Molecular single-particle excitations in heavy-ion reactions involving deformed light nuclei

PHYSICAL REVIE% C VOLUME 25, NUMBER 4 APRIL 1982

Molecular single-particle excitations in heavy-ion reactionsinvolving deformed light nuclei

Jae Young ParkDepartment ofPhysics, North Carolina State University, Raleigh, North Carolina 27650

Werner ScheidInstitut fiir Theoretische Physik der Justus Lich-ig Univ-ersitat,

Giessen, Germany

Walter GreinerInstitut fiir Theoretische Physik der Johann Wolfg-ang Goeth-e Unive-rsitat,

Frankfurt am Main, Germany

(Received 9 October 1981)

Two-center level diagrams for the neutron orbitals in the scattering of ' 0 on 'Mg and

of ' 0 on Mg are calculated by using a deformed potential for ' Mg. Possible conse-

quences of the nuclear Landau-Zener mechanism, namely the promotion of nucleons atavoided level crossings, and of the rotational coupling between crossing molecular single-

particle orbitals are studied for inelastic excitation and neutron transfer. The important ex-

citation and transfer processes, which are enhanced by the promotion process and the rota-tional coupling, are presented.

NUCLEAR REACTIONS Heavy ion scattering, theory of nucleon

transfer, molecular wave functions, asymmetric two center shell model,

single particle excitation, deformed nuclei.

I. INTRODUCTION r, /rg —QeF/(E/A ) —&30/5-2. 4 .

One of the exciting results of heavy-ion physics isthe possibility that nuclear molecules' can beformed during the scattering process. In the mi-

croscopic description of nuclear molecules one as-sumes that the outermost loosely bound nucleonsorbit around both nuclear centers and generate ahomopolar or covalent binding. Molecular single-

particle effects are not well established yet, even

though collective molecular resonances have beenobserved in several heavy-ion systems, such as' C + ' C and ' C+ ' 0 (for details see Ref. l).

Whether molecular orbitals are formed dependson the ratio of two characteristic times, the collisiontime r, -2R/v = 2R(2E/MA) 'rs and the nuclearperiod or single-particle relaxation time ~, )2R/vF, ; ——2R(2eF/M) 'r, where E is the labo-

ratory bombarding energy, A the projectile massnumber, and e~ the Fermi energy in the target orprojectile nucleus with valence nucleons. In typicallow-energy collisions this ratio is

According to this ratio we expect that molecular or-bitals and the polarization of valence nucleons bythe field of the other nucleus may have time todevelop.

Recently, new experimental works to look for evi-dence of molecular orbital phenomena have beencarried out. For example, the Yale group observedresonantlike behavior in the excitation functions ofthe ' C(' C, ' C)' C single-nucleon transfer reac-tions. From DWBA calculations, such pronouncedstructure is unlikely. Indeed it was found thatDWBA predicts little structure and is unsatisfacto-ry in reproducing oscillations in the angular distri-butions. Structure observed in the energy depen-dence of the cross section indicates that some in-

teresting processes, such as molecular orbital phe-nomena, may be occurring. A recent coupled chan-nel calculation by von Oertzen et al. confirms thatthe neutron orbits in the grazing collision of ' C on' C are of molecular nature and this prediction is

25

25 MOLECULAR SINGLE-PARTICLE EXCITATIONS IN HEAVY. . . 1903

supported by their experiment. Imanishi et al.have shown also in the case of ' 0 scattering on ' 0leading to the excited state of ' 0~ (0.87 MeV, —, )

that the neutron can be transferred back and forthat least twice, thus indicating the formation ofmolecular orbitals. In Ref. 5 we examined the pos-sibility of the nuclear Landau-Zener excitationmechanism, namely the promotion of nucleons atlevel crossings, which may cause specific structuresin the energy dependence of the heavy-ion cross sec-tions. Specifically, in the previous work, we calcu-lated level diagrams of the two-center shell modelfor the ' C + ' 0 and ' C + ' 0 systems and dis-cussed the possible consequences of level crossingsfor inelastic excitation and neutron transfer.

