carbon isotopes in the in-medium ncsm - tobias mongellic h i [f m] 0+ 1 iground state energy,...

Carbon Isotopes in the In-Medium NCSMTobias Mongelli

Carbon Isotopesin the In-Medium NCSMTobias Mongelli and Robert Roth

In-Medium NCSM

• "new" ab initio many-body methods for the description of ground and excited states inopen-shell nuclei

• No-Core Shell Model (NCSM) [1] limited by basis dimension, scaling with particlenumber, computational characteristics

• In-Medium SRG [2, 3] basic formulation limited to ground states• idea: combine NCSM with IM-SRG In-Medium NCSM[4]

NCSM

IM-SRG

NCSM

• NCSM calculation in small model space defines reference state

|Ψref〉=∑

i

ci |Φi〉

• multi-reference IM-SRG aims at decoupling the reference state |Ψref〉 fromgeneralized ph excitations in a magnus expansion formalism

ddsΩ̂(s) =

∞∑k=0

Bkk!

�Ω̂(s), η̂(s)

�k

• compute the desired operators Ô(s) using the BCH series

Ô(s) =∞∑k=0

1k!

�Ω̂(s), Ô(0)

�k

• use IM-SRG-evolved operators Ô(s) as input for subsequent NCSMcalculation

• transformation of non-scalar operators for B(E2) transition strengths and electricquadrupole moments

Uncertainty Estimation

• full uncertainty estimation with order by order convergence and many-bodyuncertainties with the newest chiral two- plus three-nucleon interactions[5]

δX (0) =Q2|X (0)|, δX (i) = max2≤ j≤i

�Qi+1|X (0)|,Qi+1− j|∆X ( j)|� for i ≥ 2

• many-body uncertainty maximum of the absolute difference between theobservable at the two largest values of Nmax and N

refmax

12C - Nmax Convergence

12C - Order by Order Convergence & Correlation

Carbon Isotopic Chain

• ground state energy, spectra, B(E2) transitions, quadrupole moments and charge radiifor carbon chain as a function of A for N refmax = 2 and Nmax = 4

Outlook

• compare B(E2) transition to pure NCSM calculations interaction or truncationproblem for 16C?

• use other cutoffs and interaction families (LENPIC collaboration)

• consistent freespace SRG for all operators (B(E2), Q, Rch)

• compute B(M1) transitions for carbon chain

• compute observables for other isotopic chains like fluorine or neon

References:[1] R. Roth, Phys. Rev. C 79, 064324 (2009).[2] K. Tsukiyama, S. K. Bogner, and A. Schwenk, Phys. Rev.

Lett. 106, 222502 (2011).

[3] H. Hergert, S. K. Bogner, S. Binder, A. Calci, et al.,Phys. Rev. C 87, 034307 (2013).

[4] E. Gebrerufael, K. Vobig, H. Hergert, and R. Roth, Phys.Rev. Lett. 118, 152503 (2017).

[5] T. Hüther, K. Vobig, K. Hebeler, R. Machleidt, et al.,arXiv:1911.04955v1 (2019).

[6] NNDC, ENSDF (https://www.nndc.bnl.gov/) (2019).[7] W. Vermeer et al., en, Phys. Lett. B 122, 23 (1983).

0 20 40 60 80s

90

80

70

60

50

40

E[M

eV]

12C

0+1

IMSRG-Flow Nrefmax = 2IM-SRGNmax = 2Nmax = 4Nmax = 6Nmax = 8

Nmax = 2, s=0 Nmax = 2, s=90

• flow evolution for 12C decoupling accelerates NCSMconvergence

• off-diagonal part becomessuppressed during the flow

• with growing Nmax, the energyat integration step zerodecreases

• interaction: N3LOEMN,500+N3LONL,500, α = 0.04 fm

4, ħhΩ= 20 MeV,natural orbital basis

0

10

20

E* [M

eV] 1+1

2+1

0+1

2+212C

Nrefmax = 0

1+1

2+1

0+1

2+212C

Nrefmax = 2

1+1

2+10+1

2+212C

Nrefmax = 4

0+1

2+1

1+1

4+1

2+2Exp

0

5

10

B(E2

) [e2

fm4 ] 2+1 0+1

12C2+1 0+1

12C2+1 0+1

12C2+1 0+1

0 2 4Nmax

4

6

8

Q [e

fm2 ]

