Structure of exotic nuclei Takaharu Otsuka University of Tokyo / RIKEN / MSU 7 th CNS-EFES summer school Wako, Japan August 26 – September 1, 2008 A presentation supported by the JSPS Core-to-Core Program “International Research Network for Exotic Femto Systems (EFES)”

Section 1: Basics of shell model Section 2: Construction of effective interaction and an example in the pf shell Section 3: Does the gap change ? - N=20 problem - Section 4: Force behind Section 5: Is two-body force enough ? Section 6: More perspectives on exotic nuclei Outline

Proton Neutron 2-body interaction Aim: To construct many-body systems from basic ingredients such as nucleons and nuclear forces (nucleon-nucleon interactions) 3-body intearction

What is the shell model ? Why can it be useful ? Introduction to the shell model How can we make it run ?

0.5 fm 1 fm distance between nucleons Potential Schematic picture of nucleon- nucleon (NN) potential -100 MeV hard core

Actual potential Depends on quantum numbers of the 2-nucleon system (Spin S, total angular momentum J, Isospin T) Very different from Coulomb, for instance 1S01S0 Spin singlet (S=0) 2S+1=1 L = 0 (S) J = 0 From a book by R. Tamagaki (in Japanese)

Basic properties of atomic nuclei Nuclear force = short range Among various components, the nucleus should be formed so as to make attractive ones (~ 1 fm ) work. Strong repulsion for distance less than 0.5 fm Keeping a rather constant distance (~1 fm) between nucleons, the nucleus (at low energy) is formed.  constant density : saturation (of density)  clear surface despite a fully quantal system Deformation of surface Collective motion

proton neutron range of nuclear force from Due to constant density, potential energy felt by is also constant Mean potential (effects from other nucleons) Distance from the center of the nucleus -50 MeV r

proton neutron range of nuclear force from At the surface, potential energy felt by is weaker Mean potential (effects from other nucleons) -50 MeV r

Eigenvalue problem of single-particle motion in a mean potential  Orbital motion Quantum number : orbital angular momentum l total angular momentum j number of nodes of radial wave function n Energy eigenvalues of orbital motion E r

Proton 陽子 Neutron 中性子

Harmonic Oscillator (HO) potential Mean potential HO is simpler, and can be treated analytically

Eigenvalues of HO potential 5h  4h  3h  2h  1h 

Spin-Orbit splitting by the (L S) potential An orbit with the orbital angular momentum l j = l - 1/2 j = l + 1/2

The number of particles below a shell gap : magic number ( 魔法数 ) This structure of single-particle orbits shell structure ( 殻構造 ) magic number shell gap Orbitals are grouped into shells closed shell fully occupied orbits

Spin-orbit splitting Eigenvalues of HO potential Magic numbers Mayer and Jensen (1949) h  4h  3h  2h  1h 

From very basic nuclear physics, density saturation + short-range NN interaction + spin-orbit splitting  Mayer-Jensen’s magic number with rather constant gaps Robust mechanism - no way out -

Back to standard shell model How to carry out the calculation ?

 i : single particle energy v ij,kl : two-body interaction matrix element ( i j k l : orbits) Hamiltonian

A nucleon does not stay in an orbit for ever. The interaction between nucleons changes their occupations as a result of scattering. mixing Pattern of occupation : configuration valence shell closed shell (core)

Prepare Slater determinants  1,  2,  3,… which correspond to all possible configurations How to get eigenvalues and eigenfunctions ? The closed shell (core) is treated as the vacuum. Its effects are assumed to be included in the single-particle energies and the effective interaction. Only valence particles are considered explicitly.

