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Time reversal symmetry broken triaxial relativistic mean field approach for magnetic moment and nuclear current in odd mass nuclei 孟 杰 北京大学物理学院 兰州重离子加速器国家实验室 中国科学院理论物理研究所 Jie Meng School of Physics/Peking University HIRFL/Lanzhou Institute for Theor.Phys./AS Contents Introduction Triaxial RMF with time-odd component Numerical details Results and discussion 1. Single particle energy 2. density distribution 3. Magnetic moment 4. Nuclear current Summary Introduction Magnetic moments are measured with high precision. Traditionally it provided a sensitive test for nuclear models. Blin-Stoyle R J 1957 Theories of Nuclear Moments (Oxford: Oxford University Press). Wilkinson D H and Rho M (Eds.) 1979 Mesons in Nuclei vol I1 (Amsterdam: North-Holland ). Arima A 1984 Prog. Part. Nucl. Phys. 11 53 Because the single particle state can couple to more complicated 2p-1h configurations and there are mesons exchange corrections caused by the nuclear medium effect, the configuration mixing provide a better foundation to describe the observed values . The mean field may not be expected to describe the magnetic moment well. Arima A Horie H 1954 Prog. Theor. Phys. 11 509 Arima A, Shimizu K, Bentz W and Hyuga H 1988 Adu. Nucl. Phys. 18 1. Introduction However, it should be appropriate for the isao-scalar magnetic moment in LS closed shell nuclei plus or minus one nucleon, as 1. LS-closure, no spin-orbit partners on both sides of the Fermi surface, therefore the magnetic moment operator can not couple to magnetic resonance. 2. Pion-exchange current contribution turned to be very small to iso-scalar current, as well as others processes. Although relativistic mean field approach has achieved great success during the last two decades: Serot & Walecka, Adv. Nucl. Phys. 16 (86) 1 Reinhard, Rep. Prog. Phys. 52 (89) 439 Ring, Prog. Part. Nucl. Phys. 37 (96) 193 Meng, Toki, Zhou, Zhang, Long & Geng, Prog. Part. Nucl. Phys. 2006, in press Straightforward application of the single-particle relativistic model does not agree with the experimental magnetic moments Introduction The Sigma and the time-component vector mesons of Omega fails to reproduce the corresponding Schmidt values: H.Ohtsubo, et. al., Prog. Theor. Phys. 49(1973 ) 877 Miller L D, Ann. Phys., NY 91 (1975) 40. Bawin M, Hughes C A and Strobel G L Phys. Reu. C 28 (1983) 456. Bouyssy A, Marcos S and Mathiot J F Nucl. Phys. A 415 (1984) 497. Kurasawa H., et. al., Phys.Lett.B165 (1985) 234 Taking into account the contribution of the back-flow to the current operator can solve this problem. This back-flow is caused by the polarization of the core by the external particle. H. Kurasawa, et. al., Phys.Lett.B165(1985)234 J. A. McNeil, et. Al., Phys. Rev. C34(1986)746 S. Ichii, W. Bentz and A. Arima, Phys. Lett. B 192(1987)11. J. R. Shepard, et al., Phys.Rev.C37(1988)1130 P. G. Blunden, Nucl. Phys. A 464 (1987)525 Introduction In these the widely investigated mean field theories there are only the time-even fields which are most sensitive to physical observables. The time-odd fields, which appear only in the nuclear systems with time-reversal symmetry broken, are very important for the description of the magnetic moments, rotating nuclei, N=Z nuclei, and pairing correlations. U. Hofmann and P. Ring, Phys. Lett. B 214, 307(1988) . J. Koenig, and P. Ring, Phys. Rev. Lett. 71, 3079 (1993). W. Satuła, in Nuclear Structure 98, edited by C. Baktash, AIP Conf. Proc. No. 481 ~AIP, Woodbury, NY, 1999!, p. 114. K. Rutz, M. Bender, P.