why do green’s functions? - github...

reactions and structure

Why do Green’s functions?• Properly executed --> answers an old question from Sir Denys

Wilkinson: “What does a nucleon do in the nucleus?”

• Nucleon self-energy —> think of potential but energy dependent • Nucleon self-energy —> elastic nucleon scattering data --> input

for the analysis of many nuclear reactions • Nucleon self-energy —> bound-state overlap functions with their

normalization --> also used in the analysis of nuclear reactions --> for exotic nuclei only strongly interacting probes available

• Nucleon self-energy—>density distribution & E/A from VNN

• Self-energy <--> data --> dispersive optical model (DOM)


Location of single-particle strength in closed-shell

(stable) nuclei

SRC

SRC theory

For example: protons in 208Pb

NIKH

EF (e,e’p) dataL. Lapikás

Nucl. Phys. A

553,297c (1993)JLab E97-006

Phys. Rev. Lett. 93, 182501 (2004) D. Rohe et al.

Elastic nucleon scattering

Reviewed in Prog. Part. Nucl. Phys. 52 (2004) 377-496


Remarks

• Given a Hamiltonian, a perturbation expansion can be generated for the single-particle propagator

• Dyson equation determines propagator in terms of nucleon self-energy

• Self-energy is causal and obeys dispersion relations relating its real and imaginary part

• Data constrained self-energy acts as ideal interface between ab initio theory and experiment


Propagator / Green’s function• Lehmann representation

• Any other single-particle basis can be used

• Overlap functions --> numerator

• Corresponding eigenvalues --> denominator

• Spectral function

• Spectral strength in the continuum

• Discrete transitions

• Positive energy —> see later

G`j(k, k0;E) =

X

m

h A0 | ak`j | A+1

m i h A+1m | a†k0`j | A

0 iE (EA+1

m EA0 ) + i

+X

n

h A0 | a

†k0`j | A1

n i h A1n | ak`j | A

0 iE (EA

0 EA1n ) i

S`j(k;E) =1

Im G`j(k, k;E) E "F

=X

n

h A1n | ak`j | A

0 i2(E (EA

0 EA1n ))

qSn`j n

`j(k) = h A1n | ak`j | A

0 i

S`j(E) =

Z 1

0dk k2 S`j(k;E)

En = EA0 EA1

n

Sn`j =

Zdk k2

h A1n | ak`j | A

0 i2 < 1


Propagator from Dyson Equation and “experiment”Equivalent to …

Self-energy: non-local, energy-dependent potential With energy dependence: spectroscopic factors < 1 ⇒ as extracted from (e,e’p) reaction

Schrödinger-like equation with:

Dyson equation also yields for positive energies

Elastic scattering wave function for protons or neutrons Dyson equation therefore provides: Link between scattering and structure data from dispersion relations

Spectroscopic factor

elE`j (r)

= h A+1

elE | a†r`j | A0 i

k2

2mn`j(k) +

Zdq q2

`j(k, q;En ) n

`j(q) = En n

`j(k)


How is it done in practice?• Dyson equation for discrete states

• with

• Work with so only 2nd derivative • Put coordinate on equidistant grid • Approximate 2nd derivative for i>1 by

• First point • Use “continuity” for r<0: for parity • using bc —> then diagonalize etc.

p2r2m

+~2`(`+ 1)

2mr2

n`j(r) +

Zdr0 r02`j(r, r

0; "n ) n`j(r

0) = "n n`j(r)

pr = i

~

@

@r+

1

r

un`j(r) = r n

`j(r)d2

dr2ri = (i 1/2) i = 1, 2, ...

u00(ri) = [u(ri+1) + u(ri1) 2u(ri)]/2

u00(r1) = [u(r2) + u(r0) 2u(r1)]/2

±1 ) u(r) = ±u(r)

u00(r1) = [u(r2) u(r1)]/2 ) +

u00(r1) = [u(r2) 3u(r1)]/2 )


Propagator• Same discretization

• Use matrix inversion now • Then spectral amplitude

• Spectral function

• and so on

G`j(r, r0;E) = G(0)

`j (r, r0;E) +

Zdr r2

Zdr0 r02G(0)

`j (r, r;E)`j(r, r0;E)G`j(r

0, r0;E)

S`j(r, r0;E) =

1

Im G`j(r, r

0;E)

S`j(r;E) =1

Im G`j(r, r;E)


Recent local DOM analysis --> towards

global0 50 100 150

R

uth

!/!

