PHYSICAL REVIEW C VOLUME 40, NUMBER 5

Fusion calculations for Ca+ ' ' Ca

NOVEMBER 1989

H. Esbensen, S. H. Fricke, and S. LandownePhysics Division, Argonne National Laboratory, A rgonne, Illinois 60439

(Received 19 June 1989)

Calculations are carried out for the Ca+ ' ' 'Ca fusion reactions, taking into account both thelow-lying inelastic excitations and the dominant single-nucleon transfer reactions which coupledirectly to the entrance channels. The results agree well with the " Ca+ Ca fusion data but theyunderpredict the low-energy Ca+" ' Ca measurements.

I. INTRODUCTION of the calculations are given in Sec. IV. Our conclusionsare summarized in Sec. V.

The study of how low-energy fusion rates depend onthe structure of the colliding nuclei is an active area ofresearch. A natural way to approach this problemtheoretically is in terms of a coupled-channels formula-tion which explicitly includes the direct reaction process-es. In this way, one calculates fusion as the difference be-tween the total reaction cross section and that due to thedirect reactions. The nuclear structure parameters insuch fusion calculations are constrained by the experi-mentally observable direct reaction cross sections.

The coupled-channels approach is dificult to imple-ment for heavy systems where many reactions are possi-ble. It is also cumbersome to include nucleon transfer re-actions in the traditional, microscopic way used in first-order perturbation calculations. Therefore, it is impor-tant to develop approximations to reduce the number ofchannels and also to simplify the treatment of transfer re-actions.

A recent analysis of the Ni+ Ni fusion reactionused newly developed techniques to specifically study theefFect of one-neutron transfer reactions. ' It was foundthat after a11owing for these transfer processes, in addi-tion to the vibrational excitations, the calculations stillunderpredicted the low-energy fusion cross sections by asignificant amount. Presumably, the discrepancy is dueto the direct transfer of two nucleons. ' A similar con-clusion was reached for the case of Si+ Ni in Ref. 2.

In the present work we apply the techniques developedin Ref. 1 to study the fusion of calcium isotopes. It hasbeen appreciated for some time that allowing for vibra-tional excitations is not sufFicient to explain the variationsthat are observed in the Ca+ ' ' Ca fusion reac-tions. ' A recent paper has reaffirmed this problem.Another work has also estimated the effect of single-nucleon transfer for the Ca+ Ca system. It is of par-ticular interest, therefore, to explore the role whichtransfer reactions play in governing the low-energy fusionrates of the asymmetric combinations.

The following section briefly reviews the formulationfor the coupled-channels calculations and the manner bywhich the single-particle transfer reactions and vibration-al excitations are incorporated. The actual parameterswhich are used are then presented in Sec. III. The results

II. FORMALISM

A. Rotating frame approximation

To reduce the number of coupled channels we use therotating frame approximation, as was done in Ref. 1.This consists of transforming to a coordinate systemwhere the z axis follows the projectile motion and thenneglecting the Coriolis couplings that such a transforma-tion generates. We only summarize the basic result here.

In the coupled-channels scheme, the radial wave func-tions are determined by solving the set of equations

p2

2pp

lp(lp+ 1)+ —kp u@ I (r)dp' r P P

(f3(IpIp )J~ V~ y(l I )J )u, (r),yl I

(2.1)

where J is the total angular momentum obtained by cou-pling the channel orbital angular momentum l& to the to-tal nuclear spin I&. The channel energy is defined by

Ak P (2.2)E+Qp=2pp

(2.3)

where the coupling element is

where E is the center-of-mass energy in the entrancechannel, Qp is the Q value for the reaction channel, and

p& is the corresponding reduced mass.We shall only consider reactions which have spinless

nuclei in the entrance channel. In that case, the rotatingframe approximation is effectively introduced by replac-ing lp on the left side of Eq. (2.1) with J. This amounts toignoring terms of the order I&/J. The equations to besolved in this approximation can then be reduced to

d + J(J+1)d„z „2 p p p

"p

g Fpl ~1 u~l (r),yr

40 2046 1989 The American Physical Society

FUSION CALCULATIONS FOR Ca+ ' ' Ca 2047

Fp =(I 0AOI Jpo) „' &Prp) v, ~~y J, &Ip, FIy Y Ip 4n(2.4)

and

~x(r)= f dq q Jx(qr)kt, (q)FI, (q) (2.10)

with I=&2I+1. The number of coupled channels hasbeen reduced considerably. There is now only one chan-nel for each physical state, characterized by its Q valueand total nuclear spin, I&. The coupled equations aresolved with the boundary conditions of ingoing waves ata radius inside the barrier in order to simulate the fusionprocess. They are matched in the usual way to appropri-ate Coulomb waves at a radius outside the range of thenuclear potential.

