Shell-Model Applications in nuclear physics and astrophysics p. 1

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1 Shell-Model Applications in nuclear physics and astrophysics Gabriel Martínez Pinedo Institut d Estudis Espacials de Catalunya Introduction Shell-Model Basics Effective Interaction Lanczos Strength Function Astrophysical Applications Shell-Model Applications in nuclear physics and astrophysics p. 1

2 Nucleus as a many body problem Nucleus is made of quarks interacting by exchange of gluons. We need to solve Quantum Chromodinamics. Shell-Model Applications in nuclear physics and astrophysics p. 2

3 Nucleus as a many body problem Nucleus is made of quarks interacting by exchange of gluons. We need to solve Quantum Chromodinamics. Nucleus is made of nucleons (neutrons and protons) that interact by meson exchange. Interaction could be described by a non-relativistic potential. Shell-Model Applications in nuclear physics and astrophysics p. 2

4 Nucleus as a many body problem Nucleus is made of quarks interacting by exchange of gluons. We need to solve Quantum Chromodinamics. Nucleus is made of nucleons (neutrons and protons) that interact by meson exchange. Interaction could be described by a non-relativistic potential. We need to solve the Schrödinger equation: HΨ = EΨ H = A i=1 p i 2m + A i<j=1 V ij + A i<j<k=1 V ijk Shell-Model Applications in nuclear physics and astrophysics p. 2

5 Basic symmetries Invariance under translations. Potential depends only on relative coordinates. Shell-Model Applications in nuclear physics and astrophysics p. 3

6 Basic symmetries Invariance under translations. Potential depends only on relative coordinates. Invariance under rotations (space, spin, isospin). States are eigenstates of total angular momentum (J = L + S) and isospin. Tensor operators (multipole operators). [J ±, T k q ] = k(k + 1) q(q ± 1)T k q+1 [J, T k q ] = q Shell-Model Applications in nuclear physics and astrophysics p. 3

7 Basic symmetries Invariance under translations. Potential depends only on relative coordinates. Invariance under rotations (space, spin, isospin). States are eigenstates of total angular momentum (J = L + S) and isospin. Tensor operators (multipole operators). [J ±, T k q ] = k(k + 1) q(q ± 1)T k q+1 [J, T k q ] = q invariance under parity and time reversal Shell-Model Applications in nuclear physics and astrophysics p. 3

8 Example of Potential Potential consistent with two-nucleon scattering data and deuteron structure Potential AV8 : V (r) = V c (r) + V τ (r)(τ 1 τ 2 ) + V σ (r)(σ 1 σ 2 ) +V στ (r)(σ 1 σ 2 )(τ 1 τ 2 ) + V t (r)s 12 + V tτ (r)s 12 (τ 1 τ 2 ) +V b (L S) + V bτ (r)(l S)(τ 1 τ 2 ) Shell-Model Applications in nuclear physics and astrophysics p. 4

9 Solution only possible for light systems Benchmark test calculations for 4 He [H. Kamada, et al., Phys. Rev. C 4, 441 (21)] E exp = MeV TABLE II. Expectation values of the eight potential operators in Eq. 24 in MeV. Method V c V V V TABLE I. The expectation values T and V of kinetic and potential energies, the binding energies E b in MeV, and the radius in fm. Method T V E b r 2 FY CRCGV SVM HH GFMC NCSM EIHH FY CRCGV SVM HH GFMC NCSM Method V t V t V b V b FY CRCGV SVM HH GFMC NCSM Shell-Model Applications in nuclear physics and astrophysics p. 5

10 GFMC calculations for light nuclei -2 Energy (MeV) He + 1/2 α+n 3/2 α+2n AV 5 He AV8 AV IL2 Exp α+d 1 He + Li Li 5/2 /2 α+t 1/2 3/2 He+2n 8 He α+α 8 Be Li+α 1 B Shell-Model Applications in nuclear physics and astrophysics p.

11 Theoretical models Limits of nuclear existence r-process protons rp-process 28 2 A=1 5 neutrons A=12 A~ 82 Density Functional Theory Selfconsistent Mean Field Ab initio few-body calculations Ñω Shell Model No-Core Shell Model G-matrix Towards a unified description of the nucleus Shell-Model Applications in nuclear physics and astrophysics p.

12 Shell-Model basics Shell-Model assumes the existence of shells. Magic numbers are obtained when a shell is completely fill. Shells results from the bunching (grouping) of levels coming from a independent particle average potential. Shell-Model Applications in nuclear physics and astrophysics p. 8

13 Independent-Particle Model Assume the existence of some single-particle wave functions that are the solution of a Schrödinger equation hφ(r) = {T + U}φ a (r)} = ε a φ a (r) The independent-particle motion hamiltonian is then: H = A k=1 T (k) + U(r k ) Eigenfunctions are the product of single-particle wave functions: Φ a1 a 2...a A (1, 2,..., A) = A k=1 φ ak (r k ) Shell-Model Applications in nuclear physics and astrophysics p. 9

14 System identical particles Wave function should be antisymmetric. For two particles: Φ ab (1, 2) = 1 2 [φ a (1)φ b (2) φ a (2)φ b (1)] = 1 2 φ a (1) φ b (2) φ a (2) φ b (1) A-particle Wave function: Φ a1 a 2...a A (1, 2,..., A) = 1 A! φ a1 (r(1)) φ a1 (r(2)) φ a1 (r(a)) φ a2 (r(1)) φ a2 (r(2)) φ a2 (r(a)) φ aa (r(1)) φ aa (r(2)) φ aa (r(a)) Shell-Model Applications in nuclear physics and astrophysics p. 1

