Time-dependent Hartree-Fock calculations for multinucleon transfer and quasifission processes in the $^{64}$Ni+$^{238}$U reaction

1.2 MB·

by  Kazuyuki Sekizawa

Time-dependent Hartree-Fock calculations for multinucleon transfer
and quasifission processes in the 64Ni+238U reaction
Kazuyuki Sekizawa1,2,∗ and Kazuhiro Yabana2,3,†
1Faculty of Physics, Warsaw University of Technology, ulica Koszykowa 75, 00-662 Warsaw, Poland
2Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571, Japan
3Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8577, Japan
(Dated: May 23, 2016)
Background: Multinucleon transfer (MNT) and quasifission (QF) processes are dominant processes in low-
energy collisions of two heavy nuclei. They are expected to be useful to produce neutron-rich unstable nuclei.
6 Nuclear dynamics leading to these processes depends sensitively on nuclear properties such as deformation and
1 shell structure.
0
Purpose: WeelucidatereactionmechanismsofMNTandQFprocessesinvolvingheavydeformednuclei,making
2
detailedcomparisons betweenmicroscopic time-dependentHartree-Fock(TDHF)calculations andmeasurements
y for the64Ni+238U reaction.
a
Methods: Three-dimensional Skyrme-TDHFcalculations are performed. Particle-number projection method is
M
used to evaluate MNT cross sections from theTDHFwave function after collision.
4 Results: Fragment masses, total kinetic energy (TKE), scattering angle, contact time, and MNT cross sections
2 are investigated for the 64Ni+238U reaction. They show reasonable agreements with measurements. At small
impact parameters, collision dynamics depends sensitively on the orientation of deformed 238U. In tip (side)

] collisions, we find a larger (smaller) TKE and a shorter (longer) contact time. In tip collisions, we find a strong
h influenceof quantumshells around 208Pb.
t
- Conclusions: It is confirmed that the TDHF calculations reasonably describe both MNT and QF processes in
cl the64Ni+238U reaction. Analyses of this system indicate the significance of the nuclear structure effects such as
u deformation and quantumshells in nuclear reaction dynamicsat low energies.
n
[

I. INTRODUCTION MNT processes was carriedout by L. Corradiet al. [18].
2
Moreover, 64Ni+238U reaction has been expected as a
v
6 It has been known that shell structure and deforma- possible candidate for synthesizing a SHE with Z =120.
5 tion, which are fundamental properties of nuclear struc- Toexaminethispossibility,E.M.Kozulinetal.measured
6 ture, play an important role in low-energy heavy ion re- fragment mass and total kinetic energy (TKE) distribu-
6 actions. For example, the barrier height for nuclear fu- tionsatseveralincident energies[19]. InRef.[19], itwas
0
sion depends on the orientation of colliding nuclei if a shown that mass-symmetric fragments are hardly pro-
.
1 deformed nucleus is involved [1, 2]. In the synthesis of ducedin64Ni+238Ureaction. Thisfactindicatesastrong
0 superheavy elements (SHEs), shell effects are crucially suppressionofthefusionreactionbyQFprocesses. Mass-
6 important, since they reduce excitation energy ofa com- angle and mass-TKE distributions including 64Ni+238U
1
: poundnucleus,andenhancing itssurvivalprobability. A were reported by J. T¯oke et al. [20].
v
remarkable example is the successful synthesis of SHEs To investigate MNT and QF processes theoretically,
i
X by the cold-fusion reactions,where 208Pb or 209Bi target various models have been developed. For MNT reac-
r is utilized [3, 4]. Recently, shell effects on multinucleon tions, semiclassical models called GRAZING[21–23] and
a transfer(MNT)andquasifission(QF)processeshavealso Complex WKB [24] have been developed with greatsuc-
beenextensivelydiscussed. Thesereactionsareexpected cesses [25]. The GRAZINGhas recently been extended
tobeusefultoproduceneutron-richunstablenuclei(see, to incorporate transfer-induced fission, which is referred
e.g., Refs. [5–17] and references therein). to as GRAZING-F [26]. To describe damped collisions,
The present study aims to elucidate reaction mecha- a dynamical model based on Langevin-type equations
nisms ofthe MNT andQFprocessesinnuclearreactions [9–13, 27–30], the dinuclear system (DNS) model [14–
involving a heavy deformed nucleus. Specifically, we fo- 17,31–38],andtheimprovedquantummoleculardynam-
cus on 64Ni+238U reaction for which abundant experi- ics model (ImQMD) [39–44] have been extensively de-
mental data of both MNT and QF processes are avail- veloped. Despite numerous successes in describing mea-
able. Since this system has a large N/Z asymmetry surements, they are to some extent empirical, contain-
[N/Z = 1.29 (64Ni) and 1.59 (238U)], MNT processes ing adjustable parameters. This fact limits their predic-
towardthechargeequilibriumofthetotalsystemareex- tive power. To further extend our understanding of re-
pected. A precise measurement of cross sections of the action mechanisms and to improve reliability to predict
cross sections, we apply the microscopic time-dependent
Hartree-Fock(TDHF)theorytoMNTandQFprocesses.
∗ [email protected] Three-dimensional simulations based on the TDHF
† [email protected] theory for low-energy nuclear reactions started around
2
70’s. They have been successful to describe various phe- investigated. In Sec. IV, we compare the TDHF results
nomena such as fusion reactions and deep inelastic col- with measurements. In Sec. V, summary and conclusion
lisions [45, 46]. Applications of the TDHF theory to are presented. A part of the results of the present anal-
MNT and QF processes are rather new. In Ref. [47], yses was reported in Ref. [62].
we applied the TDHF theory to investigate MNT pro-
cesses in 40,48Ca+124Sn, 40Ca+208Pb, and 58Ni+208Pb
reactionsatenergiesneartheCoulombbarrier,forwhich
precise experimental data are available [48–51]. Apply- II. METHOD
ing the particle-number projection (PNP) method [52]
to the TDHF wave function after collision, we evaluated A. TDHF theory
transfer probabilities and cross sections for each chan-
nel specified by the number of neutrons and protons in
Webrieflyexplainourtheoreticalframework. Westart
the reaction products. From the comparison with mea-
with an action,
surements,weshowedthatthe TDHF theoryreproduces
measured cross sections in an accuracy comparable to
otherexistingmodels. InRef.[53],wehaveextendedthe t2 A
S = ψ (t) i~∂ ψ (t) [ρ(t)] dt, (1)
PNPmethodtoevaluateexpectationvaluesofoperators. i t i
Z (cid:20) −E (cid:21)
Recently, we applied our method to an asymmetric sys- t1 Xi=1(cid:10) (cid:12) (cid:12) (cid:11)
(cid:12) (cid:12)
tem, 18O+206Pb, at energies above the barrier [54].
