Nuclear structure Anatoli Afanasjev Mississippi State University

Nuclear structure Anatoli Afanasjev Mississippi State University 1. Nuclear theory selection of starting point 2. What can be done exactly (ab-initio calculations) and why we cannot do that systematically? 3. Effective interactions 4. Density functional theory 5. Shell structure and shell effects. Their consequences. 6. Nuclear landscape: what we know and how well we extrapolate 7. Superheavy nuclei: successes and challenges

we are all made of star stuff Carl Sagan - Big Bang produces only hydrogen and helium, other elements are created in the stars in the process of nucleosynthesis All animals are equal, but some animals are more equal than others George Orwell, Animal farm - Not all hydrogen/helium nuclei will end up in uranium nuclei. It is expected that around 7000 of nuclei of different kind exist in the Universe:! around 300 of them are stable and well known,! we have some information on about 2700 unstable nuclei,! some part of remaining 4000 nuclei will be studied with FRIB (Facility for Rare Ion Beams)

The improvement in theory is critical: FRIB will allow to explore only some part of nuclear chart

1. Nuclear theory selection of starting point

Building blocks of nuclear matter The complete QCD Lagrangian 1 α µν µ α Λ = F F Ψ [ ig Ψ n A t ] µν α γ µ µ α n mnψnψn 4 δ n 6 quark fields: n=1,,6 8 SU(3) matrices numbered by the gluon-color index α=1,2,...,8" n

Proton = uud Neutron = ddu Mesons: built from quark and anti-quark Example: pion π 0 Quark-meson coupling models Quark degrees of freedom can be neglected and the description of low-energy nuclear systems can be based on the hadrons interacting by the exchange of different mesons Physics is an art of approximations

Challenges of description of many-body nuclear systems Start from Schrodinger equation E = + V ( r, t, s,...) ij ij ij 2m i p! Many-body aspect of the problem Example: exact multi-particle wave function methods 2 i< j! Nuclear forces inside of finite nuclei are poorly known Example: the need for 3-body forces " exponential wall (# of Slater determinants up to 10 9 ) A max ~12 " improvements of analytical and computational methods will lead only to modest increase of A max! Relativistic effects are PARAMETRIZED: for example, spin-orbit interaction V LS = W( r) l s

The description of the interaction of particles at short distances is extremely complicated # HARD CORE problem Polarization effects at short distances fermionic constituents do not like being put close to one another --- the Pauli exclusion principle creates additional polarization and repulsion effects if 3 particles # polarization effects depends on explicit positions of each of them THREE-BODY EFFECTS Use EFFECTIVE FORCES which truncate momenta above ~ 500-1000 MeV

Effective forces according to Brueckner theory (1958): Brueckner,Gammel, Phys. Rev. 109, 1023 (1958) The nucleons in the interior of the nuclear medium do not feel the same bare force V, as the nucleons feel in free space. They feel an effective force G. The Pauli principle prohibits the scattering into states, which are already occupied in the medium # the force G( ) depends on the density This force G is much weaker than bare force V. Nucleons move nearly free in the nuclear medium and feel only a strong attraction at the surface (shell model) 9/44

THREE-BODY EFFECTS V ( r N ) N N = V ( ) + ij rij V i j= 1 i, j, l ijl ( r ij, r il, 4- body??? r jl ) +... Axilrod-Teller potential 1 V r 12 2-body 3-body & 3cos( θ1)cos( θ2)cos( θ3) + A$ 3 3 % r12r13r23 ( 3) 1 ( r1, r2, r3 ) = 3 θ 1 r 13 θ 2 2 θ 3 r 23 3 3-body effects in liquid argon:! 10% effect on partition function! 40% effects on transport coefficients #! "

2. What can be done exactly (ab initio calculations) and why we cannot do that systematically

THREE-BODY EFFECTS Only 2-body 2-body+ 3-body

Calcium isotopes from chiral interactions! [Hagen!et!al,!!Phys.!Rev.!Le3.!109,!032502!(2012).]!! InMmedium!effecRve!3NF:!k F =0.95!fm M1,!c E =0.735,!c D =M0.2! [Holt,!Kaiser,!Weise!2009;!Hebeler!and!Schwenk!2010]!! S n ( 52 Ca)!=!5.98MeV!! (Gallant!et!al!PRL!2012)! S n ( 54 Ca)!=!3.84MeV! (Wienholtz!Nature!2013)!

