Heavy-Ion Fusion Reactions
around the Coulomb Barrier
Kouichi Hagino
Tohoku University, Sendai, Japan
hagino@nucl.phys.tohoku.ac.jp
www.nucl.phys.tohoku.ac.jp/~hagino

cf. Experimental aspects of H.I. Fusion reactions:lectures by Prof. Mahananda Dasgupta (ANU)3.11 earthquake

Sendaiafter 1 monthHeavy-Ion Fusion Reactions
around the Coulomb Barrier
Kouichi Hagino
Tohoku University, Sendai, Japan
hagino@nucl.phys.tohoku.ac.jp
www.nucl.phys.tohoku.ac.jp/~hagino

Fusion reactions and quantum tunnelingBasics of the Coupled-channels method
Concept of Fusion barrier distribution
Quasi-elastic scattering and quantum reflection
cf. Experimental aspects of H.I. Fusion reactions:lectures by Prof. Mahananda Dasgupta (ANU)

Fusion: compound nucleus formation

courtesy: Felipe CantoInter-nucleus potential
Two forces:
1. Coulomb force
Long range,
repulsive
2. Nuclear force
Short range,
attractive

Potential barrier due
to the compensation
between the two(Coulomb barrier)•above barrier•sub-barrier•deep subbarrier

Why subbarrier fusion?
Two obvious reasons:discovering new elements(SHE by cold fusion reactions)

nuclear astrophysics(fusion in stars)

Why subbarrier fusion?
Two obvious reasons:discovering new elements (SHE)nuclear astrophysics (fusion in stars)
Other reasons:reaction mechamism
strong interplay between reaction and structure(channel coupling effects)
cf. high E reactions: much simpler reaction mechanismmany-particle tunnelingcf. alpha decay: fixed energy
tunneling in atomic collision: less variety of intrinsic motions

Basic of nuclear reactions
Shape, interaction, and excitation structures of nuclei
scattering expt.
cf. Experiment by Rutherford (a scatt.)

Notationb

Target nucleus
XaProjectile（beam）Y

X(a,b)Y

208Pb(16O,16O)208Pb
208Pb(16O,16O‟)208Pb
208Pb(17O,16O)209Pb

detector

measures a particle
intensity as afunction of scattering
angles: 16O+208Pb elastic scattering: 16O+208Pb inelastic scattering: 1 neutron transfer reaction

Scattering Amplitude＝(incident wave) + (scattering wave)

Differential cross sectiondW

The number of scattered particle through the solid angle of dW
per unit time:(flux for the scatt. wave)

Scattering Amplitudepartial wave decomposition

Motion of Free particle:

In the presence of a potential:
Asymptotic form of wave function(scattering amplitude)

(note)Total incoming flux

Total outgoing fluxrr

If only elastic scattering:（flux conservation）：phase shiftOptical potential and Absorption cross section
Reaction processes
Elastic scatt.
Inelastic scatt.
Transfer reactionCompound nucleus
formation (fusion)
Optical potential(note) Gauss‟s law

Loss of incident flux
(absorption)

Total incoming flux

Total outgoing fluxrr

Net flux loss:

Absorption crosssection:In the case of three-dimensional spherical potential:(reflection coeff.)

Overview of heavy-ion reactions
Heavy-ion: Nuclei heavier than 4He
Inter-nucleus potential

Two forces:
1. Coulomb force
Long range,
repulsive
2. Nuclear force
Short range,
attractive

Potential barrier due
to the compensation
between these two(Coulomb barrier)Double Folding Potential

cf. Michigan 3 range Yukawa (M3Y)
interaction

(MeV)



Phenomenological potential

Three important features of heavy-ion reactions
1. Coulomb interaction: important
2. Reduced mass: large(semi-) classical picture
concept of trajectory

3. Strong absorption inside the Coul. barrier

rtouch
154Sm

16O

rtouchStrong absorption

Automatic compound
nucleus formation once
touched (assumption ofstrong absorption)

Strong absorption

the region of large overlap
High level density (CN)
Huge number of d.o.f.