Calculate matrix elements where  1,  2,  3 are Slater determinants,,...., Step 1: In the second quantization,  1 = ….. | 0 > a+a+ a+a+ a+a+ n valence particles  2 = ….. | 0 > a’+a’+ a’+a’+ a’+a’+  3 = …. closed shell

Step 2 : Construct matrix of Hamiltonian, H, and diagonalize it H =H =

Diagonalization of Hamiltonian matrix (about 30 dimension) c Conventional Shell Model calculation All Slater determinants diagonalization Quantum Monte Carlo Diagonalization method Important bases are selected

With Slater determinants  1,  2,  3,…, the eigenfunction is expanded as H  Thus, we have solved the eigenvalue problem :  = c 1  1 + c 2  2 + c 3  3 + ….. c i probability amplitudes

M-scheme calculation  1 = ….. | 0 > a+a+ a+a+ a+a+ Usually single-particle state with good j, m (=j z ) Each of  i ’ s has a good M (=J z ), because M = m 1 + m 2 + m Hamiltonian conserves M.  i ’ s having the same value of M are mixed.  i ’ s having different values of M are not mixed. But,

H =H = * * * * * * * * The Hamiltonian matrix is decomposed into sub matrices belonging to each value of M M=0 M=1M=-1M=2

How does J come in ? two neutrons in f7/2 orbit An exercise : M=0 M=2 M=1 m 1 m 2 7/2 -5/2 5/2 -3/2 3/2 -1/2 J+J+ m 1 m 2 7/2 -7/2 5/2 -5/2 3/2 -3/2 1/2 -1/2 m 1 m 2 7/2 -3/2 5/2 -1/2 3/2 1/2 J+J+ J + : angular momentum raising operator J + |j, m > |j, m+1 > J=0 2-body state is lost J=1 can be elliminated, but is not contained

Dimension M=0 M=1 3 M=2 M=3 M=4 M=6 M= Components of J values 4 J = 2, 4, 6 J = 4, 6 J = 6 J = 0, 2, 4, 6

By diagonalizing the matrix H, you get wave functions of good J values by superposing Slater determinants. H =H = M = 0 * * e J= e J= e J= e J=6 In the case shown in the previous page, e J means the eigenvalue with the angular momentum, J.

This property is a general one : valid for cases with more than 2 particles. H =H = M * * e J e J’ e J’’ e J’’’ By diagonalizing the matrix H, you get eigenvalues and wave functions. Good J values are obtained by superposing properly Slater determinants.

Some remarks on the two-body matrix elements

Because the interaction V is a scalar with respect to the rotation, it cannot change J or M. A two-body state is rewritten as | j1, j2, J, M > =  m1, m2 (j1, m1, j2, m2 | J, M ) |j1, m1> |j2,m2> Only J=J’ and M=M’ matrix elements can be non-zero. x =  m1, m2 ( j1, m1, j2, m2 | J, M ) x  m3, m4 ( j3, m3, j4, m4 | J’, M’ ) Two-body matrix elements Clebsch-Gordon coef.

Two-body matrix elements are independent of M value, also because V is a scalar. Two-body matrix elements are assigned by j1, j2, j3, j4 and J. Because of complexity of nuclear force, one can not express all TBME’s by a few empirical parameters. Jargon : Two-Body Matrix Element = TBME X X

Actual potential Depends on quantum numbers of the 2-nucleon system (Spin S, total angular momentum J, Isospin T) Very different from Coulomb, for instance 1S01S0 Spin singlet (S=0) 2S+1=1 L = 0 (S) J = 0 From a book by R. Tamagaki (in Japanese)

Determination of TBME’s Later in this lecture An example of TBME : USD interaction by Wildenthal & Brown sd shell d5/2, d3/2 and s1/2 63 matrix elemeents 3 single particle energies Note : TMBE’s depend on the isospin T Two-body matrix elements

USD interaction 1 = d3/2 2= d5/2 3= s1/2

Closed shell Excitations to higher shells are included effectively valence shell Partially occupied Nucleons are moving around Higher shell Excitations from lower shells are included effectively by perturbation(-like) methods ～ Effective interaction Effects of core and higher shell

Arima and Horie 1954 magnetic moment quadrupole moment Configuration Mixing Theory Departure from the independent-particle model + closed shell This is included by renormalizing the interaction and effective charges. Core polarization

Probability that a nucleon is in the valence orbit ~60% A. Gade et al. Phys. Rev. Lett. 93, (2004) No problem ! Each nucleon carries correlations which are renormalized into effective interactions. On the other hand, this is a belief to a certain extent.