15 Simple single-particle potential Empirical construction based in an harmonic oscillator potential plus a spin-orbit term to reproduce the magic numbers (M. Goeppert-Mayer and H. Jensen) U(r) = 1 2 mω2 r 2 + Dl 2 + l s ε nlj = ω[2(n 1)+l+3/2)]+Dl(l+1)+C ω = 41 MeV A1/3 { l + 1 j = l 1/2 l j = l + 1/2 Shell-Model Applications in nuclear physics and astrophysics p. 11

16 Spherical mean-field N=5 3p 2f 3p1/2 2f5/2 1h9/2 1i13/2 (14) 3p3/2 2f/2 (2) (4) () (8) (1) (12) 12 1h N=4 N=3 3s 2d 1g 2p 1f 3s1/2 2d5/2 1g9/2 1f5/2 1f/2 1h11/2 2d3/2 1g/2 2p1/2 2p3/2 (12) (2) (4) () (8) (12) (2) () (4) (8) (82) (4) (5) (4) (38) 82 5 (28) 28 N=2 2s 1d 2s 1d3/2 1d5/2 (4) (2) () (2) (14) 2 N=1 1p 1p1/2 1p3/2 (2) (4) (8) () 8 N= 1s 1s (2) (2) 2 Shell-Model Applications in nuclear physics and astrophysics p. 12

17 Solution to the many-nucleon problem HΨ(1, 2,..., A) = [ A k=1 T (k) + ] A W (k, l) Ψ(1, 2,..., A) = EΨ(1, 2,..., A) k<l=1 Hamiltonian rewritted: H = A [T (k) + U(k)] + [ A W (k, l) A ] U(k) = H + V res k=1 k<l=1 k=1 Hartree-Fock theory provides method to derive single-particle potential. The criterium is to search for the best A-particle slater determinant such us the value of H is minimum. Next, one assumes that the resulting residual interaction is small and that: Ψ(1, 2,..., A) = Φ a1 a 2...a A (1, 2,..., A) Shell-Model Applications in nuclear physics and astrophysics p. 13

18 Validity shell structure V. R. Pandharipande, I. Sick and P. K. A. dewitt Huberts, Rev. mod. Phys. 9, 981 (199) Shell-Model Applications in nuclear physics and astrophysics p. 14

19 Limits of the IPM Example 1 O h ω Π h ω ν 12 MeV 1d3/2 2s1/2 1d5/2 1p1/2 1p3/2 1s1/2 h ω h ω Shell-Model Applications in nuclear physics and astrophysics p. 15

20 Limits of the IPM Example 1 O Shell-Model Applications in nuclear physics and astrophysics p. 15

21 Limits of the IPM Example 1 O Π ν Π ν 1d3/2 1d5/2 2s1/2 1d3/2 1d5/2 2s1/2 1p1/2 1p3/2 1p1/2 1p3/2 Shell-Model Applications in nuclear physics and astrophysics p. 15

22 Two-particle wave function Let s consider the possible isospin values for two nucleons: T = 1, T z = 1 = n n, 1, 1 = p p 1, = 1 2 [ n p + p n ], = 1 2 [ n p p n ] q N-particle system: T = odd (symmetric), T = even (antisymmetric). We can also couple the angular momentum: JM = m 1 m 2 j 1 m 1 j 2 m 2 JM j 1 m 1 j 2 m 2 In N = Z nuclei protons and neutrons occupy the same orbits. Low lying states fulfill (J + T = odd number). Shell-Model Applications in nuclear physics and astrophysics p. 1

23 Example Li posible two-particle states (j 1 j 2 ) JT (p 3/2 p 3/2 ) J=1,3,5;T =, J=2,4,;T =1 (p 3/2 p 3/2 ) (p 3/2 p 1/2 ) J=1,2;T = J=1,2;T =1 (p 3/2 p 1/2 ) (p 1/2 p 1/2 ) J=1;T = J=;T =1 (p 1/2 p 1/2 ) Π p1/2 p1/2 ν 1 + T= T=1 5.3 p3/2 p3/2 2 + T= T= He Π ν 3 + T= p1/2 p1/2 p3/2 p3/2 4 He 1 + T= Li Shell-Model Applications in nuclear physics and astrophysics p. 1

24 Correlations (residual interaction) In order to incorporate the correlations, one has to go beyond mean-field Spherical mean-field breaking symmetries of the system mixing different mean-field configurations Hartree-Fock Bogoliubov Nilsson Deformed Hartree-Fock Tamm-Dancoff RPA Interacting shell-model Shell-Model Applications in nuclear physics and astrophysics p. 18

25 Possible solution Take the basics formed by the A-particle Slater determinants: Φ a = Φ a1 a 2...a A (1, 2,..., A) Shell-Model Applications in nuclear physics and astrophysics p. 19

26 Possible solution Take the basics formed by the A-particle Slater determinants: build the (infinite) matrix: Φ a = Φ a1 a 2...a A (1, 2,..., A) Φ b H Φ a Shell-Model Applications in nuclear physics and astrophysics p. 19

27 Possible solution Take the basics formed by the A-particle Slater determinants: build the (infinite) matrix: diagonalize Φ a = Φ a1 a 2...a A (1, 2,..., A) Φ b H Φ a Shell-Model Applications in nuclear physics and astrophysics p. 19

28 Shell-Model approximation The space of orbits generated by the mean-field potential are grouped in three blocks Inner core: orbits that are always full. Valence space: orbits that contain the physical degrees of freedom relevant to a given property. The distribution of the valence particles among these orbitals is governed by the interaction. External space: all the remaining orbits that are always empty. CORE Shell-Model Applications in nuclear physics and astrophysics p. 2

