Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$...

25
Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) Carlo Barbieri — University of Surrey ARIS2014 – Tokyo, June 2 nd .6 th , 2014 18 20 22 24 26 28 30 32 0 10 20 30 40 50 60 N S 2n [MeV] Ar K Ca Sc Ti Ω24 MeV Λ SRG 2.0 fm 1 15 F 17 F 23 F 25 F 29 F 180 160 140 120 100 80 13 N 15 N 21 N 23 N 27 N E g.s. MeV Exp 2N3Nind 2N3Nfull Collaborators: A. Cipollone, CB, P. Navrá7l: Phys. Rev. Le<. 111, 062501 (2013) V. Somà, A. Cipollone, CB, P. Navrá7l, T. Duguet: Phys. Rev. C 89, 061301R (2014)

Transcript of Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$...

Page 1: Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$ Collaborators:** * ... b ⇥ k Have a systematic expansion of the Hamiltonian ... trons and

Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni)

Carlo Barbieri — University of Surrey

ARIS2014!!–!Tokyo,!June!2nd.!6th,!2014!!

18 20 22 24 26 28 30 320

10

20

30

40

50

60

N

S 2n [

MeV

]

Ar

K

CaSc

Ti

!

!

!

! !

"

"

"

"

"

!

!

!!

!

"

"

"" "

#!"24 MeV#SRG"2.0 fm

$1

15F 17F 23F 25F 29F

$180

$160

$140

$120

$100

$80

13N 15N 21N 23N 27N

Eg.s.!MeV

"

Exp2N%3N#ind$2N%3N#full$

Collaborators:***

***********A.*Cipollone,*CB,*P.*Navrá7l:* ************ *Phys.*Rev.*Le<.*111,*062501*(2013)*V.*Somà,*A.*Cipollone,*CB,*P.*Navrá7l,*T.*Duguet:**Phys.*Rev.*C*89,******061301R*(2014)*

*

Page 2: Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$ Collaborators:** * ... b ⇥ k Have a systematic expansion of the Hamiltonian ... trons and

July 10-12 2010, 4th TIGRESS Science Workshop, SFU Sonia Bacca

Nuclear Forces Frontiers

34

Use effective degrees of freedom: p,n,pions

Effective Field Theory: Bridges the non-perturbative low-energy regime of QCD with forces among nucleons

L =⇤

k

ck

�Q

�b

⇥k

Have a systematic expansion of the Hamiltonian in terms of diagrams

Construct the most general Hamiltonian which is consistent with the chiral symmetry of QCD

(3NFs arise naturally at N2LO)!

Modern realistic nuclear forces

2

8 201614Neutron Number (N) Neutron Number (N)

8 201614

s 1/2

(c) G-matrix NN + 3N (!) forces

d3/2

d5/2

NN NN + 3N (!)

d3/2

d5/2

s 1/2

NN + 3N (N LO)

NN NN + 3N (!)

low k (d) V NN + 3N (!,N LO) forces 2

2

Sin

gle

-Par

ticl

e E

ner

gy (

MeV

)

4

-4

0

-8

Sin

gle

-Par

ticl

e E

ner

gy (

MeV

)

8 201614

d3/2

d5/2s 1/2

(a) Forces derived from NN theory

V

G-matrix

(b) Phenomenological forces

d3/2

s 1/2

d5/2

USD-B

SDPF-M

8 201614Neutron Number (N)Neutron Number (N)

low k

4

-4

0

-8

FIG. 2: Single-particle energies of the neutron d5/2, s1/2 andd3/2 orbitals measured from the energy of 16O as a function ofneutron number N . (a) SPE calculated from a G matrix andfrom low-momentum interactions Vlow k. (b) SPE obtainedfrom the phenomenological forces SDPF-M [14] and USD-B [15]. (c,d) SPE including contributions from 3N forces dueto! excitations and chiral EFT 3N interactions at N2LO [26].The changes due to 3N forces based on ! excitations arehighlighted by the shaded areas.

sures N = 8, 14, 16, and 20. The evolution of the SPEis due to interactions as neutrons are added. For theSPE based on NN forces in Fig. 2 (a), the d3/2 orbitaldecreases rapidly as neutrons occupy the d5/2 orbital,and remains well-bound from N = 14 on. This leadsto bound oxygen isotopes out to N = 20 and puts theneutron drip-line incorrectly at 28O. This result appearsto depend only weakly on the renormalization methodor the NN interaction used. We demonstrate this byshowing SPE calculated in the G matrix formalism [11],which sums particle-particle ladders, and based on low-momentum interactions Vlow k [12] obtained from chiralNN interactions at next-to-next-to-next-to-leading order(N3LO) [13] using the renormalization group. Both cal-culations include core polarization e!ects perturbatively(including diagram Fig. 3 (d) with the " replaced by anucleon and all other second-order diagrams) and startfrom empirical SPE [14] in 17O. The empirical SPEs con-tain e!ects from the core and its excitations, includinge!ects due to 3N forces.We next show in Fig. 2 (b) the SPE obtained from the

phenomenological forces SDPF-M [14] and USD-B [15]that have been fit to reproduce experimental binding en-

ergies and spectra. This shows a striking di!erence com-pared to Fig. 2 (a): As neutrons occupy the d5/2 orbital,with N evolving from 8 to 14, the d3/2 orbital remainsalmost at the same energy and is not well-bound out toN = 20. The dominant di!erences between Figs. 2 (a)and (b) can be traced to the two-body monopole compo-nents, which determine the average interaction betweentwo orbitals. The monopole components of a general two-body interaction V are given by an angular average overall possible orientations of the two nucleons in orbitals ljand l!j! [16],

V monoj,j! =

!

m,m!

!jm j!m!|V |jm j!m!""

!

m,m!

1 , (1)

where the sum over magnetic quantum numbers m andm! can be restricted by antisymmetry (see [17, 18] fordetails). The SPE of the orbital j is e!ectively shifted byV monoj,j! multiplied by the occupation number of the orbital

j!. This leads to the change in the SPE and determinesshell structure and the location of the drip-line [17–20].The comparison of Figs. 2 (a) and (b) suggests that the

monopole interaction between the d3/2 and d5/2 orbitalsobtained from NN theories is too attractive, and that theoxygen anomaly can be solved by additional repulsivecontributions to the two-neutron monopole components,which approximately cancel the average NN attractionon the d3/2 orbital. With extensive studies based on NNforces, it is unlikely that such a distinct property wouldhave been missed, and it has been argued that 3N forcesmay be important for the monopole components [21].Next, we show that 3N forces among two valence neu-

trons and one nucleon in the 16O core give rise to repul-sive monopole interactions between the valence neutrons.While the contributions of the FM 3N force to otherquantities can be di!erent, the shell-model configurationscomposed of valence neutrons probe the long-range partsof 3N forces. The repulsive nature of this 3N mechanismcan be understood based on the Pauli exclusion princi-ple. Figure 3 (a) depicts the leading contribution to NNforces due to the excitation of a ", induced by the ex-change of pions with another nucleon. Because this isa second-order perturbation, its contribution to the en-ergy and to the two-neutron monopole components hasto be attractive. This is part of the attractive d3/2-d5/2monopole component obtained from NN forces.In nuclei, the process of Fig. 3 (a) leads to a change of

the SPE of the j,m orbital due to the excitation of a corenucleon to a ", as illustrated in Fig. 3 (b) where the ini-tial valence neutron is virtually excited to another j!,m!

orbital. As discussed, this lowers the energy of the j,morbital and thus increases its binding. However, in nucleithis process is forbidden by the Pauli exclusion princi-ple, if another neutron occupies the same orbital j!,m!,as shown in Fig. 3 (c). The corresponding contributionmust then be subtracted from the SPE change due toFig. 3 (b). This is taken into account by the inclusion

Chiral EFT for nuclear forces:

Need at LEAST 3NF!!! (“cannot” do RNB physics without…)!

Single particle spectrum at Efermi:!!

Saturation of nuclear matter:!

[T. Otsuka et al.,!Phys Rev. Lett 105, !032501 (2010)]!

[A. Carbone et al., Phy.s Rev. C 88, 044302!!(2013)]!

SYMMETRIC NUCLEAR MATTER WITH CHIRAL THREE- . . . PHYSICAL REVIEW C 88, 044302 (2013)

Note that the N2LO potential yields a poorer reproduction ofthe phase shifts for selected partial waves compared to thericher N3LO force.

Most nuclear matter calculations using chiral forces havebeen performed within a perturbative framework startingfrom evolved interactions. In Ref. [43], convergence hasbeen analyzed order by order in many-body perturbationtheory. Results have been obtained up to third order, includingparticle-particle and hole-hole propagation [43]. In principle,the equation of state should be independent of the evolutionscales in the 2NF and the 3NF. Moreover, in the perturbativeregime, results should only be mildly dependent on the order inperturbation theory. Our nonperturbative calculations includecontributions to all orders and hence are neither limited to theperturbative regime nor dependent on the order of perturbationtheory. If the diagrammatic summation is complete, it shouldlead to scale-invariant results.