The description of heavy nuclei is still a problem. S. Binder et al, arxiv: 1312.5685v1 (2013) Add slide with 4-body results of Roth and Co

Even if these problems are resolved in future - it is not clear whether ab initio calculations will be more accurate as compared with experiment than other approaches - numerically, ab initio calculations are significantly more expensive than in other approaches Density functional theory

Collec/ve(degrees(of(freedom((deformaRon,!rotaRon,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!fission)! Angular(momentum(!!!!!!!!!!!Single6par/cle(degrees(( ((((((((((((((((((((of(freedom(( (smp!energies,!wavefuncrons)!

!!Covariant!density!funcRonal!!theory!(CDFT)! The!nucleons!interact!via!the!exchange!of!eﬀecRve!mesons!#!!!!!!!!!!!!!!!!!!!!!!!!!#!eﬀec/ve(Lagrangian(!LongMrange!!!a3racRve!!!scalar!ﬁeld!!!ShortMrange! repulsive!vector!!!!!!!!ﬁeld! Isovector!!!!ﬁeld! M!meson!ﬁelds! (Mean(( ((ﬁeld( hˆ ϕi =ε i ϕi Eigenfunc/ons(

Non-relativistic mean field and density functional theories 1. Macroscopic + microscopic method Bulk properties # liquid drop Single-particle properties # single-particle potential (Woods-Saxon, Nilsson, folded Yukawa) Very flexible but no self-consistency # extrapolation to unknown regions of nuclear chart maybe unreliable 2. Density functional theories starting from 2 - and 3 - nucleon effective interactions Finite range Gogny force Central finite range force Zero-range (contact) Skyrme forces Spin-orbit 0-range Density-dependent 0-range + Coulomb

Realistic 2-body forces Effective interactions Local: 1. Central force 2. Tensor force Many-body techniques (Bruckner G-matrix, ) V V ( r) = V e Non-local (velocity-dependent) Example, 2-body spin-orbit interaction 0 r 2 / 2 0 (1) (2) T ( 1,2) = [ VT ( r) + V ] S 0 T τ 12 τ τ r Effective nucleon-nucleon forces 1. well-behaved at short distances (no hard core problem) Hard core can be neglected: nucleons move through the nucleus most of the time as independent particles because of Pauli exclusion principle (Weisskopf, 1950) 2. Include some part of many body correlations 3. Possible to apply many-body techniques Nucleons within a nucleus do not feel the bare nn-interaction!!! FIT TO EXP. DATA # PHENOMENOLOGICAL EFFECTIVE FORCES

Mean field and Hartree+Fock(+Bogoliubov) approaches MEAN FIELD nucleons move independently in an average potential A A produced by all the nucleons $ V (1... A) = V ( i, j) V ( i) i> j= 1 H Ψ >= E Ψ > δe[ Ψ] = 0 i= 1 Exact Schrodinger eq. Transition to single-particle levels H HF = Variational eq. A i= 1 h( i) ϕ k =ε k ϕ h( i) ( i) ( i) i = { ri, si, ti} k Total HF wave function= = Slater determinant Φ k... k 1 A (1,..., A) = ϕ ϕ k 1 k A (1) (1)...... ϕ ϕ k1 k A ( A) ( A) The Hartree-Fock energy E HF HF =< Φ H Φ >

h = t + Γ Γ = kk ' ukl ' k' lρll' ll' ' % & Single-particle kinetic energy Self-consistent field ME of 2-body interaction Single-particle density matrix Pairing correlations can be accounted by BCS method or in the Hartree+Fock+Bogoliubov (HFB) method h Δ * Δ h * $ ' U " % #& V k k $ ' U " = % # & V k k $ " E # Wave function Energy of quasi-particles pp- and ph- channels of interaction are taken into account simultaneously k Pairing field Δ ρ 1 kk ' = ull' qq' κ qq' 2 qq' κ ll' ll' =< Φ + cl ' =< Φ c c Φ > l' c l l Φ > Density matrix Pairing tensor ph- pp-

