suppression of fusion by energy dissipation in heavy …suppression of fusion by energy dissipation...
TRANSCRIPT
Suppression of fusion by energy dissipation in heavy ion collisions
4th Dec. 2018
Department of Nuclear Physics, RSPE Australian National University
- Dongyun Jeung -
NN2018
Outline
§ Background - Fusion
- Energy dissipation outside the fusion radius?
§ My Project - Part 1. Experimental Signature of Energy Dissipation - Part 2. Capture Cross Sections
§ Conclusion
2
Fusion
3
HOT Compound Nucleus
Projectile Target Evaporation Residue
Neutron evaporation
dinuclear
(Statistical Model)
Fission fragment
Capture
Fusion & Energy Dissipation
4
Excitation E
nergy
• Kinetic Energy à Ex (Heat) • Irreversible E-dissipative process
( Ex à full KE )
Composite system
Ex
×
Continuum
Fusion – Coupled-Channels Models
5
Radial separation (r)
Pot
entia
l
RB
VB
E
Fusion – Coupled-Channels Models
6
Radial separation (r)
Pot
entia
l
RB
VB
Capture (Fusion)
IWBC
E
Fusion Hindrance
7
C. R. Morton, et al., Phys. Rev. C 60, 044608 (1999)
Fusion Suppression at E > VB
Fusion Hindrance
8
J. O. Newton, et. al., Phys. Rev. C 70, 024605 (2004)
Z1Z2 = 656
656
S = σexp / σmodel Fusion Suppression at E > VB
Increasing reduction in fusion with Z1Z2
Density Overlap at the Barrier Radius
9
Dasgupta M, et al., Annu. Rev. Part. Sci. 48: 401-61 (1998)
Z1Z2 64 400 2500
The density overlap increases at the barrier
Fusion – Coupled-Channels Models
10
Radial separation (r)
Pot
entia
l
RB
VB
Capture
IWBC
E
Example: Elastic scattering (No E-dissipation)
No density overlap
RB
radialseparation(r12)
RB
Density(
ρ)
RB
Fusion – Coupled-Channels Models
11
Radial separation (r)
Pot
entia
l
RB
VB
IWBC
E
Example: Fusion (Full E-dissipation inside the barrier)
RB
radialseparation(r12)
RB
Density(
ρ)
RB
Large density overlap
Capture
Fusion – Coupled-Channels Models
12
Radial separation (r)
Pot
entia
l
RB
VB
IWBC
E
Example: transfer-reaction (E-dissipation?)
Intermediate density overlap
RB
radialseparation(r12)
RB
Density(
ρ)
RB
Capture
Energy Dissipative Processes
Radial separation (r)
Pot
entia
l
RB
VB
Energy Dissipation
13
E
E-dissipated through transfer reactions?
Identifying events following E-dissipation
14
Fissile Target
ER CN
Fission fragments
Identifying events following E-dissipation
15
Target-like nucleus
Nucleons exchange
Fissile Target
CN
Projectile-like nucleus
> 6 MeV
Fission fragments
★★★
Identifying events following E-dissipation
16
Nucleons exchange
Fissile Target
CN
ç transfer-fission (E-dissipation)
é fusion-fission
Target-like nucleus
3-body system
2-body system
Research Project - Part 1
17
• Investigate an influence of the density overlap of the colliding nuclei to energy dissipative processes.
• Compare transfer-fission and capture (fusion)-fission yields as the reaction Z1Z2 is changed.
Projectile Target Z1Z2 Density Overlap E / VB 18O
232Th
720 0.93 – 1.14
30Si 1260 0.94 – 1.08
34S 1440 0.87 – 1.07
40Ca 1800 0.93 – 1.15
Increasing density overlap VB = Fusion barrier
W. J. Swikatecki, et. al., Phys. Rev. C 71, 014602 (2005)
18
SNICSIon source
Mass selection magnet
Buncher
Choppers
Linac loop
Charging chain
Foil or Gas stripper
High-voltage Terminal
SF6 Gas filled tank
CUBEchamber
Beamdump
Energy selection magnet
2nd stripper foil
External stripper foil
Energy defining slits Linac
transfer 90°
Switchingmagnet
Entry
Exit
Acceleration tube
130.