29 Shell-Model approximation The exact solution on the infinite Hilbert space spanned by the mean field orbits is approximated in the large scale shell-model calculation by the solution of the Schrödinger equation in the valence space using an effective interaction. HΨ = EΨ H eff Ψ eff = EΨ eff In general, effective operators have to be introduced to account for the restrictions of the Hilbert space Ψ O Ψ = Ψ eff O eff Ψ eff Shell-Model Applications in nuclear physics and astrophysics p. 2

30 Shell-Model calculation A shell model calculation needs the following ingredients: A valence space Shell-Model Applications in nuclear physics and astrophysics p. 21

31 Shell-Model calculation A shell model calculation needs the following ingredients: A valence space An effective interaction Shell-Model Applications in nuclear physics and astrophysics p. 21

32 Shell-Model calculation A shell model calculation needs the following ingredients: A valence space An effective interaction A code to build and diagonalize the hamiltonian matrix. Shell-Model Applications in nuclear physics and astrophysics p. 21

33 The choice of the valence space: Valence space In light nuclei the harmonic oscillator closures determine the natural valence spaces. 4 He 1 O 4 Ca 8 Zr p shell sd shell pf shell Cohen/ Brown/ Deformed Kurath Wildenthal Shell-Model Applications in nuclear physics and astrophysics p. 22

34 Valence space In heavier nuclei: jj closures due to the spin-orbit term show up N=28, 5, 82, 12 Shell-Model Applications in nuclear physics and astrophysics p. 23

35 Valence space In heavier nuclei: jj closures due to the spin-orbit term show up N=28, 5, 82, 12 the transition HO jj: occurs between 4 Ca and 1 Sn where the protagonism shifts from the 1f /2 to the 1g 9/2 Shell-Model Applications in nuclear physics and astrophysics p. 23

36 Valence space In heavier nuclei: jj closures due to the spin-orbit term show up N=28, 5, 82, 12 the transition HO jj: occurs between 4 Ca and 1 Sn where the protagonism shifts from the 1f /2 to the 1g 9/2 A valence space can be adequate to describe some properties and completely wrong for others 48 Cr (f ) 8 (f p 3 ) 8 (fp) 8 Q(2 + ) (e.fm 2 ) E(2 + ) (MeV ) E(4 + )/E(2 + ) BE2(2 + + ) (e 2.fm 4 ) B(GT) Shell-Model Applications in nuclear physics and astrophysics p. 23

37 Second quantization Creation and destruction operators: a k = k, a k k = vacuum, such a k k. For fermions antisymmetry given by anti-commutation rules {a i, a j } = {a i, a j } =, {a i, a j} = δ ij Slater determinant: Φ α1 α 2...α A (1, 2,..., A) = a α A a α 2 a α 1 Shell-Model Applications in nuclear physics and astrophysics p. 24

38 One body operators: Second quantization O = A k=1 O(k) O = αβ α O β a αa β number of particles N = α n α = α a αa α Two-body operators: O = A i<j=1 O(i, j) O = αβγδ αβ O γδ a αa β a δa γ Shell-Model Applications in nuclear physics and astrophysics p. 25

39 The interaction in second quantization H = α ε α a αa α αβ V γδ a αa β a δa γ = H + V res αβγδ Shell-Model Applications in nuclear physics and astrophysics p. 2

40 The interaction in second quantization In general it is convenient to work in a couple basis: Defining ã jm = ( 1) j+m a j m, and the coupling: [a j 1 a j 2 ] J M = m 1 m 2 j 1 m 1 j 2 m 2 JM a j 1 m 1 a j 2 m 2 H = ε α n α 1 j 1 j 2 V j 3 j 4 JT 4 α j 1 j 2 j 3 j 4 JT (2J + 1)(2T + 1)(1 + δ12 )(1 + δ 23 ) ] [[a j1 a j2 ] JT [ã j3 ãj 4 ] JT Shell-Model Applications in nuclear physics and astrophysics p. 2

41 The shell-model interaction All the information needed for a shell-model calculation are the independent particle energies (ε α ) and the two-body matrix elements j 1 j 2 ; JT V j 3 j 4 ; JT Example USD interaction (d 5/2, s 1/2, d 3/2 ) USD Shell-Model Applications in nuclear physics and astrophysics p. 28

42 Spectrum of + states in 18 O we build a 3 3 matrix: Application (d 52 ) 2 (d 32 ) 2 (s 12 ) 2 whose eigenvalues produce the spectrum: ( 2.819) ( ) ( 2.124) Shell-Model Applications in nuclear physics and astrophysics p. 29

43 Large scale Shell-Model basis and codes pf-shell valence space: 1f /2, 2p 3/2, 2p 1/2, 1f 5/2 nucleus m-scheme jj-scheme (ANTOINE) (NATHAN) 48 Cr 1,93,41 41, Fe 345,4,14 5,22,21 5 Fe 51,113,392,413,488 5 Ni 1,8,455,228 15,443,84 Impossible to store Hamiltonian matrix! Still possible to compute HΨ. Diagonalization using an iterative algorithm. Shell-Model Applications in nuclear physics and astrophysics p. 3

44 Shell-Model code ANTOINE Shell-Model Applications in nuclear physics and astrophysics p. 31

45 Lanczos algorithm Construction of a orthonormal basis: Initial vector 1. E 12 2 = (H E 11 ) 1 E 23 3 = (H E 22 ) 2 E E NN+1 N + 1 = (H E NN ) N E N 1N N 1 where E NN = N H N, E NN+1 = E N+1N Shell-Model Applications in nuclear physics and astrophysics p. 32

