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2 Outline 1 Introduction Nuclear magnetic moments Magnetic moments in non-relativistic approach Magnetic moments in relativistic approach Framework Relativistic mean-field theory Magnetic moments in relativistic approach One-pion exchange current Configuration mixing 3 Magnetic moments of odd-a nuclei with LS closed-shell core 4 Magnetic moments of odd-a nuclei with jj closed-shell core 5 Summary and Perspectives / 49

3 Outline Introduction 1 Introduction Nuclear magnetic moments Magnetic moments in non-relativistic approach Magnetic moments in relativistic approach Framework Relativistic mean-field theory Magnetic moments in relativistic approach One-pion exchange current Configuration mixing 3 Magnetic moments of odd-a nuclei with LS closed-shell core 4 Magnetic moments of odd-a nuclei with jj closed-shell core 5 Summary and Perspectives 3 / 49

4 Nuclear magnetic moments Introduction Nuclear magnetic moments Nuclear magnetic moments: one of basic nuclear properties, provides key information to understand nuclear structure. Nuclear magnetic moment of a state with spin I µ = IM ˆµ z IM M=I. (1) In extreme single-particle shell model The magnetic moment of an odd-a nucleus is determined by the magnetic moment of the unpaired nucleon in orbital j(n, l) µ = (nl) jm ˆµ z (nl) jm m=j. () 4 / 49

5 Schmidt magnetic moments Introduction Magnetic moments in non-relativistic approach Magnetic moment of a nucleon in a shell model orbital with spin j: (nls)jm ˆµ z (nls)jm m=j ˆµ = g lˆl + gs ŝ g l = 1(0), g s = 5.587( 3.86) are free orbital and spin g-factors of proton (neutron). = Schmidt magnetic moments { gl l + 1 µ S = g s, j = l + 1/ j j+1 [g l(l + 1) 1 g s], j = l 1/. (3) 1 N e u tr o n j= l-1 / P r o to n 6 5 j= l+ 1 / m -1 - j= l+ 1 / 1 / 3 / 5 / 7 / 9 / 1 1 / 1 3 / s p in j j= l-1 / 1 / 3 / 5 / 7 / 9 / s p in j m 5 / 49

6 Introduction Magnetic moments in non-relativistic approach Deviations from Schmidt magnetic moments Almost all nuclear magnetic moments are sandwiched between two Schmidt lines Blin-Stoyle, Rev. Mod. Phys. (1956). Neutron j=l 1/ Proton jl+1/ j=l+1/ j=l+1/ j=l 1/ To explain the deviations from Schmidt values in non-relativistic approach: Arima, Adv. Nucl. Phys. (1987), Towner, Phys. Rep. (1987) Meson exchange current (MEC): exchange of charged mesons. Configuration mixing (CM): correlations not included in independent-particle model. 6 / 49

7 Introduction Magnetic moments in non-relativistic approach Magnetic moments of odd-a nuclei with closed-shell core N e u tr o n P r o to n O 0 7 P b 3 9 C a j= l-1 / j= l+ 1 / 1 7 F 4 1 S c 0 9 B i m O C a P b j= l+ 1 / 1 / 3 / 5 / 7 / 9 / s p in j 0 7 T l 1 5 N 3 9 K j= l-1 / 1 / 3 / 5 / 7 / 9 / s p in j m Nuclei with LS closed-shell core 16 O, 40 Ca: CM(nd)+MEC. Nuclei with jj closed-shell core 08 Pb: CM(1st+nd)+MEC. CM (1st) Arima, Prog. Theor. Phys. (1954): j<=l-l 1 magic number j>=l Pb (M1-Giant 1 + ) h11/ h9/ protons i13/ i11/ neutrons 7 / 49

8 Introduction Isoscalar and isovector magnetic moments Magnetic moments in non-relativistic approach The nuclear magnetic moments can be divided into isoscalar part µ s and isovector part µ v For one pair of mirror nuclei µ = µ s + µ v τ 3 = Isoscalar magnetic moment: µ s = (µ p + µ n )/. Isovector magnetic moment: µ v = (µ p µ n )/. µ = 1 µ p(1 + τ 3 ) + 1 µ n(1 τ 3 ) (4) = 1 (µ p + µ n ) + 1 (µ p µ n )τ 3 (5) For nuclei with LS closed-shell core 16 O and 40 Ca Arima, Adv. Nucl. Phys. (1987), Towner, Phys. Rep. (1987) Isoscalar magnetic moments can be explained by nd. Isovector magnetic moments can be described by MEC and nd. 8 / 49