We test this hypothesis by performing calculations atdifferent evolution scales, in both the two- and the three-body sectors. We evolve the 2NF using a free-space SRGtransformation [37]. The transformation renormalizes the 2NF,suppressing off-diagonal matrix elements and giving rise toa universal low-momentum interaction. The SRG evolutionflow also induces many-body forces, which should be takeninto account to keep the calculation complete. Following thephilosophy of Ref. [43], we incorporate the effect of inducedforces through the refitting of the cD and cE LECs to the 3Hbinding energy and 4He matter radius. We use the values givenin Table I of [43]. Note that in this process we assume thatthe operatorial and momentum structures of the original andthe induced 3NFs are the same. Furthermore, we explore thedependence of our results on the 3NF cutoff, !3NF, appearingin the density-dependent 2NF. A more complete calculationwould require running a SRG evolution including the 3NF [41].

We present the results of this exploration in Fig. 8.Numerical calculations obtained using the SRG on the 2NFhave a saturation point which is much closer to the empiricalvalue when compared to the original force. Moreover, ifthe 2NF has been SRG-evolved, the results are somewhatindependent of the cutoff. Overall, one can say that themore the 2NF is evolved downward, the more attractive the

0 0.08 0.16 0.24 0.32

Density, ! [fm-3

]

-20

-15

-10

-5

0

5

10

Ene

rgy/

nucl

eon,

E/A

[M

eV]

N3LO+N2LO dd

"/#3NF=2.0/2.5 fm-1

"/#3NF=2.0/2.0 fm-1

"/#3NF=1.8/2.0 fm-1

T=5 MeV

FIG. 8. (Color online) SCGF results for the energy per nucleonof SNM as a function of the density at a temperature of T = 5 MeV.Different lines represent different choices of cutoffs for the 2NF, ",and the 3NF, !3NF.

saturation curve becomes. This effect is a consequence of theshift in importance between the 2NF and the induced 3NFassociated with the SRG. There is also a small dependence on!3NF, but the differences agree well with those presented inRef. [43].

The large differences between the results obtained withevolved and unevolved forces is striking. If correlations andinduced many-body forces had been fully taken into account,one would have expected a much closer agreement betweenthe results. This difference might indicate that the assumptionsassociated with induced 3NFs are not necessarily robust.Missing induced three-body forces, which up to now havenot been included in SNM calculations, could resolve thisdiscrepancy. Alternatively, the difference is also an indicationof missing many-body effects such as, for instance, higherorders in the treatment of the 3NF. It must be emphasized thatthe present way to proceed when applying SRG evolutionin infinite matter should be improved by carrying out theevolution on a full Hamiltonian with both two- and three-bodyforces. Recently, improvements toward the solution of thisproblem have been presented for calculations in pure neutronmatter [41], where a full Hamiltonian has been consistentlyevolved. All in all, our results seem to contradict the idea thatinduced 3NFs can be treated simply in nuclear matter.

In terms of evolved interactions, our nonperturbativecalculations can be used to check whether the perturbativeregime is actually reached. To this end, we compare, inFig. 9, our results to the perturbative calculations presentedin Ref. [43]. The BHF and SCGF calculations have beenperformed with a SRG-evolved 2NF and a 3NF with the samecut-offs, "/!3NF = 2.0/2.0 fm!1. Whereas the Bruecknerresults have been obtained with a zero-temperature code, theSCGF calculations have been extrapolated to zero temperatureby means of a simple procedure. At low temperatures,the Sommerfeld expansion indicates that the effect of tem-perature is quadratic and is the same, but with opposite sign,for the energy and the free energy [47]. Consequently, thesemi-sum of both thermodynamical potentials is an estimateof the zero-temperature energy. We obtain an extremelygood agreement between both many-body approaches and

0 0.08 0.16 0.24 0.32

Density, ! [fm-3

]

-20

-15

-10

-5E

nerg

y/nu

cleo

n, E

/A [

MeV

]Hebeler et al.BHFSCGF

T=0 MeV "/#3NF=2.0 fm-1

FIG. 9. (Color online) Comparison of results for the energy pernucleon of SNM obtained with different approaches using the sameSRG-evolved 2NF and a 3NF. Circles correspond to extrapolatedSCGF results, whereas squares are BHF calculations at T = 0 MeV.Diamonds correspond to the results of Hebeler et al. [43].

044302-7

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July 10-12 2010, 4th TIGRESS Science Workshop, SFU Sonia Bacca

Nuclear Forces Frontiers

34

Use effective degrees of freedom: p,n,pions

Effective Field Theory: Bridges the non-perturbative low-energy regime of QCD with forces among nucleons

L =⇤

k

ck

�Q

�b

⇥k

Have a systematic expansion of the Hamiltonian in terms of diagrams

Construct the most general Hamiltonian which is consistent with the chiral symmetry of QCD

N3LO (Λ = 500Mev/c) chiral NN interaction

(3NFs arise naturally at N2LO)!

N2LO (Λ = 400Mev/c) chiral 3N interaction

SRG evolution to !=2.0 fm-1!

VNN ! V3N induced !

V3N full!

“induced” Hamiltoninan!

“full” Hamiltoninan![Jurgenson,!Navrá>l,!Furnstahl,!!

Phys.!Rev.!LeE.!103,!082501!(2009);!Hebeler,!Phys.!Rev.!C!85,!021002!(2012)]!

Chiral Nuclear forces - SRG evolved

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Faddeev-RPA in two words… Particle vibration coupling is the main cause driving the distribution of particle strength—a least close to the Fermi surface…�

n� p�

≡!!!par7cle!

≡!hole!

…these modes are all resummed exactly and to all orders in a ab-initio many-body expansion.!

Page 5: Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$ Collaborators:** * ... b ⇥ k Have a systematic expansion of the Hamiltonian ... trons and

neutron*removal�

neutron*addi1on�

sca4ering�

56Ni�

One-body Green’s function (or propagator) describes the motion of quasi- particles and holes: …this contains all the structure information probed by nucleon transfer (spectral function):

2

15]. The method has later been applied to atoms andmolecules [12, 16] and recently to 56Ni [17] and 48Ca [18].The ab initio results of Ref. [18] are in good agreementwith (e, e!p) data for spectroscopic factors from Ref. [19]and also show that the configuration space needed for theincorporation of long-range (surface) correlations is muchlarger than the space that can be utilized in large-scaleshell-model diagonalizations. In Ref. [20], the FRPA wasemployed to calculate proton scattering on 16O and ob-tain results for phase shifts and low-lying states in 17F.However, the properties of the self-energy at larger scat-tering energies which are now of great interest for thedevelopments of DOM potentials was not addressed. Inparticular, one may expect to extract useful informationregarding the functional form of the DOM from a studyof the self-energy for a sequence of calcium isotopes. Itis the purpose of the present work to close this gap. Wehave chosen in addition to 40Ca and 48Ca also to include60Ca, since the latter isotope was studied with a DOMextrapolation in Refs. [8, 9]. Some preliminary results ofthese FRPA calculations for spectroscopic factors werereported in Ref. [14] but the emphasis in the present workis on the properties of the microscopically calculated self-energies. The resulting analysis is intended to providea microscopic underpinning of the qualitative features ofempirical optical potentials. Additional information con-cerning the degree and form of the non-locality of boththe real and imaginary parts of the self-energy will alsobe addressed because it is of importance to assess thecurrent local implementations of the DOM method.In Sec. II A we introduce some of the basic properties

for the analysis of the self-energy. The ingredients of theFRPA calculation are presented in Sec. II C. The choiceof model space and realistic nucleon-nucleon (NN) inter-action are discussed in Sec. III. We present our resultsin Sec. IV and finally draw conclusions in Sec. V.

II. FORMALISM

In the Lehmann representation, the one-body Green’sfunction is given by

g!"(E) =!

n

!!A0 |c!|!

A+1n "!!A+1

n |c†"|!A0 "

E # (EA+1n # EA

0 ) + i!

+!

k

!!A0 |c

†"|!

A"1k "!!A"1

k |c!|!A0 "

E # (EA0 # EA"1

k )# i!, (1)

where ", #, ..., label a complete orthonormal basis setand c! (c†") are the corresponding second quantizationdestruction (creation) operators. In these definitions,|!A+1

n ", |!A"1k " are the eigenstates, and EA+1

n , EA"1k

the eigenenergies of the (A ± 1)-nucleon isotope. Thestructure of Eq. (1) is particularly useful for our pur-poses. At positive energies, the residues of the first term,!!A+1

n |c†!|!A0 ", contain the scattering wave functions for

the elastic collision of a nucleon o" the |!A0 " ground state,

while at negative energies they give information on fi-nal states of the nucleon capture process. Consequently,the second term has poles below the Fermi energy (EF )which carry information about the removal of a nucleonand therefore clarify the structure of the target state |!A

0 "itself. Green’s function theory provides a natural frame-work for describing physics both above and below theFermi surface in a consistent manner.The propagator (1) can be obtained as a solution of

the Dyson equation,

g!"(E) = g(0)!" (E) +!

