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

Carlo Barbieri — University of Surrey

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

Eg.s. !MeV"$100

N 15 N

"

"

!

!

21

N 23 N27

N#Ω"24 MeVΛSRG "2.0 fm$1
"

$120

"
!

!

"

$160

"

!

$140

!

"
!

$180
15

F

17

F

23

F

"
!

"

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

!

"
!

25

29

F

S2n [MeV]

13

$80

60

Ti

50

Sc

40
30

K
Ar

20
10
0

F

Ca

18

20

22

24

N

26

28

30

32

r Forces Frontiers

0orders in the treatmentof the 3NF. It must be emphasized that
almost at the sames 1/2
the present way to proceed when applying SRG evolution
N = 20. The domi
in infinite matter-4 shouldbe improvedby carryings 1/2
d 5/2 out thed 5/2and (b) can be trace
evolution on a full Hamiltonian with both two- and three-bodynents, which determ
G-matrix
SDPF-M
forces. Recently,-8 improvements
towardthesolutionof
this
two orbitals. The mo
Vlow kUSD-B
problem have been presented for calculations in pure neutron
body interaction V a
8
14 16
20
8
14 16
20
matter [41], whereafull
Hamiltonianhas
been
consistently
all possible orientati
Number (N) spectrum
Neutron(N) :SingleNeutron
particle
atNumberEfermi!
and l! j ! [16],evolved. All in all,our results seem to contradict the idea that
4!
2!
(c) G-matrix
NNsimply+ 3N (∆) forces(d) Vlowmatter.
k NN + 3N (∆,N LO) forcesinduced 3NFs can betreatedin nuclear
mono[T. OtsukaVetal.,!=j,j !
In terms of 0evolved interactions, our nonperturbative
Phys Rev. Lett 105,!
m,m
d 3/2perturbatived 3/2 to check whether the
calculations can be used
032501 (2010)]!
where the sum overregime is actuallyreached. To this end, we compare,s 1/2 in
!
-4
s 1/2
m
can be restricted
d 5/2
Fig. 9, our results tothe perturbative calculations
presented
d 5/2
details). The SPE ofin Ref. [43]. The
BHF and SCGF calculations have2 been
mono
-8
Vj,j
multiplied by!
+ 3N (N LO)
NN + 3N (∆) 2NF and a 3NF NN
performed with a SRG-evolved
with
the
same
!
NN
+
3N
(∆)
j
.
This
leads to theNN
NN Brueckner
cut-offs, λ/!3NF = 2.0/2.0 fm−1 . Whereas theshell structure and t
8
14 16
20
8
14 16 the 20
results have been obtainedwitha zero-temperaturecode,
The comparison o
Neutron Number (N)Neutron Number (N)SCGF calculations have been extrapolated to zero temperaturemonopole interactio
obtained from NN thby means of a simple procedure. At low temperatures,
oxygen anomaly can
FIG. expansion
2: Single-particle
energies
the neutron
d5/2tem, s1/2 andthe Sommerfeld
indicates
thatof theeffect of
16
contributions to thedorbitalsmeasuredfromtheenergyofOasafunctionof
3/2
perature is quadratic and is the same, but with opposite sign,
neutron number
N . (a) SPE calculated
from a G matrix andwhich approximatel
Need
at
LEAST
3NF
for the energy
and the free energy [47]. Consequently, thefrom low-momentum interactions Vlow k . (b) SPE obtainedon the d3/2 orbital.
semi-sumof from
both thethermodynamical
potentials
is an [14]estimate(“cannot”
do
RNB physics
phenomenologicalforceswithout…)!
SDPF-Mand USDforces, it is unlikely
B [15]. (c,d) SPE
including
3N forces dueof the zero-temperatureenergy.
Wecontributions
obtain anfrom
extremely
have been missed, an
2to ∆ excitations
EFT 3N interactions
at NandLO [26].
good agreement
between andbothchiralmany-body
approaches
may be important fo
Single-Particle Enetheory. If the diagrammatic summation is complete, it should
lead to scale-invariant results.
We test this hypothesis by performing calculations at
different evolution scales, in both the two- and the threebody low-energy
sectors. We evolvethe 2NFofusing
a free-space
SRG
ative
regimeQCD
with forcestransformation [37]. The transformation renormalizes the 2NF,
suppressing off-diagonal matrix elements and giving rise toa universal
SRG evolution
Chirallow-momentum
EFT for interaction.
nuclearTheforces:
flow also induces many-body forces, which should be taken
into account to keep the calculation complete. Following thephilosophy of Ref. [43], we incorporate the effect of induced
forces through the refitting of the cD and cE LECs to the 3 Hbinding energy and 4 He matter radius. We use the values given
in 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 , appearing
in the density-dependent 2NF. A more complete calculation
would 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 2NF
have a saturation point which is much closer to the empiricalvalue when compared to the original force. Moreover, if
the 2NF has been SRG-evolved, the results are somewhat
independent of the cutoff. Overall, one can say that themore the 2NF is evolved downward, the more attractive theSingle-Particle Energy (MeV)