2+1

12C

2 4Nmax

2+1

12C

4Nmax

2+1

12C

2+1

• spectrum, B(E2) transitionsand quadrupole moment for12C

• error bars indicate many-bodyuncertainties

• excellent agreement withexperimental values[6, 7]

• EMN interaction performs wellfor these observables in 12C

0

5

10

15

E* [M

eV] 1+1

2+1

4+1

0+1

12CNrefmax = 0

1+1

2+1

4+1

0+1

12CNrefmax = 2

1+1

2+1

4+1

0+1

12CNrefmax = 4

0+1

2+1

1+1

4+1

Exp

0

5

10

B(E2

) [e2

fm4 ] 2+1 0+1

12C2+1 0+1

12C2+1 0+1

12C2+1 0+1

NLO N2LO N3LOChiral Order

4

6

8

Q [e

fm2 ]

2+1

12C


2+1

12C


2+1

12C

2+1

• spectrum, B(E2) transitions andquadrupole moment for 12C forNmax = 4 and Λ = 500 MeV


• error bands indicate interactionuncertainties

• excellent agreement withexperimental values also for N2LO

• uncertainty bands shrink with higherorder

3 4 5 6 7 8 9Q(2 +1 )[efm2]

4

5

6

7

8

9

B (E2

,2+ 1

0+ 1)[e

2 fm

4 ]

450MeV500MeV550MeVN3LON2LOExperimentBohr MottelsonRigid Rotor

• correlation between B(E2) transitionand quadrupole moment with combinedmany-body and interaction uncertainties

• strong correlation behaviour betweenthese observables

• rigid rotor model slightly misses thepredicted correlation

• Bohr Mottelson model matches wellwith data from In-Medium NCSM

• value for Q(2+) can be estimated byusing the Bohr Mottelson model

125

100

75

E [M

eV]

0+1

N3LON2LO

0

5

10

E* [M

eV]

2+1

0

5

B(E2

) [e2

fm4 ]

2+1 0+1

0

5

Q [e

fm2 ] 2

+1

C10 C12 C14 C16 C202.4

2.6

R ch [

fm] 0+1

• error bands indicate combinedmany-body and interactionuncertainties

• ground state energies agree withexperimental values

• 2+1 exictation energy in reasonableagreement with experiment

• B(E2) transitions for 12C and 14Cagree well with the experiment, butother isotopes show underestimation interaction problem or missinghigher-body contributions

• quadrupole moment for 12C agreeswell with the experiment

• charge radius for 12C is overestimatedbut fits well for 14C compared to theexperiment, for others noexperimental data available

March 3, 2020 | Institut für Kernphysik, TU Darmstadt | Progress in Ab Initio Techniques in Nuclear Physics | Tobias Mongelli | 1

Motivation

I "new" ab initio many-body methods for the description of groundand excited states in open-shell nuclei

I No-Core Shell Model (NCSM) limited by basis dimension, scaling with particle number

I In-Medium SRG basic formulation limited to ground states

I idea: combine NCSM with IM-SRG In-Medium NCSM

chiralinteraction

Free-SpaceSRG In-Medium

SRG

many-bodymethods


In-Medium No-Core Shell Model

NCSM

IM-SRG

NCSM


I NCSM calculation in small model space defines reference state

|Ψref〉 =∑i

ci |Φi〉


NCSM

IM-SRG

NCSM



|Ψref〉 =∑i

ci |Φi〉

I perform multi-reference IM-SRG aiming at decoupling referencestate from generalized ph-excitations in a magnus formalism

d

dsΩ̂(s) =

∞∑k=0

Bk

k!

[Ω̂(s), η̂(s)

]k, Ô(s) =

∞∑k=0

1

k!

[Ω̂(s), Ô(0)

]k


NCSM

IM-SRG

NCSM



|Ψref〉 =∑i

ci |Φi〉

I perform multi-reference IM-SRG aiming at decoupling referencestate from generalized ph-excitations in a magnus formalism

d

dsΩ̂(s) =

∞∑k=0

Bk

k!

[Ω̂(s), η̂(s)

]k, Ô(s) =

∞∑k=0

1

k!

[Ω̂(s), Ô(0)

]k

I use IM-SRG-evolved (non-scalar) operators Ô(s) as input forsubsequent NCSM calculation

I convergence of NCSM calculation massively improved w.r.t. Nmax

Interaction Details

I N3LOEMN,500 + N3LONL,500

I order by order uncertainty quantification

I interaction performs well for oxygen isotopes also for carbon isotopic chain?


