RESEARCH ARTICLE /ARAġTIRMA MAKALESĠ

ANADOLU ÜNİVERSİTESİ BİLİM VE TEKNOLOJİ DERGİSİ ANADOLU UNIVERSITY JOURNAL OF SCIENCE AND TECHNOLOGY Cilt/Vol.:8-Syı/No: 1 : 1-6 (007) RESEARCH ARTICLE /ARAġTIRMA MAKALESĠ ON THE LEVEL DENSITY PARAMETERS OF SOME SUPERDEFORMED LIGHT NUCLEI IN THE MASS REGION OF 0 A 70 SvĢ SÖNMEZOĞLU 1, Erhn ESER, ġeref OKUDUCU ABSTRACT The nucler level densities re extremely importnt for wide vriety of phenomen, rnging from nucler strophysics to rdiochemicl pplictions for stewrd-ship science. The nucler level density is lso n importnt physicl quntity both from the fundmentl point of view s well s in understnding the prticle nd gmm ry emission in vrious rections. In superdeformed light nd hevy nucleus, the gmm-ry energies drop with decresing spin in very regulr fshion. The nucler level density prmeter itself chnges with excittion energy depending on both shell effect in the single-prticle model nd different excittion modes in the collective models. In this study, the energy level density prmeters of some superdeformed light nucleus ( 0 C, 57 Co, 59 Ni, 6 Cu, 68 Zn) re determined by using energy spectrum of the interest nucleus for different bnd. In clcultion of energylevel density prmeters dependent upon excittion energy of nuclei studied, model ws considered which relies on the fct tht energy levels of superdeformed light nuclei, just like those of superdeformed hevy nuclei, re equidistnt nd which relies on collective motions of their nucleons. The present clcultion results hve been compred with the corresponding experimentl nd theoreticl results nd found to be well in greement. Keywords : Energy level density prmeters, Collective excittion bnds, Superdeformed light nuclei. 0 A 70 BÖLGESİNDEKİ BAZI SÜPERDEFORME HAFİF ÇEKİRDEKLERİN ENERJİ SEVİYE YOĞUNLUK PARAMETRELERİ ÖZ Nükleer strofizikten rdyokimysl uygulmlr kdr sırlm ve doğ olylrının geniş bir çeşitliliği için nükleer seviye yoğunluklrı son derece önemlidir. Aynı zmnd nükleer enerji seviye yoğunluğu hem prçcığın nlşılmsı hem de çeşitli reksiyonlrd gmm ışını yyınlnmsı için önemli bir fiziksel niceliktir. Hfif ve ğır deforme çekirdeklerde gmm enerjileri düzenli bir biçimde spin zlmsıyl birlikte düşmektedir. Nükleer seviye yoğunluk prmetresi, hem kollektif modelde frklı uyrılm modlrın hem de tek-prçcık modelinde shell etkisine bğlı olrk uyrılm enerjisi ile birlikte değişmektedir. Bu çlışmd, bzı süperdeforme hfif çekirdeklerin ( 0 C, 57 Co, 59 Ni, 6 Cu, 68 Zn) seviye yoğunluk prmetreleri, her çekirdeğin frklı bndlrı için, enerji spektrumlrındn yrrlnılrk hesplnmıştır. İncelenen çekirdeklerin uyrılm enerjisine bğlı olrk enerji seviye yoğunluk prmetreleri hesplnırken, süperdeforme hfif çekirdeklerinin de ğır deforme çekirdekler gibi, enerji seviyelerinin eş-rlıklı olmsını ve nükleonlrın kollektif hreketlerini temel ln bir model göz önüne lınmıştır. Elde edilen prmetre sonuçlrı diğer çlışmlrın deneysel ve teorik değerleri ile krşılştırılmış ve uyum içinde olduğu belirlenmiştir. bndlrı. Anhtr Kelimeler: Süperdeforme hfif çekirdekler, Enerji seviye yoğunluk prmetresi, Kollektif uyrılm 1, Gziosmnpş University, Fculty of Arts nd Science, Deprtment of Physics, Tokt/ Turkey e-mil: svssonmezoglu@gp.edu.tr, fx: +90 356 5 1585 Recieved: 0 December 005; Revised: 0 Mrch 006; Accepted: 13 April 006

1. INTRODUCTION Nucler level density is interesting both from purely theoreticl point of view (the problem of quntum mny-body system with contınuum excittion energy), s well s from perspective of pplictions (e.g., n essentil ingredient of sttisticl models of nucler rections) (Pezer et l., 003). Besides, the nucler level density, which is bsic ingredient in sttisticl nlysis of nucler rections, hs been the subject of mny investigtions t low excittion energy where the level density is obtined directly by counting low-lying levels (Bohr nd Mottelson, 1969). However, t incresing excittion energy, the level