nuclear reactions ii: applications to nuclear structure
Post on 29-Dec-2021
5 Views
Preview:
TRANSCRIPT
Nuclear Reactions II: Applications to Nuclear Structure
Gregory Potel Aguilar (FRIB)
Oak Ridge, June 28 2019
Oak Ridge, June 28 2019 slide 1/32
Outline
1 Why do reactions? Probing nuclear structureElastic scatteringInelastic scatteringOne-particle transfer
2 An advanced example: 2-neutron transfer and pairingPairing correlations and successive transferReaction and structure modelsA quantitative measure of pairing correlations
3 Moving forward: integrating structure and reactions(d , p) reactions: a unified approachChoosing the potential: examples of calculations
Oak Ridge, June 28 2019 slide 2/32
Outline
1 Why do reactions? Probing nuclear structureElastic scatteringInelastic scatteringOne-particle transfer
2 An advanced example: 2-neutron transfer and pairingPairing correlations and successive transferReaction and structure modelsA quantitative measure of pairing correlations
3 Moving forward: integrating structure and reactions(d , p) reactions: a unified approachChoosing the potential: examples of calculations
Oak Ridge, June 28 2019 slide 3/32
Elastic scattering
traditionally used toextract optical potentials,
rms radii, densitydistributions.
[Lapoux et al, PRC 66 (02) 034608]
Oak Ridge, June 28 2019 slide 4/32
Inelastic scattering
α
05
a
b
ground stateb a
energy loss (MeV)
num
ber α
-part
icle
s
magnet
projectile
208Pb 3
3
collective (octupole) vibration
used to extract nucleardeformations, electromagnetictransitions...
0
1
2
dσin
el/dΩ
(mb/
sr)
Full CCDWBA
208Pb(16O,16O)208Pb(3-)
A. MoroPb208
3
2
0
−
+
+
2.61
4.07
g.s.
i=2
i=3
i=12+
Oak Ridge, June 28 2019 slide 5/32
One-particle transfer
120Sn40Ca
41Ca121Snshape: angular momentum.
populates states withstrong single-particle content
L=3 L=1 L=2 L=1
magnitude: spectroscopicfactor (single-particlestrength).
Brown et al. Nucl. Phys. A 225 (1974) 267
Oak Ridge, June 28 2019 slide 6/32
Outline
1 Why do reactions? Probing nuclear structureElastic scatteringInelastic scatteringOne-particle transfer
2 An advanced example: 2-neutron transfer and pairingPairing correlations and successive transferReaction and structure modelsA quantitative measure of pairing correlations
3 Moving forward: integrating structure and reactions(d , p) reactions: a unified approachChoosing the potential: examples of calculations
Oak Ridge, June 28 2019 slide 7/32
2-neutron transfer and reactions
a
A
A B
b
b
Reaction A + a −→ (a− 2) + (A + 2).
Measure of the pairing correlations between the transferred nucleons.
Need to correctly account for the correlated wavefunction.
Oak Ridge, June 28 2019 slide 8/32
Delocalization of the pair transfer process
ξ≈36fm
(a)
(A+2)
∞ neutron matter
120Sn
p(b) A≈120
A+2
A
(e)
(c)
t
A
(d) (g)
A+2A
(f)
long4tail4Cooper4pair
irrelevant4succ.4simultaneous24fm
Cooper pair
36 fm
Oak Ridge, June 28 2019 slide 9/32
Let’s remember the Born series
|φ〉 = |φ0〉+ G0V |φ〉
Oak Ridge, June 28 2019 slide 10/32
Let’s remember the Born series
|φ〉 = |φ0〉+ G0V (|φ0〉+ G0V |φ〉)
Oak Ridge, June 28 2019 slide 10/32
Let’s remember the Born series
|φ〉 = |φ0〉+ G0V (|φ0〉+ G0V [|φ0〉+ G0V |φ〉])
Oak Ridge, June 28 2019 slide 10/32
Let’s remember the Born series
|φ〉 = |φ0〉+ G0Vφ0 + G0VG0Vφ0 + G0VG0VG0Vφ0 + . . .
