PH Y SICAL RE VIE% C VOLUME 19, %UMBER 3 MARCH 1979

Low energy one-neutron transfer reactions between heavy ions

Mahir S. HusseinInstituto de Fisica, Universidade de Sao Paulo, Sao Paulo, Brasil

(Received 21 April 1978; revised manuscript received 7 November 1978)

A constraint is imposed on the possible value of the diffusivity of the imaginary part of the optical model

potential used to fit elastic scattering data of hght heavy ions at sub-barner energies. This is done by fixing

the center of mass energy at which the optimal -Q- value one-neutron-transfer S factor crosses the fusion Sfactor.

NUCLEAR REACTIONS Sub-Coulomb one-neutron transfer between heavy ionsL discussed.

Although it is well accepted that the direct reac-tion contribution to the total reaction cross sec-tion of heavy ion collisions at energies near andbelow the barrier is quite small due to the pres-ence of the Coulomb barrier, it has been shownrecently' that the optimal- Q-value one- neutrontransfer cross section could become comparableand even larger than the fu-sion cross section atenergies well below the barrier. This suggeststhat the total nuclear S factor would rise muchmore steeply with decreasing center of mass en-ergy than the fusion S factor as was demonstratedin Refs. 1 and 2. In this note we argue that theone-neutron transfer data could supply an upperlimit for the value of the diffusivity a of the imag-inary part of the optical potential used in elasticscattering data analysis. We demonstrate this byestimating the energy E, at which the one-neutrontransfer S factor crosses the fusion S factor. Wefeel that the ambiguity inherent in the values of theparameters of the optical model potential of heavyions wouM certainly call for establishing at leastupper limits.

The total reaction cross section at very low en-ergy may be expressed in terms of the opticalmodel transmission factor as follows:

v„—= ~, T(E).

The nuclear total reaction S factor is defined asusual by

In order to estimate the relative magnitude ST„/Sr„we consider the transmission coefficient T(E)which is usually given by

I P(r) I'W(r)dr,

where p. is the reduced mass of the nuclear sys-tem, k is the asymptotic wave number, g(r) is theoptical model radial wave function, and —W(x) isthe imaginary part of the optical potential. Thefactor sp, /S'k is dictated by our implicit assump-tion that the wave function g(r) is normalized as-ymptotically to unit flux. To simulate ihe opti-mum-Q transfer reactions, which occur mainly in

the surface (barrier) region, one has to use a ra-ther large diffusivity, a, of W(r). This suggeststhat part of the contribution to T(E) comes fromabsorption under the barrier. '

Accordingly we decompose T into two parts:T, due to volume absorption, and T, due to absorp-tion in. the barrier region. Such a decompositionmay be worked out formally by treating the vol-ume absorption with an incoming wave boundarycondition model which results in a simple barrierpenetration form for T,. The difference T —Ty Tp

may then be calculated from Eq. (5) in which theintegration is performed in the barrier region. In

order to evaluate T, we assume a simpleWoods-Saxon form for ii(r). At the large distances en-countered in the barrier region W(r) behaves as

S„=Eexp(2wq)o„, (2) W(r) -Wo exp(R/ag exp( r/aJ, -

where g (the Somerfeld parameter) = Z,Z,e'/Rv, v

being the asymptotic relative velcoity of the twoions. A similar form i.s defined for the nuclearfusion S factor

SFU= E p(e2x7T g)(T . FU'The optimal-Q one-neutron transfer S factor is

thus given by

STR R FU'

where 8"„R, and a are the strength, radius,and diffusivity of the absorption potential. In ourcalculation we employ the WKB approximation for

I g(r) I' in the barrier region,

Ig(r) I'= — exp 2k(r

where k(r) is the complex local wave number de-termined from both the real part, V(r), the imag-inary part, W(r) of the optical potential, as well

