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Random probability analysis of heavy-element data
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 were developed for the determination of random correlations in a set of unrelated data. After "nding random correlations, we further process the correlations by applying nuclear property systematics. We compare the results of MCRP with methods presented in other references. The MCRP method can provide a conservative estimate of the random probability associated with observed events that takes into account the entire background observed in the experiment and 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 decay chain. 2000 Elsevier Science B.V. All rights reserved.
1. Introduction A key question relating to the discovery of a new element or isotope is the probability, P , that the event sequence observed is due to a random correlation of unrelated events. The magnitude of this probability allows readers and experimenters to judge the validity of the interpretation, and is a necessary argument for or against such a discovery. The method used for calculating this probability can cause much debate among a collaboration or within the scienti"c community. During the analysis of the data from the experiment described in Ref. [1], we had to determine the probability that the decay sequence, which is at-
* Corresponding author. E-mail address: [email protected] (N.J. Stoyer).
tributed to element 114, was, instead, due to a random correlation of unrelated events. For the event sequence shown in Fig. 1, three di!erent methods were used to calculate this probability, which include two previously published methods as well as a new method which involved a Monte Carlo technique. This paper will compare the results obtained using each of these methods and contain a thorough description of the new Monte Carlo method.
2. Description of experiment and data The calculations described in the following sections are from the data collected during the Element-114 experiment conducted at JINR in November and December 1998 by JINR in collaboration with LLNL and described more completely in Ref. [1]. In this experiment, we bombarded
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196 "les, 91 "les contained 48 800 min of data on the Ca#Pu reaction used for the production of element 114. The remainder were calibration "les and a few "les that were not analyzed because some important part of the experimental setup was not working properly.
3. Method descriptions and results
Fig. 1. This is the event sequence for the 114 event as reported in Ref. [1].
a Pu target with 236 MeV Ca ions at the U400 heavy-ion cyclotron in Dubna. At that energy, the peak cross section was expected to be the 3n evaporation channel. A total of 5.2;10 projectiles were delivered to the target. The reaction products recoil out of the target, #y through the Dubna Gas-"lled Recoil Separator, where they pass through a time-of-#ight (TOF) system, and are implanted in the focal-plane detector. The focal-plane detector is a Si solid-state detector that consists of 12 vertical strips with vertical position sensitivity. On the four sides surrounding the Si detector array, similar detectors are placed so that any decay products escaping out of the detector can also be detected. Behind this array was another detector for rejection of events which were not stopped in the focal-plane detector. The detector e$ciency is 87% for detection of a particles, averaged over the position-sensitive array. The separator suppressed unwanted reaction products a factor of 510 and scattered beam by a factor of 510 but the event rate in the detector was still &15 Hz. The data format for the experiment included information 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 experimental reaction data were collected. Of these
Each of the described methods will be applied to the data set described in Section 2. For the new Monte Carlo Random Probability (MCRP) method, external criteria based on the Geiger} Nuttall (GN) relationship between a-particle decay energies and decay lifetimes will also be applied. 3.1. GSI methods There are two methods used by GSI described in Ref. [2] by Karl-Heinz Schmidt. Both will be briefly described here, (for more details, see Ref. [2]). Provided the expectation value for the number of event chains due to background #uctuations is much less than one, a probability for the event chain being random can be easily calculated from Eq. (3) in Ref. [2]. This equation is nLK P + @ n ! K
(1)
where n is the number of event chains observed K and n is the expectation value for the complete @ event chain assuming each of the events in the chain are independent of each other. The expectation value for both methods is calculated using the counting rates for various types of events of interest, time limits for the correlation to occur, and the total time for the experiment. When there is one event sequence (i.e. n "1), then P +n . K @ The "rst method is given by Eq. (5) in Ref. [2]. This method assumes that the order of events is known. The equation to calculate n is @ ) j )\ ) G G n "¹ +1!e\ G HG RH H> , @ ( )G jG ))\ H
(2)
N.J. Stoyer et al. / Nuclear Instruments and Methods in Physics Research A 455 (2000) 433}441
where j is mean counting rate for event group G (events of same type within a de"ned energy gate) i, K is the number of di!erent event groups, *t is time interval between the successive events, and the variable ¹ is the e!ective counting time (total counting time t times f ), which includes a position factor, f, the e!ective number of pixels that can record the events. In our experiment we estimate 40 mm/strip f"12 strips; "200. 2.4 mm/pixel
(3)
With this method, a P of 0.00010 is calculated using an 8.5}10.0 MeV gate for determining the counting rate for the a groups and the appropriate energy gate for the EVRs. The counting rate for SFs was calculated from all SF events observed within an energy gate of 130}200 MeV. Because average counting rate is used, an assumption made is that the background is constant both in time and space over the whole detector system. The second method used by GSI is a related calculation given by Eq. (7) in Ref. [2]. This calculation assumes only that the "rst event is followed by the other events in an unknown order. It would make no sense to apply this equation to our event sequence since a "ssion followed by an a is as unreasonable as an a followed by an EVR. Event-rate method for entire detector system (Lazarev method) This method was used in Ref. [3] and described in a private communication [4]. This random probability calculation involves multiplying the number of EVR}a correlations within a given time and position windows by the ratio of correlations to start the events for each succeeding event, thus, for a "ve-member chain, EVR}a }a }a }SF, the expectation value is C C C ; ? \a ; ? \? ; ? \1$ n "C @ #40\a N N N ? ? ?
(4)
Throughout 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 "xed cut (e.g. 9.0}10.0 MeV).
