Table Of ContentMicroscopic analysis of sub-barrier fusion enhancement in 132Sn+40Ca vs. 132Sn+48Ca
V.E. Oberacker1 and A.S. Umar1
1Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, USA
(Dated: January 24, 2013)
We provide a theoretical analysis of recently measured fusion cross sections which show a sur-
prising enhancement at low E energies for the system 132Sn+40Ca as compared to the more
c.m.
neutron-rich system 132Sn+48Ca. Dynamic microscopic calculations are carried out on a three-
dimensionallatticewithatime-dependentdensity-constraineddensityfunctionaltheory. Thereare
no adjustable parameters, the only input is the Skyrme effective NN interaction. Heavy-ion poten-
tials V(R), coordinate-dependent mass parameters M(R), and total fusion cross sections σ(E )
c.m.
are calculated for both systems. We are able to explain the measured fusion enhancement in terms
ofthenarrowerwidthoftheion-ionpotentialfor132Sn+40Ca,whilethebarrierheightsandpositions
are approximately the same in both systems.
3 PACSnumbers: 21.60.-n,21.60.Jz
1
0
2 I. INTRODUCTION haveappliedthismethodtocalculatefusionandcapture
n crosssectionsaboveandbelowthebarrier. Asoftodate,
a Radioactive ion beam facilities enable us to study fu- wehavestudiedatotalof18differentsystems, including
J sion reactions with exotic neutron-rich nuclei. An im- 132,124Sn+96Zr[14],132Sn+64Ni[15,16],16O+208Pb[17],
3 portant goal of these experiments is to study the effects and hot and cold fusion reactions leading to superheavy
2 of neutron excess (N −Z) on fusion. In several exper- element Z = 112 [18]. Most recently, we have investi-
iments, large enhancements of sub-barrier fusion yields gated sub-barrier fusion and pre-equilibrium giant res-
]
h have been observed for systems with positive Q values onance excitation between various calcium + calcium
-t for neutron transfer. Recently, at the HRIBF facility a isotopes [19], and between isotopes of oxygen and car-
l series of experiments has been carried out with radioac- bon[20]thatoccurintheneutronstarcrust. Inallcases,
c
u tive132Snbeamsandwithstable124Snbeamson40,48Ca wehavefoundgoodagreementbetweenthemeasuredfu-
n targets [1]. It turns out that the 40Ca+Sn systems have sion cross sections and the DC-TDHF results. This is
[ many positive Q values for neutron-pickup while all the rather remarkable given the fact that the only input in
Q values for 48Ca+Sn are negative. However, the data DC-TDHF is the Skyrme effective N-N interaction, and
1
v analysis reveals that the fusion enhancement is not pro- there are no adjustable parameters.
2 portional to the magnitudes of those Q values. This paper is organized as follows: in Section II we
5 Particularly puzzling is the experimental observa- summarize the theoretical formalism and show results
5 tion of a sub-barrier fusion enhancement in the system for the ion-ion potentials calculated with the DC-TDHF
5 132Sn+40Ca as compared to more neutron-rich system method. In Section III, we discuss the corresponding
1. 132Sn+48Ca. This is difficult to understand because the total fusion cross sections, and we explain the observed
0 8additionalneutronsin48Cashouldincreasetheattrac- fusion enhancement in terms of the narrower width of
3 tive strong nuclear interaction and thus lower the fusion the ion-ion potential for 132Sn+40Ca. Our conclusions
1 barrier, resulting in an enhanced sub-barrier fusion cross are presented in Section IV.
:
v section. But the opposite is found experimentally. A
i coupled channel analysis [1] of the fusion data with phe-
X
nomenological heavy-ion potentials yields cross sections II. THEORETICAL FORMALISM AND
ar that are one order of magnitude too small at sub-barrier ION-ION POTENTIALS
energies, despite the fact that these ion-ion potentials
contain many adjustable parameters. Currently, a true quantum many-body theory of bar-
Thetime-dependentHartree-Fock(TDHF)theorypro- riertunnelingdoesnotexist. Allsub-barrierfusioncalcu-
vides a useful foundation for a fully microscopic many- lationsassumethatthereexistsanion-ionpotentialV(R)
body theory of large amplitude collective motion [2, 3] which depends on the internuclear distance R. Most of
including deep-inelastic and fusion reactions. But only the theoretical fusion studies are carried out with the
in recent years has it become feasible to perform TDHF coupled-channels(CC)method[21–24]inwhichoneuses
calculations on a 3D Cartesian grid without any symme- empirical ion-ion potentials (typically Woods-Saxon po-
try restrictions and with much more accurate numerical tentials, or double-folding potentials with frozen nuclear
methods [3–8]. In addition, the quality of effective inter- densities).