9 Introduction Magnetic moments in relativistic approach Magnetic moments in relativistic approach Relativistic mean-field (RMF) theory: great success in describing binding energy, radii, deformations etc. Ring, PPNP (1996), Vretenar, Phys. Rep. (005), Meng, PPNP (006). Magnetic moments of nuclei with LS closed-shell ±1 nucleon. Valence nucleon wavefunction cannot reproduce Schmidt values. Enhancement of Dirac magnetic moment small effective nucleon mass by scalar potential (σ meson) Miller, Ann. Phys. (1975), Serot, PLB (1981). Isoscalar magnetic moments: polarization by valence nucleon. Random-Phase Approximation (RPA): the renormalized current McNeil, PRC (1986), Ichii, PLB (1987), Shepard, PRC (1988), Self-consistent deformed RMF calculations with time-odd fields Hofmann, PLB (1988), Furnstahl, PRC (1989), Yao, PRC (006). Isovector magnetic moments: still large discrepancy. One-pion exchange current (MEC) corrections: significant but enlarge the disagreement with data Morse, PLB (1990). nd is also important? Magnetic moments of nuclei with jj closed-shell ±1 nucleon have not been reproduced. 9 / 49

10 Our goal Introduction Magnetic moments in relativistic approach Based on the magnetic moments from RMF theory with time-odd fields Including one pion exchange current 1st, nd corrections To study isoscalar and isovector magnetic moments of LS closed shell nuclei ±1 nucleon. magnetic moments of jj closed-shell nuclei ±1 nucleon. 10 / 49

11 Outline Framework 1 Introduction Nuclear magnetic moments Magnetic moments in non-relativistic approach Magnetic moments in relativistic approach Framework Relativistic mean-field theory Magnetic moments in relativistic approach One-pion exchange current Configuration mixing 3 Magnetic moments of odd-a nuclei with LS closed-shell core 4 Magnetic moments of odd-a nuclei with jj closed-shell core 5 Summary and Perspectives 11 / 49

12 Lagrangian Framework Relativistic mean-field theory Lagrangian density in point-coupling model Nikolaus, PRC (199); Büurvenich, PRC (00): L = L free + L 4f + L hot + L der + L em (6) L free = ψ(iγ µ µ m)ψ (7) L 4f = 1 α S( ψψ)( ψψ) 1 α V ( ψγ µ ψ)( ψγ µ ψ) 1 α TV ( ψ τγ µ ψ)( ψ τγ µ ψ) (8) L hot = 1 3 β S( ψψ) γ S( ψψ) γ V [( ψγ µ ψ)( ψγ µ ψ)] (9) L der = 1 δ S ν ( ψψ) ν ( ψψ) 1 δ V ν ( ψγ µ ψ) ν ( ψγ µ ψ) 1 δ TV ν ( ψ τγ µ ψ) ν ( ψ τγ µ ψ) (10) L em = 1 4 F µν F µν e 1 τ 3 ψγ µ ψa µ (11) 1 / 49

13 Framework Equations of motion for nucleons Relativistic mean-field theory Dirac equation {α [ i V(r)] + V 0 (r) + β[m + S(r)]}ψ i (r) = ε i ψ i (r), (1) with scalar potential S(r) and vector potential V µ (r) = (V 0, V). S(r) = α S ρ S + β S ρ S + γ Sρ 3 S + δ S ρ S V µ = α V j µ V + γ V (j µ V )3 + δ V j µ V + τ 3α TV j µ TV + τ 3δ TV j µ TV + e 1 τ 3 Aµ V (µ = 1,, 3): time-odd fields. ρ s = i n i ψi ψ i (13) j 0 V = i n i ψi γ 0 ψ i (14) j V = i n i ψi γ ψ i (15) j TV = i n i ψi γτ 3 ψ i (16) Time-odd fields vanish in even-even nuclei. exist in odd-a or odd-odd nuclei. 13 / 49