46 Lanczos algorithm E 11 E E 12 E 22 E E 23 E 33 E Diagonalize and obtain an approximation to the energy and wave function. Do a few more iterations Diagonalize again and check that E = E N+k E N < ɛ Shell-Model Applications in nuclear physics and astrophysics p. 32

47 Lanczos convergence RANDOM STARTING VECTOR Shell-Model Applications in nuclear physics and astrophysics p. 33

48 Lanczos convergence 48 Cr Dim (t=2) = 1 5 Dim (full space) = 2 1 f5/2 p1/2 p3/2 Π f/2 f/2 ν f5/2 p1/2 p3/2 4 Ca STARTING VECTOR :EIGENVECTOR OF A SMALLER SPACE ITER= 1 DIA= NONDIA= Shell-Model Applications in nuclear physics and astrophysics p. 34

49 Isospin symmetry Nuclear spectra are almost unchanged under the exchange of neutrons and protons Fe Cr 24 2 Shell-Model Applications in nuclear physics and astrophysics p. 35

50 Isospin representation Neutron and proton are considered different states of the same particle: the nucleon n = ( 1 ), p = ( 1 ) Isospin operators: t = 1 2 τ, t ± = t 1 ± t 2 τ 1 = ( 1 1 ), τ 2 = ( i i ), τ 3 = ( 1 1 ) States are eigenstates of total isospin (T = A i=1 ti ) T 2 = T (T + 1) T, T z, T z T, T z T T z = N Z 2 Shell-Model Applications in nuclear physics and astrophysics p. 3

51 Realistic interactions Exp. KB KLS Bonn A Bonn B Bonn C excitation energy 44 Ca Ca Ca Ca (f /2 ) 8 Ψ GS Shell-Model Applications in nuclear physics and astrophysics p. 3

52 Realistic interactions Exp. KB KLS Bonn A Bonn B Bonn C excitation energy 44 Ca Ca Ca Ca (f /2 ) 8 Ψ GS Ni model space (f p Ni (f ) 1 Ψ GS n p3/ ) 1 Shell-Model Applications in nuclear physics and astrophysics p. 3

53 Realistic interactions Exp. KB KLS Bonn A Bonn B Bonn C excitation energy 44 Ca Ca Ca Ca (f /2 ) 8 Ψ GS Ni model space (f p Ni (f ) 1 Ψ GS n p3/ ) 1 The realistic interactions do not reproduce the shell closure N or Z=28 Shell-Model Applications in nuclear physics and astrophysics p. 3

54 Realistic interactions Exp. KB KLS Bonn A Bonn B Bonn C excitation energy 44 Ca Ca Ca Ca (f /2 ) 8 Ψ GS Ni model space (f p Ni (f ) 1 Ψ GS n p3/ ) 1 The realistic interactions do not reproduce the shell closure N or Z=28 Why? Shell-Model Applications in nuclear physics and astrophysics p. 3

55 The structure of the Hamiltonian From the work of M. Dufour and A. Zuker (PRC ) Separation theorem: Any effective interaction can be split in two parts: H = H m (monopole) + H M (multipole) H m contains all the terms that are affected by a spherical Hartree-Fock variatition, hence reponsible of the global saturation properties and of the evolution of the spherical single particle field. Important property: CS ± 1 H CS ± 1 = CS ± 1 H m CS ± 1 Shell-Model Applications in nuclear physics and astrophysics p. 38

56 The structure of the Hamiltonian For all the realistic G-matrices, H m is not accurate enough. H M is almost the same. The monopole part has to be empirically corrected to reproduce the structure of the simple nuclei CS ± 1 Shell-Model Applications in nuclear physics and astrophysics p. 39

57 Multipole Hamiltonian Can be characterized as the sum of simple terms: L = isovector and isoscalar pairing Elliot s quadrupole-quadrupole force. (σ 1 σ 2 )(τ 1 τ 2 ) Octupole and hexadecapole terms r λ Y λ r λ Y λ All the interactions contains similar terms: Interaction JT = 1 JT = 1 λτ = 2 λτ = 4 λτ = 11 KB FPD GOGNY Shell-Model Applications in nuclear physics and astrophysics p. 4

58 The Monopole Hamiltonian H m contains terms that depend in n and T. Assume that we have a constant potential and single particle energy: E = nε + n(n 1) 2 V Shell-Model Applications in nuclear physics and astrophysics p. 41

59 The Monopole Hamiltonian If we consider also isospin: E = nε + n(n 1) 2 V + bt (T + 1) Shell-Model Applications in nuclear physics and astrophysics p. 41

60 The Monopole Hamiltonian In general we have different orbits each with different average interactions: E = i n i ε i + i n i (n i 1) 2 V ii + i j n i n j V ij Shell-Model Applications in nuclear physics and astrophysics p. 41

61 The Monopole Hamiltonian If we consider also the fact that we have isospin: H m = i ε i n i + ij [ δ ij a ij n i (n j δ ij ) δ ij b ij ( T i T j 3n )) i 4 δ ij Where a and b are defined from the centroids (average interaction): JT J (2J + 1)W Vij T ijij = (2J + 1) J a ij = 3V ij 1 + Vij, b ij = Vij 1 Vij 4 Shell-Model Applications in nuclear physics and astrophysics p. 41

62 Influence of monopole hamiltonian The evolution of effective spherical single particle energies with the number of particles in the valence space can be extracted from H m. In the case of identical particles the expresion is: ε j (n) = ε j (n = 1) + i V ij n i The monopole hamiltonian H m also governs the relative position of the various T-values in the same nucleus, via the terms: b ij T i T j Even small defects in the centroids can produce large changes in the relative position of the different configurations due to the appearance of quadratic terms involving the number of particles in the different orbits Shell-Model Applications in nuclear physics and astrophysics p. 42