In actual applications, the dimension of the vector space is a BIG problem ! It can be really big : thousands, millions, billions, trillions,.... pf-shell

This property is a general one : valid for cases with more than 2 particles. By diagonalizing the matrix H, you get eigenvalues and wave functions. Good J values are obtained by superposing properly Slater determinants. H =H = M * * e J e J’ e J’’ e J’’’ dimension 4 Billions, trillions, …

Dimension Dimension of Hamiltonian matrix (publication years of “pioneer” papers) Year Floating point operations per second Birth of shell model (Mayer and Jensen) Year Dimension of shell-model calculations billion

Shell model code Name Contact person Remark OXBASH B.A. Brown Handy (Windows) ANTOINE E. Caurier Large calc. Parallel MSHELL T. Mizusaki Large calc. Parallel (MCSM) Y. Utsuno/M. Honma not open Parallel These two codes can handle up to 1 billion dimensions.

Monte Carlo Shell Model Auxiliary-Field Monte Carlo (AFMC) method general method for quantum many-body problems For nuclear physics, Shell Model Monte Carlo (SMMC) calculation has been introduced by Koonin et al. Good for finite temperature. - minus-sign problem - only ground state, not for excited states in principle. Quantum Monte Carlo Diagonalization (QMCD) method No sign problem. Symmetries can be restored. Excited states can be obtained.  Monte Carlo Shell Model

References of MCSM method "Diagonalization of Hamiltonians for Many-body Systems by Auxiliary Field Quantum Monte Carlo Technique", M. Honma, T. Mizusaki and T. Otsuka, Phys. Rev. Lett. 75, (1995). "Structure of the N=Z=28 Closed Shell Studied by Monte Carlo Shell Model Calculation", T. Otsuka, M. Honma and T. Mizusaki, Phys. Rev. Lett. 81, (1998). “Monte Carlo shell model for atomic nuclei”, T. Otsuka, M. Honma, T. Mizusaki, N. Shimizu and Y. Utsuno, Prog. Part. Nucl. Phys. 47, (2001)

Diagonalization of Hamiltonian matrix (about 30 dimension) c Conventional Shell Model calculation All Slater determinants diagonalization Quantum Monte Carlo Diagonalization method Important bases are selected

Our parallel computer More cpu time for heavier or more exotic nuclei 238 U one eigenstate/day in good accuracy requires 1PFlops 京速計算機 (Japanese challenge) Blue Gene Earth Simulator Dimension Birth of shell model (Mayer and Jensen) Year Dimension of Hamiltonian matrix (publication years of “pioneer” papers) Conventional Monte Carlo Lines : 10 5 / 30 years Year Floating point operations per second Progress in shell-model calculations and computers GFlops

Section 1: Basics of shell model Section 2: Construction of effective interaction and an example in the pf shell Section 3: Does the gap change ? - N=20 problem - Section 4: Force behind Section 5: Is two-body force enough ? Section 6: More perspectives on exotic nuclei Outline

Effetcive interaction in shell model calculations How can we determine  i : Single Particle Energy : Two-Body Matrix Element

Determination of TBME’s Early time Experimental levels of 2 valence particles + closed shell TBME Example : 0 +, 2 +, 4 +, 6 + in 42 Ca : f7/2 well isolated v J = are determined directly E(J) = 2  ( f7/2) + v J Experimental energy of state J Experimental single-particle energy of f7/2

Spin-orbit splitting Eigenvalues of HO potential Magic numbers Mayer and Jensen (1949) h  4h  3h  2h  1h 

Example : 0 +, 2 +, 4 + in 18 O (oxygen) : d5/2 & s1/2,, etc. Arima, Cohen, Lawson and McFarlane (Argonne group)), 1968 The isolation of f7/2 is special. In other cases, several orbits must be taken into account. In general,  2 fit is made (i)TBME’s are assumed, (ii)energy eigenvalues are calculated, (iii)  2 is calculated between theoretical and experimental energy levels, (iv) TBME’s are modified. Go to (i), and iterate the process until  2 becomes minimum.