Relative energy is quickly lostand converted to internal energy
Formation of hot CN (fusion reaction)：can access to the strong absorption：cannot access cassicallyPartial decomposition of reaction cross section

Taken from J.S. Lilley,“Nuclear Physics”

Classical Model for heavy-ion fusion reactions
Strong absorptionb (impact parameter): can access to the strong absorption
region classicallyClassical fusion cross section is proportional to 1 / E
pRb2

1/Vb

Taken from J.S. Lilley,“Nuclear Physics”

Fusion reaction and Quantum Tunneling154Smrtouch16O

rtouchAutomatic CN formation
once touched (assumptionof strong absorption)

Strong absorption

Fusion takes place by quantum tunnelingat low energies!

Probability of fusion= prob. to access to rtouch
Penetrability of barrier

Quantum Tunneling Phenomena
V(x)
V0

-a

a

x

V(x)
x

Tunnel probability:

For a parabolic barrier……

Vbx

Energy derivativeof penetrability(note) Classical limitPotential Model: its success and failure

Asymptotic boundary condition:
Fusion cross section:
Mean angular mom. of CN:
rabs

Strong absorption

Fusion cross section:
Mean angular mom. of CN:

Wong‟s formula

C.Y. Wong, Phys. Rev. Lett. 31 (‟73)766

i) Approximate the Coul. barrier by a parabola:ii) Approximate Pl by P0:(assume l-independent Rb andcurvature)iii) Replace the sum of l with an integral

(note) For(note)

Comparison between prediction of pot. model with expt. data
Fusion cross sections calculated with a static energy independent potential

16O+27Al
14N+12C

40Ar+144Sm

L.C. Vaz, J.M. Alexander, andG.R. Satchler, Phys. Rep. 69(‟81)373

Works well for relatively light systems
Underpredicts sfus for heavy systems at low energies

Potential model:
Reproduces the datareasonably well for
E > VbUnderpredicts sfus for
E < VbWhat is the origin?
Data: J.R. Leigh et al., PRC52(„95)3151
cf. seminal work:
R.G. Stokstad et al., PRL41(„78)465
PRC21(„80)2427

Inter-nuclear Potential ispoorly parametrized?
Other origins?

With a deeper nuclear potential (but still within a potential model)…..

Potential Inversion(note)

sfus

Vb

V(r)

Semi-classical app.
E

Energy independent
localsingle-ch.


r1

r2

r



sfus

Potential inversion

V(r)Semi-classical app.

Energy independent
localsingle-ch.


64Ni + 64Ni
64Ni + 74Ge

Unphysical potentials

Beautiful demonstrationof C.C. effects
A.B. Balantekin, S.E. Koonin, andJ.W. Negele, PRC28(‟83)1565



Potential inversion

t(E)

e.g.,
t(E) = 3 +/- 0.2 fm

64Ni + 64Ni
64Ni + 74Ge

double valued potential(unphysical !!)Fusion cross sections calculated with a static energy independent potential
Potential model:
Reproduces the datareasonably well for
E > VbUnderpredicts sfus for
E < VbWhat is the origin?
Inter-nuclear Potential ispoorly parametrized?
Other origins?

Target dependence of fusion cross section

Strong target dependence at E < Vb

Low-lying collective excitations in atomic nuclei
Low-lying excited states in even-even nuclei are collective excitations,and strongly reflect the pairing correlation and shell strucuture

Taken from R.F. Casten,“Nuclear Structure from a
Simple Perspective”

Effect of collective excitation on sfus: rotational case
Excitation spectra of 154Sm
0.903
(MeV)

8+

0.544

6+

cf. Rotational energy of a rigid body(Classical mechanics)

154Sm is deformed

0.267

4+

0.082
0

2++
0
154Sm

Effect of collective excitation on sfus: rotational caseComparison of energy scales
Tunneling motion:

3.5 MeV (barrier curvature)

Rotational motion:

The orientation angle of 154Sm does not change much during fusion(note)
Ground state (0+ state) whenreaction starts
Mixing of all orientations
with an equal weight


154Sm

16O

Effect of collective excitation on sfus: rotational case
The orientation angle of 154Sm does not change much during fusion


154Sm

16O


154Sm

16O

The barrier is lowered for =0
because an attraction works from large
distances.