In the present work, we extend our considerationfor asymmetric heavy-ion collisions involving lightdeformed nuclei. We present calculations of thetwo-center level structure for the ' 0+ Mg and' 0+ Mg systems and discuss the possible im-

portant reactions involving the excitation and thetransfer of a neutron. In this calculation the Mgnucleus is described with an intrinsic deformationbuilt into the potential of the asymmetric two-center shell model (ATCSM). For large internu-

clear distances the ATCSM yields a single-particlespectrum for Mg which is in accordance with thespectrum obtained in the Nilsson model. Thechoice of the parameters of the ATCSM and thecalculation of the level diagrams are presented in

Sec. II.A systematic study of two-center level diagrams

leads to an understanding of the possible molecularsingle-particle effects without going into the detailsof coupled channel calculations. Realistic two-center level diagrams are valuable in predicting allthe reactions which should be enhanced by the pro-motion of nucleons at internuclear distances, whereavoided crossings of molecular single-particle levels

occur. Possible reactions of this type will be dis-cussed in Sec. III for ' 0+ 25Mg and ' 0+ Mgon the basis of the two-center level diagrams ob-tained in Sec. II.

P; =a; /b;, for i =1,2 .

The barrier parameter e is defined by the ratio ofthe actual barrier height to that of the asymmetrictwo-center oscillator (Fig. 1):

e=Ep/E' . (2)

In all calculations we used the empirical value@=0.68 which was found in Ref. 5 to give a properasymptotic convergence of the magnetic sublevels.The parameters of the two-center potential aredetermined by fitting the experimental single-particle energies of the separated nuclei (R= no)

and the corresponding composite nucleus (R=0)near the Fermi surface.

A. The parameters of the ATCSM potentialfor R =0 and R = 00

For zero deformations (P;=1) and internucleardistances R =0 and R = oo the ATCSM yields the

~Mg and "0+ "Mg. In this calculation we used

the asymmetric two-center shell model (ATCSM) ofMaruhn and Greiner. The potential of theATCSM along the z axis and the associated nuclear

shape are shown in Fig. 1. The deformation para-meters P; of the ATCSM are defined by the ratio ofthe axes (see Fig. 1)

II. CALCULATION OF LEVEL DIAGRAMSUSING THE ASYMMETRIC TWO-CENTER

SHELL MODEL

In order to study Landau-Zener effects in heavy-ion reactions involving deformed light nuclei wehave calculated single-particle energies as a functionof the internuclear distance for the systems ' 0+

Z2

FIG. 1. The potential of the ATCSM along the z axisand the associated nuclear shape.

1904 JAE YOUNG PARK, WERNER SCHEID, AND WALTER GREINER 25

single-particle spectrum of a shell model with aspherical oscillator potential. The correspondingsingle-particle energies can be written as follows;

1f 7/2

'Mg 17p

Ci 3/2

e(Njl)=(N+ —, )%coo

+—[j(j+1)—l(l+1)——,]

+b[l(l+1) , N(—N—+3)] V, ,—

with ficoo C/A——'~. The parameters a and b arethe strengths of the spin orbit and the I -type in-

teractions. For the spherical nuclei ' 0, ' 0, and'Ca we have used Eq. (3) to fit the parameters C,

V0, a, and b with the experimental single-particleenergies listed in Table I.

The Mg nucleus has an intrinsic quadrupolemoment of Qo ——0.616X10 fm (Ref. 7). For aspheroidal shape with semi-axes, a and b, the elec-tric quadrupole moment is given by

X1d 3/ 2

-6

2S 1/2

1CI 5/

-10

1f (0 ='/2)

1CI 5/

1d3/2(A = /2)

1dS/2(A = /2)

2S1/2 "- '

Qo ———,Z(a —b ) . (4)

Qo———ZRo P (P —1) (5)

For the experimental value of Qp and with Ro ——

1.25 A' fm we get P=1.50. This value of the ra-tio of axes corresponds to a quadrupole deformationof P2 ——0.45. Because of this large value of the de-formation parameter we have calculated the single-particle energies of Mg and Mg in the fittingprocedure numerically using the ATCSM program.