−210

1

210

410

610

810

1010

Sn112p+

Sn114p+

Sn116p+

0 50 100 150

Sn118p+

<10 MeVlabE

<20 MeVlab

10<E

<40 MeVlab

20<E

<100 MeVlab

40<E

0 50 100 150

Sn120

p+

0 50 100 150

Sn122p+

Sn124p+

[deg]cm"

0 50 100 150

#/d

!d

10

410

710

1010

1310

1510

Sn116n+

Sn118n+

0 50 100 150

Sn120n+

Sn124n+

0 50 100 150

#/d

!d

410

910

1410

1910

2410

Pb208n+

0 50 100 150

Pb208n+

[deg]cm"

[deg]cm"

0 50 100 150

R

uth

!/!

210

710

1210

1710

2210

2510

Pb208

p+

J. Mueller et al. PRC83,064605 (2011), 1-32


Elastic scattering data for protons and neutrons• Abundant for stable targets

[deg]cmθ0 50 100 150

Ω/d

σd

310

810

1310

1810

2310Fe54n+

0 50 100 150

Ω/d

σd

210

610

1010

1410

1810

2210

Ca40n+

0 50 100 150

Ca40n+

Ca48n+

[deg]cmθ0 50 100 150

1

210

410

610

810

1010

Mo92n+

0 50 100 150

210

510

810

1110

1410

1710

Ni58n+

0 50 100 150

Ni60n+

0 50 100 150

R

uth

σ/σ

210

710

1210

1710

2210

2510

Ca40p+

0 50 100 150

Ca42p+

Ca44p+

0 50 100 150

Ca48p+

<10 MeVlabE

<20 MeVlab10<E

<40 MeVlab20<E<100 MeVlab40<E

>100 MeVlabE

0 50 100 150

Rut

hσ/

σ

10

510

910

1310

1710

1910

Ni58p+

0 50 100 150

Ni60p+

0 50 100 150

Ni62p+

0 50 100 150

Ni64p+

<10 MeVlabE

<20 MeVlab10<E

<40 MeVlab20<E

<100 MeVlab40<E

0 50 100 150

R

uth

σ/σ

1

310

610

910

1210

1510

Fe54p+

0 50 100 150

Cr52p+

Ti50p+

[deg]CMθ0 50 100 150

210

710

1210

1710

2210

2610

Zr90p+

0 50 100 150

Mo92p+

[deg]cmθ


Local DOM ingredients and transfer reactions• Overlap function

• p and n optical potential

• ADWA (developed by Ron Johnson)

• MSU-WashU:-->

• 40,48Ca,132Sn,208Pb(d,p)

E 2 13

19.3 56

CH+ws

0.94 0.82 0.77 1.1

DOM

0.72 0.67 0.68 0.70

N. B. Nguyen, S. J. Waldecker, F. M. Nuñes, R. J. Charity, and W. H. Dickhoff

Phys. Rev. C84, 044611 (2011), 1-9

“SPECTR

OSC

OPIC

FACTO

RS”


132Sn(d,p)• Does it work when the potentials are extrapolated?

• Data: K.L. Jones et al., Nature 465, 454 (2010)

• Ed = 9.46 MeV 132Sn(d,p)133Sn

• CH89+ws --> S1f7/2 =1.1

• DOM --> S1f7/2 =0.72


Propagator in principle generates• Elastic scattering cross sections for p and n

• Including all polarization observables

• Total cross sections for n

• Reaction cross sections for p and n

• Overlap functions for adding p or n to bound states in Z+1 or N+1

• Plus normalization --> spectroscopic factor

• Overlap function for removing p or n with normalization

• Hole spectral function including high-momentum description

• One-body density matrix; occupation numbers; natural orbits

• Charge density

• Neutron distribution

• p and n distorted waves

• Contribution to the energy of the ground state from VNN

Understanding/Calculating Self-energy

DOM improvements

• Replace local energy-dependent HF potential by non-local (energy-independent potential) in order to calculate more properties below the Fermi energy like the charge density and spectral functions --> PRC82, 054306 (2010)

