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l Alfa-like FOur NucleOn InteractiOns in AtOmic Nuclei~ O. Dunatrescu Department of Fundamental pllysies Institute of Physics flnd Nuciear Engineering' central Institute of Physics' Magureie-Bucharest P' O' Box MG-6. Romania A moclel of the a-clusterization process in terms of the irreductible reaction amplitude of the a-cluster formation in the four particle channel within the Landau-Migdal theory of quantum liquids is proposed. By =introducing a coupling constant for the interaction between the many- body Fermi liquid state and the a-cluster state, we have calculated the a-widths for some favour- ed and unfavoured ce-transitions in different regionsof Mendeieev's table. Good a"*reemenet with the experimental data is obtained. The Pauli principle is correctly applied to obtain the a-channel wave function. ,* An interplay between the pairing and the a-1ike four nucleQn correlations in the ground and low-lying excited states ot the atomic nuclei is predicted within the BCS-1ike model. , New type of elementary excitations is predicted. These excitations may be constructed on metastable superfluid or normal states. The region ot super*'.1uid nuclei is enlarged due to the fact that the usual BCS neutron and proton superfluidity can mutually be induced via the a-like correlations. The correlations improve the descrtption of a;-clusterization processes and other clusterization processes as, e. g., the heavy cluster decay. I. Irit,rodtidtiQri . = Symmetries bring beauty and order into physical phenomena and their mathema- tical description. Some of them can be, and often are, used as tools helping us to hide our lack of deeper understanding. Yet, if absolute, they are somehow. de.ad in their rigidity. However, if a symmetry is broken-as it is well known in art or archi- tecture - it springs to life and reaveals a forceful personality. Through its breaking a• symmetry can teach us,, more about physical'system t number conservation was for a long time jyst ther~ to be used as a constraint.' Since it's breakin~ was 'found, however, it has become a gteat stimulus for experimental ahd theoretical investigations, examplified, e. g., by Soloviev's group long (20 years) search and final establishment of the validity of superfiuid nuclear model [1]. As pointed out by B o hr [2], the recognition of the deforination and its degree of ~ymrl,etry b,reaking as a central element in' defir.'ing r6tatio;nal degrees of freedom opened riew perspectives for generalized rotational spectra associated with deforma- tions in many different dimensions including spin, isospin, and gauge spaces [3], in addition to the geometrical space of our classical word. The condensates in sup~rfluid systems involve a deformation of field: that creates the condensed basons or =fefmion pairs. Thtis, the process correlated pair of electrons from a superconductof (as in ~ Josephson junction) or a nuclear pair from a superfluid nucleus constitutes a rotational mode in the gauge space in which particle number plays the role of angular momentum for a rotational mode in the ordinary geometrical space. ' presented . , * This •papei has beeritalk on the Vlth,Workshop on Nuclear Theory, Gjoletchitza 16-20 June. Bu]garia as' a 418 5cfie. ca8.•~c..J4 C198rJ. 5 1- Correlations between the nucleons are = geQerally responsible for the existence of substructures in nuclei. Of these, pairing= correl~tions are far the best understood cor- ~~relations as they aff_ect a lot of nuclear processes which have been, extensively stu.di- ed throughout all regions of nuclei. Among these processes, the two nucleon, a.-trans- fer reactions and cx-decay, display a remarkable degree of selecttvity. The pairing correlations separate among these the tavoured processes ~nd hindered one deeper witho:u=t residual interaction~. Fo, r •t,he fa than the shell model,shell model configurations in the transition, prp- rent sums =have to be taken over the babilities, while for the hindered transitions this coherence iS p~rtially or totally abS,ent. Absolute values of the a-decay and a-transfer reaction probalilities are not, how- ever., sufficiently explai•ned by * pairing and/or two nucle6n co theless, for the superfluid atomic nuclei, +the spec,troscopic:,fectors for the ce-clusteri- •zation favoured processes are enhanced by :factors of thous l,ation~ are included,, but still insufficient to deScribe . the ex~perirTl;ental probabilities. Recently a Fermi liquid model for ot-clusterization and c~-decay ha~ been pro- pqsed [4] and tested f5-6j on different Ge-transitionsi This l~l, ,o~el has ~een bQrn as a result of_ a comprehensive analysis ,[7] of the current u-decay models where is shown that the actual nuclear structure model wave functions are. not Sufficient to describe the cL-clusterization: process entering the< a-decay, and_, c~Ttransfer reactions~ .. A renorma- lizat~on of the transition operator .is necess~ry. The Fermi liquid -rylocel introduces a fotir-nucleon in'teraction for the irreductible ,reaction amplitude oi the,a.