2015. 7. 9 1

Theoretical study on nuclear structure by the electron-nucleus scattering

Jian Liu (刘健) China University of

Petroleum (East China)

Content

Introduction

Relativistic mean-field model (RMF)

Parity-violating electron scattering

Electron scattering off deformed nuclei

Summary and prospect

Sec. I Introduction

Optical microscope cellular scale (µm)

visible light 0.5 µm

Electron microscope atomic scale(nm)

electrical energy 100keV wavelength 0.004nm

The observing capability of microscope is connected with the wave length of its probe.

1.1 Elastic electron scattering

Effect method to measure at the nuclear scale: High energy electron scattering

Experimental equipment of electron scattering (Hofstadter 1954 )

Scattering cross section of electron

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.

Large scale facilities of electron scattering under construction:

• RIKEN of Japan

• GSI of Germany

Advantages of electron scattering:

The interaction between electron and nucleus is well understood.

132Sn

Experimental research progress on electron scattering

RIKEN Target project

RIKEN Verified experiment

1.2 Parity-violating electron scattering

−−=

Ω+

Ω

Ω−

Ω=

)()(sin41

22 2

22

2

QFQFQG

dd

dd

dd

dd

AP

NW

F

LR

LR θπασσ

σσ

Z0 is a good probe, which couples primary to neutron:

*

2

2Re[ ]| |

ZM MA

M

Taking into account the Coulomb distortion effects:

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.

9

The parity-violating electron scattering experiments at Jefferson Lab: PREx(208Pb)、CREx(48Ca)

PREX-I Result PRL 108 (2012) 112502 PRC 85 (2012) 03250(R)

pv 0.656 ppm0.060(stat) 0.014(syst)

A

According to the measured Apv, the neutron skin is extracted to be:

0.160.180.33 fmnp n pR R R

%1%3 =→=n

n

RdR

AdA

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

2015/7/8 12

Lagrangian of relativistic mean-field model (RMF)：

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.

1. Plane wave Born approximation (PWBA)

2.1 Theoretical methods for electron scattering

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.

2| ( ) |d fd

32

1 2( ) (2 )4

mf V

k

The wave function of incident electron is the plane wave, which is eigenfunction of Hamiltonian of free electron . The wave function of scattered electron is 。

20

ˆ P / 2H m

The differential cross section:

Eikonal approximation

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 . For the situation , we can obtain:

( ) ( )/iS xe

E V

z

From the equation, we can see that the wave function is divided into two parts: one is , which is the wave function of incident electron; the other is , which is the influence of Coulomb field of target nuclei.

( )ikze

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.

where is the radial wave function of scattered electron. is the Lengendre polynomials.

( )lR r (cos )lP

( ) ( , )

(2 1) ( ) (cos )

ii

ll l

l

ee fr

l i R r P

k rk zr

Every wave functions with angular momentum have contributions to the scattering amplitude. The total scattering amplitude can be obtained by the sum of the scattering amplitude in order.

l

l

Detailed formulas for phase-shift analysis method:

•Dirac equation of electron 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.

( ) ( ) ( )m V r E α p r r

The asymptotic behavior of the wave function:

𝑗𝑗𝑙𝑙(𝑟𝑟) →Sin(𝑘𝑘𝑟𝑟 − 𝑙𝑙 𝜋𝜋2

𝑘𝑘𝑟𝑟 r → ∞

( ) Sin( ln 2 )2EP r kr l krκ κπ η→ − − + ∆

( ) Cos( ln 2 )2EQ r kr l krκ κπ η→ − − + ∆

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).

40 60 80 10010-6

10-4

10-2

100

20 40 60 8010-5

10-3

10-1

101

40 60 80 10010-6

10-4

10-2

100

40 60 80 10010-6

10-4

10-2

100

116 Sn Theo. Expt.

112Sn Theo. Expt.

dσ/d

Ω (m

b)

Angle (degree)

118Sn Theo. Expt.

124Sn Theo. Expt.

J. Liu and Z. Ren, Nucl. Phys. A 888, 45 (2012).

Elastic cross sections of Sn isotopes

With the increase of the neutron number, the minima of the charge form factor shift inside and upside.

Elastic charge form factors of Sn isotopes

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.

3.2 Theoretical framework of parity-violating electron scattering

• The Coulomb and weak interaction between the electron and nucleus.

• The potential for electron with positive and negative parity is different

Weak density:

3 2( ) | | (1 4sin ) ( ) ( )W E W p nr d r G r r r r

about 0.08

The neutron contribute to the weak potential dominantly.

Weak potential: ( ) ( )2 2

FW

GA r r

Weak potential Coulomb potential

( ) ( ) ( )CV r V r A r

1,| |

σ p

p

1,| |

σ pp

( ) ( ) ( )CV r V r A r

During the scattering process, the wave function of polarized electron can be

divided into two parts: where .

They satisfy the Dirac equation: 5

1 (1 )2

and are cross sections of positive and negative electrons, respectively.

/d d /d d

Under the PWBA method, the parity-violating asymmetry can written as:

( )V r E α p

• The parity-violating asymmetries:

pv

d / d d / dd / d d / d

A

22

pv( ) 4sin 1( )4 2

nFW

p

F qG qAF q

Comparison betwen the PWBA and DWBA method

PWBA method can give the right changing trend. The results of Eikonal approximation and phase-shift analysis coincide with each other.

208Pb 850MeV Calculated with different method.

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 208Pb, which calculated from RMF model with different ω − ρ coupling term Λv.

The Apv of 208Pb for different ω − ρ coupling term Λv.

