Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Multi-Nucleon Transfer in Heavy-Ion Reactions
a Semiclassical Approach
Giovanni POLLAROLO
Dipartimento di Fisica Teorica, Università di TORINO
e INFN Sezione di Torino
(for the PRISMA-CLARA Collaboration)
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
HI reactions
HI reactions, a short overview
Conclusions
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
HI reactions
HI reactions, a short overview
TRANSFER REACTIONS
EVAPORATION
For stable nuclei only
NEUTRON PICK-UP and
PROTON STRIPPING
are possible for opt. Qval
Conclusions
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
HI reactions
HI reactions, a short overview
TRANSFER REACTIONS
EVAPORATION
For stable nuclei only
NEUTRON PICK-UP and
PROTON STRIPPING
are possible for opt. Qval
FUSION-FISSION
Lfus >> Lld
ORBITING
Conclusions
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
HI reactions
Macro-Microscopic approach
Conclusions
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
HI reactions
Macro-Microscopic approach
σZ (Elos) DISSIPATION
Friction (transport equation)
Conclusions
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
HI reactions
Macro-Microscopic approach
σZ (Elos) DISSIPATION
Friction (transport equation)
Transport equations
for the exchange of mass
and charge.
friction forces
for energy and angular
momentum dissipation
Conclusions
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
HI reactions
Macro-Microscopic approach
GRAZING
It uses a microscopic
approach.
σZ (Elos) DISSIPATION
Friction (transport equation)
It calculates the evolution of
the reaction by using the
intrinsic properties of the two
colliding nuclei:
Transport equations
for the exchange of mass
and charge.
surface modes
low lying modes
high lying modes
friction forces
for energy and angular
momentum dissipation
transfer channels
(microscopic formfactor
for transfer)
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Grazing
GRAZING
This is done in the semi-classical approximation where the relative
motion is treated classically and the time evolution of the intrinsic
states is described by the system of coupled equations.
X
i
i~ċβ (t) =
< β|Hint |α > cα (t)e ~ (Eβ −Eα )t+i (δβ −δα )
α
i~Ψ̇(t) = (H0 + Hint )Ψ(t)
X
i
Ψ(t) =
cβ (t)ψβ e ~ Eβ t
d β
R
where ψα are the channels wave
functions
ψα (t) = ψ a (t)ψ A (t)e
~
i δ(R)
b
A
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Grazing
GRAZING
The intrinsic Hamiltonian and the interaction are:
Ĥ0 =
(a)
X
i
i ai† ai
+
(a)
X
†
~ωλ aλµ
aλµ + (A)
λµ
V̂int (t) = V̂tr (t) + V̂in (t) + ∆UaA (t)
Vint contains the well known form-factors for inelastic excitation
and for one-particle transfer (both protons and neutrons).
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Grazing
GRAZING
The intrinsic Hamiltonian and the interaction are:
Ĥ0 =
(a)
X
i
i ai† ai
+
(a)
X
†
~ωλ aλµ
aλµ + (A)
λµ
V̂int (t) = V̂tr (t) + V̂in (t) + ∆UaA (t)
Vint contains the well known form-factors for inelastic excitation
and for one-particle transfer (both protons and neutrons).
The time dependence of the matrix elements is obtained by
solving the Newtonian equations for the relative motion in the
nuclear plus Coulomb field. For the nuclear potential we use the
Akyüz-Winther parametrisation that describes quite well elastic
scattering data for several projectile and target combinations.
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Grazing
GRAZING: What can be calculated.
GRAZING calculates how the reaction cross section is shared
among the different final mass partitions up to the ORBITING:
GRAZING
Beyond the ORBITING only the capture probability can be
estimated.
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Grazing
GRAZING: What can be calculated.
GRAZING calculates how the reaction cross section is shared
among the different final mass partitions up to the ORBITING:
GRAZING
70 %
Beyond the ORBITING only the capture probability can be
estimated.
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Grazing
GRAZING: What can be calculated.
GRAZING calculates how the reaction cross section is shared
among the different final mass partitions up to the ORBITING:
GRAZING
70 %
Beyond the ORBITING only the capture probability can be
jump orbiting
estimated.
Introduction
Data and analysis
Toward PRISMA and CLARA
Grazing
ORBITING: from order to chaos
For energies close to EB , the
travelling time in the inside
pocket τin is important for the
evolution of the reaction.
Conclusions
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Grazing
ORBITING: from order to chaos
For energies close to EB , the
travelling time in the inside
pocket τin is important for the
evolution of the reaction.
Not all the trajectories that
reach the inner pocket lead to
capture.
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Grazing
ORBITING: from order to chaos
IMPORTANCE OF iWVol (r)
Of course the dynamics of the inner pocket is
relevant only if the absorption is not to strong.
