Theoretical study on nuclear structure by the electron-nucleus scattering Jian Liu (刘健) China University of Petroleum (East China) 2015. 7. 9 1 Content Introduction Relativistic mean-field model (RMF) Parity-violating electron scattering Electron scattering off deformed nuclei Summary and prospect Sec. I Introduction The observing capability of microscope is connected with the wave length of its probe. Optical microscope cellular scale (µm) visible light 0.5 µm Electron microscope atomic scale(nm) electrical energy 100keV wavelength 0.004nm 1.1 Elastic electron scattering Effect method to measure at the nuclear scale: High energy electron scattering Experimental equipment of electron scattering Scattering cross section of electron (Hofstadter 1954 ) Theoretical development on electron scattering off nuclei • 1911 Rutherford model of atom and Rutherford scattering formula • 1929 Mott theory: electron scattering off point nuclei • 1940s Guth, Rose: study the effect of nuclear size on electron scattering with the PWBA method. • 1950s Hofstadter: experiments of high energy electron scattering off nuclei • 1950s elastic electron scattering theories Eikonal approximation Phase-shift analysis method (DWBA) • 1960s-1980s development’s peak on the electron scattering experiments Current situation of high energy electron scattering With the development of radioactive ion beams, we could produce the exotic nuclei with short lives in the laboratory and investigate its structure now. With the new discovery of the nuclear properties, such as the halo nuclei, it raises the new questions about the nuclear structure. Advantages of electron scattering: The interaction between electron and nucleus is well understood. Large scale facilities of electron scattering under construction: 132 Sn • RIKEN of Japan • GSI of Germany Experimental research progress on electron scattering RIKEN Target project RIKEN Verified experiment 1.2 Parity-violating electron scattering Z0 is a good probe, which couples primary to neutron: dσ dσ − dΩ dΩ G Q2 F (Q2 ) = R L = F − 2 θ − N A 1 4sin W 2 dσ dσ 2πα 2 FP (Q ) + dΩ R dΩ L * Taking into account the 2 Re[MM Z ] A 2 Coulomb distortion effects: ||M Parity-violating electron scattering (PVS) is considered to be a model independent method to probe the nuclear neutron properties, which was first proposed in 1989. During the scattering process, the incident longitudinally polarized electrons interact with nucleons by exchanging either a photon or a Z0 boson and the Z0 boson couples primarily to neutrons , which leads to the parity - violating experiment is sensitive to the neutron distribution. The parity-violating electron scattering experiments at Jefferson Lab: PREx(208Pb)、CREx(48Ca) Apv 0.656 ppm 0.060(stat) 0.014(syst) According to the measured Apv, the neutron skin is extracted to be: 0.16 Rnp RR n p 0.330.18 fm dA dR = 3% → n = 1% A Rn PREX-I Result PRL 108 (2012) 112502 PRC 85 (2012) 03250(R) 9 Experimental schematic of parity- violating electron scattering at JLab Report content The theoretical model of nuclear structure： • Relativistic mean-field model Theoretical method for electron scattering： • Plane-Wave Born approximation • Eikonal approximation • Phase-shift analysis Research subjects • Parity-violating electron scattering • Electron scattering off deformed nuclei Sec. II Relativistic mean-field model Lagrangian of relativistic mean-field model (RMF)： 2015/7/8 12 Apply the Euler-Lagrange equation to the wave function of nucleon： The Dirac equation for nucleon can be obtained: where the scalar and vector potential is: The Dirac spinor of nucleon: where n is the principle quantum number; m is the magnetic quantum number; s is the spin and t is the isospin Substituting the Dirac spinors into the Dirac equation of nucleon, the equations for radical wave function of nucleon can be obtained: The nuclear densities can be obtained from the radical wave functions: Apply the Euler-Lagrange equation to the wave functions of meson, the equations of motion for mesons and photon can be obtained: The nuclear energy in the RMF model: The expression for each item: Sec. III Parity-violating electron scattering The physical picture of electron scattering: The purpose of theoretical study: Calculating the scattering probability of electron in one solid angle, which is the scattering differential cross section. 2.1 Theoretical methods for electron scattering 1. Plane wave Born approximation (PWBA) The wave function of incident electron is the plane wave, which is eigenfunction of ˆ ˆ 2 Hamiltonian of free electron Hm 0 P /2 . The wave function of scattered electron is 。 The differential cross section: d 2 12m |f ( )| fV( ) (2 )3 d 4 2 k If the effect of target nuclei are not very big, the wave function of scattering can also be seen as the plane wave 。