12C Order-by-Order Convergence


0

5

10

15

E* [M

eV] 1+1

2+1

4+1

0+1

12CNrefmax = 0

1+1

2+1

4+1

0+1

12CNrefmax = 2

1+1

2+1

4+1

0+1

12CNrefmax = 4

0+1

2+1

1+1

4+1

Exp

0

5

10

B(E2

) [e2

fm4 ] 2+1 0+1

12C2+1 0+1

12C2+1 0+1

12C2+1 0+1


4

6

8

Q [e

fm2 ]

2+1

12C


2+1

12C


2+1

12C

2+1

I spectrum, B(E2)transition strength andquadrupole moment vs.interaction order

I error bars indicatemany-body uncertainties,error bands interactionuncertainties

I excellent agreement withexperiment regardingcombined many-bodyand interactionuncertainties

Carbon Chain


125

100

75

E [M

eV]

0+1

N3LON2LO

0

5

10

E* [M

eV]

2+1

0

5

B(E2

) [e2

fm4 ]

2+1 0+1

0

5

Q [e

fm2 ] 2

+1

C10 C12 C14 C16 C202.4

2.6

R ch [

fm] 0+1

I ground state energy, spectra, B(E2)transition strengths, quadrupole momentsand charge radii as function of A

I combined many-body and interactionuncertainties

I ground state energies agree well withexperimental data

I underestimation of B(E2) transitionstrengths for 10C, 16C and 20C interaction or missing higher-bodycontributions

I charge radius of 12C overestimated

Epilogue

I Thanks to my groupI S. Alexa, T. Hüther, L. Mertes, J. Müller, M. Knöll,

R. Roth, K. Vobig, T. Wolfgruber Institut für Kernphysik, TU Darmstadt

I Special Thanks toI H. Hergert, R. Wirth Facility for Rare Isotope Beams, Michigan State University

I A. Tichai Max-Planck-Institut für Kernphysik, Heidelberg

I Thank you for your attention!LICHTENBERG

COMPUTING TIME


See you at the Poster

Carbon Isotopesin the In-Medium NCSMTobias Mongelli and Robert Roth

In-Medium NCSM

• "new" ab initio many-body methods for the description of ground and excited states inopen-shell nuclei

• No-Core Shell Model (NCSM) [1] limited by basis dimension, scaling with particlenumber, computational characteristics

• In-Medium SRG [2, 3] basic formulation limited to ground states• idea: combine NCSM with IM-SRG In-Medium NCSM[4]

NCSM

IM-SRG

NCSM

• NCSM calculation in small model space defines reference state

|Ψref〉=∑

i

ci |Φi〉

• multi-reference IM-SRG aims at decoupling the reference state |Ψref〉 fromgeneralized ph excitations in a magnus expansion formalism

ddsΩ̂(s) =

∞∑k=0

Bkk!

�Ω̂(s), η̂(s)

�k

• compute the desired operators Ô(s) using the BCH series

Ô(s) =∞∑k=0

1k!

�Ω̂(s), Ô(0)

�k

• use IM-SRG-evolved operators Ô(s) as input for subsequent NCSMcalculation

• transformation of non-scalar operators for B(E2) transition strengths and electricquadrupole moments

Uncertainty Estimation

• full uncertainty estimation with order by order convergence and many-bodyuncertainties with the newest chiral two- plus three-nucleon interactions[5]

δX (0) =Q2|X (0)|, δX (i) = max2≤ j≤i

�Qi+1|X (0)|,Qi+1− j|∆X ( j)|� for i ≥ 2

• many-body uncertainty maximum of the absolute difference between theobservable at the two largest values of Nmax and N

refmax

12C - Nmax Convergence

12C - Order by Order Convergence & Correlation

Carbon Isotopic Chain

• ground state energy, spectra, B(E2) transitions, quadrupole moments and charge radiifor carbon chain as a function of A for N refmax = 2 and Nmax = 4

Outlook

• compare B(E2) transition to pure NCSM calculations interaction or truncationproblem for 16C?

• use other cutoffs and interaction families (LENPIC collaboration)

• consistent freespace SRG for all operators (B(E2), Q, Rch)

• compute B(M1) transitions for carbon chain

• compute observables for other isotopic chains like fluorine or neon

References:[1] R. Roth, Phys. Rev. C 79, 064324 (2009).[2] K. Tsukiyama, S. K. Bogner, and A. Schwenk, Phys. Rev.