actions has been substantially improved [9–12]. During While phenomenological methods provide a useful
the past several years, we have developed the Density starting point for the analysis of fusion data, it is de-
ConstrainedTime-DependentHartree-Fock(DC-TDHF) sirable to use a quantum many-body approach which
method for calculating heavy-ion potentials [13], and we properly describes the underlying nuclear shell structure
2
of the reaction system. We have developed a micro- In a typical DC-TDHF run, we utilize a few thousand
scopic approach to extract heavy-ion interaction poten- time steps, and the density constraint is applied every
tials V(R) from the TDHF time-evolution of the dinu- 10−20 time steps. We refer to the minimized energy as
clear system. The interaction potentials calculated with the “density constrained energy” E (R)
DC
the DC-TDHF method incorporate all of the dynamical
entrance channel effects such as neck formation, parti- EDC(R)=<Φρ|H|Φρ > . (6)
cle exchange, internal excitations, and deformation ef-
fects [25]. While the outer part of the potential barrier The ion-ion interaction potential V(R) is essentially the
islargelydeterminedbytheentrancechannelproperties, same as EDC(R), except that it is renormalized by sub-
the inner part of the potential barrier is strongly sen- tracting the constant binding energies EA1 and EA2 of
sitive to dynamical effects such as particle transfer and the two individual nuclei
neck formation.
V(R)=E (R)−E −E . (7)
TheTDHFequationsforthesingle-particlewavefunc- DC A1 A2
tions
In Fig. 1 we compare the heavy-ion interaction poten-
∂ tials V(R) for the systems 132Sn+40,48Ca. It should be
h({φ }) φ (r,t)=i(cid:126) φ (r,t) (λ=1,...,A) , (1)
µ λ ∂t λ noted that DC-TDHF contains particle transfer “on av-
erage” [19], but it does not describe individual transfer
canbederivedfromavariationalprinciple. Themainap-
channels. We find the unexpected result that the barrier
proximation in TDHF is that the many-body wave func-
heightsandpositionsareapproximatelythesameinboth
tionΦ(t)isassumedtobeasingletime-dependentSlater
cases, but the barrier width for 132Sn+40Ca is substan-
determinantwhichconsistsofananti-symmetrizedprod-
tially smaller.
uct of single-particle wave functions
120
Φ(r ,...,r ;t)=(A!)−1/2 det|φ (r ,t)| . (2)
1 A λ i
E = 120 MeV
TDHF
In the present TDHF calculations we use the Skyrme
SLy4 interaction [9] for the nucleons including all of the 110 132Sn + 40Ca
time-odd terms in the mean-field Hamiltonian [5]. The
132 48
Sn + Ca
numerical calculations are carried out on a 3D Cartesian
)
lattice. For 40,48Ca+132Sn the lattice spans 50 fm along V100
e
the collision axis and 30 − 42 fm in the other two di- M
rections, depending on the impact parameter. First we ) (
R
generate very accurate static HF wave functions for the ( 90
V
twonucleionthe3Dgrid. Inthesecondstep,weapplya
boostoperatortothesingle-particlewavefunctions. The
time-propagation is carried out using a Taylor series ex- 80
pansion (up to orders 10−12) of the unitary mean-field
propagator, with a time step ∆t=0.4 fm/c.