807 1979 The American Physical Society

808 MAHIR S. HUSSEIN 19

as the center of mass energy E,2p ~/

k(r) = —, [V(r) —E —iW(r)]

x (1 + jl + [W(ro)/(V(ro) —E )]'j ' ')' '

a (10)

and ap is the outer classical turning point given bythe larger root of the equation,

(9)

It is clear that the integrand Ig(r) I'W(r) attains anextremum value at a distance rp determined fromthe equation

I /22 Rek(ro) = 2 —

a ~[V(ro) -E]'

which occurs at larger energies [always consid-ering E & V(R,)]. A schematic plot showingV(r), W(r), I g(r) ~' as well as

I P(r) ~'W(r} is shownin Fig. l. It is easy to see from Eq. (11) that theminimum in jg(r) ~'W(r) occurs at distances r,smaller than R„ the-position of the maximum ofthe barrier. This is obvious since (dV(r)/dr) isalways positive at r & Ap and negative at r &80 andattains a zero value at r =Rp. Having thus estab-lished the range of values of r,' to be slightlysmaller than R„equal to Rp and larger than R„we may now proceed in calculating T, . This wedo approximately by expanding the exponent[2f", Rek(r')dr' —r/a„) that appears in the inte-grand to second order in (r- r,') and performingthe integration after approximating r,' —a,. = 0,where a, is the inner turning point [the smallerroot of Eq. (9)] indicated in Fig. 1,

Most of the contribution to T, would thus comefrom the region in the neighborhood of r,', the dis-tance at which ~(((r) ~' W(r) attains its maximumva. lue. There is another root of Eq. (9) which cor-responds to a minimum in ~g(r) ~' W(r) The. con-dition of maximum and minimum in ~f(r) ~'W(r),i.e., the sign of [(d/dr) Rek(r)]„, can be written

0as follows:

where

p(E) 22 p.W(r(')) 7T

Ik(ro)j ( "d

T2= 2 Wr r drfp

where1 r

xexp 2 Rek r' dr'a&

The first inequality in Eq. (11) corresponds to amaximum, the second to a minimum, and theequality sign corresponds to an inflection point

---:W(r): I+(r)l: (r) IV(r)l

W r r 2dr0

ap= exp —2 Rek(r)dra&

(13)

and T, is the barrier penetration factor associ-ated with volume absorption and may be given by

V(r)

0( ro Re r~+

FIG. 1. A schematic plot showing the l =0 barrier.Also shown are E, I g(r) I

', w(r), and I y(r) I', )V(r)

as well as the distances a 0, a;, 80, xo, and ~0 as definedin the text.

Recent measurement4 of the sub-Coulomb barrierfusion cross section for "C+"C indicated no ab-sorption-under-the-barrier contribution as thecross section O~„exhibited an average behaviorconsistent with a simple barrier penetration.'This indicates that one may associate T, withfusion and T, with the optimal-Q transfer reactioncontribution as well as with other possible sur-face reactions. lt has been suggested in Ref. 5that the diffusivity a that is used in optical mo-del analyses of fusion data should not exceed acritical value given by

19 LO% ENERGY ONE-NEUTRON TRANSFER REACTIONS. BKT%EEN. . . 809

STR(E,) =SOU (E,)

T, (E,) = T, (E,) .(14}

For the above to be true p(E) should be close tounity. This implies that the penetration factor

exp[+2 f,"

Re k(r) dr j

must at least be equal to, if not greater than, theabsorption factor, exp( -r', /a„), which would bepossible if the difference x', —a, attains a largevalue for a given center of mass energy.

Since a, is basically determined by a„, the dif-fusivity of the real part of the optical model po-tential, the above requirement on x,+- a, implies arather stringent condition on the value of a„. How-ever, at low energies, i.e., E«V(R, ), r, is seento be larger than R, rendering the above conditionon a„not so stringent.

From Eq. (10) we may find an approximate rela-tion between E„p and the height of the Coulombbarrier, E~ -=V(R,),

@2 ( I i'2p, (2a„j (15)

In deriving Eq. (15) we have neglected the factor

a, = k/[5 p,V(R,)]'/'.