#$

g(0)!# (E)#%#$(E) g$"(E) , (2)

in which g(0)(E) is the propagator for a free nucleon(moving only with its kinetic energy). #%(E) is the irre-ducible self-energy and represents the interaction of theprojectile (ejectile) with the target nucleus. Feshbach,developed a formal microscopic theory for the optical po-tential already in Ref. [21, 22] by projecting the many-body Hamiltonian on the subspace of scattering states.It has been proven that if Feshbach’s theory is extendedto a space including states both above and below theFermi surface, the resulting optical potential is exactlythe irreducible self-energy #%(E) [23] (see also Ref. [24]and Ref. [25] for a shorter demonstration).The above equivalence with the microscopic optical po-

tential is fundamental for the present study, since theavailable knowledge from calculations based on Green’sfunction theory can be used to suggest improvements ofoptical models. In particular, in the DOM, the dispersionrelation obeyed by #%(E) is used to reduce the number ofparameters and to enforce the e"ects of causality. Thusthe DOM potentials can also be thought of as a repre-sentation of the nucleon self-energy.

A. Self-Energy

For a J = 0 nucleus, all partial waves ($, j, %) aredecoupled, where $,j label the orbital and total angu-lar momentum and % represents its isospin projection.The irreducible self-energy in coordinate space (for ei-ther a proton or a neutron) can be written in terms ofthe harmonic-oscillator basis used in the FRPA calcula-tion, as follows:

#%(x,x!;E) =!

&jmj'

I&jmj($,&)

$

"

!

na,nb

Rna&(r)#%ab(E)Rnb&(r

!)

#

(I&jmj($!,&!))#, (3)

where x % r,&, % . The spin variable is represented by&, n is the principal quantum number of the harmonicoscillator, and a % (na, $, j, %) (note that for a J = 0 nu-cleus the self-energy is independent ofmj). The standardradial harmonic-oscillator function is denoted by Rn&(r),

2

15]. The method has later been applied to atoms andmolecules [12, 16] and recently to 56Ni [17] and 48Ca [18].The ab initio results of Ref. [18] are in good agreementwith (e, e!p) data for spectroscopic factors from Ref. [19]and also show that the configuration space needed for theincorporation of long-range (surface) correlations is muchlarger than the space that can be utilized in large-scaleshell-model diagonalizations. In Ref. [20], the FRPA wasemployed to calculate proton scattering on 16O and ob-tain results for phase shifts and low-lying states in 17F.However, the properties of the self-energy at larger scat-tering energies which are now of great interest for thedevelopments of DOM potentials was not addressed. Inparticular, one may expect to extract useful informationregarding the functional form of the DOM from a studyof the self-energy for a sequence of calcium isotopes. Itis the purpose of the present work to close this gap. Wehave chosen in addition to 40Ca and 48Ca also to include60Ca, since the latter isotope was studied with a DOMextrapolation in Refs. [8, 9]. Some preliminary results ofthese FRPA calculations for spectroscopic factors werereported in Ref. [14] but the emphasis in the present workis on the properties of the microscopically calculated self-energies. The resulting analysis is intended to providea microscopic underpinning of the qualitative features ofempirical optical potentials. Additional information con-cerning the degree and form of the non-locality of boththe real and imaginary parts of the self-energy will alsobe addressed because it is of importance to assess thecurrent local implementations of the DOM method.In Sec. II A we introduce some of the basic properties

for the analysis of the self-energy. The ingredients of theFRPA calculation are presented in Sec. II C. The choiceof model space and realistic nucleon-nucleon (NN) inter-action are discussed in Sec. III. We present our resultsin Sec. IV and finally draw conclusions in Sec. V.

II. FORMALISM

In the Lehmann representation, the one-body Green’sfunction is given by

g!"(E) =!

n

!!A0 |c!|!

A+1n "!!A+1

n |c†"|!A0 "

E # (EA+1n # EA

0 ) + i!

+!

k

!!A0 |c

†"|!

A"1k "!!A"1

k |c!|!A0 "

E # (EA0 # EA"1

k )# i!, (1)

where ", #, ..., label a complete orthonormal basis setand c! (c†") are the corresponding second quantizationdestruction (creation) operators. In these definitions,|!A+1

n ", |!A"1k " are the eigenstates, and EA+1

n , EA"1k

the eigenenergies of the (A ± 1)-nucleon isotope. Thestructure of Eq. (1) is particularly useful for our pur-poses. At positive energies, the residues of the first term,!!A+1

n |c†!|!A0 ", contain the scattering wave functions for

the elastic collision of a nucleon o" the |!A0 " ground state,

while at negative energies they give information on fi-nal states of the nucleon capture process. Consequently,the second term has poles below the Fermi energy (EF )which carry information about the removal of a nucleonand therefore clarify the structure of the target state |!A

0 "itself. Green’s function theory provides a natural frame-work for describing physics both above and below theFermi surface in a consistent manner.The propagator (1) can be obtained as a solution of

the Dyson equation,

g!"(E) = g(0)!" (E) +!

#$

g(0)!# (E)#%#$(E) g$"(E) , (2)

in which g(0)(E) is the propagator for a free nucleon(moving only with its kinetic energy). #%(E) is the irre-ducible self-energy and represents the interaction of theprojectile (ejectile) with the target nucleus. Feshbach,developed a formal microscopic theory for the optical po-tential already in Ref. [21, 22] by projecting the many-body Hamiltonian on the subspace of scattering states.It has been proven that if Feshbach’s theory is extendedto a space including states both above and below theFermi surface, the resulting optical potential is exactlythe irreducible self-energy #%(E) [23] (see also Ref. [24]and Ref. [25] for a shorter demonstration).The above equivalence with the microscopic optical po-

tential is fundamental for the present study, since theavailable knowledge from calculations based on Green’sfunction theory can be used to suggest improvements ofoptical models. In particular, in the DOM, the dispersionrelation obeyed by #%(E) is used to reduce the number ofparameters and to enforce the e"ects of causality. Thusthe DOM potentials can also be thought of as a repre-sentation of the nucleon self-energy.

A. Self-Energy

For a J = 0 nucleus, all partial waves ($, j, %) aredecoupled, where $,j label the orbital and total angu-lar momentum and % represents its isospin projection.The irreducible self-energy in coordinate space (for ei-ther a proton or a neutron) can be written in terms ofthe harmonic-oscillator basis used in the FRPA calcula-tion, as follows:

#%(x,x!;E) =!

&jmj'

I&jmj($,&)

$

"

!

na,nb

Rna&(r)#%ab(E)Rnb&(r

!)

#

(I&jmj($!,&!))#, (3)

where x % r,&, % . The spin variable is represented by&, n is the principal quantum number of the harmonicoscillator, and a % (na, $, j, %) (note that for a J = 0 nu-cleus the self-energy is independent ofmj). The standardradial harmonic-oscillator function is denoted by Rn&(r),

[CB,!M.Hjorth.Jensen,!Pys.!Rev.!C79,!064313!(2009);!CB,!Phys.!Rev.!LeE.!103,!202502!(2009)]!

Shab(!) =

1

⇡Im gab(!)

Green’s functions in many-body theory

Page 6: Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$ Collaborators:** * ... b ⇥ k Have a systematic expansion of the Hamiltonian ... trons and

! 3NF crucial for reproducing binding energies and driplines around oxygen !  cf. microscopic shell model [Otsuka et al, PRL105, 032501 (2010).]!

!

!

!

! !

"

"

"

"

"

!

!

!!

!

"

"

"" "

#!"24 MeV#SRG"2.0 fm

$1

15F 17F 23F 25F 29F

$180

$160

$140

$120

$100

$80

13N 15N 21N 23N 27N

Eg.s.!MeV

"Exp2N%3N#ind$2N%3N#full$

N3LO (Λ = 500Mev/c) chiral NN interaction evolved to 2N + 3N forces (2.0fm-1) N2LO (Λ = 400Mev/c) chiral 3N interaction evolved (2.0fm-1)!

A. Cipollone, CB, P. Navrátil, Phys. Rev. Lett. 111, 062501 (2013)

⇤⇤

⇤⇤

⇤⇤

⇤⇤⇤ ⇤

⇤ ⇤⇤

⇥⇥

⇥⇥

⇥⇥ ⇥⇥ ⇥ ⇥ ⇥ ⇥

�⇧⇤24 MeV⌅SRG⇤2.0 fm⇥1

14O 16O 22O 24O 28O⇥180

⇥160

⇥140

⇥120

⇥100

⇥80

⇥60

E g.s.�MeV

Exp2N�3N⇤full⌅2N�3N⇤ind⌅

Results for the N-O-F chains

Page 7: Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$ Collaborators:** * ... b ⇥ k Have a systematic expansion of the Hamiltonian ... trons and

Single nucleon transfer in the oxygen chain

fair agreement obtained for the calculation of the 16O rmsradii performed with the SLy4 interaction [31] compared tothe values deduced from 16O!e; e0p"15Ngs and 15N3=2#analyses [5], both states with large SFs. We thus adoptedthe HFB radii calculated for the 0p wave functions for 14Oand 18O and deduced the corresponding values of r0. Thesame calculation was done with other Skyrme interactions,always in fair agreement with the 16O!e; e0p" results, fromwhich we deduced a variance for r0.

The calculated angular distributions were normalized tothe data by a factor C2Sexp, which defines a so-calledexperimental SF. C2Sexp are mainly sensitive to the mostforward angles, and so little sensitive to the details of thenuclear potentials. C2Sexp strongly depend on radii with!SF=SF $ 6!rrms=rrms in the 14O!d; t" analysis.