0
180.0
180.
0
45°
Beam Target
Monitors(out of plane)
90°
180° 0°
Back Det.
Front
Det
.
0°
All dimensions in mm
149.
0
149.0 130.0 XRXL
90°
CUBE Binary Spectrometer 14UD ( + LINAC)
Experimental Setup
2-body Kinematic – velocity vectors in c. m.
19
x
y
z (beam axis)
CN
vCN
v1
v2
v2, c.m.
v1, c.m.
vpar - vCN
vperp
<Source velocity distribution>
(0, 0)
• vpar = vCN• vperp ~ 0
★★
Decompose a vector into perp & par components
3-body Kinematic – velocity vectors in c. m.
20
x z (beam axis)
v2, c.m.
v1, c.m.
vpar - vCN
vperp
<Source velocity distribution>
(0, 0)
• vpar > vCN or vpar < vCN • vperp ≠ 0
yvTarg-N
v1
v2
Targ -N
Decompose a vector into perp & par components
★★
Source velocity distributions – Z1Z2
21
þ Presence of transfer-fission events þ Presence of capture (FMT) -fission events þ As Z1Z2 increases, the momentum transferred to the target increases, leading to a larger recoil velocity.
★★★
2-body system 3-body system
Contribution of transfer-fission at E > VB
22
1.0
0
0.2
0.4
0.6
0.8
1.0 1.05 1.11.15 1.2 1.25 1.3
E/VB
Tra
nsfe
r-fissio
n/T
ota
lfissio
n
720
1260
1440
1800
★★★★★
18O 30Si 34S 40Ca
(R)
Increasing transfer-fission events as Z1Z2 increases.
Part 2. Cross section measurement • For comparing with CC-calculations. • From fission angular distributions:
σcapture = σfission (since σER = 0)
• 85° < θc.m.< 135° à Need extrapolation beyond experimental angular coverage
23
dσ f i s
dθc.m.∫ dθc.m.
Light system reactions
24
Fissile Target
CN
Fusion-fission
Capture
• Equilibrium process • Fission described by the statistical model
Heavy system reactions
25
Fissile Target
CN
Fusion-fission
Quasi-fission (QF)
Capture
• Non-equilibrium process
★★
Mass-Angle Distribution MAPPING QUASIFISSION CHARACTERISTICS AND . . . PHYSICAL REVIEW C 88, 054618 (2013)
180
135
90
45
0
R
0.2 0.4 0.6 0.8 1.00.01
10
10
10
10
10
2
3
5
4
M
θ c.m
.[de
g]
FIG. 3. (Color online) Measured mass-angle distribution for the16O + 196Pt reaction, expected to lead to fusion-fission. The whitesquares illustrate how the MAD is populated at two points for eachbinary event measured. For fusion-fission, the distribution is expectedto be symmetric in angle about the blue line, and symmetric in massabout the yellow line. Elastic events are not seen for the 16O beam asthey were below the MWPC timing thresholds (see text).
of the fission-like components of the MAD. The softwaregates applied in generating all the MAD are those shown inFigs. 11(b) and 12.
The second instrumental effect involves target nucleusrecoils following elastic scattering, and is relevant to thosereactions involving beams heavier than 12C and 16,18O. Kine-matic considerations show that as the elastic scattering angle ofbeam particles approaches 0◦, the corresponding target recoilsare ejected from the target with laboratory angles approaching90◦ and energies approaching zero. Clearly there is an energythreshold below which the recoils will not penetrate thegas window and cathode foil. Thus, they will not appear incoincidence with the corresponding elastic scattering event,and will consequently be absent from the MAD. This is whythe elastic-recoil coincidences corresponding to forward angleelastically scattered beam particles are absent or suppressed inthe MAD, principally for the Mg and Si beams, for which theenergies of the recoils at a given angle are low. The distributionwith angle of fission-like events in the MAD spectra show thatthis instrumental low-energy threshold has essentially no effecton the mass-angle distribution of the fission and quasifissionevents, since unlike target recoils, their laboratory energies donot approach zero.