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where N is number of x type events and C is V V\W number of position correlations between x and y type events in a given time window. P is cal culated from n as in Eq. (1). Using the described @ data set with an a energy gate of 8.5}10.0 MeV, an EVR gate as appropriate for the "le and the time intervals given in Fig. 1 for the appropriate time windows, and a position window of 1.2 mm for a}a correlations and 2.0 mm for a}SF and EVR}a correlations, we obtain a P of 0.0041. 3.3. Monte Carlo method Since other methods assume that the background is constant over the duration of the experiment, as evidenced by using overall event rates or total number of events, we set out to generate a method that would take into account background variations and possible hidden correlations within the data set. A Monte Carlo technique seemed to provide the right approach for this undertaking. If we arti"cially introduce into the data set a speci"c decay in the event sequence of interest that must be present based on either theoretical or known decay properties, we can search for event sequences correlated with that randomly introduced decay. The randomly correlated event sequences could be interpreted in the same manner as the event sequence of interest. In this manner, we determine the probability that the event sequence observed is due to the random presence of this de"ning decay. Our data consisted of events that were recorded in list mode detailing the detector(s), magnitude of signals, position in the detector, TOF start and stop, and the time of the event. For each "le of data, a list was generated that gave the event type, focal plane detector number, position in the focal plane detector, energy deposited in the focal plane detector, energy deposited in the side detector, the event time, and the TOF. Based on theoretical predictions of the decay modes for element 114 and its daughters, we assumed that any valid decay chain ends in a spontaneous "ssion. Using this list we were able to insert into the data arti"cial "ssions distributed over the duration of the experiment and the spatial extent of the focal plane detectors. Each arti"cial "ssion was generated using a Monte Carlo
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technique to determine the time, detector number, and position in the detector. We used several di!erent spatial distributions to determine if there was any correlation between the distribution of random "ssions (RF) and the calculated probability. It is this random probability method that was used to calculate the probability per "ssion of 0.006 reported in Ref. [1]. The search algorithm was based on the parameters for decay chains of interest for element 114. In order to not de"ne the parameters too narrowly, we chose them such that predicted E for elements a with Z"110}114 with t between +0.1 s and +10 000 s were not excluded and that the candidate element 114 event sequence was not excluded. We looked for three a's and an EVR preceding each RF within a given time window, energy window, and position window. Our standard parameters are given in Table 1. The position correlation determination was composed of three parts: (1) "nding a's and EVR events that were within a reasonable position window (2 mm) of the RF event, (2) checking that two sequential events met the 595% con"dence level for position correlation based on calibration data, and (3) that the entire event sequence was within the total position window (a parameter in the MCRP code). In the search for the EVR and three a's preceding a RF, we performed two di!erent types of time window searches: total event time window or separate time windows for each decay. A time window for the entire event means that from the RF only look at the preceding time window for a , a and Table 1 Parameters used for the MCRP baseline calculation Parameter
Value
E a E a E a E (First 30 "les) #40 E (Last 61 "les) #40 Total position window Total time interval Number of RFs Distribution type RNG
8.5}10.0 MeV 8.5}10.0 MeV 8.5}10.0 MeV 9}18 MeV 6}15 MeV 2.0 mm 2043.8 s 10 Flat, 3}37 mm C1
a , then EVR. If separate time windows are used, look for a in the a }SF time window before the RF, then for a in the a }a time window before a , then for a in the a }a time window before a , and last for the EVR in the EVR}a time window before a . 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 next event as soon as the "rst candidate which meets the reasonable position window and E gate requirements within the speci"ed time window is found. A simple test run was performed on the candidate `randomly correlateda chains to investigate the number of additional chains that would be found if one a were arbitrarily ignored. As this corresponds to "nding chains consisting of an EVR and four a's correlated with a RF, this is expected to be a small number of chains. Indeed, only 48 chains of an EVR, four a's, and a RF were found. Since three chains of an EVR, three a's, and a RF can be constructed from these chains depending on the a that is ignored, there is less than a 25% increase in the number of the EVR, three a's, RF chains. It should be noted that the majority of these new chains would be eliminated when GN conditions described are applied. The Geiger}Nuttall relationship between Q and a a-decay half-life was used to eliminate many a events and, thus, decay sequences, for which the implied nuclear lifetime was either much too short or much too long for the a-decay energy. When the algorithm had detected a decay sequence consisting of an RF preceded by three a's and an EVR, the signature of the decay sequence presented in Fig. 1, the Q value was calculated from the a-decay ena ergy for each a event. A time window was then constructed for each a event with which to test the validity of that a event. Using the Viola}Seaborg formula and parameters from SmolanH czuk [5], we calculated a lifetime assuming a hindrance factor of one. The lower limit of the time window was set to exclude that 15% of events with lifetimes shorter than this value. The upper limit was determined by assuming a hindrance factor of 10 for the lifetime calculated from the Q , and was set to exclude 15% a of a events whose lifetimes were longer than that limit. Any decay sequence whose a events all fell within this window was accepted as a `possiblea
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element 114 decay chain. A graphical representation of this time window is shown in Fig. 2. The baseline calculation gives a probability per "ssion of 0.0058 before the GN restrictions are applied and 0.00056 after the GN restrictions are applied. The parameters used are given in Table 1. The sensitivity of the results to a number of parameters was investigated. All parameters were kept at the baseline value except for the one for which the sensitivity was being investigated. To be sure that there was not a dependence on the density of RFs inserted into the data for the time length of our search, we used sets that had 10, 10, or 10 RFs. The results are shown in Table 2 and indicate that the density of RFs did not make a signi"cant di!erence in the calculated probability for these search parameters; therefore we performed the other parameter studies using 10 RFs unless otherwise noted. Two di!erent pseudo-random number generators (RNG) were used to test if the probability obtained was RNG dependent. The "rst RNG was the simple UNIX rand() routine which uses multiplicative congruential random number generation with a period of 2 and returns a pseudo-random number (PRN). The second was a much simpler RNG based on the modulus of a changing number divided by a very large number and also returns a PRN. We tested both RNG by looking at the distribution of RNs between 0 and 1 returned as
Fig. 2. Graphical representation of the time window determined using our GN restrictions for a nuclide whose E with a HF"1 a gives a t "1 (arbitrary unit).