We first reanalyzed published data for single nucleonpickup reactions at about the same incident energy in directkinematics [19–21] on 16O and 18O targets. The angulardistributions were well reproduced in all cases by CRCcalculations. For 16O!d; 3He" at 14 and 26 MeV=nucleon,we obtained same C2Sexp, which confirms the energy in-dependence of the analysis. For the 14O (d, 3He) and14O (d; t) transfers, the shape of the angular distributions

is nicely reproduced (Fig. 2) by the CRC calculationsassuming a !l % 1 transferred angular momentum, asexpected from the transfer of a 0p nucleon.In the second approach, we employed ab initio SFs and

OFs obtained from the single-particle Green’s function inthe third order algebraic diagrammatic constructionmethod [ADC(3)] [14,32]. Calculations were based onchiral two-body next-to-next-to-next-to leading order(N3LO) [33] plus three-body next-to-next-to leading order(N2LO) [34] interactions evolved to a cutoff ! %1:88 fm#1, as introduced in Ref. [35]. All microscopicOFs were further rescaled in coordinate space by thesame factor (i.e., introducing only one phenomenologicalcorrection) to account for differences of predicted [30] andexperimental rms radius of 16O. The OFs corresponding tothe removal of main peaks at large and small nucleonseparation energies are shown in Figs. 3(a) and 3(b),respectively, and compared to the Wood-Saxon prescrip-tion. We note very little radial difference in the removal ofthe strongly bound neutron in 14O.We give in Table I the normalizations C2Sexp for the two

kinds of OFs. From theoretical SFs inputs, either micro-scopic ab initio SFs [30] or shell-model SFs, we obtain atheoretical value "th!#" and the reduction factor Rs %"exp!#"="th!#". For shell-model SFs, we performed twocalculations with different valence space and interaction:(i) in the 0p& 2@! valence space with Oxbash [36] andthe WBT interaction [37] shown in Table I (here the activeorbitals are 0p3=2 and 0p1=2 and only 2p2h excitationstoward the sd orbitals are allowed), and (ii) in the0p1s0d valence space with Nushellx [38] and a new inter-action [39]. With the WBT interaction, we find goodagreement for the energies of the listed states, while withthe new interaction the energies of excited states in 13N and15N disagree by several MeV. Finally, we show the reduc-tion factor Rs, also plotted in Figs. 4(a) and 4(b), for WSand microscopic OFs, respectively. In the total uncertainty,we set apart in a box the uncertainties originating from theanalysis: (i) imperfect knowledge of entrance and exitpotentials, and (ii) the variance in the calculation of rms

0,0

0,1

0,2

0,3

OF

(fm

-3/2

)

0 2 4 6 8r (fm)

0 2 4 6r (fm)

-10-2

0

10-2

! (f

m-3

/2)

(a)

WS

SCGF

(b)<14O|13O > <14O|13N >

WS

SCGF

(d)(c)

FIG. 3 (color online). Radial dependence of (a), (b) the OFs forWS and microscopic (SCGF) [30] form factors normalized to 1;(c), (d) the OF difference $ (SCGF#WS).

TABLE I. The normalization C2Sexp for two OFs, phenomenological (WS) and microscopic (SCGF) [30]. For the WS OF, ther0 values were chosen to reproduce RHFB

rms , except for16O for which Rrms was taken from (e, e0p) data (see text). The SFs C2Sth are

obtained from shell-model calculations with the WBT interaction. In the second part, the analysis was performed with microscopicOFs and SFs. The two errors for C2Sexp and Rs are the experimental and analysis errors.

RHFBrms r0 C2Sexp C2Sth Rs C2Sexp C2Sth Rs

Reaction E' (MeV) J% (fm) (fm) (WS) 0p& 2@! (WS) (SCGF) (SCGF) (SCGF)

14O (d, t) 13O 0.00 3=2# 2.69 1.40 1.69 (17)(20) 3.15 0.54(5)(6) 1.89(19)(22) 3.17 0.60(6)(7)14O (d, 3He) 13N 0.00 1=2# 3.03 1.23 1.14(16)(15) 1.55 0.73(10)(10) 1.58(22)(2) 1.58 1.00(14)(1)

3.50 3=2# 2.77 1.12 0.94(19)(7) 1.90 0.49(10)(4) 1.00(20)(1) 1.90 0.53(10)(1)16O (d, t) 15O 0.00 1=2# 2.91 1.46 0.91(9)(8) 1.54 0.59(6)(5) 0.96(10)(7) 1.73 0.55(6)(4)16O (d, 3He) 15N [19,20] 0.00 1=2# 2.95 1.46 0.93(9)(9) 1.54 0.60(6)(6) 1.25(12)(5) 1.74 0.72(7)(3)

6.32 3=2# 2.80 1.31 1.83(18)(24) 3.07 0.60(6)(8) 2.24(22)(10) 3.45 0.65(6)(3)18O (d, 3He) 17N [21] 0.00 1=2# 2.91 1.46 0.92(9)(12) 1.58 0.58(6)(10)

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122503-3

fair agreement obtained for the calculation of the 16O rmsradii performed with the SLy4 interaction [31] compared tothe values deduced from 16O!e; e0p"15Ngs and 15N3=2#analyses [5], both states with large SFs. We thus adoptedthe HFB radii calculated for the 0p wave functions for 14Oand 18O and deduced the corresponding values of r0. Thesame calculation was done with other Skyrme interactions,always in fair agreement with the 16O!e; e0p" results, fromwhich we deduced a variance for r0.

The calculated angular distributions were normalized tothe data by a factor C2Sexp, which defines a so-calledexperimental SF. C2Sexp are mainly sensitive to the mostforward angles, and so little sensitive to the details of thenuclear potentials. C2Sexp strongly depend on radii with!SF=SF $ 6!rrms=rrms in the 14O!d; t" analysis.

We first reanalyzed published data for single nucleonpickup reactions at about the same incident energy in directkinematics [19–21] on 16O and 18O targets. The angulardistributions were well reproduced in all cases by CRCcalculations. For 16O!d; 3He" at 14 and 26 MeV=nucleon,we obtained same C2Sexp, which confirms the energy in-dependence of the analysis. For the 14O (d, 3He) and14O (d; t) transfers, the shape of the angular distributions

is nicely reproduced (Fig. 2) by the CRC calculationsassuming a !l % 1 transferred angular momentum, asexpected from the transfer of a 0p nucleon.In the second approach, we employed ab initio SFs and

OFs obtained from the single-particle Green’s function inthe third order algebraic diagrammatic constructionmethod [ADC(3)] [14,32]. Calculations were based onchiral two-body next-to-next-to-next-to leading order(N3LO) [33] plus three-body next-to-next-to leading order(N2LO) [34] interactions evolved to a cutoff ! %1:88 fm#1, as introduced in Ref. [35]. All microscopicOFs were further rescaled in coordinate space by thesame factor (i.e., introducing only one phenomenologicalcorrection) to account for differences of predicted [30] andexperimental rms radius of 16O. The OFs corresponding tothe removal of main peaks at large and small nucleonseparation energies are shown in Figs. 3(a) and 3(b),respectively, and compared to the Wood-Saxon prescrip-tion. We note very little radial difference in the removal ofthe strongly bound neutron in 14O.We give in Table I the normalizations C2Sexp for the two

kinds of OFs. From theoretical SFs inputs, either micro-scopic ab initio SFs [30] or shell-model SFs, we obtain atheoretical value "th!#" and the reduction factor Rs %"exp!#"="th!#". For shell-model SFs, we performed twocalculations with different valence space and interaction:(i) in the 0p& 2@! valence space with Oxbash [36] andthe WBT interaction [37] shown in Table I (here the activeorbitals are 0p3=2 and 0p1=2 and only 2p2h excitationstoward the sd orbitals are allowed), and (ii) in the0p1s0d valence space with Nushellx [38] and a new inter-action [39]. With the WBT interaction, we find goodagreement for the energies of the listed states, while withthe new interaction the energies of excited states in 13N and15N disagree by several MeV. Finally, we show the reduc-tion factor Rs, also plotted in Figs. 4(a) and 4(b), for WSand microscopic OFs, respectively. In the total uncertainty,we set apart in a box the uncertainties originating from theanalysis: (i) imperfect knowledge of entrance and exitpotentials, and (ii) the variance in the calculation of rms

0,0

0,1

0,2

0,3

OF

(fm-3

/2)

0 2 4 6 8r (fm)

0 2 4 6r (fm)

-10-2

0

10-2

! (fm

-3/2

)

(a)

WS

SCGF

(b)<14O|13O > <14O|13N >

WS

SCGF

(d)(c)

FIG. 3 (color online). Radial dependence of (a), (b) the OFs forWS and microscopic (SCGF) [30] form factors normalized to 1;(c), (d) the OF difference $ (SCGF#WS).

TABLE I. The normalization C2Sexp for two OFs, phenomenological (WS) and microscopic (SCGF) [30]. For the WS OF, ther0 values were chosen to reproduce RHFB

rms , except for16O for which Rrms was taken from (e, e0p) data (see text). The SFs C2Sth are

obtained from shell-model calculations with the WBT interaction. In the second part, the analysis was performed with microscopicOFs and SFs. The two errors for C2Sexp and Rs are the experimental and analysis errors.