Modern realistic nuclear forces!!!

The changes due to 3N forces based on ∆ excitations arehighlighted by the shaded areas.
Saturation of nuclear matter:!

(3NFs arise naturally at N2LO)!

Energy/nucleon, E/A [MeV]

Energy/nucleon, E/A [MeV]

Next, we show tha
trons and one nucleo
sive monopole intera-5et al.
10
sures N = 8, 14, 16,Hebeler
and 20.
The evolution of the SPE
While the contribut
N3LO+N2LO dd
BHFas neutrons are added. For theis due to interactions
quantities can be diff
-1
5
SCGF
λ/Λ3NF =2.0/2.5 fm
SPE
based
on
NN
forces
in
Fig.
2
(a),
the
d
orbitalcomposed of valence
3/2
-1
-10
λ/Λ3NF =2.0/2.0 fm
decreases rapidly as neutrons occupy the d5/2 orbital,of 3N forces. The re0-1
and remains well-bound from N = 14 on. This leads
can be understood b
λ/Λ3NF =1.8/2.0 fm-5
to bound oxygen isotopes out to N = 20[A.
andputs theet al.,ple. Figure 3 (a) dep
Carbone-15
neutron drip-line incorrectly at 28 O. This result
appears
forces due to theexPhy.s Rev.
C 88, 044302!!(2013)]!
-10
to depend only weakly on the renormalization
methodchange of pions wit-1
λ/Λ =2.0 fm
T=0NN
MeV
or theinteraction used.3NFWe demonstrate this by-15
a second-order pertu-20
T=5 MeV
Sonia Bacca
3
4
showingSPE
in0.24
the G matrix
ergy and to the two
00.08 calculated
0.32 formalism [11],0.16-20which sums particle-particle-3ladders, and based on lowto be attractive. Th
0
0.08
0.24
0.32
0.16
Density, ρ [fm ]-3momentum interactions Vlow k [12] obtained from chiralmonopole componen
Density, ρ [fm ]NN interactions at next-to-next-to-next-to-leading order
In nuclei, the proc
FIG. 9. (Color3 online) Comparison of results for the energy per(N LO) [13] using the renormalization group. Both calthe SPE of the j, m oFIG. 8. (Color online) SCGF results for the energy per nucleon
nucleon of SNM obtained with different approaches using the samer Forces Frontiers

Chiral Nuclear forces - SRG evolved

ative low-energy regime of QCD with forces

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

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

SRG evolution to !=2.0 fm-1!

VNN !

(3NFs
Sonia Baccaarise naturally at N2LO)!
[Jurgenson,!Navrá>l,!Furnstahl,!!
Phys.!Rev.!LeE.!103,!082501!(2009);!
Hebeler,!Phys.!Rev.!C!85,!021002!(2012)]!

4
3“induced”
Hamiltoninan!

V3N
induced !

V3N
full!

“full”
Hamiltoninan!

Faddeev-RPA in two words…
Particle vibration coupling is the main cause driving the distribution ofparticle strength—a least close to the Fermi surface…
≡!!
!par7cle!
≡!hole!