14 Framework Relativistic mean-field theory Time-reversal symmetry breaking for odd-a nuclei For odd-a nuclei, time reversal invariance is broken by the unpaired odd nucleon: polarization of the core due to the time-odd field V(r). Dirac equation with time-odd fields V(r) { iα α V(r) + V 0 (r) + β[m + S(r) ] }ψ i (r) = ε i ψ i (r), (17) with spin-dependent potential α V(r). Single-particle levels with time-odd fields: V(r) ε j = ε j ε j ε j V(r) =0 V(r) 0 The Kramers degeneracy is broken the core is polarized. 14 / 49

15 Framework Magnetic moments in relativistic approach Magnetic moments in relativistic approach Magnetic moment is defined in terms of Ampèrian currents µ = 1 dr [r J] z. (18) Electromagnetic current density J is given by J(r) = Qψ + (r)αψ(r) + κ M [ψ+ (r)βσψ(r)], (19) where the first term is Dirac current j D, and second term the so-called anomalous current j A. Covariant 4-current density, J µ (x) = Q ψ(x)γ µ ψ(x) + κ M ν[ ψ(x)σ µν ψ(x)], (0) where Q 1 (1 τ 3), κ p = 1.793µ N and κ n = 1.913µ N. Nuclear magnetic moment, µ = M dr 1 [r J(r)] z = dr[ Mc c Qψ+ (r)r αψ(r) + κψ + (r)βσψ(r)] }{{}}{{} z. (1) µ µ A D Here, and c are added to give the magnetic moment in units of nuclear magneton µ N. 15 / 49

16 Framework Corrections to magnetic moments Magnetic moments in relativistic approach The magnetic moment operator ˆµ = ˆµ free + ˆµ mec () The ground-state wave function with mixing of particle-hole (p-h) configurations: j = j + C 1p 1h j 1p-1h; j + C p h j p-h; j + (3) The magnetic moments including meson exchange current and configuration mixing. µ Total = j ˆµ free + ˆµ mec j = j ˆµ free j + j ˆµ mec j + j ˆµ free j j ˆµ free j = µ MF + µ MEC + µ CM µ MF (µ RMF ) deformed RMF theory with time-odd fields µ MEC : meson exchange current corrections j ˆµ mec (1π) j one-pion exchange current corrections µ CM : configuration mixing corrections Perturbation theory = first- and second-order CM (1st, nd) 16 / 49

17 Framework One-pion exchange current One-pion exchange current corrections to magnetic moments In Hartree approximation, pion field is zero. Exchange of virtual charged pions between two nucleons MEC corrections to magnetic moments. Morse, PLB (1990) γ π + π γ (a) (a) Seagull: a photon nucleon + pion at a single vertex. (b) In Flight: a photon pion being exchanged. µ MEC = 1 (b) dr r [ j seagull (r) + j in flight (r) ]. (4) 17 / 49

18 Framework Configuration mixing (1st, nd) Configuration mixing The ground-state wave function with mixing of particle-hole configurations, j = j + C 1p 1h j 1p-1h; j + C p h j p-h; j (5) In perturbation theory, the first-order corrections δµ cm 1st = j ˆµ j j ˆµ j = C 1p 1h [ j ˆµ j 1p-1h; j + j 1p-1h; j ˆµ j ]. (6) The second-order corrections Shimizu, NPA (1974): δµ cm nd = j V P P µ V j j µ j j V E j H 0 E j H 0 P: projection operator, 1p-1h 1p-1h, p-h p-h, etc. j and E j : unperturbed ground-state wavefunction and energy. V : two-body residual interaction. P (E j H 0 ) V j. 18 / 49

19 Numerical details Framework Configuration mixing Nuclei: LS closed-shell core: 15 O, 17 O, 15 N, 17 F, 39 Ca, 41 Ca, 39 K, 41 Sc jj closed-shell core: 09 Bi, 07 Tl, 09 Pb and 07 Pb µ total = µ RMF + µ MEC + µ cm 1st,nd. µ RMF : triaxial deformed RMF theory with time-odd fields with N f =N b =8. µ MEC : one-pion exchange current with spherical RMF theory. R box =15 fm, r step =0.1 fm, f π =1. µ 1st,nd : first-order and second-order configuration mixing corrections with spherical RMF theory. R box =15 fm, r step =0.1 fm. Configuration space, ε i < 0. Parameters: PC-F1. 19 / 49