where [ρ(t)] denotes an energy density functional
Recently, QF processes have been investigated by the
E
(EDF), which is a functional of various densities and is
TDHF theory [55–62]. First, QF dynamics in colli-
sions of two actinide nuclei such as 238U+238U [55] and constructed so as to reproduce various properties of fi-
232Th+250Cf [56] was investigated. In these studies, it nite nuclei and nuclear matter. Applying the stationary
condition, δS/δψ∗ =0, we obtain the TDHF equation,
has been suggested that QF dynamics depends sensi- i
tively on the nuclear orientations, incident energies, and
impact parameters. In Ref. [57] in which QF processes i~∂ ψ (rσq,t)=hˆ[ρ(t)]ψ (rσq,t), (2)
t i i
of 40Ca+238U were reported, it has been indicated that
shell effects reflecting Z = 82 and N = 126 magic num- where r and σ are spatial and spin coordinates, re-
bers havestronginfluence in tip collisions,while no shell
spectively. q (= n or p) denotes the isospin of i-
effect is seen in side collisions. In that work, it has also th nucleon. Single-particle wave functions, ψ (rσq,t)
i
been recognizedthat contact time is much longer in side
(i = 1, ,A), satisfy the orthonormal relation,
collisions than that in tip collisions. The mass-angle dis- ψ∗(rσ··q·,t)ψ (rσq,t)dr = δ . Single-particle
tributions (MADs), which are one of the characteristic σ i j ij
HPamRiltonian,hˆ[ρ(t)],containsamean-fieldpotentialgen-
observables of QF processes, were calculated and com-
eratedbyallthenucleonsinthesystem. Themany-body
paredwithexperimentaldata,showingreasonableagree-
wave function is given by a single Slater-determinant
ments [57, 59].
composed of the single-particle wave functions,
In this article, we report detailed investigations of
MNT and QF processes in 64Ni+238U reaction perform-
1
ing systematic TDHF calculations. Since the projectile Ψ(r σ q , ,r σ q ,t)= det ψ (r σ q ,t) .
1 1 1 A A A i j j j
and the target are open shell nuclei, pairing correlations ··· √A! (cid:8) (cid:9)
(3)
may be important in the collision dynamics. However,
Once the EDF is given,the theory contains no empirical
we ignore the effect of pairing correlation in this study,
parameters.
since the inclusion of pairing requires much more com-
putational costs which prevent systematic investigations In heavy ion reactions, the initial wave function is
for various initial conditions. Making use of the PNP a Slater determinant composed of single-particle wave
method, we are able to make detailed comparisons with functions of projectile and target nuclei in their ground
measurements, including cross sections. In the studies state. They arepreparedseparatelyby solvingthe static
reported so far [55–61], sensitive dependence of QF dy- Hartree-Fock (HF) equation and are boosted with the
namics on nuclear structure has been suggested. From relative velocity. The velocity is evaluated assuming the
our detailed analyses of this system, we expect to elu- Rutherford trajectory. For a given set of incident en-
cidate clearly those effects of deformation and quantum ergy E and impact parameter b, the solution of Eq. (2)
shells on QF processes. is uniquely determined.
The article is organized as follows: In Sec. II, we Weinvestigatereactionsinwhichbinaryreactionprod-
briefly explain the theoretical framework of the TDHF ucts are produced. To make comparisons with measure-
theory and present computational settings. In Sec. III, ments, we analyze the TDHF wave function at a cer-
we present the results of our TDHF calculations. In taintimeaftercollisionwhenthetwofragmentsaresuffi-
Sec. IIIA, we investigate 64Ni+238U reaction at E ciently separated spatially. We calculate such quantities
c.m.
≈
307.35 MeV. In Sec. IIIB, incident energy, impact pa- as fragment masses, total kinetic energy loss (TKEL),
rameter, and orientation dependence of QF dynamics is scattering angle, and MNT cross sections.
3
B. Computational settings
We use our own code of TDHF calculations for heavy
ion reactions [62]. In the code, the TDHF equation is
solved in real space and real time. Single-particle wave
functionsarerepresentedonathree-dimensionaluniform
gridwithout any symmetry restrictions. The mesh spac-
ing is set to be 0.8 fm. We employ the 11-point finite-
difference formula for spatial derivatives. The fourth-
order Taylor expansion method is utilized for the time-
evolutionoperatorwitha singlepredictor-correctorstep.
The time step is set to be ∆t = 0.2 fm/c. Hockney’s
method [63] is used to calculate the Coulomb potential
in the isolated boundary condition.
Aboxwith30 30 30gridpointsisusedtocalculate
× ×
the ground state of the projectile and target nuclei. A
box with 70 70 30 grid points is used for reaction
× ×
calculations. Wesettheincidentdirectionparalleltothe
x-axisandsettheimpactparametervectorparalleltothe
y-axis. Theinitialseparationdistancebetweencentersof
the projectile and the targetis set to be 24 fm along the
incidentdirection. WestopTDHFcalculationswhenthe
distancebetweencentersofthereactionproductsreaches
26 fm.
We use Skyrme SLy5 parameter set [64] for the EDF.
The ground state of 64Ni has an oblate shape with
β 0.12, while that of 238U has a prolate shape with FIG. 1. (Color online) Three initial configurations used for
β ≈0.27. We perform TDHF calculations for three ini- ourTDHFcalculationsof64Ni+238Ureaction. (a): Thesym-
tia≈l orientations of 238U: The symmetry axis of 238U is metryaxisof238Uissetparalleltothecollision axis(x-axis).
set parallelto the incident direction(x-axis), set parallel (b): The symmetry axis of 238U is set parallel to the impact
to the impactparametervector(y-axis),andsetperpen- parametervector(y-axis). (c): Thesymmetry axisof 238Uis
set perpendicularto the reaction plane (xy-plane).
dicular to the reaction plane (xy-plane). We call these
three cases as x-, y-, and z-direction cases, respectively.