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

In the Lehmann representation, the one-body
A.Green’s
SelfFORMALISM
function is given by

II.
Green’sfunctions in many-body theory

For"!Ψ
a A+1
J =
all
A+1
A
!
!ΨA
|c†β0|Ψnucleus,
InOne-bodythe Lehmannrepresentation,the
one-body
Green’s
0 |cα |Ψnn0"Green’s function (or propagator)
motion whereof quasidecoupled,$,j label tgαβ (E) = describes the
A+1
A
function
is
given
by
E − (E− E0 ) + iη
particles and holes:
larn momentumand τ represn†TheA−1
A+1
A+1 †irreducible
in
! !ΨA
! !ΨA
|c
|Ψ
"!Ψ
|cβ |ΨA"!ΨA−1|cαself-energy
|ΨA
α
0
n
n
0"
0 |cβ |Ψk0"kther a proton
or a neutron)gαβ (E) =
+
, (1)
A+1
A−1
A
A
E
−
(E
−
E
)
+
iη
E
−
(E
−
E
)
−
iη
n
0
0
the harmonic-oscillator
basis
k
n
k
†
A−1
A−1
tion, as follows:
A
! !ΨA
|c
|Ψ
"!Ψ
α |Ψ
0 β
0 " label a complete orthonormal basis set
k wherek α,|cβ,...,+(1) by nucleon transfer…this contains
all the structure
information, probed
!
A−1 †
A
%
! quantization
and
c
(c
)
are
the
corresponding
second
E
−
(E
−
E
)
−
iη
α
0
β
Σ (x,1x ; E) =Ikk
(spectral function):hdestruction (creation) operators.
Sab (!)In= these
Imdefinitions,gab (!)&jmj τ
56
A−1
A−1
where α, β, ..., label aNi
completeorthonormal
basis
set
⇡
A+1
A+1
"
|Ψ
",
|Ψ
"
are
the
eigenstates,
and
E
,
E

n
n
k
k
!
and cα (c†β ) are the corresponding
secondquantization%
the eigenenergies of the (A ± ×1)-nucleon
The
Rnaisotope.
& (r)Σab (E)Rndestruction (creation) operators.structure
In theseofdefinitions,Eq. (1) is particularly
usefulfor our purna ,n
b
A−1
A−1
A+1
A+1
|Ψn ", |Ψk " are the eigenstates,, Eenergies,
poses.andAt E
positivethe residues of the first term,nkAx ≡waver, σ, functions
τ . The spin
the eigenenergies of the (A ± 1)-nucleon
isotope.
The the where!ΨA+1|c†α |Ψ
scatteringforn0 ", containAσ, noffis thethe|Ψprincipal
quantumstructure of Eq. (1) is particularly
for our purstate,
the usefulelastic collisionof a nucleon
0 " groundoscillator, and a ≡ (na , $, j, τ )poses. At positive energies, the residues of the first term,
A+1 †A
cleus the self-energy is indepen!Ψn |cα |Ψ0 ", contain the scattering wave functions forradialharmonic-oscillator
funthe elastic collision of a nucleon off the |ΨA
sca4ering
0 " ground state,neutron*removal

neutron*addi1on[CB,!M.Hjorth.Jensen,!Pys.!Rev.!C79,!064313!(2009);!CB,!Phys.!Rev.!LeE.!103,!202502!(2009)]!

Results for the N-O-F chains
A. Cipollone, CB, P. Navrátil, Phys. Rev. Lett. 111, 062501 (2013)

13

⇥60
⇥80

⇥

$80

⇧⇤24 MeV⌅SRG ⇤2.0 fm⇥1

⇤

$100

⇥100

⇤
⇥
⇤

⇥120

⇥

⇤

⇥140

⇥

⇥

⇤

⇥160
⇥180

2N 3N⇤ind⌅
2N 3N⇤full⌅
Exp

⇥

⇤

⇤

14

O

16

O

22

O

⇥
⇥
⇤

⇥

⇥

⇤

⇤

24

O

⇥
⇤

⇥
⇤

28

O

Eg.s. !MeV"

Eg.s. MeV⇥⇥⇥⇤N 15 N

"

"

!