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Table 2 Comparison of RF density by varying the number of RFs used in the MCRP calculation. All other parameters were kept at the baseline value No. of RFs
P /"ssion (no GN)
P /"ssion (GN)
10 10 10
0.0066 0.0058 0.0057
0.00070 0.00056 0.00047
Table 3 Comparison of two di!erent RNGs in the MCRP calculation. All other parameters were kept at the baseline value RNG (No.)
P /"ssion (no GN)
P /"ssion (GN)
1 2
0.0058 0.0061
0.00056 0.00048
a function of the number of RNs generated; indeed, both did generate distinct sets of PRN. A comparison of the probability obtained using these two RNG is shown in Table 3. The probability obtained is shown to be RNG independent. All other calculations were performed using the "rst RNG. We know from histogramming the various types of events that di!erent species produce di!erent distributions over the detector array. We were concerned that the spatial distribution of the RFs could a!ect the probability calculation. There were not enough true SFs to determine their spatial distribution, so we studied the sensitivity of the Monte Carlo method to di!erent distributions of RFs over the detector array. We used a #at distribution over both time and space as the "rst distribution because it is simple and we had no preconceived notion of what the RF distribution should resemble. Two di!erent #at distributions were generated, the "rst was from 3 to 37 mm on the strips while the second was 0 to 40 mm on the strips. The top and bottom 3 mm were eliminated in the "rst distribution because position determination becomes problematic along the edges of the detectors. Last, we looked at how the random rate was a!ected by having the "ssions distributed #at in time and Gaussian in both strip number and
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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 at the desired number of RFs. From top-to-bottom of a strip (height of the detector), the FWHM was 35 mm, with the center at 20.0 mm, and normalized to 99.285%. These parameters were derived from Ar#Dy to make Po experiments and are typical of the performance of the separator. These normalization percentages are integral values at 2.354p (i.e. FWHM). The probabilities obtained for these various distributions are shown in Table 4. Since there is no substantial e!ect, we chose to use the simple #at distribution from 3 to 37 mm on the strips for the remaining calculations. We looked at the variation in the random rate with di!erent a energy ranges for the a-particles of interest. The probabilities obtained for these energy ranges are shown in Table 5. Wider energy windows generate higher P before GN criteria are applied but lower rates after. Since we are searching for an event sequence of EVR}a }a }a }RF, dif ferent E gates will result in di!erent event sea quences being found because di!erent a-particles will be allowed. The lower energy a events would indicate a longer lifetime which would not be reasonable given the time constraints applied using GN criteria. It is important to choose reasonable energy windows for the a events of interest. We also looked at how the random rate varied with di!erent event sequence durations. The random rate achieved its maximum at about 95 300 s with a P /"ssion of 0.045 before GN and 0.0038 after GN restrictions are applied. For this we assumed that all data "les were separate runs and that any events that occurred in one "le were not related to those in another "le. Since several "les were such that one run was ended and another was Table 4 Comparison of the RF spatial distribution in the MCRP calculation. All other parameters were kept at the baseline value Distribution
P /"ssion (no GN)
P /"ssion (GN)
Flat, 3}37 mm Flat, 0}40 mm Gauss, both
0.0058 0.0052 0.0054
0.00056 0.00050 0.00042
Table 5 Comparison of the MCRP calculation for di!erent E ranges. a All other parameters were kept at the baseline value a energies
P /"ssion (no GN)
P /"ssion (GN)
8.0}10.5 8.0}10.0 8.5}10.0 8.5}10.5
0.012 0.011 0.0058 0.0077
0.00020 0.00022 0.00056 0.00041
MeV MeV MeV MeV
Table 6 Comparison of MCRP calculation for di!erent total event time windows. All other parameters kept at baseline values Time window (s)
P /"ssion (no GN)
P /"ssion (GN)
1000.0 2043.8 3000.0 4000.0 10000.0
0.00094 0.0058 0.012 0.018 0.037
0.00005 0.00056 0.0010 0.0016 0.0031
immediately started, more elaborate algorithms could be implemented to couple and then search these two "les. This was not done in this work. The results of the probability calculations are shown in Table 6. As expected, longer times give higher P , roughly proportional to the length of the time window. The time window that is appropriate corresponds to the length of time for the event sequence of interest. If the event sequence were longer the probability would then be higher that it was a random event; conversely, if the event sequence were shorter the P would be smaller. The width of the total event sequence position window was also varied. The results from this variation are shown in Table 7. As one would expect, narrower windows give somewhat smaller P . At some point, the total event sequence position window no longer has much e!ect because of the additional criteria imposed for position correlation (see the above discussion on position correlation).