Lett. 106, 222502 (2011).

[3] H. Hergert, S. K. Bogner, S. Binder, A. Calci, et al.,Phys. Rev. C 87, 034307 (2013).

[4] E. Gebrerufael, K. Vobig, H. Hergert, and R. Roth, Phys.Rev. Lett. 118, 152503 (2017).

[5] T. Hüther, K. Vobig, K. Hebeler, R. Machleidt, et al.,arXiv:1911.04955v1 (2019).

[6] NNDC, ENSDF (https://www.nndc.bnl.gov/) (2019).[7] W. Vermeer et al., en, Phys. Lett. B 122, 23 (1983).

0 20 40 60 80s

90

80

70

60

50

40

E[M

eV]

12C

0+1

IMSRG-Flow Nrefmax = 2IM-SRGNmax = 2Nmax = 4Nmax = 6Nmax = 8

Nmax = 2, s=0 Nmax = 2, s=90

• flow evolution for 12C decoupling accelerates NCSMconvergence

• off-diagonal part becomessuppressed during the flow

• with growing Nmax, the energyat integration step zerodecreases

• interaction: N3LOEMN,500+N3LONL,500, α = 0.04 fm

4, ħhΩ= 20 MeV,natural orbital basis

0

10

20

E* [M

eV] 1+1

2+1

0+1

2+212C

Nrefmax = 0

1+1

2+1

0+1

2+212C

Nrefmax = 2

1+1

2+10+1

2+212C

Nrefmax = 4

0+1

2+1

1+1

4+1

2+2Exp

0

5

10

B(E2

) [e2

fm4 ] 2+1 0+1

12C2+1 0+1

12C2+1 0+1

12C2+1 0+1

0 2 4Nmax

4

6

8

Q [e

fm2 ]

2+1

12C

2 4Nmax

2+1

12C

4Nmax

2+1

12C

2+1

• spectrum, B(E2) transitionsand quadrupole moment for12C


• excellent agreement withexperimental values[6, 7]

• EMN interaction performs wellfor these observables in 12C

0

5

10

15

E* [M

eV] 1+1

2+1

4+1

0+1

12CNrefmax = 0

1+1

2+1

4+1

0+1

12CNrefmax = 2

1+1

2+1

4+1

0+1

12CNrefmax = 4

0+1

2+1

1+1

4+1

Exp

0

5

10

B(E2

) [e2

fm4 ] 2+1 0+1

12C2+1 0+1

12C2+1 0+1

12C2+1 0+1


4

6

8

Q [e

fm2 ]

2+1

12C


2+1

12C


2+1

12C

2+1

• spectrum, B(E2) transitions andquadrupole moment for 12C forNmax = 4 and Λ = 500 MeV


• error bands indicate interactionuncertainties

• excellent agreement withexperimental values also for N2LO

• uncertainty bands shrink with higherorder

3 4 5 6 7 8 9Q(2 +1 )[efm2]

4

5

6

7

8

9

B (E2

,2+ 1

0+ 1)[e

2 fm

4 ]

450MeV500MeV550MeVN3LON2LOExperimentBohr MottelsonRigid Rotor

• correlation between B(E2) transitionand quadrupole moment with combinedmany-body and interaction uncertainties

• strong correlation behaviour betweenthese observables

• rigid rotor model slightly misses thepredicted correlation

• Bohr Mottelson model matches wellwith data from In-Medium NCSM

• value for Q(2+) can be estimated byusing the Bohr Mottelson model

125

100

75

E [M

eV]

0+1

N3LON2LO

0

5

10

E* [M

eV]

2+1

0

5

B(E2

) [e2

fm4 ]

2+1 0+1

0

5

Q [e

fm2 ] 2

+1

C10 C12 C14 C16 C202.4

2.6

R ch [

fm] 0+1

• error bands indicate combinedmany-body and interactionuncertainties

• ground state energies agree withexperimental values

• 2+1 exictation energy in reasonableagreement with experiment

• B(E2) transitions for 12C and 14Cagree well with the experiment, butother isotopes show underestimation interaction problem or missinghigher-body contributions

• quadrupole moment for 12C agreeswell with the experiment

• charge radius for 12C is overestimatedbut fits well for 14C compared to theexperiment, for others noexperimental data available


carbon isotopes in the in-medium ncsm - tobias mongellic h i [f m] 0+ 1 iground state energy,...

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