2and

a1, then EVR. If separate time windows are used,

look for a3

in the a3}SF time window before the

RF, then for a2in the a

2}a

3time window before a

3,

then for a1

in the a1}a

2time window before a

2, and

last for the EVR in the EVR}a1

time windowbefore a

1. This latter method speci"es rather tightly

what the time history of the event sequence can be.In all cases, the search continues on to the nextevent as soon as the "rst candidate which meets thereasonable position window and E gate require-ments within the speci"ed time window is found.A simple test run was performed on the candidate`randomly correlateda chains to investigate thenumber of additional chains that would be found ifone a were arbitrarily ignored. As this correspondsto "nding chains consisting of an EVR and four a'scorrelated with a RF, this is expected to be a smallnumber of chains. Indeed, only 48 chains of anEVR, four a's, and a RF were found. Since threechains of an EVR, three a's, and a RF can beconstructed from these chains depending on thea that is ignored, there is less than a 25% increase inthe number of the EVR, three a's, RF chains. Itshould be noted that the majority of these newchains would be eliminated when GN conditionsdescribed are applied.

The Geiger}Nuttall relationship between Qaand

a-decay half-life was used to eliminate manya events and, thus, decay sequences, for which theimplied nuclear lifetime was either much too shortor much too long for the a-decay energy. When thealgorithm had detected a decay sequence consistingof an RF preceded by three a's and an EVR, thesignature of the decay sequence presented in Fig. 1,the Q

avalue was calculated from the a-decay en-

ergy for each a event. A time window was thenconstructed for each a event with which to test thevalidity of that a event. Using the Viola}Seaborgformula and parameters from SmolanH czuk [5], wecalculated a lifetime assuming a hindrance factor ofone. The lower limit of the time window was set toexclude that 15% of events with lifetimes shorterthan this value. The upper limit was determined byassuming a hindrance factor of 10 for the lifetimecalculated from the Q

a, and was set to exclude 15%

of a events whose lifetimes were longer than thatlimit. Any decay sequence whose a events all fellwithin this window was accepted as a `possiblea


Fig. 2. Graphical representation of the time window determinedusing our GN restrictions for a nuclide whose E

awith a HF"1

gives a t1@2

"1 (arbitrary unit).

Table 2Comparison of RF density by varying the number of RFs usedin the MCRP calculation. All other parameters were kept at thebaseline value

No. of RFs P%33

/"ssion (no GN) P%33

/"ssion (GN)

104 0.0066 0.00070105 0.0058 0.00056106 0.0057 0.00047

Table 3Comparison of two di!erent RNGs in the MCRP calculation.All other parameters were kept at the baseline value

RNG P%33


/"ssion (GN)(No.)

1 0.0058 0.000562 0.0061 0.00048

element 114 decay chain. A graphical representa-tion of this time window is shown in Fig. 2.

The baseline calculation gives a probability per"ssion of 0.0058 before the GN restrictions areapplied and 0.00056 after the GN restrictions areapplied. The parameters used are given in Table 1.The sensitivity of the results to a number of param-eters was investigated. All parameters were kept atthe baseline value except for the one for which thesensitivity was being investigated.

To be sure that there was not a dependence onthe density of RFs inserted into the data for thetime length of our search, we used sets that had 104,105, or 106 RFs. The results are shown in Table 2and indicate that the density of RFs did not makea signi"cant di!erence in the calculated probabilityfor these search parameters; therefore we per-formed the other parameter studies using 105 RFsunless otherwise noted.

Two di!erent pseudo-random number gener-ators (RNG) were used to test if the probabilityobtained was RNG dependent. The "rst RNG wasthe simple UNIX rand() routine which uses multi-plicative congruential random number generationwith a period of 232 and returns a pseudo-randomnumber (PRN). The second was a much simplerRNG based on the modulus of a changing numberdivided by a very large number and also returnsa PRN. We tested both RNG by looking at thedistribution of RNs between 0 and 1 returned as

a function of the number of RNs generated; indeed,both did generate distinct sets of PRN. A compari-son of the probability obtained using these twoRNG is shown in Table 3. The probability obtainedis shown to be RNG independent. All other calcu-lations were performed using the "rst RNG.

