CDDfill18HMR OfibeJlMHeHH OrO MHCTM TYTa RJle phblx MCCneAOaaHMM. AY6 Ha. Ngu yen Oinh Vin h


 Janis Chapman
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1 CDDfill8HMR OfibeJlMHeHH OrO MHCTM TYTa RJle phblx MCCneAOaaHMM AY6 Ha E'l Ngu yen Oinh Vin h lla," PI l'c OF SPHERI CA L ~ U8S Il El.LS IN Til E 3 9 lj ANn :l 7NI' l\ UCLEI 985
2 In recent years a great attention has been paid to the study o f the f ragmentation of the simplest nuclear exci t ati ons over many nuclear levels at intermediate and high excitation energies. It is wellknown that the fragmentation of fe\~quasipar t icle states plays an important role in t he def i ning many physicall y observabl. quantities such as the spreading wi dth o f giant resonances, the El  nnd Ml  radiative strength func t i ons, the neutron s trenrlh (ufwtions, etc. Thi'lt is why the study of this phenomenon hn~ rly~d much interes t of both experimenters and t heoreticians. Till ' [eclive tool for the experimental investigat ion of tilt' nn( quasiparticle state fragmentation i n oddmass nuclei. i s till om' nucicon transfer r eaction. In the stripping reactions of llw (d.p) type the particle states are popul ated, and i n the pickup f"( nctions of the ( d.t )  t ype one can observe t h e hole strenalhs. During tl'" lop.! lime the f r agmen t ation o f few quasipartic components of lll(' W::JV(> function s of nuc l ei is extensively studied in the f rarne wnrk o f the quasipar ticlephonon nuclear model / To de scribe lilt' process o f f ragmenta t ion th is model suggesls t o emp l oy t he mech3nism uf the i nter ac tion of quasipa r t i ces wi t h phonons which an' Vf'ry i mportant i n t he s tudy of the structure of l owlying liournl (Il ional s t a tes of atomi c nucl ei. Within the model, i n re f. 3' Ll e r e have been e stabl ished the genera] regulari t ies of llll' j ragmcnt a tion o f s ingl epartic l e s tates in d e formed nuc l ei which s how that the fra gmentat i on essentiall y depends on tll(> PO g i lion, quantum numbe r s of t he s inglepart i cle state and on the c ha r act erist i cs of c o l lect i ve excitations. I n / 4 ref. t he Jl'p~ndcn c e of fragmen tat i on upon the nuclear shapes is investigattd. It i s s hown t hat, as a r esult, s uhs llf..' l!> wilh orbital momenta f::. i n deformed nuclei a r e fragmented stronge r t han in spherical one s. I n ref. 5' t he fragmentation of var i ous subsho l Is i n t he rar eearth nuclei i s {~. J l cul.,tcd. Th e calcu l a ted p i ctur e o f the stre ng th d i s tributions i s i n sati s f act ory agreeme n t with t he typi cal be haviour of the ('ross section of nucleon transf er reac t ions for this region of nucle i. I n r e f. ' 6 t he fragme ntation of lowspin single particle negat i ve parity states in the i s o topes o f tungslen are i nve s t i gated. The mod e l describes r ather we ll s ome observed a noma l y i n the cross sections o f ( d. p ) r eac t ions in t hcs(> nuc l ei u s i ng a futher small limitati on on the s i ze of the hexadecapol e de f ormation of both tile neutron and proton s chemes.