At the beginning, it was a perfect  2 fit. As heavier nuclei are studied, (i)the number of TBME’s increases, (ii) shell model calculations become huge. Complete fit becomes more difficult and finally impossible. Hybrid version

Microscopically calculated TBME’s for instance, by G-matrix (Kuo-Brown, H.-Jensen,…) G-matrix-based TBME’s are not perfect, direct use to shell model calculation is only disaster Use G-matrix-based TBME’s as starting point, and do fit to experiments. Consider some linear combinations of TBME’s, and fit them.

Hybrid version - continued Some linear combinations of TBME’s are sensitive to available experimental data (ground and low-lying). The others are insensitive. Those are assumed to be given by G-matrix-based calculation (i.e. no fit). The  2 fit method produces, as a result of minimization, a set of linear equations of TBME’s First done for sd shell: Wildenthal and Brown’s USD 47 linear combinations (1970) Recent revision of USD : G-matrix-based TBME’s have been improved  30 linear combinations fitted

Summary of Day 1 1.Basis of shell model and magic numbers density saturation + short-range interaction + spin-orbit splitting  Mayer-Jensen’s magic number 2.How to perform shell model calculations 3.How to obtain effective interactions

Structure of exotic nuclei Takaharu Otsuka University of Tokyo / RIKEN / MSU 7 th CNS-EFES summer school Wako, Japan August 26 – September 1, 2008 A presentation supported by the JSPS Core-to-Core Program “International Research Network for Exotic Femto Systems (EFES)” Day 2

Section 1: Basics of shell model Section 2: Construction of effective interaction and an example in the pf shell Section 3: Does the gap change ? - N=20 problem - Section 4: Force behind Section 5: Is two-body force enough ? Section 6: More perspectives on exotic nuclei Outline

What is the shell model ? Why can it be useful ? Introduction to the shell model How can we make it run ? Day-1 lecture : Basis of shell model and magic numbers density saturation + short-range interaction + spin-orbit splitting  Mayer-Jensen’s magic number  Valence space (model space) For shell model calculations, we need also TBME (Two-Body Matrix Element) and SPE (Single Particle Energy)

Start from a realistic microscopic interaction M. Hjorth-Jensen, et al., Phys. Repts. 261 (1995) 125 –Bonn-C potential –3rd order Q-box + folded diagram 195 two-body matrix elements (TBME) and 4 single-particle energies (SPE) are calculated  Not completely good （ theory imperfect) Vary 70 Linear Combinations of 195 TBME and 4 SPE Fit to 699 experimental energy data of 87 nuclei M. Honma et al., PRC65 (2002) (R) G-matrix + polarization correction + empirical refinement Microscopic Phenomenological An example from pf shell (f 7/2, f 5/2, p 3/2, p 1/2 ) GXPF1 interaction

G-matrix vs. GXPF1 T=0 … attractive T=1 … repulsive Relatively large modifications in V(abab ; J0 ) with large J V(aabb ; J1 ) pairing 7= f 7/2, 3= p 3/2, 5= f 5/2, 1= p 1/2 two-body matrix element input output

Systematics of Shell gap N=28 N=32 for Ca, Ti, Cr N=34 for Ca ?? Deviations in E x Cr at N ≧ 36 Fe at N ≧ 38 Deviations in B(E2) Ca, Ti for N ≦ 26 Cr for N ≦ Ca core excitations Zn, Ge g 9/2 is needed

GXPF1 vs. experiment 56 Ni th. exp. 57 Ni

Probability of closed-shell in the ground state Ni 48 Cr neutron proton total 56 Ni (Z=N=28) has been considered to be a doubly magic nucleus where proton and neutron f 7/2 are fully occupied. ⇒ Measure of breaking of this conventional idea doubly magic

