Quantum Many-body Dynamics
in low-energy heavy-ion reactions
Kouichi Hagino
Tohoku University, Sendai, Japan
hagino@nucl.phys.tohoku.ac.jp
www.nucl.phys.tohoku.ac.jp/~hagino

Contents of this lecture
1. Basics of nuclear reaction: basic concepts andbrief review of quantum mechanics
2. Overview of heavy-ion collisions
3. Classical model for heavy-ion fusion
4. Heavy-ion fusion and Quantum Tunneling
5. Potential model: its success and failure
6. Low-lying collective motions in atomic nuclei
7. Influences to reaction: Coupled-channels method
8. Sub-barrier Fusion and Barrier distribution
9. Quantum reflection and Quasi-elastic scatteringReferences
Nuclear Reaction in general
z
z
z
z
z
z

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
z
z
z
z
z

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.

Basic of nuclear reactions
Shape, interaction, and excitation structures of nuclei
scattering expt.
cf. Experiment by Rutherford (α 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 cross sections

Incident beams
The number of reaction per unit
time and per one target nucleusSσ x the number of projectile nucleus＝ passed through a unit area per unit timeσ/S = probability for scattering to occur when a projectile nucleus in the beamcollides with a target nucleus

Units: 1 barn = 10-24 cm2 = 100 fm2 (1 mb = 10-3 b = 0.1 fm2)
Differential scattering cross sections:dΩScattering Amplitude
Motion of Free particle:

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

＝(incident wave)+ (scattering wave)（flux conservation）

If only elastic scattering:：phase shiftDifferential cross sectiondΩThe number of scattered particle through the solid angle of dΩper unit time:(flux for the scatt. wave)

Optical potential and Absorption cross section
Reaction processes¾Elastic scatt.
¾Inelastic scatt.
¾Transfer reaction¾Compound 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: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 the two(Coulomb barrier)z

Double Folding Potential: nucleon-nucleon interactionz

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)

Grazing angular momentum
Strong absorption

Grazing angular momentum:: can access to the strong absorption region classically: the strong absorption region is classically forbidden
Reaction is drastically changed at l=lg for a given E

i)orClassical turning point: very far away

Strong absorption

Elastic scatt. due to Coul. force(Rutherford scatt. )zExcitation due to the Coul. field(Coulomb excitation)z

Low-lying collective motion：can access to the strong absorption：cannot access cassicallyii)
Nuclear effect: important
Elastic scatt.
zInelastic scatt.
directreaction zTransfer reactionz

Quasi-elastic scattering

Dynamics as a many-body system

iii)

Strong absorption

Access to the region of large overlap
High level density (CN)zHuge number of d.o.f.
zRelative energy is quickly lostand converted to internal energy：can access to the strong absorption：cannot access cassically

Formation of hot CN (fusion reaction)iv) In the case of
Coul. Pocket: disappears at l = lg
Reaction intermediate between
Direct reaction and fusion:
Deep Inelastic Collisions (DIC)
Scattering at relatively high energy a/o for heavy systems

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
πRb2

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In the case of three-dimensional spherical potential:(reflection coeff.)Potential Model: its success and failure

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

Strong absorption

Asymptotic boundary condition:
Fusion cross section:
Inner boundary condition:(i) Method of Absorbing potential
rabs(ii) IWBC
Large W limit

Strong absorption(Incoming Wave Boundary Condition)z

Requires only Real Potential (no need for imaginary part)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)373¾Works well for relatively light systems¾Underpredicts σfus for heavy systems at low energies

Potential model:
Reproduces the datareasonably well for
E > VbUnderpredicts σfus for
E < VbWhat is the origin?
Inter-nuclear Potential ispoorly parametrized?
Other origins?

Potential Inversion(note)

Vb

E

r1

r2

r

A.B. Balantekin, S.E. Koonin, andJ.W. Negele, PRC28(’83)1565

Fusion cross sections calculated with a static energy independent potential
Potential model:
Reproduces the datareasonably well for
E > VbUnderpredicts σfus 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”

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

E

ε

In generalE

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

Harmonic oscillation

αλμλ=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

Random Phase Approximation
Hartree-Fock state

1 particle-1 hole (1p1h) state(coherent superposition of 1p1h states)

Rotational excitationz

Shell energy
zSmooth partzFluctuation part

Liquid drop model:

Taken from Bohr-Mottelson

Deformed energy surface for a given nucleus

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

β
LDM only
Shell correction

always spherical ground state
may lead to a deformed g.s.
* Spontaneous Symmetry Breaking

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
0.082
0

154Sm

4+ (note) What is 0+ state (Quantum Mechanics)?
0+: no preference of direction (spherical)+
2+0Mixing of all orientations with anequal probability

c.f. HF + Angular Momentum Projection

Evidences for nuclear deformation
zThe existence of rotational bands

zVery large quadrupole moments(for odd-A nuclei)

0.903
(MeV)

8+

0.544

6+

0.267

4+

0.082
0

2++
0
154Sm

zStrongly enhanced quadrupole transition probabilities
zHexadecapole matrix elements
zSingle-particle structure

β4

Fission fromisomer state

zFission isomers

Fission from g.s.

