Energy dependence of potential barriers and its effect on fusion cross-sections

0.5 MB·

by  A. S. Umar

Energy dependence of potential barriers and its effect on fusion cross-sections
A.S. Umar,1 C. Simenel,2 and V.E. Oberacker1
1Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, USA
2Department of Nuclear Physics, RSPE, Australian National University, Canberra, ACT 0200, Australia
(Dated: January 29, 2014)
Background: Couplings between relative motion and internal structures are known to affect fusion barriers by
dynamicallymodifyingthedensitiesofthecollidingnuclei. Theeffectisexpectedtobestrongeratenergiesnear
the barrier top, where changes in density have longer time to develop than at higher energies. This gives rise to
anenergydependenceofthebarriersaspredictedbymoderntime-dependentHartree-Fock(TDHF)calculations
[K. Washiyama and D. Lacroix, Phys. Rev. C 78, 024610 (2008)]. Quantitatively, modern TDHF calculations
are able to predict realistic fusion thresholds. However, the evolution of the potential barrier with bombarding
energy remains to be confronted with the experimental data.
Purpose: Theaimistofindsignaturesoftheenergydependenceofthebarrierbycomparingfusioncross-sections
calculated from potentials obtained at different bombarding energies with the experimental data.
4
1 Method: This comparison is made for the 40Ca+40Ca and 16O+208Pb systems. Fusion cross-sections are com-
0 puted from potentials calculated with the density-constrained TDHF method.
2
Results: The couplings decrease the barrier at low-energy in both cases. A deviation from the Woods-Saxon

n nuclear potential is also observed at the lowest energies. In general, fusion cross-sections around a given energy
a are better reproduced by the potential calculated at this energy. The coordinate-dependent mass plays a crucial
J roleforthereproductionofsub-barrierfusioncross-sections. Effectsoftheenergydependenceofthepotentialcan

8 befoundinexperimentalbarrierdistributionsonlyifthevariationofthebarrierissignificantintheenergy-range
2 spanned by the distribution. It appears to be the case for 16O+208Pb but not for 40Ca+40Ca.

Conclusions: These results show that the energy dependence of the barrier predicted in TDHF calculations is
]
h realistic. ThisconfirmsthattheTDHFapproachcanbeusedtostudythecouplingsbetweenrelativemotionand
t internal degrees of freedom in heavy-ion collisions.
-
l
c PACSnumbers: 21.60.-n,21.60.Jz
u
n
[ I. INTRODUCTION is most significant at low energy (near the barrier-top)

where the colliding partners spend enough time in the
1
v vicinity of each other with little relative kinetic energy.
Experimentally obtained fusion cross-sections are gen-
0 At high energies, however, the nuclei overcome the bar-
erallyinterpretedintermsofmodelsinvolvinganucleus-
3 rier essentially in their ground-state density. This en-
nucleus potential barrier, which results from the combi-
2 ergy dependence of the effect of the couplings on the
7 nation of the attractive nuclear force and the repulsive
density evolution was clearly shown in time-dependent
. Coulomb interaction. The reduction of the many-body
1 Hartree-Fock (TDHF) calculations by Washiyama and
fusiontoaone-dimensionalpotentialbarrierproblemre-
0 Lacroix for the same systems [7]. This naturally trans-
quires the isolation of the most important physical pro-
4
lates into an energy dependence of the nucleus-nucleus
1 cesses that contribute to the building of the correct ef-
potential, similar to what was introduced phenomeno-
: fective barrier.
v logically in the Sao-Paulo potential [8]. Consequently,
i Experimental fusion barrier distributions [1] obtained thebarriercorrespondingtonearbarrier-topenergiesin-
X
from the low-energy fusion reactions of heavy-ions shed
cludes dynamical couplings effects and can be referred
r somelightintothedetailedmicroscopicmechanismsthat
a to as a dynamic-adiabatic barrier, while at high energy
are in play during the entrance channel dynamics on
the nucleus-nucleus interaction is determined by a sud-
the way to fusion [2, 3]. In particular, they may serve
den potential which can be calculated assuming frozen
as a microscope to discern various inelastic excitations
ground-state densities.
and transfer mechanisms which couple to the relative
motion. This coupling to internal degrees of freedom Due to the dynamical nature of this effect time-
induces a splitting [4] and/or a renormalization of the dependentapproachesarewellsuitedforthisstudy. The
barrier [5]. The primary underlying mechanism is the energy-dependence of the ion-ion potentials have been
dynamical change in the density along the fusion path studied using several approaches based on the fully mi-
which modifies the potential energy. croscopicTDHFtheory[7,9]. Itisusuallyfoundthatthe
barrier heights increase with bombarding energy. How-
Obviously, this density change is not instantaneous.
ever, this increase is quite slow as the sudden potential
For instance, it was shown in Ref. [6] that the develop-
is recovered at typically twice (or more) the barrier-top
ment of a neck due to couplings to octupole phonons in
energy [7].
40Ca+40Ca could take approximately one zeptosecond.