B. Single-particle transfer

&«l~ 'j m &=Pi—(r )l I'i (r»yl/2lj. . ,

the transfer matrix element can be written as

& l2, —,',j2, m21 v,„ll),—',j),mi &

(j(m, Ap~ j2m~)A,p

(2.5)

We treat the transfer of a single nucleon from a boundstate of one nucleus with quantum numbers (l „—,',j„m, )

to a bound state of another nucleus (l2, —,', jz, m z ) in whatis essentially a no-recoil approximation. The transfer in-teraction, V,„, is taken to be the binding potential for oneof the nuclei. Using wave functions of the form

P((q)= f dr r P((r)J'((qr)0

F&(q) = f dr r Pt(r) V (r)j l(qr

(2.11)

(2.12)

2 '1/2 (2.13)

This effective coupling for each set of single-particlestates was found to work very well in Ref. 1. It is usuallydominated by the contribution which has the largest totalintrinsic spin.

In these equations, j&(x) is the spherical Bessel function.In deriving Eq. (2.10), the binding potential for the donornucleus has been used for the transfer potential. Usingthe potential of the recipient nucleus would involve anobvious modification to these equations.

We also make use of an additional approximation toreduce the number of transfer channels in the coupledequations. The total intrinsic spin k for a given nucleontransfer between a specific pair of single-particle statescan have several different values. Each one is associatedwith an independent transfer channel P which has a cou-pling F&& 0 with the spinless entrance channel o.. Thesetransfer channels all have the same Q value. In the actualcalculations we replace them by a single effective channelwith a coupling strength defined by

X (12J'2~~ Vi~( )

~~ iAJ2

(2.6)

where r is the core-core separation. In the no-recoil ap-proximation we have

I1=r+r2

A mass-dependent factor of3

4m, mz

(m +mb)(m„+mp)

(2.7)

(2.8)

(l,j,[/V, „( )/[l, j, &=( —1) '

J2

4 J1A,X — (J &A.—,'0~ j2 —,

')

1T J2

is included in the actual calculation of the matrix ele-ments to partially account for recoil effects.

In Ref. 1, the transfer matrix elements were calculatedusing an analytic approximation based on the Buttle-Goldfarb method. In the present work we evaluate thematrix elements directly by using numerical wave func-tions for each of the single-particle states. A straightfor-ward method of rendering the matrix element into theform of Eq. (2.6) is to use the Fourier transforms of thewave functions. ' The result is

C. Vibrational couplings

(A, [i VP~, (r)[iO& = —o „

Z1Z28 3

T 2A. + 1

R,

(2.14)

where A, is the multipolarity of the excited vibration, o.„and o, are the nuclear and Coulomb coupling strengths,and U(r) is the nuclear ion-ion potential. The couplingstrengths are given in terms of the deformation lengthsby

(2.15)

The couplings to surface vibrations are treated in astandard fashion. " We only include first-order vibra-tional excitations in the initial mass partition. Thesecond-order excitation processes which were found to beimportant for the Ni+Ni system in Ref. 11 are weaker inthe Ca+Ca case because the strength of the coupling in-teractions in the barrier region decreases for lighter masssystems and also the excitation energies increase. Theminor effect of the second-order couplings in the presentcase will be shown explicitly in Sec. IV. The relevant sin-gle phonon couplings are given by

where

X A~(r), (2.9)A complete calculation must include excitations in

both the projectile and target nuclei. In this work we

2048 H. ESBENSEN, S. H. FRICKE, AND S. LANDOWNE

reduce the number of inelastic channels by introducingan approximation that combines target and projectile ex-citations of identical multipolarity into a single effectivevibration. The effective coupling strengths and excitationenergies for this effective vibrational channel are definedin terms of the original strengths and energies by

Oc(fm) (MeV)

TABLE II. The Coulomb and nuclear coupling strengths (0.,and 0.„)and excitation energies E* for the low-lying vibrationalstates in the Ca isotopes. The nuclear strengths were obtainedfrom analyzing ' 0 inelastic scattering data in Refs. 14 and 15.

