PHYSICAL REVIEW C OCTOBER 1997VOLUME 56, NUMBER 4

Fusion barrier distributions in systems with finite excitation energy

K. Hagino,1,3 N. Takigawa,1,3 and A. B. Balantekin2,3

1Department of Physics, Tohoku University, Sendai 980–77, Japan2Department of Physics, University of Wisconsin, Madison, Wisconsin 53706

3Institute for Nuclear Theory, University of Washington, Seattle, Washington 98915~Received 24 June 1997!

An eigenchannel approach to heavy-ion fusion reactions is exact only when the excitation energy of theintrinsic motion is zero. In order to take into account effects of finite excitation energy, we introduce an energydependence to weight factors in the eigenchannel approximation. Using the two channel problem, we show thatthe weight factors are slowly changing functions of incident energy. This suggests that the concept of thefusion barrier distribution still holds to a good approximation even when the excitation energy of the intrinsicmotion is finite. A transition to the adiabatic tunneling, where the coupling leads to a static potential renor-malization, is also discussed.@S0556-2813~97!05410-1#

PACS number~s!: 25.70.Jj, 24.10.Eq

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It has been well recognized that cross sections of heaion fusion reactions at energies near and below the Coulobarrier are strongly influenced by coupling of the relatimotion of the colliding nuclei to nuclear intrinsic motion@1#. When the intrinsic degree of freedom has a degenespectrum, fusion cross sections can be calculated in theden approximation. In this limit, the coupling gives rise todistribution of fusion potentials. The fusion cross sectionthen given by an average over the contributions from efusion barrier with appropriate weight factors@2–4#. Basedon this idea, a method was proposed to extract the badistribution directly from the fusion excitation function usinthe second derivative with respect to the product of thesion cross section and the center-of-mass energyEs @5#.This stimulated precise measurements of the fusion csections for several systems@6#. The extracted barrier distributions have then been found to be very sensitive tostructure of the colliding nuclei, e.g., the sign of the hexacapole deformation parameter, while the fusion cross secitself is rather featureless@6,7#.

In a rigorous theoretical interpretation, the barrier disbution representation, i.e., the second derivative ofEs, has aclear physical meaning only if the excitation energy of tintrinsic motion is zero. Nonetheless, this analysis has bsuccessfully applied to systems with relatively large exction energies@6,8,9#. For example, the second derivativeEs for 16O1144Sm fusion reaction clearly shows the effecof coupling to the octupole phonon state in144Sm, whoseexcitation energy is 1.8 MeV, much more clearly than tfusion cross section itself@6,8#. Also the analysis of the fu-sion reaction between58Ni and 60Ni, where the excitationenergies of quadrupole phonon states are 1.45 andMeV, respectively, shows that the barrier distribution repsentation depends strongly on the number of phonon stincluded in coupled-channels calculations@9#. These analy-ses suggest that the representation of fusion process in tof the second derivative ofEs is a very powerful method tostudy the details of the effects of nuclear structure, irresptive of the excitation energy of the intrinsic motion.

Despite these successes, there remains the quewhether the second derivative ofEs represents a ‘‘distribu-

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tion’’ of fusion potential barriers when the excitation enerof the intrinsic motion is not zero. To address this probleone may attempt to use the constant coupling approximatwhere the coupling form factor is assumed to be consthroughout the interaction range@10#. In this approximation,the unitary transformation gives the fusion cross section asum of eigenchannel cross sections. The constant coupapproximation has, however, a serious drawback in thatresults strongly depend on the position where the strengtthe coupling Hamiltonian is evaluated. Therefore, one neto use a different approach to address the question.

In this paper we treat the weight factors as energy depdent variables. The possibility of the energy dependenceweight factors when the excitation energy is finite has besuggested in Ref.@11#. Here, we explicitly study the energdependence by performing exact coupled-channels calctions. It will be shown that the energy dependence is quweak for a wide range of excitation energy, suggesting tthe eigenchannel approximation works well even whenexcitation energy of the intrinsic motion is not small.

Let us consider, for example, the case where the intrindegree of freedom has only two levels. The coupled-chanequations then read

F2\2

2m

d2

dR2 1U~R!1S 0 F~R!

F~R! e D G S u0~R!

u1~R! D5ES u0~R!

u1~R! D , ~1!

where U(R) and m are the bare potential and the reducmass of the relative motion, respectively.F(R) is the cou-pling form factor ande is the excitation energy of the intrinsic motion, respectively. When the excitation energye iszero, the unitary matrix which diagonalizes the coupling mtrix is independent of the positionR, and the barrier penetrability is exactly given by

P~E!51

2$P0@E;U~R!1F~R!#1P0@E;U~R!2F~R!#%,

~2!