As a consequence, the dynamical change of the density Todate,thevalidityoftheTDHFapproachindescrib-
2
ing the fusion mechanism for heavy-ions has essentially Of course, microscopic approaches are much more time-
been tested by comparing TDHF fusion thresholds with consuming from a computational point of view and one
experimental barriers [6, 7, 10], although few fusion ex- has to consider approximations to the exact quantum
citationfunctionsfromdirectTDHFcalculations[10–12] many-body problem. The theoretical formalism for the
and γ-decay spectra [13] associated to pre-equilibrium microscopic description of complex many-body quantum
giant-dipole resonance [13–16] have also been compared dynamics and the understanding of the nuclear interac-
withexperimentaldata. Nevertheless,theagreementbe- tions are the underlying challenges for studying low en-
tween experimental barriers and TDHF predictions is ergy nuclear reactions.
only for near barrier-top energies, i.e., for the dynamic-
adiabatic barrier. Indeed, the predicted transition from
thedynamic-adiabaticbarriertothesuddenbarrierwith B. Time-dependent Hartree-Fock method
increasingenergyremainstobevalidatedbycomparisons
with experimental data. The purpose of this study is to The time-dependent Hartree-Fock (TDHF) theory is
accomplish this goal. a mean-field approximation of the exact time-dependent
Towards this goal, we have calculated fusion cross- many-body problem. It provides a good starting point
sections for the 40Ca+40Ca and 16O+208Pb systems us- for a fully microscopic theory of large amplitude collec-
ingpotentialsassociatedwithdifferentbombardingener- tive motion [19, 20] including fusion reactions. But only
gies. These potentials were computed using the density- in recent years has it become feasible to perform TDHF
constrainedTDHF(DC-TDHF)method[17]usingrealis- calculations on a three-dimensional Cartesian grid with-
tic TDHF trajectories. The comparison of the resulting outanysymmetryrestrictionsandwithaccuratenumer-
fusion cross-sections with experimental data is used to ical methods [21–27]. In addition, the quality of energy-
identify signatures for the energy dependence of the bar- density functionals has been substantially improved [28–
rier. 30]. One limitation of the TDHF approach is that it
InthenextsectionwegiveabriefoutlineoftheTDHF can only be used for fusion at above barrier energies
andDC-TDHFmethodsusedinthecalculations. Thisis since the theory does not allow for many-body tunnel-
followed by the calculation of barriers and fusion cross- ing. Nevertheless, the TDHF fusion threshold provides
sections for the 40Ca+40Ca system and subsequently the a prediction of the dynamic-adiabatic barrier-top energy
16O+208Pb system. The paper ends with the summary inaverygoodagreementwithexperimentaldata[7,10].
and conclusions that can be drawn from the results. The TDHF theory has then been used to study the cou-
plings between fusion and collective excitations such as
rotational motion [31–33] and vibrational modes [6, 12–
II. THEORETICAL OUTLINE 15].
Given a many-body Hamiltonian Hˆ, the action S can
A. Theoretical tools to describe fusion be constructed as
(cid:90) t2
Theoretically, the coupled-channels (CC) method is S = dt<Φ(t)|Hˆ −i(cid:126)∂t|Φ(t)> . (1)
the most commonly used approach to study fusion bar- t1
riers (see Ref. [18] for a review). The standard CC ap- Here, Φ denotes the time-dependent correlated many-
proachforcalculatingheavy-ionfusioncrosssectionscon- body wavefunction, Φ(r ,r ,...,r ;t). The variational
1 2 A
tains several adjustable parameters which determine the principleδS =0isthenequivalenttothetime-dependent
bare nucleus-nucleus potential which is often assumed to Schr¨odinger equation. In the TDHF approximation the
be of Woods-Saxon form. These potential parameters many-bodywavefunctionisreplacedbyasingleSlaterde-
are usually fitted to measured fusion cross sections or to terminantandthisformispreservedatalltimes. Thede-
elastic scattering data. In addition, experimental data terminental form guarantees the antisymmetry required
such as energies and B(Eλ) values of collective vibra- by the Pauli principle for a system of fermions. In this
tions and giant resonances are required as input for the limit,thevariationoftheactionyieldsthemostprobable
CC calculations to determine the collective coupling po- time-dependentmean-fieldpathbetweenpointst andt
1 2
tentials. This is a limitation for exotic nuclei for which in the multi-dimensional space-time phase space:
these data are not always available. A possible solution
δS =0→Φ (t), (2)
of this problem is to compute these parameters directly 0
with microscopic models [6, 7, 17] and use them in stan-
where Φ (t) is a Slater determinant with the associated
0
dard coupled channel calculations [6]. Finally, it is diffi-
single-particle states φ (r,t). The variation in Eq.(2) is
λ
cult to incorporate multi-nucleon transfer channels into
performed with respect to the single-particle states φ
λ
the CC formalism. and φ∗. This leads to a set of coupled, nonlinear, self-
Alternatively, fully microscopic theories could be used λ
consistent initial value equations for the single-particle
to overcome these limitations. In particular, they only
states
requireaneffectiveinteractionoranenergy-densityfunc-
tional to describe the interactions between the nucleons. h({φ })φ =i(cid:126)φ˙ λ=1,...,N , (3)
µ λ λ
3
and their Hermitic conjugates. These are the fully mi- as a function of time. Note that the part of the resid-
croscopic TDHF equations. As we see from Eq.(3), each ualinteractionwhichisneglectedinTDHFmayproduce
single-particle state evolves in the mean-field generated fluctuations and correlations which affect these trajecto-
by the concerted action of all the other single-particle ries. RecentbeyondTDHFdevelopmentshavebeenused
states. to investigate the effects of such fluctuations in heavy-
InstandardTDHFapplicationstoheavy-ioncollisions, ion collisions [35, 36]. However, the TDHF approach is
the initial nuclei are calculated using the static Hartree- optimized to the expectation values of one-body opera-
Fock (HF) theory and the Skyrme functional [28]. The tors [37] and is then capable to predict these quantities.
