nuclear reactions ii: applications to nuclear structure

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


Outline





Elastic scattering

traditionally used toextract optical potentials,

rms radii, densitydistributions.

[Lapoux et al, PRC 66 (02) 034608]


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+


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


Outline





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.


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


Let’s remember the Born series

|φ〉 = |φ0〉+ G0V |φ〉



|φ〉 = |φ0〉+ G0V (|φ0〉+ G0V |φ〉)



|φ〉 = |φ0〉+ G0V (|φ0〉+ G0V [|φ0〉+ G0V |φ〉])



|φ〉 = |φ0〉+ G0Vφ0 + G0VG0Vφ0 + G0VG0VG0Vφ0 + . . .


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)


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 )

)


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.


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)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)




T2NT =∑jf ji

Bjf Bji

(T (1)(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)




T2NT =∑jf ji

Bjf Bji

(T (1)(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)


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!



0 10 20 30 40 50 60 7010

0

101

102

103

104

105

θCM

dσ/d

Ω (

µ b/

sr)

successivesimultaneous




0 10 20 30 40 50 60 7010

0

101

102

103

104

105

θCM

dσ/d

Ω (

µ b/

sr)

successivesimultaneousnon−orthogonal




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




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



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


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


Outline





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.


(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.


(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


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


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).


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


g9/2 ground state resonance

dσ/d

Ω (

mb/

sr)

θCM


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.


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'


Nuclear Field Theory (NFT): 11Be, parity inversion close tothe dripline


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


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)


nuclear reactions ii: applications to nuclear structure

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