!

21

N 23 N27

N#Ω"24 MeVΛSRG "2.0 fm$1
"

$120

"
!

!

"
!

$140
$160

!

"
!

$180
15

F

17

F

23

F

"
!

"

"

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

!

"
!

25

29

F

F

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

kinematics [19–21] on 16 O and 18 O targets. The angular
distributions were well reproduced in all cases by CRC
calculations. For 16 Oðd; 3 HeÞ at 14 and 26 MeV=nucleon,
we obtained same C2 Sexp , which confirms the energy independence of the analysis. For the 14 O (d, 3 He) and
14 O (d; t) transfers, the shape of the angular distributions

15

N disagree by several MeV. Finally, we show the reduction factor Rs , also plotted in Figs. 4(a) and 4(b), for WS
and microscopic OFs, respectively. In the total uncertainty,
we set apart in a box the uncertainties originating from theanalysis: (i) imperfect knowledge of entrance and exit
potentials, and (ii) the variance in the calculation of rms

Single nucleon transfer in the oxygen chain[F. Flavigny et al, PRL110, 122503 (2013)]

TABLE I. The normalization C2 Sexp for two OFs, phenomenological (WS) and microscopic (SCGF) [30]. For the WS OF, the16O for which Rrms was taken from (e, e0 p) data (see text). The SFs C2 Sth are
r0 values were chosen to reproduce RHFB
rms , except for
14 calculations13with the WBT 14
3 the second
13 part, the analysis was performed with microscopic
obtained from shell-modelinteraction. In
OFs and SFs. The two errors for C2 Sexp and Rs are the experimental and analysis errors.

! Analysis ofO(d,t) O andReactionE' (MeV)

J%

RHFB
r0
rms
(fm) (fm)

0.00
0.00
3.50
0.00
0.00
6.32
0.00

3=2#
1=2#
3=2#
1=2#
1=2#
3=2#
1=2#

2.69
3.03
2.77
2.91
2.95
2.80
2.91

14 O (d, t) 13 O
14 O (d, 3 He) 13 N

O (d, t) 15 O
O (d, 3 He) 15 N [19,20]

18 O (d, 3 He) 17 N [21]0,81.69 (17)(20)
1.14(16)(15)0.94(19)(7)0.91(9)(8)0.93(9)(9)
1.83(18)(24)0.92(9)(12)

3.15
1.55
1.90
1.54
1.54
3.07
1.58

0.54(5)(6)0.73(10)(10)0.49(10)(4)0.59(6)(5)0.60(6)(6)0.60(6)(8)0.58(6)(10)

1.89(19)(22)
1.58(22)(2)
1.00(20)(1)0.96(10)(7)1.25(12)(5)2.24(22)(10)