resulting Slater determinants for each nucleus comprise This was demonstrated by the recent successes of TDHF
thelargerSlaterdeterminantdescribingthecollidingsys- inreproducingvariousreactionmechanismsinheavy-ion
tem during the TDHF evolution. Nuclei are assumed to collisions. Moreover, beyond TDHF calculations remain
moveonapureCoulombtrajectoryuntiltheinitialsepa- numerically difficult. We then restrict the present calcu-
ration between the nuclear centers used in TDHF evolu- lations to the TDHF level.
tion. Ofcourse,noassumptionismadeonthesubsequent OneofthemainapplicationofrecentTDHFcodeshas
trajectory in the TDHF evolution. Using the Coulomb been to study fusion reactions. For TDHF collisions of
trajectory we compute the relative kinetic energy at this light and medium mass systems, as well as highly mass-
separation and the associated translational momenta for asymmetricsystems,fusiongenerallyoccursimmediately
eachnucleus. Thenucleiarethenboostedbymultiplying abovetheCoulombbarrier. Inheaviersystems,however,
the HF states with there is an energy range above the barrier where fusion
does not occur [20, 38, 39]. This phenomenon is the mi-
Φj →exp(ıkj ·R)Φj , (4) croscopicanalogueofthemacroscopicextra-push thresh-
old[40]. In theextreme case ofactinide collisions, fusion
where Φ is the HF state for nucleus j and R is the
j becomes impossible and the fragments reseparate in few
corresponding center of mass coordinate
zeptoseconds [41–43].
The path to fusion as described in TDHF calculations
1 (cid:88)Aj is a sequence of states from dinuclear configurations to a
R= r . (5)
Aj i compact compound system. Along this path, one-body
i=1
dissipation plays a crucial role and single-particle fric-
TheGalileaninvarianceandtheconservationofthetotal tion can quickly absorb the kinetic energy of the relative
energyintheSkyrmeTDHFequationsareusedtocheck motion. As long as the average single-particle excitation
the convergence of the calculations. energy per nucleon is less than the shell energy (about
DuetothefactthatTDHFcalculationsdonotinclude 4−8 MeV) the details of the ground state potential en-
sub-barrier tunneling of the many-body wave-function, ergy surface are still felt and shell correction energies
thefusionprobability,Pfus.(L,Ec.m.),foraparticularor- influence the TDHF dynamics. It is precisely for this
bital angular momentum L at the center-of-mass energy reason that the DC-TDHF approach allows us to repro-
Ec.m. can only be PfTuDs.HF = 0 or 1. As a consequence duce ion-ion interaction barriers for heavy-ion collisions.
the quantal expression for the fusion cross-section
π(cid:126)2 (cid:88)∞
σ (E )= (2L+1)P (L,E ), (6) C. DC-TDHF method
fus. c.m. 2µE fus. c.m.
c.m.
L=0
The TDHF theory does not include quantum tun-
where µ is the reduced mass of the system, reduces to
nelling of the many-body wave function. Consequently,
π(cid:126)2 Lmax(cid:88)(Ec.m.) directTDHFcalculationscannotbeusedtodescribesub-
σ (E )= (2L+1) barrier fusion. Nevertheless, a number of approaches
fus. c.m. 2µEc.m. based on TDHF were developed to extract fusion poten-
L=0
π(cid:126)2 tials with dynamical effects [7, 17] in order to compute
= [L (E )+1]2 , (7) fusion cross-sections at sub-barrier energies.
2µE max c.m.
c.m. The density-constrained TDHF (DC-TDHF) utilizes
L being the largest orbital angular momentum lead- a novel approach of using time-dependent densities
max
ingtofusion. Thisisknownasthequantumsharpcut-off from TDHF to self-consistently calculate the underlying
formula [34]. ion-ion interaction potentials [17] and excitation ener-
Since TDHF is based on the independent-particle ap- gies [44]. These potential barriers then allow for the cal-
proximation it can be interpreted as the semi-classical culation of fusion cross-sections at both sub-barrier and
limitofafullyquantaltheorythusallowingaconnection above-barrier energies. The method was applied to cal-
to macroscopic coordinates and providing insight about culatefusionandcapturecrosssectionsaboveandbelow
the collision process. In this sense the TDHF dynamics the barrier, ranging from light systems [12, 45] to hot
can only be used to compute the semiclassical trajecto- and cold fusion reactions leading to superheavy element
ries of the collective moments of the composite system Z =112 [39]. In all cases a good agreement between the
4
measuredfusioncrosssectionsandtheDC-TDHFresults modify the inner part of the ion-ion potential, which is
was found. This is rather remarkable given the fact that important for fusion cross-sections at deep sub-barrier
the only input in TDHF is the Skyrme energy-density energies.
functional whose parameters are determined from struc-
Fusion cross-sections are calculated by directly inte-
ture information.
grating the Schr¨odinger equation
The concept of using density as a constraint for cal-
culatingcollectivestatesfromTDHFtime-evolutionwas
first introduced in Ref. [46], and used in calculating col- (cid:20)−(cid:126)2 d2 (cid:126)2(cid:96)((cid:96)+1) (cid:21)
+ +V(R)−E ψ (R)=0,
lectiveenergy surfacesinconnectionwithnuclearmolec- 2µ dR2 2µR2 c.m. (cid:96)
ular resonances in Ref. [47]. In this approach the TDHF (10)
time-evolution takes place with no restrictions. At cer- using the well-established Incoming Wave Boundary
tain times during the evolution the instantaneous den- Condition(IWBC)method[50]toobtainthebarrierpen-
sity is used to perform a static Hartree-Fock minimiza- etrabilities Pfus.(L,Ec.m.) which determine the total fu-
tion while holding the neutron and proton densities con- sion cross section [Eq. (6)].