Sincedeformationof64Niisnotverylarge,wealwaysset
the symmetry axis of 64Ni perpendicular to the reaction massenergyofEc.m. ≈307.35MeVcorrespondstoabout
27%and16%abovethebarrierforthex-andy-direction
plane, assuming that the reaction is not affected much
bythe directionofthe deformed64Ni. Figure1schemat- cases, respectively, at b = 0 fm. In our TDHF calcula-
tions at this incident energy, we always found binary re-
ically shows three cases of initial configurations. Since
actionproductsandnofusionreactionwasobservedeven
nuclear rotational motion is very slow, we assume that
in head-on collisions.
the nuclear orientation at the contact of two nuclei can
We will show scattering angle in the center-of-mass
be well specified by the configurations at the beginning
frame, θ , TKEL, and contact time. The scattering
of the TDHF calculations. c.m.
angleandtheTKELareevaluatedfromthetranslational
motionofreactionproductsasdescribedinRef.[47]. The
contacttime is defined as the durationin which the low-
III. TDHF RESULTS
est density between colliding nuclei exceeds a half of the
nuclear saturation density, ρ /2=0.08 fm−3. The same
0
A. Overview of the reaction at Ec.m. ≈ 307.35 MeV definition was also used by other authors [57].
In Fig. 2, we show θ , TKEL, and contact time in
c.m.
In this Subsection, we show results of TDHF calcula- (a), (b), and (c), respectively, as functions of the impact
tions for 64Ni+238U reaction at E 307.35 MeV. At parameter. Results for x-, y-, and z-direction cases are
c.m.
≈
around this incident energy, several measurements have shown by red circles, green crosses,and blue open trian-
been reported [18–20]. Comparisons of the TDHF re- gles connected with dotted lines, respectively. The same
sultswiththemeasurementswillbepresentedinSec.IV. symbols will be used in Figs. 4, 8, 9, and 10. In (a), the
The calculations are performed for an impact parame- scattering angle for the Rutherford trajectory is shown
ter range, 0 fm b 12 fm. We evaluate the frozen by a dotted curve. In (c), contact time is shown in zep-
HF barrier as de≤scribe≤d in Ref. [47]. The barrier height tosecond (1 zs =10−21 sec).
is evaluated to be 242.93 MeV for x-direction case and We first investigate behavior which does not depend
263.97 MeV for y-direction case. The incident center-of- muchontheinitialorientationof238U.Whentheimpact
4
180 seen in Fig. 2. As the impact parameter decreases from
) 150 Coulomb (a) b≈6fm,weobservearapiddecreaseofthescatteringan-
g gle in(a). In contrast,we observea rapidincrease ofthe
e 120
d contact time in (c). The decrease (increase) of the scat-
( 90 teringangle(contacttime) is steepestforthe x-direction
m. 60 case and becomes moderate as the orientation changes
c.
from x- to y- and from y- to z-direction. This difference
q 30
canbeunderstoodasfollows. Inthex-directioncase,the
0 symmetry axis of 238U is set parallelto the collision axis
120 (Fig.1(a)). Inthis geometry,twonucleicollidesubstan-
V) 100 (b) tially at a large impact parameter, b 5 fm, compared
≈
e to the other cases. In the z-direction case, 64Ni always
M 80
collides with the side of238U (Fig. 1 (c)). This results in
(
L 60 the slowest change of θc.m. and contact time. Results of
E 40 the y-direction case (Fig. 1 (b)) locate between those of
K
the x- and z-direction cases.
T 20
Thecontacttimeshownin(c)hasastrongorientation
0 dependence at a small-b region(b.4 fm). In the y- and
zs) 12 x-direction z-direction cases, contact time increases monotonically
( 10 y-direction as the impact parameter decreases, reaching 10–11 zs in
e head-oncollisions. On the other hand, in the x-direction
m 8 z-direction
i case, contact time takes almost a constant value (about
t
t 6 4–5 zs), even decreases slightly as the impact parame-
c
ta 4 (c) ter decreases. Because of the shorter contact time, the
n
o 2 combined dinuclear system does not rotate much. This
C explains larger scattering angles for the x-direction case
0
compared with the other cases at small impact param-
0 2 4 6 8 10
eters (b . 3 fm), seen in (a). The observed orientation
b (fm)
dependence of the contact time is consistent with the
TDHF calculations for 40Ca+238U reported in [57].
FIG. 2. (Color online) TDHF results for 64Ni+238U reaction To obtain intuitive understanding of the reaction dy-
atEc.m. ≈307.35MeV.Scatteringangleinthecenter-of-mass namics,weshowinFig.3(a-d)snapshotsofthedensityin
frame, θc.m., total kinetic energy loss (TKEL), and contact thereactionplanefortwoimpactparameters,5.5fmand
time are shown in (a), (b), and (c), respectively, as functions
2fm,andtwoorientationsof238U,thex-andy-direction
oftheimpactparameter,b. Resultsforx-,y-,andz-direction
cases. Elapsed time measured from the initial configura-
cases are shown by red circles, green crosses, and blue open
tion is indicated in zeptosecond. At b=5.5 fm shown in
trianglesconnectedwithdottedlines,respectively. In(a),the
scattering angle for the Rutherford trajectory is shown by a (a,b),wefindaformationofathinneckthroughwhicha
dotted curve. fewnucleonsareexchanged. Thereactiondynamicsdoes
notshowmuchdifference betweenthe x- andy-direction
cases at this impact parameter.