The barrier increases for =p/2.
because the rel. distance has
to get small for the attraction to work

Def. Effect: enhances sfus by a factorof 10 ~ 100
Fusion: interesting probe fornuclear structure

Two effects of channel couplings
energy loss due to inelastic excitations
no Coul-ex

with Coul-exdynamical modification of the
Coulomb barrier
large enhancement offusion cross sections
cf. 2-level model: Dasso, Landowne, and Winther, NPA405(„83)381



One warning:Don‟t use this formula for light
deformed nuclei, e.g., 28Si

4+

4.618
(MeV)

2+

1.779
(MeV)

0+

0
28Si

154Sm

16O

More quantal treatment: Coupled-Channels method
Coupling between rel. andintrinsic motions
0+

0+

0+

Entrancechannel2+0+

Excitedchannel0+}

excitedstatesgroundstate

Schroedinger equation:or

Coupled-channels equations

Coupled-channels equations

equation for yk
boundary condition:transition from fk to fk‟

Angular momentum coupling

Total ang. mom.:

Ip=8+

0.903
(MeV)
0.544

6+

0.267

4+
2+
0+

0.082
0

154Sm

Boundary condition(with ang. mom.
coupling)

0+

0+

0+

Entrancechannel2+0+

Excitedchannel0+Excitation structure of atomic nuclei

Excite the target nucleus via collision with the projectile nucleus
How does the targ. respond to
the interaction with the proj.?

Standard approach: analysis with the coupled-channels method
Inelastic cross sections
Elastic cross sections
Fusion cross sections
S-matrix SnlI

How to perform coupled-channels calculations?
1. Modeling: selection of excited states to be included

low-lyingcollective
states only

S. Raman et al.,
PRC43(„91)521

typical excitation spectrum: electron scattering data
low-lying collective excitations
GDR/GQREGDR ~ 79A-1/3 MeV
EGQR ~ 65A-1/3 MeV

M. Sasao and Y. Torizuka,
PRC15(„77)217
Ex (MeV)

low-lying non-collective excitations
Giant Resonances: high Ex, smooth mass number dependence
adiabatic potential renormalization
Low-lying collective excitations: barrier distributions,strong isotope dependenceNon-collective excitations: either neglected completely or
implicitly treated through an absorptive potential

2. Nature of collective states: vibration? or rotation?

a) Vibrational coupling
excitation operator:

0+,2+,4+2+0+Vibrational excitations
Bethe-Weizacker formula: Mass formula based on Liquid-Drop Model(A, Z)For a deformed shape,
b

a

E

e

In generalE

Harmonic oscillationalm

λ=2: Quadrupole vibration
Movie: Dr. K. Arita (Nagoya Tech. U.)
http://www.phys.nitech.ac.jp/~arita/In generalE

Harmonic oscillationalm

λ=3: Octupole vibration
Movie: Dr. K. Arita (Nagoya Tech. U.)
http://www.phys.nitech.ac.jp/~arita/

Double phonon states
1.282 MeV
4++
1.208 MeV
2+
1.133 MeV
00.558 MeV
2+
Microscopic description
Random phase approximation (RPA)

0+
114Cd

2. Nature of collective states: vibration? or rotation?

a) Vibrational coupling
excitation operator:

0+,2+,4+2+0+

2. Nature of collective states: vibration? or rotation?

b) Rotational coupling
excitation operator:

4+
2+
0+

Vibrational coupling

Rotational coupling

0+,2+,4+

4+

2+
0+

2+
0+

cf. reorientation term

3. Coupling constants and coupling potentials

Deformed Woods-Saxon model:

excitation operator

Coupling Potential: Collective ModelVibrational caseRotational case
Coordinate transformation to the body-fixed rame(for axial symmetry)

In both cases(note) coordinate transformation to the rotating frame ()