Effect of collective excitation on σfus: 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

θ
154Sm

16O

θ
154Sm

16O

The barrier is lowered for θ=0
because an attraction works from
Def. Effect: enhances σfus by a factor
large distances.
of 10 ~ 100
The opposite for θ=π/2. The barrier
is highered as the rel. distance has
Fusion: interesting probe forto get small for the attraction to work
nuclear structure

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

0+

0+

Entrancechannel2+0+

Excitedchannel0+}

excitedstatesgroundstate

Schroedinger equation:or

Coupled-channels equations

Angular momentum coupling

Total ang. mom.:

0.903
(MeV)

Iπ=8+

0.544

6+

0.267

4+
2+
0+

0.082
0

154Sm

Boundary condition

0+

0+

0+

Entrancechannel2+0+

Excitedchannel0+(note) Dynamical Polarization Potential

Entrancechannel[P space]

Other (excited)channels[Q space]

Total Hilbert space considered (conceptual figure)“Eliminate” the Q-space and project on to the P-space
Effective potential for the P-space (entrance channel) :Dynamical Polarization Potential
Energy dependent, non-local, complex potential
Optical potential VoptExample: 2 channel problem

or

(2)

Example: 2 channel problem (cont’d)(regular solution)(outgoing solution)(Wronskian)For a more general derivation: Feshbach formalism (see References)Summary of Coupled-channels method

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 method¾Inelastic cross sections¾Elastic cross sections¾Fusion cross sections
S-matrix SnlI

Coupling Potential: Collective Model(note) coordinate transformation
to the rotating frame ()¾Vibrational case

¾Rotational case
Coordinate transformation to the body-fixed rame(for axial symmetry)

In both cases

Deformed Woods-Saxon model:

Deformed Woods-Saxon model (collective model)
K.H., N. Rowley, and A.T. Kruppa,
Comp. Phys. Comm. 123(’99)143

Nuclear coupling:

Coulomb coupling:

Rotational coupling:
Vibrational coupling:Vibrational coupling

Rotational coupling

0+,2+,4+

4+

2+
0+

2+
0+

Iso-centrifugal approximation

No-Coriolis approximationzRotating frame approximationz2+

l=J-2, J, J+2

0+

l=J

J=l+I

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

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

Coupled-channels equations: two limiting cases

Calculate σfus by numerically solving the coupled-channels equations
Let us consider two limiting cases in order to understand(interpret) the numerical results
εnI: very large
z εnI = 0z

Adiabatic limit
Sudden limitTwo limiting cases: (i) adiabatic limit

In case where the rel. motion is much slower than the intrinsic motion

In case where the energy scale for intrinsic motion is much larger 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 εn(R) with respect to R
Or 2’. Consider the proton motion in a potential εn(R)Adiabatic Potential Renormalization

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

When ε 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

Two limiting cases: (ii) sudden limit

θ
154Sm

16O

Coupled-channels:
diagonalize

Slow intrinsic motion
Barrier Distribution

Barrier distribution
w

P0

B

B2

B3

B1

B2

B3 E

E

E
B

B1

Barrier distribution: understand the concept using a spin Hamiltonian
Hamiltonian (example 1)：

For Spin-up

For Spin-downxx

Wave function(general form)
Asymptotic form at(the C1 and C2 are fixedaccording to the spin stateof the system)

Tunnel probability ＝(flux at(incoming flux at))Tunneling prob. is a weighted sum of tunnel prob. for two barriers

¾Tunnel prob. is enhanced at E < Vb and hindered E > Vb
¾dP/dE splits to two peaks“barrier distribution”¾The peak positions of dP/dE correspond to each barrier height¾The height of each peak is proportional to the weight factor

Hamiltonian (example 2): in the case with off-diagonal components

If spin-up at the beginning of the reaction

Hamiltonian (example 3): more general cases

x dependent

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

(cf.) Solving the C.C. equations with WKB approximation
WKB formula for a one dimensional barrier penetrability:

Generalization to a coupledchannels problem
K.H., A.B. Balantekin, Phys. Rev. A70 (’04) 032106

01 channel

3 channel

Sub-barrier Fusion and Barrier distribution method¾Fusion takes place by quantum tunneling at low energies¾C.C. effect can be understood in terms of distribution of many barriers¾σfus is given as an average over the many distributed barriers

Tunneling of a spin system

The way how the barrier isdistributed can be clearly seen
by taking the energy derivativeof penetrability

Can one not do a similar thing with fusion cross sections?