parameter is sufficiently large (b&7 fm), the reaction is Contrarily, we find quite different reaction dynamics
governedby the Coulombinteraction,and the scattering at a small-b reaction, b=2 fm, for different orientations
angle coincides with that of the Rutherford trajectory. of 238U. Let us first look at reaction dynamics in the x-
Atthisimpactparameterregion,TKELisverysmalland directioncase at b=2 fm shownin (c). As time evolves,
contact time is zero. As the impact parameter decreases 64Nicollideswith238Uatapositionclosetothetipofthe
(b . 7 fm), TKEL increases rapidly taking maximum 238U (t = 0.67 zs). Then a thick and long neck is devel-
values at b 4–5 fm. Surfaces of two nuclei also start oped in the course of the collision, forming an elongated
≈
to touch gently, and the nuclear attractive interaction dinuclear system (t = 0.67–2.67 zs). After this stage,
distorts trajectory toward forward angles. the neck becomes thinner (t = 3.33–4 zs) and eventu-
At a small-b region (b . 5 fm), the contact time be- ally ruptures (t = 4.77 zs), producing binary reaction
comessubstantiallylong. This indicatesaformationofa products (t=5.34 zs). The produced fragments roughly
dinuclearsystemconnectedbyathick neck. Becausethe correspond to 100Zr and 202Hg . We note that we
40 60 80 122
dinuclear system rotates for a certain period, the scat- havefoundverysimilarshapeevolutiondynamicstothat
tering angle decreases noticeably as shown in (a). As shown in Fig. 3 (c) in a wide impact parameter range of
the impact parameter decreases further, the scattering b=0–4 fm, wherethe contacttime isalmostconstantas
angle increases monotonically, reaching 180◦ (backward shown in Fig. 2 (c) (see also Supplemental Material [65]
scattering) in head-on collisions. In this small-b region, for movies of the reactions).
TKEL is roughly constant. Figure3(d)showsreactiondynamicsinthey-direction
Wenextlookatdependenceontheorientationof238U caseatb=2fm. Inthiscase,64Nicollideswith238Uata
5
FIG. 3. (Color online) Snapshots of thedensity in thereaction plane in TDHF calculations for 64Ni+238U reaction at Ec.m.≈
307.35 MeV. Results for two impact parameters, b = 0.5 fm and 2 fm, and two initial orientations of 238U, the x- and y-
direction cases, are shown. Elapsed time measured from the initial configuration is indicated in each panel in zeptosecond
(1 zs =10−21 sec). See also SupplementalMaterial [65] for movies of the reactions.
position close to the side of the 238U (t=0.67 zs). After for this impact parameter range as insets in (a-d). The
the touch, a somewhat compact composite system with snapshotsofthedensityshowninFig.3(a,b)correspond
a thick neck structure is formed (t = 2 zs) (Note that to reactions in this impact parameter range.
the time of each snapshot is not the same as that shown As the impact parameter decreases further, we find a
in (c)). The dinuclear system with a thick neck struc- drastic change at around b 4–5 fm. Inside this impact
ture is maintained for a long period and rotates in the parameter, a mass equilibr≈ation process toward the di-
reaction plane (t = 2–6.67 zs). When the neck ruptures rection increasing the mass symmetry, which we call the
(t=8.89zs),fragmentswithmoresymmetricmassesare mass-drift mode, is observed. In the mass-drift mode,
generated compared with those of the x-direction case both neutrons and protons are transferred toward the
shown in (c). The produced fragments roughly corre- same direction, from the heavier nucleus to the lighter
spond to 14176Ag69 and 17835Ta112. one. Whilethefragmentmassesshowsubstantialchanges
Wenextinvestigateaveragenumbersofnucleonsinthe at b 4–5 fm, the N/Z ratios approach monotonically
≈
reaction products as functions of the impact parameter. to the fully equilibrated value. From the density pro-
Figure 4 (a) and (b) show average numbers of neutrons file during the reaction, we find that the shape evolution
and protons in the lighter fragment, which we denote and the neck rupture are responsible for the mass-drift
as N and Z , respectively. Those in the heavier frag- mode. Onceadinuclearsystemisformedinthecourseof
L L
ment, which we denote as N and Z , are shown in (c) collision, the system quickly reaches the charge equilib-
H H
and (d), respectively. N/Z ratios of the lighter and the rium,andthepositionoftheneckrupturedeterminesthe
heavier fragments are also shown in (e). In (e), the fully amountoftransfersofneutronsandprotons. InRef.[47],
equilibrated value of the system, 1.52, is indicated by a wereportedsimilartransferdynamicsinlightersystems.
horizontal dotted line. The mass-drift mode observed at b . 5 fm shows no-
When the impact parameter is sufficiently large (b & ticeable dependence on the initial orientation. In the
7fm),theaveragenumbersofneutronsandprotonscoin- z-direction case (blue open triangles), we find a gradual
cidewiththoseoftheprojectileandtargetnuclei. Asthe change of the average number of nucleons. In contrast,
impact parameter decreases (b 5–6 fm), we find that in the x- and y-direction cases, we observe an abrupt
≈
protonsaretransferredfrom64Nito238U,whileneutrons change at b 4–5 fm. In the x-direction case (red open
≈
tend to be transferred in the opposite direction. These circles),theaveragenumberofnucleonsexhibitsapromi-
directionsoftransferscorrespondtothoseexpectedform nent plateau which persists within 0 fm b . 4 fm.
≤
the initial N/Z asymmetry. We show a magnified plot In this impact parameter region, N 120–126 and
H
≈
6
75 65 do not contribute in the side collisions of 40Ca+238U.
70 (a) 60 (b)
65 55 Contrarily to it, we find another plateau behavior in the
60 40 50 30
y- and z-direction cases. In the y-direction case, at a
L 55 L 45
N 50 Z 40 small-b region, 0 fm .b.2 fm, we observe a plateau at
4405 35 5 5.5 6 3305 25 5 5.5 6 around NH 110 and ZH 72 for the heavier fragment
35 25 and N ≈70 and Z ≈48 for the lighter fragment.
30 20 L ≈ L ≈
0 2 4 6 8 10 0 2 4 6 8 10 This behavior may be influenced by the shell effect of
b (fm) b (fm)
Z = 50 in the QF process, although the fragment shows
150 100
145 95 a large deformation as shown in Fig. 3 (d). In the z-
140 90 direction case, a plateau is seen at around N 127
135 147 85 95 H ≈
H 130 H 80 and ZH 83 for the heavier fragment and NL 54
N 125 Z 75 and Z ≈ 37 for the lighter fragment. This beh≈avior
111250 142 5 5.5 6 6750 90 5 5.5 6 indicatLes≈the effect of the quantum shells of 208Pb. We
110 (c) 60 (d)
note that influence of quantum shells in QF processes
105 55
0 2 4 6 8 10 0 2 4 6 8 10 has been routinely observed experimentally [66–72] and
b (fm) b (fm)
discussed theoretically [10, 12, 28–30, 57, 73–76].