density becomes lrge nd individul levels re often not resolved in experiment (Gilbert nd Cmeron, 1965). Min prmeter of theory, which is relted to the nucler level density-the level density prmeter, is obtined from nucler resonnces nd from the nlysis of evportion spectr using the independent prticle model prediction for the level density (Shlomo nd Ntowitz, 1990). The first ttempt, the simplest expression for the nucler level density hs been obtined long time go by Bethe (1936,1937) who utilized the ssumption tht n energy independent density of single prticle sttes nd lter modified by Bloch (195) who developed generl methods to del with the mthemticl problem. There re, however, number of shortcomings in this pproch. For exmple, the lck of coupling to the collective prt of the nucler spectrum leds to n energy-independent level density prmeter. Recently there hs been considerble theoreticl ctivity in the determintion of the nucler mny-body density of sttes, tking into ccount shell, piring, nd deformtion effects (Gilbert nd Cmeron, 1965; Woosley, 1980), finite size effects (Yen nd Miller, 199), nd therml nd quntl effects (Puddu et l., 1990; Puddu et l., 1991), s well s improvements in the determintion of the spin cutoff fctors (Woosley, 1980) nd different collective modes in the excittion of nucler mtter (Igntyuk et l., 1979; Ahmdov et l., 00; Okuducu, Ahmdov, 003).. CALCULATION METHOD FOR LEVEL DENSITY PARAMETER Above-mentioned theory gives lso dependence of nucler level density on the totl ngulr momentum I of the nucleus. The most used theoreticl formul for the observble nucler level densities in the clcultion of level density prmeters cn be written s (Gilbert nd Cmeron, 1965 ; Bethe, 1936), J 1 1 U, J exp J / exp U (1) 3 1 5 U In this reltion, J is the ngulr momentum of the level with ny excittion energy U, is clled s Andolu Üniversitesi Bilim ve Teknoloji Dergisi, 8(1) spin cut-off prmeter which chrcterizes the distribution of the level density in spin. The prmeters nd, which re relted to the density of singleprticle sttes g( F ) t the Fermi energy F, cn be defined respectively s, g F () 6 m t (3) g F Here, <m > is the men squre mgnetic quntum number tht is the verge of the squre of z-projection of individul prticle ngulr momentum j nd t is the nucler thermodynmic temperture of n excited nucleus in the Fermi-gs model. These fctors given in Eqs. () nd (3) re expressed s follows: g F A 3, m 0.16A 3, F t U () where, A is the mss number of nucleus. The experimentl observtions cnnot distinguish the different vlues of J. Therefore, it is more useful to obtin the observble level density which hs form (Gilbert nd Cmeron, 1965 ; Bethe, 1936), exp U U U, J J 1 1 U 5 1 (5) Hence, substituting the Eqs. (-) into Eq. (5) one finds the observble level density s, U exp U (6) 3 3 1 0.98A 1 U The nucler level density prmeter of the Bethe theory is well estblished in number of studies (Gilbert nd Cmeron, 1965 ; Bb, 1970) on the s- wve neutron resonnce for different mss nuclei. However, this theory does not tke into ccount the collective effects of the nucler prticles in the excittion of the nuclei for determining of the nucler level density. On the other hnd, the mesured mgnetic nd qudrupole moments of the nuclei devite considerbly from the ones clculted using the single-prticle shell model in which the closed shells forming the nucler core ply no prt. In other words, the excited sttes nd the mgnetic nd qudrupole moments re the results of collective motion of mny nucleons, not just of those nucleons tht re outside the closed shell. The existence of collective energy level bnds of rottionl nd vibrtionl types cn now esily be identified from nucler spectr dt (Nucler Structure nd Decy Dt, 001) of mny deformed nuclei. In some studies such s in Refs. (Rohr, 198; Igntyuk et l., 1979), the contribution of collective motion of nucleons on the energy level density hs been considered.

