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Nuclear Instruments and Methods in Physics Research A 345 (1994) 534-537 North-Holland An automated search algorithm for superdeformed bands D .S . Haslip, G . Hackman, J .C . Waddington Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4MI (Received 4 November 1993) An algorithm is presented which searches -y-ray coincidences of arbitrarily high fold for the regular sequences of transitions which characterize the decay of superdeformed rotational bands . The algorithm was used to analyze data gathered by the EUROGAM multi-detector array . It successfully located all but one of the known superdeformed bands to 149Gd found by conventional methods and, in addition, found at least one new band as well 1 . Introduction One of the most exciting features observed in nuclei at high angular momentum [1] is superdeformed (SD) rotational bands . A -y-ray spectrum of a typical SD band shown in Fig . 1 illustrates many of the properties of SD bands. In the A = 150 mass region, SD bands consist of cascades of 10-20 regularly spaced -y-rays between the energies of- 700 and = 1700 keV. The intensities of these bands are very small ; for example, the population intensity of 149 Gd band c, shown in Fig . 1, is only - 0.1% of the total yield of the 158 MeV 124 Sn( Si, xn) fusion-evaporation reaction [2] . Superdeformed bands are studied with -y-ray energy coincidence techniques. The -y-ray spectrometers used by nuclear structure physicists today measure the ener- gies of the -y-rays associated with the decays of individ- ual nuclei . In the past, the efficiency of spectrometers was such that typically only two energies per event could be measured . Each pair of energies constitutes a "coincidence" which may be entered as a single count in a two-dimensional histogram whose ordinates are the -y-ray energies . In order to obtain the spectrum of ,y-rays in coincidence with a given transition of energy x, we select the rows of the coincidence array between energies x - S and x + S and add those rows together, where S is some width determined by the resolution of the detectors . The set of energies between x - S and x + S is called a gate . Recently, a number of spectrometers have come on-line which are consistently capable of providing 3, 4 or 5 energies per event . Such "high-fold" data make -y-ray spectroscopy far more sensitive because one can * Corresponding author. Tel . + 1 905 5259140 (ext 23635) 0168-9002/94/$07.00 © 1994 - Elsevier Science B.V . All rights reserved SSDI0168-9002(94)00236-Z 2. Method NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH Section A set multi-dimensional gates on the f-dimensional coin- cidence array to obtain the one-dimensional spectrum simultaneously in coincidence with f - 1 energies . Be- cause any given location in a four-dimensional coinci- dence array has very few counts in it, one triple-gated spectrum is often insufficient to show a weak band . One must know n transition energies and sum the triple-gated spectra obtained for each of the (3) sets of 3 transitions . This is the technique used to create the background-subtracted spectrum shown in Fig . 1 . The problem, then, is that one must know beforehand at which energies to set the gates to create a spectrum of the band . This becomes even more serious as the fold of the data increases . To search for new bands, one may proceed by testing all sets of n energies which could be indicative of a SD band . This is clearly a mammoth task which involves discarding thousands of combinations of ener- gies which generate no SD signal . An automated pro- cess that identifies sets of gates which may correspond to SD bands and rejects those which do not would simplify and accelerate the search for new bands. We present an algorithm that performs such a task . The key concept in our algorithm is the structure of the gate files . Standard gate files consist of beginning and ending channels for each of the gates . When it is possible to store a multi-dimensional coincidence array on disk, this kind of gate file makes the process of setting multi-fold gates on these data very simple . However, the storage requirements of a four- or five- dimensional coincidence array make this option unfea- sible . A four-dimensional coincidence histogram with

An automated search algorithm for superdeformed bands

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Page 1: An automated search algorithm for superdeformed bands

Nuclear Instruments and Methods in Physics Research A 345 (1994) 534-537North-Holland

An automated search algorithm for superdeformed bandsD.S . Haslip, G. Hackman, J.C . WaddingtonDepartment of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4MI

(Received 4 November 1993)

An algorithm is presented which searches -y-ray coincidences of arbitrarily high fold for the regular sequences of transitionswhich characterize the decay of superdeformed rotational bands. The algorithm was used to analyze data gathered by theEUROGAM multi-detector array. It successfully located all but one of the known superdeformed bands to 149Gd found byconventional methods and, in addition, found at least one new band as well

1. Introduction

One of the most exciting features observed in nucleiat high angular momentum [1] is superdeformed (SD)rotational bands. A -y-ray spectrum of a typical SDband shown in Fig. 1 illustrates many of the propertiesof SD bands. In the A = 150 mass region, SD bandsconsist of cascades of 10-20 regularly spaced -y-raysbetween the energies of- 700 and = 1700 keV. Theintensities of these bands are very small; for example,the population intensity of 149Gd band c, shown in Fig.1, is only - 0.1% of the total yield of the 158 MeV124Sn(3°Si, xn) fusion-evaporation reaction [2] .

