Petr VeselýNuclear Physics Institute, Czech Academy of Sciencesgemma.ujf.cas.cz/~p.vesely/

seminar at UTEF ČVUT,Prague, November 2017

Application of Equation of Motion Phonon Method to Nuclear and Exotic Nuclear

Systems

MotivationPhysics of atomic nucleus: about 2500 known isotopes, 263 stable isotopes

estimate 4000 unobserved yet

MotivationPhysics of (hyper)nuclei: we add 3rd dimension to our nuclear chart

MotivationPhysics of atomic nucleus – what do we study:

Atomic nucleus as a bound system of

nucleons...

nucleons as a bound system of

quarks

Nuclear physics – scales

< 0.0001 fm: quarks

0.1-1 fm : baryons, mesons

1 fm: nucleons

10 fm : collective modes

What we see depends on resolution:

NN interaction

Description depends on resolution:

- quarks, gluons (as degrees of freedom) … color confinement

- nucleons, mesons (as degrees of freedom) … NN interaction as exchange of one or more mesons

NN interaction

Short Intermediate Long range

Tensor force

Central force

+

+Building blocks

(138) (600)(782) (770)

scalar meson

Exchanges of different types of mesons

NN interactionExistence of more-body interactions between nucleonsAs consequence of inner structure of nucleons

Analogy between NN and Van der Waals interaction

NN force not so much

strong !

NN interactionPossible approach to derive NN interactions

Only mesons here are pions. But pion exchanges 2,3, … till any order. Multi-pion exchangesreplace presence of other types of mesons.

Diagrams of NN scatering can be divided to orders – perturbative theory (?)

Effective field theory – instead of QCD field theory with elem. degreesof freedom (quarks, gluons) we build field theory with nucleons and pions. Must obey the same symmetries as QCD –> Chiral Perturbation Theory (ChPT)

What we can tell about nuclei

Existence of “magic”numbers in atomic physic:

Consequence of movement of electrons in Coulomb field of atomic nucleus Atomic nuclei

Weizs. formula -> “average“ part of B(A,Z)

Shell corrections-> magic numbers

How shell structure occurs in nucleus?

Magic numbers 2,8,20,28,50,126

What is “mean-field“ in nucleus ?

How mean field occurs in nucleus? → changeour perspective

nucleons as non-interacting particlesin potential well

mutual interaction of nucleons creates “mean field“ → nucleons move in this field

Hartree-Fock method – mean-field is generated “by itself“ = self-consistence

mean field

= ++

Nuclear ground state propertiesNN interaction - chiral NNLO

opt

A. Ekström et al., PRL 110, 192502 (2013)

at mean-field level lot of binding missing!

strongly interacting particles cannot be described as non-interacting particles in mean field

nuclear radii too compressed with mean-field calculations based on realistic NN interactions

Nuclear ground state properties2 ways to improve description of nuclear ground states

- interaction, 3-body NNN term is missing (in HF formalism this term reflects dependence on nuclear medium)

- many-body correlations – plenty of different models to treat them

full implementation of 3-body NNN term is very demanding

corrective density dependent term added to realistic NN interaction (simulates NNN force)

D. Bianco, F. Knapp, N. Lo Iudice, P. Vesely, F. Andreozzi, G. De Gregorio, A. Porrino, J. Phys. G: Nucl. Part. Phys. 41, 025109 (2014)

DD term shrink gaps between major shells

radial density

208Pb C

= 0

C = 2000

C = 3000

Equation of Motion Phonon Method

Tamm-Dancoff phonons

1ph excitations

2ph excitations

and in general

“more“-ph excitations

Hilbert space – divided into separate n-phononsubspaces

Equation of Motion Phonon Method

Equation of Motion (EoM) – recursive eq. to solve eigen-energies on each i-phonon subspace while knowing the (i-1)-phonon solution

total Hamiltonian mixes configurations from different Hilbert subspaces

non-diagonal blocks of Hamiltonian calculated from amplitudes

Equation of Motion Phonon Method

eigen-value problem

we diagonalize the total Hamiltonian

E = diag

correlations – wave functions of each state are superpositions of many configurations from different Hilbert subspaces

e.g.

Nuclear ground state properties

Nmax

– maximal osc. shell

(defines how big basis is)h – parameter of basis

Final energy must be converged with respect to N

max and for N

max big

enough independent on h

NN interaction - chiral NNLOopt

A. Ekström et al., PRL 110, 192502 (2013)

2-phonon correlations in the g.s.

EHF

EHF

+Ecorr.

G. De Gregorio, J. Herko, F. Knapp, N. Lo Iudice, P. Veselý, PRC 95, 024306 (2017)

Nuclear ground state propertiesNN interaction - chiral NNLO

opt

A. Ekström et al., PRL 110, 192502 (2013)

2-phonon correlations in the g.s.

G. De Gregorio, J. Herko, F. Knapp, N. Lo Iudice, P. Veselý, PRC 95, 024306 (2017)

Nuclear spectra2-phonon calculation of 208Pb – see in Phys. Rev. C 92, 054315 (2015), F. Knapp, N. Lo Iudice, P. Veselý, G. De Gregorio

nuclear density distribution

C = 3000

C = 2000

C = 0

study of the dipole photoabsorption spectrum B(E1, 0+

g.s. → 1-

exc.)

most of 1- states have configurations beyond 1ph

2-phonon configurations very important to describe richness of spectrum → multifragmentation of dipole resonance...we describe width of res.