54 Fe yrast states 0p-2h configuration 0 +, 2 +, 4 +, 6 + …  (f 7/2 )  2 more than 40% prob. 1p-3h … 1 st gap One-proton excitation 3 +, ～ p-4h … 2 nd gap Two-protons excitation 12 + ～ States of different nature can be reproduced within a single framework p-h : excitation from f 7/2

58 Ni yrast states 2p-0h configuration 0 +, 2 + … (p 3/2 ) 2 1 +, 3 +, 4 + … (p 3/2 ) 1 (f 5/2 ) 1 more than 40% prob. 3p-1h … 1 st gap One-proton excitation 5 + ～ 8 + 4p-2h … 2 nd gap One-proton & one-neutron excitation 10 + ～ 12 + p-h : excitation from f 7/2

N=32, 34 magic numbers ? Issues to be clarified by the next generation RIB machines

Monopole part of the NN interaction Angular averaged interaction Effective single particle energy Isotropic component is extracted from a general interaction. In the shell model, single-particle properties are considered by the following quantities …….

Effective single-particle energy (ESPE) ESPE : Total effect on single- particle energies due to interaction with other valence nucleons Monopole interaction, v m ESPE is changed by N v m N particles

Effective single-particle energies Lowering of f5/2 from Ca to Cr - weakening of N=34 - Rising of f5/2 from 48 Ca to 54 Ca - emerging of N=34 - Why ? n-n p-n f 5/2 p 3/2 p 1/ new magic numbers ? Z=20 Z=22 Z=24

Exotic Ca Isotopes : N = 32 and 34 magic numbers ? exp. levels :Perrot et al. Phys. Rev. C (2006), and earlier papers 52 Ca 54 Ca 53 Ca ? 51 Ca

Exotic Ti Isotopes 55 Ti 56 Ti 54 Ti Ti

G GXPF1 ESPE (Effectice Single- Particle Energy) of neutrons in pf shell f 5/2 Ca Ni Why is neutron f 5/2 lowered by filling protons into f 7/2 f 5/2

Changing magic numbers ? We shall come back to this problem after learning under-lying mechanism.

Section 1: Basics of shell model Section 2: Construction of effective interaction and an example in the pf shell Section 3: Does the gap change ? - N=20 problem - Section 4: Force behind Section 5: Is two-body force enough ? Section 6: More perspectives on exotic nuclei Outline

Left-lower part of the Nuclear Chart Studies on exotic nuclei in the 80~90’s リチウム１１ neutron halo nuclei (mass number) stable exotic -- with halo A 11 Li neutron skin proton halo Stability line and drip lines are not so far from each other  Physics of loosely bound neutrons, e.g., halo while other issues like 32 Mg Neutron number  Proton number 

Neutron halo 11 Li 208 Pb About same radius Strong tunneling of loosely bound excess neutrons Nakamura’s lecture

Drip line 中性子数 （同位元素の種類） Neutron number  Proton number  In the 21 st century, a wide frontier emerges between the stability and drip lines. Stability line A nuclei (mass number) stable exotic Riken’s work huge area

Also in the 1980’s, 32 Mg low-lying 2 +

Phys. Rev. C 41, 1147 (1990), Warburton, Becker and Brown 9 nuclei: Ne, Na, Mg with N=20-22 Basic picture was energy intruder ground state stable exotic sd shell pf shell N=20 gap ~ constant deformed 2p2h state Island of Inversion

One of the major issues over the millennium was to determine the territory of the Island of Inversion - Are there clear boundaries in all directions ? - Is the Island really like the square ? Shallow (diffuse & extended) Steep (sharp) Straight lines Which type of boundaries ?