L

2 1/2~0) (2 16) "Ca 32+5

0.46500.13800.3420

0.31500.12500.1750

3.7373.9054.492

E' = g o„E; . g o „ (2.17)

This projectile-target symmetrization is obviously exactfor a reaction involving identical nuclei.

2+4+2+4+35

0.30470.14950.10440.04510.24540.1326

0.23980.08180.09030.04230.16930.0931

1.1572.2832.6563.0443.3083.914

III. PARAMETERS

R (A, )R (A~)R 2, +R A~

where the nuclear radii are given by

R (~)=(1.233~ '"—0.98~ -'") fm .

The potential radius of Ref. 12 is given by

R (A&, A2)=R (3&)+R (A2)+0. 29 fm .

(3.1)

(3.2)

Because we are concentrating on the variation in thestructure of the calcium isotopes, we have used a differentconstruction for the potential radius, namely,

Since the absorption of Aux out of the elastic channel isintroduced through the effects of coupling to the otherchannels and by the ingoing wave boundary conditions,the ion-ion potential representing the interaction betweenthe nuclei is purely real. We take this potential to have a%oods-Saxon shape essentially as determined in Ref. 12.The diffusivity is a=0.63 fm and the depth is determinedby

Ca 2+35

0.12600.25000.0490

0.19000.19000.0380

3.8324.5055.146

along with the resulting barrier heights and radii aregiven in Table I.

The coupling strengths for the excitation of thedifferent Ca isotopes are listed in Table II. These valuesare either equal to or somewhat modified from thestrengths obtained by analyzing ' 0 inelastic scattering inRef. 14. The modified values were found to reproducethe scattering of ' 0 on Ca and Ca in a recentcoupled-channels analysis which uses a purely real poten-tial with the same scaling procedure as before. ' Theeffective excitations and strengths used in the actual cal-culations, obtained as described in the preceding section,are given in Table III. For the Ca+ Ca case, the 2+projectile and target states are included exactly sincetheir energies are too different for the symmetrizationprocedure to be accurate.

The ground-state Q values for various transfer reac-R, ,( A2)

R (40, A2) =R (40)+R (40) +b, ,R, ,(40)(3.4)

where R, ,( A) are the root-mean-squared matter radiifor the calcium isotopes given in Ref. 13. A fit of the no-coupling calculations to the fusion data for Ca+" Ca atenergies above the barrier was made to determine6=0.21 fm. This value is well defined since, according tothe forthcoming calculations, the couplings have littleeffect on the Ca+ Ca fusion cross sections above thebarrier. The potential parameters for the three cases Ca+ Ca 3

2+5

I~c

(fm)

0.65760.19520.4837

IOn

(fm)

0.44550.17680.2475

gl(Mev)

3.7373.9054.492

TABLE III. The effective Coulomb and nuclear couplingstrengths and the eftective excitation energies for the vibrationalchannels used in the calculations. These are obtained from thevalues in Table II using the prescription given in Eqs. (2.16) and(2.17}.

"Ca+4'Ca4'Ca+ 44Ca

Ca 9- Ca

Vo

(MeV)

62.23563.36364.384

R(fm)

8.0708.1738.207

Vb„(MeV)

55.0354.3554.08

Rb„(fm)

9.7429.8779.931

TABLE I. The ion-ion potential parameters used in the cal-culation. Also shown are the heights and positions of the result-ing Coulomb barriers.

Ca+ Ca

"Ca+ "Ca

2+4+2+32+52+35

0.30470.15620.10440.52580.13800.36680.18690.52790.3455

0.23980.09210.09030.35760.12500.19820.22740.36790.1791

1.1572.4442.6563.6433.9054.3633.8533.9444.520

FUSION CALCULATIONS FOR ~Ca+ '4 '4'Ca 2049

TABLE IV. Ground-state Q values for various transfer reactions.

Ca+ Ca

"Ca+ "Ca

40ca+48Ca

PickupStripping

PickupStripping

PickupStripping

1n

—7.27—7.27

—2.77—8.22

—1.59—10.5

—7.24—7.24

—11.1—1.44

—14.71.29

Q (MeV)2n

—9.1—9.1

0.774—11.1

2.62—17.4

2p

—9.86—9.86

—16.82.53

—24.27.08

—1.92—1.92

—3.722.41

—9.250.637

tions are given in Table IV. For the Ca+ ' Ca cases,the Q values for single proton pickup and neutron strip-ping are large and negative. These channels are ignored.The final states for the single-nucleon transfers which areactually included in the calculations are collected inTables V—VII, together with the relevant spectroscopicfactors. For the symmetric system Ca+ Ca, the pick-up reactions are identical to the stripping reactions, so in-cluding only one type and multiplying the matrix elementby &2 accounts for both processes. In all of the calcula-tions we have used the nuclear binding potential of the

Ca projectile as the transfer potential. It should be not-ed in Table IV that there are well-matched channelsavailable for two-particle and alpha-particle transfer re-actions. Some speculations on the possible inhuence ofthese processes will be given in the following section.