Deformed Woods-Saxon model (collective model)
CCFULLK.H., N. Rowley, and A.T. Kruppa,
Comp. Phys. Comm. 123(‟99)143

http://www.nucl.phys.tohoku.ac.jp/~hagino/ccfull.html

CCFULLi) all order couplings

Nuclear coupling:

Coulomb coupling:

K.H., N. Rowley, and A.T. Kruppa,
Comp. Phys. Comm. 123(‟99)143

CCFULLi) all order couplings

K.H., N. Rowley, and A.T. Kruppa,
Comp. Phys. Comm. 123(‟99)143

CCFULL

K.H., N. Rowley, and A.T. Kruppa,
Comp. Phys. Comm. 123(‟99)143i) all order couplings

K.H., N. Takigawa, M. Dasgupta,
D.J. Hinde, and J.R. Leigh, PRC55(„97)276

K.H., N. Rowley, and A.T. Kruppa,
Comp. Phys. Comm. 123(‟99)143

CCFULLii) isocentrifugal approximation2+

l=J-2, J, J+2

0+

l=J

J=l+I

Truncation Dimension
2+
4
2
4+
9
3
6+
16
4
8+
25
5

Iso-centrifugal approximation:
λ: independent of excitationstransform to
the rotating frame“Spin-less system”

16O + 144Sm (2+)

K.H. and N. Rowley, PRC69(‟04)054610

CCFULL

K.H., N. Rowley, and A.T. Kruppa,
Comp. Phys. Comm. 123(‟99)143iii) incoming wave boundary condition (IWBC)(1) Complex potential

rabs(2) IWBClimit of large W (strong absorption)(Incoming Wave Boundary Condition)strongabsorption

Only Real part of Potential
 More efficient at low energies

cf. |Sl| ~ 1 at low E

CCFULL
K.H., N. Rowley, and A.T. Kruppa,
Comp. Phys. Comm. 123(‟99)143

http://www.nucl.phys.tohoku.ac.jp/~hagino/ccfull.html

ccfull.inp
16.,8.,144.,62.
1.2,-1,1.06,0
1.81,0.205,3,1
1.66,0.11,2,0
6.13,0.733,3,1
0,0.,0.3

105.1,1.1,0.75
55.,70.,1.
30,0.05

reaction system(Ap=16, Zp=8, At=144, Zt=62)
rp, Ivibrotp, rt, Ivibrott(inert projectile, and vib. for targ.)

ccfull.inp
16.,8.,144.,62.
1.2,-1,1.06,0

reaction system(Ap=16, Zp=8, At=144, Zt=62)
rp, Ivibrotp, rt, Ivibrott(inert projectile, and vib. for targ.)

1.81,0.205,3,1
1.66,0.11,2,0
6.13,0.733,3,1
0,0.,0.3

105.1,1.1,0.75

If Ivibrott =0: O = Ovib
Ivibrott =1: O = Orot
Ivibrott = -1: O = 0 (inert)

55.,70.,1.
30,0.05

similar for the projectileccfull.inp
16.,8.,144.,62.

(Ap=16, Zp=8, At=144, Zt=62)

1.2,-1,1.06,0(inert projectile, and vib. for targ.)

1.81,0.205,3,1

properties of the targ. excitation

1.66,0.11,2,0
6.13,0.733,3,1
0,0.,0.3

105.1,1.1,0.75
55.,70.,1.
30,0.05

E1st = 1.81 MeV
b = 0.205
l=3Nphonon = 1

1.81

3-

0

0+
144Sm

coupling to 3- vibrational state in
the target with def. parameter b = 0.205

ccfull.inp
16.,8.,144.,62.

(Ap=16, Zp=8, At=144, Zt=62)

1.2,-1,1.06,0(inert projectile, and vib. for targ.)

1.81,0.205,3,1

properties of the targ. excitation

1.66,0.11,2,0(note) if Nphonon = 2: double phonon
excitation

6.13,0.733,3,1

105.1,1.1,0.75
55.,70.,1.

(3-)2

1.81x20,0.,0.3
E1st = 1.81 MeV
b = 0.205
l=3Nphonon = 2

1.81

3-

0

0+
144Sm

30,0.05ccfull.inp
16.,8.,144.,62.

