Theoretical study on by the -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 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 :  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 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 interact with nucleons by exchanging either a photon or a Z0 boson and the Z0 boson couples primarily to , 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 0.330.18  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 αpm  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 →∞ 𝜋𝜋 𝑘𝑘𝑘𝑘 − 𝑙𝑙 � 𝑗𝑗𝑙𝑙 𝑟𝑟 → 𝑘𝑘𝑘𝑘

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 between the electron and nucleus. • The potential for electron with positive and negative parity is different σp 1, V() r V () r Ar () ||p  C σp 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 |(14sinWp ) ()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:

αp 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 4sinW 1 42Fqp () 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 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.

The relation between Apv and Rn S. Abrahamyan et. al., 0.16 PRL 108, 112502 (2012) Rnp RR n  p 0.330.18  fm Which scattering angle should be chosen?

Sensitive of Apv to neutron radius Rn for 48Ca

208Pb J. Liu and Z. Ren, Nucl. Phys. A 900, 1 (2013).

： The sensitivity of Apv to R n d lnAAR d  pv n pv Rn d lnRAnnpv d R The statistical error for measuring Rn:

R 1 n  R N AP n total pv Rn

d N IT   N total tar d

The statistical error of PVS for 48Ca 3.5 Extracting the neutron density distributions by two PVS measurements

Supposing the neutron density distribution is the Helm model distribution,   H    ()r dr fG ( r r )00 ( R r )

 2 3/2 r22/2 G where frG () 2  e F Ar() W () r 22 The density can extracted by two PVS measurements: Weak charge density Helm model has good analytical properties, whose form factors can be written as:   3 22 H iq r H  q /2 F() q e () r dr j10 ( qR ) e qR0 The analytic expression of radius of Helm model is:

23 22 rR 0 5  5

43 4 22 4 r RR0014  35  7 Statistical error for measuring W

W 1  d lnAApv W d pv    W Ntotal AP pv  W W d lnWWApv d

入射电子能量为2.2 GeV， 实验时间为30 天。测量 48Ca 中子Helm 模型密度

分布弥散系数σW 的误差。

J. Liu and Z. Ren, Nucl. Phys. A 900, 1 (2013). Under the PWBA method, we can obtain the relation between Apv and parameters of Helm model:

2 GqF QFqwk W () APV  42 ZF C () q

k0( RR 0W  0C )( WC   )

2 GqQF wk k0 10.5 ppm 42 Z

Then, take into account the Coulomb distortion corrections and calculate the Apv with the DWBA method again:

* dd 2 Re[MM Z ] Apv 2 dd || M Fitting the theoretical results with:

Akpv 0( R 0W  R 0 C )(  WC   )

Though PWBA cannot give the accurate Apv, but it can give the relation between Apv and parameters of neutron distributions.

The results of DWBA also satisfy this relation.

J. Liu, Z. Ren, and C. Xu Phys. Rev. C 88, 054321 (2013). Apv of 48Ca, calculated with the DWBA method Fitting the theoretical results from different parameter sets, we finally obtain:

APV (ppm) 6.575  4.060(RR0W  0C ) 9.948(WC  )

(48Ca, 6.5°)

Combining the weak skin thickness, the parameter of neutron density distribution can be extracted:

3 22 Rwskin RR weak  ch  R0W 5 W  R ch 5 (48Ca, 4°)

If the neutron density distribution is described by the two parameters distribution model, we can constrain the parameters by two PVS measurements at different angles.

J. Liu, Z. Ren, and C. Xu Phys. Rev. C 88, 054321 (2013). Sec. IV Effect of nuclear deformation on electron scattering

The previous considered nuclei are the spherically symmetric. Most nuclei are found to be deformed in their ground states. The electric quadropole moments can be measured from experiments. Therefore we try to take into account the nuclear deformation during the studies of electron scattering.

The nuclear quadropole moment:

1 22 Q= ρτp (r )(3z− rd ) e ∫V 4.1 Deformed density distribution model

The spherical two-parameters Fermi (2pF) distribution:

−1 rR− ρθ= ρ + (r , )0  1 exp a Introducing the nuclear surface deformation, it can be rewritten as the deformed 2pF distribution: −1 rR− (,θϕ ) ρθ= ρ + (r , )0  1 exp a

R(,θϕ )=+++ RY0 [1 β 2 20 (, θϕ ) β4 Y 40 (, θϕ ) ]

Multiply expanding the deformed 2pF model:

ρθ(,r )= ρL ()rY ( θ ) ∑ L L0 02 4 =+++ρρθρθ()r () rY20 ( ) () rY40 ( )  Nuclear surface shape for different L Nuclear proton density distribution for different L (Z=10)

The deformed nuclear density distribution can be written as the superposition of spherical part and deformed part:

0 d ρρρp()rr= pp ()r + ()

Then we calculate the deformed nuclear density distribution from the RMF model. In spherical RMF model:

A 0 1 22 ρ p()r=−+∑ ( tii )(| Gr ()| | Fr i ()|) i=1 2

The nuclear deformation is considered as the contributions of the proton hole in the outermost orbit.