Density functional theory: Hohenberg-Kohn-Sham approach! Starting point energy functional E of local densities r and currents j δe = 0 2 Ψ ρ = (...) (...) k k k vk! Maps the nuclear many-body problem for the real highly correlated many-body wave function on a system of independent particles is so-called Kohn-Sham orbitals (...) Ψk * Ψ! Variational principle # equations of motion for H Ψ (...) = Ψ (...) k ek k # # H =! The existence theorem for the effective energy functional makes no statement about its structure. Its form is motivated by ab initio theory, but the actual parameters are adjusted to nuclear structure data. Ψk (...) δe δρ In Coulombic systems the functional is derived ab initio

CEDF( r ch rms '[fm] ( NL3*( 0.0283! DD6ME2( 0.0230! DD6MEd( 0.0329! DD6PC1( 0.0253!

Theoretical uncertainties in the description of masses

TheoreRcal!uncertainRes!are!most!pronounced!!for! transironal!nuclei!(due!to!sob!potenral!energy!surfaces)!and!in!! the!!regions!!of!transiron!between!prolate!and!oblate!shapes.!! Details!depend!of!the!descripRon!of!singleMparRcle!states!

Shell structure and shell effects

Single-particle states and shell structure in spherical nuclei

ν Shell structure and shell correction energies δe = shell 2 ev 2 eg~ ( e) de ν e g ~ ( e ) shell correction energy - single-particle energies - smeared level density E tot = E LD +δ E ( p) + δe ( n) sh sh δe shell > 0 δe shell < 0

Shell effects as a function of deformation Energy TOT LD e deformation β"

Deformation dependence of the single-particle energies in a realistic Nilsson potential 1. removing of the 2j+1 degeneracy of single-particle state seen at spherical shape 2. single-particle states at deformation ε 2" not equal 0 are only two-fold degenerate 3. creation of deformed shell gaps deformation

Nuclear landscape: what we know and how well we extrapolate?

((Sources(of(uncertain/es(in(the(predic/on(of(two6neutron(drip(line( (((((((666((poorly(known(isovector((proper/es(of(energy(density(( (((((((((((((func/onals( (((((((666((inaccurate(descrip/on(of(the(energies(of(single6par/cle(states( (((((((666((shallow(slope(of(two6neutron(separa/on(energies( ((((((Posi/on(of(two6neutron(drip(line(does(not(correlate(with( ((((((nuclear(maier(proper/es(of(the(energy(density(func/onal( (((((((((AA,!S.!Agbemava,!D.!Ray!and!P.!Ring,!PLB!726,!680!(2013)! ((((((((!S.!Agbemava,!AA,!D.!Ray!and!P.!Ring,!PRC!89,!054320!(2014)!!!!!!!!!!J.Erler!et!al!et!al,!Nature!486!(2012)!509!!!

Two-neutron drip lines: the impact of slope of two-particle separation energies % S 2N = B(Z,N-2)-B(Z,N) two-neutron S.E. % S 2P = B(Z-2,N)-B(Z,N) two proton S.E. % Conditions for drip lines S 2N < 0 ; S 2P < 0

Peninsulas The presence of these gaps leading to the formation of peninsulas may be a consequence of the model limitations in some cases

Two-neutron drip lines: the impact of uncertainties in pairing Pairing strength decreased by 8%; effectively leading to pairing energies of RHB(DD-ME2) calculations Pairing strength increased by 8% Two-neutron drip line of the NL3* CEDF

The impact of single-particle states on the position of two-neutron drip line