Figure 2 shows the single-particle levels of Mgas a function of the ratio p of axes for fixed values

of C, a, b, and Vo. The "experimental" single-I

Introducing the radius Ro of a sphere with the samevolume as the spheroid (ab = Ro ), and usingp=a/b [see Eq. (1)] we obtain

-121.0 1.1 1.2 1.3 1.37 1.5 1.6

FIG. 2. The single-particle energies of ' Mg as a func-tion of the deformation parameter P defined by Eq. (1).The energies are calculated with the ATCSM for R. =20fm with the parameters listed in Table III. For com-parison the "experimental" single-particle energies of

Mg are depicted. Also the experimental single-particleenergies of ' 0 are shown.

particle levels, also shown in Fig. 2, are obtainedfrom the spectrum of Mg as follows: The spec-trum of Mg can be explained by the strong cou-pling of the neutron motion to the deformed Mgcore and described in the framework of the Nilssonmodel and the aligned coupling scheme. ' The

Mg states can be grouped into rotational bandsbuilt on single-particle excitations. The energies ofthe levels are given by the formula' '"

EI x =En lr+ [I(I+1)—2&'+&x, )g2d( —1)'+'"(I+1/2)1 .

TABLE I. Experimental neutron single-particle energies (in MeV) of ' 0 (Ref. 15) and 'Ca(Ref. 7).

Level. 1d5/2 2s &/2 1d3/2 2p 3/2 2p

17O

'Ca—4.14 —3.27 0.94

—8.36 —6.26 —4.46

MOLECULAR SINGLE-PARTICLE EXCITATIONS IN HEAVY. . . 1905

TABLE II. The characteristic quantities of the lowest rotational bands of Mg (from Ref.11): the energies of the band heads, measured with respect to the ground state and to the

negative value of the neutron separation energy ( —7.332 MeV), their spin I, spin projection Kon the body-fixed symmetry axis, and parity m, the inertia parameter A /2J of the rotational

energy and decoupling parameter d, the single particle energies e~, calculated according to Eq.(6), and the quantum numbers of the corresponding spherical states for zero deformation.

E (MeV)

Far (Mev)

I, E, m

fi /2J (MeV)

d(Me V)

e~ ~ (Mev)

Spherical

state

—7.3325 5 +2' 2'

0.231

—6.47

1d5'

0.585—6.747

1 1 +2' 27

0.163—0.202—6.82

2S 1'

2.562—4.770

1 1 +0.150

—0.473—4.88

1d3'

3.414—3.918

3 1

0.113—3.54—3.48

(f7/2

Here, e~ is the single-particle energy, d the decou-pling parameter, and E and 0 the components ofthe total and single-particle angular momenta alongthe body-fixed axis, respectively. In order to obtainthe absolute values of the single-particle energies wehave set the energy of the ground state of Mg tobe equal to the negative value of the neutron separa-tion energy, i.e., EI 5', ~ 5' —— —7.332 MeV.

The single-particle energies are obtained accordingto Eq. (6) by using the experimental energies EI xand the values of A/2J and d given in Table II.These "experimental" single-particle energies, listed

in Table II, can be specified by the quantum num-

bers of the states in a spherical potential, to which

they approach for zero deformation.Comparing the "experimental" single-particle en-

ergies with the calculated ones (see Fig. 2) we con-

clude that a realistic order of levels near the Fermi

surface can be obtained for P=1.37. In this case1

the energies of the levels lf7/2 (0 2 ) and ld3/21 5(0=—, ) are fitted exactly. The levels Id5/2 (0=—, )

and 2s1/2 (Q= —, ) have the correct order, and their

energies agree roughly with the "experimental"ones.

The final parameters of the ATCSM at R =0 andR = ao which we used in the calculations are listed

in Table III. The values of %co, a, b, and Vo at in-

ternuclear distances between these limits are proper-

ly interpolated by using the values given in TableIII.

B. The level diagramsfor 60 + 25Mg and 0 + Mg

Figures 3 and 4 present the level diagrams calcu-lated with the set of parameters listed in Table III.

TABLE III. The parameters of the ATCSM potential {in MeV) for E. =0 and R = oo. The parameters a and b are the

strengths of the spin-orbit and 1 -type interactions [for definition see Ref. 5 and Eq. (3)].