B. Reaction outcome features
Having covered the instrumental influences on the experi-mental MADs, the physical mechanisms responsible for theappearance of each MAD can be explored. An idealizeddiagram of the features seen in an experimental MAD ispresented in Fig. 4. It shows the location in mass and angle ofthe different reaction outcomes, and also illustrates the originof the correlation between the mass and angle seen in thequasifission events. Each of these features is discussed below.
1. Quasielastic and deep inelastic collisions
The two green shaded regions marked QE in Fig. 4 aremainly populated through elastic and inelastic scattering of the
0.2 0.4 0.6 0.8 0.10.0
45
90
135
0
180
c.m
. θ
[deg
]
MR
I
II
III
EQEQ
QF
QF
FF
FIG. 4. (Color online) Schematic illustration of the MAD show-ing the regions corresponding to different reaction processes.Quasielastic and deep inelastic scattering are denoted by QE, fusion-fission by FF, and quasifission by QF. The curved red and blue linescorrespond to average quasifission trajectories for a single angularmomentum (see text).
projectile and target nuclei, with some contribution from trans-fer and deep inelastic reactions. By deep-inelastic reactions,we mean that class of events where (i) energy is increasinglydissipated into heating the colliding nuclei until completedamping of relative motion occurs, (ii) mass exchange canoccur, but without significant mass drift away from theentrance channel mass-asymmetry [36,37], and (iii) deviationsfrom Coulomb trajectories occur due to nuclear orbiting. Theangular dependence of the elastic and quasielastic scatteringyield essentially follows Rutherford scattering, except atthe backward angles, where scattering events are depleted,being transformed into fission-like events as a result ofcapture, sticking, and mass flow towards symmetry. For elasticscattering, the expected mass ratio MR should correspond tothe value expected from the entrance channel projectile andtarget masses mp and mt ; i.e., MR = mp/(mp + mt ) if theprojectile is detected in MWPC1, or MR = mt/(mp + mt ) ifthe target nucleus reaches MWPC1.
2. Fusion-fission and quasifission
The processes of interest in this work are quasifission andfusion-fission. Both result in fission fragments consistent witha two-body full-momentum transfer (FMT) fission event [9].Most kinematic properties of the fission fragments resultingfrom these two processes will be similar, if not identical. How-ever, heavy ion fusion-fission generally shows the peak yield atmass symmetry (MR = 0.5), and a standard deviation in MR of∼ 0.05 to 0.08. Low-energy mass-asymmetric actinide fissiontypically has a peak mass yield around A = 139; thus the massratio is typically not more asymmetric than MR = 0.60/0.40.The gray shaded fusion-fission band (marked FF in Fig. 4) thusis drawn as extending from MR = 0.4 to 0.6. Although bothfusion-fission and quasifission can contribute to this shadedregion, quasifission alone gives significant population to the
054618-7
26
R. du Rietz, et. al., Phys. Rev. C 88, 054618 (2013)
Sticking time (sec)
~ 5×10-21 ~10-19
MADs – light vs heavy systems
27
• Slow QF/Fusion-fission. • Weak mass-angle correlation. • Symmetric distribution. • σcap ≈ σSlow QF (σFF)
• 18O + 232Th, 30Si + 232Th
• Fast QF. • Strong mass-angle correlation. • Asymmetric distribution. • σcap = σFF + σFast QF
• 34S + 232Th, 40Ca+ 232Th
0 0
Determination of capture cross sections
Fission events identification Extrapolation of fission angular distribtuionsClassifying FMT-fission
FMT-fission
Quasifission
Simulation
Transition State Model
calculation
Extrapolated
Angular
distributions
Total capture
cross sections
No mass-anglecorrelation
- slow fission
Mass-anglecorrelation
- fast quasifission
Exp. angular
distributions
Light system
Heavy system
28
Method 1
Method 2
Extrapolation of fission angular distributions
Method 1. Transition State Model
§2.3 Transition state model (TSM) for fusion-fission 15
transition state model (TSM) [47, 48]. In this approach the highly deformed nucleus atthe fission saddle-point configuration is considered to be a transition state between theequilibrium CN and the scission configuration.