Superdeformed bands are studied with -y-ray energycoincidence techniques. The -y-ray spectrometers usedby nuclear structure physicists today measure the ener-gies of the -y-rays associated with the decays of individ-ual nuclei . In the past, the efficiency of spectrometerswas such that typically only two energies per eventcould be measured . Each pair of energies constitutes a"coincidence" which may be entered as a single countin a two-dimensional histogram whose ordinates arethe -y-ray energies . In order to obtain the spectrum of,y-rays in coincidence with a given transition of energyx, we select the rows of the coincidence array betweenenergies x - S and x + S and add those rows together,where S is some width determined by the resolution ofthe detectors . The set of energies between x - S andx +S is called a gate .

Recently, a number of spectrometers have comeon-line which are consistently capable of providing 3, 4or 5 energies per event. Such "high-fold" data make-y-ray spectroscopy far more sensitive because one can

* Corresponding author. Tel. + 1 905 5259140 (ext 23635)

0168-9002/94/$07.00 © 1994 - Elsevier Science B.V . All rights reservedSSDI0168-9002(94)00236-Z

2. Method

NUCLEARINSTRUMENTS& METHODSIN PHYSICSRESEARCH

Section A

set multi-dimensional gates on the f-dimensional coin-cidence array to obtain the one-dimensional spectrumsimultaneously in coincidence with f- 1 energies . Be-cause any given location in a four-dimensional coinci-dence array has very few counts in it, one triple-gatedspectrum is often insufficient to show a weak band .One must know n transition energies and sum thetriple-gated spectra obtained for each of the (3) sets of3 transitions . This is the technique used to create thebackground-subtracted spectrum shown in Fig. 1 . Theproblem, then, is that one must know beforehand atwhich energies to set the gates to create a spectrum ofthe band . This becomes even more serious as the foldof the data increases.

To search for new bands, one may proceed bytesting all sets of n energies which could be indicativeof a SD band . This is clearly a mammoth task whichinvolves discarding thousands of combinations of ener-gies which generate no SD signal . An automated pro-cess that identifies sets of gates which may correspondto SD bands and rejects those which do not wouldsimplify and accelerate the search for new bands. Wepresent an algorithm that performs such a task .

The key concept in our algorithm is the structure ofthe gate files . Standard gate files consist of beginningand ending channels for each of the gates. When it ispossible to store a multi-dimensional coincidence arrayon disk, this kind of gate file makes the process ofsetting multi-fold gates on these data very simple .However, the storage requirements of a four- or five-dimensional coincidence array make this option unfea-sible . A four-dimensional coincidence histogram with

Page 2: An automated search algorithm for superdeformed bands

300

250

200

1500U

100

50

D.S. Hashp et al. /Nucl. Instr. and Meth .

i , i , T J I I . , f I I rl"I tn 1 .

800 1000 1200 1400 1600

Energy (keV)

Fig. l . A -y-ray spectrum of an excited superdeformed band inthe nucleus 149Gd, made by triple-gating on four- and higherfold data . A double-gated background has been removed, as

described in the text .

an energy resolution of 1 keV per channel covering theregion of interest for SD bands in the A = 150 massregion (700-1700 keV) with 1 byte per element wouldrequire 1 terabyte of storage, far more disk space thanis available on modern systems. One solution to thisproblem is to develop new methods of storing coinci-dence information to reduce storage requirements [3].Another alternative is to adopt a new method of analy-sis. For instance, high-fold data may be analysed in anevent-by-event manner, each event consisting of aninteger denoting the fold of the event and a sequenceof integers representing the -y-ray energies observed inthe event. This style of analysis may be facilitated bythe use of a very different form of gate file .