C =trna/I '/' Nucleus flop Vp Fitted levels

16O + 25Mg 30.0

160

25Mg

"Ca

11.90

10.26

8.70

—2.898—1.260

—1.200 —0.090 45.44

0.338 43.23

0.165 43.23

1d5/2~ 2$1//2

1d3/2 (0=—), 1f7/2 (0=—)1 1

1f7/2» 2J/ 3/2» F1/2

"O+ "Mg 30.0

17p

Mg

"'Ca

11.67

10.40

8.70

—1.247—1.270

—1.200 —0.090 45.44

0.063 43.79

0.155 43.791dsn, 2s1n

1d3/2 (0= )» 1f7/2 (0= )

lf 7/2» 273/2» 217 I/2


'CQ = Mg+ 0 Mg '0

— 2p3 (A=&/2)

1 g g/ —

1f7/2 (~= /2)

1f5/2 -42p

-62p 3/2

EQJ 1 f 7/2

1d3/2 (~= /2)

2S1/2—1f 7/2 (0='/2)1ds12—ids, (ti='12l

1dsliiA=Q}

2si/, (A ='/2)

1d5/2 (9= /2)

-10

-12

1ds/, (0='/2)

1p)/

1d 3/2

2S ]/2

-14

1p3/—1p, (0=' )

-161d5/2

R Irmj12

FIG. 3. The neutron level diagram as a function of the internuclear distance R, as calculated with the ATCSM for thereaction ' 0+ ~'Mg~4'Ca. The values of the parameters which are used in the calculations are given in Table III.

It should be noted that the potential of the ATCSMconsidered by Maruhn and Greiner is sphericallysymmetric about the axis connecting the twocenters. Therefore, the level diagrams, presented inthe figures, correspond to the special case in whichthe symmetry axis of Mg lies parallel to the internu-clear axis. One expects that the single-particle lev-els depend on the relative position of the symmetryaxis of Mg and internuclear axis in the touchingzone of the nuclei. In order to treat the dependenceof the single-particle levels on the relative positionof the nuclear deformations with respect to the in-ternuclear axis, one has to develop a new parametri-zation of the ATCSM potential which has not yetbeen considered in literature. Such a new ATCSMpotential is needed in dynamical calculations of thescattering of deformed nuclei in the framework ofmolecular single-particle wave functions.

Since we have fitted the parameters of the

ATCSM with the experimental single-particle levelswe have achieved the correct order of the levels inthe diagrams near the Fermi level. The correct or-der of levels is important for the discussion of pos-sible reactions on the basis of level diagrams.

III. DISCUSSION OF POSSIBLE REACTIONS

The level diagrams, shown in Figs. 3 and 4, canbe used for the discussion of possible reactions.There are two main excitation mechanisms betweenmolecular single-particle states, the radial and rota-tional couplings. Let yn i(r,R) be the single-particle wave functions of the ATCSM, where 0 isthe quantum number of the angular momentumcomponent in the direction of the internuclear axisand A, the remaining quantum numbers of thestates. Then the transition matrix elements of the

25 MOLECULAR SINGLE-PARTICLE EXCITATIONS IN HEAVY. . . 1907

41CQ

2d 5/2 2

41C 24 g 17O 24'

2p3/2t A = /2)

17O

199/ 1f7/2 (A= /2)

}= 1d3/2

2 p1/2

-62 p3/2

O

X-8~ 1f7,2

-10

1d3/2 (0-"/2)

1f7/2 (0 ='/2)

) = &4si,1d3/2 IA ='/2)

1d 5/2 (0 =5/2)

2S1/2 ~~ = /2)

1d5/2(& ='/2)

-12 1d5/2 IA='/2)

1 CI 3/2

-142S1/2

-161d 5/

1P1/2

1p3/1p1,, n='/2)

R ffmJ

FIG. 4. The neutron level diagram as a function of the internuclear distance R, as calculated with the ATCSM for thereaction ' 0+ Mg —+ 'Ca.

radial and rotational couplings can be written as: where the dimensionless "interaction parameter" 6is given by

and

a%ox

&&%ox'

(V&nx I J+ I Pn+t, v ~ .