It is worth emphasizing that, in the y-direction case,
1.6 x-direction the average number of nucleons changes dramatically
y-direction
1.5 z-direction when the impact parameter becomes a tiny but a finite
Z
N/ value. For instance, from b = 0 to 0.25 fm, the average
1.4
(e) number of nucleons changes as large as 25. We consider
1.3 that the observed behavior is related to the symmetry
that appears only at b=0 fm in which the colliding sys-
0 2 4 6 8 10
b (fm) tem has a rotationalsymmetry aroundthe collisionaxis.
This symmetry disappears once the impact parameter
FIG. 4. (Color online) TDHF results for 64Ni+238U reac- becomes finite.
tion at Ec.m. ≈ 307.35 MeV. Average numbers of nucleons We note that the behavior at around b = 0 fm is dif-
in lighter (a, b) and heavier (c, d) fragments are shown as ferent between y- and z-direction cases. To understand
functions of theimpact parameter, b. Left panels (a,c) show the origin of the difference, let us consider shape of the
those of neutrons, while right panels (b, d) show those of systemviewedfromaframerotatingwiththevectorcon-
protons. Insets are magnified plots of an impact parameter necting centers of the two colliding nuclei, R(t), in the
region, b = 5–6 fm. The neutron-to-proton ratios of lighter
adiabatic limit neglecting currents. In the z-direction
and heavier fragments are shown in (e). Results for x-, y-,
case, the system always persists a reflection symmetry
and z-direction cases are shown by red circles, green crosses,
withrespecttotheplanewhichcontainsR(t)andisper-
and blue open triangles connected with dotted lines, respec-
tively. In(e),thefully equilibrated valueofthesystem,1.52, pendicular to the reaction plane. On the other hand, in
is indicated by a horizontal dotted line. the y-direction case at a nonzero impact parameter, the
systemdoesnothavethesymmetrymentionedabovedue
to the deformed shape of 238U. Thus the system may go
Z 78–82areobserved. Weconsiderthatthequantum throughmorecomplexshapeevolutiondynamics. Infact,
H
shel≈lsof208Pbmakeasignificantcontributiontothisbe- once the impact parameter becomes nonzero in the y-
havior. A similar shell effect of 208Pb has been reported directioncase,wefindtheprojectile-likesubsystemmov-
in the tip collisions of 40Ca+238U in TDHF calculations ing along the elongated direction of the 238U-like sub-
[57]. Wenotethat,inourcalculations,thelighterpartner system forming a very thick neck, which results in the
has N 55–60 and Z 37–42. A production of sim- abrupt change of the average number of transferred nu-
L L
≈ ≈
ilar fragments has been observed in TDHF calculations cleons (See Supplemental Material [65] for movies of the
for the side collisions of 40,48Ca+238U at b = 0 fm [58]. reactions).
A possible influence of stabilization by strongly bound The orientation dependence is also clearly seen in the
Zr isotopes with large prolate deformation in this mass TKEL at a small-b region (b . 4 fm) in Fig. 2 (b). In
region has been advocated [58]. the y- and z-direction cases, TKEL takes almost con-
In the y-direction case (green crosses), the behavior is stantvalues, 70–80MeV. We observesomewhatlarger
≈
quite different. As in the x-direction case, we observe values of TKEL in the y-direction case compared with
an abrupt change of the average number of nucleons at those of the z-direction case. This difference may re-
b 4–5 fm. However,the plateau around N 126 and flect the reflection symmetry mentioned above which re-
H
≈ ≈
Z 82doesnotappear. Thecompositesystemtendsto strictsreactiondynamicsinthez-directioncase. Inthex-
H
≈
split into more mass-symmetric fragments. It indicates direction case, we observe smaller values, 50–60 MeV.
≈
that the quantum shells of 208Pb are not significant in In Ref. [77], fission dynamics of 258Fm was investigated
thiscase. Asimilarinterplaybetweenthequantumshells by TDHF+BCS approach. It was shown that the TKE
and the nuclear orientation was reported in 40Ca+238U exhibits clear dependence on the shape of the fissioning
[57]. In Ref. [57], it was reported that quantum shells nucleus,andthat the differentshape evolutiondynamics
7
90 80
Figure5(a)and(b)showaveragenumbersofneutrons
80 (a) 70 (b)
and protons in the lighter fragment, respectively. Those
70 60
in the heavier fragment are shown in (c) and (d). In (e),
NL 60 ZL 50 contacttime is also presented. The horizontalaxis is the
50 40
center-of-mass energy, E .
40 30 c.m.
First,weconsiderthex-directioncase(opensymbols).
30 20
200 300 400 500 200 300 400 500 Asthecenter-of-massenergyincreases,wefindanabrupt
E (MeV) E (MeV)
c.m. c.m. change in the fragment masses when the energy exceeds
150 100 the barrier height, V 242.93 MeV. Just above the
B
140 90 ≈
barrier, the fragment masses are about N 58 and
L
130 80 ≈
Z 40 and N 124 and Z 80 for both b = 0.5-
NH 120 ZH 70 anLd≈2-fm cases.H ≈ H ≈
110 60
For b = 2 fm case (red open circles), the fragment
100 (c) 50 (d)
masses are almost independent of the center-of-mass en-
90 40
200 300 400 500 200 300 400 500 ergyforanenergyrange,290MeV.E .500MeV.It
c.m.