Thus a„ that appears in 7, must necessarily belarger than a, . At the center of mass energy atwhich the integrand tI/(r)

iW(r) exhibits an inflec-

tion point, T, and T, have comparable values. Wedenote this energy by E,. At E» E„T,dominatesover T, since the two turning points, a, and a„getcloser to each other rendering the barrier pene-tration easier. At E «E, most of the contributionto T comes from absorption under the barriersince the barrier penetration probability T, be-comes much smaller. These features are closeto the behavior of the experimentally deducedS» and S~U as functions of the center of mass en-ergy. In Ref. 4 it was found that at E= V(R,),ST„«S~„whereas at very small energies S»» SF„. The energy at which the experimentallydeduced S»——-SF„may be associated with E,above, i,.e. ,

( )IV(R0)

E~-Ein Eq. (10}which is a rea.sonable approximation atthe energies and distances considered. We havealso set V(ro) = V(RO).

In Table I we exhibit the values of a calculatedfrom Eq. (15) for three nuclear systems studiedrecently. " These values are to be considered asupper limits of the a that enter in the analysis ofelastic scattering data at low energies as long asother reactions (besides the one-neutron transfer)have negligible contribution to the total reactioncross section at these energies which is apparentlythe case.

It was found in Ref. 1 that the optimal-Q one-neutron transfer S factor at low energies behavesas

STa- exp(- AE)

3/2A =0.0]

I

i 2 —i ( c2}i/2 MeI/ i

(E,'/' mi

where E, is the binding energy and m is the massof the transferred neutron.

The above fprmula clearly indicates that therate of rise of S» with decreasing center of massenergy depends critically on the binding energyE, of the transferred neutron. Since the Q„„.is given by

mQ

one would expect that for subbarrier one-neutrontransfer reactions between nuclei with closed neu-tron shell Q„, is larger and A is smaller than therespective values of these quantities in reactionsbetween heavy ions which contain loosely boundneutrons. As has already been stressed above,since the value of a is closely related to the type .

of nuclear system being investigated, one mayconclude that for nuclei with closed neutron shell,E, would be quite small, making an unambiguousdetermination of the one-neutron transfer cross

TABLE I. The vaIue of az estimated from Eq. (15}for three different systems studied inRefs. (1}and (2}.

System Z, (MeV} Z& (MeV} a~ (fm} a, (fm}

'4N+'4N, Ref. 1O+ Be, Ref. 2

~OB+~80, Ref. 2

4.94.1

0

8.615.897.09

0.450.710.45

0.290.390.34

810 MAHIR S. HUSSEIN

section at these very low energies very difficult.We conclude by saying that the measurement of

the one-neutron optimal-Q-value transfer crosssection at sub-Coulomb barrier energies would

yield information about the value of the diffusivitya of the imaginary part of the optical model po-tential used to fit elastic scattering data. 'This isdone by finding E„ the center of mass energy at

which S~ and STR cross and utilizing the approxi-mate formula (15). This should serve as a con-straint on the values of the parameters of the ion-ion optical model potential, and it emphasizesonce again the point that even at very low energiesquasielastic reactions must be considered togetherwith elastic data in order to find a rather lessambiguous ion-ion optical model potential.

'Z. E. Switkowski, B. M. Wieland, and A. Winther, Phys.Rev. Lett. 33, 84G (1974).

2Z. K. Svitkowski, S. C. Wu, J. C. Overley, and C. A.Barnes, Caltech, Lemon Aid Report No. 157, 1976(unpublished) .

G. Michaud and Z Vogt Phys Bev C 3 864 (1970)4K. U. Kettner et al. Munster report, 1976; H. Feshbach,

Report No. MIT-CTP No. 671, 1977 (unpublished).5M. S. Hussein, Phys. Lett. 718, 149 (1977).