Coulomb dissociation of the exotic nuclei using Coulomb dynamical polarization potential

Abstract

Coulomb dissociation of the exotic nuclei using Coulomb dynamical polarization potential

Full text

Coulomb dissociation of the exotic nuclei using
Coulomb dynamical polarization potential
Presentation by
Hasan Maridi
Heavy ion Laboratory (HIL), University of Warsaw, Poland
Funded by: the Ulam Programme from the Polish National Agency for
Academic Exchange (NAWA)
Collaborators: Prof. K. Rusek (HIL), Prof. N. Keeley (NCBJ)
Based on: arXiv:2206.07546 (2022) & Phys. Rev. C 104, 024614 (2021).
For: the 11th conference on Direct Reactions with Exotic Beams
(DREB2022) Santiago de Compostela, Spain
29 June 2022 (12:20 CET)
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Coulomb dynamical polarization potential- Introduction I
For heavy ion targets, the projectile may become polarized by the
electric field of the target and the charge distribution is distorted
inducing an additional long-range interaction which is referred to as
the Coulomb dynamical polarization potential (CDPP).
A weakly-bound projectile with low break-up threshold can become
polarized and may break up and giving rise to a strong Coulomb
dipole excitation to the low-lying continuum.
The real classical CDPP is given as δVR=−1
2α0
Z2
Te2
R4,α0:polarizability
CDPP by Andr´es & G´omez-Camacho, Phys. Rev. Lett. 82, 1387 (1999)
δU(R) = 4π
9
Z2
te2
~v
1
(r−a0)2rZdεex
dB(E1)
dεex gr
a0
−1, ξ+if r
a0
−1, ξ
The CDPP may be extracted from continuum-discretized coupled
channels (CDCC) calculations by means of the trivially equivalent
method [Franey, Ellis, Phys. Rev. 23, 787 (1981)]. Such CDCC
calculations can be difficult and time-consuming.
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Coulomb dynamical polarization potential-Formalism I
In this work, a weakly-bound projectile moving in the
Coulomb field of a heavy target The projectile is
considered to be a two-body cluster structure with a
core (c) (with mass mcand charge Zc) and valence
neutron(s) (v) with mass mv=nmn.
By using adiabatic approximation Ψ(r,R) = ψ(R)φ(r,R), following the
model of Ref. Borowska et al., Phys. Rev. C 76, 034606 (2007)., we solve the
Schr¨odinger equation for the internal motion of the projectile and the
resulting CDPP δUcan be expressed as [Phys. Rev. C 104, 024614 (2021)]
δV(R) = ε∗
0"QG0F0+Q2G0F0G0
0F0
0+Q2F2
0F02
0
F4
0+G2
0F2
0
−1#
δW(R) = ε∗
0"Q2F0F0
0−QF 2
0
F4
0+G2
0F2
0#.
where F0,G0are the Coulomb functions in ρ=kR,Q(R) = µp
mc
k(R)
κ0
k(R) = s2m2
c
µp~2(VC(R) + ε∗
0+δU(R))
where κ0=q−2µpε∗
0/~2,ε∗
0=ε0+εIπ
c,ε0is the separation energy, εIπ
c
the core level energy, VC(Rthe Coulomb potential.
3/15

Continuum based CDPP I
Now we include the excitations to the continuum with energy εas
ε∗
0→ε∗
0+ε, and k(R)→k(R, ε), and then δU(R)→δU(R, ε)as
continuum based CDPP
δVIπ
c(R, ε) = ε∗
0QG0F0+Q2G0F0G0
0F0
0+Q2F2
0F02
0
F4
0+G2
0F2
0−ε∗
0−ε
δWIπ
c(R, ε) = ε∗
0Q2F0F0
0−QF 2
0
F4
0+G2
0F2
0
we include Iπ
cto refer to a definite core state.
Let us consider an Eλ-transition (E1,E2, ...)from `0bound state to
`-wave continuum with `=`0+λ
φIπ
c
`0j0(r)represents the initial bound state and describes the wave
function of the valence neutron in `0j0orbital relative to the core Iπ
c,
and φε`j(r)is the final state (continuum) wave function.
δU(R, ε)can be expanded in Legendre polynomials as
δU(R,r, ε) = PλδU(R, ε)Pλ(cos(θr))
4/15