strained to be the corresponding instantaneous TDHF In writing Eq. (8) we have introduced the concept of
densities. Inessence,thisprovidesuswiththeTDHFdy- an adiabatic reference state for a given TDHF configu-
namical path in relation to the multi-dimensional static ration. The adiabatic reference state is the one obtained
energy surface of the combined nuclear system. The ad- viathedensityconstraintcalculation. ItistheSlaterde-
vantages of this method in comparison to other mean- terminant with lowest energy for the given density with
field based microscopic methods such as the constrained vanishing current. It is then used to approximate the
Hartree-Fock (CHF) method are obvious. First, there collective potential energy [46]. We would like to em-
is no need to introduce external constraining operators phasize that this procedure does not affect the TDHF
whichassumethatthecollectivemotionisconfinedtothe time-evolution and contains no free parameters or nor-
constrainedphasespace. Second,thestaticadiabaticap- malization.
proximationisreplacedbythedynamicalanaloguewhere
Finally, ion-ion interaction potentials calculated using
the most energetically favorable state is obtained by in-
DC-TDHFcorrespondtotheconfigurationattaineddur-
cluding sudden rearrangements and the dynamical sys-
ing a particular TDHF collision. For light and medium
tem does not have to move along the valley of the po-
mass systems as well as heavier systems for which fu-
tential energy surface. In short we have a self-organizing
sion is the dominant reaction product, DC-TDHF calcu-
system which selects its evolutionary path by itself fol-
lations at near barrier-top energy give a fusion barrier
lowing the microscopic dynamics. All of the dynamical
which is expected to match the TDHF fusion threshold.
features included in TDHF are naturally included in the
In practice, due to the underlying numerical approxima-
DC-TDHF calculations. These effects include neck for-
tionsintheDC-TDHFmethod,small(typicallylessthan
mation,massexchange,internalexcitations,deformation
0.5MeV)underestimationoftheTDHFfusionthreshold
effects to all order, as well as the effect of nuclear align-
are sometime observed.
ment for deformed systems.
In the DC-TDHF method the ion-ion interaction po-
tential is given by
VDC(R)=EDC(R)−EA1 −EA2 , (8) III. RESULTS
where E is the density-constrained energy at the in-
DC
stantaneous separation R(t), while E and E are the TDHF calculations for the DC-TDHF computation of
bindingenergiesofthetwonucleiobtaAi1nedwithAt2hesame microscopic potential barriers for the 40Ca+40Ca system
effective interaction. This ion-ion potential V (R) is weredoneinaCartesianboxwhichis50fmalongthecol-
DC
asymptoticallycorrectsinceatlargeinitialseparationsit lisionaxisand25fmintheothertwodirections. Thenu-
exactly reproduces V (R ). In addition to the cleiwereplacedataninitialseparationof20fm. Forthe
Coulomb max
ion-ion potential it is also possible to obtain coordinate 16O+208PbsystemwehavechosenaCartesianboxwhich
dependent mass parameters. One can compute the “ef- is 60 fm along the collision axis and 30 fm in the other
fective mass” M(R) using the conservation of energy two directions. The two nuclei are placed at an initial
separationof24fm. CalculationsusedtheSLy4Skyrme-
2[E −V (R)] functional [28] as described in Ref. [26]. Static calcula-
M(R)= c.m. DC , (9)
R˙2 tions aredone usingthe damped-relaxationmethod[51].
The numerical accuracy of the static binding energies
where the collective velocity R˙ is directly obtained from and the deviation of the computed DC-TDHF potential
the TDHF evolution. This coordinate dependent mass from the point Coulomb energy in the initial state of the
can be exactly incorporated into the ion-ion potential, collision dynamics is of the order of 50−150 keV. We
whichwecallV(R),byusingapoint-transformation[48, have performed density constraint calculations at every
49]. The effect of the coordinate-dependent mass is to 10−20 fm/c interval.
5
A. 40Ca+40Ca Fusion Barriers In the DC-TDHF method the energy dependence of
thebarriersarisesfromthechangingdynamicalbehavior
of the system. At energies close to the barrier-top the
Recently, particular experimental attention has been
onset of neck dynamics is slow and allow ample time for
given to fusion reactions involving Ca isotopes [52–55].
density rearrangements for the system, whereas as the
Thesenewexperimentssupplementolderfusiondata[56]
energy is increased there is less and less time for rear-
and extend them to lower sub-barrier energies. In
rangements to occur and a long-lived neck to form, thus
Ref. [55] a comprehensive CC calculation for this system
approachingthefrozen-densitylimit[7]. Thebarriercor-
has also been presented utilizing the shallow potential
respondingtothelowestTDHFenergymaybecalledthe
approach [57]. These calculations use M3Y+repulsion
dynamic-adiabaticbarrierasopposedtoastatic-adiabatic
potential and the excitations of collective phonons. In
barrier that could be obtained by using the constrained
particular, octupole vibrations have been shown to play
Hartree-Fock approach or a prescription like the folding-
animportantroleonthedynamicsinthissystem[6,58].
model. The barrier corresponding to TDHF energies
much higher than the dynamic-adiabatic barrier may be
labeled as the sudden barrier. We see from Fig. 1 that
1. Nucleus-nucleus potentials this leads to an increasing barrier height with increasing
collision energy and quickly saturates for energies that
The 40Ca+40Ca system was investigated in Ref. [59] are considerably higher than the lowest energy barrier.