Oak Ridge, June 28 2019 slide 10/32
Two-nucleon transfer: stick to second order
In order to account for the successive transfer of two nucleons, we stick to2nd order, A + a → (A + 1) + (a− 1) → (A + 2) + (a− 2)
|φ〉 = |φ0〉+ G0Vφ0 + G0VG0Vφ0
A+a
(A+2)+(a-2)
(A+1)+(a-1)
(A+2)+(a-2)
A+a A+a
V → one-nucleon transfer
G0 → Green’s function of each one of the many intermediate states(A + 1) + (a− 1)
Oak Ridge, June 28 2019 slide 11/32
Reaction and structure models
Structure:
Φi (r1, σ1, r2, σ2) =∑ji
Bji
[ψji (r1, σ1)ψji (r2, σ2)
]00
Φf (r1, σ1, r2, σ2) =∑jf
Bjf
[ψjf (r1, σ1)ψjf (r2, σ2)
]00
mean field potentialsradial wave functionsuji (r)
radial wave functionsujf (r)
Reaction:
T2NT =∑jf ji
Bjf Bji
(T (1)(ji , jf ) + T
(2)succ(ji , jf )− T
(2)NO(ji , jf )
)
Oak Ridge, June 28 2019 slide 12/32
Introducing T (1)(ji , jf ), T(2)succ(ji , jf ) and T
(2)NO(ji , jf )
very schematically, the first order (simultaneous) contribution is
T (1) = 〈β|V |α〉,
while the second order contribution can be separated in a successive and anon-orthogonality term
T (2) = T(2)succ + T
(2)NO
=∑γ
〈β|V |γ〉G 〈γ|V |α〉 −∑γ
〈β|γ〉〈γ|V |α〉.
If we sum over a complete basis of intermediate states γ, we can apply the
closure condition and T(2)NO cancels T (1)
the transition potential being single particle, two-nucleon transfer is asecond order process.
Oak Ridge, June 28 2019 slide 13/32
Two particle transfer in 2–step DWBA
Potel et al., PRL 107 092501 (2011)Potel et al., PRL 105 172502 (2010)
T2NT =∑jf ji
Bjf Bji
(T (1)(ji , jf ) + T
(2)succ(ji , jf )− T
(2)NO(ji , jf )
)
Simultaneous transfer
T (1)(ji , jf ) = 2∑σ1σ2
∫drfFdrb1drA2[Ψjf (rA1, σ1)Ψjf (rA2, σ2)]0∗0 χ
(−)∗bB (rbB)
× v(rb1)[Ψji (rb1, σ1)Ψji (rb2, σ2)]00χ(+)aA (raA)
Oak Ridge, June 28 2019 slide 14/32
Two particle transfer in 2–step DWBA
Potel et al., PRL 107 092501 (2011)Potel et al., PRL 105 172502 (2010)
T2NT =∑jf ji
Bjf Bji
(T (1)(ji , jf ) + T
(2)succ(ji , jf )− T
(2)NO(ji , jf )
)Successive transfer
T(2)succ(ji , jf ) = 2
∑K ,M
∑σ1σ2σ′1σ
′2
∫drfFdrb1drA2[Ψjf (rA1, σ1)Ψjf (rA2, σ2)]0∗0
× χ(−)∗bB (rbB)v(rb1)[Ψjf (rA2, σ2)Ψji (rb1, σ1)]KM
×∫
dr′fFdr′b1dr
′A2G (rfF , r
′fF )[Ψjf (r′A2, σ
′2)Ψji (r′b1, σ
′1)]KM
× 2µfF~2
v(r′f 2)[Ψji (r′b2, σ′2)Ψji (r′b1, σ
′1)]00χ
(+)aA (r′aA)
Oak Ridge, June 28 2019 slide 14/32
Two particle transfer in 2–step DWBA
Potel et al., PRL 107 092501 (2011)Potel et al., PRL 105 172502 (2010)
T2NT =∑jf ji
Bjf Bji
(T (1)(ji , jf ) + T
(2)succ(ji , jf )− T
(2)NO(ji , jf )
)
Non–orthogonality term
T(2)NO(ji , jf ) = 2
∑K ,M
∑σ1σ2σ′1σ
′2
∫drfFdrb1drA2[Ψjf (rA1, σ1)Ψjf (rA2, σ2)]0∗0
× χ(−)∗bB (rbB)v(rb1)[Ψjf (rA2, σ2)Ψji (rb1, σ1)]KM
×∫
dr′b1dr′A2[Ψjf (r′A2, σ
′2)Ψji (r′b1, σ
′1)]KM
× [Ψji (r′b2, σ′2)Ψji (r′b1, σ
′1)]00χ
(+)aA (r′aA)
Oak Ridge, June 28 2019 slide 14/32
Contributions to the 112Sn(p,t)110 total cross section
0 10 20 30 40 50 60 7010
0
101
102
103
104
105
θCM
dσ/d
Ω (
µ b/
sr)
simultaneous
Transfer driven by single-particle potential (mean field) → essentially asuccessive process!
Oak Ridge, June 28 2019 slide 15/32
Contributions to the 112Sn(p,t)110 total cross section
0 10 20 30 40 50 60 7010
0
101
102
103
104
105
θCM
dσ/d
Ω (
µ b/
sr)
successivesimultaneous
Transfer driven by single-particle potential (mean field) → essentially asuccessive process!