(Ap=16, Zp=8, At=144, Zt=62)

1.2,-1,1.06,0(inert projectile, and vib. for targ.)

1.81,0.205,3,1

properties of the targ. excitation

1.66,0.11,2,0(note) if Ivibrott = 1 (rot. coup.)

6.13,0.733,3,1

the input line would look like:
0.08,0.306,0.05,3 instead of 1.81,0.205,3,1

0,0.,0.3

E2+

b2

b4

105.1,1.1,0.75
55.,70.,1.
30,0.05

3 excitatedstates (Nrot=3)+ g.s.

Nrot
6x7x0.08/6

6+

4x5x0.08/6
0.08
0

4+
2+
0+

ccfull.inp
16.,8.,144.,62.

(Ap=16, Zp=8, At=144, Zt=62)

1.2,-1,1.06,0(inert projectile, and vib. for targ.)

1.81,0.205,3,1

properties of the targ. excitation

1.66,0.11,2,0

same as the previous line, but thesecond mode of excitation in thetarget nucleus (vibrational coupling
only)

6.13,0.733,3,1
0,0.,0.3

105.1,1.1,0.75
55.,70.,1.
30,0.05

Nphonon = 0  no second mode

ccfull.inp
16.,8.,144.,62.

(Ap=16, Zp=8, At=144, Zt=62)

1.2,-1,1.06,0(inert projectile, and vib. for targ.)

1.81,0.205,3,1

properties of the targ. excitation

1.66,0.11,2,1

second mode in the targ.
(note) if Nphonon = 1:
the code will ask you while you
run it whether your coupling
scheme is (a) or (b)(b)

6.13,0.733,3,1
0,0.,0.3

105.1,1.1,0.75
55.,70.,1.
30,0.05

1.81
+1.66

3 - x 2+

32+

1.81
1.66

32+

0+

0

(a)
1.81
1.66

b3

b2

0
144Sm

b3

b2

0+
144Sm

ccfull.inp
16.,8.,144.,62.

(Ap=16, Zp=8, At=144, Zt=62)

1.2,-1,1.06,0(inert projectile, and vib. for targ.)

1.81,0.205,3,1

properties of the targ. excitation

1.66,0.11,2,1

second mode in the targ.

6.13,0.733,3,1

properties of the proj. excitation(similar as the third line)

0,0.,0.3

105.1,1.1,0.75
55.,70.,1.
30,0.05(will be skipped for an inert
projectile)ccfull.inp
16.,8.,144.,62.

(Ap=16, Zp=8, At=144, Zt=62)

1.2,-1,1.06,0(inert projectile, and vib. for targ.)

1.81,0.205,3,1

properties of the targ. excitation

1.66,0.11,2,1

second mode in the targ.

6.13,0.733,3,1

properties of the proj. excitation(similar as the third line)

0,0.,0.3transfer coupling (g.s. to g.s.)105.1,1.1,0.75
55.,70.,1.
30,0.05(Ap + At)Qtr = +3 MeV(Ap‟ + At‟)* no transfer coup. for F = 0

ccfull.inp
16.,8.,144.,62.

(Ap=16, Zp=8, At=144, Zt=62)

1.2,-1,1.06,0(inert projectile, and vib. for targ.)

1.81,0.205,3,1

properties of the targ. excitation

1.66,0.11,2,1

second mode in the targ.

6.13,0.733,3,1

properties of the proj. excitation(similar as the third line)

0,0.,0.3transfer coupling (g.s. to g.s.)105.1,1.1,0.75

potential parameters

55.,70.,1.
30,0.05

V0 = 105.1 MeV, a = 0.75 fm
R0 = 1.1 * (Ap1/3 + At1/3) fm

ccfull.inp
16.,8.,144.,62.

(Ap=16, Zp=8, At=144, Zt=62)

1.2,-1,1.06,0(inert projectile, and vib. for targ.)

1.81,0.205,3,1

properties of the targ. excitation

1.66,0.11,2,1

second mode in the targ.