63 Effect of monopole corrections E (MeV) /2 5/2 /2 1/2 5/2 5/2 9/2 /2 3/2 5/2 /2 1/2 9/2 /2 3/2 5/2 5/2 /2 1/2 5/2 3/2 /2 + (1/2,3/2 ) 5/2 9/2 + 1/ /2 /2 1 3/2 3/2 3/2 KB KB KB3 3/2 Expt. Shell-Model Applications in nuclear physics and astrophysics p. 43

64 Evolution SPE Shell-Model Applications in nuclear physics and astrophysics p. 44

65 Limits of IPM Example 1 O h ω Π h ω ν 12 MeV 1d3/2 2s1/2 1d5/2 1p1/2 1p3/2 1s1/2 h ω h ω Shell-Model Applications in nuclear physics and astrophysics p. 45

66 Limits of IPM Example 1 O Shell-Model Applications in nuclear physics and astrophysics p. 45

67 Limits of IPM Example 1 O Π ν Π ν 1d3/2 1d5/2 2s1/2 1d3/2 1d5/2 2s1/2 1p1/2 1p3/2 1p1/2 1p3/2 Shell-Model Applications in nuclear physics and astrophysics p. 45

68 Monopole explanation Π ν Π ν 1d3/2 1d5/2 2s1/2 1d3/2 1d5/2 2s1/2 1p1/2 1p3/2 1p1/2 1p3/2 E ph = ε p + V pp 2 E 1ph = 11ε p + ε r + 55V pp + 11V pr E 2ph = 1ε p + 2ε r + 45V pp + 2V pr + V rr E 4ph = 8ε p + 4ε r + 28V pp + 32V pr + V rr Shell-Model Applications in nuclear physics and astrophysics p. 4

69 Monopole explanation Π ν Π ν 1d3/2 1d5/2 2s1/2 1d3/2 1d5/2 2s1/2 1p1/2 1p3/2 1p1/2 1p3/2 1ph = ε r ε p + 11(V pr V pp ) 2ph = 2(ε r ε p ) + V pp + 2(V pr V pp ) + (V rr V pp ) 4ph = 4(ε r ε p ) + 32(V pr V pp ) + (V rr V pp ) Shell-Model Applications in nuclear physics and astrophysics p. 4

70 Evolution SPE far off stability Given H m, the effective single particle energies [from Otsuka et al, Phys. Rev. Lett. 8, 8252 (21)], Shell-Model Applications in nuclear physics and astrophysics p. 4

71 Evolution SPE far off stability N=2 effective SPE (MeV) 1 2 1s1/2 d5/2 1p3/2 f/2 d3/ Proton number Shell-Model Applications in nuclear physics and astrophysics p. 4

72 Vanishing of shell closure at N=2 5 4 E ph E 2p2h O F Ne Na Mg Al Si N Shell-Model Applications in nuclear physics and astrophysics p. 48

73 Computation of transition operators Given a one-body transition operator O, how do we compute Ψ f O Ψ i here Ψ i and Ψ f are many-body wave functions obtained from shell-model diagonalization One body operators: O = A i=1 o(r(i)) O = i,j i O j a i a j We need to know the value of our one body operator between single particle wave functions i O j the one body density matrix elements Ψ f a i a j Ψ i Shell-Model Applications in nuclear physics and astrophysics p. 49

74 Computation of transition operators for i O j, one needs (eventually) to know the radial part of the wave function: usually harmonic oscillator, sometimes wood-saxon. i O j = d 3 r φ i (r)o(r)φ j (r) for the one body density matrix elements (same procedure as for the hamiltonian): a 5a = 111 now we know the procedure to compute: EL transitions: r L Y L β decay : Fermi decay : τ ± Gamow-Teller decay: στ ± Shell-Model Applications in nuclear physics and astrophysics p. 5

75 Example calculation β decay half-life calculation Determine initial state Ψ i. Determine all posible final states Ψ f. Compute matrix elements Ψ f O Ψ i λ f = ln 2 K f(z, W f )[B f (F ) + B f (GT )] Determine total decay rate: λ = ln 2 T 1/2 = f λ f Shell-Model Applications in nuclear physics and astrophysics p. 51

76 Lanczos Strength Functions Ψ ground state given nucleus. Sum rule state (or doorway): Ω = Ω Ψ. Total sum rule (sum over all final states of the matrix element squared) is the norm of state Ω : Ω Ω = Ψ Ω Ω Ψ = i i Ω Ψ 2 We can think of the state Ω as a (probability) distribution over the eigenvalues i of H with values i Ω 2 = i Ω Ψ 2 Shell-Model Applications in nuclear physics and astrophysics p. 52

77 Lanczos Strength Functions Any distribution can be characterized by the moments of the distribution. Ē = Ω H Ω = i E i i Ω Ψ 2 m n = Ω (H Ē)n Ω = i (E i Ē)n i Ω Ψ 2 Gaussian distribution characterized by two moments (Ē, σ2 = m 2 ) g(e) = 1 σ 2π exp( (E Ē)2 2σ 2 ) g(e) 2σ E Shell-Model Applications in nuclear physics and astrophysics p. 52

78 Lanczos Strength Functions In general we only need a finite number of momenta. We can define a basis of α states. m n = Ω (H Ē)n Ω = N (E α Ē)n α Ω Ψ 2 ( n M) α E α α H α With N states we can reproduce 2N moments of the distribution. Shell-Model Applications in nuclear physics and astrophysics p. 52

