Quantum%20Many-body%20Dynamics

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Quantum Many-body Dynamics in low-energy
heavy-ion reactions
Kouichi Hagino Tohoku University,
Sendai, Japan
hagino_at_nucl.phys.tohoku.ac.jp
www.nucl.phys.tohoku.ac.jp/hagino
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Heavy-Ion Fusion Reactions around the Coulomb
Barrier
Key Points

• Fusion and quantum tunneling
• Basics of the Coupled-channels method
• Concept of Fusion barrier distribution
• Quasi-elastic scattering and quantum reflection

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Basic of nuclear reactions
Shape, interaction, and excitation structures of
nuclei scattering expt. cf.
Experiment by Rutherford (a scatt.)
Notation
b
Target nucleus
detector
X
Y
a
measures a particle intensity as a function of
scattering angles
Projectile (beam)
X(a,b)Y
208Pb(16O,16O)208Pb 16O208Pb elastic
scattering 208Pb(16O,16O)208Pb
16O208Pb inelastic scattering 208Pb(17O,16O)209Pb
1 neutron transfer reaction
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Scattering Amplitude
Motion of Free particle
In the presence of a potential
Asymptotic form of wave function
(scattering amplitude)
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(incident wave) (scattering wave)
(flux conservation)
If only elastic scattering
phase shift
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Differential cross section
dW
The number of scattered particle through the
solid angle of dW per unit time
(flux for the scatt. wave)
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Optical potential and Absorption cross section
Reaction processes

• Elastic scatt.
• Inelastic scatt.
• Transfer reaction
• Compound nucleus
• formation (fusion)

Loss of incident flux (absorption)
Optical potential
(note) Gausss law
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Total incoming flux
Total outgoing flux
r
r
Net flux loss
Absorption cross section
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Overview of heavy-ion reactions
Heavy-ion Nuclei heavier than 4He
Two forces 1. Coulomb force Long range,
repulsive 2. Nuclear force Short range,
attractive
Inter-nucleus potential
Potential barrier due to the compensation between
the two (Coulomb barrier)
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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
rtouch
Automatic Compound nucleus formation once
touched (assumption of strong absorption)
Strong absorption
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Strong absorption
Access to the region of large overlap

• High level density (CN)
• Huge number of d.o.f.

Relative energy is quickly lost and converted to
internal energy
can access to the strong absorption cannot
access cassically
Formation of hot CN (fusion reaction)
(note) 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
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Fusion reaction and Quantum Tunneling
154Sm
16O
rtouch
rtouch
Automatic CN formation once touched
(assumption of strong absorption)
Probability of fusion prob. to access to
rtouch
Strong absorption
Penetrability of barrier
Fusion takes place by quantum tunneling at low
energies!
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Quantum Tunneling Phenomena
V(x)
V0
x
a
-a
Tunnel probability
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For a parabolic barrier
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Energy derivative of penetrability
(note) Classical limit
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Potential Model its success and failure
Asymptotic boundary condition
Fusion cross section
Mean angular mom. of CN
rabs
Strong absorption
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Wongs 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 and curvature)
iii) Replace the sum of l with an integral
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(note)
For
(note)
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Comparison between prediction of pot. model with
expt. data
Fusion cross sections calculated with a static
energy independent potential
16O27Al
40Ar144Sm
14N12C
L.C. Vaz, J.M. Alexander, and G.R. Satchler,
Phys. Rep. 69(81)373

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

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Potential model
Reproduces the data reasonably well for E gt
Vb Underpredicts sfus for E lt Vb
What is the origin?
Inter-nuclear Potential is poorly
parametrized? Other origins?
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Potential Inversion
(note)
Vb
E
r1
r
r2
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A.B. Balantekin, S.E. Koonin, and J.W. Negele,
PRC28(83)1565
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Fusion cross sections calculated with a static
energy independent potential
Potential model
Reproduces the data reasonably well for E gt
Vb Underpredicts sfus for E lt Vb
What is the origin?
Inter-nuclear Potential is poorly
parametrized? Other origins?
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Target dependence of fusion cross section
Strong target dependence at E lt Vb
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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
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Effect of collective excitation on sfus
rotational case
Comparison 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) when reaction
starts
16O
154Sm
Mixing of all orientations with an equal weight
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154Sm
16O
The barrier is lowered for q0 because an
attraction works from large distances.
Def. Effect enhances sfus by a factor
of 10 100
The opposite for qp/2. The barrier is highered
as the rel. distance has to get small for the
attraction to work
Fusion interesting probe for
nuclear structure
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More quantal treatment Coupled-Channels method
Coupling between rel. and intrinsic motions
0
Entrance channel
0
0
0
Excited channel
2
0
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Schroedinger equation
or
Coupled-channels equations
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Boundary condition
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Coupling Potential Collective Model
(note) coordinate transformation to the
rotating frame ( )