3.17
1.58
1.90
1.73
1.74
3.45

0.60(6)(7)
1.00(14)(1)
0.53(10)(1)0.55(6)(4)0.72(7)(3)0.65(6)(3)16

O data18

O data14

0,4O(d, t)week ending
HYSICAL
22 MARCHP2013

REVIEW LETTERS

14 13
14 13
(a)
(b)
Despitedifferent OFs
and< O| SFs,
O > the analysis< O| N >122503-30,3 ab initio OF [30] provides very
performed with theWS
WS
similar results with Rs ð!S ¼ 0Þ ¼ 0:636ð34Þð42Þ
and ! ¼
0,2 &1
SCGF
SCGF&0:0042ð28Þð36Þ MeV , with calculated!S ¼ 17:6 MeV[Fig. 4(b)].
0,1
In summary, we measured exclusive differential cross
0,0
sections at 18 MeV=nucleon
for the 14 Oðd; tÞ13 O and
-2
(c)
(d)
14 Oðd; 3 HeÞ13 N transfer
10
reactions and elastic scattering.
0
WS OFs with a constraint
on HF radii and microscopicOFs (obtained from SCFG
theory) have been compared for-2-10
the first time for symmetric and very asymmetric nuclei
0
2
4
6
0
2
4
6
8
and gave similar results.
We extracted
the reduction
factorsr (fm)r (fm)
Rs over a high asymmetry range, !S ¼ '18:5 MeV, foroxygen isotopes.
theonline).
good agreement
between ofthe(a), (b) the OFs for
FIG.From
3 (color
Radial dependence
CRC calculations
and microscopic
the set of transfer
datahighlighted
in normalized to 1;
WS and(SCGF)[30]
form factors
our work, the(c),asymmetrydependence
is
found
to
be
non(d) the OF difference $ (SCGF # WS).
existent (or weak), within the error bars. This result is inagreement with ab initio Green’s function and coupled16
fair agreement
for the calculation
cluster calculations[7,14], obtained
but contradicts
the trend of the O rms
radii performed
SLy4 interaction
observed in nucleon
knockoutwith
datatheobtained
at incident[31] compared to16Oðe; withe0 pÞ15theNgs and 15 N3=2#
the100valuesdeduced andfromenergies below
MeV=nucleon
analyzedsudden-eikonal
formalism.
The disagreementof thetwoWe thus adopted
analyses[5], bothstates with largeSFs.
systematic trends
fromknockout
andtransfer
calls
for
a
the HFB radii calculated for the 0p wave functionsfor 14 O
18 so-called direct reaction mechanisms
better description of)

O data0,2

Rs = σexp(θ) / σth(θ)

Rs(SCGF)

-3/2

14

0,6

(b) SCGF

0,8

is nicely reproduced (Figassuming a !l ¼ 1 transexpected from the transfer
In the second approach,OFs obtained from the sin
the third order algebra
method [ADC(3)] [14,32]
chiral two-body next-to(N3LO) [33] plus three-bo(N2LO) [34] interaction
1:88 fm#1 , as introduced
OFs were further rescale
same factor (i.e., introduci
correction) to account for dexperimental rms radius of
the removal of main pea
separation energies are s
respectively, and comparetion. We note very little radthe strongly bound neutron
We give in Table I the nokinds of OFs. From theor- Overlap functions and strengths from GF

0,6
0,4
0,2

C2 Sth(SCGF)OF (fm3

O(d, He)

1

C2 Sexp(SCGF)

δ (fmRs = σexp(θ) / σth(θ)(a) WS + SM
14

Rs
(WS)

P H Y S I C A L R E V I E W L E T TPRL
E R 110,S122503 (2013)

PRL 110, 122503 (2013)
1

C2 Sth0p þ 2@!

)

16

1.40
1.23
1.12
1.46
1.46
1.31
1.46

C2 Sexp
(WS)

-3/2

16

O(d, He) N transfer reactions @ SPIRAL!

-20

-10

0

10

20

∆S = ε (Sp - Sn) (MeV)

FIG. 4 (color online). Reduction factors Rs obtained with (a) a
WS OF and the SLy4 interaction [31], averaged over four

- Rs independent of asymmetry!

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

EA
0 [MeV]

-300
-350
-400

40

Experiment
NN + 3N (full)
NN + 3N (ind.)
SM (NN + 3N)
CC35

S2n [MeV]GGF [NN + 3N (ind.)]
GGF [NN + 3N (full)]
ADC(3) "corrected"
IM-SRG [NN + 3N (full)]

30
25
20
15

-450

Nmax=13

36

38

10

40

42

44

46

48

50

52

5

38

40

ACa

42

44

46
A

48

50

52

Ca

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

Neighbouring Ar, K, Ca, Sc, and Ti chains
V.!Somà,!CB!et*al.!Phys.!Rev.!C89,!061301R!(2014)!

S2n [MeV]

Two-neutron separation energies predicted by chiral NN+3NF forces:!
60

Ti

50

Sc

40
30

Ca
K
Ar

20
10
0

18

20

22

24

26

28

30

32

N

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

S2n [MeV]

Two-neutron separation energies predicted by chiral NN+3NF forces:!
60

Ti

50

Sc

40
30

Works well in
the pf shell!

