Why do Green’s functions? - GitHub...
Transcript of 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)
reactions and structure
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
reactions and structure
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
reactions and structure
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
reactions and structure
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)
reactions and structure
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 )
reactions and structure
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)
reactions and structure
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
reactions and structure
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
reactions and structure
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”
reactions and structure
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
reactions and structure
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
Understanding/Calculating Self-energy
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
Understanding/Calculating Self-energy
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.
Understanding/Calculating Self-energy
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
Understanding/Calculating Self-energy
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)
Understanding/Calculating Self-energy
Tensor force
Understanding/Calculating Self-energy
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)
reactions and structure
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
Understanding/Calculating Self-energy
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
Understanding/Calculating Self-energy
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
`
`
Understanding/Calculating Self-energy
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
Understanding/Calculating Self-energy
Ab initio calculation of elastic scattering n+40Ca
• ONLY treatment of short-range and tensor correlations
DOM & data
DOM l≤4
CDBonn l≤4
reactions and structure
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)
reactions and structure
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)
reactions and structure
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
reactions and structure
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
reactions and structure
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)
reactions and structure
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)
reactions and structure
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)
reactions and structure
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
reactions and structure
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
reactions and structure
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!!
reactions and structure
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
reactions and structure
• 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
reactions and structure
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%
reactions and structure
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)
reactions and structure
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
reactions and structure
Critical experimental data
r [fm]0 2 4 6 8
]-3
[fm
ρ
0
0.02
0.04
0.06
0.08
0.1experimentcalculated
reactions and structure
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)
reactions and structure
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
reactions and structure
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)
reactions and structure
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!
reactions and structure
Adding an s1/2 neutron to 40Ca
Multiplied by r2
reactions and structure
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]
reactions and structure
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
]
reactions and structure
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)
reactions and structure
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)
reactions and structure
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
reactions and structure
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)
reactions and structure
Neutron spectral function in 48Ca• Neutrons in 48Ca less correlated <-> 40Ca but qualitatively similar
Preliminar
y!
reactions and structure
Proton spectral function in 40Ca• Learning how to deal with Coulomb in momentum space
Preliminar
y!
reactions and structure
Protons in 48Ca• Protons in 48Ca more correlated than in 40Ca
Preliminar
y!
reactions and structure
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 ↑
reactions and structure
Comparison for d3/2 and s1/2 protons
d3/2
reactions and structure
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
reactions and structure
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
reactions and structure
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?
reactions and structure
reactions and structure
Hagen et al. based on PRL109,032502(2012)• Coupled-cluster ab initio
• Chiral forces have limitations
• Coupled-cluster method also
• 48Ca
reactions and structure
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
reactions and structure
Mean-field for 208Pb neutrons B. Shufang, C J Horowitz and R Michaels J. Phys. G: Nucl. Part. Phys. 39 (2012) 015104
reactions and structure
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
reactions and structure
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