N semi-magic intruder Small gap vs. Normal gap For smaller gap, f is smaller  diffuse boundary For larger gap, f must be larger  sharp boundary normal The difference  v is modest as compared to “semi-magic”. open-shell v=0  v=smaller  v=large v ~ Inversion occurs for semi-magic nuclei most easily Max pn force

Na isotopes : What happens in lighter ones with N < 20 Original Island of Inversion

Electro-magnetic moments and wave functions of Na isotopes ― normal dominant : N=16, 17 ― strongly mixed : N=18 ― intruder dominant : N=19, 20 Onset of intruder dominance before arriving at N=20 Q  Config. Monte Carlo Shell Model calculation with full configuration mixing : Phys. Rev. C 70, (2004), Utsuno et al. Exp.: Keim et al. Euro. Phys. J. A 8, 31 (2001)

Level scheme of Na isotopes by SDPF-M interaction compared to experiment N=16 N=17 N=18N=19

Major references on MCSM calculations for N~20 nuclei "Varying shell gap and deformation in N~20 unstable nuclei studied by the Monte Carlo shell model", Yutaka Utsuno, Takaharu Otsuka, Takahiro Mizusaki and Michio Honma, Phys. Rev. C60, (1999) “Onset of intruder ground state in exotic Na isotopes and evolution of the N=20 shell gap”, Y. Utsuno, T. Otsuka, T. Glasmacher, T. Mizusaki and M. Honma, Phys. Rev. C70, (2004), Many experimental papers include MCSM results.

Monte Carlo Shell Model (MCSM) results have been obtained by the SDPF-M interaction for the full-sd + f7/2 + p3/2 space. Expansion of the territory Effective N=20 gap between sd and pf shells Ca O Ne Mg SDPF-M (1999) WBB (1990) ~2MeV ~5MeV Neyens et al Mg Tripathi et al Na Dombradi et al Ne Terry et al Ne

Phys. Rev. Lett. 94, (2005), G. Neyens, et al. Strasbourg unmixed USD (only sd shell) Tokyo MCSM 2.5 MeV 0.5 MeV 31 Mg 19

New picture energy intruder ground state stable exotic sd shell pf shell N=20 gap changing deformed 2p2h state Conventional picture energy intruder ground state stable exotic sd shell pf shell N=20 gap ~ constant deformed 2p2h state spherical normal state ?

Steep (sharp) Straight lines Shallow (diffuse & extended) Expansion of the territory Island of Inversion ? ? Ca O Ne Mg SDPF-M (1999) ~2MeV ~6MeV Effective N=20 gap between sd and pf shells Island of Inversion is like a paradise constant gap

Why ?

Section 1: Basics of shell model Section 2: Construction of effective interaction and an example in the pf shell Section 3: Does the gap change ? - N=20 problem - Section 4: Force behind Section 5: Is two-body force enough ? Section 6: More perspectives on exotic nuclei Outline

From undergraduate nuclear physics, density saturation + short-range NN interaction + spin-orbit splitting  Mayer-Jensen’s magic number with rather constant gaps (except for gradual A dependence) Robust mechanism - no way out -

One pion exchange ~ Tensor force Key to understand it : Tensor Force

Key to understand it : Tensor Force  meson (~  +  ) : minor (~1/4) cancellation  meson : primary source Ref: Osterfeld, Rev. Mod. Phys. 64, 491 (92)  Multiple pion exchanges  strong effective central forces in NN interaction (as represented by  meson, etc.)  nuclear binding One pion exchange  Tensor force This talk : First-order tensor-force effect (at medium and long ranges)

How does the tensor force work ? Spin of each nucleon is parallel, because the total spin must be S=1 The potential has the following dependence on the angle  with respect to the total spin S. V ~ Y 2,0 ~ 1 – 3 cos 2  attraction  =0 repulsion  =  /2  S relative coordinate

Deuteron : ground state J = 1 proton neutron Total spin S=1 Relative motion : S wave (L=0) + D wave (L=2) Without tensor force, deuteron is unbound. The tensor force is crucial to bind the deuteron. Tensor force does mix No S wave to S wave coupling by tensor force because of Y 2 spherical harmonics

Monopole part of the NN interaction Angular averaged interaction Effective single particle energy Isotropic component is extracted from a general interaction. In the shell model, single-particle properties are considered by the following quantities …….