16=Av, d

dR[e&(R)—eq(R)]

R=RC

(10)

Here, j+ ——j„+ijz are the angular momentumoperators of the single particle. At avoided cross-ings of levels with the same value of 0, the molecu-lar wave functions change rapidly as a function ofthe internuclear distance. At these distances thematrix elements (7) have the largest values and,therefore, the processes, induced by the radial cou-pling, are strongly enhanced. According to theLandau-Zener formula, ' the probability P of aquantum jump from one level to another one at anavoided crossing (pseudocrossing) at R, is given by

p ~—2mG (9)

Here, H&2 is the matrix element for the coupling be-tween the unperturbed states 1 and 2 with energiese&(R) and e2(R ), which cross at R, without pertur-

bation. The probability P for excitation increaseswith increasing relative velocity v, of the two nu-clei. In Figs. 3 and 4 we recognize many pairs oflevels between which the radial and rotational cou-plings can cause the excitation and transfer of nu-

cleons. The locations, i.e., the internuclear dis-tances, where these reactions can proceed are indi-cated in Figs. 5 —8. The distances where avoidedlevel crossings occur are fairly independent of varia-


25' 16O 25' 16O 24' 170Mg 0

1d 5/2

1d32(A= /2)

-6Ol

X

LU

1d 5/2(A = 5/2)

2S1/ {A= /2}

-10

1d5/2(A= /2)

! I t ~l ~+ I II I~ ——————1d3 (A= /2)1! 6.10 /2

6, 10-4—

~ ~ M4.5, 6,7.9,10'

-6—X

-10—

1d3»(n= »)

1f7/ (n ='/2)

1d3/2(n = ' 2)

1ds/2(n= /2)

2S1/2(n= 2)

1d5»(n = '/2)

1d3/

2,7, 8,9, 10I I I I I I I I

6 9 12R [fm)

1d5/(A= '/2)

-12—I i i I i i I

6 9 12R(fmI

1d~,,(n='/ )

FIG. 5. Inelastic excitation in the reaction ' 0+ 'Mg.The figure illustrates the processes listed in Table IV.The numbers are the process numbers given in Table IV.The solid, dashed, and dashed-dotted lines represent the

1 3 5levels with Q= —,—,and —.The straight and wavy ar-

rows indicate transitions induced by the radial and rota-tional couplings, respectively.

tions in the parameter sets of the ATCSM, as alsofound in the previous work. In the following dis-cussion we will not consider level crossings whichare located well inside the interaction region, sayR & 3 fm, since they are hidden by the centrifugalpotentials due to the radial relative motion and bythe strong absorption occurring inside. Also wewill discuss only one-nucleon excitation andtransfer.

A. The ' 0+ Mg reactions

FIG. 7. Inelastic excitation in the reaction ' 0+ Mg.The figure illustrates the processes listed in Table VI.For further explanations see the caption of Fig. 5.

level diagram shown in Fig. 3. In all these process-es a single neutron is promoted to a higher level bythe radial and rotational couplings at localized rela-tive distances. The number of steps, given in thetables, counts the number of localized couplingsacting between the initial and final states of theneutron. For simplicity we mainly consider one-and two-step processes.

The tables contain an assignment number for theprocess, the number of steps, the final and initialstates of the neutron, the intermediate state in thecase of two step processes, the approximate loca-tions of the couplings, the type of excitationmechanism, and the 0-quantum numbers of the

In Tables IV and V we have listed possible inelas-tic and neutron transfer processes which can occurin the scattering of ' 0 on Mg according to the

24 g 17O 24'g 17O

—-6)ILI

ELU

25m g ~ "'O

I /I

I II

I II

I

25 qg 16O

) = 1d5/2

1d3/2{A = /2)

1d5„(A = 5/2)

2s1,,(A = '/2)

ELU

1d 3/2(A = 3/2)

1f7/2 (A='/2)}= 1d5,,1d 3/2(A= /2)

2s1, (A= '/2)

-10

I

3 6 9 12R (fm]

1d5/2(A =3/2)

1d5/2(A ='/2)

-10—

1d5„( =' )

! I I I

'"3 6 9 12RI fm)

FIG. 6. Transfer of a neutron in the reaction' 0+ Mg. The figure illustrates the processes listed inTable V. For further explanations see the caption of Fig.5.