RHFBrms r0 C2Sexp C2Sth Rs C2Sexp C2Sth Rs

Reaction E' (MeV) J% (fm) (fm) (WS) 0p& 2@! (WS) (SCGF) (SCGF) (SCGF)

14O (d, t) 13O 0.00 3=2# 2.69 1.40 1.69 (17)(20) 3.15 0.54(5)(6) 1.89(19)(22) 3.17 0.60(6)(7)14O (d, 3He) 13N 0.00 1=2# 3.03 1.23 1.14(16)(15) 1.55 0.73(10)(10) 1.58(22)(2) 1.58 1.00(14)(1)

3.50 3=2# 2.77 1.12 0.94(19)(7) 1.90 0.49(10)(4) 1.00(20)(1) 1.90 0.53(10)(1)16O (d, t) 15O 0.00 1=2# 2.91 1.46 0.91(9)(8) 1.54 0.59(6)(5) 0.96(10)(7) 1.73 0.55(6)(4)16O (d, 3He) 15N [19,20] 0.00 1=2# 2.95 1.46 0.93(9)(9) 1.54 0.60(6)(6) 1.25(12)(5) 1.74 0.72(7)(3)

6.32 3=2# 2.80 1.31 1.83(18)(24) 3.07 0.60(6)(8) 2.24(22)(10) 3.45 0.65(6)(3)18O (d, 3He) 17N [21] 0.00 1=2# 2.91 1.46 0.92(9)(12) 1.58 0.58(6)(10)

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! Analysis of 14O(d,t)13O and 14O(d,3He)13N transfer reactions @ SPIRAL!

-  Overlap functions and strengths from GF

-  Rs independent of asymmetry!

[F. Flavigny et al, PRL110, 122503 (2013)]

radii (and consequently of r0) due to different Skyrmeinteractions, provided the rms radii of 15N extracted from(e, e0p) [5] are reproduced. All the other experimentaluncertainties are accounted for by the error bars displayedon Fig. 4. A rather flat trend is found without the needfor the large asymmetry dependence suggested by inter-mediate energy knockout data analyzed with the eikonalformalism [10]. For a quantitative evaluation, we fittedthe reduction factor with a linear dependence Rs!!"!S#". We obtained mean values for ! and " withassociated errors from a minimization over the 48 data sets,considering (i) eight combinations of optical potentials forthe entrance and exit channels, (ii) three Skyrme interac-tions to calculate the rms radii, and (iii) the two above-mentioned shell-model calculations.

For the WS OF, the reduction factor Rs ! 0:538$28%$18%(for !S ! 0 nuclei) is in agreement with Ref. [9] and theslope parameter ! ! 0:0004$24%$12% MeV&1, thereforeconsistent with zero. The first standard error obtainedover one data set depends on the experimental uncertain-ties; the second one comes from the distribution over the 48data sets. Within the error bars, the data do not contradictthe weak dependence found by ab initio calculations, with!0 ! &0:0039 MeV&1 between the two 14O points inRef. [7], although the calculated !S is much reducedcompared to the experimental value.

Despite different OFs and SFs, the analysisperformed with the ab initio OF [30] provides verysimilar results with Rs$!S!0%!0:636$34%$42% and !!&0:0042$28%$36%MeV&1, with calculated !S!17:6MeV[Fig. 4(b)].In summary, we measured exclusive differential cross

sections at 18 MeV=nucleon for the 14O$d; t%13O and14O$d; 3He%13N transfer reactions and elastic scattering.WS OFs with a constraint on HF radii and microscopicOFs (obtained from SCFG theory) have been compared forthe first time for symmetric and very asymmetric nucleiand gave similar results. We extracted the reduction factorsRs over a high asymmetry range, !S ! '18:5 MeV, foroxygen isotopes. From the good agreement between theCRC calculations and the set of transfer data highlighted inour work, the asymmetry dependence is found to be non-existent (or weak), within the error bars. This result is inagreement with ab initio Green’s function and coupled-cluster calculations [7,14], but contradicts the trendobserved in nucleon knockout data obtained at incidentenergies below 100 MeV=nucleon and analyzed with thesudden-eikonal formalism. The disagreement of the twosystematic trends from knockout and transfer calls for abetter description of so-called direct reaction mechanismsin order that a consistent picture of nuclear structureemerges from measurements at different incident energies.The authors thank N. T. Timofeyuk and N. Alamanos for

enlightening discussions and P. Navratil for providingevolved two- and three-body interactions relevant to thisstudy. This work was supported by LIA COPIGAL andPOLONIUM PHC under Grant No. 22470XA. Theoreticalwork was supported by the UK’s STFC Grant No. ST/J000051/1.

[1] W.H. Dickhoff and C. Barbieri, Prog. Part. Nucl. Phys. 52,377 (2004).

[2] S. Boffi et al., Electromagnetic Response of AtomicNuclei, Oxford Studies in Nuclear Physics Vol. 20(Clarendon Press, Oxford, 1996).

[3] M. Bernheim et al., Nucl. Phys. A375, 381 (1982).[4] L. Lapikas, Nucl. Phys. A553, 297 (1993).[5] M. Leuschner et al., Phys. Rev. C 49, 955 (1994).[6] C. Barbieri, Phys. Rev. Lett. 103, 202502 (2009).[7] Ø. Jensen, G. Hagen, M. Hjorth-Jensen, B. A. Brown, and

A. Gade, Phys. Rev. Lett. 107, 032501 (2011).[8] G. J. Kramer, H. P. Blok, and L. Lapikas, Nucl. Phys.

A679, 267 (2001).[9] J. P. Schiffer et al., Phys. Rev. Lett. 108, 022501 (2012).[10] A. Gade et al., Phys. Rev. C 77, 044306 (2008).[11] C. Louchart, A. Obertelli, A. Boudard, and F. Flavigny,

Phys. Rev. C 83, 011601(R) (2011).[12] F. Flavigny, A. Obertelli, A. Bonaccorso, G. F. Grinyer, C.

Louchart, L. Nalpas, and A. Signoracci, Phys. Rev. Lett.108, 252501 (2012).

[13] J. Lee et al., Phys. Rev. C 83, 014606 (2011).

0,2

0,4

0,6

0,8

1

Rs =

!ex

p(")

/ !th

(")

14O data16O data18O data

-20 -10 0 10 20

#S = $ (Sp - Sn) (MeV)

0,2

0,4

0,6

0,8

1

Rs =

!ex

p(")

/ !th

(")

14O(d, t)

14O(d, 3He)

(b) SCGF

(a) WS + SM

FIG. 4 (color online). Reduction factors Rs obtained with (a) aWS OF and the SLy4 interaction [31], averaged over fourentrance and two exit potentials, and compared to shell-modelcalculations performed with the WBT interaction [37] in the0p# 2@! valence space; (b) a microscopic (SCGF) form factor[30]. The detail of error bars is given in text.

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!  induced and full3NF investigated ! genuine (N2LO) 3NF needed to reproduce the energy curvature and S2n

! N=20 and Z=20 gaps overestimated! ! Full 3NF give a correct trend but over bind!

Ab-initio calculation of the whole Ca: induced and full3NF investigated !

36 38 40 42 44 46 48 50 52

-450

-400

-350

-300GGF [NN + 3N (ind.)]GGF [NN + 3N (full)]ADC(3) "corrected"IM-SRG [NN + 3N (full)]

ACa

EA [M

eV]

Nmax=13

0Calcium isotopic chain

38 40 42 44 46 48 50 525

10

15

20

25

30

35

40ExperimentNN + 3N (full)NN + 3N (ind.)SM (NN + 3N)CC

ACaS 2n

[MeV

]

V.!Somà,!CB!et*al.!Phys.!Rev.!C89,!061301R!(2014)!

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V.!Somà,!CB!et*al.!Phys.!Rev.!C89,!061301R!(2014)!

! First ab-initio calculation over a contiguous portion of the nuclear chart—open shells are now possible through the Gorkov-GF formalism

Neighbouring Ar, K, Ca, Sc, and Ti chains

18 20 22 24 26 28 30 320

10

20

30

40

50

60

N

S 2n [

MeV

]

Ar

K

CaSc

Ti

Two-neutron separation energies predicted by chiral NN+3NF forces:!

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V.!Somà,!CB!et*al.!Phys.!Rev.!C89,!061301R!(2014)!

Neighbouring Ar, K, Ca, Sc, and Ti chains

Works well in the pf shell!

18 20 22 24 26 28 30 320

10

20

30

40

50

60

N

S 2n [

MeV

]

Ar

K

CaSc

Ti

Two-neutron separation energies predicted by chiral NN+3NF forces:!

! First ab-initio calculation over a contiguous portion of the nuclear chart—open shells are now possible through the Gorkov-GF formalism

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18 20 22 24 26 28 30 320

10

20

30

40

50

60

N

S 2n [

MeV

]

Ar

K

CaSc

Ti

V.!Somà,!CB!et*al.!Phys.!Rev.!C89,!061301R!(2014)!

Neighbouring Ar, K, Ca, Sc, and Ti chains

Over estimated N=20 and Z=20 gaps!

Two-neutron separation energies predicted by chiral NN+3NF forces:!

! First ab-initio calculation over a contiguous portion of the nuclear chart—open shells are now possible through the Gorkov-GF formalism

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V.!Somà,!CB!et*al.!Phys.!Rev.!C89,!061301R!(2014)!