Let us concern ourselves with an energy range of1000 keV in the SD region, with an energy resolutionof 1 keV per channel. We express a set of gates as a1000-channel spectrum with ones in the gated channelsand zeroes elsewhere. We store this in memory as a1000-bit word, the gate word . Now consider the analy-sis of a single four-fold event with all y-ray energies inthe region of interest . This event may be entered into a1000-channel spectrum which will consist mostly of

in Phys. Res. A 345 (1994) 534-537

535

zeroes but will contain ones in four locations. Thisspectrum, too, may be temporarily stored in memory asa 1000-bit word, the event word . To determine howmany of the energies lie within the gates of a given SDband, we perform a binary AND operation betweenthe gate word and the event word (i .e . : 0011 AND0101 = 0001, but with one thousand bits rather thanfour). The number of ones in the result is equal to thenumber of energies inside the gates. Since the ANDoperation is a low-level operation, this method of com-parison should be very fast .

As a first approximation, the Compton backgroundof a triple-gated spectrum could be taken as a sum ofdouble-gated spectra generated with the same gate list .Our experience with the EUROGAM data [2] indi-cates that such a background subtraction gives reason-able results for known bands. This technique has there-fore been used in our automated routine.

In the simultaneous analysis of many thousands ofbands, however, one cannot expect to store two 1000-channel spectra, a triple-gated spectrum and a double-gated spectrum, for each band . Instead, one should beable to get a reasonable idea of what the background-subtracted spectrum looks like by recording the totalnumber of counts outside the gates and the countsinside each of the n gates for each of the two spectra.Each band, therefore, requires only the storage of2(n + 1) numbers.

Ideally, the actual analysis would proceed as fol-lows . A file is prepared which contains the gate wordsfor N possible SD bands. These words are loaded intoan N-element array. Events are read sequentially ei-ther off of a magnetic tape or from a hard disk . Weanalyse only four- and five-fold events with all -y-rayenergies in the region of interest, higher-fold eventshaving been unpacked into five-fold events. Each event,in turn, is stored in an event word as described beforeand compared to each of the N sets of gates. A score iscalculated representing the number of energies fallinginside the gates (a number between zero and five).Knowing the fold of the event, m, and the value of thescore, r, it is a simple matter to decide how manycounts (p) should go into each element of the peakarrays and how many counts (b) should go into thebackgrounds of the x-fold spectra for each band . Theseresults are summarized in the following equations, forr>_x :

P = ( r )(r -x)

>x r

b=(x)(m-r) .

(lb)

In practice, the analysis is slightly different. Bitarithmetic cannot be performed with 1000-bit words, sowe divide the region into thirty 32-bit words. The gate

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536

D.S. Haslsp et al. /Nucl. Instr. and Meth

words and event words, therefore, are handled as 30-element arrays of 32-bit words. In the analysis of anm-fold event with a single set of gates, we perform theAND operation only with the m non-zero words in theevent array and the corresponding words in the gatearray. We count the number of non-zero results fromthe m calculations to determine the number of ener-gies inside the gates. We analyse only the events whereno two energies are separated by less than 32 keV sothat two energies are never in the same word in theevent array. Finally, it is in fact far more efficient toimplement the n-element peak array discussed aboveas a 30-element peak array, where the 30 elementscorrespond to the 30 elements of the event and gatearrays .

3. Results

Analysis was performed on data gathered in anexperiment performed with the EUROGAM multi-de-tector array. The nucleus studied was 149Gd [2] . Weknow that SD bands in this mass region are character-ized by -y-ray cascades between 700 and 1700 keV withregular spacings between y-rays of 40-60 keV. In addi-tion, most changes in the spacing along the cascadeoccur in a regular fashion. As such, we characterize the-y-ray energies, E� of a typical SD band by the formu-lae

E, = E,-, + AE,_,

(1 > 0),

(2a)

AE, =AE,_,+AZE

(i >0) .

(2b)

In the initial search, we tested sets of 15 energies .An obvious disadvantage of testing 15 energies is thatthere are many more sets of energies to test than if onewas testing only 5 or 10 energies . The advantage ofusing 15 transitions comes from the fact that we areusing high-fold data . It is clearly important to makeevery effort to maximize the statistics in the spectrumof a weak band . The use of 15 transitions does this,since in the analysis of four-fold data there are (45 ) _1365 combinations of 4 energies to choose from,whereas in the case of 10 transitions, there are only(a°) = 210 combinations, a factor of 6.5 smaller. Thesearch we have performed tested over 60000 bandswith different values of Eo , AE, and 02E . We believethat, in this way, we have tested all possible smoothlyvarying SD candidates .