In our DC-TDHF approach, the time-evolution takes
70
place with no restrictions. At certain times t or, equiva- 6 8 10 12 14 16 18 20
lently, at certain internuclear distances R(t) the instan- R (fm)
taneous TDHF density
FIG. 1. DC-TDHF calculation of the heavy-ion potentials
ρ (r,t)=<Φ(t)|ρ|Φ(t)> (3) V(R) for the systems 132Sn+40,48Ca. The ion-ion potentials
TDHF
are energy-dependent and have been calculated at E =
TDHF
isusedtoperformastaticHartree-Fockenergyminimiza- 120 MeV.
tion
(cid:90) Using TDHF dynamics, it is also possible to compute
δ <Φ |H − d3r λ(r) ρ(r) |Φ >=0 (4) the corresponding coordinate dependent mass parame-
ρ ρ
ter M(R) [17]. At large distance R, the mass M(R) is
equal to the reduced mass µ of the system. At smaller
while constraining the proton and neutron densities to
distances, when the nuclei overlap, the mass parame-
be equal to the instantaneous TDHF densities
ter generally increases. We find that the structure of
<Φ |ρ|Φ >=ρ (r,t) . (5) M(R)forthe132Sn+40Careactionisfairlysimilartothe
ρ ρ TDHF
mass parameter calculated for 132Sn+48Ca (see Fig. 6 of
These equations determine the state vector Φ . This Ref. [26]), and it is therefore not shown here.
ρ
means we allow the single-particle wave functions to re- Instead of solving the Schr¨odinger equation with coor-
arrange themselves in such a way that the total energy dinate dependent mass parameter M(R) for the heavy-
is minimized, subject to the TDHF density constraint. ion potential V(R), it is numerically advantageous to
3
use the constant reduced mass µ and to transfer the at low energy (E =120 MeV) has a barrier of only
TDHF
coordinate-dependence of the mass to a scaled potential E =112.3 MeV located at R¯ =12.4 fm.
B
U(R¯) using a scale transformation
120
(R,M(R),V(R))−→(R¯,µ,U(R¯)) . (8) 132 40
Sn + Ca
Details are given in Ref. [17]. DC-TDHF
In Fig. 2 we display the transformed potentials U(R¯) ) Point Coulomb
V100
which correspond to the constant reduced mass µ. e
M
A comparison of Fig. 1 and Fig. 2 reveals that the (
)
coordinate-dependent mass changes only the interior re- R
(
U
gion of the potential barriers, and this change is most ),
pronounced at low Ec.m. energies. Note that the trans- V(R 80 ETDHF = 120 MeV
formation to a constant mass parameter preserves the E = 136 MeV
TDHF
basic features found for the original potentials, i.e. the
narrower width of the ion-ion potential for 132Sn+40Ca, ETDHF = 180 MeV
whilethebarrierheightsandpositionsareapproximately
60
the same in both systems. 8 10 12 14 16 18
R,R (fm)
120
E = 120 MeV FIG. 3. (Color online) Solid lines: original heavy-ion poten-
TDHF tials V(R). Dashed lines: transformed potentials U(R¯) cor-
132 40
110 Sn + Ca responding to the reduced mass µ. The potentials have been
calculated at three different energies.
132 48
Sn + Ca
V)100 The Schr¨odinger equation corresponding to the con-
e stant reduced mass µ and the scaled potential U(R¯) has
M
( the familiar form
)
R
U( 90 (cid:20)−(cid:126)2 d2 + (cid:126)2(cid:96)((cid:96)+1) +U(R¯)−E (cid:21)ψ (R¯)=0.
2µ dR¯2 2µR¯2 c.m. (cid:96)
(9)
80
By numerical integration of Eq. (9) using the
well-established Incoming Wave Boundary Condition
(IWBC) method [21] we obtain the barrier penetrabil-
70 ities T which determine the total fusion cross section
6 8 10 12 14 16 18 20 (cid:96)
R (fm)
π(cid:126)2 (cid:88)∞
σ (E )= (2(cid:96)+1)T (E ) . (10)
FIG. 2. Transformed heavy-ion potentials U(R¯) correspond- fus c.m. 2µEc.m. (cid:96) c.m.
(cid:96)=0
ing to the constant reduced mass µ.
