REVIS1)\ MEXICANA DE FlSICA 48 SUPLEMENTO 2. 108-116 l':OVIE~BRE 2002

Study of quasimolccular statcs in 24Mg via d-a angular correlatioll analysis in the12C(14N,d)24Mg *(a?ONe rcaction

T.L. BelyaevaUniversidad AUlonoma del Estado de México. ooסס5 To/uca. Múieo.

N.S. Zelenskaya/J. V.Skobeltsyn Institute o/ Nuclear l'hysies. M. V.Lomonosov Moscow State University. 119899 Moseow. Russia.

Recibido el 19 de febrero de 2002; aceptado el 25 de febrero de 2002

Thc mcthods oC calculating the angular corrc!alion functions in Ihe rcaclions induccd by Jight and semihcavy ¡ons ofenergy up lo 10 MeV/nucleonhave been discussed. The difTerential cross scctions and cr-d angular corrclation funclions in the reaction 12 Ce4N, d)24Mg. (a:)20Nc involv-ing high-lying slates in 24Mg al 14N bcam cnergies E1ab = 29-45 McV in the model of direct 12C transfer the framework of the distort-wavemelhod with finitc inleraelion rangc and the statistical compound-nucleus model have beco analyzed. The reduced width amplitudes for thehigh-excitcd states in 24Mg with 1208l12C quasimolccular structure are extracted and the ¡nfluence of relative motion of 12C+12C. arestudied.

Keywords: Angular correlation functions; quasimolecuJar struclure

Métooos teóricos de estudios de correlación angular de partfcula-partícula en reacciones nucleares iniciadas por los iones ligeros y scmipesa-dos con energía hasta 10 MeV por nucleón son discutidos. Las secciones tmnsversales diferenciales y las funciones de correlación angular dealfa-deuler6n en la reacción 12C( 14N, d)24Mg. (O')20Ne involucran los estados de la excitación alta en el modelo de traspaso directo de 12Cbasado en el método de ondas distorcionadas con interacci6n de radios finilo y en el modelo estadístico de núcleo compuesto son analizadas.Las amplitudes de anchuras reducidas para los estados de alta excitación en 24Mg con estructura cuasi moleculares 1208l12C están extraídosy la influencia de movimiento relativo de 12C+12C. es estudiada.

Descriptores: Funciones de correlacion angular; estructuras cuasimolcculares

I'ACS: 25.70.lIi. 25.70.Gh. 27.30.+1

1. Introduction

Experimental and theoretieal Sludies of nuclear molecularstates or nuclear quasimolccules in the heavy ion reactionshave been eondueted sinee lhe beginning 1960's, when lheywere expcrimenlally discovered [1 J in the first studies of12C+12C inleraclions. These unexpecledly narrow and well-separated resonances in the interaction of heavy ions wcrefound well correlated in different ehanncls and had a widlhmueh greater than the eompound-nucleus (CN) one for givenexeilation energy of Ihe CN syslem around 15-50 MeV. Dur-ing Lhela,1 decade nlher light sySlems which fenlure even N.even Z. N = Z nuclei have been inlensively studied [2,31primarily in elastic and inclao;;ticscattering, and in rcactionsleading lo n-particle transfer ehannels (Lhemodero Slate-of-arl one can find in reviews [4]). The light-heavy-ion grazingcollisions can fonn SlaleS of very high angular momenlumlhat are associated wiLhquasimoleeular configurations of thelwO rolating nuclei [5,61.

The study of particle-particle angular correlations histor-ically bcgan wiLh invcstigations oCthe qua'iiela<;tic knockoutreaetions A(P,2p)B" on Iighl nuclei [7J and lhe c1usleringphenomena in lighl nuclei by Lhe (a, ax) reaetion [8, 9J. Im-portant quantitative spectroscopic infonnation and sorne ncwthenretical resull, were oblained reeenIly in ReL !O. wherethe qua,iela,tic knockoUl of n-c1uslers by intermediate en-ergy protons and ultrarclativistic electrons wao;;swdicd.

As early a, in 1984 [11] experimental dala of Lhemea-surement of lhe d-n angular correlalion function (ACF) inIhe 12C(I'N, d)21Mg(a)20Ne reaetion were oblainedwhiehhave altraeled considerable interesl to lhe study of quasi-molecular states of nuclei via parlicle-particle angular cor-relalions. 11was found lhat for Lheexcited state 13.45 MeV(.1' = 6+) in 21Mg there are oseillations in ACF at forwarddeuteron emission angles. whose shape is well described bythe square of the Sixlh-order Legendre polynomial and couldnot be explained in lhe CN mode!. The authors inlerpretedthis facl as Lhe proof of direct transfer of 12 nucleons andsuggested the presence of quasimolecular 12C @12C config-urations in this state of24Mg. This result was confirmed onesagain [121 in 1994.