-cluster for- mation in the' four-particle channel. This idea is suggested by M i g d a:1 [8] following the L,andau theory, of quantum liquids. , . , Such an interaction is probably that o,ne necessitates to descripe the early Mori- naga's suggestion of the condensation oi. two p~oton,s and two. neutrons into the a-clusters. However, it is important to recognize that an a-cluster in the nucleus is not the same as a free ce-particle. It is naturally distorted by the fields of surrounding nucleons and may be violently changed by close interactions between them. By a-clu- i sters one should = probably understand aggregates formed _ from two protons and two neutrons with relatively strong ' spatial correlations, such that the spatial localization of these aggregates to be smaller than the spatial l,ocaii,zation of the nucleus itself. Some more grounds to work- with the Fermi liquid four-nucleon interaction [4-6] could be the following. The nucleus is a system o,f, strongly interacting particles, where the mean distance between the nucleons is of the same order of magnitude as the range of nuclear forces. This fact tells us = that the nucleus has to be treated as a Permi liquid, drop with strdng correlations among the nucleons r8-10]. The possibi- lity of clusterization is equivalent to the coe.xistence of {wo different kinds oi states in nuclei. One kind of states describes the nucleus as a Fermi ~1iquid drop in which =the particles are more , or less uniformly distributed over the whole nuclear volume. The other kind of states describes two Fermi liquid fragments, - ce-cluster and the core in relatively weak interaction (cluster st~te). The transtition -between these,states _ can take place with significicant,probab.ility only if large enough momenta are trans- ferred to the four nuclebns, which ate the a-cluster constituents. At the Same time the energy involved in the a-decay for instance is quite small. The Ge-decay process takes place in two steps,:. 1) the ct-clusterization (the formation of the ce-cluster state) and 2) the decay of the cluster state, which is allowed or forbidden mainly by the barrier. The' cdexistence of ,the abov,e-mentioned two states can be also found in the stable nuclei [11, 12]. The four nucleon _interactions may generate, howey~r, another type of a-aggre- . gate formed from two correlated Cooper ,pairs : ,(one proton pai,r and one ,neutron pair)-the a-superfluid , a.ggregate. This aggregate surely has weaker spatial correla.- tions as compared with ce-clusters but stronger correlat,ions in* the. angular momentum space. This ce-superfluid aggregate in coordinate space is ~pread throughout the nti- cleus in the same way as t,he Cooper pair in a superconductor. 27 Bulg. J, phys. 14 el987), 5 419 Recently [12] similar c'orrelatiofis~ Ieading to an a-1ike "condensate" have been proposed in the framework of the interacting boson model (IBM), each boson corres- ponding to a pair of nucleons. However, the IBW greatly conceals the mechanism or '•f6rination of the superfluid a-aggregate *by approximati ",bosons. Such an approximation is only valid ior weakly~ interacting fermion systems in low density regime, a eondition which is far to be met in the nuclear systems. 'The underlying fermionic structure of the paired bosons certainly plays an important ,role in the formation of the a-superfluid aggregates in nuclei, a role which the IBM .does not take care of. On the other hand, the odd-even staggering of the experimental '(average) one pair binding energies brought up [12] as evidence for the a-like con- ' ~densate in nuclei 'includes to a ',large extent the effect of the nuclear symmetry enefgy [13] and it is difficult to assess [14] at this level how much of the staggering amplitude comes exlcusively from the aLlike correlations. In this respect it would be desirabld to have a more nonambiguous. quantitative measure [15] of these ce-like cor- relations in nuclei. The underlyin.g fermionic structure of the bosons ignored in the IBM approach [12] seems fo be of great importance as far as genuine four-nucleon correlations ,are looked for. Such an investigation raises the question of whether a condensed state could directly be obtained by starting with purely fermionic Hamiltonian and incorpo- rating the fout-riucleon corfelations just at the outset. Doing so, the ensuing con- ?densed state might be viewed as corresponding to genuine iour-f,ermion correlations rather too attractive correlationS between already condensed pairs of fermions. , Early attempts [16-22] to account for a-superfiuid aggregates in nuclei