For energies closeNotice
to EB ,that
the most of the coupled-channel
codes
impose
the incoming-wave
boundary
Not all the trajectories
that
travelling time in the inside
conditions
at
the
pocket
location.
reach
the
inner
pocket
lead
to
pocket τin is important for the
capture.
evolution of the reaction.
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Total Reaction Cross section
Fusion and Total Reaction cross sections
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Total Reaction Cross section
Fusion and Total Reaction cross sections
DIC collisions (quasi-fission, incomplete fusion,...)
are the most likely candidates to explain the HINDRANCE
to FUSION at the higher energies
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Total Reaction Cross section
Fusion and Total Reaction cross sections
DIC collisions (quasi-fission, incomplete fusion,...)
are the most likely candidates to explain the HINDRANCE
to FUSION at the higher energies
Show data and Ca+Zr
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Phenomenology
Why multinucleon transfer
Role of transfer channels in fusion. (Fusion reaction should be
discussed not alone but with the other reaction channels,
elastic, inelastic and transfer, and...
The study of the residual interaction, in particular PAIRING
CORRELATION. (For reactions heavier then t only inclusive
measurements are available and they are nor suited to study
correlations. Exclusive measurements are needed)
The structure of the single particle states (shell closure).
With PRISMA+CLARA we will pursue this task at least for some
systems.
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Phenomenology
Why multinucleon transfer
Role of transfer channels in fusion. (Fusion reaction should be
discussed not alone but with the other reaction channels,
elastic, inelastic and transfer, and...
The study of the residual interaction, in particular PAIRING
CORRELATION. (For reactions heavier then t only inclusive
measurements are available and they are nor suited to study
correlations. Exclusive measurements are needed)
The structure of the single particle states (shell closure).
With PRISMA+CLARA we will pursue this task at least for some
systems.
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Phenomenology
Why multinucleon transfer
Role of transfer channels in fusion. (Fusion reaction should be
discussed not alone but with the other reaction channels,
elastic, inelastic and transfer, and...
The study of the residual interaction, in particular PAIRING
CORRELATION. (For reactions heavier then t only inclusive
measurements are available and they are nor suited to study
correlations. Exclusive measurements are needed)
The structure of the single particle states (shell closure).
With PRISMA+CLARA we will pursue this task at least for some
systems.
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Phenomenology
Why multinucleon transfer
Role of transfer channels in fusion. (Fusion reaction should be
discussed not alone but with the other reaction channels,
elastic, inelastic and transfer, and...
The study of the residual interaction, in particular PAIRING
CORRELATION. (For reactions heavier then t only inclusive
measurements are available and they are nor suited to study
correlations. Exclusive measurements are needed)
The structure of the single particle states (shell closure).
With PRISMA+CLARA we will pursue this task at least for some
systems.
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Phenomenology
Why multinucleon transfer
Role of transfer channels in fusion. (Fusion reaction should be
discussed not alone but with the other reaction channels,
elastic, inelastic and transfer, and...
The study of the residual interaction, in particular PAIRING
CORRELATION. (For reactions heavier then t only inclusive
measurements are available and they are nor suited to study
correlations. Exclusive measurements are needed)
The structure of the single particle states (shell closure).
With PRISMA+CLARA we will pursue this task at least for some
systems.
Introduction
Data and analysis
Toward PRISMA and CLARA
Phenomenology
Some Phenomenology of MNT
Conclusions
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Phenomenology
Some Phenomenology of MNT
The system does not reach charge equilibration. The
population in the (N,Z) plane is dictated by the Q opt
For each transferred neutron the cross section drops by a
constant factor (∼3.5) (sequential transfer)
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Phenomenology
Some Phenomenology of MNT
The ONE-neutron transfer channel is much larger than the
ONE-proton transfer channel
The pure TWO-proton transfer is as large as the ONE-proton
transfer (pair-transfer mode ?)
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Phenomenology
Some Phenomenology of MNT
EVAPORATION may strongly influence the isotopic
distribution of the final fragments, these are indeed produced
quite HOT (at high excitation energies)
go to evaporation
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Theory
Theory: results with GRAZING or WKBM
The 40 Ca +208 Pb
Channels incuded in the
calculations:
inelastic:
2+ ,3− states (L/H)
transfer
all s.p. transitions
MULTINUCLEON is included in the successive approximation,
Akyüz-Winther parametrization is used for the nuclear potential.
Introduction
Data and analysis
Theory
Isotopic distribution
Toward PRISMA and CLARA
Conclusions
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Theory
Isotopic distribution
1pt
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Theory
Isotopic distribution
1pt+2pt
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Theory
Isotopic distribution
+Evap.
go to Ni+Pb reaction
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Theory
Pure neutron, proton transfer channels
Simultaneous transfer:
Is achieved by adding a PAIR-MODE (δn fi = ±2, δzfi = ±2)
∂V opt (r )
∂A
where the strength βp is adjusted to the experiment.