20 Outline Magnetic moments of odd-a nuclei with LS closed-shell core 1 Introduction Nuclear magnetic moments Magnetic moments in non-relativistic approach Magnetic moments in relativistic approach Framework Relativistic mean-field theory Magnetic moments in relativistic approach One-pion exchange current Configuration mixing 3 Magnetic moments of odd-a nuclei with LS closed-shell core 4 Magnetic moments of odd-a nuclei with jj closed-shell core 5 Summary and Perspectives 0 / 49

21 Magnetic moments of odd-a nuclei with LS closed-shell core Isoscalar magnetic moments.0 S c h. A = 4 1 S c h.+ M E C S c h.+ M E C + C M E x p..0 A = 4 1 R M F R M F + M E C R M F + M E C + C M E x p A = 1 7 A = 1 7 m s [m N ] m s [m N ] A = 3 9 A = A = 1 5 A = Figure 1: Non-rel. results from Ref. [Arima1987]. 0.0 Figure : Present rel. calculations. With relatively small MEC and nd corrections, both the non-rel. and rel. results are in reasonable agreement with data. The relativistic descriptions of isoscalar magnetic moment become slightly worse with nd. 1 / 49

22 Magnetic moments of odd-a nuclei with LS closed-shell core Isovector magnetic moments S c h. S c h.+ M E C S c h.+ M E C + C M E x p R M F R M F + M E C R M F + M E C + C M E x p. A = A = A = 1 7 m v [m N ] 3.0 A = 1 7 m v [m N ] A = A = 1 5 Figure 3: Non-rel. results from Ref. [Arima1987]. 0.0 A = A = 1 5 Figure 4: Present rel. calculations. Non-rel. : the net effect of positive MEC and negative nd gives the right sign for the correction except A = 39. Rel. : the negative nd corrections improve the description of isovector magnetic moments, especially for A = 17 and A = 41. / 49

23 Outline Magnetic moments of odd-a nuclei with jj closed-shell core 1 Introduction Nuclear magnetic moments Magnetic moments in non-relativistic approach Magnetic moments in relativistic approach Framework Relativistic mean-field theory Magnetic moments in relativistic approach One-pion exchange current Configuration mixing 3 Magnetic moments of odd-a nuclei with LS closed-shell core 4 Magnetic moments of odd-a nuclei with jj closed-shell core 5 Summary and Perspectives 3 / 49

24 Magnetic moments of odd-a nuclei with jj closed-shell core First-order corrections Table 1: First-order (1st) corrections to magnetic moments of 09 Bi. Non-rel. Rel. Interactions KK Gillet KR. I KR. II Brueckner HJ Kuo M3Y PC-F1 DD-ME1 PC-F1* DD-ME1* Ref. [1] [] [3] with π (1h91h 1 11 ) π (1i111i 1 13 ) ν Total [1] Mavromatis, NP (1966). [] Mavromatis, NPA (1967). [3] Arima and Huang-Lin, PLB (197). The interaction matrix elements are taken from Bertsch and Schaeffer, NPA (1977). PC-F1 and DD-ME1 give small 1st corrections. When π is included, PC-F1* and DD-ME1* give considerable 1st corrections, which are in reasonable agreement with non-relativistic calculations. 4 / 49

25 Magnetic moments of odd-a nuclei with jj closed-shell core First-order corrections Relativistic calculations Without π, the 1st are negligible. With π, 1st corrections are in reasonable agreement with non-relativistic calculations for all present nuclei. 5 / A r im a P C -F 1 + p P C -F T l 0 7 P b m 1 s t (µ N ) B i 0 9 P b

26 Magnetic moments of odd-a nuclei with jj closed-shell core Second-order corrections A r im a P C -F 1 + p P C -F B i 0 7 T l 0 7 P b m n d (µ N ) P b With π, in general, the nd corrections are more close to non-relativistic results. 6 / 49