FIG. 8. Transfer of a neutron in the reaction"0+ Mg. The figure illustrates the processes listed inTable VII. For further explanations see the caption ofFig. 5.


g 0 ~0g

ed

0 Pes ~~

0

chch

ch(yo cd

cd

cd

g0

0ch ~ & bQch ~ ~ g9

Ch

Pa 8 Q~0

Vch

QbQ Wg % Ck g~ u

~ vH

C4

o~)8",ch 0o o.

Vch

bQ ClCk ~ chch

bQ O

g g O

74

D. q)g/1cd g cd

bQ g

~ m ch Vch~ P9~crt 0V

og

I

0C4

0~ ~~ Wo

S4

0o00

'~ 'pH

0

V

cd~ f~~l

~ H

0

z

8~ ~cd

o8

bQ

~ W

0

0

0~ ~

cJd

bQ

0o

0 0 0 0

00

ERR

mI~II

0

~IceII

0

~ I've

II

0

Iru ~IceII II

C C

ct}

~I@4II II

ggg+ + +0 0 0

eIeq aI(v eI+ ~Ice Irv

~J

cu

~jew ~I've ~Ice ~Ice ~IceT

~ /eu

0 0

II

tV

0

bQ

+0

O

II

CV

I

II

~I

+0

0 0

O

II

0

~I

fV

II

bQ

I

II

CV

bQ

f4

+0

O

II

0

II

CV

1910 JAE YOUNG PARK, WERNER SCHEID, AND WALTER GREINER

nfrv v /~1'

00

~ W

05

8

c5

00

~ W

05

a5

Y4

0

I

0

0~ W

~ W

0V

00

~ W

8

bQ

~ W

00 0

OO

II

~I

bQ

II

CV

O

0 0

oo

bQ

03V

0

V

V$~ W~ 'W

0

0~ W

bQ

0O

+ +0 0

25 MOLECULAR SINGLE-PARTICLE EXCITATIONS IN HEAVY. . .

~frv ~lml~

-Ic4 ~l~ -Icv -l~l

~Ice -Irq-irv

~IH wlc4

0~ 'I+~l

~ ~V

6~ 'I~+I

05

8

00

~ rt805OV

V

00

~ W

05

crt

74V

V

0

0V

00

~ vH

05

0$~ ~

M

ao"

~ Ieu ~ Ice

II II

~I II

CV

0

u5 oo

II II

~I

hl

0 0

-I f4

II

0

II

0

II

CV

ao

CV

0

II

CV

bQ

II

bQ

bQ

8

bQ

CV

+0

~ ~

5

O~ &

V0O

0~ ~

bO

OV

+ +0 0

bO

+

0

C4

+

0 0

Ice

II

Cg

+0

A

I

Icu

II

fV

+0

II

I

II

+0


ice ice ico

00

~ W

K',05

8

cd

0~ M

05

Y4

0

bP

V

bQ

I

0

0~ tH

~ &V

r4

8~ ~

O

8

S0

05V

05~ &~ W

0

0

~ I++I e ~OO

~j~II

C4

0

V

z z

vH g)ooX

00 IA

II

0

0 0 05 0

O

II

A

"a

Cj

OO

II

CV

00

~ ~V05V

tQ

~ &

0

VCcd

0

cd

0~ 't+I

bQ

0O

+0

j~ ~ jew

pA

ajarII

gg~fV hl

+ +0 0 0

+0

+0

+

II

Cl


levels involved. In the case of the radial couplingthe quantum number 0 is unchanged, whereas 0 ischanged by one unit by the rotational coupling.The processes are schematically shown in Fig. 5 (in-elastic excitation) and Fig. 6 (transfer processes). Inthese figures the radial coupling at an avoidedcrossing is indicated by a straight arrow and the ro-tational coupling by a wavy arrow.