The application condition of PWBA is the energy of incident electron is high enough and influence of target nuclei is weak. Eikonal approximation z If the energy of incident electron is high and scattering angle is not large, the particle behavior of electron is strong. The semiclassical path concept is applicable. The wave function of scattering electron can be replace by the semiclassical wave function ( ) e iS ( x )/ . For the situation EV , we can obtain: From the equation, we can see that the wave function () is divided into two parts: one is e ikz , which is the wave function of incident electron; the other is , which is the influence of Coulomb field of target nuclei. Phase-shift analysis method (DWBA) Another method to deal with the Coulomb distortion is the phase-shift analysis method which expands the wave function of scattered electron to the sum of components of angular momentum. eikr ()r efikz (, ) r l (2l 1) iRll ( r ) P (cos ) l where Rr l () is the radial wave function of scattered electron. P l (cos ) is the Lengendre polynomials. Every wave functions with angular momentum l have contributions to the scattering amplitude. The total scattering amplitude can be obtained by the sum of the scattering amplitude in l order. Detailed formulas for phase-shift analysis method: •Dirac equation of electron αpm Vr() () rr E () Coulomb potential •If V(r) is symmetric, the wave functions can be described with Dirac spinor: •Expansion coefficents P(r),Q(r) satisfy: •Combining the boundary conditions, the phase-shift analysis can be obtained. The asymptotic behavior of the wave function: Sin( 2 ( ) r →∞ − � → π P( r )→ Sin( kr − l −η ln 2 kr +∆ ) Eκκ2 π Q( r )→ Cos( kr − l −η ln 2 kr +∆ ) Eκκ2 Direct scattering amplitude Spinflip scattering amplitude Cross section: Charge form factor: where Mott cross section is: The comparison between the experimental data and theoretical calculations of DWBA The charge density distributions of Sn isotopes from the RMF model With the decrease of the neutron number, the spatial extension becomes larger. J. Liu and Z. Ren, Nucl. Phys. A 888, 45 (2012). 101 100 116 112Sn Sn Theo. Theo. -1 -2 Expt. 10 Expt. 10 10-3 10-4 (mb) 10-5 10-6 Ω 20 40 60 80 40 60 80 100 0 0 /d 10 10 118 124 σ Sn Sn d Theo. Theo. 10-2 Expt. 10-2 Expt. 10-4 10-4 -6 -6 10 40 60 80 100 10 40 60 80 100 Angle (degree) Elastic cross sections of Sn isotopes J. Liu and Z. Ren, Nucl. Phys. A 888, 45 (2012). Elastic charge form factors of Sn isotopes With the increase of the neutron number, the minima of the charge form factor shift inside and upside. 3.2 Theoretical framework of parity-violating electron scattering Research background • Neutron, being electrically neutral particles, cannot be measured from the elastic electron scattering. The experiments with hadronic probes always have large error bars and the interpretation is model-dependent. • The meanings of accurate neutron distribution. The input of neutron reaction model. Testing the nuclear structure model. • In 1989, Donnelly, Dubach and Sick propose the parity-violating electron scattering to study the neutron density distribution. • The Coulomb and weak interaction between the electron and nucleus. • The potential for electron with positive and negative parity is different σp 1, V() r V () r Ar () ||p C σp 1, VrVrAr() () () ||p C Coulomb potential Weak potential GF Weak potential: Ar() W () r 22 about 0.08 Weak density: 32 W()r d rGE |rr |(14sinWp ) ()r n () r The neutron contribute to the weak potential dominantly. During the scattering process, the wave function of polarized electron can be 1 divided into two parts: where (1 ) . 2 5 They satisfy the Dirac equation: αp Vr() E d /d d /d • The parity-violating asymmetries: Apv d /d d /d dd / and dd / are cross sections of positive and negative electrons, respectively. Under the PWBA method, the parity-violating asymmetry can written as: Gq2 Fq() F n 2 Apv 4sinW 1 42Fqp () Comparison betwen the PWBA and DWBA method 208Pb 850MeV Calculated with different method. PWBA method can give the right changing trend. The results of Eikonal approximation and phase-shift analysis coincide with each other. 3.3 Effect of ω − ρ meson coupling term in RMF model to parity-violating electron scattering Lagrangian of RMF model: The proton and neutron densities of 208 208Pb, which calculated from RMF The Apv of Pb for different ω − model with different ω − ρ coupling ρ coupling term Λv. term Λv. J. Liu and Z. Ren, Phys. Rev. C 84, 064305 (2011). Λv for different parameter sets Apv of 208Pb for different parameter set The neutron density distributions obtained from three different parameter sets. J. Liu and Z. Ren, Phys. Rev. C 84, 064305 (2011). 3.4 Statistical error of PVS measurement at different scattering angle The assumption for extracting Rn: There are linear relation between Apv and Rn. Therefore, from one PVS measurement, the Rn can be extracted.