-G. Reinhard, and J. A. Maruhn, Phys.Lett. B 468, 1 (1999) The broken time reversal symmetry a non-vanishing vector part of the ω-field a magnetic potential and changes the nuclear wave function and the resulting magnetic moments. The magnetic field created by magnetic potential will influence the magnetic moment, single-particle spin and angular momentum. Introduction The core polarization is always neglected in Spherical cases, For the axial deformed case, the RMF with time-odd components are developed and the isoscalar magnetic moment are well reproduced: U. Hofmann and P. Ring, Phys. Lett. B 214, 307(1988) . R. J. Furnstahl, C. E. Price, Phys. Rev. C40 (1989) 1398. Time-even triaxial RMF have been developed to investigate the triaxial deformation and MD D. Hirata, et al.., Nucl. Phys. A609, 131 (1996). J. Meng, et al., Phys. Rev. C 2006 Purpose: developing the time reversal symmetry broken triaxial RMF approach, investigating the non-vanishing vector part of the ω-field, nuclear current,, magnetic potential and magnetic moments Starting point of RMF theory Nucleons are coupled by exchange of mesons via an effective Lagrangian s (Jp T)=(0+0) Sigma-meson: attractive scalar field S(r) gss(r) r w (Jp T)=(1-0) (Jp T)=(1-1) Omega-meson: Short-range repulsive Rho-meson: Isovector field 1 - 3 0 V( r ) gww ( r ) + gr 3r ( r ) + e A (r) 2 Serot & Walecka, Adv. Nucl. Phys. 16 (86) 1 Ring, Prog. Part. Nucl. Phys. 37 (96) 193 0 0 Reinhard, Rep. Prog. Phys. 52 (89) 439 Meng, Toki, Zhou, Zhang, Long & Geng, Prog. Part. Nucl. Phys. 2005, in press Lagrangian of RMF theory 1 - 3 L i i - M - gs s - gw w - gr r - e A i 2 1 1 + s s - U(s) - + U(w) 2 4 1 1 1 1 1 - RR + U(r) - FF U (s ms2s 2 + g 2s 3 + g 3s 4 2 3 4 4 4 2 1 1 U (w mw2w w + c3 (w w p Jp 0- T 1 s 0+ 0 w r 11- 0 1 meson 2 4 2 1 1 U ( r mr2 r r + d 3 ( r r 2 4 w - w R r - r - gr (r r F A - A Equations of Motion ( p - V ( r ) + V( r ) + M + S( r ) i ii Magnetic field V( r) g w0 ( r) + g r0 ( r) + e 1 - 3 A0 ( r) Same footing for w r 3 2 Deformation 1 - 3 Rotation A( r) V( r) gw ω( r) + gr 3 ρ( r) + e 2 Pairing (RHB,BCS,SLAP) Magnetic potential S(r) gs s( r) … -space-like components of vector mesons Nuclear magnetism -behaves in Dirac equation like a magnetic field ( + ms2 )s - gsr s - g2 s2 - g3 s3 μ ( + mw2 ) w gw jv - c3 w w w μ 2 ( + m ) g j r r r R A ( r) ejP r s ( r) A i ( r)i ( r) i1 + j ( r) A ( r ) i ( r) v i1 i A + j ( r ) ( r ) i ( r) R i1 i 1- A + 3 j ( r ) ( r ) i ( r) c i1 i 2 Numerical techniques for time reversal invariance violation Nucleon 1 (r , s) nx ( x)ny ( y )nz (Tz ) (r , s) nx +- 1 (r , s) 2 ( -1) T in y(r , s) nx + ny +1 (r , s1) (r , s) n x ( x)n y ( y )n z ( z ) ( -1) nx 2 ( 1) T -is K i Expanded on 3D HO Basis nf j f ( r , s ) (t ) j ( r , s, t ) n g t j ig ( r , s ) nf f j ( r , s, t ) ng ig y T j (r , s, t ) - j (r , s, t ) ( r , s ) (t ) t j ( r , s ) j ny T j (r , s, t ) j ( r , s, t ) Dirac equation meson ( x) n ( x)n ( y )n ( z ) x y z where s , w , r Coulomb field: the standard Green function method Nuclear Magnetic potential: vector part of the ω-field Nuclear Magnetic Fields due to the vector part of the ω-field Magnetic field B = w at y=z=1.29 fm Single nucleon levels with time reversal invariance violation Single nucleon levels with time reversal invariance violation Density distribution of the last odd nucleon The other degree of freedom was integrated. Density distribution for proton, neutron and