2104 © 1997 The American Physical Society

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56 2105FUSION BARRIER DISTRIBUTIONS IN SYSTEMS . . .

whereP0@E;V(R)# is the penetrability of potentialV(R) atenergyE. In this limit, the original single barrier splits intotwo effective potential barrier with equal weights.

If the excitation energye is not zero the unitary matrixdepends explicitly on the positionR. The unitary matrix,therefore, does not commute with the kinetic energy opetor, and hence simple eigenchannel approach does notAn approximation which is often made is to assume thatcoupling form factorF(R) is a constant. In this approximation, the unitary matrix becomes independent ofR, and thepenetrability is given by a formula similar to Eq.~2! but withdifferent eigenvalues and weight factors@10#. In order to takeinto account the radial dependence of the coupling form ftor, an approximation was proposed, where the couplingtrix is diagonalized to obtain the eigenbarriers at each ption of the relative motionR while the weight factors arecalculated at a chosen positionRw @13#. The computer codeCCMOD uses this approximation@14#.

Even when we introduce this approximation, the weigfactors are still functions of the chosen positionRw , and theresults might strongly depend on that choice. UsuallyRw ischosen to be the position of maximum of the bare potenbarrier@13,14#. However, there is no theoretical justificatiothat this is the optimum choice, and furthermore, it is nobvious whether one can determine the weight factors inpendent of the incident energy.

In order to avoid these drawbacks and examine the endependence of the weight factors, we parametrize the petrability as

P~E!5v1~E!P1~E!1v2~E!P2~E!, ~3!

and evaluate the weight factorsv6 at each incident energE. HereP6(E) are the penetrabilities of the eigenpotentiaU(R)1l6(R), respectively,l6(R) being the eigenvalueof the coupling matrix at each position ofR, which are givenby

l6~R!5@e6Ae214F~R!2#/2. ~4!

If the weight factors slowly vary as functions of the incideenergy, the effects of channel coupling can be interpreteterms of the barrierdistribution, even when the excitationenergy is nonzero. In the Appendix, we generalize Eq.~3! toa multichannel case using the path integral approach. Sthe weight factors have to satisfy the unitarity condition@3#,they can be uniquely determined in the present two leproblem, and are given by

v1~E!5@P~E!2P2~E!#/@P1~E!2P2~E!#,

v2~E!5@P1~E!2P~E!#/@P1~E!2P2~E!#. ~5!

We performed coupled-channel calculations to examthe energy dependence of the weight factors. To this endcoupled-channel equations have to be solved with goodcuracy, since the penetrabilitiesP(E) andP6(E) in Eq. ~5!are exponentially small quantities at energies below therier. The incoming wave boundary condition, which is oftused in coupled channels calculations for heavy-ion cosions @15#, can bring some numerical errors, though thwould be small enough for the purpose of calculating fus

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cross sections. In order to avoid this, we use here a schemmodel of heavy-ion fusion reactions by Dassoet al. @10#,where the radial motion between colliding nuclei is treateda one-dimensional barrier penetration problem. FollowRef. @10#, we assume a Gaussian shape for both the bpotential and the coupling form factor:

U~R!5U0e2R2/2s2, F~R!5F0e2R2/2sf

2. ~6!

The parameters are chosen following Ref.@10# to be U05100 MeV, F053 MeV, and s5sf53 fm, respectively,which mimic the fusion reaction between two58Ni nuclei.We have checked that our conclusions do not significanchange as long as the value ofsf is not too small. The massm and the excitation energye are taken to be 29mN , mNbeing the nucleon mass, and 2 MeV, respectively.

The upper panel of Fig. 1 shows potential barriers forpresent problem. The dotted line is the bare potentialU(R)and the solid and the dashed lines are the eigenpotenU(R)1l2(R) andU(R)1l1(R), respectively. If we adoptthe eigenchannel picture, Fig. 1 means that the potentialriers are ‘‘distributed’’ and the original bare potential~thedotted line! splits into two potentials~the solid and thedashed lines!. Because of the noncommutativity of the untary matrix which diagonalizes the coupling matrix, and tkinetic energy operator, the standard eigenchannel picdoes not apply. If we ignore this noncommutativity~i.e., ifwe adopt theCCMOD prescription! the weight factors aregiven by

w6~R!5F~R!2/@F~R!21l6~R!2#. ~7!