3 In the lying and and g 7 mentation nucleus. present paper we s tudy the damping of several highl ow lying neutron subshells such as 3 р 3, i, j 5 in the 39 U. We also calculate the fragof а large number of proton subshells in the 3 7Np The model and methods for solving its main equations were described in ref.' 3. In this paper we write out onl y the formulae rej:uired for the understanding of the obtained results. In ref. 3 it was shown that the fragment ation of onequasiparticle states in deformed nuclei may Ье studied within а simplified model. The wave function of the nonrotational state with the angular momentum projection into t he nucleus symmetry axis К and parity " of an odda deformed nucleus has the form: i + i ++ Ч'. (К ) = ~!~Срара + ~D (aq) }'' 0,,[ а Р g g g where '' 0 is the wave function of the ground state of а douьly even nucleus having one nucleon less than the studied one; i is the number of the state, g : vл~j, j being the number of the root of the secuar equation for. the onephonon state of multipolarity Л~. The set of quantum numbers for any singleparticle state is denoted Ьу (va) and for states with fixed К" Ьу (ра) with а=+. The wave function () is normalized to unity. The structure of onephonon states is calculated in the RPA. The quantity (С~) is de termined f r om the wave function normalization (). The secular equation for defining the energies 77i symbolically can Ье written as F (77. ) ~ О. р ( ) () Ср(77) = ~. ( С~) p ( 77 i 77) averaged with t he weight ~ р ( 7) 7)) = с 77. _ 77 ) +(М ) The energy interval of averaging ~ specifies the type of representation of calculational results. The сюiсе of ~ is discussed in ref J 3 ; here we take ~ = 0.4 MeV. One can write (3) in the form '3/ : с~ (77) =... Im! J. " Fp(7J + im) In the simplified model te function C (r )  р has the following form: ~ Г(77) с (7)) = Р w ( f(p) у(77) .,.,) + (М) Г(.,.,) were f (р) г (7)) = + ~ g s the ener gy of the onequasiparticlc s tate г Е_! (p(g).,.,) +(i\/ ) у(.,.,) = ~ Гpg(p(g) ) g (р (g) .,., )  ( м ) Р (3) (4) (5) (6) and (7) (8) То descri be highly excited states within the model.the phonons of multipoletype with Л =,..., 7 as well as а large number of phonons of each multipolarity are taken into account. Alongside with the known lowlying collective quadrupole and octupole phonons we consider таnу weakly collectivized phonons as well as highlying phonons like the giant resonances. The phonon space of the model is large enough and the good description of the density of highly excited states 7 proves its completeness. We determine the singleparticle fragmentation in deformed nuclei using the direct calculational method of averaged characteristics without а detailed calculation of each individual state. Following refs. 3, we construct the s~rength function of the energy distribution of the onequasiparticle state р in the form "" "' Here p(g) = с ( ') + w л~j is tle cncr gy of thc phonon state, Грg defines the iпt e r ac ti on of s t ates р and v through phonon Л~j With the str cngth di s tribution of the singlepartic l e state с: (77) we can now cal culate the fragmentation of the spherical subshell denoted Ьу the quantum numbers nfj over many nuclear levels in а deformed nuc l eu s. It is known that the static deformation resul ts in а splitting of the subshell nfj on j + / suьlevels with the momentum projections into the nuclear symmetry axis К= /, 3/,..., j. The subshell strength is spread over many singleparticle states with different К. Every singleparticle state р comes in this distribution with the weight determined Ьу the expansion coefficient ар~ of the singleparticle state wave function Фрк of the naxialsymmetric ~  " 3 у