79 Lanczos Strength Functions Example: Beta half-life λ i f(z, W i ) i Ω Ψ 2 f(z, W i ) W i 1 W 2 (W i 1) 2 dw (W i ) 5 3 Shell-Model Applications in nuclear physics and astrophysics p. 52

80 Lanczos Strength Functions Lanczos method provides a natural way of determining the basis α. Initial vector 1 = Ω. Ω Ω E 12 2 = (H E 11 ) 1 E 23 3 = (H E 22 ) 2 E E NN+1 N + 1 = (H E NN ) N E N 1N N 1 where E NN = N H N, E NN+1 = E N+1N Each Lanczos iteration gives information about two new moments of the distribution. E 11 = 1 H 1 = Ē E 2 12 = Ω (H E 11) 2 Ω = m 2 E 22 = m 3 m 2 + Ē E 2 23 = m 4 m 2 m2 3 m 2 2 m 2 Diagonalizing Lanczos matrix after N iterations gives an approximation to the distribution with the same lowest 2N moments. Shell-Model Applications in nuclear physics and astrophysics p. 52

81 Evolution Strength Distribution M1 Strength on 5 Fe 12 1 Iteration B(M1) (µ N 2 ) E (MeV) Shell-Model Applications in nuclear physics and astrophysics p. 53

82 Evolution Strength Distribution M1 Strength on 5 Fe 12 2 Iterations B(M1) (µ N 2 ) E (MeV) Shell-Model Applications in nuclear physics and astrophysics p. 53

83 Evolution Strength Distribution M1 Strength on 5 Fe 12 5 Iterations B(M1) (µ N 2 ) E (MeV) Shell-Model Applications in nuclear physics and astrophysics p. 53

84 Evolution Strength Distribution M1 Strength on 5 Fe 8 1 Iterations B(M1) (µ N 2 ) E (MeV) Shell-Model Applications in nuclear physics and astrophysics p. 53

85 Evolution Strength Distribution M1 Strength on 5 Fe 8 2 Iterations B(M1) (µ N 2 ) E (MeV) Shell-Model Applications in nuclear physics and astrophysics p. 53

86 Evolution Strength Distribution M1 Strength on 5 Fe 8 5 Iterations B(M1) (µ N 2 ) E (MeV) Shell-Model Applications in nuclear physics and astrophysics p. 53

87 Evolution Strength Distribution M1 Strength on 5 Fe 2. 1 Iterations 1.5 B(M1) (µ N 2 ) E (MeV) Shell-Model Applications in nuclear physics and astrophysics p. 53

88 Evolution Strength Distribution M1 Strength on 5 Fe Iterations 1.5 B(M1) (µ N 2 ) E (MeV) Shell-Model Applications in nuclear physics and astrophysics p. 53

89 Evolution Strength Distribution M1 Strength on 5 Fe 2. 4 Iterations 1.5 B(M1) (µ N 2 ) E (MeV) Shell-Model Applications in nuclear physics and astrophysics p. 53

90 Evolution of Strength Distribution GT Strength on 48 Sc Shell-Model Applications in nuclear physics and astrophysics p. 54

91 Evolution of Strength Distribution GT Strength on 48 Sc Shell-Model Applications in nuclear physics and astrophysics p. 54

92 Fermi matrix elements B(F ) = 1 2J i + 1 M i,m f J f M f ; T f T zf A k=1 t k ± J i M i ; T i T zi 2 B(F ) = [T i (T i + 1) T zi (T zi ± 1)]δ Ji,J f δ Ti,T f δ Tzf,T zi ±1 Shell-Model Applications in nuclear physics and astrophysics p. 55

93 Fermi matrix elements B(F ) = 1 2J i + 1 M i,m f J f M f ; T f T zf A k=1 t k ± J i M i ; T i T zi 2 B(F ) = [T i (T i + 1) T zi (T zi ± 1)]δ Ji,J f δ Ti,T f δ Tzf,T zi ±1 Energetics: E IAS = Q β + sign(t zi )[E C (Z + 1) E C (Z) (m n m H )] Shell-Model Applications in nuclear physics and astrophysics p. 55

94 Fermi matrix elements B(F ) = 1 2J i + 1 M i,m f J f M f ; T f T zf A k=1 t k ± J i M i ; T i T zi 2 B(F ) = [T i (T i + 1) T zi (T zi ± 1)]δ Ji,J f δ Ti,T f δ Tzf,T zi ±1 Energetics: E IAS = Q β + sign(t zi )[E C (Z + 1) E C (Z) (m n m H )] Selection rule: J = T = π i = π f Shell-Model Applications in nuclear physics and astrophysics p. 55

95 Fermi matrix elements B(F ) = 1 2J i + 1 M i,m f J f M f ; T f T zf A k=1 t k ± J i M i ; T i T zi 2 B(F ) = [T i (T i + 1) T zi (T zi ± 1)]δ Ji,J f δ Ti,T f δ Tzf,T zi ±1 Energetics: E IAS = Q β + sign(t zi )[E C (Z + 1) E C (Z) (m n m H )] Selection rule: J = T = π i = π f Sum rule (sum over all the final states): S(F ) = S (F ) S + (F ) = 2T zi = (N Z) Shell-Model Applications in nuclear physics and astrophysics p. 55

96 Gamow-Teller matrix elements back B(GT ) = g2 A 2J i + 1 J f; T f T zf A k=1 σ k t k ± J i ; T i T zi 2 Selection rule: g A = 1.22 ±.18 J =, 1 (no J i = J f = ) T =, 1 π i = π f Ikeda sum rule: S(GT ) = S (GT ) S + (GT ) = 3(N Z) Shell-Model Applications in nuclear physics and astrophysics p. 5