DOModel --> DOMethod-->DSelf-energyMethod


40Ca spectral functionRecent theoretical development: nonlocal “HF” self-energy --> below the Fermi energy WD, Van Neck, Charity, Sobotka, Waldecker, PRC82, 054306 (2010)

Old (p,2p) data from Liverpool

Below εF


Spectral functions and momentum distributions• 40Ca

0 1 2 3 4 5

k [fm-1

]

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

S(k

;E)

[MeV

-1 f

m3] s

1/2

0 1 2 3 4 5

k [fm-1

]

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

S(k

;E)

[MeV

-1 f

m3] d

3/2

0 1 2 3 4 5

k [fm-1

]

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

n(k

) [f

m3]

0 1 2 3 4 5

k [fm-1

]

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

k2 n

(k)

[fm

]

PRC 82, 054306 (2010)

Nucleon correlations

Charge density

Not a good reproduction of charge density even though mean square radius was fitted.

Related to local representation of the imaginary part of the self-energy --> independent of angular momentum --> must be abandoned to represent particle number correctly as well.


DOM extensions linked to ab initio FRPA

• Employ microscopic FRPA calculations of the nucleon self-energy to gain insight into future improvements of the DOM -->

• FRPA = Faddeev RPA --> Barbieri for a recent application see e.g. PRL103,202502(2009)

• Most important conclusions – Ab initio self-energy has imaginary part with a substantial non-locality

– Tensor force already operative for low-energy imaginary part

– Absorption above and below Fermi energy not symmetric

Phys. Rev. C84, 034616 (2011), 1-11S. J. Waldecker, C. Barbieri and W. H. Dickhoff


Volume integrals from microscopic FRPA relevant up to ~ 75 MeV

Volume integral for local imaginary potentials

Microscopic potentials: nonlocal --> depend strongly on l Here averaged

JW (E) = 4

Zdr r2 W (r, E)


Tensor force


Comparison with DOM for 40,48Ca

0

100

200

-100 0 100 200E - EF [MeV]

0

100

200

| JW

/ A

| [M

eV fm

3 ]

40Ca (p)

48Ca (p)


DOM extensions linked to ab initio treatment of SRC

• Employ microscopic calculations of the nucleon self-energy to gain insight into future improvements of the DOM -->

• CDBonn --> self-energy in momentum space for 40Ca • Most important conclusions

– Volume absorption below the Fermi energy is also nonlocal

– Reaction cross section comparable with DOM above ~ 80 MeV

H. Dussan, S. J. Waldecker, W. H. Dickhoff, H. Müther, and A. PollsPhys. Rev. C84, 044319 (2011), 1-16


Ab initio with CDBonn for 40Ca

• Dussan et al. PRC84, 044319 (2011); spectral functions available

0 1 2 3 4 5

k [fm-1

]

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

k2n(k

) [f

m]

DOM

CDBonn

4

Zdk k2 n(k) = 1


Non-locality of imaginary part• Fit non-local imaginary part for =0

• Integrate over one radial variable

• Predict volume integrals for higher Parameters

WNL(r, r0) = W0

pf(r)

pf(r0)H

r r0

`

`


Ab initio description of elastic scattering• Must be done much better

0.001

0.01

0.1

1

10

100

1000

0 20 40 60 80 100 120 140 160 180

dσ

/Ω [

mb

/sr]

θc.m. [deg]

Elab=95 MeV

Full DOM

0.001

0.01

0.1

1

10

100

1000

0 20 40 60 80 100 120 140 160 180

dσ

/Ω [

mb

/sr]

θc.m. [deg]

Elab=65 MeV

Full DOM

0.1

1

10

100

1000

0 20 40 60 80 100 120 140 160 180

dσ

/Ω [

mb

/sr]

θc.m. [deg]

Elab=26 MeVFull DOM

1

10

100

1000

0 20 40 60 80 100 120 140 160 180

dσ

/Ω [

mb/s

r]

θcm [deg]