IV. CALCULATIONS

The fusion cross sections obtained when only vibra-tional excitations are included are shown in Fig. 1. Ineach case, the subbarrier cross sections are significantlylarger than those obtained from a no-coupling calcula-tion. These enhancements generally agree with those ob-tained in Ref. 4 using a matrix diagonalization approxi-mation and with recent calculations where the effects ofvibrational excitations were included through an effectivepolarization potential. As shown by the dashed curves inFig. 1, including second-order vibrational effects" only

slightly increases the fusion cross sections over the first-order results. These effects will not be included in thecalculations which follow. It is clear that vibrational ex-citations alone cannot account for the subbarrier fusioncross sections for the Ca+ ' Ca cases. It should alsobe noted in these cases that the no-coupling calculationsslightly overpredict the cross sections at energies abovethe barrier. The potential radii could be reduced toachieve agreement with these higher-energy data points.This would cause a corresponding decrease in the calcu-lated subbarrier cross sections, thereby increasing thediscrepancy with the lower-energy data.

The results of the calculations where both vibrationalexcitations and single-nucleon transfer channels are in-cluded are shown by the solid curves in Fig. 2. Alsoshown by the dashed curves are the results obtained whenonly the transfer channels are included. A comparison ofFig. 2 with Fig. 1 shows that the enhancement obtaineddue to transfer alone is similar in magnitude to that ob-tained with vibrational excitations. However, the com-bined effect is not a simple addition of the two. Figure 2shows that the full calculation for Ca+ Ca agreesrather well with the data (see also Ref. 6). For instance,the small discrepancy could be attributed to the higher-order vibrational couplings (see Fig. I). However, thesubbarrier fusion cross sections in the Ca+ ' Ca casesremain well below the data when the single-nucleontransfer channels are included. It is clear for these casesthat an additional mechanism must be invoked.

TABLE V. Particle and hole states generated in one-neutronand one-proton pickup or stripping reactions on Ca. The exci-tation energies, spins and parities, and spectroscopic factors aregiven for the different states of the final nucleus. The spectro-scopic factors for Ca, 'Ca, K, and 'Sc are from Refs. 16—19,respectively.

44CaE

{MeV) CS

TABLE VI. Particle and hole states generated in one-neutron and one-proton pickup or stripping reactions on Ca.The spectroscopic factors for 'Ca and 'Sc are from Refs. 20and 21, respectively.

4'Ca

"Ca"'Ca

4iS

3 +272323+2

] +272

{MeV)

0.00

0.00

1.94

0.00

2.52

0.00

CS3.70

0.95

0.70

4.23

1.62

1.12

4'Ca

4'Sc

72

323+2

723+232

0.00

0.37

0.59

0.99

0.00

0.0124

0.3761

3.50

0.27

0.14

2.50

0.71

0.53

0.14

40D S. LANDOR(CKE, AND s.EspE»SE»2050

b examiningunderstood yaction in

is can be uon transfer r

tude l~rger.t single proton

sections fordominan

nsfer crotai]s of the

1 ely low «anb]e transfer

es. The re atthe unfavora

e

4oCa+ CaV jn this case.

s section is

40 canbea.

On t ef —7.2 Me '"e fusion cross ~

g va&ne oh nel on th

]'g fo&In f e

of this c annnsfer coup ing

the effect o

C I tfor Ca+1

vel large

binding en-about twice as

weak proton in'

nnsnaDs u its from t e ud from the

f'hergy tfactor due to t e sy

here the single-processes.

ctions, w eremetric nb"---In the asymr e, itcan

these channelsou lings toe trans ert check ther means o

nts exist onyide anot e

l.,ti. .,...In ig. , alculated e asec idata at three en

1 channels arelculation, w i e

in1 However,'

geffects o e

e transfer cr10 b the r 1tdes am

fhthe full calcu a

'

channels are reasona

fer and vi-

40

cleon trans eCa+ Ca.Since the

'

ations as " a+ ~ a,rational excita e.f,h. f.„.. .planation o

in one-hole states generated'

k or stripp gneutron an n ic pThe spectroscopic acand 23, respectively.