97 SN198A Type II supernova in LMC ( 55 kpc) neutrinos from SN198A Kamiokande II IMB Energy (MeV) E grav 1 53 erg E rad erg E kin 1 51 erg = 1 foe Time (seconds) E ν erg Shell-Model Applications in nuclear physics and astrophysics p. 5

98 Evolution Massive Stars Evolution 2 solar mass star Shell-Model Applications in nuclear physics and astrophysics p. 58

99 Late stages core evolution He H Fe Si, S O, Ne C, O PRE-SUPERNOVA ν e shock core ν e ν-sphere NEUTRINO BREAKOUT ν e ν e ν e ν e ν e ν e Fe core ν e COLLAPSE ν e ν e ν e ν e ν ν ν shock EXPLOSION ν e ν e ν e ν ν ν ν ν envelope ν ν ν ν ν NEUTRINO TRAPPING νν νν νν proto n-star νν COOLING neutrinosphere shock core CORE BOUNCE νν νν νν νν outlying matter Shell-Model Applications in nuclear physics and astrophysics p. 59

100 * + * + * + &' &' &',-,-,- () () () "# "# "# $ % $ % $ %!!! Semileptonic Weak Processes in Stars λ ν e ν ν e e ν e λ k orbital e capture q = k ν λ bound-state decay q = + k λ ν λ λ β λ λ k λ λ λ e e k λ β decay q = + k ν λ ν e λ ν λ ν l l ν λ λ ν λ e + e λ k β + decay q = + k λ ν λ λ ν ν e l λ ν ν l l + λ λ ν l ν ν k k λ l + (anti)neutrino capture kλ kλ qλ = kλ ν λ continuum charged (anti)lepton capture q = k ν l λ λ λ ν λ ν l νl ν ν ν λ ν (anti)neutrino scattering qλ = ν λ ν λ l ν λ l λ O F e iqr τ O GT e iqr στ Shell-Model Applications in nuclear physics and astrophysics p.

101 / / / / / / / / / / / / / / / / / / / / / / / / / /.. / /. Presupernova evolution T =.1.8 MeV, ρ = g cm 3. Composition of iron group nuclei (A = 45 5) Important processes: electron capture: e + (N, Z) (N + 1, Z 1) + ν e β decay: (N, Z) (N 1, Z + 1) + e + ν e Dominated by allowed transitions (Fermi and Gamow-Teller) Evolution decreases number of electrons (Y e ) and Chandrasekar mass (M Ch 1.4(2Y e ) 2 M ) / (Z 1,A) Laboratory Low lying Strength σ τ + (Z,A) (Z 1,A) Supernova Gamow Teller Resonance (Z,A) electron distribution Fenomenological model (Fuller, Fowler, Newman, 1985) Charge exchange reactions (n, p), (d, 2 He) Microscopic model (Shell-Model) (Langanke & Martínez-Pinedo, 21) Shell-Model Applications in nuclear physics and astrophysics p. 1

102 GT in charge exchange reactions GT strength could be measured in CE reactions: GT proved in (p, n), ( 3 He, t). GT + proved in (n, p), (t, 3 He), (d, 2 He). Mathematical relationship (E p 1 MeV/nucleon): dσ dωde ( ) S(E x )B(GT ) B(GT ) = ( ga g V ) 2 f k σk t k ± i 2 2J i + 1 Ikeda sum rule: S S + = 3(N Z) Shell-Model Applications in nuclear physics and astrophysics p. 2

103 Independent Particle Model GT + strength in 58 Ni measured in (n, p). Independent particle model (FFN) Ni (El-Kateb et al, 1994) p 1/2 f 5/2 FFN p 3/2 GT Strength 1..5 p 1/2 f 5/2 f /2 ν π στ + p 3/2 f /2 ν π p 1/2 f 5/2 p 3/ E (MeV) 58 Ni f /2 ν 58 Co π Shell-Model Applications in nuclear physics and astrophysics p. 3

104 Gamow-Teller strength GT + strength measured in charge-exchange (n, p) experiments (TRIUMF) Fe FFN Exp SM V Fe. 55 Mn GT Strength Ni. 59 Co E (MeV) E (MeV) Shell-Model Applications in nuclear physics and astrophysics p. 4

105 GT + strength measured in (d, 2 He) dσ/dω[mb/(sr 5 kev)] 51 V(d, 2 He) 51 Ti E lab =11 MeV Θcm< MeV 3.2 MeV 4.88 MeV Ex [MeV] High resolution Gamow-Teller distributions on 51 V, 58 Ni ( 4 Ni,... ) measured at KVI (Groningen) by EuroSupernova Collaboration V(n,p) Alford et al. (1993).4 B(GT+) B(GT+).3.2 Shell-Model Calculation G. Martinez-Pinedo.1 Ex [MeV] Shell-Model Applications in nuclear physics and astrophysics p. 5

106 Shell-model (LMP) vs FFN rates λ ec (s 1 ) LMP FFN 1 5 Fe 5 Ni ρ = Co 59 Co 54 Mn ρ = Co ρ =4.32 ρ = ρ =1. ρ = T 9 T 9 Shell-Model Applications in nuclear physics and astrophysics p.

107 Most important nuclei Most important nuclei to determine the electron capture rate Fe 5 Co 5 Fe 1 Ni WW LMP Fe 55 Co WW LMP 5 Fe 55 Fe Ye Fe 53 Cr 5 Fe 53 Cr Ye Fe 5 Fe 1 H 53 Cr 15 M 25 M Time till core collapse (s) Time till core collapse (s) Shell-Model Applications in nuclear physics and astrophysics p.