Elab=11 MeV

DOM full


Ab initio calculation of elastic scattering n+40Ca

• ONLY treatment of short-range and tensor correlations

DOM & data

DOM l≤4

CDBonn l≤4


DOM predictions

• Use non-local “HF” potential and dispersive DOM potential to extrapolate to unstable Sn isotopes and predict (e.g.) properties of the last proton (based on the analysis of elastic scattering data on STABLE Sn nuclei)


Asymmetry dependence of imaginary potentials

• Volume –> small asymmetry dependence determined in 208Pb

• Neutron surface –> no strong dependencies on A or (N-Z)/A

• Proton surface absorption –> increases with increasing neutron number

Wvolume = W 0volume ±

N Z

AW 1

volume

W [

MeV

]

0

5

10

15Ca

48(a)

[MeV]FE−E−100 0 100 200

W [

MeV

]

0

5

10

15Ca

40(d)

Sn124(b) p

n

[MeV]FE−E−100 0 100 200

Sn116

(e)

volumesurface

[MeV]FE−E−100 0 100 200

Pb208

(c)

[MeV]FE−E−100 0 100 200

Mo92

(f)


Last proton in Sn nuclei (g9/2)Spectral function for different Sn isotopes

Sn 102 106 112 124 130 132

S 0.80 0.68 0.63 0.50 0.48 0.56

n 0.91 0.85 0.83 0.78 0.78 0.81


People involved

Wim Dickhoff Bob Charity

Lee Sobotka Helber Dussan Seth Waldecker Hossein Mahzoon Dong Ding

Carlo Barbieri, Surrey Arnau Rios, Surrey Arturo Polls, Barcelona Dimitri Van Neck, Ghent Herbert Müther, Tübingen

N.B. Nguyen & F. Nuñes


Dispersive Optical Model• Claude Mahaux 1980s

– connect traditional optical potential to bound-state potential

– crucial idea: use the dispersion relation for the nucleon self-energy

– smart implementation: use it in its subtracted form

– applied successfully to 40Ca and 208Pb in a limited energy window

– employed traditional volume and surface absorption potentials and a local energy-dependent Hartree-Fock-like potential

– Reviewed in Adv. Nucl. Phys. 20, 1 (1991)

• Radiochemistry group at Washington University in St. Louis: Charity and Sobotka propose to use it for a sequence of Ca isotopes —> data-driven extrapolations to the drip line - First results 2006 PRL

- Subsequently —> attention to data below the Fermi energy related to ground-state properties —> Dispersive Self-energy Method (DSM)


Optical potential <--> nucleon self-energy• e.g. Bell and Squires --> elastic T-matrix = reducible self-energy • Mahaux and Sartor

– relate dynamic (energy-dependent) real part to imaginary part

– employ subtracted dispersion relation

General dispersion relation for self-energy:

Calculated at the Fermi energy

Subtract

F = 12

(EA+1

0 EA0 ) + (EA

0 EA10 )

Re (E) = HF 1

PZ 1

E+T

dE0 Im (E0)

E E0 +1

PZ E

T

1dE0 Im (E0)

E E0

Re (F ) = HF 1

PZ 1

E+T

dE0 Im (E0)

F E0 +1

PZ E

T

1dE0 Im (E0)

F E0

Re (E) = Re gHF (F )

1

(F E)P

Z 1

E+T

dE0 Im (E0)

(E E0)(F E0)+

1

(F E)P

Z ET

1dE0 Im (E0)

(E E0)(F E0)

Adv. Nucl. Phys. 20, 1 (1991)


Nonlocal DOM implementation PRL112,162503(2014)

• Particle number --> nonlocal imaginary part • Microscopic FRPA & SRC --> different nonlocal properties above

and below the Fermi energy • Include charge density in fit • Describe high-momentum nucleons <--> (e,e’p) data from JLab

Implications

• Changes the description of hadronic reactions because interior nucleon wave functions depend on non-locality

• Consistency test of the interpretation of (e,e’p) possible • Independent “experimental” statement on size of three-body

contribution to the energy of the ground state--> two-body only:

E/A =1

2A

X

`j

(2j + 1)

Z 1

0dkk2

k2

2mn`j(k) +

1

2A

X

`j

(2j + 1)

Z 1

0dkk2

Z "F

1dE ES`j(k;E)