with the DC-TDHF method using TDHF c.m. ener- In this sense, we obtain a distribution of barriers as a
gies 55, 60, and 65 MeV. The resulting potential bar- function of collision energy.
riers are reported in Fig. 1. In the present work, ad- An important dynamical effect is due to the
ditional calculations have been performed to study in coordinate-dependenceofthemass,M(R). InFig.1this
more details the energy dependence of the barrier and effect is demonstrated by plotting the direct DC-TDHF
itseffectonthefusioncross-sections. Wehaveperformed potentials,VDC(R)(dashedlines),andthosethatinclude
TDHFcalculationsin1MeVintervalsinthe53−65MeV themodificationofthecoordinate-dependentmass,V(R)
range and computed the corresponding DC-TDHF po- (solid lines). For TDHF collisions of symmetric systems
tentials. As a result, barrier heights are in the range of the net particle transfer is zero and cannot affect M(R).
52.6−53.6MeValllocatedinthevicinityofnuclearsep- However,thedynamicalneckformationandcollectiveex-
arationR=10.2fm. Weobservethatforthe40Ca+40Ca citations are possible and can change the effective mass.
system DC-TDHF potential barriers do not show an ap- The potentials shown in Fig. 1 should not be directly
parent strong energy dependence. compared with nucleus-nucleus potentials entering CC
calculations. Indeed, the latter are un-coupled poten-
tials with various couplings and particle transfer added
54 40 40 Point Coulomb on subsequently. In cases were double-folding method is
Ca + Ca
used the densities are frozen as the nuclear separation R
52 changes. Thisusuallyimpliesahigherun-coupledbarrier
heightasitwasfoundtobeintherange54.1−54.7MeV
50 in Refs. [6, 7, 55, 60].
)
V
e
M 48
R) ( VDC(R) 2. Fusion cross-sections
(46
V
The corresponding fusion cross-sections calculated
from the potentials V(R) shown in Fig. 1 are plotted
44
E = 55 MeV inFig.2inlogarithmicscaleandinFig.3inlinearscale.
TDHF
E = 60 MeV The experimental points are from Refs. [55, 56]. The
42 TDHF
E = 65 MeV cross-sections clearly depend on the TDHF energy used
TDHF to extract the DC-TDHF potential. The interaction po-
40
7 8 9 10 11 12 13 14 15 tential corresponding to the lowest TDHF energy leads
R (fm) tofusioncross-sectionswhichareingoodagreementwith
the sub-barrier fusion data but overestimate the cross-
sections at higher energies. On the other hand, the po-
FIG. 1. (Color online) DC-TDHF ion-ion interaction poten-
tials V(R) (solid lines) including coordinate dependence of tential corresponding to the highest energy reproduces
theeffectivemassM(R)for40Ca+40CaobtainedfromTDHF the highest energy data but underestimates the data at
calculations at various center-of-mass energies. The poten- lower energies.
tials VDC(R) obtained without the coordinate dependence of In principle, each set of cross-sections σn(E) is valid
M(R) are plotted with thin dashed lines. Shown also is the only near the TDHF energy E used to calculate the
n
corresponding point-Coulomb potential (thick dashed line).
potential. One can then generate a unique function
6
(cid:80)
σ¯(E) = σ (E)f (E) where f (E) is a weighting agreement between sub-barrier data and the theoretical
n n n n
function peaked at E = E . In practice, σ¯(E) has been cross-sections calculated with the dynamic-adiabatic po-
n
generated using tential indicates that this energy dependence is likely to
 be small.
1 E <E
 0
f0 = cos2[π2 E∆−EE0] E0 ≤E ≤E1 600
0 E >E
1 55 MeV
0 E <E 500 60 MeV
 n−1
f0<n<N = cos2[π2 E∆−EEn] En−1 ≤E ≤En+1 65 MeV
0 E >E 400 E-dependent
n+1 )
0 E <E mb Montagnoli et al.
 N−1 ( 300 Aljuwair et al.
fN = cos2[π2 E∆−EEN] EN−1 ≤E ≤EN . us
1 E >E f
N
s 200
E isthelowestTDHFenergyatwhichfusionisobserved
0
and from which a potential can be extracted, while EN 100
is the maximum TDHF energy considered in this work.
∆EistheconstantenergystepintheTDHFcalculations.
0
(The generalization to non-constant ∆E is trivial.) 50 55 60 65
E (MeV)
FIG.3. (Coloronline)SameasFig.2inlinearverticalscale.
100
)
b
m 10 55 MeV 3. Fusion barrier distributions
(
s 60 MeV
u
f 65 MeV Toinvestigatepossiblesignaturesoftheenergydepen-
s E-dependent
dence at energies close to the barrier, we have calculated
1
Montagnoli et al. the following quantity [3]
Aljuwair et al.