According to the TSM, the angular distribution of fission fragments resulting fromthe fissioning system passing over the saddle configuration can be determined by threequantum numbers: J , the total angular momentum; K, the projection of J on the nuclearsymmetry (fission) axis, as shown in Fig. 2.3, and M the projection on the space-fixed(beam) axis.
The model assumes that the fission fragments separate along the nuclear symmetry axis(= fission axis) and that the direction of this axis is unchanged as the system evolves fromsaddle to scission and separation. Although the composite system experiences dynamicchanges in shape, the quantum numbers J and M are conserved in the entire fissionprocess (neglecting particle evaporation) and K is taken to be determined at the saddlepoint. The fission angular distribution W J
M,K(◊) is expressed as a function of the quantum
numbers through |DJ
M,K(◊)|2, the symmetric-top wave functions, which depends on ◊:
W J
M,L(◊) = [(2J + 1)/2]|DJ
M,K(◊)|2, (2.7)
where M is the projection of J on the spaced-fixed axis, which generally considered as thebeam axis (z).
For the simplest application of Eq. 2.7, it is assumed that the projectile and targetnuclei have zero spins (true for all the reactions considered here) and no particle emissionfrom the initial CN occurs before scission (i.e. M = 0). A reduced equation for the fissionangular distribution for the fission of spin zero nuclei including the contribution of all J
leading to CN formation is given by [47]:
W (◊) =Œÿ
J=0(2J + 1)TJ
Jÿ
K=≠J
flJ(K)|DJ
0,K(◊)|2 (2.8)
where TJ is the transmission coe�cient for fusion of partial wave J and flJ(K) are thedensity of levels at the transition state with the assumption of a Gaussian K distributiongiven by
flJ(K) =
Y___]
___[
exp(≠K2/2K
20 )
JqK=J
exp(≠K2/2K20 )
, if K Æ J.
0, K > J.
(2.9)
where K0 is the variance of the K-distribution. The distribution of K, characterised byK2
0 , is estimated from the properties of the fissioning system at the saddle-point, taken to29
• Angular distribution of fission fragments (if M = 0) :
Saddle configuration
§2.3 Transition state model (TSM) for fusion-fission 15
transition state model (TSM) [47, 48]. In this approach the highly deformed nucleus atthe fission saddle-point configuration is considered to be a transition state between theequilibrium CN and the scission configuration.
According to the TSM, the angular distribution of fission fragments resulting fromthe fissioning system passing over the saddle configuration can be determined by threequantum numbers: J , the total angular momentum; K, the projection of J on the nuclearsymmetry (fission) axis, as shown in Fig. 2.3, and M the projection on the space-fixed(beam) axis.
The model assumes that the fission fragments separate along the nuclear symmetry axis(= fission axis) and that the direction of this axis is unchanged as the system evolves fromsaddle to scission and separation. Although the composite system experiences dynamicchanges in shape, the quantum numbers J and M are conserved in the entire fissionprocess (neglecting particle evaporation) and K is taken to be determined at the saddlepoint. The fission angular distribution W J
M,K(◊) is expressed as a function of the quantum
numbers through |DJ
M,K(◊)|2, the symmetric-top wave functions, which depends on ◊:
W J
M,L(◊) = [(2J + 1)/2]|DJ
M,K(◊)|2, (2.7)
where M is the projection of J on the spaced-fixed axis, which generally considered as thebeam axis (z).