Ffipair (r ) = βp
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Pairing
Energy Spectra: pairing vibrations
40 Ca+208 Pb
EL =235 MeV
40 Ca+208 Pb
(only (+2n))
go to (p,t) data
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Pairing
42
Ca
0+
E2 E2
2
23 75
E=3.4MeV
E1
5.87 MeV
E=4.3MeV
Decay of the 0+ states at
∼5.8 MeV
1−
3.88 MeV
E1
E1
60
40
4+
2+
0+
2.75 MeV
E2
99
E2 98
2+
0+
E2
E2+M1
30
70
2.42 MeV
1.84 MeV
1.52 MeV
E2
E0
100
2
0.0 MeV
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Pairing
The
40
Ca +
96
Zr reaction
40
40
42
41
39
38
39
38
Introduction
Data and analysis
Toward PRISMA and CLARA
Pairing
The
40
Ca +
96
Zr reaction
Conclusions
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Pairing
The
40
Ca +
96
Zr reaction
in/out of plane
42
Ca
0+
23 75
E=3.4MeV
E2 E2
2
5.87 MeV
E=4.3MeV
E1
1−
3.88 MeV
E1
E1
60
40
4+
2+
0+
2.75 MeV
E2
99
E2 98
2+
0+
E2
E2+M1
30
70
2.42 MeV
1.84 MeV
1.52 MeV
E2
E0
100
2
0.0 MeV
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Excitation energy
Which nuclei and states are populated via MNT
The amplitude for the transfer of a nucleon between the states
(j,k) is given by (first order perturbation theory):
Z
1 +∞
dt < ψβ |Hint |ψα >t
cjk =
i~ −∞
1
=
i~
Z
+∞
− ~i
~
dtfjk (R(t))e
−∞
(k −j )−Qopt t
where
ψα = ψa (ξa )ψA (ξA )e i δaA is the channels wave function
δaA the semi-classical phase
~
fjk (R(t))
the transfer form-factors
Qopt the Q-optimun of the reactions
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Excitation energy
Selectivity in energy
The amplitude cjk is maximized if the distance of closest approach
in entrance (α) and exit (β) channels are!the same.
Za ZA
Qopt =
− 1 Ecm
Zb ZB
A better expression
Qopt =
+
Zd
Zd
−
ZA Zb
!
EB
!
md
md
md r̈0
−
(RA mb − Ra mB )
(E − EB ) +
mb
mA
ma + m A
+µ~φ̇0
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
Excitation energy
Selectivity in energy
The amplitude cjk is maximized if the distance of closest approach
in entrance (α) and exit (β) channels are!the same.
Za ZA
Qopt =
− 1 Ecm
Zb ZB
A better expression
Qopt =
+
Zd
Zd
−
ZA Zb
!
EB
!
md
md
md r̈0
−
(RA mb − Ra mB )
(E − EB ) +
mb
mA
ma + m A
+µ~φ̇0
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
angular momentum transfer
Selectivity in angular momentum
At low relative motion velocities the exchange of a nucleon
occur in a plain (the ORBITALS in projectile and target match
smoothly)
V rel << V
d
F
l1
b
A
l2
and the transfer angular momentum acquire its maximum value
|~`tr | = |~`2 − ~`1 | = |~`2 | + |~`1 |
Since the transfer interaction has a very weak dependence from the
spin the intrinsic spin keeps its alignment.
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
angular momentum transfer
Vrel ~ VF
Vrel << VF
yra
st
Excitation energy
Regions in the (I,E) plane, populated by MNT reactions
Angular Momentum (I)
Appendix
Introduction
Data and analysis
In the end
Toward PRISMA and CLARA
Conclusions
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
In the end
ET
G
AR
T
E
IL
T
EC
OJ
PR
Conclusions
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
Appendix
In the end
ftr (r ) = e −r /atr
1pt: a=∼ 1.2 fm
2pt: a=∼ 0.6 fm
α transfer ???
ET
G
AR
T
E
IL
T
EC
OJ
PR
WITH STABLE BEAMS
neutron-rich projectile-like
proton rich target-like
Drop factor: ∼ 3.5 for each neutron
transfer
Introduction
Data and analysis
In the end
Toward PRISMA and CLARA
Conclusions
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
In the end
ftr (r ) = e −r /atr
1pt: a=∼ 1.2 fm
2pt: a=∼ 0.6 fm
α transfer ???
NEUTRON RICH NUCLEI
Only by using neutron-rich
radioactive beams on heavy
target (all stripping and
pich-up channels are open)
Appendix
Introduction
Data and analysis
Toward PRISMA and CLARA
Conclusions
In the end: a nice mass-spectra from PRISMA
Appendix