18 20 22 24 26 28 30 320

10

20

30

40

50

60

N

S 2n [

MeV

]

Ar

K

CaSc

Ti

Neighbouring Ar, K, Ca, Sc, and Ti chains

Two-neutron separation energies predicted by chiral NN+3NF forces:!

Lack of deformation due to quenched cross-shell quadrupole excitations!

! First ab-initio calculation over a contiguous portion of the nuclear chart—open shells are now possible through the Gorkov-GF formalism

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Inversion of d3/2—s1/2 at N=28

37 39 41 43 45 47 49 51 53

0

1

2

3

4

5

6ExperimentLowest peakESPE

AK

E(1/

2+ - 3/

2+ ) [M

eV]

7

imental data on the energy of the first-excited state isneeded to further test the validity of both models.

Very recently, ab initio calculations of open-shell nu-clei have become possible in the Ca region [48] on thebasis of the self-consistent Gorkov-Green‘s function for-malism [49]. State-of-the-art chiral two- (NN) [50, 51]and three-nucleon (3N) [52] interactions adjusted to two-, three- and four-body observables (up to 4He) are em-ployed, without any further modification, in the com-putation of systems containing several tens of nucleons.We refer to Ref. [48] for further details. In the presentstudy, Gorkov-Green’s function calculations of the low-est 1/2+ and 3/2+ states in 43�51K have been performedby removing a proton from 44�52Ca. Similarly to Fig. 5,the upper panel of Fig. 6 compares the results to exper-imental data. The inversion of the states at N = 28 isnot obtained in the calculation, because odd-A spectraare systematically too spread out [48]. This shortcom-ing actually correlates with the systematic overbinding ofneighboring even-A ground-states. Still, one observes thecorrect relative evolution of the 1/2+ state with respect tothe 3/2+ when going from 43K to 47K and then from 47Kto 49K. As a matter of fact, rescaling the theoretical re-sults to the experimental ones at, e.g. 47K, demonstratesthat the relative evolution of the two states is quantita-tively well reproduced. This result is very encouraging forthose first-ever systematic ab initio calculations in mid-mass nuclei. Indeed, it allows one to speculate that cor-recting in the near future for the systematic overbindingproduced in the Ca region by currently available chiralEFT interactions, and thus the too spread out spectra ofodd-A systems, might bring the theoretical calculation ingood agreement with experiment. Although this remainsto be validated, it demonstrates that systematic spec-troscopic data in mid-mass neutron-rich nuclei providea good test case to validate/invalidate specific featuresof basic inter-nucleon interactions and innovative many-body theories.

To complement the above analysis, the lower panelof Fig. 6 provides the evolution of proton 1d

3/2 and2s

1/2 shells. These two e↵ective single-particle energies(ESPEs) recollects [49] the fragmented 3/2+ and 1/2+

strengths obtained from one-proton addition and removalprocesses on neighboring Ca isotones. Within the presenttheoretical description, the evolution of the observable(i.e. theoretical-scheme independent) lowest-lying 1/2+

and 3/2+ states does qualitatively reflect the evolutionof the underlying non-observable (i.e. theoretical-schemedependent) single-particle shells. As such, the energy gapbetween the two shells decreases from 4.81MeV in 43K to2.39MeV in 47K, which is about 50% reduction. Adding4 neutrons in the ⌫2p

3/2 causes the energy di↵erence toincrease again to 4.49MeV.

FIG. 6. (color online) Upper panel: energy di↵erence betweenthe lowest 1/2+ and 3/2+ states obtained in 43�51K from abinitio Gorkov-Green‘s function calculations and experiment.Lower panel: ⇡d

3/2 and ⇡s1/2 e↵ective single-particle energies

in 43�51K.

B. Even-A

The configuration of the even-K isotopes arises fromthe coupling between an unpaired proton in the sd shellwith an unpaired neutron. Di↵erent neutron orbits areinvolved: starting from 38K where a hole in the ⌫1d

3/2

is expected, then gradually filling the ⌫1f7/2 and finally,

the ⌫2p3/2 for 48,50K.

In order to investigate the composition of the ground-state wave functions of the even-K isotopes, we first com-pare the experimental magnetic moments to the empiri-cal values. Based on the additivity rule for the magneticmoments (g factors) and assuming a weak coupling be-tween the odd proton and the odd neutron, the empiricalmagnetic moments can be calculated using the followingformula [53]: µ

emp

= gemp

· I, with

gemp

= g(j⇡)+g(j⌫)2

+ g(j⇡)�g(j⌫)2

j⇡(j⇡+1)�j⌫(j⌫+1)

I(I+1)

, (5)

where g(j⇡) and g(j⌫) are the g factors of the nuclei withan odd proton or neutron from the corresponding orbitand I the total spin. The calculations were performedusing the measured g factors of the neighboring isotopeswith the odd-even and even-odd number of particles in j⇡and j⌫ , respectively. For the empirical values of unpairedprotons, results from Table III were used. The g factorsfor the odd neutrons were taken from the correspondingCa isotones [54–57]. The obtained results with the list ofisotopes used for di↵erent configurations are presented inTable VI.A comparison between the experimental and empiri-

cal g factors is shown in Fig. 7. For 38K, the empiricalvalue calculated from 39K and 39Ca provides excellent

2

⇡2s1/2 and ⇡1d

3/2 levels is predicted to occur at N = 28and returns to a ”normal” ordering approaching N = 40(Fig 1(c) in Ref. [11]). The reordering of the orbitals isdriven by the monopole part of the proton-neutron inter-action which can be decomposed into three components:the central, vector and tensor. Initially Otsuka et al. [12]suggested that the evolution of the ESPEs is mainly dueto the tensor component. However, in more recent publi-cations [11, 13, 14] several authors have shown that boththe tensor term as well as the central term have to beconsidered.Regarding the shell model, potassium isotopes are ex-

cellent probes for this study, with only one proton lessthan the magic number Z = 20. Nevertheless, little andespecially conflicting information is available so far forthe neutron-rich K isotopes. Level schemes based on thetentatively assigned spins of the ground state were pro-vided for 48K [15] and 49K [16]. In addition, an exten-sive discussion was presented by Gaudefroy [17] on theenergy levels and configurations of N = 27, 28 and 29isotones in the shell-model framework and compared tothe experimental observation, where available. However,the predicted spin of 2� for 48K, is in contradiction withI⇡ = (1�) proposed by Krolas et al. [15]. In addition,the nuclear spin of the ground state of 50K was proposedto be 0� [18, 19] in contrast to the recent � decay stud-ies where it was suggested to be 1� [20]. The groundstate spin-parity of 49K was tentatively assigned to be(1/2+) by Broda et al. [16], contrary to the earlier ten-tative (3/2+) assignment from beta-decay spectroscopy[21]. For 51K, the nuclear spin was tentatively assignedto be 3/2+ by Perrot et al. [19].Our recent hyperfine structure measurements of K iso-

topes using the collinear laser spectroscopy techniquegave the answer to the question as to what happens withthe proton sd orbits for isotopes beyond N = 28. By es-tablishing the nuclear spins of 49K and 51K to be 1/2 and3/2 [22] respectively, the evolution of these two states inthe K isotopes is firmly ascertain. This is presented inFig. 1 for isotopes from N = 18 up to N = 32 wherethe inversion of the states is observed at N = 28 fol-lowed by the reinversion back at N = 32. In addition,we have confirmed a spin-parity 1� for 48K and 0+ for50K [26]. The measured magnetic moments of 48�51Kwere not discussed in detail so far and will be presentedin this article. Additionally, based on the comparisonbetween experimental data and shell-model calculations,the configuration of the ground-state wave functions willbe addressed as well.

II. EXPERIMENTAL PROCEDURE

The experiment was performed at the collinearlaser spectroscopy beam line COLLAPS [27] atISOLDE/CERN. The radioactive ion beam was producedby 1.4-GeV protons (beam current ⇠1.7µA) imping-ing on a thick UC

x

target (45 g/cm2). Ionization of

39K 41K 43K 45K 47K 49K 51K

2522

980

561 474 35992

1371

37K

1/2+

1/3+

Ene

rgy

(keV

)

FIG. 1. (color online) Experimental energies for 1/2+ and3/2+ states in odd-K isotopes. Inversion of the nuclear spinis obtained in 47,49K and reinversion back in 51K. Results aretaken from [16, 23–25]. Ground-state spin for 49K and 51Kwere established [22].