They are shown in the lower panel of Fig. 1 as a functionthe distanceR. Although they do not vary so much near thbarrier position, their changes are appreciable throughout

FIG. 1. The eigenpotentials~the upper panel! and the associatedweight factors~the lower panel! as a function of the relative distanceR. The solid and the dashed lines in the lower panel areweight factors for the lower and the higher eigenpotentials, resptively. The dotted line in the upper panel represents the bare potial barrier.

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2106 56K. HAGINO, N. TAKIGAWA, AND A. B. BALANTEKIN

barrier region. Therefore, the prescription to fix the weigfactors to the values at the barrier position may not be safactory.

The results of exact coupled-channel calculationsshown in Fig. 2. The first and the second panels arepenetrabilities and their first derivative with respect to tenergy, respectively. The latter corresponds to the secderivative ofEs @5#. We use the point difference formulwith DE52 MeV to obtain the first derivative of the penetrbility, as is often done in the analyses of heavy-ion fusreactions@6#. The first derivative of the penetrability hasclear double-peaked structure, which could be associwith the two eigenpotential barriers and could thus be inpreted in terms of a ‘‘barrier distribution.’’ In order to sewhether this is the case, we plot in the last panel of Fig. 2optimum weight factors defined by Eq.~5!. The solid and thedashed lines in the figure correspond to the weight factorsthe lower and the higher potentials, respectively. We obsethat the optimum weight factors change only slightly as futions of the incident energy; their change is barely 2.6% fr10 MeV below the barrier to 10 MeV above the barrieAnother important result of this calculation is that the weigfactors are considerably different from those estimated inusual way, i.e., at the barrier position. When the constcoupling approximation was first introduced, it was expecthat the choice of the position where the weight factorsestimated is not critical to calculate tunneling probabilitbecause the weight factors vary slowly across the bar

FIG. 2. The penetrability~the first panel! and its first derivative~the second panel! for the two level problem as a function of thincident energy. The third panel shows the optimum weight facobtained according to Eq.~5!. The solid and the dashed lines corespond to the weight factors for the lower and the higher potials, respectively.

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region @10,13#. The position was then chosen to be the brier position of the uncoupled barrier. Contrary to this expetation, our calculations show that determining the weight ftors at the barrier position does not give proper weigfactors even though the weight factors do not have a strradial dependence. The situation would be much more sous in realistic calculations, since the weights are still chaing at the barrier position due to the fact that the couplextends outside the Coulomb barrier@11#.

The same calculations were repeated for different valof the excitation energye and the results are shown in Fig.The quantities shown in each panel are the same as thoFig. 2 except for the third panel, where only the optimuweight factors for the lower barrier are plotted. We agaobserve that the weight factor changes only marginally afunction of the incident energy, even when the excitatienergy is finite. We thus conclude that the eigenchannelproach is still applicable even when the excitation energythe intrinsic motion is finite.

When the excitation energye is zero, the barrier distribu-tion, i.e., the first derivative of the penetrability has two symetric peaks and the weight factor has no incident enedependence~the solid line!. As the excitation energy in-creases, some strength is transfered from the higher peathe lower peak, and the barrier distribution becomes asmetric. As the excitation energy significantly exceedscurvature of the bare barrier, which is about 4 MeV in oexample, one expects to reach the adiabatic limit. To illtrate this, Fig. 4 shows the influence of the coupling toexcited state whose energy is 8 MeV. The figure also ctains the result for the the no coupling case~the dotted line!

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FIG. 3. Same as Fig. 2, but for different values of the excitatenergy of the intrinsic motion which are denoted in the inset. Tthird panel shows only the weight factors for the lower potentia

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56 2107FUSION BARRIER DISTRIBUTIONS IN SYSTEMS . . .

for comparison. In this case, the weight for the higher peaconsiderably smaller than that for the lower peak, andbarrier distribution has essentially only a single peak. Tpeak position, however, is shifted towards a lower enerThis is consistent with the adiabatic picture, i.e., the meffect of the coupling to a state whose excitation energymuch larger the barrier curvature is to introduce an eneindependent shift of the potential accompanied with the mrenormalization@12,16#. Correspondingly, the barrier distrbution is only shifted without significantly changing ishape unless the coupling form factor has a strong radependence@17#. It is thus clear that the second derivativeEs can represent a wide range of situations between theextreme limits, i.e., from the adiabatic limit where the copling leads to an adiabatic potential renormalization, tosudden limit where the coupling gives rise to a barrier dtribution.