In Fig. 5, for example, we note that the 2sl&2 lev-el of Mg, which is occupied initially by thevalence neutron, has an avoided crossing with theld3/2 (0—

2 ) level of Mg near 5.3 fm. This cancause the single-particle excitation to the 1d3/2(A = —, ) state by radial coupling (process number 1).Other inelastic and transfer processes can easily betraced in Figs. 5—8 by foBowing the correspondingprocess numbers.

It should be noted in the level diagrams that nolevel crossings exist for one- and two-step excita-tions of ' O. In all of the listed transfer processesthe final ' 0 nucleus is in its ground state. Further-more, it is remarkable that not only the looselybound neutron in the 2sI~2 level, but also the 1d5~2

3 1(0= —, ) and Id&&2 (0=—, ) neutrons, which arebound strongly in the Mg nucleus, are involved in'3these processes The . ld5~2 (0=—, ) and Id&&2

(0=—, ) levels have crossings with unoccupied lev-

els at internuclear distances of 4—5 fm and, there-fore, the promotion of these neutrons to excitedstates can also occur, leading to a particle-hole exci-tation of Mg.

B. The ' 0+ ~Mg reactians

In the same manner as discussed in Sec. III A onecan apply the level diagram shown in Fig. 4 topredict possible inelastic excitations and neutrontransfer reactions in the case of the scattering of' 0 on Mg. The processes are summarized inTables VI and VII and schematically sketched inFigs. 7 and 8.

Among the possible neutron transfers we findtwo types of processes in which the loosely boundneutron of ' 0 or one of the neutrons in the 1d5&2(0= —, ) levels of Mg is transferred. In the firstcase the final nuclei are ' 0+ Mg, while in thesecond case they are ' 0+ Mg.

In Tables IV —VII, we note that both radial androtational couplings are equally frequent in these re-actions. In general, the radial coupling has a largerstrength than the rotational one, as found in thework of Terlecki et al. ' Hence we expect that those

processes have the largest yields in which the radialcoupling is involved. Therefore, the Landau-Zenereffect, ' inducing the transitions at avoided levelcrossings, is the most important excitation mechan-ism for the inelastic excitation and neutron transferwhen the reaction is described in terms of molecularsingle-particle states of a two-center shell model.

IV. CONCLUDING REMARKS

Realistic two-center level diagrams give usgreater microscopic understanding for the selectivi-ty of specific reaction channels in the nucleus-nucleus collision and for the role played by thevalence nucleons in heavy-ion reactions. Studies,such as the present one, help us to select importantinelastic excitation and reaction channels and sim-plify coupled channel calculations which are tediousand difficult when a complete molecular Hamiltoni-an is used. The present study also sheds light onthe sequence of configurations of the heavy-ion sys-tems during the course of multistep reactions.Since the enhanced excitation at level crossings isdistinctly a molecular effect, it is hoped that themeasurement of excitation functions for inelasticscattering and neutron transfer reactions for the' 0+ Mg and '70+ Mg systems will give anexperimental proof for molecular single-particle or-bitals.

Level crossings between an occupied and unoccu-pied levels can destroy the internal structures, andhence lead to a damping of the elastic scatteringprocess and coupling to the inelastic and transferchannels. Single-particle excitations induced by thenuclear Landau-Zener mechanism' at avoidedcrossings thus have an influence on the imaginarypart of the scattering potential as discussed by Glasand Mosel. ' A systematic study of the promotioneffects at crossings for various projectile target sys-tems should be done on the basis of realistic two-center shell models in order to understand micros-copically the role of individual molecular nucleonicorbitals in the mechanism of absorption that mayproduce characteristic signatures in the excitationfunctions and angular distributions.

ACKNOWLEDGMENTS

One of us (J. P.) would like to gratefully ack-nowledge the hospitality of Professor John Blairand Professor Larry Wilets and the stimulating at-

JAE YOUNG PARK, WERNER SCHEID, AND WALTER GREINER

mosphere at the Topical Research Institute ofTheoretical Physics at the University of %ashing-ton, Seattle, where a large portion of the presentwork was carried out. Discussions with many par-

ticipants at the Institute were helpful. This workwas supported by the Deutsche Forschungsgemein-schaft (DFG) and the Bundesministerium furForschung und Technologie (BMFT).

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Molecular single-particle excitations in heavy-ion reactions involving deformed light nuclei