Andolu University Journl of Science nd Technology, 8 (1) 3 However, these studies nturlly involve messy equtions nd mke the model complex for clcultion of the nucler level density prmeters of deformed nuclei. Mny superdeformed light nuclei, especilly with mss rnging from A=30 to A=70, hve stble deformtion in their ground-sttes. Such these nuclei my rotte due to interctions with n externl incident prticle or emitting the prticle. Rottionl energy of n xilly symmetric deformed even-even nucleus is given s (Bohr nd Mottelson, 1969), E rot, I I 1 1 1 I K K J 0 J 3 J 0 (7) where I nd K re the totl ngulr momentum nd its projection on the xis of symmetry, respectively, of nucleus; J 3 nd J 0 re the principl moments of inerti bout symmetry xis nd n rbitrry xis perpendiculr to the symmetry xis, respectively. Authors of Ref. (Dvidson, 1968) hve used the hydrodynmic moments of inerti restricting the deformed nucler surfce by qudrupole term only. A further ssumption of this model is tht these nuclei re, on the verge, symmetric: tht is J 3 0. Therefore, Eq.(7) will be meningful only if the vlue of K is tken identiclly zero. Then we come to the following rottionl energy eqution: E rot J 0 I I 1, K=0 (8) Note here tht, the bove expression is in good greement with the observed low-lying energy levels of the even even deformed nuclei, which is the vlues of ngulr momentum I, I = 0,,, 6,. As mentioned bove the energy level sequence in such cse is clled s ground-stte rottionl bnd hving positive-prity. The so-clled β nd γ excited bnds introduced in Ref. (Dvidson, 1968) re lso well identified the observed energy levels of the collective nture in the lrge deformed even even nuclei. The β bnd is ssocited with vibrtions tht preserve the xis of symmetry nd therefore is K = 0 bnd with the level sequence given by Eq. (8) nd the bnd hed hω β. The γ bnd is ssocited with the vibrtions not preserving the symmetry xis nd hving the levels given by Eq. (7). The spin sequence of γ bnd with K= is I=+,3+,+,5+ In such cse, the rottionl bnd with given K vlue nd spin sequence I K, K 1, K,, where K nd re the projections of the totl ngulr momentum nd odd nucleon ngulr momentum, respectively on the nucler symmetry xis, hs level sequence nd spcing which re given by (Nilsson, 1955), E J I, K E E I I 1 KK 1 IK KK 0 (9) A good exmple of this simple level structure given by Eq (9) is to be found in the odd-neutron nucleus 59 8Ni 31. In the deformed xilly symmetric odd-odd nuclei quntum number K is lso determined s K p n, where p nd n re projections of proton nd neutron ngulr momentums, respectively on the symmetry xis. The ground stte spins of these nuclei re determined with the sme coupling rules. Ech bnd with given K is built upon the proton nd neutron intrinsic sttes of the Nilsson model (Nilsson, 1955). Some superdeformed light nuclei, such s eveneven, odd-odd nd odd-a nuclei, with their different corresponding bnds for which the level density prmeters re estimted in the present work re listed in Tble. Tble 1. The clculted nd compiled vlues of the nucler level density prmeters for some superdeformed light nuclei Nucleus Bb Gilbert-Cmeron BSFG Model,, MeV -1, MeV -1 MeV -1 Clculted 0, MeV -1 Corresponding bnds 0 0C 5, 5, 3,6 5,31 β-vibrtionl bnd 57 7Co 5,70 5,95-5,51 Octupol Bnd 59 8 Ni 6,97 5,97 5,07 6,6 Octupol bnd 6 9 Cu 8,78 8,09 6,55 8,1 γ-vibrtionl positive prity 68 Zn 30 9,13 9,75 7,8 8,7 γ-vibrtionl positive prity