Differential cross sections and analyzing powers

0 50 100 150

A0

5

10

15

20

25

Ca40p+

0 50 100 150

Ca40n+

[deg]cmθ

0 50 100 150

[mb/

sr]

Ω/d

σd

510

1210

1910

2610

3310Ca40n+

< 10lab 0 < E < 20lab10 < E < 40lab20 < E < 100lab40 < E

> 100labE

0 50 100 150

Ca40p+

[deg]cmθ


Reaction (p&n) and total (n) cross sections

[MeV]LabE0 50 100 150 200

[mb]

σ

0

1000

2000

3000

4000

Ca40n+

totσreactσ

[mb]

σ

0

500

1000

1500

Ca40p+

totσ

Nonlocal imaginary self-energy: proton number --> 19.88 neutron number -> 19.79 S0d3/2= 0.76 S1s1/2 = 0.78 0.15 larger than NIKHEF analysis!

` 5


40Ca spectral function

Old (p,2p) data from Liverpool or (e,e’p) from Saclay

Below εF

-140 -120 -100 -80 -60 -40 -20 0

-310

-210

-110

1

1/2s

-140 -120 -100 -80 -60 -40 -20 0

1/2p

-140 -120 -100 -80 -60 -40 -20 0

]-1

Spec

tral

func

tion[

MeV

-310

-210

-110

1

3/2d

-140 -120 -100 -80 -60 -40 -20 0

3/2p

E [MeV]-140 -120 -100 -80 -60 -40 -20 0

-310

-210

-110

1

5/2d

E [MeV]-140 -120 -100 -80 -60 -40 -20 0

7/2f

Not part of fit!!


Correlations from nuclear reactions

Different optical potentials --> different reduction factors for transfer reactions Spectroscopic factors > 1 ??? PRL 93, 042501 (2004) HI PRL 104, 112701 (2010) Transfer

(e,e’p)

Recent summary —> Jenny Lee

Different reactions different results???

In (e,e’p) proton still has to get out of the nucleus —> optical potential Nucl. Phys. A553,297c (1993)

Consistency study in progress

Linking nuclear reactions and nuclear structure —> DOM


• Extracting information on correlations beyond the independent particle model requires optical potentials in (e,e’p), (d,p),(p,d),(p,pN), etc.

• Quality of ab initio to describe elastic scattering or optical potentials should be improved substantially and urgently

Linking nuclear reactions and nuclear structure

Coupled cluster calculation using overlap functions PRC86,021602(R)(2012) Probably limited to low energy

Green’s function result —> optical potential with emphasis on SRC only PRC84,044319(2011)

40Ca


High-momentum protons have been seen in nuclei!Jlab E97-006 Phys. Rev. Lett. 93, 182501 (2004) D. Rohe et al.

12C

• Location of high-momentum components • Integrated strength agrees with theoretical prediction Phys. Rev. C49, R17 (1994)

⇒ ~0.6 protons for 12C ⇒ ~10%


High-momentum componentsRohe, Sick et al. JLab data for Al and Fe (e,e’p) per proton

0 0.1 0.2 0.3 0.4 0.5Em [GeV]

10-14

10-13

10-12

10-11S no

rm( E

m, p

m )

[MeV

-4sr

-1] 0.650

0.5700.4900.4100.3300.250

pm (GeV / c)


Jefferson Lab data per proton• Pion/isobar contributions cannot be described • Rescattering contributes some cross section (Barbieri, Lapikas)

[MeV]mE0 100 200 300 400 500

]-1

sr

-4) [

MeV

m,p

mS(

E

-1410

-1310

-1210

-1110

-1010= 250 [MeV/c]mp= 330 [MeV/c]mp= 410 [MeV/c]mp= 490 [MeV/c]mp= 570 [MeV/c]mp= 650 [MeV/c]mp

0 2 4 6

r [fm]

0

0.02

0.04

0.06

0.08

0.1

0.12

!ch(r

) [f

m-3

]

Local version Charge density 40Ca radius correct… Non-locality essential PRC82,054306(2010) PRL112,162503(2014)

High-momentum nucleons —> JLab can also be described —> E/A


Critical experimental data

r [fm]0 2 4 6 8

]-3

[fm

ρ

0

0.02

0.04

0.06

0.08

0.1experimentcalculated


Historical perspective...• The following authors identify the single-particle propagator (or

self-energy) as central quantities in many-body systems

• but apart from qualitative features, they don’t answer what it looks like for a real system like a nucleus!