27 Magnetic moments of odd-a nuclei with jj closed-shell core Magnetic moments of 09 Bi, 07 Tl, 09 Pb and 07 Pb B i T l m (m N ) n d M E C 1 s t N o n -r e l. E x p. n d M E C 1 s t R M F w ith tim e -o d d fie ld s R e la tiv is tic. m (m N ) N o n -r e l. S c h m id t M E C 1 s t n d E x p. n d R e la tiv is tic. M E C 1 s t R M F w ith tim e -o d d fie ld s.5 S c h m id t P b P b R M F w ith tim e -o d d fie ld s 1 s t 0.8 M E C m (m N ) s t M E C n d S c h m id t N o n -r e l. E x p. n d M E C 1 s t R e la tiv is tic. R M F w ith tim e -o d d fie ld s m (m N ) S c h m id t 1 s t n d M E C N o n -r e l. E x p. n d R e la tiv is tic. In the 1st and nd, the residual interaction provided by π is included. Magnetic moments of all four nuclei are greatly improved by including 1st, MEC and nd, in agreement with non-relativistic results. 7 / 49

28 Outline Summary and Perspectives 1 Introduction Nuclear magnetic moments Magnetic moments in non-relativistic approach Magnetic moments in relativistic approach Framework Relativistic mean-field theory Magnetic moments in relativistic approach One-pion exchange current Configuration mixing 3 Magnetic moments of odd-a nuclei with LS closed-shell core 4 Magnetic moments of odd-a nuclei with jj closed-shell core 5 Summary and Perspectives 8 / 49

29 Summary and Perspectives Summary and Perspectives Summary Based on the magnetic moments from RMF theory with time-odd fields, the one-pion exchange current, 1st and nd corrections are taken into account to study the magnetic moments of nuclei with one valence nucleon (or hole) outside a closed-shell core. For nuclei with LS closed-shell core 16 O and 40 Ca: ➀ isoscalar moments are reasonably reproduced. ➁ isovector moments are reproduced in full relativistic theory for the first time. For nuclei with jj closed-shell core 08 Pb: ➀ magnetic moments are well reproduced. ➁ π is important. 9 / 49

30 Summary and Perspectives Summary and Perspectives Summary Based on the magnetic moments from RMF theory with time-odd fields, the one-pion exchange current, 1st and nd corrections are taken into account to study the magnetic moments of nuclei with one valence nucleon (or hole) outside a closed-shell core. For nuclei with LS closed-shell core 16 O and 40 Ca: ➀ isoscalar moments are reasonably reproduced. ➁ isovector moments are reproduced in full relativistic theory for the first time. For nuclei with jj closed-shell core 08 Pb: ➀ magnetic moments are well reproduced. ➁ π is important. Perspectives Other nuclei with closed-shell core: 13 Sn Dirac sea Relativistic Hartree-Fock theory: Fock term, tensor force effects. MEC CM, RPA type CM, isobar current / 49

31 Summary and Perspectives ÜŠöµ Š # Ç ŒÆ ô² ÜHŒÆ Ê ÊUŒÆ a3œˆ P!ÓÆ 30 / 49

32 Summary and Perspectives ÜŠöµ Š # Ç ŒÆ ô² ÜHŒÆ Ê ÊUŒÆ a3œˆ P!ÓÆ 30 / 49

33 appendix Magnetic moments in other non-relativistic nuclear models The above single-particle shell model calculations of corrections to magnetic moments were restricted to the nuclei with one valence nucleon (or hole) outside a closed-shell core. However for nuclei with open shell core, large deformation, far from β-stability line, etc. Shell model Caurier, Rev. Mod. Phys. (005): 55 Ni, 57,59,71,73,75 Cu, 7,9,31,33 Mg, 31,33 Al, etc., connected with hot topic as magic number, island of inversion, shell evolution Neyens, Phys. Rev. Lett. (005), Yordanov, Phys. Rev. Lett. (006)... Landau-Migdal theory for finite Fermi system Migdal, (1967): Systematic calculations for more than 100 odd spherical nuclei and long isotopic chains Cu, Sn, Pb good agreement with experimental data. Borzov, Eur. Phys. J. A (010) Particle plus phonon model Hamamoto, Phy. Lett. B (1976): The correction from the admixture of states featuring a virtually excited low-lying collective state (phonon). Collective models Bohr and Mottelson, (1975): particle rotor model and interacting boson model, etc. The nuclear magnetic moments of deformed nuclei from collective and single-particle motion / 49