48C CS(MeV)

72

3 +2

Ca 0.00 6.70

2.58 3.604'Sc 0.00 0.9788

Iluransfer cross sec-neutron trans e c-ted proton andith the fusion d3 together witwn in Fig.

ss sections apptudes of the rabehavior wi vTheasonable.be quite re

It is interest g

1 08 M V)cl gweak p roton bin inand t eh more favvorable

s V—VII) The magn'-ni-a ' reactionsnd on

sfer reac-oP rs determint calcu a epi~ons.

ren ths w ict effects due oks at the en

oupling str g

s in Fig.As one loo h

n . , notes a noniintuitiveeffect for

a o

hs TheCa is aboutCa

corresponding cro

I I I I I I I I I I I I I I II

~ I I'''I''''I''''I'''40Ca+

I I I I»

Ca+

I I I I «I I I II II

I I I

40C

I I I ~

II I I I I I I

II I I I »

»

10

iOO

»

I I I I I I I I I ~ ~ I I I I I~. . . . , . . . ~ . ) ;.. . i. . . ,. I. . . . I. . . , I. . . .60 65 45 50

I I I I1045 50

E, (Me V)

55

'g 1 s (dotted line),ling results

f.pffects of vibrationa sg

results from first-order exci a3 ~

,48CSFOR C'+ALCULATIOFUSION CA

~ I I~ I II ~

2051

10 I

- "Ca+'»

I I I

: 4'Ca+

I ~IIII-"c

102

»

100~s

I ~ I10 45' '50 555 60 65I I I I I III' I I I I I I I I I I I

55 6045 50 55

E, (Me V)

50 55 6065 45

nels. Shown are ththe no-ransfer channels.ared to the

1 and sing e-nulation {so id hlt{d h dli )

cross sec ioonl resuh. --f-. ,F

coupling resu tdata from RRef. 3.

itional channels arewhether additionat resting to pr'

s eculate w eer

nsfer reac-h

c~ h 1 ft

acroscopi 'c approacd s' th mo-nucleon

veen estimate u'

there are two-tions has een

metric cases7. As Ta e

r the asymmhannels avavalues athat are favor-p g

n. To invith ad an additiona cintroduced an a

of theltot eh derivative olin proportionaV and a coupling pMe aion-ion- on potentia

dU(4.1)V, = —o,

as obtained by re-05fmw40 48

o p g

4'Ca+4'Caa fit to the ow-

obtained forh't '1'"dthe samh arne quality as t ain Fig. 2.

i03 ~ I I

40- "Ca+ Ca

I I

- "Ca+ Ca

II I ~ I

II ~ I I ~ I I~ I I I

II ~aa ~ I I I

I~

- 4'Ca+ "Ca~

' I I»

102» aa

100

101 //

lt/

, . . . I.. . . . I, ~ ~ .10 '50' 5545

/. z, I. . . , I.

60 65

» /~ . i.t, . l. .I I I

45 50, I, . . . I I I ~ ~ II I I ~I I I I I I I I I I ~ I I I I I I I I ~ I I ~I I I I I I I I I I I I ~

55 60

E, ~ (MeV)

and dashed lines sn-II. The dotted anded in Tab — . ales V—V~ gdicate the proton an neu

ND S. LANDOH FRICKE AH ESBENSEN~ S.2052

1O4

1O3

102

1O4

103

10

1O4

10

102

I ~ I I~ I ~ I ~ I ~ I II

~ I ~ ~

I

I ~ I ~

II

eon transferbable single-nucleopthe low-energy

citations an ba e

h 1. For each case we avtions. o er cross sections.

Ca+ Ca fusion da'

data.1 t'ons also compar

for this system.ble elastic sca e'

ovides a goo re e td eference pointb'nin the asic

terac ition.n ss sections are pre ic e

Cg

to in

t'b'lar "thIt is interes

nfort eCa

h their strengths'

is redicte oincrease,

vibrational exci a ith those of t e viwhen combined wit~ I I I I i a s

40 5010 s I s s ~ s I s s s s I s s ~ a

0 80 9060 70(deg)

I 5 ~10 ~ a ~ ~

~~ I ~ ~

s calculated forross sectiong

6 The solid curves arerom Ref. 2 .uded, wh~ eh 1

di o 1 h ibtrans er c

'rationa c anobtained by inclu ingcurves are o tain

a)