(cid:20)d2(Eσ (E))(cid:21)
D(E )= fus (11)
0.1 i dE2
50 55 60 65 i
E (MeV) (cid:18)(Eσ) −2(Eσ) −(Eσ) (cid:19)
(cid:39) i+1 i i−1 ,
∆E2
FIG. 2. (Color online) Fusion cross-sections for 40Ca+40Ca
obtainedfromtheDC-TDHFpotentialsshowninFig.1. The whichisknownasthefusionbarrierdistribution[1]. Itis
dashedlinerepresentsthecombinedcross-sectionsσ¯(E). The essentially zero except in the energy range of the barrier
data points are from Refs. [55, 56]. and has then been widely used to study the effect of the
couplingsbetweenrelativemotionandinternalstructures
Theresultingσ¯(E)islabelled”E-dependent”inFigs.2 on fusion barriers. As it was discussed in some detail in
and 3. Considering the experimental error bars and the Ref. [3] the calculation of the barrier distribution using
fluctuationsbetweenthedatasets,weseethatthereisan the above formula is sensitive to the value of the en-
overallagreementbetweenσ¯(E)andtheexperimentalfu- ergy separation ∆E used in the finite-difference formula.
sion cross-sections, despite a slight overestimation of the Commonly, a value between ∆E =1−2 MeV is used.
more recent data from Ref. [55] in the barrier region. It Selected barrier distributions obtained from different
is then reassuring to observe that the energy-dependent TDHF energies are shown in Fig. 4 together with ex-
DC-TDHF potentials lead to reasonable reproduction of perimental data from Refs. [55, 56]. The barrier distri-
the data in the energy-range of the TDHF energy used butions were calculated with ∆E = 1 MeV. The dis-
fortheircalculation. Thiscomparisonwithexperimental tributions corresponding to different TDHF energies are
data also confirms that the potential barrier ”seen” by generally smooth but the centroids shift to a higher en-
the system at high energy is effectively higher than the ergy with increasing TDHF energy and the heights of
one at low-energy. It is unfortunate, however, that this the distributions become lower. This change can be in-
energy-dependencecannotbeinvestigatedbelowthebar- terpretedasbeingduetothedifferenceinthedynamical
rier. This is due to the fact that the TDHF calculations processes that are more prevalent at barrier-top ener-
at sub-barrier energies do not lead to fusion and, then, gies in comparison to higher energies where we approach
the DC-TDHF method cannot be applied to extract the the frozen-density limit. Despite fluctuations in the ex-
potential in this energy regime. Nevertheless, the good perimental data, it is clear that the distributions asso-
7
ciated with the high TDHF energies (E = 60 and 1. Nucleus-nucleus potentials
TDHF
65 MeV) do not reproduce the experimental barrier dis-
tribution. This is of course not a problem as the com- We have performed TDHF calculations in 1 MeV in-
parison should be made at energies close to 60-65 MeV, tervals between 75 and 80 MeV center-of-mass energies,
for which D(E) (cid:39) 0. Nevertheless, this indicates that as well as at 90 and 100 MeV. The corresponding DC-
the measured barrier distributions provide information TDHF barriers are shown in Fig. 5 for E = 75,
TDHF
on the dynamic-adiabatic barrier, but not on the poten- 80, and 100 MeV. Barrier heights are in the range of
tial seen by the system at higher energies. 73.7 − 75.0 MeV all located in the vicinity of nuclear
separation R = 12 fm. As it was in the previous study
1600 thebarrierthicknessatsub-barrierenergieschangeswith
changing collision energy due to the fact that at lower
1400 Montagnoli et al. energies the system has more time to rearrange its den-
Aljuwair et al.
) sity, which would manifest itself as the formation of a
1
-ev 1200 ETDHF = 53 MeV neck followed by nucleon transfer [7, 10, 63] and collec-
.mbM 1000 EETTDDHHFF == 5650 MMeeVV thiivgeheexstciftoartihoingsh.eSstimeinlaerrglyy,tahpepernoearcghyinogftthheebsaurdrdieern-tloimpiist
2 ( 800 ETDHF = 65 MeV at high energies. Moreover, we observe that as we move
E downfromthepotentialpeaktheinnerpartofthebarrier
d
)/ 600 usually deviates from the Wood-Saxon+Coulomb form,
)
E
σ(fus 400 wwhitihcohuitstthheeccoaosredifnoarted-edeeppesnudbe-bnatrmriearsse.nergies, with or
E
2( 200 76
d
0 16 208
O + Pb Point Coulomb
74
-200
50 52 54 56 58 60
E (MeV)
c.m. 72
)
FIG. 4. (Color online) Fusion barrier distributions for V
40Ca+40Ca obtained from DC-TDHF potentials calculated Me70
withdifferentTDHFenergies. Thedatashownassolid-filled (
circlesarefromRef.[55],andthesquaresarefromRef.[56]. R) 68 V
( DC
V
66
E = 75 MeV
TDHF
E = 80 MeV
64
TDHF
B. 16O+208Pb Fusion Barriers E = 100 MeV
TDHF
62
10 11 12 13 14 15
Thesecondsystemwehavestudiedis16O+208Pb. The
R (fm)
choiceofthissystemispartlymotivatedbythefactthat
its fusion barrier is affected by early charge equilibration
FIG. 5. (Color online) DC-TDHF ion-ion interaction po-
dynamics [10, 61]. Quantitative reproductions of fusion
tential V(R) (solid lines) including coordinate dependence of
cross-sectionsforthissystemwouldthenbeanindication theeffectivemassM(R)for16O+208PbobtainedfromTDHF
that the TDHF approach is able to treat the interplay
calculations at various center-of-mass energies. The poten-
between nucleon transfer and fusion. This system is also tials V (R) obtained without the coordinate dependence of
DC
one for which fusion hindrance at deep sub-barrier ener- M(R) are plotted with thin dashed lines. Shown also is the
gies has been observed [62]. Standard coupled-channels corresponding point-Coulomb potential.