Ca
K
Ar

20
10
0

18

20

22

24

26

28

30

32

N

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

S2n [MeV]

Two-neutron separation energies predicted by chiral NN+3NF forces:!
60

Ti

50

Sc

40
30

Ca
K
Ar

Over estimated
N=20 and Z=20 gaps!

20
10
0

18

20

22

24

26

28

30

32

N

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

S2n [MeV]

Two-neutron separation energies predicted by chiral NN+3NF forces:!
60

Ti

50

Sc

40
30

Ca
K

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

Ar

20
10
0

18

20

22

24

26

28

30

32

N

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

Enerd that the evolution of the ESPEs is mainly due
980
nsor component. However, in more recent publi561
474
11, 13, 14] several authors have shown that both359
92
r term as well as the central term have to be
39
41
43
45
47
49
51
37
d.
2
K
K
K
K
K
K
K
K
ding the shell model, potassium isotopes are ex= 28 for this study, with only one proton less1/2
FIG. 1. (color online) Experimental energies for 1/2+ and
robes
1/3
= 40
3/2+ states in odd-K isotopes. Inversion of the nuclear spin
magic
number Z2522= 20. Nevertheless, little andalsis
is obtained in 47,49 K and reinversion back in 51 K. Results areynterconflicting information is available so far for
taken
from et[16,
Ground-statespin for 49 K and 51 K
ents:
on-rich K isotopes. Level schemes based on the
J. Papuga,al.,23–25].
PRL 110,172503 (2013)
were established [22].
ly[12]
assigned1371
spins of the ground state were produe
980
48
49
K [16]. In addition, an extenAK isotopes!
ubli-K [15] andPhotomultiplier
561
474
Laser
both
Post-acceleration359
ussion was presented by Gaudefroy [17] on the
tubes
voltage
92
o be
vels
and configurations of N = 27, 28 and29
Laser spectroscopy@ ISOLDE!
K
K
K
K
K
K
K
K
Electrostatic
Atoms
n
the
shell-model
frameworkandcomparedtoe exContinuous ion beam
deflectors
from HRS
FIG.
1. (color online)
Experimental
energies
for 1/2+ andlessimentalobservation,
whereavailable.
However,+
Mirror
3/2
statesin
odd-K
isotopes.
Inversionofthenuclearspin
48
and spin of 2 for
cted
with
47,49 K, is in contradiction
51
is obtained in
K and reinversion back in K. Results are 7
r for
49
proposedby [16,KrólasetGround-state
al. [15]. spin
In foraddition,
taken
from
23–25].
K and 51 K
ISCOOL
Changein separation
described
by chiral
NN+3NF:!
n)thewere
established[22].state of 50 K was proposed
ar
spin
of
the
groundproCharge exchange
e is
Ions
xtencell[18, 19] in contrast to the recent decay studPhotomultiplier
Laser
Post-acceleration
n the
6
tubes
Experiment
ed it29 was suggested to be 1 [20].voltageThe groundLowest peak
49
Electrostatic
n-parityof K was tentatively
assignedAtomsto be
5
ed to
ESPE
Continuous ion beam
deflectors
FIG. 2. (color online) Shematic representation of the
setup
HRS
ever,
ynuBrodafromet
al. [16],
contrarytotheearliertenMirrorforcollinearlaser
spectroscopyat
ISOLDE.
4
with+/2) assignment from beta-decay spectroscopy
the
tion,
ISCOOL
51
orK, the nuclear spin was tentatively
assigned3
osed
Charge exchange
Ions
+
cell
51]
studby Perrot et al. [19].
the resulting fragments
was achieved by the surface ion
2wooundcent
hyperfinestructuremeasurementsofK
isosource.
Thetargetandthe ionizing tube were heatedo be
emFIG. 2. (color online) Shematic representation of the setup
1ng
thecollinearlaser
spectroscopy
technique0
tenomto ⇠2000 C. The accelerated ions (up to 40 kV) were
for collinear laser spectroscopy at ISOLDE.
copy
answerto the question as to what happens withns.
mass separated0 by the high resolution separator (HRS).
gned
ent
n sd orbits for isotopes beyond N = 28. By esThe gas-filled Paul
trap
(ISCOOL)
[28,
29]
was
used
for53
51
the resulting
surface
37
39
41
43
45
47
49
51
gowthe nuclear
spinsfragmentsof 49 K was
andachieved
K tobybethe1/2
andion cooling and bunching
K
isoof
the
ions.
MultiplebunchesperAsource. The target and the ionizing tube were heatedmed
K
espectively,
the
nique
0 evolution of these two states into ⇠2000
C. The accelerated ions (up to 40 kV) were proton pulse were created by closing ISCOOL for ⇠90 ms
. 5,
with
otopesisfirmlyascertain.
is presentedin
mass
separatedby the highThis
resolution
separator (HRS).
ESPE:“centroid”
energies!
perper bunch. The
bunched calculations
ions were guided atto the(Gorkov
2ndsetuporder)!
y es- FIG.
(colorN
online)Upperpanel:
energydi↵erence
The 6.
gas-filled
Paul
trap
(ISCOOL)
[28,
29]
was
usedbetween
forisotopes
from
=
18
up
to
N
=
32
where
8 and
is
+
+
43 51
for collinear laser spectroscopy where they were superthe
lowest1/2
and
3/2
statesobtained
in
K
from
ab
coolingandbunching
of
the
ions.
Multiplebunchespertrain of the states is observed at N = 28 folsion
esinitioGorkov-Green‘sfunction
calculations
andprotonpulse were createdby closing
ISCOOLforexperiment.
⇠90 ms imposed with the laser. A schematic representation ofomedthein reinversion
back
at⇡sN
=
32. guided
In addition,Lowerpanel: The⇡dande↵ectivesingle-particle
energies
per bunch.
bunchedions
wereto the setup
3/2
1/2