E (MeV) E (MeV)
c.m. c.m. indicates a significant influence of the quantum shells of
25 208Pb, even above barrier energies. On the other hand,
(zs) 20 (e) bb==22..00 ffmm,, yx for b = 0.5 fm case (green open squares), the amount
e of transferred nucleons decreases as the center-of-mass
m 15 b=0.5 fm, x
act ti 10 b=0.5 fm, y eenffeercgtyisinwcreeaakseense.dTahsisthbeehinacviidoerntimepnleiergsythinactrethaseesshfeolrl
ont 5 b = 0.5 fm case. In the x-direction case, an elongated
C
0 dinuclear system is observed even at energies well above
200 300 400 500
E (MeV) the barrier (See also Supplemental Material [65]). Be-
c.m.
cause of the large elongation, the dinuclear system splits
inarelativelyshortperiod( 4–5zs)asseeninFig.5(e),
FIG. 5. (Color online) TDHF results for 64Ni+238U reaction ≈
and no fusion reaction was observed for all incident en-
at b=0.5 fm and 2 fm for x- and y-direction cases. Average
ergies examined here.
numbers of nucleons in lighter (a, b) and heavier (c, d) frag-
ments are shown as functions of the center-of-mass energy, Next,weconsiderthey-directioncase(filledsymbols).
Ec.m.. Left panels (a, c) show those of neutrons, while right For both b = 0.5- and 2-fm cases, we observe similar
panels (b, d) show those of protons. In (e), contact time is behavior as a function of the center-of-mass energy. As
presented. Results for b = 0.5- and 2-fm cases are shown by in the x-direction case, we find an abrupt change in the
squaresandcircles,andthoseforx-andy-directioncasesare fragmentmasseswhenthe center-of-massenergyexceeds
shown by open and filled symbols, respectively. the barrier height, V 263.97 MeV. In contrast to the
B
≈
x-directioncase,the fragmentmassescontinuetochange
asthecenter-of-massenergyincreases,uptoE 338
c.m.
isassociatedwithdifferentvalleysinthepotentialenergy ≈
(386) MeV for b = 0.5 (2) fm. We also find an abrupt
surface(PES).AlthoughwehavenotconductedPEScal-
change in the contact time in (e). In the y-direction
culations of302Ubn composite system,we expect that
120 182 case,thecompositesystemshowsacompactshape,which
there exists a valley in the PES of the composite system
becomes a mononuclear shape as the center-of-mass en-
associatedwiththedoublymagic208Pbandthattheval-
ergyincreases. Themononuclearsystemsplitsintomass-
ley causes the small TKEL and the short contact time.
symmetricfragments. Asaresultofthemononuclearsys-
We note thatanexperimentallymeasuredTKEdistri- temformation,the contacttime becomes muchlongerin
butionof64Ni+238Ureactionatasmallerincidentenergy, the y-direction case than that in the x-direction case, as
Ec.m. 282.13 MeV, was reported [19]. In the measure- shown in (e).
≈
ment, a two-peaked structure of TKE was observed. Al-
We note that, in the y-direction case at higher center-
thoughthe plotwasconstructedfromselectedfragments of-mass energies, E & 338 (386) MeV for b = 0.5
c.m.
havingA /2 20,itisexpectedthatdifferentdynamics
CN (2) fm, a capture process takes place, forming a su-
±
associatedwith the large deformation of 238U affects the
perheavy composite system with Z = 120. We contin-
measured trends.
ued time-evolution calculations up to 40 zs (60,000 time
steps). Similarcriteriafor fusionwerealsousedby other
authors [57, 58]. In this period, the composite system
B. Incident energy dependence exhibits a compact mononuclear shape (See also Supple-
mental Material [65]). In Ref. [19], measured fragment
In this Subsection, we examine incident energydepen- mass distributions in 64Ni+238U reaction were reported
dence of QF processes in 64Ni+238U reaction. We inves- at several incident energies. They showed that mass-
tigate reactions at two impact parameters, b = 0.5 fm symmetricfragmentsarehardlyproducedinthereaction.
and2fm, fortwoorientationsof238U,x- andy-direction Ourresultsareconsistentwiththeexperimentalobserva-
cases. tion, since the highest incident energy of the experiment
8
x-direction y-direction z-direction s (mb2) ΘV(V¯)(r)denotesaspacedivisionfunctionwhichisequal
10 to 1 inside V (V¯) and 0 elsewhere. V¯ is the complement
60
100 of V. In practice, the integral in Eq. (5) is evaluated us-
ing the trapezoidalrule discretizing the interval into 300
ZL 40 10-2 equal grids. The production cross section for a reaction
-4
20 (a) (b) (c) 10 product composed of N neutrons and Z protons is given
-6 by
10
20 40 60 20 40 60 20 40 60 80
N N N ∞
L L L σ(N,Z)=2π bP (b)db, (7)
N,Z
Z
0
100
(n) (p)
whereP takesaproductfrom,P P ,intheTDHF
H 80 N,Z N Z
Z theory.
In Fig. 6, we show production cross sections, σ(N,Z),
60 (d) (e) (f)
for 64Ni+238U reaction at E 307.35 MeV. Upper
c.m.
≈
100 120 140 100 120 140 100 120 140 160 panels (a-c) show cross sections for lighter fragments,
N N N whilelowerpanels(d-f)showthoseforheavierfragments.
H H H
We show cross sections for x-, y-, and z-direction cases
FIG. 6. (Color online) Primary production cross sections, in left, middle, and right panels, respectively. To com-
σ(N,Z), for 64Ni+238U reaction at Ec.m. ≈ 307.35 MeV in parewithmeasurements,weshouldtakeaproperaverage
TDHF calculation. Upper panels (a-c) show cross sections over the orientations of 238U. We did not do it, since it
for lighter fragments, while lower panels (d-f) show those for
requires too much computational costs.
heavierfragments. Contributionsfromx-,y-,andz-direction
Fromthe figure,we find that the crosssections extend
casesareshowninleft,middle,andrightpanels,respectively.
widely in the N-Z plane. There is a peak of σ(N,Z) at
The contour lines correspond to σ = 100, 10, 1, 0.1, and
around (N ,Z )=(36,28) in (a-c) for lighter fragments
0.01 mb. L L
and (N ,Z ) = (146,92) in (d-f) for heavier fragments.
H H
Theyarecontributedfromalarge-bregion,b&5fm. We
was Ec.m. ≈ 301.05 MeV, and is much smaller than the alsofindapeakinσ(N,Z)locatedinsidearegionofNL >
present threshold energy for fusion in our TDHF calcu- 50, Z > 30 in (a-c) and N < 130, Z < 90 in (d-f).