We know from histogramming the various typesof events that di!erent species produce di!erentdistributions over the detector array. We were con-cerned that the spatial distribution of the RFscould a!ect the probability calculation. There werenot enough true SFs to determine their spatialdistribution, so we studied the sensitivity of theMonte Carlo method to di!erent distributionsof RFs over the detector array. We used a #atdistribution over both time and space as the "rstdistribution because it is simple and we had nopreconceived notion of what the RF distributionshould resemble. Two di!erent #at distributionswere generated, the "rst was from 3 to 37 mm onthe strips while the second was 0 to 40 mm on thestrips. The top and bottom 3 mm were eliminatedin the "rst distribution because position determina-tion becomes problematic along the edges of thedetectors. Last, we looked at how the random ratewas a!ected by having the "ssions distributed #atin time and Gaussian in both strip number and


Table 4Comparison of the RF spatial distribution in the MCRP calcu-lation. All other parameters were kept at the baseline value

Distribution P%33


/"ssion (GN)

Flat, 3}37 mm 0.0058 0.00056Flat, 0}40 mm 0.0052 0.00050Gauss, both 0.0054 0.00042

Table 5Comparison of the MCRP calculation for di!erent E

aranges.

All other parameters were kept at the baseline value

a energies P%33


/"ssion (GN)

8.0}10.5 MeV 0.012 0.000208.0}10.0 MeV 0.011 0.000228.5}10.0 MeV 0.0058 0.000568.5}10.5 MeV 0.0077 0.00041

Table 6Comparison of MCRP calculation for di!erent total event timewindows. All other parameters kept at baseline values

Time window (s) P%33


/"ssion (GN)

1000.0 0.00094 0.000052043.8 0.0058 0.000563000.0 0.012 0.00104000.0 0.018 0.0016

10000.0 0.037 0.0031

top-to-bottom of a strip. Across the strip numbers(width of the detectors), the FWHM used was 12,with the center at 6.5, and normalized to 98.122%,a value which keeps the area under the Gaussian atthe desired number of RFs. From top-to-bottom ofa strip (height of the detector), the FWHM was 35mm, with the center at 20.0 mm, and normalized to99.285%. These parameters were derived fromAr#Dy to make Po experiments and are typical ofthe performance of the separator. These normaliz-ation percentages are integral values at 2.354p (i.e.FWHM). The probabilities obtained for these vari-ous distributions are shown in Table 4. Since thereis no substantial e!ect, we chose to use the simple#at distribution from 3 to 37 mm on the strips forthe remaining calculations.

We looked at the variation in the random ratewith di!erent a energy ranges for the a-particles ofinterest. The probabilities obtained for these energyranges are shown in Table 5. Wider energy win-dows generate higher P

%33before GN criteria are

applied but lower rates after. Since we are searchingfor an event sequence of EVR}a

1}a

2}a

3}RF, dif-

ferent Ea

gates will result in di!erent event se-quences being found because di!erent a-particleswill be allowed. The lower energy a events wouldindicate a longer lifetime which would not be rea-sonable given the time constraints applied usingGN criteria. It is important to choose reasonableenergy windows for the a events of interest.

We also looked at how the random rate variedwith di!erent event sequence durations. The ran-dom rate achieved its maximum at about 95 300s with a P

%33/"ssion of 0.045 before GN and 0.0038

after GN restrictions are applied. For this we as-sumed that all data "les were separate runs andthat any events that occurred in one "le were notrelated to those in another "le. Since several "leswere such that one run was ended and another was

immediately started, more elaborate algorithmscould be implemented to couple and then searchthese two "les. This was not done in this work. Theresults of the probability calculations are shown inTable 6. As expected, longer times give higher P

%33,

roughly proportional to the length of the time win-dow. The time window that is appropriate corres-ponds to the length of time for the event sequenceof interest. If the event sequence were longer theprobability would then be higher that it was a ran-dom event; conversely, if the event sequence wereshorter the P

%33would be smaller.