6.13,0.733,3,1

properties of the proj. excitation(similar as the third line)

0,0.,0.3transfer coupling (g.s. to g.s.)105.1,1.1,0.75

potential parameters

55.,70.,1.

Emin, Emax, DE (c.m. energies)

30,0.05

Rmax, Dr

ccfull.inp

OUTPUT

16.,8.,144.,62.
1.2,-1,1.06,0
1.81,0.205,3,1
1.66,0.11,2,1
6.13,0.733,3,1
0,0.,0.3
105.1,1.1,0.75
55.,70.,1.
30,0.05

16O + 144Sm Fusion reaction------------------------------------------------Phonon Excitation in the targ.: beta_N= 0.205, beta_C= 0.205,
r0= 1.06(fm), omega= 1.81(MeV), Lambda= 3, Nph= 1------------------------------------------------Potential parameters: V0= 105.10(MeV), r0= 1.10(fm),a= 0.75(fm),power= 1.00
Uncoupled barrier: Rb=10.82(fm), Vb= 61.25(MeV),
Curv=4.25(MeV)------------------------------------------------Ecm (MeV) sigma (mb)<l>------------------------------------55.00000 0.97449E-02 5.87031
56.00000
0.05489 5.94333
57.00000
0.28583 6.05134
58.00000
1.36500 6.19272
59.00000
5.84375 6.40451…………………
69.00000
70.00000427.60179 17.16365
472.46037 18.08247

In addition, “cross.dat” : fusion cross sections only

Coupled-channels equations and barrier distribution

Calculate sfus by numerically solving the coupled-channels equations
Let us consider a limiting case in order to understand(interpret) the numerical results
enI: very large
 enI = 0Adiabatic limit
Sudden limitComparison of two time scales
spring on a boardkDl

mg sin 
mg cos mg

static case: mg sin  = kDl  Dl = mg sin  / kmove very slowly? or move instantaneously?

Comparison of two time scales
similar related example: spring on a moving board

move very slowly? or move instantaneously?
keep the original length (Dl =0) “sudden limit”
always at the equilibrium length (Dl = mg sin  / k ) “adiabatic limit”

fast reaction

large fluctuation

slow reaction+ small fluctuation
around the adiabatic path

relative distanceTwo limiting cases: (i) adiabatic limit

much slower rel. motion than the intrinsic motion

much larger energy scale for intrinsic motion than
the typical energy scale for the rel. motion（Barrier curvature v.s. Intrinsic excitation energy）

c.f. Born-Oppenheimer approximation for hydrogen molecule
er
p

p
R

1. Consider first the electron motion for a fixed R

2. Minimize en(R) with respect to R
Or 2‟. Consider the proton motion in a potential en(R)Adiabatic Potential Renormalization

16O(3-) + 144Sm(3-)

When e is large,

where

Fast intrinsic motion
Adiabatic potential renormalization
Giant Resonances, 16O(3-) [6.31 MeV]

K.H., N. Takigawa, M. Dasgupta,
D.J. Hinde, J.R. Leigh, PRL79(‟99)2014

typical excitation spectrum: electron scattering data
low-lying collective excitations
GDR/GQREGDR ~ 79A-1/3 MeV
EGQR ~ 65A-1/3 MeV

M. Sasao and Y. Torizuka,
PRC15(„77)217
Ex (MeV)

low-lying non-collective excitations
Giant Resonances: high Ex, smooth mass number dependence
adiabatic potential renormalization
Low-lying collective excitations: barrier distributions,strong isotope dependenceNon-collective excitations: either neglected completely or
implicitly treated through an absorptive potential

Two limiting cases: (ii) sudden limit


154Sm

16O

Coupled-channels:
diagonalize

Slow intrinsic motion
Barrier Distribution

Barrier distribution
w

P0

B

B2

B3

B1

B2

B3

E

E

B

B1

E

Barrier distribution: understand the concept using a spin Hamiltonian

Hamiltonian (example 1)：

For Spin-up

For Spin-downxx

Wave function(general form)