Continuous ion beamfrom HRS

ISCOOL

Mirror

Laser

Electrostatic deflectors

Charge exchange cell

Photomultiplier tubesPost-acceleration

voltage

Ions

Atoms

FIG. 2. (color online) Shematic representation of the setupfor collinear laser spectroscopy at ISOLDE.

the resulting fragments was achieved by the surface ionsource. The target and the ionizing tube were heatedto ⇠2000 0C. The accelerated ions (up to 40 kV) weremass separated by the high resolution separator (HRS).The gas-filled Paul trap (ISCOOL) [28, 29] was used forcooling and bunching of the ions. Multiple bunches perproton pulse were created by closing ISCOOL for ⇠90msper bunch. The bunched ions were guided to the setupfor collinear laser spectroscopy where they were super-imposed with the laser. A schematic representation ofthe beam line for collinear laser spectroscopy is shown inFig. 2.A cw titanium:sapphire (Ti:Sa) laser was locked to the

4s 2S1/2 ! 4p 2P

1/2 transition at 769.9 nm, providing⇠1mW power into the beam line. An applied voltageof ±10 kV on the charge exchange cell (CEC) providedthe Doppler tuning for the ions, which were neutralizedthrough the collisions with K vapor. Scanning of the hfswas performed by applying an additional voltage in arange of ±500V. The resonance photons were recordedby four photomultiplier tubes (PMT) placed immediatelyafter the CEC. By gating the signal on the PMTs to thefluorescence photons from the bunches, the signal wasonly recorded for ⇠6µs when the bunches were in frontof the PMTs. Consequently, the background related tothe scattered laser light was suppressed by a factor ⇠ 104

(6µs/90ms). More details about the setup can be found

J. Papuga, et al., PRL 110, 172503 (2013)

2

⇡2s1/2 and ⇡1d

3/2 levels is predicted to occur at N = 28and returns to a ”normal” ordering approaching N = 40(Fig 1(c) in Ref. [11]). The reordering of the orbitals isdriven by the monopole part of the proton-neutron inter-action which can be decomposed into three components:the central, vector and tensor. Initially Otsuka et al. [12]suggested that the evolution of the ESPEs is mainly dueto the tensor component. However, in more recent publi-cations [11, 13, 14] several authors have shown that boththe tensor term as well as the central term have to beconsidered.

Regarding the shell model, potassium isotopes are ex-cellent probes for this study, with only one proton lessthan the magic number Z = 20. Nevertheless, little andespecially conflicting information is available so far forthe neutron-rich K isotopes. Level schemes based on thetentatively assigned spins of the ground state were pro-vided for 48K [15] and 49K [16]. In addition, an exten-sive discussion was presented by Gaudefroy [17] on theenergy levels and configurations of N = 27, 28 and 29isotones in the shell-model framework and compared tothe experimental observation, where available. However,the predicted spin of 2� for 48K, is in contradiction withI⇡ = (1�) proposed by Krolas et al. [15]. In addition,the nuclear spin of the ground state of 50K was proposedto be 0� [18, 19] in contrast to the recent � decay stud-ies where it was suggested to be 1� [20]. The groundstate spin-parity of 49K was tentatively assigned to be(1/2+) by Broda et al. [16], contrary to the earlier ten-tative (3/2+) assignment from beta-decay spectroscopy[21]. For 51K, the nuclear spin was tentatively assignedto be 3/2+ by Perrot et al. [19].

Our recent hyperfine structure measurements of K iso-topes using the collinear laser spectroscopy techniquegave the answer to the question as to what happens withthe proton sd orbits for isotopes beyond N = 28. By es-tablishing the nuclear spins of 49K and 51K to be 1/2 and3/2 [22] respectively, the evolution of these two states inthe K isotopes is firmly ascertain. This is presented inFig. 1 for isotopes from N = 18 up to N = 32 wherethe inversion of the states is observed at N = 28 fol-lowed by the reinversion back at N = 32. In addition,we have confirmed a spin-parity 1� for 48K and 0+ for50K [26]. The measured magnetic moments of 48�51Kwere not discussed in detail so far and will be presentedin this article. Additionally, based on the comparisonbetween experimental data and shell-model calculations,the configuration of the ground-state wave functions willbe addressed as well.

II. EXPERIMENTAL PROCEDURE

The experiment was performed at the collinearlaser spectroscopy beam line COLLAPS [27] atISOLDE/CERN. The radioactive ion beam was producedby 1.4-GeV protons (beam current ⇠1.7µA) imping-ing on a thick UC

x

target (45 g/cm2). Ionization of

39K 41K 43K 45K 47K 49K 51K

2522

980

561 474 35992

1371

37K

1/2+

1/3+

Ener

gy (k

eV)

FIG. 1. (color online) Experimental energies for 1/2+ and3/2+ states in odd-K isotopes. Inversion of the nuclear spinis obtained in 47,49K and reinversion back in 51K. Results aretaken from [16, 23–25]. Ground-state spin for 49K and 51Kwere established [22].

Continuous ion beamfrom HRS

ISCOOL

Mirror

Laser

Electrostatic deflectors

Charge exchange cell

Photomultiplier tubesPost-acceleration

voltage

Ions

Atoms

FIG. 2. (color online) Shematic representation of the setupfor collinear laser spectroscopy at ISOLDE.

the resulting fragments was achieved by the surface ionsource. The target and the ionizing tube were heatedto ⇠2000 0C. The accelerated ions (up to 40 kV) weremass separated by the high resolution separator (HRS).The gas-filled Paul trap (ISCOOL) [28, 29] was used forcooling and bunching of the ions. Multiple bunches perproton pulse were created by closing ISCOOL for ⇠90msper bunch. The bunched ions were guided to the setupfor collinear laser spectroscopy where they were super-imposed with the laser. A schematic representation ofthe beam line for collinear laser spectroscopy is shown inFig. 2.A cw titanium:sapphire (Ti:Sa) laser was locked to the

4s 2S1/2 ! 4p 2P

1/2 transition at 769.9 nm, providing⇠1mW power into the beam line. An applied voltageof ±10 kV on the charge exchange cell (CEC) providedthe Doppler tuning for the ions, which were neutralizedthrough the collisions with K vapor. Scanning of the hfswas performed by applying an additional voltage in arange of ±500V. The resonance photons were recordedby four photomultiplier tubes (PMT) placed immediatelyafter the CEC. By gating the signal on the PMTs to thefluorescence photons from the bunches, the signal wasonly recorded for ⇠6µs when the bunches were in frontof the PMTs. Consequently, the background related tothe scattered laser light was suppressed by a factor ⇠ 104

(6µs/90ms). More details about the setup can be found

Change in separation described by chiral NN+3NF:!

AK isotopes!

(Gorkov calculations at 2nd order)!ESPE: “centroid” energies!

Laser spectroscopy @ ISOLDE!

Page 14: Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$ Collaborators:** * ... b ⇥ k Have a systematic expansion of the Hamiltonian ... trons and

V.!Somà,!CB!et*al.,!in!prep.!

!  Error bar in predictions are from extrapolating the many-body

expansion to convergence of the model space. 47 49 51 53

0

5

10

15

Experimenthw28 rampAhw32 rampAhw36 rampAGGF extr. rampAhw28 rampD'

AK

S2n

[MeV

]Two-neutron separation energies

for neutron rich K isotopes

AME extrapolation!

Page 15: Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$ Collaborators:** * ... b ⇥ k Have a systematic expansion of the Hamiltonian ... trons and

! Large J in free space SRG matter (must pay attention to its convergence) ! Overall conclusions regarding over binding and S2n remain but details change

Calculations based on ramps D, N1, N2:!

48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80-700

-650

-600

-550

-500

-450

-400

-350 ExperimentFull 3NF (ramp N1)Full 3NF (ramp N2)

ANiE

[MeV

]

IM-SRG results from H. Hergert!

Ca and Ni isotopic chains

18 20 22 24 26 28 30 32 34 36 38 40-500

-450

-400

-350

-300 ExperimentGGF λ = 2.0IM-SRG λ = 1.88IM-SRG λ = 2.23

N

E [M

eV]

prelimi

nary

Ca

d3/2 f7/2 p3/2

p1/2 f5/2

Preliminary!

Page 16: Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$ Collaborators:** * ... b ⇥ k Have a systematic expansion of the Hamiltonian ... trons and

! Large J in free space SRG matter (must pay attention to its convergence) ! Overall conclusions regarding over binding and S2n remain but details change

Calculations based on ramps D, N1, N2:!

IM-SRG results from H. Hergert!

50 52 54 56 58 60 62 64 66 68 70 72 74 76 780

10

20

30

40 ExperimentFull3NF (ramp N1)

ANiS 2n

[MeV

]

Ca and Ni isotopic chains

18 20 22 24 26 28 30 32 34 36 38 40

05

101520253035 Experiment

GGF λ = 2.0IM-SRG λ = 1.88IM-SRG λ = 2.23

N

S 2n [

MeV

]

prelimi

nary

Ca

f7/2 p3/2d3/2 p1/2

f5/2

Preliminary!

Page 17: Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$ Collaborators:** * ... b ⇥ k Have a systematic expansion of the Hamiltonian ... trons and

! In general over binding per nucleon (E/A) appear to stabilize above A~40-50 but more investigations are required.

Difference of calculated BEs to the experiment for different masses:!

14 22 30 38 46 54 62 70 78

-0.2

0

0.2

0.4

0.6

0.8

1

OCaNi

A

ΔE/

A [M

eV]

prelimi

nary

Ca and Ni isotopic chains

Page 18: Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$ Collaborators:** * ... b ⇥ k Have a systematic expansion of the Hamiltonian ... trons and

�  What to did we learn about realistic chiral forces from ab-initio calculation ?

!  Leading order 3NF are crucial to predict many important features that are observed experimentally (drip lines, saturation, orbit evolution, etc…)

!  Experimental binding is predicted accurately up to the lower sd shell (A≈30) but deteriorates for medium mass isotopes (Ca and above) with roughly 1 MeV/A over binding.

!  This hints to the need of more repulsion in future generations of chiral realistic forces.