The width of the total event sequence positionwindow was also varied. The results from this vari-ation are shown in Table 7. As one would expect,narrower windows give somewhat smaller P

%33. At

some point, the total event sequence position win-dow no longer has much e!ect because of the addi-tional criteria imposed for position correlation (seethe above discussion on position correlation).

4. Comparison

For two speci"c sets of parameters, we compareall methods. For this comparison, all data "les that


Table 7Comparison of the MCRP calculation as a function of positionwindow width for the total event sequence. All other parameterskept at baseline values

Position window(mm)

P%33

/"ssion(no GN)

P%33

/"ssion(GN)

1.6 0.0051 0.000461.8 0.0055 0.000502.0 0.0058 0.000562.2 0.0058 0.00056

Table 8Parameters for the comparison of the P

%33calculations. Set 1 is

the baseline parameters

Parameter Set 1 Set 2

Ea1

8.5}10.0 MeV 9.1}10.5 MeVE

a28.5}10.0 MeV 8.1}9.9 MeV

Ea3

8.5}10.0 MeV 8.1}9.7 MeVEEVR

(First 30 "les) 9}18 MeV 9}18 MeVEEVR

(Last 61 "les) 6}15 MeV 6}15 MeVTotal position window 2.0 mm 2.0 mmEVR}a

1time interval! * 30.4 s

a1}a


a2}a


a3}Fission time interval! * 992.8 s

!Used total time from EVR to Fission of 2043.8 s.

Table 9Comparison of the methods described in this paper. The param-eters are detailed in Table 8. An attempt was made to have theP%33

calculations for each method based on the same parameterswhenever possible for each of the parameter sets; however,approximations had to be made in some cases because of meth-odology

Method Parameter Set 1 Parameter Set 2

GSI Method! 0.00010# 0.00012Lazarev Method" 0.0041# 0.0032MCRP without GN 0.0058 0.000048$

MCRP with GN 0.00056 410~6 $

!No position window speci"ed except through f."Position windows where 1.2 for a}a and 2.0 mm for EVR}a

1or a

3}SF correlations.

#Time intervals necessary, used Parameter Set 2 values.$Used 106 RFs to get reasonable statistics for MCRP withoutGN.

contained 48Ca#244Pu reaction data from 1998were used. See Table 8 for the parameters used andTable 9 for the results.

MCRP can have either an event window for theentire event sequence or time intervals for eachevent. The former is more representative of P

%33for

a reasonable event sequence, while the latter fora similarly timed event sequence. The change inparameters from Set 1 to Set 2 is largely a timewindow change; in MCRP there is two or moreorders of magnitude di!erence in the probabilities.Thus, specifying the time history is very limiting.Likewise, restricting the E

ato tight energy gates

also de"nes the event sequence as having similarenergies rather than having reasonable energies.With more than one event sequence, such limita-tions can be imposed.

For parameter Set 1, we "nd that MCRP with-out GN and Lazarev Method are comparable whileMCRP with GN and GSI Methods are comparableand about an order of magnitude lower. Since theGSI Method uses products of poisson distribu-tions, while Lazarev Method uses products of num-bers of correlations and event types, it is reasonablethat the former would approximate the GN limita-tions used in MCRP better than the latter. Al-though the GSI method does not include the GNcriteria explicitly, it does include it super"ciallywhen the event sequence under analysis meets GNcriteria by virtue of the poisson function that utiliz-es the time interval between events. Since the GNcriteria is such a good correlation between thea-energy and hal#ife for even}even isotopes anda good guide for even Z-odd A isotopes, events thatfall too far o! the correlation will be met with much

more suspicion and likely be viewed as unreason-able without considerable explanation. It is inter-esting to note that the two very di!erent methodsgive similar results for the P

%33.