• Vibrational case

• Rotational case

Coordinate transformation to the body-fixed rame
(for axial symmetry)
In both cases
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Deformed Woods-Saxon model
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Deformed Woods-Saxon model (collective model)
CCFULL
K.H., N. Rowley, and A.T. Kruppa, Comp. Phys.
Comm. 123(99)143
Nuclear coupling
Coulomb coupling
Rotational coupling
Vibrational coupling
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Vibrational coupling
Rotational coupling
4
2
0
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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
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C.C. in the sudden limit
Coupled-channels
diagonalize
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Barrier distribution
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Barrier distribution understand the concept
using a spin Hamiltonian
Hamiltonian (example 1)
For Spin-up
For Spin-down
x
x
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Wave function (general form)
Asymptotic form at
(the C1 and C2 are fixed according to the spin
state of the system)
(flux at )
Tunnel probability
(incoming flux at )
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Tunneling prob. is a weighted sum of tunnel prob.
for two barriers
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• Tunnel prob. is enhanced at E lt Vb and hindered E
  gt 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

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Hamiltonian (example 2) in case with
off-diagonal components
If spin-up at the beginning of the reaction
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Hamiltonian (example 3) more general cases
x dependent
E dependent
K.H., N. Takigawa, A.B. Balantekin, PRC56(97)2104
(note) Adiabatic limit
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Sub-barrier Fusion and Barrier distribution method

• Fusion takes place by quantum tunneling at low
  energies
• C.C. effect can be understood in terms of
  distribution of many barriers
• sfus is given as an average over the many
  distributed barriers

Tunneling of a spin system
The way how the barrier is distributed can be
clearly seen by taking the energy derivative of
penetrability
Can one not do a similar thing with fusion cross
sections?
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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
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Classical fusion cross section
Fusion Test Function
Tunneling effect smears
the delta function

• Fusion test function
• Symmetric around EVb
• Centered on EVb
• Its integral over E is
• Has a relatively narrow width

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Barrier distribution measurements
Fusion barrier distribution
Needs high precision data in order for the 2nd
derivative to be meaningful
(early 90s)
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Experimental Barrier Distribution
Requires high precision data
M. Dasgupta et al., Annu. Rev. Nucl. Part. Sci.
48(98)401
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Investigate nuclear shape through barrier
distribution
Nuclear shapes
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By taking the barrier distribution, one can very
clearly see the difference due to b4!
Fusion as a quantum tunneling microscope for
nuclei
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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
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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)2
943
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Quantum reflection and quasi-elastic scattering
In quantum mechanics, reflection occurs even at E
gt Vb
Quantum
Reflection
Reflection prob. carries the same information as
penetrability, and barrier distribution can be
defined in terms of reflection prob.
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Quasi-Elastic Scattering
Fusion
A sum of all the reaction processes other than
fusion (elastic inelastic transfer )
Quasi-elastic
Detect all the particles which reflect at the
barrier and hit the detector
Related to reflection
In case of a def. target
Complementary to fusion
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Quasi-elastic barrier distribution
Quasi-elastic barrier distribution
H. Timmers et al., NPA584(95)190
(note)
Classical elastic cross section in the limit of
strong Coulomb field
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Quasi-elastic Test Function
Classical elastic cross section (in the limit of
a strong Coulomb)
S. Landowne and H.H. Wolter, NPA351(81)171 K.H.
and N. Rowley, PRC69(04)054610
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Quasi-elastic test function
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Fusion Quasi-elastic
Comparison of Dfus with Dqel
H. Timmers et al., NPA584(95)190
A gross feature is similar to each other
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Experimental advantages for Dqel
less accuracy is required in the data (1st vs.
2nd derivative) much easier to be measured
Qel a sum of everything
a 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-beam energy
relation between a scattering angle and an impact
parameter
measurements with a cyclotron accelerator
possible
Qel will open up a possibility to study the
structure of unstable nuclei
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16O 144Sm
Expt. impossible to perform at q p
Relation among different q?
Effective energy
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16O 144Sm
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Future experiments with radioactive beams
Fusion barrier distribution requires high
precision measurements for sfus
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
Quasi-elastic scattering
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Dqel measurements with radioactive beams
Low intensity radioactive beams
High precision fusion measurements still
difficult Quasi-elastic measurements may be
possible
(0,2,4)
(0,3-)
Example 32Mg 208Pb
E4/E2 2.62
Investigation of collective excitations unique
to n-rich nuclei
K.H. and N. Rowley, PRC69(04)054610
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16C
Recent expt very small B(E2)
16C

• Different (static) deformation
• between n and p?
• Neutron excitation of
• spherical nuclei?

16C
N. Imai et al., PRL92(04)062501
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Does break-up hinder/enhance fusion cross
sections?
Reference 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?
R. Raabe et al. Nature(04)
ii) Measurement of average fusion barrier
Fusion barrier distribution 9Be
208Pb 6,7Li 209Bi
M. Dasgupta et al. PRL82(99)1395
Neutron-rich nuclei
Dqel(E)
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Surface diffuseness problem
VN(r) -V0/1exp((r-R0)/a)
Scattering processes a 0.63 fm Fusion a
0.75 1.5 fm
C.L. Jiang et al., PRL93(04)012701
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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
16O 154Sm
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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,70Zn20
8Pb,209Bi
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Preliminary data
S. Mitsuoka, H. Ikezoe, K. Nishio, K.
Tsuruta, S.C. Heong, Y.X. Watanabe (05)
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Comparison
Present data
Evaporation residue cross section by GSI and RIKEN
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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.