For the simplest application of Eq. 2.7, it is assumed that the projectile and targetnuclei have zero spins (true for all the reactions considered here) and no particle emissionfrom the initial CN occurs before scission (i.e. M = 0). A reduced equation for the fissionangular distribution for the fission of spin zero nuclei including the contribution of all J
leading to CN formation is given by [47]:
W (◊) =Œÿ
J=0(2J + 1)TJ
Jÿ
K=≠J
flJ(K)|DJ
0,K(◊)|2 (2.8)
where TJ is the transmission coe�cient for fusion of partial wave J and flJ(K) are thedensity of levels at the transition state with the assumption of a Gaussian K distributiongiven by
flJ(K) =
Y___]
___[
exp(≠K2/2K
20 )
JqK=J
exp(≠K2/2K20 )
, if K Æ J.
0, K > J.
(2.9)
where K0 is the variance of the K-distribution. The distribution of K, characterised byK2
0 , is estimated from the properties of the fissioning system at the saddle-point, taken to
R. Vandebosh, et. al., Nuclear Fission (1973).
Angular distributions – light systems
10-1
100
101
102
103
90 100 110 120 130 140 150 160 170 180
dσ/dΩ(θ)[mb/sr]
θc.m.[deg]
18O+
232Th
Ec.m.=89.6MeV
Ec.m.=87.0MeV
Ec.m.=84.5MeV
Ec.m.=79.9MeV
Ec.m.=75.1MeV
Ec.m.=72.6MeV
100
101
102
90 100 110 120 130 140 150 160 170 180
dσ/dΩ(θ)[mb/sr]
θc.m.[deg]
30Si+
232Th
Ec.m.=146.6MeV
Ec.m.=143.0MeV
Ec.m.=139.4MeV
Ec.m.=135.7MeV
Ec.m.=131.6MeV
Ec.m.=128.0MeV
30
18O + 232Th 30Si + 232Th
★★★
Method 2. Quasi-fission Simulation
31
Example: 34S + 232Th at Ec.m. = 150.5 MeV
Quasi-fission Simulation
32
• Scattering angle:
R. du Rietz, et. al., Phys. Rev. C 88, 054618 (2013)
• Rotation angle:
θ(ts ) = π −Θin −Θs (ts )−Θout
MAPPING QUASIFISSION CHARACTERISTICS AND . . . PHYSICAL REVIEW C 88, 054618 (2013)
180
135
90
45
0
θc.
m.[
deg]
0.2 0.4 0.6 0.8 1.00.01
10
10
10
102
3
4
MR
FIG. 13. (Color online) MAD scatter plot for the reaction40Ca + 238U at Ebeam = 240.5 MeV.
d. Determination of the mass-angle distributions
Once the velocity and RTKE conditions shown inFigs. 11(b) and 12 have been applied, the mass ratio MR
and center-of-mass scattering angle (θc.m.) for fragmentsdetected in the back detector can be used to generate the finalmass-angle-distribution (MAD) scatter plots. Figure 13 showsthe final experimental MAD for the reaction 40Ca + 238U atEbeam = 240.5 MeV, where the event selection criteria de-scribed above have been applied. The geometrical acceptanceof the detector is about 100◦ in θc.m., independent of mass ratio.The angular coverage shifts with mass ratio by about 15◦.
The vertical bands around MR ∼ 0.14 and 0.86 are princi-pally from quasielastic events, associated with projectile-likeand target-like nuclei, respectively. Fission-like events areclearly seen in the region between the two elastic bands. Theyshow a strong correlation between mass and angle, indicatingthe short timescale of this reaction.
APPENDIX B: QUASIFISSION SIMULATION
The basis of the Monte Carlo model developed [34]to simulate quasifission mass-angle distributions is showngraphically in Fig. 14. The figure shows the projectile nucleus(red) incident from the top, which follows a incoming Coulombtrajectory from infinity. The interaction with the target nucleus
θ = ωS tS
IIII
II
FIG. 14. (Color online) Schematic illustration of the basic physicsof the Monte Carlo model used to simulate quasifission. Threedifferent quasifission outcomes (I to III), depending on sticking time(ts) and rotation speed (ω), are indicated.