Inversion of d3/2—s1/2 at N=28+

Energy (keV)

+

39

41

43

45

47

49

51

E(1/2+ - 3/2+) [MeV]37

Two-neutron separation energies
for neutron rich K isotopes
15

S2n [MeV]AME extrapolation!
10

Experimenthw28 rampA
hw32 rampA
hw36 rampA
GGF extr. rampA
hw28 rampD'

5

0
47

49

51

53

! Error bar in predictions are fromextrapolating the many-body
expansion to convergence of themodel space.

AK

V.!Somà,!CB!et*al.,!in!prep.!

Ca and Ni isotopic chains

!
y
r
a
n
i
m
i
l
e
Pr

Calculations based on ramps D, N1, N2:!

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

Ca

E [MeV]

-350
-400

p

-450
-500

d3/2

i
rel

n
mi

a ry
p1/2

f5/2

Experiment
Full 3NF (ramp N1)
Full 3NF (ramp N2)

-350
-400

E [MeV]

-300

-450
-500
-550
-600

f7/2

p3/2

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

N

-650
-700
48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80
ANi! Large J in free space SRG matter (must pay attention to its convergence)! Overall conclusions regarding over binding and S2n remain but details change
IM-SRG results from H. Hergert!

Ca and Ni isotopic chains

!
y
r
a
n
i
m
i
l
e
Pr

35
30
25
20
15
10
5
0

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

Ca

ry
a
in

m
eli

pr

f5/2

40

S2n [MeV]

S2n [MeV]

Calculations based on ramps D, N1, N2:!

Experiment
Full3NF (ramp N1)

30
20
10

d3/2

f7/2

p3/2

p1/2

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

N

050 52 54 56 58 60 62 64 66 68 70 72 74 76 78
ANi! Large J in free space SRG matter (must pay attention to its convergence)! Overall conclusions regarding over binding and S2n remain but details change
IM-SRG results from H. Hergert!

Ca and Ni isotopic chains
Difference of calculated BEs to the experimentfor different masses:!