Continuum based CDPP II
we define the Eλ-transition CDPP as
δUIπ
c,Eλ
`0j0→ε`j(R) = ZdεDφε`j(r, ε)|δU(R,r, ε)|φIπ
c
`0j0(r)E
Let us define the excitation energy distribution of the Eλ-transition
`0j0→ε`jas
ρEλ,Iπ
c
`0j0→ε`j(ε) = Dφε`j(r, ε)|Pλ(cos(θr))|φIπ
c
`0j0(r)E
Now, we can write CDPP for the Eλ-transition as
Eλ-transition CDPP
δUIπ
c,Eλ
`0j0→εlj (R) = ZdερEλ,Iπ
c
`0j0→ε`j(ε)δUIπ
c(R, ε)
this potential will be useful in the study of nuclear reactions that
include excitation to the continuum.
5/15

Continuum based CDPP III
The bound-state radial wave function φIπ
c
`0j0(r) = u`0j0(r)/rwhere u`0j0(r)
is the single-particle wave function that is obtained by adjusting the
Woods-Saxon potential parameters to reproduce ε∗
0=ε0+εIπ
c.
The extracted spectroscopic factor can differ by more than 20% by
choosing different parameters. We choose (r0= 1.15 fm, a= 0.5 fm)
Sauvan et al., Phys. Lett. B 491 1 (2000)
The continuum is calculated assuming the plane wave approximation
φε`j(r) = p2µpκ/~2πj`(κr) = uε`j(r)/rwhere κ=p2µpε/~.
Considering expansions of φIπ
c
`0j0(r)and φε`j(r)in spherical harmonics
ρEλ,Iπ
c
`0j0→ε`j(ε) = p4π(2l0+ 1) h`00λ0|`0i2Z∞
0
dr uε`j(r)u`0j0(r)
For `0= 0,φIπ
c
`0j0(r)can be given by Yukawa form , so for E1transition
ρE1,Iπ
c
s→p(ε) = rµpκ0κ
~2
4
κ2−κ0κ
κ2
0+κ2+ tan−1κ
κ0
which will be useful since the valence neutron(s) are in a relative
s-state in the ground state of many exotic light projectiles of interest.
6/15

Coulomb dissociation at high energies I
Coulomb dissociation can take
place when a projectile moving
with high energy (several
hundred of MeV/u) passes a
heavy ion target.
It may be excited by absorbing virtual photons from the Coulomb
field and the electromagnetic excitation is dominated by dipole
excitations and the core is assumed as a spectator so the core state
in the projectile remains the same after removal of one neutron
We start from the usual formula for the absorption cross section
σabs =−2
~vDψ(+)
K(R)|W(R)|ψ(+)
K(R)E
where W(R)is the imaginary potential and ψ(+)
Kthe usual distorted
wave function, vthe relative velocity, and Kits wavenumber.
In our case W(R) = δWIπ
c,Eλ
`0j0→lj (R)
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Coulomb dissociation at high energies II
the total absorption due to the Coulomb dissociation is given by
σCD =−2
~vZdεX
Iπ
cX
EλX
`0j0X
lj
ρEλ,Iπ
c
`0j0→ε`j(ε)Dψ(+)
K(R)|δWIπ
c(R, ε)|ψ(+)
K(R)E
and the differential cross section for Eλ-transition, `0j0→ε`j, at Iπ
cis
The differential Coulomb breakup/dissociation cross section
dσIπ
c
dε(Eλ, `0j0→ε`j) = −2
~vρEλ,Iπ
c
`0j0→ε`j(ε)Dψ(+)
K(R)|δWIπ
c(R, ε)|ψ(+)
K(R)E
This is similar to the Hussein-McVoy formula for the inclusive breakup
At high energies, we can use the eikonal approximation and the
optical limit of the Glauber theory, [Glauber, in Lectures in Theoretical
Physics (Interscience, New York, 1959), Vol. 1, p. 315. ]; and we can write
−2
~vDψ(+)
K(R)|δWIπ
c(R, ε)|ψ(+)
K(R)E=Zdb1−e2
~vR∞
−∞ δWIπ
c(b,´z,ε)d´z
where R= (x,y,z)=(b,z)and bis the impact parameter
8/15