The nucler energy level density depending on the excittion energy, U tking into ccount different excittion modes cn be written in the following form, U i i U (10) i where i (U ) is the prtil energy level density for i th excittion bnd nd i is the weighting coefficient stisfying the condition 1. As shown in this i work, to derive the universl expression for i we follow the work in Ref. (Bohr nd Klckr, 1937) expressing the excittion energy U by number of mny different combintions of the unit energy nd use simple expression for the energy level density which considered the collective excittion modes. Here we remin the importnt properties of observed energy spectrum of nuclei considered. These properties cn pproximtely be verified for the energies of the collective rottionl nd vibrtionl bnds in the even-even nd of the coupled stte bnds in the odd-odd nd odd-a superdeformed light nuclei s being the rtios given by, R R : R : R :... 1: r : r :3 :... (11) 1 : 3 r Here, R 1, R, R 3, R,... re the rtios of sequentil level energies to the pproprite energy unit of corresponding bnd. In our present study the nucler level density formul introduced depending on the excittion energy U nd energy unit o for i th excittion bnd cn be represented s (Ahmdov et l., 00; Okuducu, Ahmdov, 003), oi i U, oi exp oiu (1) 3 3 U oi which re firly simple nd contins only one prmeter oi defined s, oi (13) 6 oi nd represents collective level density prmeter corresponding to the i th bnd with the unit energy oi. The unit energy oi is the energy difference of the low-lying energy levels. For exmple, the unit energies re E( ), E( ) E(0 ) 0GS i 0 nd 0oct E(3 ) E(1 ) for ground-stte, nd octupole bnds, respectively. Since the nd octupole bnds hve not ground-stte energy levels, we hve used suitble unit energy E( + )- E(0 + ) for bnd nd E(3 - ) - E(1 - ) for octupole bnd. In the oddodd nd odd-a nuclei it hs been shown tht the unit energy is either energy of first excited stte (for ground stte bnds) or the energy seprtion between Andolu Üniversitesi Bilim ve Teknoloji Dergisi, 8(1) the second nd first excited sttes (for excited bnds) of the corresponding bnd with given the projection of the totl ngulr momentum K. As mentioned before, these bnd energies clerly should, t lest pproximtely, stisfy Eq.(11). 3. RESULTS AND DISCUSSION Now, it cn be compred the observble level density expression of Eq. (6) nd Eq. (1) which hve similr dependence on the energy, even they hve been obtined from different pproches. Eq. (6) obtined from Bethe theory hs been bsed on single-prticle nucler model wheres Eq. (1) hs been extrcted from symmetry properties of the nucler spectr dt expressed by Eq.(11). So, our pproch which re succesfully used erlier for the clssifiction of the level density prmeter using Ref. (Ahmdov et l., 00; Okuducu nd Ahmdov, 003) in this study tkes into considertion different collective-excittion modes of light deformed nuclei considered in the nucler level density prmeter clcultion. Tht is to sy the investigted of the prmeters oi in our prescription hs been mde simply from nucler spectr dt obtined from in Ref. (Nucler Structure nd Decy Dt, 001) by the use of