a.s aris- ing from four-nucleon correlations use a trial' wave function with such correlations iricluded, thereby sinrulating the' f'olrr-fermion condensate, the interaction femaining-of A different point of view is assumed ' =' two-fermion type. ~in:'our recent works [15, 23] essentially by using the well-known BCS-like pairing trial wave function and accounting for four- fenuion correlations 'by two pair (proton and neutron) interactions. It is shown that within this simple ~airing approximation the four-fermion correlations lead to con- 'densed ground and/or metastable states of a Fermi gas model which consists of cor- related fermidn pairs. Considering a simplified model in which the single_ partic]e part of our Hamiltonian for a deformed axially sym_metric nucleus has equidistant ener,gy ing conclusions. Taking a for two single particle levels (49-) as' usual (protons and•neutr level spectralarge number of ,types of fermions ,[1] for sufficiently large coupling constants Gp, G* - for pairing interactions a.nd G4 -_for aLlike four nucleon interaction, the'-condensed symmetry ,broken ground state mode ~.of the a*superfluid aggregates is energetically favoured with 'respect to the normal fluid one. Consequently, under these circumstances, a no transition (of first or second order) is predicted. >, ' Por a region of interinediate coupling constants normal fluid and superfluid meta- stable phases are predicted~+ the possibility of introdticing the concept of ndrmal, superfluid ~nd metastable normal o'r. superfluid bands can be also discussed. Anal,og- ous to the ivell-known rotational bands, all the elementary excitations built up on normal or superfluid ground ,and tnetastable state const,itute a normal or super fluid band. ' , . + ' ,*~j 'In Ref. [15, 23] the procedure, for extracting the pai.ring and a-like correlations strengths from the experimental binding energie~ is discussed. With such strengths we calcttlated the enhancement factors- for Cooper pair transfer processes as, e. g., two nticleon and d-transfer reactions and a.-decay; The result show an enhancement of. this factor~ for favoured transfer processes when the a-1ike correlati,ons are included to- gether with usual pairing correlations. , ~ 420 50fa ~u3. ~c. 14 (1~7), 6 i II. Fermi Liquid Mode] for ct-Clusterization In the framework of th~ many body theory applied to the atomic nucleus we may separate a Hamiltonian generating the basis of the eigenstates of the a-cluster problem (interaction switched off')'- Ho from the residual interaction H/. Thus the Hamiltonian of the system H i s ' ' "iH~;Ho;+H ;(2.1) Ho~spectrum intb' iour part~ 17] : 1) bound states embedd~d in continuum We split the (BSEC); four projection oper~tors P. Q, q and A, respectively, in the above- and .introduce of 2) active open channels;-3)iPassive, • open chan ~ ' The a-d~=qay -width subspaces. . i of ' m~trix ' . ~rrientioried qan be e~'~.'re,'ssed in t~r'ms ~ the,." ~elements of a re- _ p6.rni'alized':;tt~n'sition d~efatof: < = ' (2.2) : : ,..,.' . _R,= ~+ W W, . EIHO~QWQ q v. = (2 .3) ; W = V+ V ** E~HoLqVq A (2.4) V H +f~' ELHO~~H'A ' H between one BSEC state <0 and the active open channel states [24] ((pec) as fol- 2'c .(,2.5)._ ,. ; r= 2lc I <~ec IR 0> 1 2. Introducing the correlated' initial ' * (2:6) ' , ' :, Ii)~~ 1+ E_H~q qVq ( V) I O ) and final-ch,anne.1 : _ . ' ' ' '*' ' (2 .7) ~ IWec> - l+ E 7~:Q'QWQ W/\l~c8> ~( - o~ wave functions. and the 'a-tr'ansitjoh oP erator " , Ta : "- " = i A =(2'8)~~ V=H' '+H'H' Eo~ lt:, o ~ AH'A ' tri~ aid~c~j width (2.5) bec.o, .p:e'~;:;:' ; : :,= : ' (2:9) {~-iii: - '~' I r= 2lc I (VGC I Ta I i) ja. The Eq. (2.g) cah be ~pi~roxiineted wi~h an equation for the four-particle Green's', ftinc tion i,n th_~ usual., way , _ . . .; - '. '_ . ' ( l' ~' ~ (2.10~. ' . Ta"== + . c~ Bulgl J' Phys. 14 (1987). 5 42 1 where [8J (2. I l') ' T{to) ~ ~ ce is the irreductible amplitude for the a-cluster formation in the four-particle channe,1' For this amplitude we have used , [4] a contaet interaction form (~.,12) , ' ~ ~ ' ~ . T(o)e~,e8 (~l)'6(~i) 6 (~3) t ~ '-'. in the position coordihai~s repiesentation. ~/ are the lacobi coordinate,s of tri~ joi-cltis- ter ; the 6 (~i) = Dirac delta. functions describe the, packi~rg p,roce ss ~of the four nu- cleons in a small voluhle of the a-clusfer ' volum6,' proce<~s iri ' whic~ =ia, rg~ momen=ts are transferred ; t-operator selects the terms containing two neut}ons arid' two prot6ns among the four fermion orbitals. The constant 8( is assumed to have a unique value for all processes in which the'a-clusters are transferred (ce-decay, a-transfer reactions, . . .). The nuclear states (2.6) and (2.7) should be generated by nuclear structure models The a-channel state wave function should be cofrected by using the P auli K ernel [5-6] as follows : , ;wher.e, * +;+ f'+ (2. 