34 Magnetic moment operator appendix Def. I: Magnetic pole Magnetic multipole operator of order λ M λ = e Mc (r λ P λ ) [g l λ + 1 l + g ss]. (7) Magnetic dipole moment operator (λ = 1), i.e., the magnetic moment operator The magnetic moment ˆµ = g l l z + g s s z. (8) µ = g l l z + g s s z. (9) Def. II: Ampèrian current Magnetic moment defined in terms of Ampèrian currents µ = 1 dr[r J] z, (30) where J is the current density vector. µ r J 3 / 49

35 appendix Dirac and anomalous magnetic moment The Dirac magnetic moment, corresponding to tree-level Feynman diagrams (classical result), calculated from the Dirac theory. In quantum electrodynamics, the anomalous magnetic moment of a particle is a contribution of effects of quantum mechanics, expressed by Feynman diagrams with loops. = + + Tree Loop Vertex Function Γ µ γ µ κ iσµν m q ν E.M. current J µ ψγ µ ψ (Dirac) κ ψ iσµν m q νψ (Anomalous) µ = 1 d r r J l + s (gs = ) κ s Point particle Internal structure 33 / 49

36 appendix The enhanced Dirac magnetic moment in RMF theory Relativistic valence nucleon magnetic moments Schmidt values. Dirac current = orbital (convection) current + spin current Gordon identity, j D = Q ψ(r)γψ(r) = Q M ψ(r)pψ(r) + Q M [ψ+ (r)βσψ(r)], (31) where M = M + S 0.6M is the effective (scalar) mass of nucleon. Thus, µ = µ D + µ A = Q dr M M ψ[l + Σ]ψ + drκ ψσψ (3) }{{ }}{{} µ D µ A Magnetic moment operator defined using magnetic poles in non-relativistic theory µ = g l l + g s s = g l l + g s σ = g l(l + σ) + ( g s }{{} g l)σ. (33) µ D }{{} µ A µ D (rel.) µ D (non-rel.) M = M (free motion) µ D (rel.) > µ D (non-rel.) M > M (large scalar potential) enhanced Dirac magnetic moment Polarization effect by valence nucleon deformed RMF theory with time-odd fields 34 / 49

37 appendix How to construct the one-pion exchange current? The Lagrangian for a system of nucleons and pions is a sum of three terms L = L ππ + L NN + L πnn, (34) where L πnn is the interaction Lagrangian describing the emission and absorption of a pion by a nucleon. γ π + π γ π L πnnγ L ππγ L πnn L ππ By the minimal substitution, four-point vertex involving the nucleon, pion, and photon fields, and photon-pion vertex can be obtained. L πnn L πnnγ = j µ A µ seagull (35a) L ππ L ππγ = j µ A µ in-flight (35b) 35 / 49

38 appendix One-pion exchange current correction to magnetic moments The space-like component of seagull current j µ seagull is j seagull (r) = eg πnn M [ ψ p (r)γγ 5 ψ n (r) ψ n (r)γγ 5 ψ p (r) The space-like component of in-flight current j µ in flight is j in flight (r) = ieg πnn M dyd π (r, y) ψ n (y) M M γ 5ψ p (y) dyd π (r, y) ψ p (y) M M γ 5ψ n (y)]. (36) dxdyd π (r, x) ψ p (x)m γ 5 ψ n (x) r D π (r, y) ψ n (y)m γ 5 ψ p (y). (37) The one-pion exchange current correction to magnetic moments µ MEC = 1 dr {r [j seagull (r) + j in flight (r)]} z, = eg πnn dr M ψ p (r)[ r γ ] z γ 5 ψ n (r) dyd π (r, y) ψ n (y) M M γ 5ψ p (y) iegπnn drdxdy drdxdy ψ p (x) M M γ 5ψ n (x)d π (x, r)[ r r ] z D π (r, y) ψ n (y) M M γ 5ψ p (y). (38a) 36 / 49

39 appendix Rayleigh-Schrödinger perturbation theory H 0 = i (T i + U i ), V = i>j V ij i U i is the residual interaction known as the perturbation and φ n is the complementary projection operator. H = H 0 + V H 0 n (0) = E n (0) n (0) (H 0 + V ) n = E n n φ n = 1 n (0) n (0). (39) Using perturbation expansion, up to second order, n = n (0) + φ n E (0) n H 0 V n (0) + As the perturbed ket n is not normalized, E (0) n φ n H 0 V φ n E (0) n H 0 V n (0). (40) n n = n (0) n (0) + n (0) V E (0) Finally the corresponding core polarization correction becomes φ n φ n V n (0) + 0(V 3 ) = 1 + α. (41) n H 0 E n (0) H 0 δµ nd mn = m µ n m (0) µ n (0) α m + α n m (0) µ n (0). (4) 37 / 49