11 ρd (r )= β (|Gr ( )|22 +− | Fr ( )| )( | Y (θϕ , )|2 ) p dr2 j j 4π lm

For example, there exists a proton hole in the 1p1/2 14 27 level for N and in the 1d5/2 level for Al. 2 m |Ylm (θϕ , ) | =(-1) YYl,, m l− m

2 1/2 m (2l + 1) =(-1) ∑lm, l− m L 0 l 0, l 0 L 0 YL0 L 4π (2L + 1)

Deformed nuclear proton density distribution:

2l 0 L ρρp()r = p ()r + ∑ ρ pL () rY0 L=2

A 0 1 22 ρ p()r=−+∑ ( tii )(| Gr ()| | Fr i ()|) i=1 2

2 1/2 Lm(2l + 1) 1 22 ρβp()r =−+ (-1) dlm, l m L 0 l 0,0 l L 02 (| Gjj ()| r | F ()|) r 4π (2Lr+ 1) 4.2 Electron scattering off deformed nuclei

b target nuclei Scattering angle θ incident electrons electron  Orientation angle β b θ (,r ) β z β  ()r

(a) (b)

Sketch of electron scattering off deformed nuclei The nuclear proton form factor under the PWBA:

1 ⋅ F() q= dr3 ρ ()r eiqr ppZ ∫

iLqr⋅ ' e =∑ 4π (2L '+⋅⋅ 1) i jLL' ( qr ) ⋅ Y '0 (θϕ , ) L' ππ 2 ∗ ddϕ θSin θ YL'' m (, θϕ ) YLm (, θϕ )= δL,' L δ m , m ' ∫∫00

222 |Fq ()|= Fsphe.. () q + Fdefo () q

ll2 2 2 1  =+I0 () q ∑∑(2l +− 1) lm , l m L 0 l 0,0 l L 0 IL () q 21l + mlL=−=2

1 I() q= d30 rρ () r j ( qr ) 00Z ∫ p 1 I( q )= drj ( r )( G22+ F ) L Z ∫ L jj Taking into considerations of the Coulomb distortion effect:

2DW 22 2 |FqC ()|= Fsphe.. () q + ξ Fdefo () q

Finally, the scattering cross sections can be obtained: 2 dσ qµ = ση F 2 dqΩ mC2 2l 0 L ρρp()r = p ()r + ∑ ρ pL () rY0 L=2

100 Expt. 10-1 ξ=0.0 ξ=1.0 -2 2 10 ξ=1.6

-3

10 |F(q)| 10-4

-5 14 10 N 10-6 0.5 1.0 1.5 2.0 2.5 3.0 q (fm-1)

Nuclear charge form factors of 14N

2DW 22 2 |FqC ()|= Fsphe.. () q + ξ Fdefo () q 2l 0 L ρρp()r = p ()r + ∑ ρ pL () rY0 L=2

-1 Expt. 10 ξ=0.0 ξ=1.0 ξ=0.5 2 10-3

|F(q)|

10-5 27Al 10-7 0.5 1.0 1.5 2.0 2.5 3.0 q (fm-1)

Nuclear charge form factor of 27Al

2DW 22 2 |FqC ()|= Fsphe.. () q + ξ Fdefo () q By including the nonspherical contribution in the scattering model, the theoretical results coincide with the experimental data much better. The introduction of the nonspheriacal contribution modifies the theoretical charge form factors at the positions of diffraction minima and high momentum transfers, but does not change the theoretical results at other positions.

This conclusion is obtained where the nuclear deformation is considered as the contribution of proton hole; however it is also applicable for other deformed nuclei. In the following work, we will further study electron scattering off deformed nuclei where the proton density distributions are obtained from the deformation-constrained mean-field calculations based on microscopic energy density functionals. Summary

1. The parity-violating electron experiment is sensitive to the neutron density distribution. If the neutron density distribution is described by the two parameters distribution model, we can constrain the parameters by two PVS measurements at different angles.

2. The introduction of the nonspheriacal contribution modifies the theoretical charge form factors at the positions of diffraction minima and high momentum transfers, but does not change the theoretical results at other positions.

Thanks for your attention

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