1 4) <R 1 1 -k I R'> = ~J~f(6(R-Ra)~a (~~daug) I ~f(6 (R'- Ra)~a tpdaug)>, wheTe (pa and Rdaug describe the intrinsic motiori of free a-particle and the daughter nuc~leus. The explicit expressions of the Pauli Kernel k wi,thin different appr.o~imations are discussed in Ref. [5, 6]. The relative GL-daughter motion wave function entering the I V(cOs)>-wave func. t,io,n is generated by the double-folding in:odel potential [25]. The self consistency of our Fermi liquid model is upheld by the fact that ,ive.,do not use "free" parameters as in the R-nratrix [24] theory (the channel radius), except those usually used in the nuclear stru,cture rQodels or in the' nuclear scattering folding models (e. g., the Y ukawa Ad8Y pzirameters [25]), which are fixed by reproducing many other nuclear properties. The only pararyleter in our mQdel is the s,tr.ength .of the ce-4N v'..'rtex which is taken to be equal td ,e=1.43XI06 MeV fm The results of the cal~ulations reported in T a b I e I are obtained 'using differel~t nuclear structure wave fundtioris (t,he Z ucker-Buck-'Mc G r o ry large scale' sbell model wave functions 16] for 160, the L e n i ngrad RPA wa.ve f~nctions for=Po-~i region [5], and BCS-wave functions for Rn-Ra region [5]). ' ' = = ' : "' ~ .= ' ' ^_ . In T a ble I therc are reported for comp.aris.on other calculations too, taken from lc.O and from Ref. [27] is taken much Ref. [26] forthe experiment in Ref. [26] the channel radiusfo}= Po•Bi isotopes. On good agreement with too, Iarge,,:Vhile ~he ,Ppbna "Fer~li t gas" ,model does no~ repro~.~ce t~e, experime,n- ' ' ' ' ' ~'~ ' ' ' ' ' nuclear ? tal data. "~'I we see' that for different ~' ' =:structtire' wav~ 'fundiohs From Table the Fermi liquid model of a-decay describes rather well the experimental data. Some exceptions are for a.-tra'nsitions for what rexp/rth-values are less than unity among 210Bi-206T1 a-transitions. The involved riuclear states of 206Tl are not sufficiently well described by the choosen RPA•,model. This fact can b of other nuclear observables involving those states -[27]. The general trend is that with the same constant ,e (see Eq. 2.12) for the a-4N vertex the difference between 422 5 ,ee cu8, uc. 14 (l~7J, 5 , Table 1 Ratios of the expeirmental and theoretical a-~'idths obtained in the F e r m i I i q u i d m o d e I of aT~<~qay comp~,=red, to R-matrix mode; (Ref. [2ql for 160) and p u b n a m o d e I (,Reif. j[~7] for, = t~q rest) Fexpi ~ex pi Ei a•tratl ~it••ion (MeV) Ef (MeV) f,i Ti /"f T E* (MeV) ff rth(other) rth(Fermi l) 160~'l2C 0+ o 2 .69 1 .040 1 'OO 9.85 o 2+ O 1 i .52 O 2+ O 0+ o 4.36 0.986 1 .02 ~5.86 0.265 . 3.30 : 1 8.02 2+ O 0+ o , .7 1 1 .897 = 2.36~, , 8.87 o 2-; O, 0+ o aIOBi->e06Tl o o 0.304 0.265 l- 1=- 1- 2- = 4.649 4 ! 686 158 1.lO,~O 51 0.3s3 0.304 9~ 1- ;4.908< =107. *=' 0.3 1 0.353 0.634 9- 2- 4.568 415 2.39 0.353 Q,80Q 9- 3Ti. , 2- ' 4.4,.13. , 4.224 92, 0.40 1 .60 0.353 0.221 9L 0+ ~ i 5.304 1 34 1,88 210po->206pb O o 0•+ o 0.780 0+ 2+ 4.525 114 1 .37 08Ra~2Q4 R n o O 0+ 0+ 7. 131 0,56 7.018 1 .OO 10R a~tzo6Rn o O.+ 0+ 12~ a~:'2b~;Rh o 0' d+ 0+ 6.869 ' l":56 7: 1 36' 2 .95 214R a->8roRn o o 0+ 0+ 6 .3,3 6 0.3 l z08Rn~198Po O O 0+ O* 06Rn->202po o o 0+ 0+ 6.258 l .60 6:04 1 1 .56 10Rn->ao6po o O Ot 0+ ~ the experimental and theoretical a-widths in different regions of nuclei is less than 1000/0 from the experimental value. The discrepancy experiment - theory can now be removed b,y ,improvjng the nu~lear struct.ur~ m9del,s. . , , . III. Alpha-1ike Four Nucleon Correlations in the Superfluid Phase of ~tomic Nuclei We this section we present the results in In consider a system of nucleons.(protons and neutrons) which are movingofaRefs^ [15,,23]. certain single particle self consistent field as, e. gL, a deformed S a x o n - W o o d s one [1]. As basic functions of the second quantization representation we choose the wave functions of a nucleon in this field. States which differ only in the signs of the projections of the angular momentum , along the symmetry axis are degenerate and conjugate with respect to the time reversal operation [1]. The Hamiltonian for the system of interacting nucleons is pair pair (3.1) H=Hav+Hialv+Hp +Hn +H4' where (~•2) . lf:f, ptn)~7= Esas+ciaso' s* (3.3) HSp(ani; ~ = qp(n) ~rr b~bs ' ss ( Bulg. J. Phys.'l4'(les7), 5 423' lr (3.4) = -G4 /' btbtbR bv H4 vv 'toco ' ' * Here= as+c(as") are the Fermi operators which cr~aie (de~trb~) t'he nucleon in (frbhi) the single particle state I sa>, where c; is ihe i sign of the projection of the angular that label the single particle energy the nuclear symmetry momentum onto levels (SFV fot protons and s=(o for neutrons). axis, s bein,g th The nucleon pair operators are here denofed by = the Eq. (3.1) is : ( 3.5) b~(3.4) inas-as+an' effective, coherent two'-pair (four nu- The ,1ast term cleon) 