4 SaxonWoods potential in the shell functions Фn~ symmetric potential ф рк Р К= nj.а nfj ф nfj of spherical Thus, the subshell nfj strength comprises all these state strengths taking into account their fragmentation due to the nteraction with phonons. The subshell strength function has the form 4 : snf/77) = Jк <<~ ) С~(77) (0) with the normalization condition of type ~ ( рк.., а f ) = J +  рк n J То compare with the spectroscopic data obtained in experiments in spite of Snf/77) one must use for the reactions of (d,t) type рк S nfj (7) = k (anfj ) vp С Р (7) рк and for the reactions of (d,p)type (9) ( ) () 0.6. sg7,.}~) меv SзPJ;}~)MeVo~ 0. Or== ~ MeV Fig.l~ Fragment ation of the g7;  and 3р 3 ; subsheu s in the 3 9u nuc'leus. n.t Sjst/~)Mev n of,, ". 0.8 Si (?) меv 0.6 n 0. Fig.. Fragmentation of the lj 5 ;  and li ; sub ~ О ~ sheus in the 39 U nuc'leus. 4 6?MeV S nfj (7) k (ар К рк nfj) upcp(7), (3) where Up and vp are the Bogolubov transformati on coefficients. It is obvious that  = Snfj (77) = S nfj (77) +Snfj (77). (4) In numerical calculations we use the singleparticle energies and wave functions of the axialsymmetric Saxon\voods potential. The potential parameters, pairing the multipolemultipole interaction constants, and phonon number are taken from refs. /,3/ We have calculated the fragmentations of the li ; , lj 5 ; , 3р ; ja  3 and g 7 ; neutron subshells in the 39 tt nucleus. We investgated the same proьlem for several proton subshells in the 37 Np nucleus. The particle and the hole strength functions are obtained for some of these subshells. We show the strength functions S nfj (7) for the highlying neutron subshells g 7 and 3р 3 ; in fig. and the strength functions Snfj (7) for the lowlying neutron subshells i ; 4 ]: EMeV о Р n 3 f7~  'Лn g~ hg; === ~5/ Лр Зs/~g dз; h;  L Зр~ Fig.J. The r e'lative posit ions of suъshe'l'ls in the neutron and the proton schemes of the region of А = 39. The Л n and Л р are the Fermi surface in the neutron and the proton schemes. 5
5 and lj5; in fig.. Figure shows that the strength distribution of g 7 ;  and 3р 3 ; subshells inside their strength location regions is rather uniform, i.e., the distribution is characterised Ьу а large second moment. The strength location region of g7/ subshell amounts to 7 MeV whereas such а quantity for the 3р 3 ;  subshell has the range of  9 MeV. Thus, the 3р 3 ; subshell is said to Ье fragmented stronger than the g 7 ; subshell. In fig. it is easily s een that the ;  subshell fragments stronger than the lj 6 ; one (the location region of strength for li ; subshell amounts to 8 MeV, and for the j 5 ; to б MeV only). Figure shows an interacting feature: the strength distribution of these subshells is charact e rised Ьу а big concentration of strength in the low region of t he excitation ener gies. Thus, almost 40% of the total strength of li / subshell and up to 50% of the total lj 5/ strength are exhausted in the region of :S: MeV of exci tat.ion energies, whereas, it is clear from fig., only  0% of the total s trength f or the 3р 3 ;  and up to 30% for the g 7 ; subshells is in the same region of excitation energies. Such а behaviour of the distribution of high lying and lowl ying subshells strength may Ье easily understood on the bases of the general regularities of fragmentation of singleparticle s tates in deformed nuclei. In fig.3, we present t!e calculated relative positions of subshells to the Fermi level in the 39 u nucleus (in the righthand side of energy axis) and in the 37 Np nucleus (in the left part of fig.3). Our calculations show that, as а result, t!e higlying