Oak Ridge, June 28 2019 slide 15/32
Contributions to the 112Sn(p,t)110 total cross section
0 10 20 30 40 50 60 7010
0
101
102
103
104
105
θCM
dσ/d
Ω (
µ b/
sr)
successivesimultaneousnon−orthogonal
Transfer driven by single-particle potential (mean field) → essentially asuccessive process!
Oak Ridge, June 28 2019 slide 15/32
Contributions to the 112Sn(p,t)110 total cross section
0 10 20 30 40 50 60 7010
0
101
102
103
104
105
θCM
dσ/d
Ω (
µ b/
sr)
successivesimultaneoussimultaneous+non−orthogonalnon−orthogonal
Transfer driven by single-particle potential (mean field) → essentially asuccessive process!
Oak Ridge, June 28 2019 slide 15/32
Contributions to the 112Sn(p,t)110 total cross section
0 10 20 30 40 50 60 7010
0
101
102
103
104
105
θCM
dσ/d
Ω (
µ b/
sr)
totalsuccessivesimultaneoussimultaneous+non−orthogonalnon−orthogonal
Transfer driven by single-particle potential (mean field) → essentially asuccessive process!
Oak Ridge, June 28 2019 slide 15/32
Probing pairing with 2–transfer: 112Sn(p,t)110Sn @ 26 MeV
0 10 20 30 40 50 60 7010
0
101
102
103
104
θCM
dσ/d
Ω (
µ b/
sr)
experiment
pure (d5/2
)2 configuration
Shell ModelBCS
112Sn(p,t)110Sn, Elab
=26 MeV
enhancement factor withrespect to the transfer ofuncorrelated neutrons:ε = 20.6
Experimental data and shell model wavefunction from Guazzoni et al.PRC 74 054605 (2006)
experiment very well reproduced with mean field (BCS) wavefunctions
Oak Ridge, June 28 2019 slide 16/32
Examples of calculations
0 30 60 90 120
100
101
θCM
dσ/d
Ω(m
b/sr
)
7Li(t,p)9Li, Et=15 MeV
0 20 40 60 8010
−1
100
101
10Be(t,p)12Be, Et=17MeV
θCM
dσ/d
Ω(m
b/sr
)
good results obtained for halo nuclei,population of excited states,superfluid nuclei,normal nuclei (pairing vibrations),heavy ion reactions...Potel et al., Rep. Prog. Phys. 76(2013) 106301
Absolute cross sections reproduced
Oak Ridge, June 28 2019 slide 17/32
Outline
1 Why do reactions? Probing nuclear structureElastic scatteringInelastic scatteringOne-particle transfer
2 An advanced example: 2-neutron transfer and pairingPairing correlations and successive transferReaction and structure modelsA quantitative measure of pairing correlations
3 Moving forward: integrating structure and reactions(d , p) reactions: a unified approachChoosing the potential: examples of calculations
Oak Ridge, June 28 2019 slide 18/32
Extracting the structure information: a standard approach
many-body Hamiltonian
many-body wfs and energies
"spectroscopic amplitudes"
r s-p wfs interactions
Ec
R
scattering wfs optical potentials
STRUCTURE REACTIONS
σ=Si2σ
Uc(R)
Rr
Factorization of structure and reactions.
Can suffer from inconsistency between the two schemes.
Extracted spectroscopic factor S2i = σ/σ problematic.
Oak Ridge, June 28 2019 slide 19/32
(d , p) reactions: a unified approach
1 Describe the structure of the 2-body subsystems in some givenframework of choice.
2 Employ the same quantum many-body methods to work out theinteractions UAn,UAp,Upn (in general, non-local andenergy-dependent).
3 Write down the resulting 3–body Hamiltonian H.4 Obtain cross sections from H using controlled approximations.
3–body Hamiltonian
H = T + UAn(rn, r′n,En, Jn, πn) + UAp(rp, r
′p,Ep, Jp, πp)
+ Upn(rpn, r′pn,Epn, Jpn, πpn)
Disclaimer
Still not the end of the story! 3–body forces UAnp not taken into accountat this stage.