Abrikosov, Gorkov, Dzyaloshinski (Methods of Quantum Field Theory in Statistical Physics, 1963 Dover Revised edition 1975),

Pines (The Many-body Problem, 1961 Addison Wesley reissued 1997),

Nozieres (Theory of Interacting Fermi Systems, 1964 Addison-Wesley reissued 1997),

Thouless (The Quantum Mechanics of Many-body Systems, 1972 Dover reissue of second edition, 2014),

Anderson (Concepts in Solids, Benjamin 1963; World Scientific reissued 1998),

Schrieffer (Theory of Superconductivity, 1964 Benjamin revised 1983),

Migdal (Theory of Finite Fermi Systems and Applications to Atomic Nuclei (Interscience, 1967),

Fetter and Walecka (Quantum Theory of Many-particle Systems, 1971 Dover reissued 2003)


Energy of the ground state• Energy sum rule (Migdal, Galitski & Koltun)

• Not part of fit because it can only make a statement about the two-body contribution

• Result: – DOM ---> 7.91 MeV/A T/A ---> 22.64 MeV/A

– 10% of particles (momenta > 1.4 fm-1) provide ~⅔ of the binding energy!

– Exp. 8.55 MeV/A

– Three-body ---> 0.64 MeV/A “attraction” —> 1.28 MeV/A “repulsion”

– Argonne GFMC ~ 1.5 MeV/A attraction for three-body <--> Av18

E/A =1

2A

X

`j

(2j + 1)

Z 1

0dkk2

k2

2mn`j(k) +

1

2A

X

`j

(2j + 1)

Z 1

0dkk2

Z "F

1dE ES`j(k;E)

EN0 = h N

0 | H | N0 i

=1

2

Z "F

1dE

X

↵,

h↵|T |i+ E ↵, Im G(,↵;E) 1

2h N

0 | W | N0 i

with three-body interaction


Do elastic scattering data tell us about correlations? • Scattering T-matrix

• Free propagator

• Propagator

• Spectral representation

• Spectral density at positive energy

• Coordinate space

• Elastic scattering explicit

`j(k, k0;E) =

`j(k, k0;E) +

Zdqq2

`j(k, q;E)G(0)(q;E)`j(q, k0;E)

G(0)(q;E) =1

E ~2q2/2m+ i

Gp`j(k, k

0;E) =X

n

n+`j (k)

hn+`j (k0)

i

E EA+1n + i

+X

c

Z 1

Tc

dE0cE0

`j (k)hcE0

`j (k0)i

E E0 + i

G`j(k, k0;E) =

(k k0)

k2G(0)(k;E) +G(0)(k;E)`j(k, k

0;E)G(0)(k;E)

Sp`j(r, r

0;E) =X

c

cE`j (r)

cE`j (r

0)

elE`j (r) =

2mk0~2

1/2 j`(k0r) +

Zdkk2j`(kr)G

(0)(k;E)`j(k, k0;E)

Sp`j(k, k

0;E) =i

2

hGp

`j(k, k0;E+)Gp

`j(k, k0;E)

i=

X

c

cE`j (k)

cE`j (k

0)


How is it done?• Solve

• with

• See discussion by Arturo Polls • Note: irreducible self-energy is a complex quantity • Cross section as shown in first lecture

`j(k, k0;E) =

`j(k, k0;E) +

Zdqq2

`j(k, q;E)G(0)(q;E)`j(q, k0;E)

G(0)(q;E) =1

E ~2q2/2m+ i

QMPT 540

Elastic nucleon scattering• Scattering from potential

• Potential real —> phase shift real

• Scattering amplitude

• Elastic nucleon scattering – Involves reducible self-energy (see also later)

– Scattering amplitude

– Phase shift now includes imaginary part when potential is absorptive

hk0| S`j(E) |k0i e2i`j = 1 2i

mk0~2

hk0|`j(E) |k0i

k0|S(E)|k0 =1 2i

mk0

2

k0|T (E)|k0

e2i

fm0s,ms(,) = 4m2

~2 hk0m0s|(E)|kmsi

f(,) =

l

2l + 1k0

mk0

2

k0|T (E)|k0P(cos )