subshells, such as g, 3р of the neutron scheme and r 7 ;, d 3 ;, lh,' of the proton scheme, concentrate their strength mainly in some ( or 3) singl eparticle states lying far from the f e rmi level. At the same time the lowlying subshells sucl as lj 5 i, li of the neutron scheme and h 9 : of the proton scleme lave their str ength concentrating in several (4 or 5) stat es lying close to the Fermi level. According to the ge neral r egularit i es o f fragmentation of singleparticle states in deformed nuclei, the farther the state is from the Fermi level the stronger it fragmeпts. That is why such а behaviour of subshell strengtl distributions is observed in figs. and. We present in figs.4 and 5 the str e ngth functions S 0 ~(~ for the lh 9 ;  and f 7 /  subshe s i n the 3 7 Np nuc l eus. The f 7 ; subshell is the highly ing and the h 9 ; subshel l is the l owlying ones. It is c lear that the f 7 / subshell is fragmented stronger than the lh 9/ At the end we show our calculated r esul t s for the spectroscopic factors Snfj (7) for several sbshells in the 39U and 37 Np nuclei in the taьle. These spectroscopic factors may Ье experimentally obtained in the nucleon transfer reactions of (d, t) type. We should like to note that, as it is easily seen 6 The spectroscopic factors 8 0 ~ Nucleus nfj. g 7 / 3!J 3 Рз : lj 5/ li / Зs / 37Np d3/ lh /.4, s,h9/ ( ~) меv р о о ~~~~~~~~ Fig.4. The fragmentation of the lh 9 ; subshell in the 37 Np nucleus о for subshells ile х, MeV 478 о о о t7. (?)MeV У 4 р Tab le о Fig.Б. The fragmentation of the f 7 ; suьsheu in the 3 7Np nucleus. from (), factor vp leads up to excludin$ the contributithe fмnction S nfj (7). These ons of the particle states in states manif~ st themselves in the Snfj (7) functions. Thus, to observe the total strength of the nfj subshell it is 7
6 necessary to use not only the the ( d,p ) reaction. ( d,t )  reaction but a l so HryeH AiHb BliHb E aTyxaHtle cmepnljeckhx no,[logonoqek B a.u;pax U H Np The author is deeply i ndeb ted t o Prof. V. C. Soloviev and Dr. L.A.Malov for useful discussion and help. REFERENCES. Soloviev V. G. Elem. Part. and Atom.Nucl., 978, v.9, p.580. Sol oviev V.C. Nucleonica, 978, v.3, p.49.. Malov L.A., Soloviev V.C. Elem. Part. and Atom. Nucl., 980, v., p Ha l ov L. A., Soloviev V.C. Nucl.Pbys., 9 76, A 70, p.87. Ma l ov L. A., Soloviev V. C. Yad. Fiz., 977, v. 6, p Vdov i n A.I. et a l. JINR, P , Dubna, Nguyen Dinh Vinh. JINR, P , Uubna, Nguyen Dinh Vi nh, Sol oviev V. G..TINR, E , Dubna, Vdovi n A.I. et al. Elem.Part. and Atom.Nutl., 976, v.7, p.95. H P<l~l(,'IK KH:I'lt lcl:nj'lnot!i"!icjihll(' ~0HeJH flapa pac~hitaha!jjpar ~JeHTal{I{ psl,[la aej!'i'pohiibix Ii npotolihblx BblcoKonelKallUi'X H HH3KonelKatlIRX nogo50no'iek 6 flgpax 3\lU H 37Np. PaCC4.li'l'aHbl cnektpo C.<OflJflIeCKRe tjjaktopbl lui" 3'LX no,[lo50no'lek, J(OTopble MorYT 5blTb nony4.eflbl peaklllul "YKJIOIllMX nepcab'i THTIa (d, t ) Pa~o'ra nhm')[lir~ II:J II Jl'::U"hpn'l'OPlfH 'Teope'li4ecKol I}!H9KH OHID!.. Coo6~eHKe 06~CWIHeHHoro HKCTHTYTa R~ep~x Hccne~oBaBHA. Dy6HB 985 Nguyen Dinh Vinl! E Damping of Sphericn.! SlIhf;hells in the 39 U and 87Np Nuclei The fragment<)!: i Oil of sc'vcra l neutron and protod highlying and lowlying SUhHhcl Is in the 39U and e37 Np nuclei is calculated within the 'lllilsipartidepbonon nuclear model. The speclroscopic factor rur Liles!! subshel ls are obtained. They may be observed in till! nuc'll'(l transfer reactions of the (d.t) type. The inv~sligatiol lios bc! per formed al the Laboratory of Theoretical Physics, JINR. Received by Publishing Departme nt on June 3, 985. Communication of the Joint Institute for Nuclear Research. Oubna 985 8