calculations including low lying vibrational states and
one-neutron transfer channels could not consistently re- In Ref. [64] a method was developed to extract the
produce the high and low-energy fusion data. While the ion-ion potential directly from the experimental sub-
shallow-potential approach of Ref. [57] had some suc- barrier cross-sections in an attempt to understand the
cess in reproducing the low-energy part of the data it reason for CC calculations not reproducing sub-barrier
required an imaginary potential to reproduce the high- and high-energy part of the data with a single poten-
energy part of the data. Furthermore, inconsistencies in tial model. These calculations showed that the form
the shallow-potential approach for simultaneously repro- of the potential deviated from the Wood-Saxon shape
ducing the low and high-energy fusion data was pointed and one of the possible reasons to account for this de-
out in Ref. [62]. viation was suggested to be the coordinate-dependent
8
mass. The potential barrier extracted directly from the 1000
sub-barrierdatawascalledtheadiabatic potentialandis
plotted in Fig. 6 (solid line, the shaded region indicates
uncertainty) together with the DC-TDHF potential at 100
75MeVwith(dashedline)andwithout(dottedline)the
)
b
coordinate-dependent mass. The potential with coordi- m
nate dependent mass is in much better agreement with (s 10 75 MeV
the one extracted from data using the inversion method. fu 80 MeV
100 MeV
We can conclude from these calculations that indeed the s
E-dependent
coordinate-dependent mass, which is really a byproduct 1 E-dependent (raw)
ofheavy-ionandneckdynamics,islargelyresponsiblefor TDHF direct
Morton et al.
thethickeningofthebarrierfordeepsub-barrierenergies
as shown in Fig. 5. 0.1
70 80 90 100 110
E (MeV)
7755
1166OO++220088PPbb FIG. 7. (Color online) Fusion cross-sections for 16O+208Pb
obtainedfromtheDC-TDHFpotentialsshowninFig.5. The
dashedanddottedlinesrepresentthecombinedcross-sections
σ¯(E)obtainedwithandwithoutcoordinatedependentmass,
V)V) 7700
ee respectively. The data points are from Ref. [65].
MM
((R) R)
V(V(
same amount (see Fig. 8). As the TDHF calculations
6655 Inversion reproduce well the centroid of the experimental barrier
V(R)
distribution [7, 10], it is then likely that beyond mean-
V (R)
DC field effects are responsible for the observed discrepancy
above the barrier. For instance, the transfer of a proton
99 1100 1111 1122 1133 1144 1155 1166
pair, and, to a lesser extent, of an α-cluster, which are
RR ((ffmm))
not included in TDHF calculations, have been shown to
beanimportantmechanisminthissystem[66,67]. Nev-
FIG. 6. (Color online) The adiabatic potential obtained in
ertheless, this discrepancy is relatively small considering
Ref.[64]comparedwiththeDC-TDHFpotentialreproducing
the fact that there are no free parameters.
the sub-barrier cross-sections.
1200
16 208
O + Pb
2. Fusion cross-sections
1000
As in the 40Ca+40Ca reaction, the energy dependence b) 800
of the ion-ion interaction potentials observed in Fig. 5 m
(
leads to the corresponding change in the calculated fu- us 600 75 MeV
sioncross-sectionsasshowninFigs.7(logarithmicscale) f 80 MeV
s 100 MeV
and 8 (linear scale). Also shown in the same figures 400 E-dependent
are the experimental cross-sections from Refs. [65]. The E-dependent (raw)
general trends observed in the energy-dependence of the 200 TDHF direct
Morton et al.
cross-sections is similar to the 40Ca+40Ca case. Indeed,
0
thepotentialobtainedatthelowestTDHFenergyrepro- 70 80 90 100 110
duces the sub-barrier cross-sections. In addition, the ex- E (MeV)
perimentalcross-sectionsathighenergyarebetterrepro-
duced by potentials calculated at similar energies. How- FIG.8. (Coloronline)SameasFig.7inlinearverticalscale.
ever,itisnoticeablethat,evenintheenergyrangewhere
they are supposed to be valid, the energy-dependent po-
tential overestimates the experimental data. Note that Let us now investigate the effect of the coordinate de-
this is not a drawback of the method used to extract pendenceoftheeffectivemassM(R)onthefusioncross-
the potential as the problem can be traced back to the sections. In Figs. 7 and 8 we have also plotted the cal-
TDHF approximation itself. Indeed, direct TDHF cross- culated E-dependent cross-sections without the use of
sections computed at above barrier energies by finding the coordinate-dependent mass (dotted curve). Figure 8
the maximum impact parameter for fusion at each en- shows that, above the barrier, the inclusion of the co-
ergy overestimate the experimental cross-sections by the ordinate dependence of the effective mass does not play
9
animportantroleasbothenergy-dependentcalculations energies. This qualitative observation was also made in
leadtosimilarcross-sections. However,Fig.7showsthat the 40Ca+40Ca case. In addition, each theoretical bar-
this is not the case below the barrier. Here, the effect of rier distribution is narrower than the experimental one.
M(R) on the low-energy cross-section is seen to be es- This observation could be attributed to the energy de-
sential. Indeed, without the coordinate dependence of pendence of the potential. Indeed, the high energy tail
the mass, the cross-sections are overestimated below the of the experimental barrier distribution extends up to
barrier. Including this dependence widens the barrier ∼ 80 MeV. The barrier distribution computed from the
(see Fig. 5) and consequently reduces the cross-sections, E =75 MeV potential naturally fails to reproduce
TDHF
providing a much better agreement with the data (see the high energy part of the experimental barrier distri-
Fig. 7). bution. The latter is much better reproduced by the
E =80MeVpotential. Thetailinthe75−80MeV
TDHF
regioncanthenbeinterpretedasaneffectofthegradual
3. Fusion barrier distributions increase of the barrier height in this energy range. Note
that this effect is not visible in the 40Ca+40Ca data due
to the fact that the change in barrier height is not no-
Finally,westudytheeffectoftheenergydependenceof
ticeable in the limited energy range span by the barrier
thepotentialonthefusionbarrierdistributionD(E). So
distribution.