40 First-order core polarization appendix The ground-state wave function with mixing of 1p-1h configurations, j = j + α j 1p-1h; j. (43) In perturbation theory, the first-order corrections δµ 1cp = j ˆµ j j ˆµ j = [α j ˆµ j 1p-1h; j + α j 1p-1h; j ˆµ j ]. (44) ˆμ V j ˆμ p h + + j V j p h ˆμ j ˆμ j + C 1p 1h j ˆμ j ph 1 (1 + ); j + C 1p 1h j ph 1 (1 + ); j ˆμ j } {{ } δμ 1cm j is valence particle state, and V residual interaction. δµ 1st cp = { j h µ j p ( 1) jh+j+j ĵ 1 j jh j (J +1) p 1 E j j + 1 j j J j p j h J } jj p ; JM V jj h ; JM. (45) For LS closed-shell nuclei ±1 nucleon, the 1st doesn t contribute. 38 / 49

41 appendix Second-order Corrections for nuclei with one particle (a) (b) N(p-1h) = N(3p-h) = S(p-1h) = S(3p-h) = C(p-1h) = C(3p-h) = Figure 5: Diagrams representation of second-order corrections for nuclei with one particle: (a), p 1h, (b), 3p h. N(p-1h) = C jj jj10 j µ j j + 1 j 1 j j h,j 1 (J + 1) Ej (j + 1) jj h, J V j 1 j, J, (46) where j is the valence particle state, j 1, j, and h corresponding to the p-1h intermediate state, and V is the residual interaction. 39 / 49

42 appendix Second-order Corrections for nuclei with one particle C jj jj10 S(p-1h) = j + 1 j 1 j,j h j h,j ( 1) j h+j+j (J + 1) { jh j h 1 j j J 1 jj h, J V j 1 j, J j 1 j, J V jj h, J j h µ j h (47) E E } C(p-1h) = C jj jj10 j + 1 ( 1) j1+j+jh+j (J + 1)(J + 1) j 1 j,j 1 j h JJ { } { } J 1 J J 1 J j 1 j j 1 j j h j jj h, J V j 1 j, J j 1j, J V jj h, J j 1 µ j 1 1 (48) E E 40 / 49

43 appendix Second-order Corrections for nuclei with one particle Formulas for 3p-h state are obtained by interchanging p and h in formulas for p-1h. C jj jj10 N(3p-h) = j µ j j + 1 C jj jj10 S(3p-h) = j + 1 j 1 j,j h1 j h,j j h1 j h j 1,J 1 (J + 1) Ej (j + 1) jj 1, J V j h1 j h, J, (49) { } 1 ( 1)j1+J+j j1 j (J + 1) 1 j j J 1 jj 1, J V j h1 j h, J j h1 j h, J V jj, J j 1 µ j E E, (50) C(3p-h) = C jj jj10 ( 1) jh1 +j h +j 1 +j (J + 1)(J + 1) j + 1 j h1 j h,j h j 1 JJ 1 1 { } { } J 1 J J 1 J E E j h 1 j h j h1 j j 1 j jj 1, J V j h1 j h, J j h 1 j h, J V jj 1, J j h1 µ j h 1. (51) 41 / 49

44 appendix Second-order Corrections for nuclei with one hole (a) (b) N(h-1p) = N(3h-p) = S(h-1p) = S(3h-p) = C(h-1p) = C(3h-p) = Figure 6: Second-order Corrections for nuclei with one hole: (a), h-1p, (b), 3h-p. N(h-1p) = C j hj h j h j h 10 jh + 1 j h µ j h j h1 j h j,j 1 (J + 1) Ej (j + 1) j hj, J V j h1 j h, J. (5) where j h is the valence hole state, j h1, j h, and j corresponding to the h-1p state. The expression of their contributions for N(h-1p) can be obtained by interchanging p and h in the equations for N(p-1h) etc. 4 / 49