(blue) starts as the two nuclei touch, and mass can betransferred between them. While the dinuclear system stickstogether, for a time ts , it rotates around its center of mass withan angular velocity ω, assumed to be constant. Once the systemseparates the fragments again move on Coulomb trajectoriestowards infinity. The figure illustrates three outcomes withincreasing sticking time (I, II, and III) during the firstrevolution of the system. To calculate the MAD accordingto the above picture the simulation model is divided into twoparts, dealing with calculation of the observation angle of thefragments, and with the evolution of their masses during thesticking time.
The observed scattering angle, which depends on thesticking time ts , is determined from
θ (ts) = π − $in − $s(ts) − $out. (B1)
Here $s(ts) is the rotation angle of the system betweencontact and scission, and $in and $out are the relevant anglesderived from classical Coulomb trajectories for the incomingand outgoing nuclei, respectively. $in is the angle betweenthe vector joining the centers of the two colliding nucleiat infinity (before the collision) and that at the distance ofclosest approach. Thus the relationship between the $in andthe Coulomb scattering angle θin with respect to the beamdirection is 2$in = π − θin. θin is calculated in the usual wayfrom the expression tan(θin/2) = Din/2bin, where bin is theimpact parameter associated with a given incident angularmomentum. Thus $in is calculated according to
$in = π
2− tan− 1 Din
2bin. (B2)
Evaluation of the equivalent angle $out for the outgoingchannel is slightly more complicated, involving estimation ofthe parameters Dout and bout. To determine Dout we assumeno radial velocity at scission, and that the fragment kineticenergies are fully damped, and follow Viola’s total kineticenergy [53] systematics, accounting for asymmetric masssplits [54]. To evaluate bout we assume that 2/7 of theinitial angular momentum Jh is converted to intrinsic angularmomentum in the two fragments.
It is assumed that the two colliding nuclei touch andstick at the distance of closest approach Din, evaluated forthe incoming Coulomb trajectory. This is reasonable for theenergies of interest near the capture barrier. The rotation angleduring time ts between contact and scission is obtained from
θs(ts) =√
J (J + 1) h⟨I ⟩
ts . (B3)
Angular momenta are chosen randomly, weighted by theangular momentum distributions calculated using the couple-channels code CCFULL. This is the angular momentum dis-tribution corresponding to capture in the entrance-channelpotential pocket. As such it implicitly includes fusion-fission,quasifission, and any deep inelastic reaction processes thatresult after capture. The average moment of inertia ⟨I ⟩ isestimated as the average of the entry Iin and exit Iout momentsof inertia. In the similar picture of Toke et al. [2], it wasassumed the two nuclei were touching spheres at scission. Forconsistency with the mass equilibration time constant extracted
054618-19
θs (ts ) =J(J +1)!< I >
ts
Θin
Θout
Toke, et. al., Nucl. Phys. A 440, 327-365 (1985)
Sticking time distributions
33
§2.4 Dynamical approach for quasifission 19
2.4.2 Quasisim: mapping quasifission timescale
The simulation of QF populating the MAD is achieved through the Monte Carlos method,parameterizing tje variables that influence the MAD. Below, the physics ingredients ofthe model and parameterizations used in this work are presented. Simulated MADs alsodescribe the angular distributions of QF allowing extrapolation to angels close to the beamdirection. The adjustment of parameters and a detailed quantitative comparison betweenmeasured and calculated MAD will be given in Chapter 6.
2.4.2.1 Determination of the emission angle in the centre of mass frame:
The observed emission angle of a projectile-like fission fragment based on [27] is determinedas:
◊QF (ts) = fi ≠ [�in + ◊rot(ts) + �out] (2.11)
where ◊rot(ts) is the rotation angle of the system while they stick together, and �in and�out are the angles associated with the classical Coulomb trajectories for the incomingand outgoing nuclei.