In summary, we have discussed the energy dependenthe weight factors in the eigenchannels approximation. Pforming exact coupled-channel calculations, we showedthe energy dependence is very weak, regardless of the etation energy of the intrinsic motion. Therefore, it maksense to call the second derivative ofEs as the fusion barrierdistribution, even when the excitation energy is finite. Walso discussed a transition from the sudden to the adiabtunneling limits, and showed that the fusion barrier distribtion can represent these two limits in a natural way.

The authors thank M. Dasgupta for useful discussioThey also thank the Institute for Nuclear Theory~INT! at theUniversity of Washington for its hospitality and the Depament of Energy for partial support during the course of twork. The work of K.H. was supported by the Japan Socifor the Promotion of Science for Young Scientists. Thwork was supported by Monbusho International ScientResearch Program: Joint Research, Contract No. 09044and the Grant-in-Aid for General Scientific Research, C

FIG. 4. Effects of the coupling to a high excitation energy staThe dotted line is the result without channel coupling, while tsolid line is obtained by taking into account the coupling toexcited state whose excitation energy is 8 MeV.

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tract No. 08640380, from the Japanese Ministry of Edution, Science and Culture, and by U.S. National ScienFoundation Grant No. PHY-9605140.

APPENDIX A: EIGENCHANNEL APPROXIMATIONIN THE PATH INTEGRAL APPROACH

Consider a system where a macroscopic degree of fdom R couples to an intrinsic degree of freedomj. We as-sume the following Hamiltonian for this system:

H~R,j!52\2

2m

]2

]R2 1U~R!1H0~j!1V~R,j!, ~A1!

wherem is the mass for the macroscopic motion,U(R) thepotential in the absence of the coupling,H0(j) the Hamil-tonian for the internal motion, andV(R,j) the coupling be-tween them. The barrier transmission probability from tinitial position Ri on the right side of the barrier to the finapositionRf on the left side is then given by@18#

P~E!5 limRi→`

Rf→2`

S Pi Pf

m2 D E0

`

dTe~ i /\!ET

3E0

`

dTe2~ i /\!ETE D@R~ t !#

3E D@R~ t !#e~ i /\![St~R,T!2St~R,T!]rM@R~ t !,T;R~ t !,T#,

~A2!

whereE is the total energy of the system, andPi andPf arethe classical momenta atRi andRf , respectively.St(R,T) isthe action for the macroscopic motion along a pathR(t), andis given by

St~R,T!5E0

T

dtS 1

2mR~ t !22U@R~ t !# D . ~A3!

The effects of the internal degree of freedom are includedthe two time influence functionalrM , which is defined as

rM@R~ t !,T;R~ t !,T#

5(nf

^ni uu†@R~ t !,T#unf&^nf uu@R~ t !,T#uni&.

~A4!

Here, ni and nf are the initial and the final states of thinternal motion at positionRi andRf , respectively. The timeevolution operatoru@R(t),t# of the intrinsic motion obeys

i\]

]tu@R~ t !,t#5$H0~j!1V@R~ t !,j#%u@R~ t !,t#.

~A5!

with the initial conditionu(R,t50)51.We introduce the eigenchannel~or the adiabatic! basis, as

@H0~j!1V~R,j!#wn~R,j!5ln~R!wn~R,j!. ~A6!

Expanding the intrinsic wave function at timet in this basis

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2108 56K. HAGINO, N. TAKIGAWA, AND A. B. BALANTEKIN

u~R,t !uni&5(n

an~ t !expF2i

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t

dt8ln@R~ t8!#G3uwn@R~ t !#&, ~A7!

Equation~A2! for the barrier penetrability reads

P~E!5 limRi→`

Rf→2`

S Pi Pf

m2 D E0

`

dTe~ i /\!ETE0

`

dTe2~ i /\!ET

3E D@R~ t !#

3E D@R~ t !#(n,m

an~T!am~ T!*

3^wm@R~ T!#uwn@R~T!#&

3ei /\ *0Tdt@1/2mR22U~R!2ln~R!#

3e2 i /\ *0Td t @1/2mR8 22U~R!2lm~R!#. ~A8!

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We use the semiclassical approximation, and for energwell below the barrier, where the single path dominatevaluate the path integral along the classical path

R~ t !5R~ t !5Rcl~ t !, T5T* 5Tcl . ~A9!

In this case, the orthogonalitiy of the adiabatic basis lead

P~E!5(n

vn~E!P0@E;U~R!1ln~R!#, ~A10!

where the weight factors are given by

vn~E!5uan~Tcl!u2. ~A11!

The weight factors depend implicitly on the energyEthrough the time evolution of the intrinsic system along tclassical path.

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