L H H
lations. Our results indicate that more mass-symmetric They are producedby the QF processesaccompanyinga
fragments will be produced after forming a mononuclear large mass-drift toward the mass symmetry, which take
systemathigherincidentenergies,althoughitshouldac- place in a small-b region, b . 4 fm. The appearance of
company substantial excitation energy. We note that re- separatedpeaksintheN-Z planeiscausedbytheabrupt
centexperimentaldata[78,79]showthatthesuperheavy change of the reaction mechanism from quasielastic and
element with Z =120could be formedby 64Ni+238U re- MNT to QF at b 4–5 fm. The peak positions are
action at Ec.m. 332.88 MeV, which lived longer than consistent with the≈observation in Fig. 4.
10−18 s. ≈ In Ref. [18], experimentally measured transfer cross
sections for 64Ni+238U reaction at E 307.35 MeV
c.m.
≈
werereported. In Fig.7,we showa comparisonoftrans-
IV. COMPARISON WITH MEASUREMENTS
fercrosssectionsbetweenourTDHFresultsandthemea-
surements as a function of the mass number of lighter
A. Production cross sections fragments. Each panel shows cross sections for different
proton-transfer channel. The number of transferredpro-
TocomparewithmeasuredcrosssectionsofMNTpro- tons is indicated by ( xp; X), where X stands for the
±
cesses, we employ the PNP method [47, 52, 53]. We use correspondingelement. The plus signis for transfer pro-
the PNP operator, cesses from 238U to 64Ni (pickup), while the minus sign
is for the opposite direction (stripping). Measured cross
Pˆn(q) = 21π Z02πei(n−NˆV(q))θdθ, (4) sinecxti-o,nys-,araendshzo-wdnirebcytiornedcafislelesdacreircslheosw. nTbDyHrFedressoulildts,
where Nˆ(q) is the number operator in a volume V. The green dashed, and blue dotted histograms, respectively.
V Cross sections calculated by the GRAZINGcode [23] us-
probabilitythat n nucleons are included inV is givenby
ing standard input parameters 1 are also shown by filled
1 2π areas.
P(q) = einθdet (q)(θ)dθ, (5)
n 2π Z B
0
where
1 InputparametersthatweusedfortheGRAZINGcalculation:
Bi(jq)(θ)=Xσ Z ψi∗(rσq)ψj(rσq)(cid:0)ΘV¯(r)+e−iθΘV(r)(cid:1)dr. F(1o2r.0lo9w)-el2ybin2g,Eex3ci=ta3ti.o5n6s(:0E.723)=M1e.3V5,(B0.(0E43))M=eV0.,02B2(E(02.)58=)0e2.0b726;
(6) for giant resonances: E2 =57 (94) A−1/3 MeV, strength = 0.8
9
3
Expt. L. Corradi et al. 10 (+2p; Zn) (+1p; Cu)
{ 102
x-direction ) 1
TDHF b 10
y-direction m 0
w/o evap. 10
z-direction ( -1
10
GRAZING w/o evap. s 10-2
-3
10
64Ni+238U (E » 307.35 MeV) 50 60 70 50 60 70
c.m.
3
10 (-6p; Ti) (-5p; V) (-4p; Cr) (-3p; Mn) (-2p; Fe) (-1p; Co) (0p; Ni)
2
10
) 1
b 10
m 0
10
( -1
10
s 10-2
-3
10
50 60 70 50 60 70 50 60 70 50 60 70 50 60 70 50 60 70 50 60 70
MASS NUMBER
FIG. 7. (Color online) Transfer cross sections for 64Ni+238U reaction at Ec.m. ≈307.35 MeV. Each panel shows cross sections
for different proton-transfer channel indicated by (±xp; X), where X stands for the corresponding element. The horizontal
axis is the mass number of lighter fragments. Experimental data [18] are shown by red filled circles. TDHF results in x-, y-,
and z-direction cases are represented by red solid, green dashed, and bluedotted histograms, respectively. Wealso show cross
sections calculated by theGRAZINGcode [23] with standard parameter sets 1.
We note that experimental data are suffered by ef- failure indicates a necessity of descriptions beyond the
fects of particle evaporation from excited reaction prod- TDHF theory. Beyond mean-field theories such as the
ucts, whereas the TDHF results correspond to primary method of Balianand V´en´eroni[82, 83] and the stochas-
cross sections just after the reseparation. In addition, ticmean-fieldapproach[84–90]haverecentlybeendevel-
the measurement was performed for an angular range of oped, which are expected to remove the discrepancies.
50◦ θ 105◦ to cover the main transfer channels in
lab InFig.7,wefindthatthecrosssectionsdependrather
≤ ≤
grazingreactions,whereastheTDHFresultsareobtained weakly on the initial orientation of 238U. Difference is
by Eq.(7)without filteringby the scatteringangle,asin
substantial only for ( 5p), ( 6p), and (+2p) channels.
Fig. 6. − −
Differences in these channels are associatedwith the dif-
Fromthefigure,wefindthattheTDHFresultsreason-
ferent trends of nucleon transfer. The proton-stripping
ably reproduce measured cross sections for (0p), ( 1p),
processes are originated from an impact parameter re-
±
and ( 2p) channels. As the number of removed pro-
gion, b 5–6 fm, as shown in Fig. 4. From the insets
−
tons increases (( xp) with x 3), the peak position of ≈
showninFig.4(b,d), wefindthatproton-strippingpro-
− ≥
experimental cross sections shifts towardless mass (neu-
cesses are favored in the x-direction case. This trend
tron) numbers, which is not reproduced by the TDHF
resultsin the difference in ( 5p)and( 6p)channels. In
results. Thedisagreementmaypartlybeoriginatedfrom − −
Fig. 4, a gradual change of the average number of nu-
the evaporation effect. We note that, although the peak
cleons was observed in the z-direction case. This change
position is different, the height of the peaks of the cross
brings a large contributionto (+1p) and (+2p) channels
sections for proton-stripping channels is in good agree-
from a wide impact parameter range.
ment with the experimental data, up to ( 4p) channels.