Intuitive picture of monopole effect of tensor force wave function of relative motion large relative momentum small relative momentum attractive repulsive spin of nucleon TO et al., Phys. Rev. Lett. 95, (2005) j > = l + ½, j < = l – ½

j>j> j<j< j’ < proton neutron j’ > Monopole Interaction of the Tensor Force Identity for tensor monopole interaction (2j > +1) v m,T + (2j < +1) v m,T = 0 ( j’ j > ) ( j’ j < ) v m,T : monopole strength for isospin T T. Otsuka et al., Phys. Rev. Lett. 95, (2005)

Major features Opposite signs T=0 : T=1 = 3 : 1 (same sign) Only exchange terms (generally for spin-spin forces) tensor proton, j > neutron, j’ < spin-orbit splitting varied

j>j> j<j< j’ < proton neutron j’ > Tensor Monopole Interaction : total effects vanished for spin-saturated case Same Identity with different interpretation (2j > +1) v m,T + (2j < +1) v m,T = 0 ( j’ j > ) ( j’ j < ) v m,T : monopole strength for isospin T no change

j>j> j<j< s 1/2 proton neutron Tensor Monopole Interaction vanished for s orbit For s orbit, j > and j < are the same : (2j > +1) v m,T + (2j < +1) v m,T = 0 ( j’ j > ) ( j’ j < ) v m,T : monopole strength for isospin T

Monopole Interaction of the tensor force is considered to see the connection between the tensor force and the shell structure

Tensor potential tensor no s-wave to s-wave coupling differences in short distance : irrelevant

Proton effective single-particle levels (relative to d 3/2 ) f 7/2 neutrons in f 7/2 d 3/2 d 5/2 proton neutron  meson tensor exp. Cottle and Kemper, Phys. Rev. C58, 3761 (98) Tensor monopole

Spectroscopic factor for -1p from 48 Ca: probing proton shell gaps w/ tensorw/o tensor d 3/2 -s 1/2 gap d 5/2 -s 1/2 gap Kramer et al. (2001) Nucl PHys A679 NIKHEF exp.

Example : Dripline of F isotopes is 6 units away from O isotopes Sakurai et al., PLB 448 (1999) 180, … Tensor force d 5/2 d 3/2 N=16 gap : Ozawa, et al., PRL 84 (2000) 5493; Brown, Rev. Mex. Fis (1983) only exchange term

Monte Carlo Shell Model (MCSM) results have been obtained by the SDPF-M interaction for the full-sd + f7/2 + p3/2 space. Expansion of the territory Effective N=20 gap between sd and pf shells Ca O Ne Mg SDPF-M (1999) WBB (1990) ~2MeV ~5MeV Neyens et al Mg Tripathi et al Na Dombradi et al Ne Terry et al Ne

1h 11/2 neutrons 1h 11/2 protons 1g 7/2 protons Exp. data from J.P. Schiffer et al., Phys. Rev. Lett. 92, (2004 ) + common effect (Woods-Saxon) Z=51 isotopes h 11/2 g 7/2 51 Sb case No mean field theory, (Skyrme, Gogny, RMF) explained this before. Opposite monopole effect from tensor force with neutrons in h 11/2. Tensor by  meson exchange

Single-particle levels of 132 Sn core Weakening of Z=64 submagic structure for N~90 64

Weakening of Z=64 submagic structure for N~90 8 neutrons in 2f 7/2 reduces the Z=64 gap to the half value 8 protons in 1g 7/2 pushes up 1h 9/2 by ~1 MeV 1h 9/2 64 2d 3/2 2d 5/2 Proton collectivity enhanced at Z~64

Mean-field models (Skyrme or Gogny) do not reproduce this reduction. Tensor force effect due to vacancies of proton d 3/2 in Ar 29 : 650 (keV) by  +  meson exchange. f 5/2 f 7/2 Neutron single-particle energies

A city works its magic. … N.Y. RIKEN RESEARCH, Feb Magic numbers do change, vanish and emerge. Conventional picture (since 1949) Today’s perspectives

Effect of tensor force on (spherical) superheavy magic numbers Woods-Saxon potential Tensor force added Occupation of neutron 1k17/2 and 2h11/2 1k17/2 2h11/2 Neutron N=184 Proton single particle levels Otsuka, Suzuki and Utsuno, Nucl. Phys. A805, 127c (2008)