Eq. (13). In the present study we hve seen tht the energy levels of different excittion bnds (in prticulr, the bnds given in Tble) for even-even, odd-odd nd odd-a superdeformed light nuclei lso pproximtely stisfies Eq. (11). Thus, Eq. (13) cn be pplied for estimtion of the corresponding level density prmeters. The clculted vlues of the level density prmeters due to different excittion bnds for some superdeformed light nuclei hve been shown in Tble. In figure, we illustrte the comprison of the single-prticle level density prmeters versus the mss number with our clculted vlues of 0 for corresponding to the different bnds in the region of some deformed light nuclei. The demonstrted vlues of the prmeters were compiled by Refs. (Gilbert nd Cmeron, 1965; Bb, 1970; Belgy et l., 005) for s-wve neutron resonnces ner the neutron binding energy. From figure, it is cler tht the vlues of the level density prmeters 0 clculted by Eq. (13) for corresponding to the different bnds re in good greement with the compiled vlues of the prmeters. In the light of dtbse bove, we cn conclude tht the nucler level density prmeters of superdeformed light nuclei, with mss rnging from A= 0 to A= 70, my identified different excittion bnds (octupole,, ), just like those in the region of lrge deformtions. In other words, no dominnt bnd lone is responsible for identifiction of level density prmeters for even-even, odd-odd nd odd-a nuclei of the region of interest, nd the nucler level density for such nuclei involves combintion of the prtil level densities corresponding to the

Level Density Prmeters nd o (MeV -1 ) Andolu University Journl of Science nd Technology, 8 (1) 5 10 Bb () Clculted (o) 8 BSFG Model () Gilbert-Cmeron () 6 0 57 59 6 68 Mss Number, A Figure. Mss dependence of nucler level density prmeters 0 nd for some superdeformed light nuclei. The vlues of prmeters 0 re clculted for different bnds, nd the theoreticl results for BSFG Model re tken from Ref.( Belgy et l., 005) different bnds, which is given by Eq. (10). The energy level popultion t ny excittion ner the neutron binding energy my clerly hve different chrcter such s collective rottionl, collective vibrtionl, intrinsic nd so on. This property of nucler excittions, s it is well estblished especilly for superdeformed light nuclei, chnges from nucleus to nucleus. Consequently, we remrk tht the present level density prmeter clcultions bsed on the properties of mesured nucler low-lying level spectr should prove productive re of study tht should override the inherent experimentl difficulties involved, t lest in the region of superdeformed light nuclei.. REFERENCES Ahmdov, H., Zorb, I., Yilmz, M., Gonul, B. (00). Nucl. Phys. 706(A), 313. Bb, H. (1970). Nucl. Phys. 159(A), 65. Belgy, T., Bersillon, O., Cpote, R., Fukhori, T., Zhigng, G., Goriely, S., Hermn, M., Igntyuk, A.V., Kils, S., Koning, A., Oblozhinsky, P., Plujko, V. nd Young, P.(005). Hndbook for Clcultions of Nucler Rection Dt: Reference Input Prmeter Librry,http://www-nds.ie.org/RIPL-/, IAEA, Vienn.(BSFG Model). Bethe, H. A. (1936). Phys. Rev. 50, 33. Bethe, H.A. (1937). Mod. Phys. Rev. 9(), 69. Bloch, C. (195). Phys. Rev. 93, 109. Bohr, N., Klckr, F. (1937). Mth.