ΔE/A [MeV]

1
0.8
0.6
0.4

l
e
r
p

0.2
0

ry
a
in

im

O
Ca
Ni

-0.2
14

22

30

38

46

54

62

70

78

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

Conclusions
 What to did we learn about realistic chiral forces from ab-initio calculation ?
! Leading order 3NF are crucial to predict many important features thatare 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) withroughly 1 MeV/A over binding.

! This hints to the need of more repulsion in

Ti

50

Sc
Ca

eV]

60

40

S 2n [Mfuture generations of chiral realistic forces.

K
Ar

30
20
10

0

18

20

22

24

N

26

28

30

32

Conclusions
 What to did we learn about realistic chiral forces from ab-initio calculation ?
! Leading order 3NF are crucial to predict many important features thatare 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) withroughly 1 MeV/A over binding.

! This hints to the need of more repulsion in

Ti

50

Sc
Ca

eV]

Thank you for
yourattention!!!

60

40

S 2n [Mfuture generations of chiral realistic forces.

K
Ar

30
20
10

0

18

20

22

24

N

26

28

30

32

Results for the N-O-F chains

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

3

6
2

!

!

1d3!2

"
!

"

0

!6

!

"

!2
!4

2.8

!

!
"
"
!
"
!

"

2s1!2

"
!

!
"

1d5!2

"
!

"

R_rms [fm]ΕiA#1 "MeV#4

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

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

2.6

!
"

"
!

!8

!
"

2.4

!
"
2.2
14

16

18

20

22

A (Oxygen isotopes)! 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)!

24

26

28

Ca spectral distributions – at 2nd orderneu lSRG =2 L=13,16,16 RAMP Dpp FULL

1ê23ê23ê2+
7ê2-

Esp @MeVD

-10

1ê2+

-30
36

Ca

38

Ca

40

Ca

42

Ca

44

Ca

46

Ca

48

Ca

50

Ca

52

Ca

54

Ca

56

Ca

58

Ca

60

Ca

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
SF

!

5ê2+

-20

ry
mina
Preli

5ê2-

0

particle propagator one can as well derive a perturbativeV(9)
expansion, which translates into an infinite series of diagrams and has the advantage of providing systematic
Pairing gapsways of approximating the solution. The connection between the diagrammatic expansion and the equation of
motion for G leads to the definition of the (irreducible)-body forces! Three-point massdifferences
self-energy⇥ and the derivation of Dyson’s equationNN¯ ,|1:c̄; 2:d⌅Pairing gaps

RMALISM

(
(3)
(A)
=
n

superfluid systems

he fundamental object
ingle-particle propaga-† ⇥b (t )

⌃

t t)

|⇤N
0 ⌅,

Gab (t, t⇥ )cΔn(3) (A) [MeV]

(10)

3

(18)

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

d
2

b
1

⇥R
cd (⌅)

(11)
0

(12)

2 E0A + E0A 1 ]a4

ate of the system dedenoted by |⇤N
0 ⌅, so-⇤Nk ⌅1)A A+1[E023840(0) 42
Gab (⌅) = Gab (⌅) +

(19)

44(0)
46
48
50
Gac
(⌅) ⇥⇥cd (⌅) Gdb (⌅) (20)

cd

ACa

where G (0) is the single-particle propagator in the unper-

Pairing gaps

Pairing gaps

! Inversion of odd-even staggering
2NF (HFB)
3NF (HFB)
3NF (2nd)exp.

3

ΔE [MeV]

2
1
0
-1
-2
40

42

44
ACa

46

48

" Second order and 3NF necessary to invert the staggeringNi spectral distributions – at 2nd order

100

10

100

10

100

10

100

10

100

3/2-

2
1
0

61Ni

63Ni

65Ni

67Ni

5/21/2-

pre

li

na
mi

69Ni

ry
9/2+

9/2+

10

100

71Ni

!

Ex [MeV]

10

5/2-

5/2+

ry
mina
Preli

3

3/2-

1/2-

SF [%]Collaborators
A. Cipollone, A. Rios
V. Somà, T. Duguet
A. Carbone
P. Navratil
A. Polls
W.H. Dickhoff, S. Waldecker
2/19/2014

D. Van Neck, M. Degroote

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