All programs were written in ANSI C [4]. They weredeveloped and run on a Sun SPARC-10, model 41 .Four- and five-fold events with all energies in theregion of interest were written to the hard disk prior tooperation for the use of the program. The hard diskhad an 11 ms seek time and a SCSI-2 interface.

in Phys Res. A 345 (1994) 534-537

Initial tests of the program showed that 60000 can-didates could be checked against our entire data set inapproximately 30 days . This is considerably longer thandesirable . As a result, a scheme was developed tospeed up this process. After scanning a considerablysmaller fraction of the data (10%, say), one should beable to see many cases of candidate bands whose signalis so small that it is clear that further analysis of thecandidate is unwarranted. In fact, we estimate thatapproximately two-thirds of the candidates ought to beso distinguished after reading 10% of the data . Analy-sis can then continue on a smaller subset of the candi-date bands at a faster pace . This brings analysis timesdown to the order of 10 days .

The ranking of candidate bands for this preliminarycut and for the final analysis may be performed in anumber of ways . The data output by the program afterrunning is sufficient to allow us to calculate the num-ber of counts in each of the gates and in the back-ground for both the double- and triple-gated spectra.Using these data, we have performed the followinganalysis .

The assumption is made that the spectrum for a SDband consists of a series of peaks of nearly equalintensities . The test we use to determine if such aseries of peaks exists is similar to that described in ref.[5] . We calculate a pseudo-spectrum by subtractingfrom every element in the triple-gated peak array afraction of the corresponding quantity in the double-gated peak array. The fraction is equal to the ratio ofthe number of counts in the triple-gated background tothat in the double-gated background, so that, in princi-ple, background counts are completely removed fromthe peak arrays . We call these differences x, . Wecalculate an arithmetic mean, z, and a geometric mean,x, of the x,'s, defined as

For an ideal superdeformed band these numbers wouldbe large and nearly equal. A combination of thesecriteria is used to rank band candidates . Due to thefact that we are dealing with high-fold data over alarger number of transitions, we find that this methodis relatively insensitive to the effect of contaminatingnormal-deformed transitions, even if there are a num-ber of them in coincidence . Corrections may be madeto the x,'s to account for the intensity profile in thefeeding region .

Prior to the running of this program, eight SDbands were known to exist in 149Gd [2] . This programhas identified all of these bands as well as yrast and

15ix- ~`

15 r x � (3a)=1

1s 1~15

z=l~

x,l

(3b)i=1

Page 4: An automated search algorithm for superdeformed bands

4. Summary

D.S. Hashp et al. /Nucl. Instr and Meth . in Phys. Res. A 345 (1994) 534-537

excited bands in the neigbouring nuclei, 148Gd and150Gd. The lone exception is the least intense of theknown bands in 149Gd, band h. Most importantly, how-ever, this program has identified a new SD band in149Gd, the characteristics of which will be reported at alater date . The intensity of this band appears to be ofthe order of 5% of the yrast SD band, whose intensity,in turn, is approximately 2% of the 149Gd reactionchannel. It is clear, therefore, that this algorithm iscapable of identifying weak bands in superdeformednuclei .

An algorithm has been developed to perform anautomated search for SD bands in high-fold data . Theprogram identified the known SD bands in the nucleusstudied and has located an additional weak band . Run-ning time for the program is of the order of a week toten days for a data set containing = 3 x 108 4-foldevents, a considerable achievement given that over60000 SD candidates were simultaneously studied.

537

Although the difficulties associated with conven-tional search techniques increase with fold, the presentmethod may be extended trivially for application tohigher-fold data . Of course, the rewards of taking allpossible combinations of gates will be even larger forsuch data than they are for four-fold data .

Acknowledgement

This work was partially supported by the NaturalSciences and Engineering Research Council of Canada(NSERC).

References

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[21 S. Flibotte et al ., Phys . Rev. Lett . 71 (1993) 688.[31 S. Flibotte et al., Nucl. Instr. and Meth . A320 (1992) 325.[41 A. Kelley and 1 . Pohl, A Book on C (Benjamin/Cum-

mings, Redwood City, CA, 1990).[51 J.K . Johansson et al ., Phys . Rev. Lett . 63 (1989) 2200 .