Oak Ridge, June 28 2019 slide 20/32
(d , p) reaction as a 2-step process
p
n
d
A A
non elastic breakupelastic breakup
p
n
A
G
p
B*
to detector
step1
step2
breakup
propagation of n in the field of A
n
Oak Ridge, June 28 2019 slide 21/32
UAn potential and neutron states
E-E
Fx9
MeV
8
scatteringxstates
weaklyxboundxstates
deeplyxboundxstates
Wx9MeV8
EF
Imaginaryxpartxofxopticalxpotential neutronxstates
narrowxsingle-particle
scatteringxandxresonances
broadxsingle-particle
Mahaux,xBortignon,xBrogliaxandxDassoxPhys.xRep.x120x919858x1x
Oak Ridge, June 28 2019 slide 22/32
Dispersive Optical Model (DOM): Calcium isotopes
-8 -6 -4 -2 0 2 4 6 8 10 12 140
50
100
150
200
250
300
-6 -4 -2 0 2 4 6 8 10 12 140
20
40
60
80
100
totalEBNEB
0 2 4 6 8 10 12 140
100
200
300
400
-8 -4 0 4 8 12 16 20 24 28 320
10
20
30
40
50
-8 -4 0 4 8 12 16 20 24 28 320
5
10
15
20
25
30
0 4 8 12 16 20 24 28 320
10
20
30
40
50
40Ca(d,p)48Ca(d,p) 60Ca(d,p)
60Ca(d,p)48Ca(d,p)
40Ca(d,p)Ed=20 MeV
Ed=40 MeV
Ed=20 MeV Ed=20 MeV
Ed=40 MeV
Ed=40 MeV
dσ/d
E (
mb/
MeV
)
En (MeV)
continuumbound
GP et al., Eur. Phys. J. A 53 (2017) 178.
DOM used to compute (d , p) cross sections on Ca isotopes.
Both bound and continuum neutron states described.
DOM can be extrapolated to unknown territory (60Ca).
Oak Ridge, June 28 2019 slide 23/32
Dispersive Optical Model (DOM): Calcium isotopes
0 10 20 30 40 500.1
1
dσ/d
Ω (
mb/
sr)
θCM
48Ca(d,p), Ed=56 MeV
g9/2 resonance
0 20 40 60 80 100 120 140 160 1800.01
0.1
1
10
dσ/d
Ω (
mb/
sr)
θCM
60Ca(d,p), Ed=40 MeV
g9/2 ground state resonance
dσ/d
Ω (
mb/
sr)
θCM
40Ca(d,p), Ed=10 MeV
0 20 40 60 80 100 120 140 1600
1
2
3
4
5
f7/2 (gs)data
-8 -6 -4 -2 0 2 4 6 8 10 12 140
50
100
150
200
250
300
40Ca(d,p)
-8 -4 0 4 8 12 16 20 24 28 320
5
10
15
20
25
30
48Ca(d,p)
0 4 8 12 16 20 24 28 320
10
20
30
40
50
60Ca(d,p)
total
EBNEB
dσ/d
E (
mb/
MeV
)
Absolute transfer cross sections without spectroscopic factors.
Oak Ridge, June 28 2019 slide 24/32
Nuclear Field Theory (NFT): 11Be, parity inversion close tothe dripline
-7
-6
-3
-2
-1
0
1
2
3
E (MeV)
6.50
0.14
-3.01
-6.39
1d5/2
2s1/2
1p1/2
1p3/2
mean field (NFT)ren exp.
5/2+ 5/2+
1/2 1/2
1/2+ 1/2+
1.45 1.28
-0.18
-0.50 -0.50
-0.18
3/2-6.39
-6.81
6
7
2+
p1/2
p1/2
p3/2-1
s1/2
s1/2
d5/2
2+
s1/2
d5/2
2+
d5/2
structure (energies, widths) reproducedwithin an EFT-like framework (NFT)
self energies and Green's functions obtainedwithin the same framework
G(E,l,j)=G0(E) +G0(E)Σ(E,l,j)G(E,l,j)
Σ(E,l,j)= + + ...l,j
l,j l,j
l,j
l',j'
Oak Ridge, June 28 2019 slide 25/32
Nuclear Field Theory (NFT): 11Be, parity inversion close tothe dripline
Oak Ridge, June 28 2019 slide 25/32
Nuclear Field Theory (NFT): 11Be, parity inversion close tothe dripline
Oak Ridge, June 28 2019 slide 25/32
Nuclear Field Theory (NFT): 11Be, parity inversion close tothe dripline
Oak Ridge, June 28 2019 slide 25/32
Coupled-Cluster (CC): Ca isotopes
(Eexp)
-2.20
-3.88
-4.35
-4.56
(Eexp)
structure calculated withinab initio coupled clusterframework by J. Rotureau (MSU)
remarkable agreement with experimental data
coupled cluster+(d,p)+continuum spectroscopy
powerful tool for exoticmedium-mass nuclei
Eexp=-8.36 MeV
Eexp=-5.15 MeV
Oak Ridge, June 28 2019 slide 26/32
Coupled-Cluster (CC): Ca isotopes
structure calculated withinab initio coupled clusterframework by J. Rotureau (MSU)
Rotureau, GP, Li, Nunes Submitted to PRL
(Eexp)
-2.20
-3.88
-4.35
-4.56
(Eexp)
Eexp=-8.36 MeV
Eexp=-5.15 MeV
(Eexp)
(Eexp)
Oak Ridge, June 28 2019 slide 26/32
top related