4. Comparison For two speci"c sets of parameters, we compare all methods. For this comparison, all data "les that
N.J. Stoyer et al. / Nuclear Instruments and Methods in Physics Research A 455 (2000) 433}441 Table 7 Comparison of the MCRP calculation as a function of position window width for the total event sequence. All other parameters kept at baseline values Position window (mm)
P /"ssion (no GN)
P /"ssion (GN)
1.6 1.8 2.0 2.2
0.0051 0.0055 0.0058 0.0058
0.00046 0.00050 0.00056 0.00056
contained Ca#Pu reaction data from 1998 were used. See Table 8 for the parameters used and Table 9 for the results. MCRP can have either an event window for the entire event sequence or time intervals for each event. The former is more representative of P for a reasonable event sequence, while the latter for a similarly timed event sequence. The change in parameters from Set 1 to Set 2 is largely a time window change; in MCRP there is two or more orders of magnitude di!erence in the probabilities. Thus, specifying the time history is very limiting. Likewise, restricting the E to tight energy gates a also de"nes the event sequence as having similar energies rather than having reasonable energies. With more than one event sequence, such limitations can be imposed. For parameter Set 1, we "nd that MCRP without GN and Lazarev Method are comparable while MCRP with GN and GSI Methods are comparable and about an order of magnitude lower. Since the GSI Method uses products of poisson distributions, while Lazarev Method uses products of numbers of correlations and event types, it is reasonable that the former would approximate the GN limitations used in MCRP better than the latter. Although the GSI method does not include the GN criteria explicitly, it does include it super"cially when the event sequence under analysis meets GN criteria by virtue of the poisson function that utilizes the time interval between events. Since the GN criteria is such a good correlation between the a-energy and hal#ife for even}even isotopes and a good guide for even Z-odd A isotopes, events that fall too far o! the correlation will be met with much
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Table 8 Parameters for the comparison of the P calculations. Set 1 is the baseline parameters Parameter
Set 1
Set 2
E a E a E a E (First 30 "les) #40 E (Last 61 "les) #40 Total position window EVR}a time interval a }a time interval a }a time interval a }Fission time interval
8.5}10.0 MeV 8.5}10.0 MeV 8.5}10.0 MeV 9}18 MeV 6}15 MeV 2.0 mm * * * *
9.1}10.5 MeV 8.1}9.9 MeV 8.1}9.7 MeV 9}18 MeV 6}15 MeV 2.0 mm 30.4 s 925.8 s 94.7 s 992.8 s
Used total time from EVR to Fission of 2043.8 s.
Table 9 Comparison of the methods described in this paper. The parameters are detailed in Table 8. An attempt was made to have the P calculations for each method based on the same parameters whenever possible for each of the parameter sets; however, approximations had to be made in some cases because of methodology Method
Parameter Set 1
Parameter Set 2
GSI Method Lazarev Method MCRP without GN MCRP with GN
0.00010 0.0041 0.0058 0.00056
0.00012 0.0032 0.000048 410\
No position window speci"ed except through f. Position windows where 1.2 for a}a and 2.0 mm for EVR}a or a }SF correlations. Time intervals necessary, used Parameter Set 2 values. Used 10 RFs to get reasonable statistics for MCRP without GN.
more suspicion and likely be viewed as unreasonable without considerable explanation. It is interesting to note that the two very di!erent methods give similar results for the P . For parameter Set 2, neither the GSI Method nor Lazarev Method change much; both essentially have only minor changes to the E 's used for the a three di!erent a particles. MCRP has a dramatic change because individual time windows are used in addition to the E changes. The MCRP results a are much lower than both the GSI and Lazarev Methods. One major di!erence between the GSI
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Method and MCRP is correlation determination; GSI Method uses event rates for the various events to determine `time correlationsa and the f factor to determine `position correlationsa while MCRP looks for actual position and time correlations between events in the data. The Lazarev Method looks for time and position correlations between various pairs of events and assumes that the ratio of correlations to start events is constant over the duration of the experiment. One interpretation of this apparent discrepancy between MCRP and the GSI and Lazarev Methods for parameter Set 2 is that while GSI and Lazarev Methods are at most "nding true correlations between event pairs, MCRP is looking for a correlation between "ve events in a speci"c time sequence with speci"c energies. In more popular terms, we have gone from looking for a winning bridge hand to a bridge hand that contains the top 2, 3, or 4 cards in each suit; there are many more combinations that make a winning bridge hand than there are with the top 2, 3, or 4 cards in each suit. These speci"c constraints should not be applied without some scienti"c justi"cation. Such speci"city will allow one to determine the probability of "nding an event sequence exactly like the one which exists rather than one that would be interpreted as arising from the decay of the same nuclide. MCRP is much more sensitive to such changes because of the "ve-fold correlation that is needed. Because MCRP is looking for actual correlations in the data, it is also more sensitive to background #uctuations than the other two methods that use average event rates or total event counts.
5. Conclusion During the analysis of the event sequence represented in Fig. 1, the question of random probability was broached. Since the running time was long, the magnitude of background #uctuations was of concern. Some dissatisfaction with the current published methods and their handling of backgrounds was expressed. We decided to tackle the background #uctuation problem by using the actual data to construct decay sequences using RFs inserted into the data. From these sequences it was
clear that many would not be reasonable candidate decays, so we added the pertinent physics by using GN criteria. We created a method that works with the background present and includes the physics that the event sequence will be judged with; therefore, in Ref. [1] we reported the MCRP (with GN included) value of 0.006. MCRP is a useful tool in evaluating the P for a set of data, where the event sequence has a unique de"ning feature. Hidden correlations should be discovered for the event sequence of interest. Having one event sequence, means that the criteria for E and time interval(s) need to be somewhat wider, a since you are trying to determine what the probability is of "nding an event sequence randomly that would be interpreted in the same way as the event sequence found. If more event sequences are found, narrower energy and time criteria can be applied since both the lifetime and decay energies would be de"ned better. This would result in lower P . MCRP has the advantage that the data set with all of its hidden correlations is used during the random probability calculation. It does not assume, like the GSI method, that the background is constant at the average counting rate for the duration of the experiment. A disadvantage of MCRP is the necessity of having a uniquely de"ned decay in the event sequence. Both the GSI and Lazarev Methods have no such requirement and would be useful even in situations where a de"ning decay is not present. MCRP, like most tools, needs to be applied with knowledge of its limitations and usefulness. It has the potential to "nd hidden correlations in the data set being analyzed while giving a reasonable measure of the random probability associated with the decay sequence of interest. The minimum P , which can be calculated using this method, has as limiting factors the computer speed and storage capabilities available.
Acknowledgements The work at LLNL was performed under the auspices of the US Department of Energy under Contract No. W-7405-ENG-48. This work has
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been performed with the support of the Russian Foundation for Basic Research under Grant No. 96-02-17377 and of INTAS under Grant No. 96662. These studies were performed in the framework of the Russian Federation/US Joint Coordinating Committee for Research on Fundamental Properties of Matter.