=

2 + 1k0

ei sin P(cos )

QMPT 540

Spin-orbit physics included• Scattering amplitude

• Rewrite – with

– then

– Unpolarized differential cross section

fm0s,ms(,) = 4m2

~2 hk0m0s|(E)|kmsi

[f(,)] = F()I + · nG()

F() =1

2ik

1X

`=0

(`+ 1)

e2i`+ 1

+ `

e2i` 1

P`(cos )

G() = sin

2k

1X

`=1

e2i`+ e2i`

P 0`(cos )

d

d

unpol

= |F|2 + |G|2

n =k k0

|k k0| =k k0

sin

0

50

1000 1 2 3 4 5 6 7

0

0.05

0.1

0.15

0.2

0.25

r [fm]

E [MeV]

S −

|χ |2 [

MeV

−1 fm

−1]

• Inelastically! • Zero when there is no absorption!


Adding an s1/2 neutron to 40Ca

Multiplied by r2


d3/2

• One node now

0

50

1000 1 2 3 4 5 6 7

0

0.1

0.2

0.3

0.4

r [fm]

l = 2

E [MeV]

S−| χ

|2 [MeV

−1 fm

−1]


No nodes• Asymptotically determined by inelasticity

0

50

100

150 0 1 2 3 4 5 6 7

0

0.1

0.2

0.3

0.4

0.5

0.6

r [fm]

l = 4 ( g92)

E [MeV]

S −

| χ

|2 [M

eV−1

fm−1

]


Determine location of bound-state strength• Fold spectral function with bound state wave function

• —> Addition probability of bound orbit • Also removal probability

• Overlap function

• Sum rule

Sn+`j (E) =

Zdr r2

Zdr0 r02n

`j (r)Sp`j(r, r

0;E)n`j (r0)

Sn`j (E) =

Zdrr2

Zdr0r02n

`j (r)Sh`j(r, r

0;E)n`j (r0)

qSn`j

n`j (r) = h A1

n | ar`j | A0 i

1 = nn`j + dn`j=

Z "F

1dE Sn

`j (E)+

Z 1

"F

dE Sn`j (E)


Spectral function for bound states• [0,200] MeV —> constrained by elastic scattering data

-100 -50 0 50 100

E [MeV]

10-4

10-3

10-2

10-1

100

Sn(E

) [M

eV-1

]0s

1/2

0p3/2

0p1/2

0d5/2

0d3/2

1s1/2

0f7/2

0f5/2

εF

40Ca

proton number --> 19.88 neutron number -> 19.79 S0d3/2= 0.76 S1s1/2 = 0.78 0.15 larger than NIKHEF analysis!

PRC90, 061603(R) (2014)


Compared with ab initio —> SRC only• CDBonn probably too soft • SRC relevant at higher energy

0 20 40 60 80 100 120 140 160 180 200E [MeV]

10-4

10-3

10-2

Sn(E

) [

MeV

-1]

0s1/2

0p3/2

0p1/2

0d5/2

0d3/2

1s1/2

DOM

CDBonn

PRC90, 061603(R) (2014)

• Orbit closer to the continuum —> more strength in the continuum

• Note “particle” orbits

• Drip-line nuclei have valence orbits very near the continuum


Quantitatively

Table 1: Occupation and depletion numbers for bound orbits in 40Ca.dnlj [0, 200] depletion numbers have been integrated from 0 to 200 MeV. Thefraction of the sum rule that is exhausted, is illustrated by nn`j + dn`j ["F , 200].Last column dnlj [0, 200] depletion numbers for the CDBonn calculation.

orbit nn`j dn`j [0, 200] nn`j + dn`j ["F , 200] dn`j [0, 200]DOM DOM DOM CDBonn

0s1/2 0.926 0.032 0.958 0.0350p3/2 0.914 0.047 0.961 0.0361p1/2 0.906 0.051 0.957 0.0380d5/2 0.883 0.081 0.964 0.0401s1/2 0.871 0.091 0.962 0.0380d3/2 0.859 0.097 0.966 0.0410f7/2 0.046 0.202 0.970 0.0340f5/2 0.036 0.320 0.947 0.036

PRC90, 061603(R) (2014)


Neutron spectral function in 48Ca• Neutrons in 48Ca less correlated <-> 40Ca but qualitatively similar

Preliminar

y!