The analysis of the deuleron angular distributions a, wellas d-n-angular enrrelation funclions [13] pcrformed in theframework of the eompound nucleus Hauser-Feshbach for-malism cnnfirmed Lhe importance of CN eontribution. but,at lhe same lime. il demonstrated lhe smooth behavior ofd-o-angular corrclation functions for 13.45 1vleV. 6+ statcin 21Mg for all forward deuleron emission angles. It wasfound [14, 15J lhat calculalions of Lhedifferential eross sec-lions pcrfonncd in thc exact finite-range dislortcd-wave Bohrapproximalion (EFR DWBA) for Lhe direcl carbon-transferrnechanism in sorne cao;;csreproduce the experimental datamore successfully.

STlJDY 01' QUASIMOLECULAR STATES IN 24Mg VIA d-n ANGULAR CORRELATION ANALYSIS Ir-;'THE ... 109

Nevertheless, a number of prineipal questions still re-mained: Why was a polynomial strueture of the ACFs ob.served only for one seleeled state in 24Mg and was nol ob-served for o!her s~,les with high spins near this state? Is itonly exeeption rather lhan the rule? What is the origin of en.ergy dependenee of the ACFs fm !his state? What is a reae-tion mcchanism, which makcs the main contribution to thereaction cross scclions and ACFs?

The preselll study focuses on !he lhcoretieal analysis ofthe quasimoleeular slates in 24Mg and !heir infiuenee onthereaetion eharaeteristies. We ealculate the differenlial erossseetions as well as d.a-ACFs in 12C(14N,d}24Mg'(afONe10 obtain !he full deseription of experimental data knownlO-day and to answer the questions melllioned aboye. We ex.traet quantilative values of !he redueed wid!h amplitudes forthe 12C @12C eonfigurations and we investigate how doesthe relative motion of two 12C clusters cffeet on the reactioncharacteristics.

2. Methods of calculating the partide-partide angular correction functions based on the spindensity l1Iatrix and its spin tensors

We shall analyze lhe differential eross seetions and the angularemrelalion funetions in reaetions indueed by light andsemiheavyions using the generaltheory of angular eorrelations in terms of the spin density matrix (one can find ilS detailed deseriptionin the classic papers of Biedenharm and Rose [16], and Goldfarb [17)). For a binary nuclear reaction A(a, b)B' in a case ofunpolarized incident particle and target the spin densily matrix PII (M,M') of!he nucleus B' in !he state with spin 1, andprojection M, may be wrillen as

(1)

where T;,(Ob) is the transilion amplitude of reaction.For practical applications, it is convenielll to inlroduce !he spin-tensors Pk. (1, ) of the density matrix !hat can be defined in

the standard manner [171 wi!h O< k < 21, and 1< = -k, ... ,k.We normalize lhe densily malrix (1) 10differential cross section and monopole spin-lensor

(2)

Let us consider a two slage nuclear reaclion,!he firsl stage ofwhich, a(1.) + A(1A) -t B'(1,) + b(1b), is the binaryreaction where!he nuclcus B' is produced in !he excited stale. The sccond stagc, B'(1,) -tC(1o) + C(12), is lhc decay oflhe nucleus B' lo lhe nuclcus C generally in the exciled state wilh spin lo and lhe second emission, for exampl~, particlc c(!he lotal nucleus spins are shown in brackels). Accmding to !he general formalism [16, 17], the angular correlation functionW(Ob, Oc) is the probability for simullaneous delection of particle b in the direction Ob and emitted particle c in the dircctionOc' It is defined in terms of lhe spin-tensms Ph(1,; Ob)of the density matrix and the tensors "'h(l,; Oc) of!he efficiencymatrix a'i

W(l,; 0b, Oc) = I>h(1,; Ob)",•• (1,; Oc).h

(3)

In a spccial ca", of zero spin le of emilled particle (for example, a-particle decay of exciled nuclear state) and zero spin loof!he residual nuclcus C !he efficiency tensor "'k.(1, ) is of!he most simple form

(4)

The expressions (1) and (4) allow one to wrile the ACF (3) as [l8, 19]

(21, + 1)1/2 '" 1/. '"W(1,;Ob,Oe) = (21n+ l)(21A + 1) ~ (-1) YI/M/(Oe)YI/M/(Oe) ~

M¡Mí M"MAM~

(5)

Let us consider!hc me!hods of!he spin.tensm of densily matrix and !he ACF calculation in !he frameworks of two com-plementary nuclear reaction models: EFR DWBA and !he Hauser-Feshbach (HF) formalism of statistical CN mode\.