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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 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 were developed for the determination of random correlations in a set of unrelated data. After "nding random correlations, we further process the correlations by applying nuclear property systematics. We compare the results of MCRP with methods presented in other references. The MCRP method can provide a conservative estimate of the random probability associated with observed events that takes into account the entire background observed in the experiment and 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 decay chain. 2000 Elsevier Science B.V. All rights reserved.
1. Introduction A key question relating to the discovery of a new element or isotope is the probability, P , that the event sequence observed is due to a random correlation of unrelated events. The magnitude of this probability allows readers and experimenters to judge the validity of the interpretation, and is a necessary argument for or against such a discovery. The method used for calculating this probability can cause much debate among a collaboration or within the scienti"c community. During the analysis of the data from the experiment described in Ref. [1], we had to determine the probability that the decay sequence, which is at-
* Corresponding author. E-mail address: [email protected] (N.J. Stoyer).
tributed to element 114, was, instead, due to a random correlation of unrelated events. For the event sequence shown in Fig. 1, three di!erent methods were used to calculate this probability, which include two previously published methods as well as a new method which involved a Monte Carlo technique. This paper will compare the results obtained using each of these methods and contain a thorough description of the new Monte Carlo method.
2. Description of experiment and data The calculations described in the following sections are from the data collected during the Element-114 experiment conducted at JINR in November and December 1998 by JINR in collaboration with LLNL and described more completely in 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
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196 "les, 91 "les contained 48 800 min of data on the Ca#Pu reaction used for the production of element 114. The remainder were calibration "les and a few "les that were not analyzed because some important part of the experimental setup was not working properly.
3. Method descriptions and results
Fig. 1. This is the event sequence for the 114 event as reported in Ref. [1].
a Pu target with 236 MeV Ca ions at the U400 heavy-ion cyclotron in Dubna. At that energy, the peak cross section was expected to be the 3n evaporation channel. A total of 5.2;10 projectiles were delivered to the target. The reaction products recoil out of the target, #y through the Dubna Gas-"lled Recoil Separator, where they pass through a time-of-#ight (TOF) system, and are implanted in the focal-plane detector. The focal-plane detector is a Si solid-state detector that consists of 12 vertical strips with vertical position sensitivity. On the four sides surrounding the Si detector array, similar detectors are placed so that any decay products escaping out of the detector can also be detected. Behind this array was another detector for rejection of events which were not stopped in the focal-plane detector. The detector e$ciency is 87% for detection of a particles, averaged over the position-sensitive array. The separator suppressed unwanted reaction products a factor of 510 and scattered beam by a factor of 510 but the event rate in the detector was still &15 Hz. The data format for the experiment included information 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 experimental reaction data were collected. Of these
Each of the described methods will be applied to the data set described in Section 2. For the new Monte Carlo Random Probability (MCRP) method, external criteria based on the Geiger} Nuttall (GN) relationship between a-particle decay energies and decay lifetimes will also be applied. 3.1. GSI methods There are two methods used by GSI described in Ref. [2] by Karl-Heinz Schmidt. Both will be briefly described here, (for more details, see Ref. [2]). Provided the expectation value for the number of event chains due to background #uctuations is much less than one, a probability for the event chain being random can be easily calculated from Eq. (3) in Ref. [2]. This equation is nLK P + @ n ! K
(1)
where n is the number of event chains observed K and n is the expectation value for the complete @ event chain assuming each of the events in the chain are independent of each other. The expectation value for both methods is calculated using the counting rates for various types of events of interest, time limits for the correlation to occur, and the total time for the experiment. When there is one event sequence (i.e. n "1), then P +n . K @ The "rst method is given by Eq. (5) in Ref. [2]. This method assumes that the order of events is known. The equation to calculate n is @ ) j )\ ) G G n "¹ +1!e\ G HG RH H> , @ ( )G jG ))\ H
(2)
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where j is mean counting rate for event group G (events of same type within a de"ned energy gate) i, K is the number of di!erent event groups, *t is time interval between the successive events, and the variable ¹ is the e!ective counting time (total counting time t times f ), which includes a position factor, f, the e!ective number of pixels that can record the events. In our experiment we estimate 40 mm/strip f"12 strips; "200. 2.4 mm/pixel
(3)
With this method, a P of 0.00010 is calculated using an 8.5}10.0 MeV gate for determining the counting rate for the a groups and the appropriate energy gate for the EVRs. The counting rate for SFs was calculated from all SF events observed within an energy gate of 130}200 MeV. Because average counting rate is used, an assumption made is that the background is constant both in time and space over the whole detector system. The second method used by GSI is a related calculation given by Eq. (7) in Ref. [2]. This calculation assumes only that the "rst event is followed by the other events in an unknown order. It would make no sense to apply this equation to our event sequence since a "ssion followed by an a is as unreasonable as an a followed by an EVR. Event-rate method for entire detector system (Lazarev method) This method was used in Ref. [3] and described in a private communication [4]. This random probability calculation involves multiplying the number of EVR}a correlations within a given time and position windows by the ratio of correlations to start the events for each succeeding event, thus, for a "ve-member chain, EVR}a }a }a }SF, the expectation value is C C C ; ? \a ; ? \? ; ? \1$ n "C @ #40\a N N N ? ? ?
(4)
Throughout 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 "xed cut (e.g. 9.0}10.0 MeV).