Coulomb dissociation of 11Be
11Be (1/2+) as (10 Be+n) on lead at 520 MeV/u.
Iπ
c` σexp σth Our SF factors Spectroscopic factors
s.p. Yuk s.p. Yuk shell exp.
0+s1/2605(30) 1570 1111 0.4 0.57 0.74 0.36,0.60,0.61(5),
0.72(4),1.0(2)
2+d3/232.5
data: Palit et al., Phys. Rev. C 68, 034318 (2003).
0123456
0
100
200
300
400
1 1 B e + P b
5 2 0 M e V / n u c l e o n
11B e ( 1 / 2 +) a s 10B e ( 0 +) + νs1 / 2
dσ/ d ε ( m b / M e V )
ε ( M e V )
P a l i t 2 0 0 3
Y u k a w a
s . p .
9/15

Coulomb dissociation of 15C
15C (1/2+) as (14 C+n) on lead at 605 MeV/u.
Iπ
c` σexp σth Our SF factors Spectroscopic factors
s.p. Yuk s.p. Yuk shell exp.
0+s1/2324(15) 570 350 0.56 0.93 0.98 0.91(6),0.72(5);
0.73(5),0.97(8)
1−p1/2
36(3)
8.2 1.03
p3/28.4 0.16
0−p1/24.8 1.41
2−p3/23.9 0.016
data: Pramanik et al., Phys. Lett. B 551, 63 (2003).
012345678910
0
5 0
100
150
1 5 C + P b
6 0 5 M e V / n u c l e o n
15C ( 1 / 2 +) a s 14C ( 0 +) + νs1 / 2
dσ/ d ε ( m b / M e V )
ε ( M e V )
D a t t a P r a m a n i k 2 0 0 3
Y u k a w a
s . p .
10/15

Coulomb dissociation of 17C
17C (1/2+) as (16 C(2+)+n) on lead at 496 MeV/u.
Iπ
c` σexp σth Our SF factors Spectroscopic factors
s.p. Yuk s.p. Yuk shell exp.
0+d3/29+15
−9607 0.015 0.035
2+d5/262(7) 96.8 0.23 1.41 0.6(4),1.6(6)
s1/2119.1 60 0.28 0.86 0.16 0.23(8),26(14)
0+d3/225(7) 33.9
4+d5/215.6 0.76
data: Pramanik et al., Phys. Lett. B 551, 63 (2003).
0 1 2 3 4 5 6 7 8 9 1 0
0
5
1 0
1 5
2 0
2 5
1 7 C + P b
4 9 6 M e V / n u c l e o n
17C ( 3 / 2 +) a s 16C ( 2 +) + n
dσ/ d ε ( m b / M e V )
ε ( M e V )
D a t t a P r a m a n i k 2 0 0 3
s 1/2 Y u k a w a
s 1/2 s . p .
d 5/2 s . p .
s u m s . p .
11/15

