Threshold energy for sub-barrier fusion hindrance phenomenon

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by  V. V. Sargsyan

Threshold energy for sub-barrier fusion hindrance phenomenon
V.V.Sargsyan1,2, G.G.Adamian1, N.V.Antonenko1, W. Scheid3, and H.Q.Zhang4
1Joint Institute for Nuclear Research, 141980 Dubna, Russia
2International Center for Advanced Studies, Yerevan State University, 0025 Yerevan, Armenia
3Institut fu¨r Theoretische Physik der Justus–Liebig–Universit¨at, D–35392 Giessen, Germany
4China Institute of Atomic Energy, 102413 Beijing, China
(Dated: January 31, 2013)
Therelationshipbetweenthethresholdenergyforadeepsub-barrierfusionhindrancephenomenon
andtheenergyat whichtheregimeofinteractionchanges(theturning-offofthenuclearforces and
friction) in thesub-barriercaptureprocess, is studied within thequantumdiffusion approach. The
quasielastic barrier distribution is shown to be a useful tool to clarify whether theslope of capture
cross section changes at sub-barrierenergies.
3
1
0 Theexperimentswithvariousmedium-lightandheavy at given angular momentum and orientations of the in-
2 nuclei have shown that the experimental slopes of the teracting nuclei) where the nucleon density of colliding
complete fusion excitation function keep increasing at nuclei approximately reaches 10% of saturation density.
n
a low sub-barrier energies and may become much larger IfthecollidingnucleiapproachthedistanceRintbetween
J than the predictions of standard coupled-channel calcu- their centers, the nuclear forces start to act in addition