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where N is number of x type events and C is V V\W number of position correlations between x and y type events in a given time window. P is cal culated from n as in Eq. (1). Using the described @ data set with an a energy gate of 8.5}10.0 MeV, an EVR gate as appropriate for the "le and the time intervals given in Fig. 1 for the appropriate time windows, and a position window of 1.2 mm for a}a correlations and 2.0 mm for a}SF and EVR}a correlations, we obtain a P of 0.0041. 3.3. Monte Carlo method Since other methods assume that the background is constant over the duration of the experiment, as evidenced by using overall event rates or total number of events, we set out to generate a method that would take into account background variations and possible hidden correlations within the data set. A Monte Carlo technique seemed to provide the right approach for this undertaking. If we arti"cially introduce into the data set a speci"c decay in the event sequence of interest that must be present based on either theoretical or known decay properties, we can search for event sequences correlated with that randomly introduced decay. The randomly correlated event sequences could be interpreted in the same manner as the event sequence of interest. In this manner, we determine the probability that the event sequence observed is due to the random presence of this de"ning decay. Our data consisted of events that were recorded in list mode detailing the detector(s), magnitude of signals, position in the detector, TOF start and stop, and the time of the event. For each "le of data, a list was generated that gave the event type, focal plane detector number, position in the focal plane detector, energy deposited in the focal plane detector, energy deposited in the side detector, the event time, and the TOF. Based on theoretical predictions of the decay modes for element 114 and its daughters, we assumed that any valid decay chain ends in a spontaneous "ssion. Using this list we were able to insert into the data arti"cial "ssions distributed over the duration of the experiment and the spatial extent of the focal plane detectors. Each arti"cial "ssion was generated using a Monte Carlo
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technique to determine the time, detector number, and position in the detector. We used several di!erent spatial distributions to determine if there was any correlation between the distribution of random "ssions (RF) and the calculated probability. It is this random probability method that was used to calculate the probability per "ssion of 0.006 reported in Ref. [1]. The search algorithm was based on the parameters for decay chains of interest for element 114. In order to not de"ne the parameters too narrowly, we chose them such that predicted E for elements a with Z"110}114 with t between +0.1 s and +10 000 s were not excluded and that the candidate element 114 event sequence was not excluded. We looked for three a's and an EVR preceding each RF within a given time window, energy window, and position window. Our standard parameters are given in Table 1. The position correlation determination was composed of three parts: (1) "nding a's and EVR events that were within a reasonable position window (2 mm) of the RF event, (2) checking that two sequential events met the 595% con"dence level for position correlation based on calibration data, and (3) that the entire event sequence was within the total position window (a parameter in the MCRP code). In the search for the EVR and three a's preceding a RF, we performed two di!erent types of time window searches: total event time window or separate time windows for each decay. A time window for the entire event means that from the RF only look at the preceding time window for a , a and Table 1 Parameters used for the MCRP baseline calculation Parameter
Value
E a E a E a E (First 30 "les) #40 E (Last 61 "les) #40 Total position window Total time interval Number of RFs Distribution type RNG
8.5}10.0 MeV 8.5}10.0 MeV 8.5}10.0 MeV 9}18 MeV 6}15 MeV 2.0 mm 2043.8 s 10 Flat, 3}37 mm C1
a , then EVR. If separate time windows are used, look for a in the a }SF time window before the RF, then for a in the a }a time window before a , then for a in the a }a time window before a , and last for the EVR in the EVR}a time window before a . 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 next event as soon as the "rst candidate which meets the reasonable position window and E gate requirements within the speci"ed time window is found. A simple test run was performed on the candidate `randomly correlateda chains to investigate the number of additional chains that would be found if one a were arbitrarily ignored. As this corresponds to "nding chains consisting of an EVR and four a's correlated with a RF, this is expected to be a small number of chains. Indeed, only 48 chains of an EVR, four a's, and a RF were found. Since three chains of an EVR, three a's, and a RF can be constructed from these chains depending on the a that is ignored, there is less than a 25% increase in the number of the EVR, three a's, RF chains. It should be noted that the majority of these new chains would be eliminated when GN conditions described are applied. The Geiger}Nuttall relationship between Q and a a-decay half-life was used to eliminate many a events and, thus, decay sequences, for which the implied nuclear lifetime was either much too short or much too long for the a-decay energy. When the algorithm had detected a decay sequence consisting of an RF preceded by three a's and an EVR, the signature of the decay sequence presented in Fig. 1, the Q value was calculated from the a-decay ena ergy for each a event. A time window was then constructed for each a event with which to test the validity of that a event. Using the Viola}Seaborg formula and parameters from SmolanH czuk [5], we calculated a lifetime assuming a hindrance factor of one. The lower limit of the time window was set to exclude that 15% of events with lifetimes shorter than this value. The upper limit was determined by assuming a hindrance factor of 10 for the lifetime calculated from the Q , and was set to exclude 15% a of a events whose lifetimes were longer than that limit. Any decay sequence whose a events all fell within this window was accepted as a `possiblea
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element 114 decay chain. A graphical representation of this time window is shown in Fig. 2. The baseline calculation gives a probability per "ssion of 0.0058 before the GN restrictions are applied and 0.00056 after the GN restrictions are applied. The parameters used are given in Table 1. The sensitivity of the results to a number of parameters was investigated. All parameters were kept at the baseline value except for the one for which the sensitivity was being investigated. To be sure that there was not a dependence on the density of RFs inserted into the data for the time length of our search, we used sets that had 10, 10, or 10 RFs. The results are shown in Table 2 and indicate that the density of RFs did not make a signi"cant di!erence in the calculated probability for these search parameters; therefore we performed the other parameter studies using 10 RFs unless otherwise noted. Two di!erent pseudo-random number generators (RNG) were used to test if the probability obtained was RNG dependent. The "rst RNG was the simple UNIX rand() routine which uses multiplicative congruential random number generation with a period of 2 and returns a pseudo-random number (PRN). The second was a much simpler RNG based on the modulus of a changing number divided by a very large number and also returns a PRN. We tested both RNG by looking at the distribution of RNs between 0 and 1 returned as
Fig. 2. Graphical representation of the time window determined using our GN restrictions for a nuclide whose E with a HF"1 a gives a t "1 (arbitrary unit).