far, standard CC calculations have not been able to re-
produce the fusion barrier distributions consistently for
low and high energies [65]. Improved barrier distribu-
tionsatlowerenergieswerecalculatedusingCCwiththe IV. SUMMARY AND DISCUSSION
shallow-potential method [57]. However, above barrier
cross-sections could only be explained with addition of Ion-ion potentials are sensitive to the excitation and
an imaginary absorbing potential. We have constructed transfermechanisminplayonthewaytoformingacom-
pound nucleus. However, these couplings between rela-
tivemotionandinternaldegreesoffreedomhavetimeto
1200 16O + 208Pb Morton et al. affect the nucleus-nucleus potential only at low energy,
Dasgupta et al.
1) leading to a “dynamic-adiabatic” potential. At high en-
-V 1000
ergy, the system does not have enough time to rearrange
e
M its density, leading to a “sudden” potential. As a result,
mb. 800 ETDHF= 75 MeV this leads to an energy dependence of the potential and,
E = 80 MeV
2E] ( 600 ETTDDHHFF= 100 MeV iwnaspatrotiicduelnatri,fyofsiigtnsabtuarrreisero.f tThhiseepnuerrpgoysedeopfenthdiesncweorink
d
)/ experimentalfusioncross-sectionsbycomparingwiththe
)
E predictions of microscopic calculations.
( 400
us Fusion potentials around the barrier have been calcu-
f
sE lated for the 40Ca+40Ca and 16O+208Pb systems using
( 200
2 the DC-TDHF method based on TDHF density evolu-
d
[ tions. It is shown that, as we go to above barrier en-
0
ergies, the energy dependence of the potential increases
65 70 75 80 85 90 the barrier height and consequently slows down the in-
E (MeV) crease of the fusion cross-sections with increasing bom-
c.m.
barding energy. This effect happens in a large energy
range until the sudden potential is reached (according
FIG. 9. (Color online) Fusion barrier distributions for
16O+208Pb obtained for various TDHF energies. The data to Ref. [7], this can occur at about twice the energy of
shown as solid-filled circles are from Refs. [62, 65] the barrier). As a result, the dynamic-adiabatic and the
. sudden barriers can be very different. The former repro-
ducessub-barrierdata, whilethelatterprovidesabetter
fusion-barrier distributions from the DC-TDHF cross- agreement at well above barrier energies than at low en-
section by using an energy spacing of ∆E =2.0 MeV as ergies. Discrepancies remain, however, at above barrier
shown in Fig. 9 for TDHF bombarding energies 75, 80, energies for the 16O+208Pb system, which could be due
and100MeV.AlsoshownarethedatafromRefs.[62,65]. to proton-pair and α-cluster transfer not included in the
A first observation is that the DC-TDHF barrier dis- theory. It should also be noted that signatures of the
tributions suffer from the overestimation of the barrier- energy-dependence of the potential are less visible in the
distributions at intermediate energies. The difficulty in experimental barrier distributions due to the fact that
reproducing this region is shared with standard CC ap- these distributions usually span a small energy range in
proaches. Theoriginofthisdiscrepancyarestillunclear. which the variation of the barrier is not always very sen-
Another observation is that, as the TDHF energy is in- sitive.
creased, the corresponding distributions peak at higher Finally, let us compare the energy-dependence of the
10
potentialsinbothsystems. ThisisdoneinFig.10where dynamics present in TDHF does properly account for
we plot the ratio of the barrier heights obtained from many of the excitation and transfer mechanisms. Nat-
DC-TDHF, VDC−TDHF, and direct TDHF, VTDHF, urally, this is achieved in an average way as opposed to
B B
calculations as a function of the dimensionless variable a fully quantal theory. The present calculations are an-
E /VTDHF. Itisinterestingtonotethattheenergy other testament to a growing number of TDHF calcu-
TDHF B
dependence of the barriers are found to be very similar lations, both in the small amplitude limit for low-lying
for both systems. It is then not surprising that the same and collective state calculations and in the large ampli-
behavior is obtained in the fusion cross-section plots. tude limit of reaction dynamics, finding good compar-
isons with experimental observations. This progress is
1.02 partially due to the increased computational capabilities
that allow such calculations to be performed without us-
ing any symmetry restrictions and with modern energy
HF density functionals. This suggests that for low-energy
TD 1.01 heavy-ion reactions TDHF remains as an ever more use-
B ful theoretical tool.
V
/
F
H
D
T
C
D 1 40 40
B Ca+ Ca
V
16 208 ACKNOWLEDGMENTS
O+ Pb
0.99 Useful discussions with M. Dasgupta and D. J. Hinde
1 1.1 1.2 1.3
TDHF are acknowledged. This work has been supported
E /V
TDHF B by the U.S. Department of Energy under Grant No.
DE-FG02-96ER40975 with Vanderbilt University, and
FIG. 10. (Color online) The ratio of barrier heights ob- by the Australian Research Council under Grants No.
tainedfromDC-TDHFanddirectTDHFcalculationsplotted FT120100760, No. FL110100098, and No. DP1094947.
against the dimensionless scale variable ETDHF/VBTDHF. PartofthecalculationshavebeenperformedontheNCI
National Facility in Canberra, Australia, which is sup-
The quality of the results suggests that the mean-field ported by the Australian Commonwealth Government.
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