0 lations. This was identified as the fusion hindrance with to the Coulomb interaction. Thus, at R<Rint the rela-
3 the threshold energy E [1–3]. More experimental and tive motion may be more coupled with other degrees of
s
theoreticalstudiesofsub-barrierfusionhindrancearere- freedom. AtR>Rint therelativemotionisalmostinde-
]
quired to improve our understanding of its physical rea- pendentoftheinternaldegreesoffreedom. Dependingon
h
t son, which may be especially important in astrophysical whether the value of externalturning point Rex is larger
-
l fusion reactions [4]. or smaller than interaction radius Rint, the impact of
c
Asshownwithinthequantumdiffusionapproach[5–9], coupling with other degrees of freedom upon the barrier
u
duetoachangeoftheregimeofinteraction(theturning- passageseemstobedifferent. So,tworegimesofinterac-
n
[ off of the nuclear forces and friction) at deep sub-barrier tionatsub-barrierenergiesdifferbytheactionofnuclear
energies,thecurverelatedtothecapturecrosssectionas forces and, respectively, of nuclear friction. Due to the
3
afunctionofbombardingenergyhassmallerslope. Inthe switching-off the nuclear interaction at external turning
v
2 presentpaperwetry todemonstratethe relationshipbe- point Rex, the cross sections falls with the smaller rate
1 tween the threshold energy Es for a deep sub-barrier fu- at a deep sub-barrier energies.
3 sionhindrancephenomenonandtheenergyEch atwhich
2 the regime of interaction changes in the sub-barrier cap-
1. ture process. 180 Eth Y Zr
21 nuIcnleithise tqrueaatnetdumindtifferumsisonofapaprsoinagclhe tchoelleccatipvteurvearoi-f eV)114600 Es 6089 Ni+ 9090 Zr+
1 able: the relative distance between the colliding nuclei. M
Xiv: Ttnauhkceelennueisun-nttoruoccnloetunrssaindpseofretaertniaotninadltnhournocluteghaherdtisheofeotordmpeipacetnicoodnmenepffcoeesciottfsiotanhrsee, MeV), E (th11028000 16208 O+Pb 64100 Ni+Mo
ar dasciopepufpaoprtrolimioanncaghteiwffotaeintkcshtesasvniianndrticoooourclilsoeisnncithsoainadtnseionronaeftslhsioeo(anffvoitnyrhteeieoxrflnaausmcctwtpiunhlaegic,thnicouomnculoapednil.iednlOgdthiuoser-f E (s 246000 4208 He+Pb 48 Ca+Ca 16238 O+U 54544080FCNCeaia++++55998806NNZZirri 54648848NCNCiaia++++664994N06NZZiirr
the relative motion with low-lying collective modes such 0 48 58Ni+60Ni
as dynamical quadrupole and octupole modes of target
0 2000 4000 6000 8000 10000
1/2
and projectile [10]). We have to mention that many Z1Z2[A1A2/(A1+A2)]
quantum-mechanical and non-Markovian effects [11–13] FIG. 1: The experimental threshold energy E for a deep
s
accompanying the passage through the potential barrier sub-barrier fusion hindrance phenomenon [1] and the calcu-
are taken into considerationin our formalism [5–9]. The lated energy Ech at which the regime of interaction changes
detailsofusedformalismarepresentedinourpreviousar- intheindicatedsub-barriercapturereactionsasafunctionof
ticles [5,6]. With this approachmany heavy-ioncapture Z1Z2[A1A2/(A1+A2)]1/2.
reactions at energies above and well below the Coulomb
barrier have been successfully described. As seen in Fig. 1, for the reactions 4He + 208Pb, 58Ni
Within the quantum diffusion model [5–9] the nuclear + 54Fe, 48Ca + 48Ca,90,96Zr, 40Ca + 90,96Zr, 58Ni +
forces start to play a role at relative distance R = 58,60,64Ni, 60Ni + 89Y, 64Ni + 64Ni,100Mo, 90Zr+90Zr,
int
R +1.1 fm (R is the position of the Coulomb barrier and 16O + 208Pb,238U, there is a good agreement be-
b b
2
(a) 16O+120Sn (a) 16O+208Pb
0.15
-1 Mev )00..1105 -1 MeV) 0.10
D (qe0.05 D (qe0.05
0.00
0.00
-1 ev ) 10--21 (b) 16O+120Sn -1 MeV)1100--21 (b) 16O+208Pb
D (Mqe1100-3 D (qe10-3
-4
10
-4
10
-5
10
-5 65 70 75 80 85
10
45 50 55 Ec.m. (MeV)
Ec.m. (MeV) FIG. 3: The same as in Fig. 2 but for the16O + 208Pb reac-
FIG. 2: The calculated (solid line) Dqe(Ec.m.) = tion. Theexperimental data (symbols) are from Ref. [19].
dPcap(Ec.m.,J =0)/dEc.m.forthe16O+120Snreaction. The
experimental data (symbols) are from Ref. [22]. The values
0
ofDqe(Ec.m.)areshown in thelinear (a)and logarithmic (b) 10
4 208
scales. He+ Pb
tween the threshold energy Es for a deep sub-barrier fu- -1 eV ) 10-1
M
tshioenrheignimdreanocfeinptheerancotmioennochnaanngdesthineethneersguybE-bcahrariterwchaicph- D (qe
tureprocess. ThevaluesE andE almostcoincideand
s ch -2
linearly increase with Z1Z2[A1A2/(A1+A2)]1/2. 10
and the capture cross section is the sum of the fu-
sionandquasifissioncrosssections,fromthe comparison
of calculated capture cross sections and measured fusion -3
10
cross sections one can extract the hindrance factor and 18 21 24 27
the threshold incident energy for a deep sub-barrier fu- Ec.m. (MeV)
sion hindrance phenomenon. The small fusion cross sec- FIG. 4: The calculated (solid line) Dqe(Ec.m.) =
tion at energies well below the Coulomb barrier may in- dPcap(Ec.m.,J =0)/dEc.m. for the4He + 208Pb reaction.
dicate that the quasifissionchannel is preferable and the
system goes to this channel after the capture [5–9]. So,
vation of the reaction flux):
theobservedhindrancefactormaybeunderstoodinterm
of quasifission. At deep sub-barrier energies, the quasi-
fission event corresponds to the formation of a nuclear- Pqe(Ec.m.,J)+Pcap(Ec.m.,J)=1
molecular state or dinuclear system with small excita-
and
tion energy that separates (in the competition with the
compound nucleus formation process) by the quantum dPcap/dEc.m. =−dPqe/dEc.m.,
tunneling throughthe Coulombbarrierinabinaryevent
withmassandchargeclosetothecollidingnuclei. Inthis
whereP isthereflectionprobabilityandP isthecap-
qe cap
sense the quasifission is the general phenomenon which
ture(transmission)probability. Thequasielasticscatter-
takesplaceinthereactionswiththemassive[14–17],and
ingisthesumofelastic,inelastic,andtransferprocesses.
medium-mass nuclei [6].
The reflection probability
Sincethe quasielasticmeasurementsareusuallynotas
complex asthe capture(fusion) measurements,andthey Pqe(Ec.m.,J =0)=dσqe/dσRu
arewellsuitedtosurveythedecreasingrateoffallofthe
sub-barrier capture cross section. There is a direct rela- for angular momentum J = 0 is given by the ratio of
tionship between the capture and the quasielastic scat- the quasielastic differential cross section and Rutherford
tering processes, because any loss from the quasielastic differentialcrosssectionat180degrees[18–23]. Thebar-
channel contributes directly to the capture (the conser- rierdistributionisextractedbytakingthefirstderivative
3
of the Pqe with respect to Ec.m., that is, effect. The reactions 4He,16O + ASn,144Sm,208Pb and
48,40Ca,36S+90Zr withthe sphericalnucleiarepreferable
Dqe(Ec.m.)=−dPqe(Ec.m.,J =0)/dEc.m. = for the experimental study of Dqe(Ec.m.).
In conclusions, employing the quantum diffusion ap-
=dPcap(Ec.m.,J =0)/dEc.m.. proach, we demonstrated the relationship between the
threshold energy for a deep sub-barrier fusion hindrance
Thus, one can observe the change of the fall rate of phenomenon and the energy at which the regime of in-
Pcap(Ec.m.,J = 0) at sub-barrier energies by measuring teractionchangesinthe sub-barriercaptureprocess. We
the barrier distribution Dqe. By employing the quan- predicted the sharp change of the slope of the quasielas-
tum diffusion approachand calculating dPcap(Ec.m.,J = tic barrier distribution below the threshold energy. This
0)/dEc.m., one canobtainDqe(Ec.m.). In additionto the isexpectedtobetheexperimentalindicationofachange
meanpeakpositionoftheDqe aroundthebarrierheight, of the regime of interaction in the sub-barrier capture.
we predict the sharp change of the slope of Dqe below One concludes that the quasielastic technique could be
the threshold energy because of a change of the regime an important tool in capture (fusion) research.
ofinteractioninthesub-barriercaptureprocess(Figs.2–
4). The effect seems to be more pronounced in the col- This work was supported by DFG, NSFC, and
lisions of spherical nuclei (Figs. 3 and 4). The collisions RFBR. The IN2P3(France)-JINR(Dubna) and Polish -
of deformed nuclei occurs at various mutual orientations JINR(Dubna) Cooperation Programmes are gratefully
on which the value of R depends. Thus, the defor- acknowledged.
int
mation and neutron transfer effects can smear out this
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