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Table 2 Comparison of RF density by varying the number of RFs used in the MCRP calculation. All other parameters were kept at the baseline value No. of RFs
P /"ssion (no GN)
P /"ssion (GN)
10 10 10
0.0066 0.0058 0.0057
0.00070 0.00056 0.00047
Table 3 Comparison of two di!erent RNGs in the MCRP calculation. All other parameters were kept at the baseline value RNG (No.)
P /"ssion (no GN)
P /"ssion (GN)
1 2
0.0058 0.0061
0.00056 0.00048
a function of the number of RNs generated; indeed, both did generate distinct sets of PRN. A comparison of the probability obtained using these two RNG is shown in Table 3. The probability obtained is shown to be RNG independent. All other calculations were performed using the "rst RNG. We know from histogramming the various types of events that di!erent species produce di!erent distributions over the detector array. We were concerned that the spatial distribution of the RFs could a!ect the probability calculation. There were not enough true SFs to determine their spatial distribution, so we studied the sensitivity of the Monte Carlo method to di!erent distributions of RFs over the detector array. We used a #at distribution over both time and space as the "rst distribution because it is simple and we had no preconceived notion of what the RF distribution should resemble. Two di!erent #at distributions were generated, the "rst was from 3 to 37 mm on the strips while the second was 0 to 40 mm on the strips. The top and bottom 3 mm were eliminated in the "rst distribution because position determination becomes problematic along the edges of the detectors. Last, we looked at how the random rate was a!ected by having the "ssions distributed #at in time and Gaussian in both strip number and
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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 at the desired number of RFs. From top-to-bottom of a strip (height of the detector), the FWHM was 35 mm, with the center at 20.0 mm, and normalized to 99.285%. These parameters were derived from Ar#Dy to make Po experiments and are typical of the performance of the separator. These normalization percentages are integral values at 2.354p (i.e. FWHM). The probabilities obtained for these various distributions are shown in Table 4. Since there is no substantial e!ect, we chose to use the simple #at distribution from 3 to 37 mm on the strips for the remaining calculations. We looked at the variation in the random rate with di!erent a energy ranges for the a-particles of interest. The probabilities obtained for these energy ranges are shown in Table 5. Wider energy windows generate higher P before GN criteria are applied but lower rates after. Since we are searching for an event sequence of EVR}a }a }a }RF, dif ferent E gates will result in di!erent event sea quences being found because di!erent a-particles will be allowed. The lower energy a events would indicate a longer lifetime which would not be reasonable given the time constraints applied using GN criteria. It is important to choose reasonable energy windows for the a events of interest. We also looked at how the random rate varied with di!erent event sequence durations. The random rate achieved its maximum at about 95 300 s with a P /"ssion of 0.045 before GN and 0.0038 after GN restrictions are applied. For this we assumed that all data "les were separate runs and that any events that occurred in one "le were not related to those in another "le. Since several "les were such that one run was ended and another was Table 4 Comparison of the RF spatial distribution in the MCRP calculation. All other parameters were kept at the baseline value Distribution
P /"ssion (no GN)
P /"ssion (GN)
Flat, 3}37 mm Flat, 0}40 mm Gauss, both
0.0058 0.0052 0.0054
0.00056 0.00050 0.00042
Table 5 Comparison of the MCRP calculation for di!erent E ranges. a All other parameters were kept at the baseline value a energies
P /"ssion (no GN)
P /"ssion (GN)
8.0}10.5 8.0}10.0 8.5}10.0 8.5}10.5
0.012 0.011 0.0058 0.0077
0.00020 0.00022 0.00056 0.00041
MeV MeV MeV MeV
Table 6 Comparison of MCRP calculation for di!erent total event time windows. All other parameters kept at baseline values Time window (s)
P /"ssion (no GN)
P /"ssion (GN)
1000.0 2043.8 3000.0 4000.0 10000.0
0.00094 0.0058 0.012 0.018 0.037
0.00005 0.00056 0.0010 0.0016 0.0031
immediately started, more elaborate algorithms could be implemented to couple and then search these two "les. This was not done in this work. The results of the probability calculations are shown in Table 6. As expected, longer times give higher P , roughly proportional to the length of the time window. The time window that is appropriate corresponds to the length of time for the event sequence of interest. If the event sequence were longer the probability would then be higher that it was a random event; conversely, if the event sequence were shorter the P would be smaller. The width of the total event sequence position window was also varied. The results from this variation are shown in Table 7. As one would expect, narrower windows give somewhat smaller P . At some point, the total event sequence position window no longer has much e!ect because of the additional criteria imposed for position correlation (see the above discussion on position correlation).