45 appendix Residual interaction in the point-coupling RMF model Finite-range Zero-range S V TV J T)=(0 0) The residual interaction is self-consistent J T)=(1 0) J T)=(1 1) J=0, T=0 J=1, T=0 J=1, T=1 V αβα β = The corresponding interaction matrix elements V ABab δ E(ˆρ) δ ˆρ αβ δ ˆρ α β. (53) = AB V ab, VABab S = drψ + A γ 0ψ a (α S + β S ρ S + 3γ S ρ S + δ S )ψ + B γ 0ψ b, VABab V = drψ + A ψ a[α V + 3γ V ρ V + δ V ]ψ + B ψ b drψ + A αψ a[α V + γ V ρ V + δ V ] ψ + B αψ b, VABab TV = drψ + A γ 0 τγ µ ψ a (α TV + δ TV ) ψ + B γ 0 τγ µ ψ b, VABab C e 1 τ 3 = dr 1 dr r 1 r {[ψ+ A ψ a ] 1 [ψ + 1 τ 3 B ψ b ] [ψ + 1 τ 3 A αψ a ] 1 [ψ + B VABab π = dr 1 dr ψ + A (1)ψ+ B ()[ f π τγ 0 γ 5 γ k k ] 1 [ f π τγ 0 γ 5 γ l l ] D π (1, )ψ b ()ψ a (1) m π m π + dr 1 dr ψ + A (1)ψ+ B ()1 3 [ f π τγ 0 γ 5 γ] 1 [ f π τγ 0 γ 5 γ] δ(r 1 r )ψ b ()ψ a (1). m π m π (54a) (54b) (54c) 1 τ 3 αψ b ] }, (54d) (54e) 43 / 49

46 appendix The total magnetic moments m (m N ) S c 1 7 F 3 9 C a 1 5 O 3 9 K 1 5 N 1 7 O 4 1 C a R M F R M F + M E C R M F + M E C + C P E x p. MEC: positive corrections for odd proton nuclei and negative corrections for odd neutron nuclei. CP: negative corrections for odd proton nuclei and positive corrections for odd neutron nuclei, except 15 O and 39 Ca. The descriptions of magnetic moments for most nuclei are greatly improved with both MEC and CP, with the exception of 15 O and 39 Ca. 44 / 49

47 First-order core polarization appendix Table : The reduced matrix elements j h µ j p and excitation energies E for configurations (1h9 1h 1 11 ) π and (1i11 1i 1 13 ) ν obtained from non-relativistic and relativistic calculations. PC-F1 effective interaction is used for the relativistic calculations. Excitation (1h9 1h i 1 13 ) ν Non-rel. PC-F1 ( ) Non-rel. PC-F1 ( ) ) π (1i11 j h µ j p (µ N ) E (MeV) Although there are some differences for the matrix elements and excitation energies, these won t give large difference for the 1cp between Non-rel. and PC-F1 in Table / 49

48 First-order core polarization appendix Table 3: The M3Y and PC-F1 interaction matrix elements, which are responsible for the first-order core polarization in 09 Bi. π1h9π1h9; J V π1h9π1h11; J π1h9ν1i11; J V π1h9ν1i13; J J M3Y PC-F1 PC-F1* M3Y PC-F1 PC-F1* The differences between M3Y and PC-F1 interaction matrix elements π1h9π1h9; J V π1h9π1h11; J are responsible for the large difference of 1cp between Non-rel. and PC-F1 in Table / 49

49 First-order core polarization appendix Table 4: The M3Y and PC-F1 interaction matrix elements, which are responsible for the first-order core polarization in 09 Bi. π1h9 π1h9 ; J V π1h9 π1h11 ; J J M3Y V S V V V V TV V TV V V C V π V δπ V Total / 49

50 First-order core polarization appendix Table 5: The M3Y and PC-F1 interaction matrix elements, which are responsible for the first-order core polarization in 09 Bi. ν1h9ν1h9; J V ν1h9ν1h11; J ν1h9π1i11; J V ν1h9π1i13; J J M3Y PC-F1 PC-F1* M3Y PC-F1 PC-F1* / 49

51 appendix Particle-particle and particle-hole interaction The Pandya particle-hole transformation Pandya, Phys. Rev. (1956) j 1 j4 1 ; J V j 3j 1 ; J = { } ˆK j1 j K j j 3 j 4 J 1 j ; K V j 3 j 4 ; K. (55) K 49 / 49