- Fys. Medd. 1, 10. Bohr, A. nd Mottelson, B. (1969). Nucler Structure, Volume I, Benjmin, Reding, M.A. Dvidson, J. P. (1968). Collective models of the Nucleus, Acdemic Press, NY. Gilbert, A. nd Cmeron, A.G.W. (1965). Cn. J. Phys. 3, 16. Igntyuk, A.V., Istekov, K. K., Smirenkin G.N. (1979). Sov. J. Nucl. Phys. 9, 50. Nilsson, S. G. (1955). Kgl. Dnske. Videnskb. Selskb. Mt. Fys. Medd. 9, 16. Nucler Structure nd Decy Dt. (001). Ntionl Nuclr Dt Center, Brookhven Ntionl Lbortory, ENSDF (Evluted Nucler Structure Dt File), Upton, NY. Okuducu, Ş., Ahmdov, H. (003). Phys. Lett. 565(B), 10 Pezer, R., Ventur, A., Vretenr, D. (003). Nucl. Phys. 717(A), 1. Puddu, G., Bortignon, P. nd Brogli, R. A. (1990). Phys. Rev., 1830. Puddu, G., Bortignon, P. nd Brogli, R. A. (1991). Ann. Phys. 06, 09. Rohr, G. (198). Z. Phys. 318(A), 99

6 Shlomo, S. nd Ntowitz, J.B. (1990). Phys. Lett. 187(B). Woosley, S. E. (1980). In Theory nd Applictions of Moment Methods in Mny Fermion Systems, edited by B. J. Dlton, S. M. Grimes, J. P. Vry, nd S. A. Willims, Plenum, New York, p.61. Yen, G. D. nd Miller H. G. (199).Mod. Phys. Lett. 7(A), 1503. SvĢ Sönmezoğlu, 1980 yılınd Khrmnmrş ilinde doğdu. İlkokul ve ortokul eğitimini Khrmnmrş t tmmldıktn sonr, Ankr Attürk Andolu Lisesi nden 1998 yılınd mezun oldu. Aynı yıl kzndığı Krdeniz Teknik Üniversitesi Eğitim Fkültesi Fizik Öğretmenliği Bölümü nden 003 yılınd bşrıyl mezun oldu. 003 yılı Eylül yınd Gziosmnpş Üniversitesi Fizik Bölümü Nükleer Fizik Anbilim Dlınd yüksek lisns öğrenimine bşldı ve 006 yılınd yüksek lisns öğrenimini tmmldı. 006 yılınd ynı üniversitenin Fizik Anbilim Dlınd doktor öğrenimine bşldı. Hlen öğrencisi olduğu Fen Bilimleri Enstitüsünde 00 yılındn beri Arştırm Görevlisi olrk çlışmktdır. Evli ve bir çocuk bbsıdır. Andolu Üniversitesi Bilim ve Teknoloji Dergisi, 8(1) ġeref Okuducu, 1968 yılınd Khrmnmrş d doğdu. İlk ve ort öğrenimini K.Mrş Ytılı İlköğretim Bölge Okulund tmmldıktn sonr, Adn Düziçi Öğretmen Lisesi nden 1987 yılınd mezun oldu. Aynı yıl Ort Doğu Teknik Üniversitesi Eğitim Fkültesi Fizik Bölümü nü kzndı. Bir yıl hzırlık okuduktn sonr, Fizik Bölümü nden 199 yılınd mezun olup, ynı yıl Emniyet Genel Müdürlüğü ne bğlı Ankr Polis Koleji nde Fizik öğretmeni olrk göreve bşldı. 1993 yılınd Gzi Üniversitesi Fizik Bölümü Nükleer Fizik Anbilim Dlınd Yüksek Lisns öğrenimine bşldı ve 1995 yılınd yüksek lisns öğrenimini tmmldı. 1996 yılınd ynı üniversitenin Nükleer Fizik Anbilim Dlınd doktor öğrenimine bşldı ve 1999 yılınd tmmldı. Ksım 000 yılınd Gziosmnpş Üniversitesi Fen Edebiyt Fkültesi Fizik Bölümü Nükleer Fizik Anbilim Dlınd Yrdımcı Doçent olrk göreve bşldı. 00 yılınd ynı üniversitede Doçentlik ünvnını lmış ve 006 yılındn beri Gzi Üniversitesi, Fen Edebiyt Fkültesi, Fizik Bölümü, Nükleer Fizik Anbilim Dlınd görevine devm etmektedir. Evli ve iki çocuk bbsıdır. Erhn ESER, 1979 yılınd Ankr ilinde doğdu. İlk ve ort öğrenimini Ankr Üçkızlr ilköğretim okulu, lise öğrenimini Ankr Uluğbey Lisesi nde tmmldı. 1998 yılınd Gziosmnpş Üniversitesi Fen Edebiyt Fkültesi Fizik Bölümü nü kzndı ve 00 yılınd d mezun oldu. 00 yılınd Gziosmnpş Üniversitesi Fen Bilimleri Enstitüsü Fizik Anbilim Dlınd yüksek lisns progrmın girdi ve 006 yılınd yüksek lisns öğrenimini tmmldı. 006 yılınd ynı üniversitenin Fizik Anbilim Dlınd doktor öğrenimine bşldı. Hlen öğrencisi olduğu Fen Bilimleri Enstitüsünde 00 yılındn beri Arştırm Görevlisi olrk çlışmktdır.