matter Magnetic Moment in Relativistic approach 2Mc 2 1 m ( r ) + c 2 Magnetic moment A i 1 f (r , s) ( r , s, t ) t (t ) ig (r , s) nucleon wave function i i Relativistic effect 2Mc2 1 + + d r i (r)r i (r) + i (r)i (r) hc 2 3 A A 2Mc2 1 D d r r ji (r) + jiA (r) i 1 i 1 hc 2 3 D A Dirac current j (r ) + (r ) (r ) Anomalous current j (r ) + (r ) (r ) Magnetic Moment magnetic moments of light nuclei near closed shells (N) 15O 17O 39Ca 41Ca 15N 17F 39K 41S Exp. 0.72 -1.89 1.02 -1.60 -0.28 4.72 0.39 5.43 Schmidt 0.64 -1.91 1.15 -1.91 -0.26 4.79 0.12 5.79 Spher. 0.66 -1.91 1.17 -1.91 -0.03 5.05 0.72 6.32 Axial 0.65 -2.03 0.96 -2.13 -0.29 4.99 0.33 6.07 Triaxial 0.57 -2.00 0.98 2.13 0.19 4.89 0.37 6.04 Spherical and axial RMF results with NL1 taken from Hofmann 1989 Triaxial RMF with PK1 Magnetic Moment Iso-scalar magnetic moment (N) (Z, N + (Z + 1, N - 1/ 2 s A Schmidt Landau Spher. Axial RHA Triaxial Exp. 15 0.19 0.19 0.32 0.18 0.20 0.19 0.22 17 1.44 1.41 1.57 1.48 1.44 1.45 1.41 39 0.64 0.64 0.94 0.64 0.66 0.67 0.71 41 1.94 1.91 2.21 1.97 1.95 1.96 1.92 Landau: taking into account the current by linear response theory Magnetic Moment Iso-vector magnetic moment (N) (Z, N - (Z + 1, N - 1/ 2 V A Schmidt s-w+config. Spher mixing Triaxial Exp. 15 0.451 0.357 0.345 0.376 0.501 17 -3.353 -3.487 -3.480 -3.446 -3.303 39 0.512 0.217 0.225 0.305 0.312 41 -3.853 -4.141 -4.115 -4.086 -3.513 s-w including config. mixing: Y Nedjadi and J R Rook, J. Phys. G: Nucl. Part. Phys. 15 (1989) 589 U. Hofmann, P. Ring, Phys. Lett. B214 (1988) 307 Dirac and Anomalous parts of Magnetic Moment (N) D 15O 17O 39Ca Schmidt - - - Spher. - - 41Ca 15N 17F 39K 41S - 0.33 3.00 1.20 4.00 - - 0.59 3.26 1.81 4.54 Axial -0.13 -0.13 -0.30 -0.22 0.44 3.21 1.50 4.29 Triaxial -0.11 -0.13 -0.16 -0.28 0.46 3.15 0.64 4.31 A 15O 17O 39Ca 41Ca 15N 17F 39K 41S Schmidt 0.64 -1.91 1.15 -1.91 -0.60 1.79 -1.08 1.79 Spher. 0.66 -1.91 1.17 -1.91 -0.62 1.79 -1.09 1.79 Axial 0.78 -1.90 1.26 -1.87 -0.73 1.78 -1.17 1.79 Triaxial 0.68 -1.86 1.13 -1.85 -0.64 1.75 -1.00 1.73 Nuclear current in 17F and 17O in y-z plane Dirac current Anomalous nuclear current in 17F and 17O in y-z plane Dirac and anomalous current in 17F Dirac current Anomalous current Summary and perspective Triaxial RMF without time reversal symmetry is developed Ground-state properties of light odd mass nuclei near double-closed shells, i.e., E/A, single-particle energy, density distribution, etc., are calculated self-consistently The broken time reversal symmetry leads to a non-vanishing vector part of the ω-field, which creates a magnetic potential and changes the nuclear wave function and the resulting magnetic moments. The first calculated nuclear magnetic moments of light LS- closed shells nuclei plus or minus one nucleon agree well with the Schmidt values and the data. 奇核子系统问题-时间反演对称性破缺 正确确定激态及价核子组态-绝热与非绝热约束计算 constraints MD 奇核子系统问题-时间反演对称性破缺 s.p. levels in 106Rh 奇核子系统问题-时间反演对称性破缺 和 40Ca 的中子(左)和 质子(右)的单粒子能级 41Ca 考虑磁势后41Ca 中互为时 间反演态的能级劈裂 How to define the single-particle property in dense, strongly interacting many-body system? Laudau and Migdal answer: The responds of the system as a whole when a quasi-particle is removed. Thus, the single quasi-particle current is defined as the difference in the total baryon current when the particle is removed. ji JB , ni J B n j j j j j nj Ej ( k j - gw J B ) Relativistic extension of Landau’s Fermi-liquid theory based on sigma-omega model J. A. McNeil, et. Al., Phys. Rev. C34(1986)746 s -w