108 Collapse phase Important processes: Neutrino transport (Boltzman equation): ν + A ν + A (trapping) ν + e ν + e (thermalization) cross sections E 2 ν electron capture on protons: e + p n + ν e What is the role of electron capture on nuclei? e + (N, Z) (N + 1, Z 1) + ν e What is the role of inelastic neutrino-nucleus scattering? ν + A ν + A Shell-Model Applications in nuclear physics and astrophysics p. 8

109 Collapse abundances 4 T= 1.84 GK, ρ= 3.39e+11 g/cm 3, Y e =..39 Z (Proton Number) Log (Mass Fraction) N (Neutron Number) Shell-Model Applications in nuclear physics and astrophysics p. 9

110 (Un)blocking electron capture at N=4 Independent particle treatment (Bruenn) N=4 Blocked g 9/2 f 5/2 p GT 1/2 p 3/2 f /2 Unblocked Correlations Finite T f 5/2 p GT 1/2 neutrons protons g 9/2 p 3/2 f /2 neutrons protons Core Core Weak Rates for nuclei with A=5 112 computed using the Shell Model Monte Carlo plus RPA approach Shell-Model Applications in nuclear physics and astrophysics p.

111 Electron capture: nuclei vs protons λec (s 1 ) Electron capture rates ρ (g cm 3 ) Abundances 1 H 8 Ni 9 Ni Ga 9 Ge 89 Br (MeV) Energetics µ e Q = µ n µ p Q p ρ (g cm 3 ) Abundance Y n Y h Y α R h = i Y iλ i = Y h λ h Y p R p = Y p λ p ρ (g cm 3 ) Shell-Model Applications in nuclear physics and astrophysics p. 1

112 Reaction rates Rec (s 1 ) Eνe (MeV) protons nuclei protons nuclei ρ (g cm 3 ) ρ (g cm 3 ) Electron capture on nuclei dominates over capture on protons Shell-Model Applications in nuclear physics and astrophysics p. 2

113 Consequences With Rampp & Janka (General Relativic model) 15 M presupernova model from A. Heger & S. Woosley T 1 5 Bruenn LMS Ye,c, Ylep,c Bruenn LMS Y e Y lep ρ c (g cm 3 ) sc (kb), Tc (MeV) Bruenn LMS ρ c (g cm 3 ) s Eν emission (MeV s 1 ) Eν (MeV) ρ c (g cm 3 ) ρ c (g cm 3 ) Shell-Model Applications in nuclear physics and astrophysics p. 3

114 Consequences With Hix, Liebendörfer, Mezzacappa, Messer (Newtonian Gravity) Ye.4.3 ρ (g cm 3 ) Bruenn LMS velocity (1 4 km s 1 ) Enclosed Mass (M ) T (MeV) Enclosed Mass (M ) Shell-Model Applications in nuclear physics and astrophysics p. 3

115 Shock evolution With Rampp & Janka (General Relativic model) 2 15 Rshock (km) 1 5 Bruenn LMS t (s) Shell-Model Applications in nuclear physics and astrophysics p. 4

116 Neutrino interactions in the collapse Bruenn and Haxton (1991) Based on results for 5 Fe Elastic scattering: ν + A ν + A (trapping) Absorption: ν e +(N, Z) e +(N 1, Z +1) ν-e scattering: ν + e ν + e Inelastic ν-nuclei scattering: ν + A ν + A Shell-Model Applications in nuclear physics and astrophysics p. 5

117 ν e absorption cross section on 5 Fe 5 Fe(ν e, e ) 5 Co measured by KARMEN collaboration: σ exp = 2.5 ± 1.8(stat) ±.43(syst) 1 4 cm 2 σ th = cm 2 σ (1-42 cm 2 ) σ (1-42 cm 2 ) Energy (MeV) Energy (MeV) Shell-Model Applications in nuclear physics and astrophysics p.

118 Neutrino nucleosynthesis Neutrinos interact with abundant nuclear species Neutral current (ν, ν ): Nucleus excited to particle unbound states that decay by particle emission. Charged current (ν e, e ) and ( ν e, e + ) Neutron rich matter ν ν ν x (E ~ 25 MeV) e ν (E ~ 1 MeV) ν e (E ~ 11 MeV) ν e + n p + e ν e + p n + e + R ~ 1/T Product Parent Reaction 11 B 12 C (ν, ν n), (ν, ν p) 15 N 1 O (ν, ν n), (ν, ν p) 19 F 2 Ne (ν, ν n), (ν, ν p) 138 La 138 Ba (ν, e ) 139 La (ν, ν n) 18 Ta 18 Hf (ν, e ) 181 Ta (ν, ν n) 11 B/ 1 B traces galactic evolution: Big ratio ( 1) predicted by ν-process. Cosmic rays spallation reactions on C give a ratio of 2. Solar system ratio is La, 18 Ta production is sensitive to neutrino oscillations. Shell-Model Applications in nuclear physics and astrophysics p.

119 Nucleosynthesis with and without ν With Heger (Los Alamos) Shell-Model Applications in nuclear physics and astrophysics p. 8

120 The production of 138 La Produced by 139 La(γ, n), 138 Ba(ν e, e ) Shell-Model Applications in nuclear physics and astrophysics p. 9

121 The production of 18 Ta Produced by 181 Ta(γ, n), 18 Hf(ν e, e ) Shell-Model Applications in nuclear physics and astrophysics p. 8

122 Presupernova abundances T= 9.1 GK, ρ=.8e+9 g/cm 3, Y e = Z (Proton Number) 3 2 A=5 (LMP) 1 Log (Mass Fraction) N (Neutron Number) Shell-Model Applications in nuclear physics and astrophysics p. 81