'interaction term, which is expected to induce the ct-]'ike four 'nucleon correla- tions in the superfluid phase of the atomic nucleus. The other terms in Eq. (3.1) des- cribe the usual BCS-superflu,idity. The Gp, Gn' G4-quantiti,es. ar.e positive valued coupl- ing strengths constants non.vanishing in a certain energy 'range (the cut-off energy range). As trial wave function we use the :BCS-like wave funct, <i'o.ti, ; ' I. B(s>= H (3.6) (us + vsbt), I 0>, s=v, co where u~+v~=1 and (0> denotes the absolute vacuum. ~ Thus, the energy functibnal in' the tirst approxi~lation (dropping otit, for ~~ample~, as usually the selfconsistent field corrections) is , . . , ' W= <BCS I H,-~pZ~'-~n~ I BCS> (3.7) ~;2:12(Ev~~p)~~+2;r' 2(~:;;_'~n)v~ ' (D i, GpX~ = d nic~Id 4X2x2 ' -' 1) n' where ~p(n) denotes the proton (neutron) new ~ermi level~ (chemical botential), Z"(~) is the proton (neutron) number operator and (3.8) I Xp(n) =<BCSI ,b+. IBCS>=2~' = ' ' ' ' " , . vv(o))uy(co) v((o) ' v((o) * < ' *v(o))< ' ' = The minimization of W given by. Eq.:' (3.7) Ieads to the following gap ' equations •1 (bp+c2C)2:" 1 1 * ., 2 , v~ ~ FZ ~( 1 E)p<, + ~ (3.9) I ' = ' 1 (c1 8.co , +G4X~)I-= -2 . =n a) e)( E~~~ 27 1 t~L =N 8~ for doubly even mass nuclei. For od'd- or odd-odd-mass nuclei Eqs (3.9) are modified according to the blocking effect [1]. The new quantiti,es in Eq. (3.9) are defined as follows : ' 8s = [(Es~ ~p(:1))~ + A2p(n)]1/2 424 50/e. eetl3. ~c. l4 (les7), .5 ~pln] s (3.11) I I + E ( 2)=( _~. ). ~ us v 2 Eqs (3.9) represent a set of coupled nonlinear eqs for the nontrivial (superfluid) solu- tions. The original gap Eqs including the trivial solutions are : Ap =xp(Gp+ G x ) Z= ~~ 2v2 (3. 1 2) An = xn(Gn + G4X~), N= 27 2 vco; o) where Xp(n) = ~v ~(*) 28+(~) In the Eqs (3.9) and (3.12) Z(A) is the number of protons (neutrons) taken into account in the cut-off energy range. , '- For a sphe.rical nucleus the labels v and co stand tor the single particle (n, l, j) shell model quantum numbers and Eqs. (3.7) and (3.9) should be replaced by W z,~ 2~ (E ~p)v~2+~:2~~(E~ ~ )v2(o -Gpix2p=Gn~Xn2~ G4X~X~ , and 1 (G +G4X~2n)27 ~ = TP "1 ~ ~ ~- ~;~~~(1 E~ p) =Z, .(3.15) I (G.+G4~2p)lJ~7 ~~ ~F1, ~ ~~ ) , 2;1~~(1 E~-~n =N ~ 8~ where (3. 1 6) ~~(~)Ap(") ~ -2' Xn(p) =~ "(~) 2 8~(~) with 1) (3.17) ~ ='L (2js+ s 2 Ap(n)' 8~(~), Z(N), us' vs have the same expressions as those given by eqs (3.10 --3.12) in which the correlation functions (3.16) should be used. The gap Eqs (3.9, 3.15) have nonirivial (supe'rfluid) solutions for coupling con! stants Gp(n) exceeding the critical values given by B e I y a e v's [28] conditions Bulg J. Phys. l4 (1es7). 5 425 ( (3.18) I Gp(n) I -=1 ~~ (~) 2 crv(~) I E~(*)-~p(n) l with ~v(*)=1 for deformed, axially symmetric -nucleus ~nd ~' ' 1 (2j~(~) + i) = ' "(*)=T , for a spherical one. The phase structure of this model relativ~ to Gp(") control parameters reduces (independently for protons and neutrons) to~ normal phases for Gp(,,) Gp(,,)c' and super- fluid ones in the opposite cases, the phase transitions being of the second order. In the case of our model a corrlplete discussion relative to the three control pa- rameters Gp, O*. G4 and an arbitrary single particle spectrum is practically a very difficult task. In order to grasp the character of the phase structure and to identify specific features associated to the new G4-coupling constant, we consider a simplified model which proves to be enough to rich attention by itse]f and to suggest the highly the realistic (3.1) for defor~ned, nontrivial behaviour of part of our Hamiltonian model. ' Let us assume that the single particle axially symmetric nucleus, has equidistant energy level spectra for two types of fer- mions and introduce the following notations : E~=E (p)+ . E~~ Bpermi (n)+ , Fermi Pp * *, ; = Pn ~ EFerrni (i) + : GFi' , gi = piGi, xi =(piAi) , i= P, n, . 1 g pG4' pF ~ (Pp+pn)' Pp = P* = P, gp = gn = g2' (3.20) ~p=~n=~, , = a_p=0nFFci and look for "symmetric" solutions x=xp=xn of the gap eqs in the case of the half- filling of the A-shells [29] N1' n N2, n+1 Z=N=Z(n+1) in which case, the constraint Eqs (from (~.9) =have ~olutions 1 a cr (;=T', The correlation energy of our model' becomes E pEco" P( Wi(x)'L- W(O)) (3.21) = 2(n + I )~- 4 S1(x) + 4xS_1(x) - 2g2xS~l(x) - g4x2S4_1(x) with n+ 1 S (x)1 ~ 1 )2ljk/2 1 ( 2 x+T m ~ m=- which has to be studied on solutions of the gap 'Eq. (3.22) F(x) = [g2 + g4 X S2_1(x)]S'~i(x) - I = O. 426 Eode. cua. ~c.~l4~(1es7),'5 In order to fi,nd the number and character 'of the solutions of Eq. (3.22) the ' fol- ' lowing curves in (gg,, g4)-plane prov,e to be useful : (3.23) f(x) o d E =0,dx ' = O, (3.24) F(O) F(x) ='O. (3.25) ~D:(x) = O, Curve (3.2~) (~oint-dash'h~ .Pig. 1) separ~tes regions in' which' the number of solu tions of the ga~ ~q. (3.22) differs by two curves' (3.24) ,(solid and horizontal in Fig, 1) separate regions in which the number of soltitions differs by one and reduc~s tq the ' case g4=0 to the critical value given by Belyaev:s condition (3.18),. The ~ro.~sing of. th.e cur,ve, (3.2~) (dashed' in= Fig. i) change.S, the sigu_ of the correlaiiQn energ~ for (~.~2). , < _ ' ' ' one solutibn of the,gap. Eq., minimum of the correlation,energy (3~21), Taking into 'Co'nsideration that the global shotild b,e identified to the gro~nd state of the system, while the local mininrum ~ray' describe ~ the metastabl,e 'state anc. ,the= f~llQwing derivatives of ~~ : ,: ; " (3.26) dEn ' > - -~XS2_1(X) fF(x)~o <0 dg2 F(x)=0 ~ (3.27). dEn - L-XaS4_1(X) IF(x)=0 < O, dg4 F(x) =0 g2 c o:26, " 0.24 \ '~~\.\~\~( ~_ : : ~ \~ 0~2 '\. B \ t\\\~-~-\.~ ~~- ' Il ' ~t 't.' '- tV IVb '~'~*~~~: :~*~l~- _i 1:: _J~ :~~- o.20, A _a :L *'~~~w \ l \.\\ b.18 0.16 * \\ ~'\ 0.1'4 ,\\ l\ 1).12 ,~\ ~ nl'\ IT~ 0.1 O \\.>-\. \,,• , 0.08 .l i~ l t o.os 1 t. \ o~04 l } a02 " 4 5 Is '7 ~. 4xIO~ o 1 2 6 Fig. l Bulg J Phys. 14 (19~7), 5 427' l which' es~entially ' say that ' the derivatives dE~ d E~ , jump when the *gap for!;the ground state has a discontinuity, we reach the following conclusions. In region I (see Fig. 1) the gap Eq. (3.22) has no superfluid solutions and the ground state of the system is in the normal fluid phase. In region H the gap Eq. (3.22) has one solution, which corresponds to a negative minimum of E* (3.21). The ground state of the system is in the superfluid phase. In region lll the gap Eq. (3.22) has ,two superfluid solutions, one of them being .. Region 111-a has a ground state in the normal fluid phase, w, hile the local minimum niay correspond to a metastable state in a superfluid phas'e. In region. 111-b the ground state is in the superfluid phas~ and in the normal phase there is ~ rfietast-abie- state of th~- s_jsfem. : = '~ " ~ I~'~~gion IV' the gap Eq. (3.22) has three solutions, two of them corresp9ndirig to minitnai'~ One of "the minima desdribes the' stperfluid ground state, th.~ .gther .'is. asso- ciated to a superfluid metastable state. The value of the gap in th~' tninimun~ c6rres- pondirig to _ the ground state is less than' the value of the gap corfcS~bndirig td the meta~stable' S'tate. The jump from ' 6ne minim~m to the other occur~ '6n cufYd CB in Fig. 1, which may be found by deteriuinirig thd<selfintersections of cutves (3.28) E.(x) = Fonst <0, F(x) = O. At the right side of this curve the ground state corresponds to the greater of the minimum solutions of the gap Eq. (3.22) and converseiy on the left. . These results are consistent to the phase diagram schematically presented in Fig. I for n=20. Crossing of the segment AB corresponds to a second order phase transition and crossing of the curve CBD corresponds to a first order phase tran- A few more comments may be in order here. First of ali, our mean fiel~ approach is not to be taken too seriously for gaps (x) which are not significantly smaller than the cut-off. In our case this corresponds to x much less than (n+1), otherwise, the situation is one of the strong coupling. In particular, the +transition through CB' curve appears at x~~;20, which may rise some doubts about its fr*eal'ity. In the phase diagram from Fig. I n=20, close to the number of the energy letels used in the most of the calculations [1]. It may be that a renormalization group-type arguinent -improving our correlation energy confirms or eliminates this transition. In any case, a large part of regions 11 and IV Iies deep inside the domain of validity of our approach, so that at least part of the richness of the behaviour we found is to be taken as granted. Another interesting aspect is that the crossing 9f the CB curve corresponds to the transition from a region in which the ground stat~ has evolved continuously from that at g4=0 (which we call a g2-dominated one) to that with large g4 (a>g4-domi- nated one). This transition may be a phase transition from a pair 7 superfluid phase to a quadruplet (co-like) superfluid phase. The crossing of the BD curve may corres- pond to a phase transition from a normal fluid phase to a quadruplet (ce-like) super- fluid phase. As to the difficulties in identifying the four-fermion superfluid phase we remark that an investigation in the framework of the lattice gauge theory of a non- abelian model [30] has failed to find a stable phase (at finite scale) in which the sym- metry is dynamically broken by a four-fermion condensate despite the fact that the necessity of this phase was strongly motivated from a theoretical point of view. Even if this failure may be traced back to the mean field approach which was used, our treatment (perhaps renormalization group improved, quantum corrections included) seems to be .troQ1: pragmatic point of view a better candidate for describing