− From a comparison between cross sections by TDHF
Similar disagreement was observed in lighter systems
and those by GRAZING, we find that the TDHF results
[47,54,80,81]. InRefs.[54,80,81],weinvestigatedpar-
show a better overallagreementwith experimental data.
ticle evaporation effects on MNT cross sections using a
ItisremarkablethattheTDHFcalculationprovidessub-
statistical model. Although the inclusion of the evapo-
stantial cross sections for the proton-pickup channels.
rationeffect improved the agreementbetween TDHF re-
The GRAZINGcalculation underestimates cross sections
sults and measurements, there remain disagreements for
forthosechannels,especiallyfor(+2p)channel,forwhich
channels involving a number of transferred nucleons far
the TDHF calculation overestimates. In Ref. [18], it was
from the average values in the TDHF calculation. This
mentioned that lighter fragments with proton numbers
up to Z 40 were observed experimentally, especially
≈
at forward angles, although quantitative cross sections
were not shown. The TDHF calculation provides sub-
(0.4) % of sum rule, width = 2.5 (6); for single-particle states:
δν =8,δπ =8,leveldensity=2.455(2.053)MeV−1 (neutron), stantial cross sections for lighter fragments with Z 40
10.527 (8.298) MeV−1 (proton); these values are for projectile (cf. Fig. 6 (a-c)) as a result of the QF processes≈at a
(target). small-b region, b.4 fm.
10
FIG. 9. (Color online) Mass-angle distribution (MAD) plots.
FIG. 8. (Color online) Wilczyn´ski plots. (a): TDHF results
for 64Ni+238U reaction at Ec.m. ≈ 307.35 MeV. Results for (a): TDHF results for 64Ni+238U reaction at Ec.m. ≈
307.35 MeV. Results for x-, y-, and z-direction cases are
x-, y-, and z-direction cases are shown by red circles, green
shown by red circles, green crosses, and blue open trian-
crosses, and blue open triangles connected with dotted lines,
gles connected with dotted lines, respectively. (b): Experi-
respectively. (b): Experimental data of Wilczyn´ski plots for
main transfer channels, (+1n), (+2n), (−1p), and (−2p), mentally measured MAD in 238U+64Ni reaction at Ec.m. ≈
in 64Ni+238U reaction at Ec.m. ≈ 307.35 MeV. The figures 302.62 MeV. The figureshown in (b) is taken from Ref. [20].
shown in (b) are taken from Ref. [18].
tween TDHF calculations and measurements have been
B. Wilczyn´ski plot reported for 40Ca+238U [57] and 50,54Cr+180,186W [59].
In Fig. 9 (a), we show the MAD plot in the TDHF
calculation, which is constructed from the results shown
Combining θ in Fig. 2 (a) and TKEL in Fig. 2 (b),
c.m. in Figs. 2 (a) and 4 for 64Ni+238U reaction at E
we obtain the Wilczyn´ski plot which is shown in c.m.
307.35 MeV. In (b), the measured MAD for 238U+64N≈i
Fig. 8 (a). In Ref. [18], experimental data of Wilczyn´ski
reaction at E 302.62 MeV is shown. Because the
plotsforvarioustransferchannelswerereportedforgraz- c.m.
≈
ingreactionof64Ni+238UatE 307.35MeV.In(b), inverse kinematics was employed in the experiment, the
c.m. ≈ angle of 180◦ θ is used in the plot of (a).
we show the experimental data for main transfer chan- c.m.
−
As seen from Figs. 2 (a) and 4, both scattering an-
nels, (+1n), (+2n), ( 1p), and ( 2p).
− − gle and fragment masses change substantially when two
The experimental data show a peak at around θ
c.m.
90◦, which shifts toward Q 60 MeV (lower TKE≈), nuclei start to overlap in the course of collision. These
≈ − trends induce correlated behavior in (a) showing an
as the number of transferred protons increases. Our re-
oblique distribution from A = 64 (238) to A
sults agree with the observed trend. At the scattering L(H) L(H)
≈
angle of θ 80◦–85◦ (b 5–6 fm), our TDHF calcu- 100 (200). We note that the TDHF calculation provides
c.m. ≈ ≈ no contributionsto θ . (&)90◦ atA 64 (238),due
lation describes proton-stripping processes, as shown in c.m.
≈
to the classical nature of the trajectory.
Sec. IIIA. In this regime, two nuclei touched gently at
the distance of closest approach, forming a subtle neck InourTDHFcalculations,collisionsatasmall-bregion
whichpersistsonlyforashortperiod. Dueto the forma- inwhichthe mass-driftmodetowardthe masssymmetry
tion of the subtle neck, nucleons are exchanged between isobservedcontribute tocertainfragmentmasses. Reac-
the projectile and target nuclei and the TKE decreases tions of the x- and z-direction cases produce fragments
rapidly, while the scattering angle is kept almost con- of AL 90–100 and AH 200–210, while those of the
≈ ≈
isnta(nbt),,θwc.em.fi≈nd8a0◦t–a8il5◦o.f tIhnethyieeledxspwerhimichenetxatlednadtsatsohwoawrnd yA-Hdirec1ti8o0n.cTasheeprerofodruec,ewferaegxmpeencttstahraotunthdeAyLie≈lds12i0natnhde
≈
forward angles up to θ 75◦, as the energy loss in- MADofthefragmentswithsymmetricmassesarecaused
c.m.
creasesuptoQ 75MeV≈.IntheTDHFresultsshown by the collisions in the y-direction case.
in (a), a similar≈tre−nd is observed at θ 75◦.
c.m.
≈
D. Mass-TKE distribution
C. Mass-angle distribution
TheTKEofoutgoingfragmentsisalsoacharacteristic
QF processes are known to show a characteristic cor- observable of QF processes. In Fig. 10 (a), we show the
relation between fragment masses and scattering angle, mass-TKE distribution in the TDHF calculation, which
and thus, MADs in QF processes have been measured is constructed from the results shown in Figs. 2 (b)
extensively [20, 57, 59, 91–93]. The MAD for 238U+64Ni and 4. In (b), the measured mass-TKE distribution
reactionatE 302.62MeVwas reportedby J.T¯oke for 64Ni+238U reaction at E 301.05 MeV [19] is
c.m. c.m.
≈ ≈
et al. [20]. We compare the TDHF results with the shown. In these plots, two prominent peaks at around
experimental data. Similar comparisons of MADs be- A 64 and A 238 are seen, which correspond to
L H
≈ ≈