18 20 22 24 26 28 30 32

0

10

20

30

40

50

60

NS 2

n [M

eV]

ArK

CaSc

Ti

Conclusions

Page 19: Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$ Collaborators:** * ... b ⇥ k Have a systematic expansion of the Hamiltonian ... trons and

�  What to did we learn about realistic chiral forces from ab-initio calculation ?

!  Leading order 3NF are crucial to predict many important features that are observed experimentally (drip lines, saturation, orbit evolution, etc…)

!  Experimental binding is predicted accurately up to the lower sd shell (A≈30) but deteriorates for medium mass isotopes (Ca and above) with roughly 1 MeV/A over binding.

!  This hints to the need of more repulsion in future generations of chiral realistic forces.

18 20 22 24 26 28 30 32

0

10

20

30

40

50

60

NS 2

n [M

eV]

ArK

CaSc

Ti

Conclusions

Thank you for your

attention!!!�

Page 20: Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$ Collaborators:** * ... b ⇥ k Have a systematic expansion of the Hamiltonian ... trons and

!

!

!!

!

"

"

""

"

! !

! !

!

"

"

""

"

!

!

!!

!

!!

"

"

"

"

"

2s1!21d5!2

1d3!2

!8

!6

!4

!2

0

2

4

6

" iA#1"MeV

# 2N$3N$full%2N$3N$ind%

!  d3/2 raised by genuine 3NF

!  systematic underestimation of radii!

N3LO (Λ = 500Mev/c) chiral NN interaction evolved to 2N + 3N forces (2.0fm-1) N2LO (Λ = 400Mev/c) chiral 3N interaction evolved (2.0fm-1)!

A. Cipollone, CB, P. Navrátil, Phys. Rev. Lett. 111, 062501 (2013)

14 16 18 20 22 24 26 28A (Oxygen isotopes)

2.2

2.4

2.6

2.8

3

R_rm

s [fm

]

O - Gkv-2nd: D l=2 hw=28 IND 13,16,16 e=13O - Gkv-2nd: D l=2 hw=28 FULL 13,16,16 e=13O - Dys-ADC(3): D l=2 hw=28 IND 13,16,16 e=13O - Dys-ADC(3): D l=2 hw=28 FULL 13,16,16 e=13r_rmp (charge) exp

Results for the N-O-F chains

Page 21: Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$ Collaborators:** * ... b ⇥ k Have a systematic expansion of the Hamiltonian ... trons and

Preliminary!

Ca spectral distributions – at 2nd order

5ê2-1ê2-3ê2-7ê2-3ê2+

1ê2+5ê2+

36Ca 38Ca 40Ca 42Ca 44Ca 46Ca 48Ca 50Ca 52Ca 54Ca 56Ca 58Ca 60Ca

0 0.5 1 0.5 1 0.5 1 0.5 1 0.5 1 0.5 1 0.5 1 0.5 1 0.5 1 0.5 1 0.5 1 0.5 1 0.5 1

-30

-20

-10

0

SF

E sp@Me

VD

neu lSRG=2 L=13,16,16 RAMP Dpp FULL

Page 22: Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$ Collaborators:** * ... b ⇥ k Have a systematic expansion of the Hamiltonian ... trons and

Pairing gaps

! Three-point mass differences

3

the antisymmetrized matrix element of V NN expandedin terms of direct-product state matrix elements, denotedby |1, 2). The antisymmetrization of the NNN potentialcan be carried out by introducing the operator

A123 ⇥ (1 + P12 P23 + P13 P23)(1� P23)

= 1� P12 � P13 � P23 + P12 P23 + P13 P23 , (5)

where P12 exchanges nucleons 1 and 2, such that an an-tisymmetrized matrix element reads

V NNNabcdef ⇥ ⇤1:a; 2:b; 3:c|V NNN |1:d; 2:e; 3:f⌅

⇥ (1:a; 2:b; 3:c|V NNNAdef |1:d; 2:e; 3:f)= (1:a; 2:b; 3:c|V NNN |1:d; 2:e; 3:f)� (1:a; 2:b; 3:c|V NNN |1:e; 2:d; 3:f)� (1:a; 2:b; 3:c|V NNN |1:f ; 2:e; 3:d)� (1:a; 2:b; 3:c|V NNN |1:d; 2:f ; 3:e)+ (1:a; 2:b; 3:c|V NNN |1:e; 2:f ; 3:d)+ (1:a; 2:b; 3:c|V NNN |1:f ; 2:d; 3:e) . (6)

Every time an index x appears in a matrix element (3),(4) or (6), the associated annihilation (creation) operatoris to be intended of the form (1), i.e. ax (a†x). It followsthat

tab ⇥ ⇥a ⇥b (1:a|Tkin|1:b) , (7)

V NNabcd ⇥ ⇥a ⇥c ⇤1:a; 2:b|V NN |1:c; 2:d⌅ , (8)

V NNabcd ⇥ ⇥a ⇥b ⇥c ⇥d ⇤1:a; 2:b|V NN |1:c; 2:d⌅ , (9)

etc.

C. Treatment of three-body forces

III. GORKOV FORMALISM

A. Standard propagator and superfluid systems

Let us consider the ground state of the system de-scribed by the Hamiltonian (2), denoted by |⇤N

0 ⌅, so-lution of

H |⇤Nk ⌅ = EN

k |⇤Nk ⌅ (10)

with the lowest eigenvalue EN0 . The fundamental object

of Green’s function theory is the single-particle propaga-tor, defined as

iGab(t, t⇥) ⇥ ⇤⇤N

0 |T⇧aa(t)a

†b(t

⇥)⌃|⇤N

0 ⌅ , (11)

Gab(⌅) =

⇥d (t� t⇥) ei�(t�t�) Gab(t, t

⇥) (12)

⇤O⌅ =�

ab

⇥d⌅

2⇤Oab Gab(⌅) (13)

O =�

ab

Oab a†a ab (14)

⇤T ⌅ =�

ab

⇥d⌅

2⇤tab Gab(⌅) (15)

⇤H⌅ = E0 =�

ab

⇥d⌅

2⇤[tab + ⌅ �ab] Gab(⌅) (16)

where the operator T orders a and a† according to theirtime argument (larger times to the left) and where an-nihilation and creation operators are in the Heisenbergrepresentation

ab(t) = a(H)b (t) ⇥ exp[iHt] ab exp[�iHt] , (17a)

a†b(t) =⇤a(H)b (t)

⌅†⇥ exp[iHt] a†b exp[�iHt] . (17b)

The knowledge of G enables us to compute expecta-tion values of all one-body operators plus the two-bodyground-state energy, i.e. the expectation value of theHamiltonian if only two-body forces are considered. Onecan define two-, three-, ..., N -particle propagators in asimilar way, to evaluate up to N -nucleon observables.

The equations of motion for the Green’s functions takethe form of a set of N coupled integro-di⌅erential equa-tions, each of them involving the (i� 1)-, i- and (i+ 1)-particle propagators. In order to compute the single-particle propagator one can as well derive a perturbativeexpansion, which translates into an infinite series of di-agrams and has the advantage of providing systematicways of approximating the solution. The connection be-tween the diagrammatic expansion and the equation ofmotion for G leads to the definition of the (irreducible)self-energy ⇥ and the derivation of Dyson’s equation

�(3)n (A) =

(�1)A

2[EA+1

0 � 2EA0 + EA�1

0 ] (18)

a

c

d

b

⇥Rcd(⌅) (19)

Gab(⌅) = G(0)ab (⌅) +

cd

G (0)ac (⌅)⇥⇥

cd(⌅)Gdb(⌅) (20)

where G(0) is the single-particle propagator in the unper-turbed system defined by a one-body Hamiltonian H0 ofchoice.

38 40 42 44 46 48 500

1

2

3

4 Exp.GGF NNGGF NN+3NF (Indc)GGF NN+3NF (Full)

Δn(3

) (A)

[MeV

]

ACa

Pairing gaps

Page 23: Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$ Collaborators:** * ... b ⇥ k Have a systematic expansion of the Hamiltonian ... trons and

Pairing gaps

! Inversion of odd-even staggering

" Second order and 3NF necessary to invert the staggering

40 42 44 46 48

-2

-1

0

1

2

32NF (HFB)3NF (HFB)3NF (2nd)exp.

ACa

ΔE

[MeV

]

Pairing gaps

Page 24: Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$ Collaborators:** * ... b ⇥ k Have a systematic expansion of the Hamiltonian ... trons and

Preliminary!

0 1 2 3

10 100

5/2-

3/2-

E x [M

eV]

SF [%]

1/2-

61Ni

10 100

63Ni

1/2-

10 100

65Ni

3/2-

10 100

67Ni

5/2+

prelimi

nary

10 100

9/2+

69Ni

5/2-

10 100

71Ni

9/2+

Ni spectral distributions – at 2nd order

Page 25: Three-Nucleon Forces in Neutron Rich Isotopes (from O to Ni) · 2N%3N#ind$ 2N%3N#full$ Collaborators:** * ... b ⇥ k Have a systematic expansion of the Hamiltonian ... trons and

D. Van Neck, M. Degroote

A. Carbone

W.H. Dickhoff, S. Waldecker

M. Hjorth-Jensen

A. Cipollone, A. Rios

P. Navratil

V. Somà, T. Duguet

A. Polls

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