One important fact: experimental observable is not penetrability, but
fusion cross section(Fusion barrier distribution)
N. Rowley, G.R. Satchler,
P.H. Stelson, PLB254(’91)25(note) Classical fusion cross section

Fusion Test Function

Classical fusion cross section:

Tunneling effectsmears the delta function
Fusion test function:¾ Symmetric around E=Vb
¾ Centered on E=Vb
¾ Its integral over E is
¾ Has a relatively narrow width

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

θT
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 β4!
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

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

θT
154Sm

16O

Quasi-elastic barrier distribution

θT
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 function¾The peak position slightly deviatesfrom Vb¾Low energy tailz Integral over E: unityz Relatively narrow width
Close analog to fusion b.d.

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

Comment on the iso-centrifugal approximation
Inelastic excitations: Coupled-channels method
Inherent problem
dimensionality2+

l=J-2, J, J+2

0+

l=J

J=l+I

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

Iso-centrifugal approximation:
λ: independent of excitations¾Works well for fusion¾Not successful for scattering

16O + 144Sm (2+)

16O + 144Sm (2+)

Backward scattering
Small ang. mom.
Iso-centrifugal approximation
works good enough¾Simplifies C.C. calculations¾Ensures the similarities between
Dfus and Dqel in CC systems

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
measurements with a cyclotron accelerator: possible

Suitable for low intensity exotic beams
Qel: will open up a possibility to study the structure of unstable nuclei

Scaling property of Dqel
Expt.: impossible to performat θ = πRelation among different θ?

Effective energy:

16O + 144Sm

16O + 144Sm

Future experiments with radioactive beams
Fusion barrier distribution: requires high precision measurements for σfus
Radioactive beams: much lower beam intensity
than beams of stable nuclei
Unlikely for high precision data at this moment

Possible to extract barrier distribution in other ways?
Exploit reflection prob.
instead of penetrability
P+R=1

FusionQuasi-elastic

Quasi-elastic scatteringDqel measurements with radioactive beams
Low intensity radioactive beams:(0+,2+,4+) (0+,3-)High precision fusion measurements
still difficult
Quasi-elastic measurements
may be possible
Example: 32Mg + 208Pb
32Mg: breaking of the N=20 shell?

Expt. at RIKEN and GANIL:
large B(E2) and small E2+
deformation?
MF calculations
spherical?
E4+/E2+ = 2.62
Investigation of collective
K.H. and N. Rowley, PRC69(’04)054610 excitations unique to n-rich nuclei

16C
16C

16C

N. Imai et al., PRL92(’04)062501

Expt. at RIKEN:
E2+ = 1.766 MeV
B(E2) = 0.26 0.05W.u.
Mn/Mp=7.6 1.7

Recent expt: very small B(E2)¾Different (static) deformation
between n and p?
¾Neutron excitation of
spherical nuclei?

Reference cross sections

Does break-up hinder/enhancefusion cross sections?

How to choose reference cross sections?
Fusion enhancement/hindrance compared to what?
i) Comparison to tightly-bound systems
11Be + 209Bi

10Be + 209Bi

6He + 238U

4He + 238U

Separation between static and dynamical
effects?
ii) Measurement of average fusion barrier
Fusion barrier distribution
9Be + 208Pb
6,7Li + 209Bi
Neutron-rich nuclei

Dqel(E)

R. Raabe et al.
Nature(’04)

M. Dasgupta et al.
PRL82(’99)1395

Surface diffuseness problem

VN(r) = -V0/[1+exp((r-R0)/a)]
Scattering processes: a ~ 0.63 fm
Fusion: a = 0.75 ~ 1.5 fm

VC.b.

C.L. Jiang et al., PRL93(’04)012701

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

Application to SHE
Synthesis of superheavy elements: extremely small cross sections

Important to choose the optimum incident energy
Absence of the barrier height systematics

Determine the fusion barrier height for SHE using Dqel

Future plan at JAERI
Cold fusion reactions: 50Ti,54Cr,58Fe,64Ni,70Zn+208Pb,209Bi

Preliminary data

S. Mitsuoka, H. Ikezoe, K. Nishio, K. Tsuruta,
S.C. Heong, Y.X. Watanabe (’05)

Present data

Evaporation residue cross section by GSI and RIKEN
GSI
VBass

RIKEN

GSI

GSI

RIKEN

GSI

RIKEN

RIKEN

Comparison

Summary
Heavy-Ion Fusion Reactions around the Coulomb Barrier

Fusion and quantum tunneling
Fusion takes place by quantum 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