four-fer- mion correlations than considering four-fermion 'tondensation. ' To improve our approach we may introduce corrections using models that treat correlations in nuclei without violating particle number conservation [31-34]. The pro- 428 50fie. e~u3. ~c. 14 (1es7).,5, : . blem of restoring broken symmetries arises in various approaches to the nuclear many- body problems, as, for example, the Hartree-Fock-Bogoliubov theory, and ,ha,S received a great deal of attention over the years. Consequently, the projection tedhni~u~ has reached a high degree,,of sophistication through, the work, pt many au,thors. To touch this topic is ciearly beyond the scope of our present research. For, a comprehensive discussion, including references, we refer the reader to the monograph [29] by R i ng and Schak. .. = , . . For an application ' of our model to realistic atomic, nuclei w~ have to determine the coupling strengths Gp, G*, G4 from the experiment. We adopt the same strategy as usually [1], i. e., \ve use the nuciear odd-e.ven mass differences : (3.29) 1 . P..= T {2e(Zi,.1 , N)i~(Z, N)-e)'"(Z=2, N)}. (3.30) 1 PN,~ T {2e(Z, .N- I )i~(Z:, N)-~(Zf ,< N- 2)} and because we have the four•nue,eQn interaction part ,w= " ' P4=(~(Z N)-e(Z-2 N-2)-,~(Z+ I N) +e(Z- ~. N)-e(Z, N+ 1)+~(Z. N- 1) to fix all Gp, G* and G4 constants: , ,* The results of the numerical determination of the pairing and a-type co,'re- lations coupling strengths Gh,~,Gn' G4 (see Eqs 3.36.-3.37). The ga'p parame- ters Ap and An and Pz, PN and P4 (see Eq. (3.29-3.31) - quantities Nucleus Cp C11 C4 Af (MeV) A~ (MeV) 152 60Ndg2 1 9.340 27.72b 23,644 1 7: f 68 1 . 1 70 1 . 1 60 0.920 1 . 1 1 O Is6,: 62v mg4 1 8 .3 1 O 1 8. 090 29.200 0.777 0.889 24.•130 2 1 .870 O 3 1 .043 +21,652. - i ' 27.060 0.685 0.850 1 60 64Cdg6 0.832 0.991 26..140 - -24..93'5 ' ' • O 0.752 0.925 164 v g8 66DJ 24.200 21 .820 = , 22.78e 0.818 0.833 27.781 24.693 ' = ' O 0.739 0.800 168t:* 68, ut 100. 25 .538 22. 1 83 27.688 0.949 - 0.887 29A47~t ~ = 26,lOO . ;* O " ~' ,0.832 ' , = 0.839 l.7762N:fl04 =, t9.1=96 ~3:~80 : ,:: ','19.911,. _' 0.755 , , O*8+ 4 3 27:86'f6 22.3~~ ' ' O 0.737 0.840 18701 WI08 26.839 - 19.884 22.2gO 0.908 0.803 31.32~ , , . ,~~...04q . O, O.S78 0.786 184n* 7 Iv*108 22. 24. 1 6426.390, ' *1 84 25.293 0.688 1 . 1 75 31.6.10 ' - ' ' O 0.670 1 .080 24 o 94Pul46 34.074 2 1 .590= 1 5.229 0.900 0.685 24968Cfl48 3 1 .708 19.994 14.555. . ; ;?~ O.609. oL355 B4lg$ J, Phys.~14-(1987), 5 429i ~~*~__ ~ ~.I~ Tab'l e 2 (contintled, ' I P (MeV) I P (Mev) P '(MeV) P (Mev) I Nucleus P ( Mev) Z,th N'ex p' N.th 4'ex p 4.th' P z'exp O. I b43 ; 152 N d o..675 = 0.676 0.9715 0.9735 o. 1 63 O*6745 O:97 14 " L0.736 1 56 0.474 0,7.213 *;{ _0.177 -O. 1 8 1 62Smg4 o,.474 = 0.72 l ' 0.473 0,7207 > -0.6 1 7 d.7525= * _0.2 16 16604Gd96 0•5175 0.5 1 72 0.5 1 75 0.75 45 0.7545 -0.2 1 85 -0.766 164 v 66DJ g8 0.4285 0.4293 o:6845 0.6845 _0.~82 -0.576 0.4286 0.6847 -0.825 i0.40 i ' " ' f 168 = ; 0.5045 '~ ; .0.66i6 _0.40 1 68Errco 0.5045 0.5046 0.6676 0.6677 -0.883 17762Hf 104 0.594 0.5938 0.728 Q'7273 - -0.348 -0.340 ~ 0.5939, 0.7283 18704Wro6 0.68 1 0.6813 0.74 1 5 0.7240 " ~ ' ' ' J0.076 = ~0.070 0.6807 0.74 1 9 -0.508 1~410sro8 0.4395 O. 4405 0.9 1 05 0.9 1 25 O. 9 1 07 ~0.823 -0.830 0.4393 - 0.222 240pu 0.591 0.603 0.443 , 0.419 :* -0.313 -0.32 6 94 146 2g~C f i48 0.588 0.5426 0.5 46 0.543 _0.266 -0.253 He re e(Z,N) I = 2E~V5~ + Z ~E~V2 (3.32) -GpX~- C*X~ -G4X~ X~ for a doubly even 'mass nucleus and (~(Z+1 N)=E~~+/~' v 2E~V~2 2E. V2 + ~7 , - Gp~~~'G*X~-GiX~X~ for an odd-mass one. The experimental Pz' PN, and P4 '~uantities are obtained . from (3.3i-3.33) by replacing (~ by B, B being the experimental ~inding energieS. The P4-quantity has been chosen in the same spirit as the analogous quantity f=0r pairing vibfations [lO]. (3.34) P4 = ,~ (Z, N)= ~: (Z-2, N-2) -2~p~~~n' In the first approximation this quatity is determined by P4 ~~ Ap (G4 + O) - A~ '(C4 = O) (3.35) + A* (G4+0)-An (G4 = O) analogous [21] to Pz~~Ap, P*~~An' The quantities Pz' PN. P4 involve 8 nuclei. 430 EoJ'e. cus~ 'e~c. 14 (1987ti 5 a For each nucleus of this set we have T ~ b I e 3 , . , tb = soiv~ th6 gap Eqs; (3:9) ~'r (.~.15). Superfluid enhance,1~ent. factors= for fu,voured two- Thu, s! w,~:h. ~~re ' i~1 ~~il' -~ nonlit;ia~ ~yst~Sl : and fotir n~ciebn trdn~iti~ons , ~ ' of'~~5 > ~qti~tiohs ~ with 35 unknbwh~ f~r ' i = , . ' . ,, , lj x ij - ',' ..Choosinig, . , . ~,+ +, (~:. ,N) ' ' , , , , . nticleus' Z+N, p2 , each =as vsually.[1,~,th,e A< Nucleu~.' , 'I 2 n C. ,c : ddcehddnce.! of the~ couplih~ sfrength Gp, 152 ' ' ~ ' 37.725 53.675 ,:C a~ ipll9Vs,: eoNdg2 ,n. _ , =~ . , ': : ' ' >' . ~ 51.500 o 27 975 ~ Cp MeV _ G~<~ 'A qn;:MeV 38.750 ..(=~.36) . Cp= ' 1 1 56Q 29.20.0, o =22.4. 75 1 9.1,25 38.345=': ".