In general, our DC-TDHF calculations show that ion- III. FUSION CROSS SECTIONS
ion potentials for heavy systems are strongly energy-
dependent. By contrast, DC-TDHF calculations for light In Figures 4 and 5 we show fusion cross sections mea-
ion systems such as 16O+16O show almost no energy- sured at HRIBF [1] for the systems 132Sn+40,48Ca. A
dependence even if we increase E by a factor of comparison of the fusion cross sections at low energies
c.m.
four [27]. Even in reactions between a light and a very yieldsthesurprisingresultthatfusionof132Snwith40Ca
heavy nucleus such as 16O+208Pb, we see only a rela- yields a larger cross section than with 48Ca. For exam-
tively weak energy dependence of the barrier height and ple, at E = 110 MeV we find an experimental cross
c.m.
width [17]. In Fig. 3 the original potentials V(R) (solid section of ≈ 6 mb for 132Sn+40Ca as compared to 0.8
lines) and the transformed potentials U(R¯) are shown at mb for the more neutron-rich system 132Sn+48Ca. If the
three different TDHF energies. We notice that in these data are scaled for trivial size effects (nuclear radii) the
heavy systems the potential barrier height increases dra- difference between the “reduced” cross sections is found
matically with increasing energy E , and the bar- to be even larger, see Fig.6 of Ref. [1]. The experimen-
TDHF
rier peak moves inward towards smaller internuclear dis- talists have carried out a coupled channel (CC) analysis
tances. The potential U(R¯) calculated at high energy of the fusion data with phenomenological Woods-Saxon
(E = 180 MeV) has a barrier E = 115.3 MeV lo- potentials which generally underpredict the data at low
TDHF B
cated at R¯ = 11.8 fm, whereas the potential calculated E energies. Inthecaseof132Sn+40Ca,theCCmodel
c.m.
4
calculations yield cross sections which differ by a fac- vibrations. AlloftheseeffectsareincludedinDC-TDHF,
tor of 10 or more from the data, despite the fact that andapparentlytheyreducethefusionbarrierheightand
the optical model potentials contain 7 adjustable pa- stronglymodifytheinteriorregionoftheheavy-ioninter-
rameters. Interestingly, it is possible to get a good fit actionpotential. Asaresultofthedecreaseinthefusion
to the data using the empirical Wong model (tunneling barrier, onefindsstronglyenhancedfusioncrosssections
through a single parabolic barrier, with 3 adjustable pa- at low sub-barrier energies. In particular, we observe
rameters). In this case the analysis reveals an unusually thatthedatapointsmeasuredatthetwolowestenergies
largecurvatureofthebarrier,(cid:126)ω =13.13±1.09MeV,for E = 108.6 and 111 MeV are described well by the
c.m.
132Sn+40Caascomparedtothe132Sn+48Casystemwith heavy-ion potential calculated at E = 115 MeV.
TDHF
only (cid:126)ω =5.77±0.52 MeV. A large curvature implies a This is the lowest-energy potential U(R¯) we have been
narrowparabolicbarrier. Ofcourse,thesevaluesaresim- able to calculate using the DC-TDHF method, with a
plyfitstothemeasureddata;themodeldoesnotexplain potential barrier E = 111.5 MeV located at R¯ = 12.4
B
why the barrier curvatures are so dramatically different. fm.
As we will see, our microscopic DC-TDHF theory de-
scribes these results naturally in terms of the underlying
1000
mean-fielddynamics,withoutanyadjustableparameters.
1000
132 40
Sn + Ca
100
b) 132Sn+48Ca
100 m
exp. (HRIBF) σ (
mb) DC-TDHF potentials 10 DC-TDHF
σ ( E = 115 MeV exp. (HRIBF)
TDHF
E = 120 MeV
10 TDHF
E = 125 MeV 1
TDHF
E = 136 MeV
TDHF
110 120 130 140 150
E = 180 MeV
TDHF E (MeV)
c.m.
1
110 120 130 140
E (MeV) FIG. 5. (Color online) Total fusion cross section obtained
c.m.
with the DC-TDHF method for 132Sn+48Ca. Cross sections
FIG. 4. (Color online) Total fusion cross sections for calculatedforseveralenergy-dependention-ionpotentials[26]
132Sn+40Ca. The cross sections have been calculated with have been interpolated in this case (single blue line). The
the DC-TDHF method for several energy-dependent ion-ion experimental data are taken from Ref. [1].
potentials U(R¯) some of which are displayed in Fig. 3. The
experimental data are taken from Ref. [1].