The spin direction does not change during tunneling:Tunneling prob. is a weighted sum of tunnel prob. for two barriersTunnel prob. is enhanced at E < Vb and hindered E > Vb
dP/dE splits to two peaks“barrier distribution”
The peak positions of dP/dE correspond to each barrier heightThe height of each peak is proportional to the weight factor

simple 2-level model (Dasso, Landowne, and Winther, NPA405(„83)381)
entrance channel

excited channelsimple 2-level model (Dasso, Landowne, and Winther, NPA405(„83)381)

Sub-barrier Fusion and Barrier distribution method(Fusion barrier distribution)
N. Rowley, G.R. Satchler,
P.H. Stelson, PLB254(‟91)25

N. Rowley, G.R. Satchler,
P.H. Stelson, PLB254(‟91)25

centered on E= Vb

Barrier distribution measurements
Fusion barrier distribution
Needs high precision data in order for the 2nd derivative to be meaningful(early 90‟s)

Experimental Barrier Distribution
Requires high precision data

T
154Sm

16O

M. Dasgupta et al.,Annu. Rev. Nucl. Part. Sci. 48(‟98)401

Investigate nuclear shape through barrier distribution

Nuclear shapes

By taking the barrier distribution, one can very clearly see
the difference due to b4!
Fusion as a quantum tunneling microscope for nuclei

Advantage of fusion barrier distribution
Fusion Cross sections

Very strong exponential energy
dependence
Difficult to see differences due
to details of nuclear structure
Plot cross sections in a different way: Fusion barrier distribution
N. Rowley, G.R. Satchler,
P.H. Stelson, PLB254(‟91)25

Function which is sensitive to details of nuclear structure

Example for spherical vibrational system
16O + 144Sm

Anharmonicity of octupole
vibration

3-

1.8

0+
144Sm

Quadrupole moment:

K.Hagino, N. Takigawa, and S. Kuyucak,
PRL79(‟97)2943

Barrier distribution
16O + 144Sm

3-

1.8 MeV

0+
144Sm

K.Hagino, N. Takigawa, and S. Kuyucak,
PRL79(‟97)2943

Coupling to excited states

distribution of potential barrier

multi-dimensional potential surface
single barriera collection ofmany barriers

relative distancerx(intrinsic coordinate)Representations of fusion cross sectionsi) sfus vs 1/E (~70‟s)
Classical fusion cross section is proportional to 1 / E :

pRb2

1/Vb
Taken from J.S. Lilley,“Nuclear Physics”ii) barrier distribution (~90‟s)

M. Dasgupta et al.,Annu. Rev. Nucl. Part. Sci. 48(‟98)401

iii) logarithmic derivative (~00‟s)cf.

R. Vandenbosch,Ann. Rev. Nucl. Part. Sci. 42(„92)447

M. Dasgupta et al., PRL99(„07) 192701

deep subbarrier hindrance of fusion cross sections

C.L. Jiang et al., PRL89(„02)052701; PRL93(„04)012701

Systematics of the touching point energy and deep subbarrier hindrance

mechanism of deep subbarrier
hindrance:
not yet been fully understoodT. Ichikawa, K.H., A. Iwamoto,
PRC75(‟07) 064612 & 057603

how to model the dynamics after
touching?

Quantum reflection and quasi-elastic scatteringVbx

In quantum mechanics, reflection occurs even at E > Vb
Quantum Reflection
Reflection prob. carries the same information as penetrability, andbarrier distribution can be defined in terms of reflection prob.

Quasi-Elastic Scattering

FusionA sum of all the reaction processes
other than fusion (elastic + inelastic+ transfer + ……)

Quasi-elastic

Detect all the particles whichreflect at the barrier and hit thedetector
In case of a def. target……

Related to reflection
Complementary to fusion

T
154Sm

16O

Subbarrier enhancement offusion cross sections

Quasi-elastic scattering(elastic + inelastic)

16O + 154Sm

16O + 154Sm


154Sm

16O

Quasi-elastic barrier distribution

T
154Sm

16O

Quasi-elastic barrier distribution:
H. Timmers et al.,
NPA584(‟95)190(note) Classical elastic cross section in the limit of strong Coulomb field:

Quasi-elastic test function
Classical elastic cross section (in the limit of a strong Coulomb):

Nuclear effects

Semi-classical perturbation theory

S. Landowne and H.H. Wolter, NPA351(‟81)171
K.H. and N. Rowley, PRC69(‟04)054610

Quasi-elastic test function

The peak position slightly deviatesfrom Vb
Low energy tail
 Integral over E: unity Relatively narrow width
Close analog to fusion b.d.