Rev. Mex. FIs. 4552 (2002) 108-116

110 T.L. BELYAEVA AND N.S. ZELENSKAYA

(7)

In !he EFR DWBA lhe transition amplilude Ti/ of!he reaelion A(a, b)B may be written as 120, 21J1 Kb

Ti/(flb) = .¡E;E¡ KEiE/ a

X L (_I)'+M, [(2!¡ + 1)(212 + 1)]'/2 (lA MAl, M, I l,M,) (1oMbl,M21 laMa) (1, - M,l,M, Ilm,)11 MI 12M'llml

X L (-1)A, +A, e'A, A, 1, 1,lx 13'm,A, A,Ix Ex (flb). (6)A1A'JlxEx

The cssence of Lhcrcaclion mechanism is contained in the kincmatical amplitudes 13lmlAI '\21 x Ex (Ob). which are the ovcrlapintcgrals of the incoming and outgoing dislorted waves, the wavc [unclioos of lhe rclalivc molioo, and the intcraclion polen.tial V. For !he one-slep direcr rerms ("break-up" of!he projeclile) V represenls lhe imeraetion potential for lhe direet strippingand heavy knock-on meehanisms. The s!ruelure faetors

e = (_I)lx u(l,A,l,A2 : lxl) e1l ...•x+ Aea ...•X+bAtA2lthlx J2Ix+1 Al1Jlx A'Jhlx'

may be expressed [21] in terms of lhe redueed wid!h ampli-tudes (RWAs) en ...•X+A and ea ...•X+b whieh describe !he

Al/l/X A212lx •probability of formation of !he cluster eonfigurations X + Aand X +b eonlaining the intermediate nucleus X in !he nueleiB and a, respeelively.

Here we take inlo aeeounl !he total angular-momenlumeoupling sohemes

1= A,+A2 = 1, + 1"

la= lx+A, + lb = lx+J2 = 12 + lb,

1/= lx+A, + lA = lx+J, = 1, + lA (8)

and use !he eoupling seheme for orbital angular momema :

(9)

where 1 is lhe transferred angular momenlum, A, and A2 areI

Ithe orbilal angular momenta of relative motion of the eluslersA + X and b+ X in the nuelei B and a, respeetively, and !¡and 1,are lhe 10lallransferred angular momenta.

We use the eonvenlional definilion of!he clusler speetro-seopie amplilude of redueed wid!h for decay probabilily oflhe nuclei a and B via lhe ehannels X +b and X +A, respec-livcly, in lerms of lhe fraelional parentage eoeffieienls (FPC)in the translationally invariant shell model technique (sec, forexample, Refs. 22-26). Withoul ehanging the formal slrue-lure of the EFR DWBA formalism lel us introduce RWAs inthe vertexes B --t X + A and a --t X + b, whieh describe apo.ssibilily of formalion of a massive cluster X possessing itsown cxcitcd sta les as

( 10)

whcrc

1/'=a...•x+b_(Na) (!:.)NA'/' "\' aJ.'I~ ah7'x aJ •.r,-A,lx1, - Nx X ~ [/.II••S. [/x]1>xSx '''IL.S.

II/.IL.S. T.If.]L,S,II/ x)I>xSx'I'x

X (_I)A,+lx +1. -1.u(A,LbJ,Sb; L21o)K(A2Lb; L,) (Tx M'I'x TbM-r.ITaM7'.) (t ~:~:)La Sa la

X (Na[fa]LaSaTaINx[fx ]LxSxTx; NbA2[fb]LbSbTb{L2}) (11)

In (11) a(i.ii.s. are !he inlermediate eoupling eneffieients, K(A,Lb : L,) are the general Talmi eoeffieienls !hat extraet theintrinsic wavc funetiao of "ucleus b in the given state with orbital momcntum A2 [rom the wavc function of b nucleoos withangular momenlum L2 [22,23]. We wrole !he expressions (10) and (11) for vertex a --t X + b (we refer 10 subseript 2 forverlex a --t X + b and subseript l for vertex B --t X + A) taking in the mind that for vertex B --t X + A !he expressions arethe same wilh lhe substitution 2 H 1, a H B 1 b H A.

Finally, substituting (6) into (5) we write!he expression for!he angular eorrelation fllnetion in lhe EFR DWBA for lhe casewhen lhe spins of emitted partiele e and the residual nucleus C are zeros

2

LM::Z;:M.(flb) X Y'~MI(flc)MI

(12)

Rev. Ma. F'ls. 4S 52 (2002) 108-1 t6

STUDY OF QUASIMOLECULAR STA1T,~ IN "Mg VIA d'Q ANGULAR CORRELATION ANALYSIS IN THE ...

whcrc

M:;~;M.•(Ob) = L (2I¡ + 1)'/' (lA MAl, M¡[l/M/) L il (1, - M,l,M¡[lml)11M¡ lm¡

111

x L (-1)l+A,+A'0IA,A,I,I,lx13lm¡A,A,lxEx(Ob)' (13)/'qA"lxEx

Le! us now eonsider the method of the spin-tensor of the density matrix and ACF ealeulation in the modified statistiealCN model [271, whieh al!ows to investigate interaetion between nuelei with arbilIary spins ineluding the spin-orbit in!eraetion.Writing the transition amplitude Ti/ of the reaetion A(a, b)B in the usual manner

(14)

we use the redueed width teehnique and express the wave funetions >Ji" >Ji/' and >Jieof initial, final. and eompound quasista-tionary state A with spin le, leve! energy E, and IOtal width f, in the forro

>Ji,= >JiA>JiaX(+)(Kara),

>Jif = >JiB>JibXH(Kbrb),

1>Jic(r, E,) = (E, _ E) + tf, >Jic(r),

>Jie(r) = L (laMI.!AMI .• II¡ MI,) (I¡ MI, lamalleMle) O~~:)j;'A+a>Ji A>Jia>Jidra)11 MIl/eMe

L (hMIJf MI,IM,f¡, > (l,MI,lbmblleMle) o~}~t'B+b>Ji B>Jib>Jil,(rb),I'2M,,,lcMc

(15)

where X(+) (Kar a) and X(-) (Kbrb) are the distoned waves in the entranee and exit reaetion ehannels, >JiB, >Jib, >JiA, and>Jia are the internal wave funetions of corresponden! nudei, >Jil. (r a) and >Jil,(rb) are the wave functions of relative motion,O~}:)j;,A+a and O~}~;;B+b are the reduced width amplitudes of the deeay of the CN quasistationary state A via the channels

C-+ A + a and C-+ B + b.Here we define the fol!owing angular-momentum eoupling seheme

II = la + lA, 1,= Ib+lf

le = 1, + (,= 1, + 4, (16)

where (, and 4 are the orbital angular momenta of the panide relative motion in the enlIance and exit channels, and 1, and 1,are the total spins of the enlIance and exit channels.

lnserting the partial-wave expansion of the distorted waves, we can obtain the lIansition amplitude and the spin-tensorsPh (I/ ) of the density matrix in the case of forroing only one CN resonance. In the region of the quasieontinuous speetrum,where resonances overlap strongly, it is necessary to average the contribution of eaeh resonanee over the energy and to sum al!resonanccs in the avcraging intervalo As a result, wc havc the transmission cocfficienl~7í:71 , which are related 10 the averagepanial width amplitudes

(17)

and the average spaeing DIe between levels as

(18)

Rev. Moc. Fis. 45 52 (2002) 108-tt6

112 T.L. BELYAEVA AND N.S. ZELENSKAYA

Finally, the spin-tensors of the density matrix in the statistieal Jimit of lhe CN model take the form

7;Jc T.lcx '¿;ir~'/,?"m, (Bb)F\,m; (Bb)[(21+ 1)(21' + 1)(21. + 1)(210 + 1)]1/' (Im,I' - m;lk,,) (I.Olb - mJllm,)

x (1~01; - m;ll'm;) x w(J,1,1,1~: hk)w(J,II~I': /¡k)w(/¡l.hh: 1el)w(JII~/~lb: lel')} (19)

The tO~11decay widlh G(Jc) ineludes all the energetieallyallowed dccay channels of lJie CN and lhe transmission cocf-ficients T¡;c can be delennined lhrough the clastie scalteringmatrices of the entranee and exil channcls, and may be ealeu-laled, for example, in the optical model of elastic scaltering.

Finally. wc are able to calculate ACFs by using lhe spin-tensors of the density matrix (19) and the appropriale expres-sion for the efficiency lensors [ •• (J,).

3. 12C(14N,dj24 Mg'(et)20Nc rcactioncalculations and quasimolccular 12C~ 12C'statcs in 24Mg

3.1. DilTcrcnliall'ross scclions

The first resulls of studies of nuclear reactions induced by ni-lrogen ions at incidenl energies up to 100 MeV on 1p-, 28-1d-shell nuclei perfonned in lhe early 1970s [28,29) sug-gested thal these reaetions could be lrealed as ones involvingthe transfer of large group of nueleons. Nevertheless, aflerfurther analysis of the spectra, the exeitation functions, and,in particular, lhe observat;ons of correlations of resonanee-likc structurc in the cxcitation funclions [rom clastic seat-tering and from reaclions. it was concluded thal lhe mech-anism of CN formalion plays lhe dominant role in these reae-lions 130-321. However, afler analyzing the differential erosssections for these rcactions, in Ref. 32 and later io Ref. 14.it was noticcd mal the angular distributions have a clear os-eillated slructure. This indicales that the statistical CN modeldeseribes the smoolh, but substant;al background, on whichoscillations oc irregularitics a'isociatcd with othcr rcactionrncchanisms are supcrimposcd. lhe conlribulion oC which canbe comparable to thal of CN formation.

We perfonned ealculalions of the differential eross see-tions al 14N bombarding energies of 29 and 35 MeV forthe diffcrcnt cxCilCd statcs in 24Mg, using the statislical eNmodel and diccet massivc cluster transfer model in the fmme.work of the formalism presenled in See. 2.

The computer code CNCOR 133) based on the modifiedHF fonnalism of lhe CN mooel was used lO oblain a corre-spondent conlribution of lhe statistical CN rnechanism to thereuction cross section. The CNCOR code allows one to cal-culute a1l lhe components of the spin.tensors of the densitymatrix for any state of the final nucleus B', the differenlialcross-section of the reaction, particle.particle and particle.,-

TABLE 1. Reduced width amplitudes e~:~;;:2C'"+d.A. Ix = 0, g.5.

¡, O ~A, O ~e14K-+12C+d O.3~6 0.736A21x 12

B. Ix = 2, Ei = 4.44 McV

1, 2 O 2

/\, [) 2 2 2e14K-+12C+d 0.885 0.415 -0.533 0.198A21X12

quantum angular correlation functions. The detailcd discus.sion of parameter et.oice in the CN calculation was presentedin our previous work [131.

Let us now considcr the deuleron angular distrihutionsin the 12C(l1N,d)21Mg" reaction in the dircct massivc clus-ter transfer model and lel us stan with the RWA (10) and(11) ealeulation for the 14j1,'--+1' e + d vertex. We use theshell modcl wave functions of nuclei with lhe inlennedialeeoupling eocffieienl' at;:í7.. s.' obtained in Ref. 35 by thediagonalization of the Hamiltonian with nuclcoll-nucleoTl in.

. 14N 12c dteraellon. The calculated R\VAs eA;';'2 + for lhe trans-fer of massive cluster 12C possessing two intcnnediate slalcs[g.s. (0+) and 4.33 MeV (2+}I are represenled in Tablc 1.

24M -f'12c+12cThe RWAs 0A, I,glx were considered as adjustedparameters. whieh thcn wcrc uscd for thc stmclurc factor01A1A2ltl2lx (7) calculation. Reaction cross section calcu.lations for lhe direet transfer mcchanism wcrc carricd outin the EFR DWBA using the OI.Y\1I'-5 eomputer code [331.which we modi (jea to perfonn summation over spin Ix 01'lhe intennediatc nuclcus ami tOlal angular momcnta 11 ami1,. The optieal-pOlential and interaet;on-polelllial paramelersare shown in Tablc 11.

Figure 1 represents lhe comparison of the results of eNand EFR DWBA ealculal;ons wilh lhe experimenlal differen-tial cross sections for the 12C(l1N 1 d)24 Mg'" rcaclion invnlv-ing differenl cxcitcd statcs in 21Mg at hcam cncrgy 29 McV.The theorctical curves are lhc incoherent sum of lhe CNmodel and direct massivc transfcr calculutions. We look ioto

Rev. Me:<. Fls. 45 52 (~OO~)108-116

STtJDY o¡: QUASIMOLECULAR STATES IN 24Mg VI¡\ d-Ct ANGULAR CORRELATION ANALYSIS IJ'\'TI lE . _. 113

"C("N,d)"Mg', 29 MeV 29 MeV

8.11 MeV, 6'

...'-

13.45 MeV, 6+

13.21 MeV, 8'o '.

14.15 MaV, 8+

....::_~.--_.---_.,"

.... ..•

!. ..••_~ - •.- ~: •..-- •....•---

......

10" --.... ._0 ........•_~102

10'

102

137 MeV, 2"

02040 W M l001~1401M1M8cm(dcg)

101O 20 40 60 80 100 120 140 160 180

8<.", (deg)

FiGURE 1. 'fhe dcutcron angular disUibutions from the 12C(14;.¡, d)24Mg'" rcaction al beam cncrgy Elab = 29 McV calculalcd [oc the dircct12C lmnsfcr mcchanism (lhe dashcd curves), eN rncchanism (lhe dotlcd curves) and fOf lhe incohercnl sum of (hese two rncchanisms (lhesalid curves). The experimental poinls are [rom Rcf. 14.

TAHI.E 11.Oplica! model and inlcraction patemial paramctcrs.

Channel V R,. av IV Rw aw Re(McV) (fm) (fm) (MeV) (fm) (fm) (1m)

14N+12C 100.0 5.6 0.48 27,0 5.n 0.26 6.58d+"Mg 50.0 4.33 0,59 16.0 4.33 0.59 4,33

d+"C 2.97 0.65 2,9712C+12C 4,235 0.7 4.235

accounl a stripping rncchanism of direct transfcr of lhe 12C'"clusler in !he g.s. and in lhe firsl exeilcd 4.44 MeY (2+) stale.

Qur analysisshowcd !hal a direcltransfcrof Ihe 12C clus-Icr providcs approximalely 80-85% of the eross seelion alforward angles for lhc 9.8. (0+), 70% for the 1.37 MeV(2+) level, 50-70% for thc 4.12 McV (4+) and 8.11 McV(G+) level, 55--60% for the 13.45 MeY (8+) level al 35 and29 Mev. The CN meehanism is a Icading one al big anglcs.For lhe 13.45 MeY (G+) slale a direel transfer meehanismdominates al aU angles. hccausc lhe eN cross sectjo" is smallduc 10 lhe smalll~nax and l¡;ax valucs.

24M --+12C+12CWC could clo;tirnatc 0AI/Ig/X valucs by compar-

¡son of !he experimental and calculatcd diffcrcmial crossseetions and ASFs. The cstimalcd values of !he RWDs

24M -+I2C+12C0A,I,glx are shown in Table Ill. Nalurally, Ihey de-pend on a choice of Ihe modcl paramelers and, in particular.on lhe radius of 12C+12C intcraction variation. For examplc.a sumcienl inerease of the 12C+12C interaclion radius (fol-

:14M 1:le 12Clowing lORef. 36) leads lOa decrease of!he 0 A, I,gi; +values by faclor 3-5. However, !hese RWA values are one-lwo order larger Ihanlhe analytical ones calculaled in Ref. 36.This faet givcs r¡se a qucslion of whcthcr lhe conventionaltcchnique of clusler spcctroseopic amplitude calcu!alion isapplieablc lo deserihe multinucleon quasimoleeular SpeClrO-scopic ampliludes. It should also he nOled, lhal !he angulardistribulions, lheir fonns and absolule values, depend on lherelative conlribulion of lhe 120Z)12C configurations wi!h dif-

24M 12C "cferclll0 g-> +A¡lllx

The angular distributions for lhe low-Iying levels Ig.s.,1.37 and 4.24 MeV (2+») have lhe mosl pronounced Slrue-lure. In particular, Ihe best agrcemelll wilh lhe experimenlaldata for !he states 1.37 MeV and 4.24 MeY (2+) was ob-lained by proposing lhal!he componenls wi!h Al = 11 = 2,Ix = 0,2 illlroduce lhe main contribution, whereas IheAl = O. Ix = 2 componcnl is negligiblc. Duc lO t.he con.lribulion of lhe major componenl with A, = 4 (Ix = 2) aposilion of lhc firsl angular distribulion maximum becomcsmore corrccl.

Rev. Mex. Fh. 45 S2 (2002) 108-116

114 T.L. BELYAEVA ANO N.S. ZELENSKAYA

, o" 24 12 12IABLE 111.Rcduccd wldth amplitudes el''1~lgí-; c+ e

1345 MeV, 6"

E, •••= 33 MeV(a)

(1 RWAs for the 13.45 McV(6+) state wcrc obtaincd al Elab =33 McV aod 42 MeV by comparison of calculalcd and experimen-

tal data from ReL 12.

120

1:1:01: 1: 1

0:1:1

0:1:0

0:1:2

1:1: 10:1:1

0:1:01:1:0

(b)

100 120

'00

40 60 80

O,.m(deg)

40 60 80

1',..- (degl

13.45 MeV, 6'

EL.t>=42 MeV

20

200.0

O

05

ª 04

"""E 0,3oz~g 02€«

00o

0.1

02

~ 08

""Eo 06z<-,:e 04 \

< "\.

FIGURE 2. d-a-Angular correlation functions from the12C(14Nld)24Mg.(o:)20Nc rcaction al bcam cncrgies a) E1ab =33 McV and b) 42 McV for lhe 13.24 McV (6+) stale in "Mgal deuteron aoglc Oc.m. = O°. Thc experimental points are fromRcf. 12. The curves represent dirccl12C'. (Ix = 2) transfer calcu-latiaos roe 1208l12C. configurations correspondent lo the diffcrentrelativccontributionof8At=4: 8"\=6: e"1=8-

OO0.447

0.447

OO0.28

0.56

O

0.35

0.35

0.84

0.66

O

0.66

-0.2

0.66

0.2

0.66

-0.2

0.35

0.35

0.35

035

O0.447

0.447

0.447

O0.28

0.28

O

0.84

0.66

O0.66

-0.20.66

O

0.66

-0.2

0.35

0.35

0.35

OO

0.447

0447

0.447

O0.114

0.114

OO

035

0.35

O

29 MeV 35 MeVa 42 McV

O2

22

O2

2

2O

2

2

2

OO2

2

2

O2

2

2

O

22

2

4

4

44

2

2

2

2

o2

O2

42

O

2

4

4

2

4

6

6 66 46 66 86 66 46 66 88 88 68 88 10

o2

2

2

2

8+ ,13.21;

and

8+ ,14.15

6+ ,13.45

6+,8.11;

and

6+,8.44

2+ ,4.24

g"Mg,~"(MeV) 11 Al Ix

O+,g.s.

2+ ,1.37

The angular distributions for the high-Iying levels (E" 2:8 MeV) do nol have any dislinguish s!rUeture, and so lo ex-trael the RWAs we analyzed lhem simullaneously with thecorrespondent angular carrclalion functions.

3.2. d-", An¡:ular eorrelalion funelions

The direet 12C transfer ealeulalions for ACF in the frame-work 01' lhe formalism developed above were earried OUlforthe exeiled states in 24Mg beginning from lhe 4.12 MeV (4+)statc. Making a comparison of the ealculatcd ACFs and thecxpcrimcnL.'l1anes al diffcrcnl incident cncrgies wc wcre ah le10restore the cluster slructurc of the givcn statc. In Fig. 2 wcshow lhe ealculaled and experimenlal [11, 12) ACFs for the13.45 MeV (6+) slale in 24Mg al 14N bombarding energies01' 33 (a) and 42 MeV (b) at deuleron angle e'ab = Oo. Qnecan sce lhnt the experimental ACFs al E'ab = 33 manifest

elear polynomial angular dependency. A fcrm of lhe ACF an-gular dislribulion is determined by lhe relalive populations 01'the Aff projcctions. Al zcro-angle dcutcron cmission Lhean-gular momenlum projcetions MI = M2 = 0, oi 1, oi 2 onlyare possiblc. Meanwhile. lhe ACF ealeulalions performed

24M -+12C+12Cwith differenl 0A,I,'Ix values have demonslralOOslrong dependenee 01' the MrProjeelion popolntions on lherelativecontributionofdifferenl [12C 0'2 C"JA,I,Ix config-uralions, whieh are eharacterized by lhe angular momenlumcoupling seheme (8). Qur present caleulations show lhal lheagrccmcnt with lhe experimental data al 1-iN bombarding en-ergy 01' 33 MeV can be reached only if a coherenl summation01'lwo 12C0'2C" eonfigurations wilh A, = 4,6 and Ix = 2(i.e .• "C-clusler is lransferred in lhe firsl excitOOslale) is ear-ned. The 12C (g.s.) transfer with Ix = O should be limitOObecause lhe only A, = 11 is allowed in this case. Ir lhe beamenergy inereases up lo 42 MeV, lhe experimental ACF s!rUc-

Rev. Mex. Fls. 45 S2 (2002) 108-116

STUDY OF Qt..:ASIMOLECULAR STATES I~ 24YJg VIA d-Q; ANGULAR CORRELATION ANALYSIS IN TlIE ... 115

90

9C

(a)

(b)

1:1:0

1:1:1

1:1(1,,"'0,2):00;1:0

,-

29 MeV

33 MeV.•.•. 42MeV

42 MeV with

(.)"06:B"••:e~I'lo =O: 1:.

30 60

0c.m. (deg)

1321 MeV, 8'E¡•• "'29 MeV

1321 MeV,8'

El'" =29-42 MeV

"

,,,,. '. ,

1.0

00o

00o

08

I~re~ 04

t-~ 0,2

1

Finally, we can eonclude !hal lhe study of !he energydependenee of ACFs gives !he mosl importanl inforrnationaboul !he relative eontribution of lhe different 12C0l2C'eonfigurations lo the quasimoleeular stales in 24Mg. The fulland delail presentation of lhe obtained resulls is published inRef.37.

Ba~ed on !he general !heory of lhe spin density matrix andit~ spin-tensoes we have presenled !he methods for ACF eal-eulations. The !heorelieal analysis of !he differential erossseetions and ASFs in !he l2C(14N,d)24Mg(a)20Ne reaetionwa~ perforrned under lhe assumption lhal direcl transfer ofl2C' cluster and CN meehanisms are realized in lhe reae-lion. Using!he tOlal transferred angular momenlUm eouplingseheme we have ealeulaled !he RWAs for !he lighl vertex"N --+12C' + d involving g.S. and 4.44 MeY (2+) inter-mediare slates in 12C•. Ilmade possible lo obrain quantitativeagreement belween !he !heoretieal ealeulations and lhe exper-imem••1dala. The eontribution of Ihe CN mcchanism ealeu-lated with lhe eritieal CN angular momentum le" whieh sat-

FIGURE 4. d.Q.ACFs for lhe 13.21 MeV (a+) state in "Mg. <alDirect I2C• (Ix = 2) transfer calculations at E14N = 29 MeVfor 12C012C. configurations correspondent to the differcnt rel-ativc contribution of 81\1=6 : 81\1=8 : 8A1=10. (b) Energydependencc of calculated ACFs for 120~12C configuration witheA1=(l == 8A1=8 = 0.35; 8A1=10 = o.