In Fig. 5 we show total fusion cross sections for
In Fig. 4 we show the excitation function of the total 132Sn+48Cawhichcontains8additionalneutrons. Inthis
fusion cross section for 132Sn+40Ca. The cross sections case, we have interpolated the theoretical cross sections
havebeencalculatedwithfivedifferentenergy-dependent obtained with the energy-dependent DC-TDHF poten-
ion-ionpotentials;thecorrespondingDC-TDHFenergies tials [26]. We can see that our theoretical cross sections
are listed in the figure. The main point of the chosen agree remarkably well with the experimental data. The
display is to demonstrate that the energy-dependence of main experimental puzzle, i.e. the fact that the low-
the heavy-ion potential is crucial for an understanding energy sub-barrier fusion cross section for 132Sn+40Ca is
of the strong fusion enhancement at subbarrier energies: substantiallyenhancedascomparedtothemoreneutron-
At very high energy (E = 180 MeV) the potential rich system 132Sn+48Ca, can be understood by examin-
TDHF
approachesthelimitofthefrozendensityapproximation: ing the transformed ion-ion potential shown in Fig. 2.
the collision is so fast that the nuclei have no time to Both systems are found to have approximately the same
rearrange their densities. We observe that the measured barrierheightsandpositions,butthebarrierfor40Cahas
fusion cross sections at energies E > 118 MeV are anarrower width,resultinginenhancedfusion. OurDC-
c.m.
well-described by this high-energy potential. TDHF approach naturally explains the results of the ex-
As we have seen in Fig. 3, the potential barrier height perimental data analysis which used the empirical Wong
decreases dramatically as we lower the energy E . model fit (see remarks at the beginning of this section).
TDHF
Due to the slow motion of the nuclei at sub-barrier en- However,themicroscopicpotentialbarrierisnotasimple
ergies, the nuclear densities have time to rearrange re- parabola,andtheDC-TDHFion-ionpotentialisfoundto
sulting in neck formation, neutron transfer, and surface depend strongly on the energy E for heavy systems.
c.m.
5
IV. SUMMARY fusion enhancement in terms of the narrower width of
the ion-ion potential for 132Sn+40Ca, while the barrier
heightsandpositionsareapproximatelythesameinboth
In this paper, we calculate heavy-ion interaction po- systems.
tentials and total fusion cross sections for 132Sn+40Ca Whileforthefusionoflightnucleithemicroscopicion-
at energies E below and above the Coulomb barrier, ionpotentialsarealmostindependentofthec.m.energy,
c.m.
and we compare these results to the more neutron-rich for heavier systems a strong energy-dependence is ob-
system 132Sn+48Ca studied earlier [26]. The ion-ion po- served. With increasing c.m. energy, the height of the
tential calculations are carried out utilizing a dynamic potentialbarrierincreases,andthebarrierpeakmovesin-
microscopic approach, the Density Constrained Time- ward towards smaller internuclear distances (see Fig. 3).
Dependent Hartree-Fock (DC-TDHF) method. The This behavior of the ion-ion potential has a dramatic
single-particle wave functions are generated on a 3D influence on the sub-barrier fusion cross sections. For
Cartesian lattice which spans 50 fm along the collision thesystem132Sn+40Ca,wehavecalculatedheavy-ionin-
axis and 30 − 42 fm in the other two directions. The teraction potentials at 8 different energies ranging from
only input is the Skyrme N-N interaction, there are no ETDHF = 115 MeV to ETDHF = 180 MeV. The time-
adjustable parameters. dependent and density constraint calculations are com-
putationally very intensive. The total CPU time in this
The main objective of this paper is to give a theoreti- case amounts to 192 days on a single Intel Xeon proces-
calanalysisoffusioncrosssectionswhichweremeasured sor. Our calculations are performed on a Dell LINUX
recently at HRIBF. The experimental data show a sur- workstationwith12processorsusingOpenMP,whichre-
prisingenhancementatlowE energiesforthesystem duces this time to about 16 days.
c.m.
132Sn+40Ca as compared to the more neutron-rich sys-
tem132Sn+48Ca. Basedongeometricconsiderations,one
would expect the opposite: as a result of the increased ACKNOWLEDGMENTS
nuclearradiusfor48Ca,thefusionbarrierfor132Sn+48Ca
should be reduced which in turn should increase the fu- ThisworkhasbeensupportedbytheU.S.Department
sion cross section. Using the microscopic DC-TDHF ap- of Energy under Grant No. DE-FG02-96ER40975 with
proach we are able to explain the measured sub-barrier Vanderbilt University.
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