Scaling property
Expt.: impossible to performatRelation among different

Effective energy:

16O + 144Sm?

16O + 144Sm

Comparison of Dfus with Dqel

FusionQuasi-elastic

H. Timmers et al., NPA584(‟95)190

A gross feature is similar to each other
K.H. and N. Rowley, PRC69(‟04)054610

Experimental barrier distribution with QEL scattering
H. Timmers et al.,
NPA584(‟95)190
70Zn + 208Pb

70Zn : E = 0.885 MeV,2

2 phonon, 208Pb: E3 = 2.614 MeV, 3 phonon

Muhammad Zamrun F., K. H., S. Mitsuoka, and H. Ikezoe, PRC77(‟08)034604.

Experimental Data: S. Mitsuoka et al., PRL99(‟07)182701

Experimental advantages for Dqel・ less accuracy is required in the data (1st vs. 2nd derivative)・ much easier to be measured
Qel： a sum of everythinga very simple charged-particle detector
Fusion： requires a specialized recoil separator
to separate ER from the incident beam
ER + fission for heavy systems

・several effective energies can be measured at a single-beamenergyrelation between a scattering angle and an impact
parameterDeep subbarrier fusion and diffuseness anomaly

Scattering processes:
Double folding pot.
Woods-Saxon (a ~ 0.63 fm)
successful

A. Mukherjee, D.J. Hinde, M. Dasgupta, K.H., et al.,
PRC75(‟07)044608

Fusion process: not successful
a ~ 1.0 fm required (if WS)

scattering

fusionHow reliable is the DFM/WS?
What is an optimum potential?


deduction of fusion barrier from exp. data?
(model independent analysis?)

Quasi-elastic scattering at deep subbarrier energies?
K.H., T. Takehi, A.B. Balantekin,and N. Takigawa, PRC71(‟05) 044612
K. Washiyama, K.H., M. Dasgupta,
PRC73(‟06) 034607

QEL at deep subbarrierenergies: sensitive only
to the surface region
16O + 154Sm

Summary
Heavy-Ion Fusion Reactions around the Coulomb
BarrierFusion and quantum tunneling
Fusion takes place by tunnelingBasics of the Coupled-channels method
Collective excitations during fusion

Concept of Fusion barrier distribution
Sensitive to nuclear structure

Quasi-elastic scattering and quantum reflection
Complementary to fusion

Computer program: CCFULL
http://www.nucl.phys.tohoku.ac.jp/~hagino/ccfull.html

References
Nuclear Reaction in general







G.R. Satchler, “Direct Nuclear Reactions”
G.R. Satchler, “Introduction to Nuclear Reaction”
R.A. Broglia and A. Winther, “Heavy-Ion Reactions”“Treatise on Heavy-Ion Science”, vol. 1-7
D.M. Brink, “Semi-classical method in nucleus-nucleus collisions”
P. Frobrich and R. Lipperheide, “Theory of Nuclear Reactions”

Heavy-ion Fusion Reactions






M. Dasgupta et al., Annu. Rev. Nucl. Part. Sci. 48(‟98) 401
A.B. Balantekin and N. Takigawa, Rev. Mod. Phys. 70(‟98) 77
Proc. of Fusion03, Prog. Theo. Phys. Suppl. 154(‟04)
Proc. of Fusion97, J. of Phys. G 23 (‟97)
Proc. of Fusion06, AIP, in press.

Hamiltonian (example 3): more general cases

x dependent

E dependent
K.H., N. Takigawa, A.B. Balantekin, PRC56(‟97)2104(note) Adiabatic limit: