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The Nuclear Shell Model

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The Nuclear Shell Model The Nuclear Shell Model N. A. Smirnova CEN Bordeaux-Gradignan, France Bridging Methods in Nuclear Theory IPHC, Strasbourg, June 26–30, 2017 N. A. Smirnova The Nuclear Shell Model The Nuclear Shell Model Table of contents 1 Introduction to the nuclear many-body problem: variational against diagonalization methods (lecture of Ph. Quentin on self-consistent theories, Wednesday) 2 Shell model Basis construction Solution of the eigenvalue problem Matrix elements of transition operators Effective nucleon-nucleon (NN) interactions (lecture of L. Bonneau on NN interaction, Thursday lecture of H. Molique on group theory, Friday morning) 3 No-Core Shell Model for Light Nuclei (lecture of R. Lazauskas on few-body problems, Tuesday) 4 Shell-model code ANTOINE (E. Caurier, F. Nowacki, IPHC Strasbourg) afternoon session Lecture of A. P. Zuker “Beyond the Shell Model” on Friday afternoon N. A. Smirnova The Nuclear Shell Model Structure of complex nuclei Nuclear charge (matter) density distribution ρch(~r) (ρm(~r)) with sharp 1=3 radius R ' r0A . Empirical evidence on the existence of an average potential and the corresponding shell structure (From masses, nucleon separation energies, low-energy spectra, etc.) Independent-particle motion near Fermi-level. (From nucleon transfer reactions: nuclear mean-field with strong spin-orbit splitting and large shell gaps) Pairing (superfluid behavior) at low excitation energy. (Sn versus S2n; two-nucleon spectra, comparison of spectra of even-even and even-odd, odd-even nuclei, etc.) Low-lying multipole (quadrupole) modes, vibrational or rotational energy structures. (Coulomb excitation; scattering of charge particles; heavy-ion fusion-evaporation reactions, etc) The aim of the microscopic theory is to describe these motions starting from a NN force. N. A. Smirnova The Nuclear Shell Model Nuclear many-body126 problem Limits of nuclear existence 82 ss ce ro r‐p 50 s es c y ro or protons 82 ‐p The ld p al Fie r on n 28 cti ea 20 un t M 50 y F en it ist ens ns D ‐co 8 28 elf 2 20 neutrons s 0 2 8 10 ~6 A= 12 A A= Toward s a unified ab initio 0 h Shell description of the nucleus few‐body Model calc ulations No ‐Core Shell Model N. A. Smirnova The Nuclear Shell Model Nuclear many-body problem Non-relativistic Hamiltonian for A nucleons A A X ~p2 X H^ = i + W (~r −~r ) 2m i j i=1 i<j=1 A A A X ~p2 X X H^ = i + U(~r ) + W (~r −~r ) − U(~r ) ; 2m i i j i i=1 i<j=1 i=1 | {z } | {z } H^ (0) V^ Mean-field theories Search for the most optimum mean-field potential starting from a given two-body interaction + correlations Shell model Schematic average potential + residual interaction N. A. Smirnova The Nuclear Shell Model Variational approach to the nuclear many-body problem Starting point: antisymmetric product wave function φα1 (1) φα1 (2) : : : φα1 (A) 1 φα2 (1) φα2 (2) : : : φα2 (A) Ψ(1; 2;:::; A) = p . .. A! . φαA (1) φαA (2) : : : φαA (A) The best wave function is determined via a variational principle: Z ~ 2 ~ δhΨjHjΨi = hδΨjHjΨi = 0 with jφαi (r)j dr = 1: N. A. Smirnova The Nuclear Shell Model Self-consistent mean-field potential Hartree-Fock equations 2 ! Z − ~ ∆ + U (~r) φ (~r) + U (~r; r~0)φ (r~0)dr~0 = " φ (~r) 2m H i F i i i Direct (Hartree) term Exchange (Fock) term Z X X ∗ ~0 ~0 ~0 ~0 U (~r; r~0) = φ∗(~r)W (~r; r~0)φ (r~0) UH (~r) = φb (r )W (~r; r )φb(r )dr F b b b2F b2F Iterative solution of Hartree-Fock equations: (0) ~ (1) ~ (1) (HF) ~ (HF) φi (r) φi (r);"i : : : φi (r);"i #%#%#% (0) ~ (0) ~ ~0 (1) ~ (1) ~ ~0 (HF) ~ UH (r); UF (r; r ) UH (r); UF (r; r ) ::: U (r) N. A. Smirnova The Nuclear Shell Model Self-consistent mean-field potential Direct (Hartree) term Z X ∗ ~0 ~0 ~0 ~0 UH (~r) = φb(r )W (~r; r )φb(r )dr b2F R ~0 ~0 ~0 UH (~r) = ρ(r )W (~r; r )dr P 2 ρ(~r) = jφb(~r)j b2F Example: a delta-force W (~r; r~0) / δ(~r − r~0) UH (~r) / ρ(~r) N. A. Smirnova The Nuclear Shell Model Hartree-Fock ground state properties From the product HF wave function, a number of ground-state properties can be calculated: A 1 Y Ψ (1; 2;:::; A) = p A φHF (i) HF αi A! i=1 ^ hΨHF jHjΨHF i = E0 A P 2 2 hΨHF j ^ri jΨHF i = hr i i=1 A P hΨHF j ρ^(~ri )jΨHF i = ρ(~r) i=1 Application to open-shell nuclei and calculation excitation spectra require inclusion of correlations (pairing, etc) ! beyond mean-field techniques (lecture of Ph. Quentin). N. A. Smirnova The Nuclear Shell Model Shell model: energy matrix diagonalization Non-relativistic Hamiltonian for A nucleons A A A X ~p2 X X H^ = i + U(~r ) + W (~r −~r ) − U(~r ) 2m i i j i i=1 | {z } i<j=1 i=1 | {z } h^ i V^ Construction of a basis from single-particle states ^ hφα(~r) = "αφα(~r) ! f"α; φα(~r)g Spherical potential U(~r) = U(r): α = fnα; lα; jα; mαg (j) Rnlj (r) h i φnljm(~r) = Yl (θ; ') × χ 1 r 2 m | {z } P 1 (lml msjjm)Ylm (θ;')χ 1 2 l ms ml ms 2 N. A. Smirnova The Nuclear Shell Model Single-particle wave functions Radial differential equation 2 2 l(l + 1) − ~ R00(r) + ~ R(r) + [U(r)R(r) + a f (r)] R(r) = "R(r) 2m 2m r 2 ls ls Normalization condition Z Z 2 X 1 0 1 0 jφnlsjm(~r)j d~r= (lml msjjm)(lml msjjm) Ylm (θ; φ)Y 0 (θ; φ)dΩ l lml 0 0 2 2 ml msml ms 1 1 Z Z 2 2 ×hχ 1 jχ 1 0 i jRnl (r)j dr = jRnl (r)j dr = 1 2 ms 2 ms 0 0 Parity ^ ^ l Pφnlsjm(~r) = Pφnlsjm(−~r) = (−1) φnlsjm(~r) N. A. Smirnova The Nuclear Shell Model Examples of spherically-symmetric potentials . Square-well potential + strong spin-orbit term J (kr) l+1/2 U(r) s l=0 (100) r d l=2 (93) ex. N = 4 g l=4 (75.5) . Harmonic oscillator potential + orbital + spin-orbit term l +1/2 22 U(r) e 2 L n-1 (ν r) . r . mω . (ν = ) √ h N : degenerate in n, l N. A. Smirnova The Nuclear Shell Model Harmonic oscillator potential N=5 2p1/2 2 2 2p 1f5/2 2p m! r 1f 3/2 ~ ~ ~ 0h9/2 U(r) = + αl · l + βl · ~s 1f7/2 0h 82 2 0h11/2 N=4 2s 1d3/2 2s1/2 1d 0g7/2 1d 3 3 0g 50 5/2 "N = ! 2n + l + = ! N + ; 0g9/2 ~ ~ N=3 1p 2 2 1p 0f1/2 1p5/2 0f 28 3/2 0f7/2 N = 0; 1; 2;:::; 20 N=2 1s 0d3/2 l = N; N − 2;:::; 1 or 0 0d 1s1/2 0d5/2 n = (N − l)=2 : 8 N=1 0p 0p 1/2 0p3/2 2 Harmonic oscillator potential N=0 0s 0s possesses many symmetry 1/2 properties which make it a preferable choice as a basis. M. Mayer (1949) O. Haxel, H. Jensen, H.E. Suess (1949) N. A. Smirnova The Nuclear Shell Model Isospin The idea of W.Heisenberg: proton and neutron are considered as two states of a nucleon. 0 1 vππ ≈ vνν ≈ vπν π = ; ν = 1 0 Isospin operators (in analogy with the Pauli matrices): 1 0 1 0 −i 1 0 ~t = ~τ; τ = ; τ = ; τ = 2 x 1 0 y i 0 z 0 −1 Single-particle wave functions 1 φ (r) = φ(~r) = φ(~r) θ ; ν 0 t=1=2;mt =1=2 0 φ (r) = φ(~r) = φ(~r) θ : π 1 t=1=2;mt =−1=2 N. A. Smirnova The Nuclear Shell Model Isospin and classification of nuclear states A A ^ X ^ X T = ^ti ; Tz = ^tzi : i=1 i=1 h i Charge independent Hamiltonian: H^ ; T^ = 0 : 1 1 A M = (N − Z); (N − Z) ≤ T ≤ : T 2 2 2 Realistic situation Z 2 ^ X e mp ≈ mn; VCoulomb = j~ri −~rj j i<j=1 ^ ^ (T =0) ^ (T =1) ^ (T =2) VCoulomb = V + V + V : 2 E(T ; MT ) = a(T ) + b(T )MT + c(T )MT ; N. A. Smirnova The Nuclear Shell Model Two-particle wave function for identical fermions Angular-momentum coupled state (ja 6= jb) X (J) Φ (~r ;~r ) = (j m j m jJM)φ (~r )φ (~r ) = φ (~r ) × φ (~r ) ja(1)jb(2);JM 1 2 a a b b jama 1 jbmb 2 ja 1 jb 2 M mamb ja ≡ (nalaja), J = jja − jbj; jja − jbj + 1;:::; ja + jb, while M = −J; −J + 1;:::; J − 1; J. Not antisymmetric with respect to permutation of two identical fermions ! Normalized and antisymmetric state 1 X Φ (~r ;~r ) = (ja ma j m jJM) φ (~r )φ (~r ) − φ (~r )φ (~r ) jajb;JM 1 2 N b b jama 1 jbmb 2 jbmb 1 jama 2 mamb Since the Clebsch-Gordan coefficients have the following property: ja+jb−J (ja ma jb mb jJM) = (−1) (jb mb ja majJM) 1 X Φ (~r ;~r ) = (ja ma j m jJM)φ (~r )φ (~r ) jajb;JM 1 2 N b b jama 1 jbmb 2 mam b i −(− )ja+jb−J ( j )φ (~ )φ (~ ) 1 jb mb ja ma JM jbmb r1 jama r2 N.
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