Nucleon: model Meson fields: Total current w g 2w / m 2w Self-consistent Dirac equation: m* Landau quasiparticle current: Enhancement is reduced Especially T=0 K: j i |T 0 0.6m Backflow effect Renormalization of current k 2i -1 k i [m + w r0 - s r s + ] r0 + , rs 2m Remark: The cancellation of the scalar enhancement due to the vector meson Spin ½ particle One-body matrix element of current -ig Vector fields Vertex correction g g Renormalized current Transfer momentum Electric form factor magnetic form factor q p '- p 0 Remark: Dirac current is related to the electric form factor! Relativistic extension of Landau’s Fermi-liquid theory J. A. McNeil, et. Al., Phys. Rev. C34(1986)746 The relativistic wave functions are obtained from a relativistic Woods-Saxon well with parameters adjusted to give the separation energy and elastic electron scattering form factor. The interaction vertex is renormalized by consideration of backflow effect in nuclear medium, namely, 3 2 R {1 + w r0 [( p 2 r0 )3/ 2 + (m*) 2 ]-1/ 2 }-1 Effective (renormalized) Dirac current j D R (r ) (r ) R (r ) Remark1: the wave function and the interaction vertex are not consistent! Remark2: The anomalous current is not renormalized in this paper. Remark3: The renormalization are considered without the consideration of iso-vector meson fields, i.e., rho and pi, thus the iso-vector current and magnetic moment are still enhanced. Even if rho meson is considered, the enhancement of iso-vector current still can not be reduced significantly because of the small rho-N coupling constant. P. G. Blunden, Nucl. Phys. A 464 (1987)525 Comment: mw mr r / w 1/ 4 - 1/ 9 Remark4: In additional, the anomalous iso-vector moment, which is much larger that the Dirac moment, does not get affected by the scalar field, so that the total iso-vector spin moment will not be enhanced much. Magnetic Moment in Relativistic approach Dirac current D j (r ) + (r ) (r ) can be decomposed into an orbital current and a spin current 1 (r ) (r ) * (r )[( p '+ p) - is q ] (r ) 2m spin convection Anomalous current A j (r ) + (r ) (r ) Iso-scalar and iso-vector magnetic moment 1 Mc 2 s (r ) + (0) 2 c 3 Mc 2 v - [ (r ) + (1) ] 2 c (0) -0.12 N (1) 3.706 N 3 +1, n 3 -1, p Introduction Extensive shell model calculations within the full Ohw shell-model space show good agreement between theoretical and observed values. The remaining deviations arising from higher order corrections, i.e. meson exchange currents, isobar currents and higher-order configuration mixing, are removed through the use of effective operators to be determined empirically: Arima A, Shimizu K, Bentz W and Hyuga H 1988 Adu. Nucl. Phys. 18 1. Brown B A and Wildenthal B H 1983 Phys. Reu. C 28 2397. Relativistic s-w model + the configuration mixing within one major shell for the mirror pairs, 150-15N, 170-17F, 39K-39Ca and 41Ca-41S, removes most of the discrepancies for isovector moments while leaving the isoscalar moments unaltered, i.e. also in agreement with experiment when vertex corrections are included. For isovector moments, this agreement is better than in similar non-relativistic calculations: Y Nedjadi and J R Rook, J. Phys. G: Nucl. Part. Phys. 15 (1989) 589-600. 奇核子系统问题-时间反演对称性破缺 奇核子系统: 未配对核子破坏时间反演对称性, 从而导致矢量介 子场的空间部分不为零, Dirac 方程中出现磁势 球对称: 奇A核处理成偶偶核额外加入一个核子, 体系核子波函 数仍具有球对称性. 无法考虑时间反演对称性破缺对整个原子核 的影响 轴对称: Hofmann等人(88) 和 Furnstahl等人(89)自洽地考虑了 磁势, 研究了核芯极化效应对整个原子核性质的影响. 这种核芯 极化效应能抵消标量场引起的相对论效应对同位旋标量磁矩的 增强, 给出与 Scnmidt 值一致的原子核磁矩. 本工作: 三轴形变框架下研究时间反演对称性破缺