Details of angles related to Coulomb trajectory calculations are given in Appendix A.The rotation angle during the sticking time (ts) between contact and scission is estimatedfrom the relationship between the angular velocity Ê and the sticking time ts, ◊rot = Ê◊ts.An expansion of the angular velocity gives the rotation angle:
◊rot(ts) =
J(J + 1)h< I >
ts (2.12)
where J is the angular momentum carried by the projectile and < I > is the averagemoment of inertia during the sticking time. Each of the components in Eq. 2.12 isdetermined as follows to reproduce the experimental MAD:
1. Angular momentum distributionAngular momentum distributions were obtained the capture partial wave cross sec-tions ‡l calculated with a CC code such as CCFULL [6] or CCMOD [52]. Thesecodes can provide the angular momentum distributions for each beam energy1. InQuasisim, the angular momentum of each events is randomly chosen and weightedby the angular momentum distributions.
2. Sticking time distributionThe sticking time is parameterized as a distribution rather than a single value. Theassumed general form of the sticking time distribution is a half-Gaussian rise followedby an exponential decay [50], as shown in Fig. 2.6. This sticking time distribution
1We acknowledge using the coupled channels codes to explore physics outside of CC framework, butsome reasonable assumptions about the angular momentum distribution must be made. This is currentlythe best approach available.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0 5 10 15 20
σ
τ
μ
Probability
time[zs]
Gaussian
Expoentialdecay
• Parameterisation of a sticking time distribution: (peak centre, width of Gaussian, life-time)
§2.4 Dynamical approach for quasifission 21
symmetry with time, as shown in Fig. 2.4b. The initial MR(0) is the ratio of target massto the CN mass. The mass flow between the two constituents can be defined in terms ofthe mass ratio MR as follows:
MR(ts) = [MR(0) ≠ 0.5] exp≠ts/tm +0.5 (2.13)
where tm is the mass equilibrium time constant, determined 5.2 ◊ 10≠21 s by [27], and ts
is the sticking time.In the simulation, the final mass ratio is calculated when each event is randomly
assigned a sticking time from the distribution. Once the system evolves at a particularsticking time ts, the two constituent fragments are populated in the MAD according toMR(ts) and ◊c.m.(ts).
2.4.2.3 Specific conditions:
In some systems, additional ingredients must be added to the simulation to account forlocalised physics phenomena. Following conditions are particularly considered for thiswork.
Figure 2.7: Potential energy surface for 274Hs. Fission path depends on the collidingangle of 36S with the symmetrical axis of 238U. [Ref]. Fission outcomes depends on thetrajectories of the dinuclear system travel over the potential energy surface.
1. Shell energy correctionOnce the nuclei are in contact, forming the dinuclear system, the system evolvesin shape towards the minimum of the potential energy surface (PES) [55] beforeundergoing scission. For example, some events may trapped in the shell closure
★★★
Quasisim output MADs – 34S + 232Th
34
Quasisim output MADs – 40Ca + 232Th
35
Angular distributions – heavy systems
36
0
20
40
60
80
100
120
140
160
0 20 40 60 80 100 120 140 160 180
dσ/dθ(θ)[mb/rad]
θc.m.[deg]
34S+
232Th
Ec.m.=166.7MeV
Ec.m.=158.4MeV
Ec.m.=150.5MeV
Ec.m.=147.9MeV
Ec.m.=145.7MeV
Ec.m.=143.6MeV
0
50
100
150
200
250
300
0 20 40 60 80 100 120 140 160 180
dσ/dθ(θ)[mb/rad]
θc.m.[deg]
40Ca+
232Th
Ec.m.=222.7MeV
Ec.m.=210.2MeV
Ec.m.=203.2MeV
Ec.m.=197.0MeV
Ec.m.=193.1MeV
Ec.m.=185.5MeV
40Ca + 232Th 34S + 232Th
Coupled channels calculations
• a0 =0.65 fm
• Add relevant couplings - Rotational coup. - Vibrational coup.
• Exp. VB
0
100
200
300
400
500
600
4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5
10-3
10-3
1/Ec.m.[MeV-1]
σ[mb]
30Si+
232ThExp.
