1. 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)”

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

3. 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

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

5. 0.5 fm 1 fm distance between nucleons Potential Schematic picture of nucleon- nucleon (NN) potential -100 MeV hard core

6. 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)

7. 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

8. 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

9. 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

10. 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

11. Proton 陽子 Neutron 中性子

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

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

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

15. 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 2 20 8 Orbitals are grouped into shells closed shell fully occupied orbits

16. Spin-orbit splitting Eigenvalues of HO potential Magic numbers Mayer and Jensen (1949) 126 8 20 28 50 82 2 5h  4h  3h  2h  1h 

18. 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 -

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

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

21. 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) 配位

22. 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. 配位

23. 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

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

25. 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

26. 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

27. 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 3 +..... Hamiltonian conserves M.  i ’ s having the same value of M are mixed.  i ’ s having different values of M are not mixed. But,

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

29. 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

30. Dimension M=0 M=1 3 M=2 M=3 M=4 M=6 M=5 3 2 2 1 1 Components of J values 4 J = 2, 4, 6 J = 4, 6 J = 6 J = 0, 2, 4, 6

31. By diagonalizing the matrix H, you get wave functions of good J values by superposing Slater determinants. H =H = M = 0 * * e J=0 0 0 0 0 e J=2 0 0 0 0 e J=4 0 0 0 0 e J=6 In the case shown in the previous page, e J means the eigenvalue with the angular momentum, J.

32. This property is a general one : valid for cases with more than 2 particles. H =H = M * * e J 0 0 0 0 e J’ 0 0 0 0 e J’’ 0 0 0 0 e J’’’ By diagonalizing the matrix H, you get eigenvalues and wave functions. Good J values are obtained by superposing properly Slater determinants.

33. Some remarks on the two-body matrix elements

34. 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.

35. 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

36. 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)

37. 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

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

39. 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

40. 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 配位混合理論

41. Probability that a nucleon is in the valence orbit ~60% A. Gade et al. Phys. Rev. Lett. 93, 042501 (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.

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

43. 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 0 0 0 0 e J’ 0 0 0 0 e J’’ 0 0 0 0 e J’’’ dimension 4 Billions, trillions, …

44. 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

45. 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.

46. 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 負符号問題 - 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 補助場（量子）モンテカルロ法

47. 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, 1284-1287 (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, 1588-1591 (1998). “Monte Carlo shell model for atomic nuclei”, T. Otsuka, M. Honma, T. Mizusaki, N. Shimizu and Y. Utsuno, Prog. Part. Nucl. Phys. 47, 319-400 (2001)

48. 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

49. 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) Lines : 10 5 / 30 years Year Floating point operations per second Progress in shell-model calculations and computers GFlops Monte Carlo Conventional

50. 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

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

52. 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

53. Spin-orbit splitting Eigenvalues of HO potential Magic numbers Mayer and Jensen (1949) 126 8 20 28 50 82 2 5h  4h  3h  2h  1h 

54. 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.

55. 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

56. 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.

57. 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

58. 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