4. Surnmary and conclusion

12090

29 MeV33 MeV

. - - •• 3!i MeV

42 MeV

60

Oc ".,(deg)

30

1345 MeV, 6'Ellltl

=29~42 MeV

05

00o

FIGURE 3. d - o-ACFs for the 13.24 MeV (6+) sla(c in 24Mg for12C012C configuration with eAl=4 = 0.114; 8A1=6 ::::;0.114;e/,!::8 = o al ditTcrent bcam cncrgics.

lUre beeomes smoother and does nOl show elear oseillalions.To explain !his behavior of experimental ACF, we proposedlhal allhe high ineidenl energy !he eontribulion of!he higherrclative mOlioo angular morncnta Al = 6,8 becorncs moreimportanl, and il causes a smoO!hing of lhe ACFs. Assuminglhal !he physieal motivation for lhe unexpccledly big RWAvalues is lhe orbital relative motion of lwo l2C nuelei, il isnO! a surprise lhal ACFs exhibit !he pronounee energy de-pendenee, whieh is shown in Fig. 3. Moreover, il shouldbe stressed, lhat it is !he energy dependenee of lhe ACFs,whieh indieales lhal lhe orbital mOlion of lWOl2C-elusleesis responsible for lhe fonnation of lhis qua~imolccular stale.Indeed, for the reaelion involved !he given exciled slale lhetransferred momenlUm t:J.K = Ka - K, inerea~s a~ thebeam energy inerea~es. Sinee when mulliplied by !he inter-action radius, it givcs the maximum transfcrred angular rno-menlUm I = A, + A2, we see !hal al !he high ineidenl energylhe relevanl eontribulion of lhe higher allowed transferred an-gular momemum / and!he higher relative mO!ion angular mo-menta Al beeomes more important.

The absenee of oseillations in ACFs foe lhe levels differ-cm from lhe 13.45 MeY (6+) stale, whieh wa~ diseussed inReL 13, requires reasonable explanation and justifieation. Lelus eonsider lhe results of dirccl-transfer model ACF ealcula-lion for !he o!her high-Iying exeiled stales in 24Mg belween4 and 14 MeY exeitation energy. Figure 4a shows lhe resulls

24M -+12C 12cof ACF ealculalions oblained wilh different eA, I,glx +values for !he 13.21 MeY (8+) Slale. One can seo lhat !heACFs ealculaled wi!h equivalent eontribution of lhe eon-figurations wilh Al = 6,8,10 and Ix = 2 do nO! eon-lain any oseillaled s!ruelure and are suffieiently smoo!h. TheACFs presented in Fig. 4b were ealeulated at differenl'4Nbombarding energies EI•b = 29,33,35 and 42 MeY using24M -+12C+12CeA, 1,"lx values from Table JIl. One can sce !hal c1ear

oseillalions in ACFs for!he 13.21 MeY (8+) stale may ap-pear al high ineidenl energies, for example, al42 MeY.

Rey. Mex. Fls. 45 S2 (2002) 108-116

116 T.L. BELYAEVA A;-';D :\'.5. ZELEXSKAYA

¡slles the sclected mies in the entrance and cxil rcaction chan-ncls, docs nol excced 50% of the total eross seetion for all thestudicd excilcd statcs in the final 2'1Mg* nuclcus al bcam CTl-

ergics 29 and 35 MeV. Our ealeulations eonfinncd thatthemechanism of I'C-c1uster direet transfer plays the dominantrole in this rcaction bccausc of!he presence of qua"imolccu-lar 120S;12C conflgurations in fcw statcs in 21Mg.

Thc consistenl calculations of the diffcrcntial cross scc-tions ami ACFs allowed us to estimate RWAs for the2.1l\.1g* -+ 12C+12C" vertex, assuming diffcrcnt cOTllributionsof various [l:le 012C.] 1\11

1/ x configurations. The cxtractcd

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24Mg-+12C+1:lCe A

1/1¡x values are found 10 be sufficicntly laeger (by

a factor of lO') lhan ealeulaled inthe framework of lhe clus-ter model 1361. The big RWA values as well as !he energydependenee of lhe ACFs are eonsistent wi!h the suggestionabout the orbital rclativc motion of two 12e with angular roo-menta Al in thc certaio quasimolccular configuration.

Acknowledgments

This work was partially supported by the CONACyT,México.

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Re\'. Mex. Fís. 45 S2 (2002) 108-116