“Quasiparticles“ in nuclei

nuclei with semi-closed shell

nucleons “jumping“ between energetically very close levels

“smearing“ of Fermi energy

occupations of levels 0 < V

i

2 < 1

becomes probabilistic

quasiparticle states - partially occupied orbits

Nuclear spectra – open-shell nucleiquasiparticle formulation of multiphonon model – useful for description of nuclei which are not doubly magic – see in Phys. Rev. C 93, 044314 (2016)P. Veselý, F. Knapp, N. Lo Iudice, G. De Gregorio

20O

much richer low lying spectrum (with 2-phonon configs.)

density of states in agreement with experiment

composition of states

Nuclear spectra – open-shell nuclei

first two 1- levels clearly have dominantly 2-phonon origin

11

- … 91% 2-phonon1

2- … 78% 2-phonon

G. De Gregorio, F. Knapp, N. Lo Iudice, P. Veselý, Phys. Rev. C 93, 044314 (2016)

recent experimental measurement on 20O N. Nakatsuka et al., Physics Letters B 768, 387–392 (2017)

Ex (MeV) ISD EWSR(%)

5.660 0.59

6.617 2.19

[exp.] E. Tryggestad et al., Phys. Rev. C 67, 064309 (2003)

Nuclear spectra – odd nuclei

explicit coupling of nucleon to general excitations of the nuclear core

then diagonalization of complete Hamiltonian

Hilbert space – divided into separate n-phononsubspaces

Tamm-Dancoff phonons

Nuclear spectra – odd nuclei

application of EMPM on the odd nuclear systems:G. De Gregorio, F. Knapp, N. Lo Iudice, P. Vesely, Phys. Rev. C94, 061301(R) (2016)

NNN forceNN+NNN interaction - NNLO

sat (Ekström et al. Phys. Rev. C 91 (2015) 051301R )

HO basis N = (2n + l)

h = 20 MeV Nmax

up to 4

charged radii HF energy

test calculations in minimal configuration spacebut most of qualitative effect from NNN already there!

NNN interaction shrinks gaps between major shells

radial density

NNN forceNN+NNN interaction - NNLO

sat (Ekström et al. Phys. Rev. C 91 (2015) 051301R )

HO basis N = (2n + l)

h = 20 MeV Nmax

up to 4

charged radii HF energy

test calculations in minimal configuration spacebut most of qualitative effect from NNN already there!

TDA calculation of photoabsorption cross section – shrinked s.p. spectra shifts giant resonance down in energy

NNN interaction shrinks gaps between major shells → important for correct description of giant resonance

Description of hypernucleiapplication of approach HF(B)+(Q)TDA+EMPM on exotic nuclear systems → single hypernuclei

NN+NNN interaction - NNLOsat

(Ekström et al. Phys. Rev. C 91 (2015) 051301R )

ΛN interaction - LO (H. Polinder, J. Haidenbauer, U. Meissner, Nucl. Phys. A 779 (2006) 244)cut-off = 550 MeV

extension of HF+TDA formalism on hypernuclei → proton-neutron- HF + N TDA

(replacement of the nucleon by )

formalism derived also for 3-body NN forces – but these forces not present yet (only leading order N interaction used) … alternatively NN may appear indirectly as SRG induced from N

main effect of 3-body NNN force on single particle energies of:

interacts with nuclear core via N interaction

NNN force modifies nuclear core (distribution of density, s.p. energies of nucleons)

modification of nuclear core modifies single particle energies of

so far implemented:

NNN force – effect on hypernuclei

NN+NNN interaction - NNLOsat

(Ekström et al. Phys. Rev. C 91 (2015) 051301R )

ΛN interaction - LO (H. Polinder, J. Haidenbauer, U. Meissner, Nucl. Phys. A 779 (2006) 244)cut-off = 550 MeV

HO basis

N = (2n + l)

Nmax

up to 4

h = 20 MeV

only qualitative study → we need to enlarge configuration space

sd-shell in hypercarbon not realistic due to small space

main effect:

NNN force shrinks gaps between major shells also for !

relative energies between s- & p- shells realistic but absolute scale wrong

Problems:

- wrong order spin-orbit partners 0p

3/2 & 0p

1/2 in

hyperoxygen

- strong dependence on cut-off of N force

s.p. energies:

NNN force – effect on hypernuclei

NN+NNN interaction - NNLOsat

(Ekström et al. Phys. Rev. C 91 (2015) 051301R )

ΛN interaction - LO (H. Polinder, J. Haidenbauer, U. Meissner, Nucl. Phys. A 779 (2006) 244)cut-off = 550 MeV

NN+NNN interaction - NNLOsat

ΛN interaction - LO

NN interaction - NNLOopt

ΛN interaction - LO

NNN force – effect on hypernuclei

NN+NNN interaction - NNLOsat

(Ekström et al. Phys. Rev. C 91 (2015) 051301R )

ΛN interaction - LO (H. Polinder, J. Haidenbauer, U. Meissner, Nucl. Phys. A 779 (2006) 244)cut-off = 550 MeV

to solve problems – highly desirable to improve ΛN interaction: NLO N includes the tensor term which may solve problem of 0p

3/2 & 0p

1/2

documented by toy model calculation with purely phenomenological tensor term

also necessity to implement mixing

NLO N interaction

where

17

0p3/2

0p1/2

0s1/2

Outlook - further study on effect of NNN interaction (revision of results of EMPM) - further improvements of multiphonon calcs. (3-phonon in Ca isotopes) - study on odd-odd nuclei - application of EMPM on other effects – beta or double-beta decays - calculations of deformed nuclei - in hypernuclei: - Effect of - coupling and improvment of N interaction - Coupling of single particle with phonon and multi-phonon configurations - Any other ideas for further improvements are welcome

List of Collaborators

Thank you!!

Nuclear Physics Institute, Czech Academy of Sciences

Petr VeselýJan PokornýGiovanni De GregorioPetr Bydžovský

Institute of Nuclear and Particle Physics, Charles University

František Knapp

Universita degli Studi Federico II, Napoli

Nicola Lo Iudice