V. CONCLUSIONS

e coupe-1 d-channels ap-f vibrational

's wok we apl oac 0h t calculate the corn ine

t . 5(n are shown in F g.this calculation are

h h'h

th'n 'h' """'h'b ations in Ta

s onding cross secle-nucleon tran

dd' io 1'

Q-

ce Table setric corn ina

instance,d f r the asymmer matche or

he revious calust ate t po

fr et by a largethe resu

'gitin cross sec

'

c annels in addi-t}1

~

~

re 1 coupled channe she vibrational excita

'f in fact t ere

tions an eh' h we have inclu eions w ic

c scatteringhave a signi'

cesection. This p tse . oint is

ns correspg

h dd 1r e modificationss of the eas icmeasurements o

h strongly cou-dicate that me

whether suc sinshould be a e wpe1 d channels are presen .

10

10»

100

. . . ir. I10 45''

50 55 60K (M~v)

65 70

~ ~ ~

- (b)

I ~ ~ ~~

I~ ~ ~ ~

I~ ~ ~ I

102

»O'

10

I a J ~ I

45 5O 55 60E, (Mev)

65 70

s obtaine d by including an. 5. The fusion crfor the a

FIG.

f rm factor oft ong vlbrat1' lt (d

ed line) using a s rotted line) an eo. . The no-coup i

1' ) are also s ow .g

without the e6'ective c ass sections oropen poin s

n art(a) t ehe Q value is d cr

=0.9 fm.= —3 MeV and o-, =while for part ibi Q =—

FUSION CALCULATIONS FOR Ca+ ' ' Ca

110 C ~ ~ ~ ~

[l I ~ ~

ff ~ I I

)~ % I I

)I ~

-=()

100 --- E 70 MeV

110 g ~ ~ ~ ~)

~ ~ ~ ~

i~ ~ ~ Ijl ~

= (b)

100

~

/~ ~ ~ ~

10—'

100E, =65 MeV

101

100K~=65 MeV

10 10—1

10'

10E, =60 MeV

10 g

1.00E~=-60 MeV

10—'

10

Q = +1,0 MeV

s s I I I I I ~ I I I i I I s a I a I I a I I

Q = —3MeV

I ~ I ~ I ~ ~ I ~ ~ ~ ~ ~ I I ~ ~ ~ I i ~ I ~

60 70 80 90 1008 (deg)

50 60 70 80 90 100ec rnonde&&

FIG. 6. The elastic scattering cross sections for Ca+ Ca corresponding to the two sets of calculations shown in Figs. 5(a) and5(b).

are not sufhcient to explain the low-energy Ca+ ' Cafusion data. This discrepancy is judged to be particularlyserious for the Ca+ Ca case since here both nucleihave simple closed-shell structures.

The options to explain this discrepancy appear to berather restricted. Simply invoking a larger change in theion-ion potential in going from Ca+ Ca to " Ca+ Cato improve the agreement with the low-energy fusiondata would result in a severe overprediction of thehigher-energy fusion cross section. The strengths of thevibrational excitations and single-nucleon transfers whichwe have included are constrained by other types of data.One can argue that there are single-particle transfers tohigher-lying states which have not been included andtheir combined effect might significantly enhance the sub-barrier fusion. The relatively small effects we have ob-tained from the more favorable transfer reactions do notlend support to this argument. The low-energy fusioncross section can be increased specifically by introducingadditional channels which couple directly to the initialstate. Our estimates in the present case require a strongcoupling resulting in a cross section which is comparable

to that of the single-nucleon transfer. This seems to beunreasonably large for two-particle transfer reactionsalone. A combination of two-particle and alpha-particletransfer processes seems to be a more likely possibility.

It is necessary to have additional measurements for theCa+ Ca system in order to better understand this

problem. The elastic scattering cross section can provideuseful information. It woold further constrain the pa-rameters of the ion-ion potential for the Ca+ Ca sys-tem. It should also reveal the presence, or absence, ofany significant directly coupled channe1 not already in-

cluded in the present calculations. Measurements of thesingle-nucleon transfer cross sections would directlycheck the conventional shell-model description which wehave used, and measurements of the multip articletransfer reactions would limit the present freedom onehas to parametrize these processes.

This work was supported by the Department of Ener-gy, Nuclear Physics Division, under Contract %'-31-109-ENG-38.

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~8A calculation similar to the one shown in Fig. 5(b) was madepreviously in Ref. 4. However, the implication in Ref. 4 thatthe cross section should be associated with single-nucleontransfer is not supported by the present results.