Anatomy of shell-model interaction

Start from a realistic microscopic interaction M. Hjorth-Jensen, et al., Phys. Repts. 261 (1995) 125 –Bonn-C potential –3rd order Q-box + folded diagram 195 two-body matrix elements (TBME) and 4 single-particle energies (SPE) are calculated  Not completely good （ theory imperfect) Vary 70 Linear Combinations of 195 TBME and 4 SPE Fit to 699 experimental energy data of 87 nuclei M. Honma et al., PRC65 (2002) (R) G-matrix + polarization correction + empirical refinement Microscopic Phenomenological Shell evolution by realistic effective interaction : pf shell GXPF1 interaction

G-matrix vs. GXPF1 T=0 … attractive T=1 … repulsive Relatively large modifications in V(abab ; J0 ) with large J V(aabb ; J1 ) pairing 7= f 7/2, 3= p 3/2, 5= f 5/2, 1= p 1/2 two-body matrix element input output

T=0 monopole interactions in the pf shell GXPF1A G-matrix (H.-Jensen) Tensor force (  +  exchange) f-fp-p f-p “Local pattern”  tensor force

T=0 monopole interactions in the pf shell GXPF1A G-matrix (H.-Jensen) Tensor force (  +  exchange) Tensor component is subtracted

The central force is modeled by a Gaussian function V = V 0 exp( -(r/  ) 2 ) (S,T dependences) with V 0 = -166 MeV,  =1.0 fm, (S,T) factor (0,0) (1,0) (0,1) (1,1) relative strength Can we explain the difference between f-f/p-p and f-p ?

GXPF1 G-matrix (H.-Jensen) Central (Gaussian) - Reflecting radial overlap - Tensor force (  +  exchange) T=0 monopole interactions in the pf shell f-fp-p f-p

T=1 monopole interaction

T=1 monopole interactions in the pf shell j = j’ Tensor force (  +  exchange) G-matrix (H.-Jensen) GXPF1A Basic scale ~ 1/10 of T=0 Repulsive corrections to G-matrix

j = j’ Tensor force (  +  exchange) G-matrix (H.-Jensen) GXPF1A Central (Gaussian) - Reflecting radial overlap - T=1 monopole interactions in the pf shell

(Effective) single-particle energies Lowering of f5/2 from Ca to Cr : ~ 1.6 MeV = 1.1 MeV (tensor) MeV (central) KB3G n-n p-n KB interactions : Poves, Sanchez-Solano, Caurier and Nowacki, Nucl. Phys. A694, 157 (01) Rising of f5/2 from 48 Ca to 54 Ca : p3/2-p3/2 attraction p3/2-f5/2 repulsion

Major monopole components of GXPF1A interaction T=0 - simple central (range ~ 1fm) + tensor - strong (~ 2 MeV) - attractive modification from G-matrix T=1 - More complex central (range ~ 1fm) + tensor - weak ~ -0.3 MeV (pairing), +0.2 MeV (others) - repulsive modification from G-matrix even changing the signs Central force : strongly renormalized Tensor force : bare  +  meson exchange Also in sd shell….

T=0 monopole interactions in the sd shell SDPF-M (~USD) G-matrix (H.-Jensen) Tensor force (  +  exchange) Central (Gaussian) - Reflecting radial overlap -

T=1 monopole interactions in the sd shell SDPF-M (~USD) G-matrix (H.-Jensen) Tensor force (  +  exchange) j = j’ Basic scale ~ 1/10 of T=0 Repulsive corrections to G-matrix

This is not a very lonely idea  Chiral Perturbation of QCD S. Weinberg, PLB 251, 288 (1990) Tensor force is explicit Short range central forces have complicated origins and should be adjusted.

Section 1: Basics of shell model Section 2: Construction of effective interaction and an example in the pf shell Section 3: Does the gap change ? - N=20 problem - Section 4: Force behind Section 5: Is two-body force enough ? Section 6: More perspectives on exotic nuclei Outline

To be continued