4. Comparison For two speci"c sets of parameters, we compare all methods. For this comparison, all data "les that
N.J. Stoyer et al. / Nuclear Instruments and Methods in Physics Research A 455 (2000) 433}441 Table 7 Comparison of the MCRP calculation as a function of position window width for the total event sequence. All other parameters kept at baseline values Position window (mm)
P /"ssion (no GN)
P /"ssion (GN)
1.6 1.8 2.0 2.2
0.0051 0.0055 0.0058 0.0058
0.00046 0.00050 0.00056 0.00056
contained Ca#Pu reaction data from 1998 were used. See Table 8 for the parameters used and Table 9 for the results. MCRP can have either an event window for the entire event sequence or time intervals for each event. The former is more representative of P for a reasonable event sequence, while the latter for a similarly timed event sequence. The change in parameters from Set 1 to Set 2 is largely a time window change; in MCRP there is two or more orders of magnitude di!erence in the probabilities. Thus, specifying the time history is very limiting. Likewise, restricting the E to tight energy gates a also de"nes the event sequence as having similar energies rather than having reasonable energies. With more than one event sequence, such limitations can be imposed. For parameter Set 1, we "nd that MCRP without GN and Lazarev Method are comparable while MCRP with GN and GSI Methods are comparable and about an order of magnitude lower. Since the GSI Method uses products of poisson distributions, while Lazarev Method uses products of numbers of correlations and event types, it is reasonable that the former would approximate the GN limitations used in MCRP better than the latter. Although the GSI method does not include the GN criteria explicitly, it does include it super"cially when the event sequence under analysis meets GN criteria by virtue of the poisson function that utilizes the time interval between events. Since the GN criteria is such a good correlation between the a-energy and hal#ife for even}even isotopes and a good guide for even Z-odd A isotopes, events that fall too far o! the correlation will be met with much
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Table 8 Parameters for the comparison of the P calculations. Set 1 is the baseline parameters Parameter
Set 1
Set 2
E a E a E a E (First 30 "les) #40 E (Last 61 "les) #40 Total position window EVR}a time interval a }a time interval a }a time interval a }Fission time interval
8.5}10.0 MeV 8.5}10.0 MeV 8.5}10.0 MeV 9}18 MeV 6}15 MeV 2.0 mm * * * *
9.1}10.5 MeV 8.1}9.9 MeV 8.1}9.7 MeV 9}18 MeV 6}15 MeV 2.0 mm 30.4 s 925.8 s 94.7 s 992.8 s
Used total time from EVR to Fission of 2043.8 s.
Table 9 Comparison of the methods described in this paper. The parameters are detailed in Table 8. An attempt was made to have the P calculations for each method based on the same parameters whenever possible for each of the parameter sets; however, approximations had to be made in some cases because of methodology Method
Parameter Set 1
Parameter Set 2
GSI Method Lazarev Method MCRP without GN MCRP with GN
0.00010 0.0041 0.0058 0.00056
0.00012 0.0032 0.000048 410\
No position window speci"ed except through f. Position windows where 1.2 for a}a and 2.0 mm for EVR}a or a }SF correlations. Time intervals necessary, used Parameter Set 2 values. Used 10 RFs to get reasonable statistics for MCRP without GN.
more suspicion and likely be viewed as unreasonable without considerable explanation. It is interesting to note that the two very di!erent methods give similar results for the P . For parameter Set 2, neither the GSI Method nor Lazarev Method change much; both essentially have only minor changes to the E 's used for the a three di!erent a particles. MCRP has a dramatic change because individual time windows are used in addition to the E changes. The MCRP results a are much lower than both the GSI and Lazarev Methods. One major di!erence between the GSI
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Method and MCRP is correlation determination; GSI Method uses event rates for the various events to determine `time correlationsa and the f factor to determine `position correlationsa while MCRP looks for actual position and time correlations between events in the data. The Lazarev Method looks for time and position correlations between various pairs of events and assumes that the ratio of correlations to start events is constant over the duration of the experiment. One interpretation of this apparent discrepancy between MCRP and the GSI and Lazarev Methods for parameter Set 2 is that while GSI and Lazarev Methods are at most "nding true correlations between event pairs, MCRP is looking for a correlation between "ve events in a speci"c time sequence with speci"c energies. In more popular terms, we have gone from looking for a winning bridge hand to a bridge hand that contains the top 2, 3, or 4 cards in each suit; there are many more combinations that make a winning bridge hand than there are with the top 2, 3, or 4 cards in each suit. These speci"c constraints should not be applied without some scienti"c justi"cation. Such speci"city will allow one to determine the probability of "nding an event sequence exactly like the one which exists rather than one that would be interpreted as arising from the decay of the same nuclide. MCRP is much more sensitive to such changes because of the "ve-fold correlation that is needed. Because MCRP is looking for actual correlations in the data, it is also more sensitive to background #uctuations than the other two methods that use average event rates or total event counts.
5. Conclusion During the analysis of the event sequence represented in Fig. 1, the question of random probability was broached. Since the running time was long, the magnitude of background #uctuations was of concern. Some dissatisfaction with the current published methods and their handling of backgrounds was expressed. We decided to tackle the background #uctuation problem by using the actual data to construct decay sequences using RFs inserted into the data. From these sequences it was
clear that many would not be reasonable candidate decays, so we added the pertinent physics by using GN criteria. We created a method that works with the background present and includes the physics that the event sequence will be judged with; therefore, in Ref. [1] we reported the MCRP (with GN included) value of 0.006. MCRP is a useful tool in evaluating the P for a set of data, where the event sequence has a unique de"ning feature. Hidden correlations should be discovered for the event sequence of interest. Having one event sequence, means that the criteria for E and time interval(s) need to be somewhat wider, a since you are trying to determine what the probability is of "nding an event sequence randomly that would be interpreted in the same way as the event sequence found. If more event sequences are found, narrower energy and time criteria can be applied since both the lifetime and decay energies would be de"ned better. This would result in lower P . MCRP has the advantage that the data set with all of its hidden correlations is used during the random probability calculation. It does not assume, like the GSI method, that the background is constant at the average counting rate for the duration of the experiment. A disadvantage of MCRP is the necessity of having a uniquely de"ned decay in the event sequence. Both the GSI and Lazarev Methods have no such requirement and would be useful even in situations where a de"ning decay is not present. MCRP, like most tools, needs to be applied with knowledge of its limitations and usefulness. It has the potential to "nd hidden correlations in the data set being analyzed while giving a reasonable measure of the random probability associated with the decay sequence of interest. The minimum P , which can be calculated using this method, has as limiting factors the computer speed and storage capabilities available.
Acknowledgements The work at LLNL was performed under the auspices of the US Department of Energy under Contract No. W-7405-ENG-48. This work has
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been performed with the support of the Russian Foundation for Basic Research under Grant No. 96-02-17377 and of INTAS under Grant No. 96662. These studies were performed in the framework of the Russian Federation/US Joint Coordinating Committee for Research on Fundamental Properties of Matter.
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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.