Coulomb dissociation at 70 MeV/u
Coulomb dissociation at 70 MeV/nucleon.
0123456
0
500
1000
1500
2000
2500 1 1 B e + P b
7 0 M e V / n u c le o n
11B e ( 1 / 2 +) a s 10B e ( 0 +) + νs1 / 2
dσ/ d ε ( m b / M e V )
ε ( M e V )
Fukoda 2004
N a k u m o r a 1 9 9 4
Y u k a w a
s . p .
0 1 2 3 4 5 6 7 8 9
0
100
200
300
400
500 1 5 C + 208P b
6 8 M e V / n u c le o n
15C ( 1 / 2 +) a s 14C ( 0 +) + νs1 / 2
dσ/ d ε ( m b / M e V )
ε ( M e V )
N a k u m o r a 2 0 0 9
Y u k a w a
s . p .
0123456
0
200
400
600
800
1000
1200 1 9 C + 208P b
6 7 M e V / n u c le o n
19C ( 1 / 2 +) a s 18C ( 0 +) + νs1 / 2
dσ/ d ε ( m b / M e V )
ε ( M e V )
N a k u m o r a 1 9 9 9
Y u k a w a
s . p .
data: Nakamura et al., Phys. Lett. B 331, 296 (1994) & Fukuda et al., Phys. Rev.
C70, 054606 (2004) for 11Be;
Nakamura et al., Phys. Rev. C 79, 035805 (2009) for 15 C;
Nakamura et al., Phys. Rev. Lett. 83, 1112 (1999) for 19C.
11Be Rσ=0.8 from s.p., ∼1 from Yukawa
15CRσ=0.8 from s.p., ∼1 from Yukawa
19CRσ=0.7 from s.p., ∼1 from Yukawa, Shell model 0.58,
experiment 0.67
12/15

Coulomb dissociation for Borromean nuclei
Coulomb dissociation spectra for 19B and 22C (as core and two neutrons)
on lead target using Yukawa wave function for s-bound state.
0123456
100
101
102
103
104
105
0 1 2 3 4
0
200
400
600
1 9 B + P b
2 2 0 M e V / n u c l e o n
17B g . s . ( 0 +) + ν( s 1/2)2
dσ/ d ε ( m b / M e V )
ε ( M e V )
C o o k 2 0 2 0
ε0= 0 .0 8 9 M e V
ε0= 0 .5 M e V
ε0= 0 .6 5 M e V
σexp = 1160(70)mb, ε0= 0.089+0.56
−0.089 MeV
ε0σth(b) SF
0.089 10
0.5 1.5 0.7
0.65 1.08 1.0
data: Cook et al., Phys. Rev. Lett. 124,
212503 (2020)
012345678
100
101
102
103
104
105 2 2 C + P b
5 0 0 M e V / n u c l e o n
20C g . s . ( 0 +) + ν( s 1/2)2
dσ/ d ε ( m b / M e V )
ε ( M e V )
ε0= 0 .0 3 5 M e V
ε0= 0 .1 5 0 M e V
ε0= 0 .5 0 0 M e V
no exp. data, ε0= 0.035+0.20
−0.035 MeV
ε0σth(b)
0.035 17
0.15 4.8
0.5 1.25
13/15

Conclusions I
In this work
Breakup of light exotic projectiles with a two-body deuteron-like cluster
structure incident on heavy ion targets was studied.
A new Coulomb dynamical polarization potential (CDPP) was presented.
A continuum-based CDPP that includes core excitations and excitations to the
continuum into account was presented.
A new model for the Coulomb dissociation using this continuum-based CDPP
was applied to data for selected weakly-bound projectiles at high energies.
In progress
Use the new CDPP to study
the breakup effect on elastic
scattering below the Coulomb
barrier. Interesting results from
application to 6He+208Pb at 14
MeV [S´anchez-Ben´ıtez et al.,
Nucl. Phys. A803, 30 (2008).]
0 20 40 60 80 100 120 140 160
0 . 8
0 . 9
1 . 0
6H e + 208P b
1 4 M e V
s/sR
qc . m . ( d e g )
e x p d a ta
n o c o u p l i n g
C D P P
C D P P E 1 s - p
CDCC
14/15

Conclusions II
Future of this work
Can this idea (including the continuum energy) be applied to the
optical potential?
Calculate the inclusive cross section for transitions between two
bound states σ(Eλ, `0j0→`j)
B(Eλ, `0j0→`j) as well
Possible improvements:
taking into account target excitations
accounting for all couplings to the initial and final states
include three-body projectiles and their nucleon-nucleon correlations
extend calculations to proton-halo nuclei (8B). Is this possible?
Thank you
15/15