For parameter Set 2, neither the GSI Methodnor Lazarev Method change much; both essentiallyhave only minor changes to the E

a's used for the

three di!erent a particles. MCRP has a dramaticchange because individual time windows are usedin addition to the E

achanges. The MCRP results

are much lower than both the GSI and LazarevMethods. One major di!erence between the GSI


Method and MCRP is correlation determination;GSI Method uses event rates for the various eventsto determine `time correlationsa and the f factor todetermine `position correlationsa while MCRPlooks for actual position and time correlations be-tween events in the data. The Lazarev Methodlooks for time and position correlations betweenvarious pairs of events and assumes that the ratio ofcorrelations to start events is constant over theduration of the experiment. One interpretation ofthis apparent discrepancy between MCRP and theGSI and Lazarev Methods for parameter Set 2 isthat while GSI and Lazarev Methods are at most"nding true correlations between event pairs,MCRP is looking for a correlation between "veevents in a speci"c time sequence with speci"c ener-gies. In more popular terms, we have gone fromlooking for a winning bridge hand to a bridge handthat contains the top 2, 3, or 4 cards in each suit;there are many more combinations that makea winning bridge hand than there are with the top2, 3, or 4 cards in each suit. These speci"c con-straints should not be applied without some scien-ti"c justi"cation. Such speci"city will allow one todetermine the probability of "nding an event se-quence exactly like the one which exists rather thanone that would be interpreted as arising from thedecay of the same nuclide. MCRP is much moresensitive to such changes because of the "ve-foldcorrelation that is needed. Because MCRP is look-ing for actual correlations in the data, it is alsomore sensitive to background #uctuations than theother two methods that use average event rates ortotal event counts.

5. Conclusion

During the analysis of the event sequence repre-sented in Fig. 1, the question of random probabilitywas broached. Since the running time was long, themagnitude of background #uctuations was of con-cern. Some dissatisfaction with the current pub-lished methods and their handling of backgroundswas expressed. We decided to tackle the back-ground #uctuation problem by using the actualdata to construct decay sequences using RFs in-serted into the data. From these sequences it was

clear that many would not be reasonable candidatedecays, so we added the pertinent physics by usingGN criteria. We created a method that works withthe background present and includes the physicsthat the event sequence will be judged with; there-fore, in Ref. [1] we reported the MCRP (with GNincluded) value of 0.006.

MCRP is a useful tool in evaluating the P%33

fora set of data, where the event sequence has a uniquede"ning feature. Hidden correlations should be dis-covered for the event sequence of interest. Havingone event sequence, means that the criteria forE

aand time interval(s) need to be somewhat wider,

since you are trying to determine what the prob-ability is of "nding an event sequence randomlythat would be interpreted in the same way as theevent sequence found. If more event sequences arefound, narrower energy and time criteria can beapplied since both the lifetime and decay energieswould be de"ned better. This would result in lowerP%33

.MCRP has the advantage that the data set with

all of its hidden correlations is used during therandom probability calculation. It does not as-sume, like the GSI method, that the background isconstant at the average counting rate for the dura-tion of the experiment. A disadvantage of MCRP isthe necessity of having a uniquely de"ned decay inthe event sequence. Both the GSI and LazarevMethods have no such requirement and would beuseful even in situations where a de"ning decay isnot present.

MCRP, like most tools, needs to be applied withknowledge of its limitations and usefulness. It hasthe potential to "nd hidden correlations in the dataset being analyzed while giving a reasonablemeasure of the random probability associated withthe decay sequence of interest. The minimum P

%33,

which can be calculated using this method, has aslimiting factors the computer speed and storagecapabilities available.

Acknowledgements

The work at LLNL was performed under theauspices of the US Department of Energy underContract No. W-7405-ENG-48. This work has


been performed with the support of the RussianFoundation for Basic Research under Grant No.96-02-17377 and of INTAS under Grant No. 96-662. These studies were performed in theframework of the Russian Federation/US Joint Co-ordinating Committee for Research on Funda-mental Properties of Matter.

References

[1] Yu.Ts. Oganessian et al., Phys. Rev. Lett. 83 (1999) 3154.[2] K.-H. Schmidt, C.-C. Sahm, K. Pielenz, H.-G. Clerc, Z.

Phys. A, Atoms Nucl. 316 (1984) 19}26.[3] Yu.A. Lazarev et al., Phys. Rev. C 54 (1996) 620.[4] Yu.A. Lazarev, private communication, March 1995.[5] R. SmolanH czuk, Phys. Rev. C 56 (1997) 812.


random probability analysis of heavy-element data

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