Proton spectral function in 40Ca• Learning how to deal with Coulomb in momentum space

Preliminar

y!


Protons in 48Ca• Protons in 48Ca more correlated than in 40Ca

Preliminar

y!


Quantitative comparison of 40Ca and 48Ca

Spectroscopic factors

40Ca p 48Ca n 48Ca

0d3/2 0.76 0.65 ↓ 0.80 ↑

1s1/2 0.78 0.71 ↓ 0.83 ↑

0f7/2 0.73 0.59 ↓ 0.84 ↑


Comparison for d3/2 and s1/2 protons

d3/2


In progress• 48Ca —> charge density has been measured • Recent neutron elastic scattering data —> PRC83,064605(2011) • Local DOM OLD Nonlocal DOM NEW

0 50 100 150

Ω/d

σd

510

1210

1910

2610

3310Ca48n+

0 < Elab < 1010 < Elab < 2020 < Elab < 4040 < Elab < 100Elab > 100

0 50 100 150

Ca48p+

[deg]cmθ

0 2 4 6 8r [fm]

0

0.02

0.04

0.06

0.08

0.1

ρ

DOM p-foldedExp chargeDOM n-folded

48Ca


Results for 48Ca• Density distributions • DOM —> neutron distribution —> Rn-Rp

0 2 4 6 80

0.02

0.04

0.06

0.08Neutron matter distributionDOMExperiment

48Ca nuclear charge distribution

r [fm]

48Ca Densities

Eur. Phys. J. A (2014) C.J. Horowitz, K.S. Kumar, and R. Michaels

DOM 0.249±0.023


Rn-Rp for 48Ca

• Charge density for 40Ca • Charge density for 48Ca • Neutrons in 40Ca well constrained • 8 extra neutrons in 48Ca constrained by new elastic scattering

data at low energy and total cross sections up to 200 MeV, level structure, and particle number

• neutron skin “large” • neutron distribution smooth like the charge density

Question • How important is the “straightjacket effect” for the relation

between the slope of the symmetry energy and Rn-Rp?


Hagen et al. based on PRL109,032502(2012)• Coupled-cluster ab initio

• Chiral forces have limitations

• Coupled-cluster method also

• 48Ca


Can we get L from the DOM?• Perhaps…

• We could calculate energy density as a function of r for both nuclei…

• Identify the normal density part from the interior…

0 2 4 6 80

0.02

0.04

0.06

0.08Neutron matter distributionDOMExperiment

48Ca nuclear charge distribution

40Ca 48Ca


Mean-field for 208Pb neutrons B. Shufang, C J Horowitz and R Michaels J. Phys. G: Nucl. Part. Phys. 39 (2012) 015104


Conclusions• It is possible to link nuclear reactions and nuclear structure • Vehicle: nonlocal version of Dispersive Optical Model (Green’s

function method) pioneered by Mahaux —> DSM • Can be used as input for analyzing nuclear reactions • Can predict properties of exotic nuclei • “Benchmark” for ab initio calculations: e.g. VNNN —> binding • Can describe ground-state properties

– charge density & momentum distribution

– spectral properties including high-momentum Jefferson Lab data

• Elastic scattering determines depletion of bound orbitals

• Outlook: reanalyze many reactions with nonlocal potentials... • For N ≷ Z exhibits sensitivity to properties of neutrons —> weak

charge —> neutron skin and perhaps more


M. van Batenburg & L. Lapikás from 208Pb (e,e´p) 207Tl NIKHEF 2001 data (one of the last experiments)

Up to 100 MeV missing energy and 270 MeV/c missing momentum

Covers the whole mean-field domain!!

Occupation of deeply-bound proton levels from EXPERIMENT

Confirms predictions for depletion

SRC LRC

n(0) ⇒ 0.85 Reid 0.87 Argonne V18 0.89 CDBonn/N3LO

Nuclear matter

why do green’s functions? - github...

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