34S+
232ThExp.
40Ca+
232ThExp.
48Ca+
232ThExp.
0
100
200
300
400
500
600
10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5
10-3
10-3
1/Ec.m.[MeV-1]
σ[mb]
18O+
232ThExp.
37
§2.1 Single barrier penetration model 11
V0 , and the radius parameter r0. It is given by:
Vnuc(r) = ≠V01 + exp[(r ≠ R0)/a0] (2.2)
R0 can be related to the sum of the colliding nucleus radii, R0 = r0(A1/31 +A1/3
2 ), and r0 isthus expected to be ≥ 1.2 fm [31, 32]. A typical di�useness parameter a0 = 0.65 fm is wellaccepted to reproduce elastic scattering [32]. However, a relatively higher a0 is needed tofit heavy ion capture data [33, 34].
The simplest expression for the Coulomb potential between two nuclei separated fromeach other (r > R0) is given by:
VCoul(r) = e2
4fi‘0
Z1Z2r
(2.3)
where Z1,2 are the nuclear charges of the colliding nuclei.
For non-head on collisions where the collision motion involves not only radial motionbut also angular motion, the centrifugal potential is non-zero and is expressed as:
Vl(r) = l(l + 1)h2
2µr2 (2.4)
where lh is the orbital angular momentum and µ is the reduced mass.
The sum of the three potentials forms a potential barrier - a local peak at a barrierradius RB as shown in Fig. 2.1a. This is commonly known as the Coulomb (or capture)barrier (VB); inside the barrier exists a potential pocket where the projectile is irreversiblycaptured by the target. The height of the barrier varies depending on the angular momen-tum, as shown in Fig. 2.1b; as l increases, the barrier height increases and the potentialpocket gets shallower, eventually vanishing at the maximum value of lcrit.
2.1.2 Capture cross section
The yield of capture is usually expressed in terms of the cross section. The classical pictureof the single barrier passing model is a simplistic way to approach a complex heavy ionfusion reaction and define the cross section. It neglects the e�ects of the intrinsic degreesof freedom between two colliding nuclei and the influence of competing reaction channels.The classical expression for the fusion cross section is:
‡(E) =
Y]
[fiR2
B(1 ≠ VB
E) for E Ø VB
0 for E > VB
(2.5)
where RB is the radius of the l = 0 barrier (it neglects the e�ect of angular momentumon RB) and fusion is forbidden at energies below the barrier. It predicts that capture cross
VB
Cross sections – Light systems
38
0
100
200
300
400
500
600
700
0.85 0.9 0.95 1 1.05 1.1 1.15
σ[mb]
Ec.m.[MeV]
18O+
232Th
CC:a0=0.65fm
CCx0.77
18O+
232ThExp.
0
100
200
300
400
500
600
700
0.85 0.9 0.95 1 1.05 1.1 1.15
σ[mb]
Ec.m.[MeV]
30Si+
232Th
CC:a0=0.65fm
CCx0.75
30Si+
232ThExp.
Cross sections – Heavy systems
0
100
200
300
400
500
600
0.85 0.9 0.95 1 1.05 1.1 1.15
σ[mb]
Ec.m.[MeV]
34S+
232Th
CC:a0=0.65fm
CCx0.70
34S+
232ThExp.
0
100
200
300
400
500
600
700
800
0.85 0.9 0.95 1 1.05 1.1 1.15
σ[m
b]
Ec.m.[MeV]
40Ca+
232Th
CC:a0=0.65fm
CCx0.60
40Ca+
232ThExp.
39
Capture suppression
40
★★★★★
Summary § Study E-dissipative process
à Transfer followed by fission increases with Z1Z2
à Correlates with increasing fusion suppression with Z1Z2
Promising signature that E-dissipation suppresses fusion
§ Absolute capture cross section measurements
à Quasi-fission important as Z1Z2 increases
à Simulations done for extrapolation to get σcap
An enhancement of capture suppression by energy dissipation with an increase in Z1Z2
41