(3: .37) G4= ~2 C41 MeV "27.060 * ' 60 > 64Gdg6 , ,;O*' i ='= 23.4-10 = _20*260 .; . . , , 35.275 iv~ ' ac, tihid,e nticlei =tlpd , s0<hae rare-earth 164 '~ . ~' . = <28.~5,0 , i ~hd ,have dalctil:atdd forCp. Cn . and C.4 '66D. 98 ' ' 19.045 .22.080 29.775 , ="O = : c<dristants. The results are - given= ,in . 168Er >< ' 8L250 : 27.6. 88 , , 26.970 , t ~ ~ I e 2. In the ,performec .qale,ulations = 68 Ioo . 29.225 _ ~~~0= , . . 22.225 ~iih~mact'~uf;~f4fo ennuecrlgeyo~aenngeer~cy~nlt.Leavienl~s = ~~Ptrhoe = l~~HflQ4 1 : 4 4,025i : ', 22.37Cr'= 1 9. 1 96 die. fdrm'e'd S~xon-W, o. . dZs, p~n~ehtiiai [~5]. , o:= , 2 1 :925 : * = = 43L:86a ' i h the deteirirtiried ~frehaths ive 180,,r 38.900 26.650 haveWcaltculated the superfluid enha~5ncement 74 ~ I06 37.730 22.28 O 25.450 184 5 .9 1 o factors x2 (see Eqs 3.13, 3.16) for trans-22, 1 84 1 63.120 ro8 ferring a proton (neutron) C o o p e r 71 o 56.975 15.2,0. Q pair [1]. The superfluid enhancement factor 240pu ior transferping an ~-cluster. ., may j be 94 =5 146 47.61 l .229 33.74 ide~tified with the 3. ' ' 'restilts ar~' given' in Table p,roduct ,[1]1X~X~• The " From T a b I e 2 we learn that the pairing strengths become a little smaller com- paring w,,ith the usual ~BCS-superf.luidity (G4=0).. The 6verall conclusion from T a b I e 3 is that 'the superfltiid enhancement factors increase with G4 with 5-20 o/o ..for two ,Qpcleon transfer, reactions and with l0-500/0 fbr a-clusterization processes in t;he stperiitiid region' of atomic nuclei. Concerning the new constant C4 We may conclude it has* almost•the same thata~ --20 o/o increase when G4 is switched on. v:ar!iation as Cp and 'Cn" The Ap su~fer - : ' IV. ConcluSionS The analysis oi the calbulated~ a~widths' in the framework of the Fermi liquid model of cL-clusterization 'process~s 'e'bmpared 'that it i;s neces~aty 'fo u~e a ~ operator 'whrch Incluces s.trqng correlations that 7'~', ~ ' = ' ' • to other mod are not present in the actual nuclear st,ruet~re.., mQdels; ,Ou, r '.model for Tu seems,,t, o ~b, e~a goQd app~rQ~;imation because relatively,' ~ood ~ agreement with the e'xperiment ;is obtained. The remaining_ discrepanqies between theory and experiment may be re- moved, e. g., by includi'hgi'the seqo~c'.term o,f _the Eq. (~.10) and/dr by 'impr9y,ing the One way :,to structure model w~v~ is to incotpo- nuclear improve the nuclear_' structure model wave functions'ftinction. > ~ ' : ; ,, i rate the a-1ike four nucleon c6rrelations as discuss,ed in Section "I!J. From this s~ctibn we leairn that new; tyb~s _ ~if ,stip~fflui id pha:~e~i=iriduic~d by a-like' foti.r riuc.leon.correla- tions seem to be pi,edict~d i:~ ~'the' heavy def6rpi~d nuclei. The modej ipclude~, , iu ,addi- tion to ,the ,-=usuai BCS-pairihg correlatio,ns, _two-pairs a-like four, nucleon correlations, Balg Phys. J. l4 (1SC7,. 5 431 which are assumed to be responsibie fo'r the forinatiori of' th~ alsup~rfluid aggregates. Further on, a BCS4ike trial ~waVe fti'nction is used, which dontain~ Correlated pairs Qf protons and neutrons.' The 'main' advarifage of this ~p,proach con~ist~'in t.he'fact~ that',~it is 'able to take into account the underlying fermioriic sfrufdttiie df the ci!stiperfltiid aggiegates which are no longer apprbximated [12] by interacting ipQso, ns. Por sufficiently 'large coupltng constants the condensed, symmetry bro.ken grbtit]dLstate of stich,<'eorrelated c~lsuperfluid aggregates are energeticallypha~e nti'ciear phase.'Consequently,' :tinder these circumstances, a nor,nal fluid-superfluid favotired• • with res transition (of first and second order) is predjcted.. In adcition dominated superfluid in Section transition a•likeare' fixed from the experiment as describedphase111. The proton andis also~ predict c6hstants neutron gap ,p~~ameters ,.~~r.e slightly increased by the presence of the o.-like correla'- tions, which = induces a further enhancement of the probabilities of two-nucleon. a-transfer reactions and favoured a-decay. In addition, the proton and neutron BCS pairin,g, superfluidity can ttiutually be ' induced by one aQotber via the a-like correla- tiohs." This point might be of relevance in explaining the ~ "anomalou~,, a-d~cay rates of light Pb isotope.s [36] ' and not to conclud'e that ~~~82 ceases to bc 'a magic number. The 0+ states, identified in Ref. [37] to be'>>ptotori paiting vibratipnal stateS .in Sn.iso,topes, could be superfluid. metastable states predicted by' bur. ' I~l'odel;" becau~e e,alculated"cross-sectioris within the the ex- thewhile similar calculations for ground-ground tr'ahsitipns underestimateBCS-mod•el .oYdrest lues, periment. :. :' = . : ' . ' i . , References 1. SQI ov i e v, V. G. Theory of Complex Nucl~i. Nauka, Moscow, 'i97i; pergamon Pr~ss; N. Y. 2. Boh r, A. Rotattonal Motion in Nuclei. Nobel Lectur,e's. , .Elsevier Publ. Co. 1976, . , 3. Zh a n g - J i n g - Y e. 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