J. Liu and Z. Ren, Phys. Rev. C 84, 064305 (2011).

The neutron density distributions obtained from three different parameter sets.

Apv of 208Pb for different parameter set

Λv for 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

0.160.180.33 fmnp n pR R R

There are linear relation between Apv and Rn. Therefore, from one PVS measurement, the Rn can be extracted.

S. Abrahamyan et. al., PRL 108, 112502 (2012)

The relation between Apv and Rn

The assumption for extracting Rn:

Which scattering angle should be chosen?

Sensitive of Apv to neutron radius Rn for 48Ca

The sensitivity of Apv to ： nR

pv pv

pv

d ln dd ln dn

nR

n n

A ARR A R

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

The statistical error for measuring Rn:

total pv

1

n

n

n R

RR N A P

total tardN I T Nd

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,

The weak charge density can extracted by two PVS measurements:

( ) ( )2 2

FW

GA r r

Weak charge density

Helm model has good analytical properties, whose form factors can be written as:

The analytic expression of radius of Helm model is:

0 0( ) ( ) ( )HGr dr f r r R r

2 23/22 /2( ) 2 rGf r e

where

2 2 /21 0

0

3( ) ( ) ( )H iq r H qF q e r dr j qR eqR

2 2 20

3 55

r R

4 4 2 2 40 0

3 14 357

r R R

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。

Statistical error for measuring W

total pv

1

W

W

W N A P

pv pv

pv

d ln dd ln dW

W

W W

A AA

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

2wk

0 0W 0C

( )( )4 2

( ) ( )

WFPV

C

W C

Q F qG qAZ F q

k R R

2wk

0 10.5 ppm4 2

FG q QkZ

Under the PWBA method, we can obtain the relation between Apv and parameters of Helm model:

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

*

pv 2

2Re[ ]| |

ZM Md dA

d d M

Apv of 48Ca, calculated with the DWBA method

Fitting the theoretical results with:

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.

pv 0 0 0( ) ( )W C W CA k R R

J. Liu, Z. Ren, and C. Xu Phys. Rev. C 88, 054321 (2013).

Fitting the theoretical results from different parameter sets, we finally obtain:

0W 0C(ppm) 6.575 4.060( ) 9.948( )PV W CA R R

(48Ca, 6.5°)

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

2 2wskin weak ch 0W W ch

3 55

R R R R R (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:

2 21 ( )(3 )pVQ z r d

eρ τ= −∫ r

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

0

1

( , ) 1 exp r Rra

ρ θ ρ−

− = +

Introducing the nuclear surface deformation, it can be rewritten as the deformed 2pF distribution:

0

1( , )( , ) 1 exp r Rraθ ϕρ θ ρ

−− = +

0 2 20 4 40( , ) [1 ( , ) ( , ) ]R R Y Yθ ϕ β θ ϕ β θ ϕ= + + +

Multiply expanding the deformed 2pF model:

0

0 2 420 40

( , ) ( ) ( )

( ) ( ) ( ) ( ) ( )

LLL

r r Y

r r Y r Y

ρ θ ρ θ

ρ ρ θ ρ θ

=

= + + +

∑

4.1 Deformed density distribution model

Nuclear surface shape for different L

0( ) ( ) ( )dp p prρ ρ ρ= +r r

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

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

Nuclear proton density distribution for different L (Z=10)

0 2 2

1

1( ) ( )(| ( ) | | ( ) | )2

A

p i i ii

r t G r F rρ=

= − +∑

2 2 22

1 1( ) (| ( ) | | ( ) | )( | ( , ) | )4

dp d j j lmG r F r Y

rρ β θ ϕ

π= + −r

In spherical RMF model:

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

For example, there exists a proton hole in the 1p1/2 level for 14N and in the 1d5/2 level for 27Al.

20

02

( ) ( ) ( )l

Lp p p L

Lr r Yρ ρ ρ

=

= + ∑r

2, ,

1/22

0

| ( , ) | =(-1)

(2 1)(-1) , 0 0, 0 04 (2 1)

mlm l m l m

mL

L

Y Y Y

l lm l m L l l L YL

θ ϕ

π

−

+= − +

∑

1/222 2

2(2 1) 1( ) (-1) , 0 0, 0 0 (| ( ) | | ( ) | )

4 (2 1)L mp d j j

lr lm l m L l l L G r F rL r

ρ βπ

+= − + +

0 2 2

1

1( ) ( )(| ( ) | | ( ) | )2

A

p i i ii

r t G r F rρ=

= − +∑

Deformed nuclear proton density distribution:

(b)

βb

β

b

z

θelectron Orientation angle β

incident electronstarget nuclei

(a)

( , )r

( )r

Scattering angle θ

Sketch of electron scattering off deformed nuclei

4.2 Electron scattering off deformed nuclei

31( ) ( ) ip pF q d r e

Zρ ⋅= ∫ q rr

'' '0

'= 4 (2 ' 1) ( ) ( , )i L

L LL

e L i j qr Yπ θ ϕ⋅ + ⋅ ⋅ ⋅∑q r

2

' ' , ' , '0 0Sin ( , ) ( , )L m Lm L L m md d Y Y

π πϕ θ θ θ ϕ θ ϕ δ δ∗ =∫ ∫

The nuclear proton form factor under the PWBA:

3 00 0

1( ) ( ) ( )pI q d r r j qrZ

ρ= ∫2 21( ) ( )( )L L j jI q drj r G F

Z= +∫

2 2 2. .

222

02

| ( ) | ( ) ( )

1( ) (2 1) , 0 0, 0 0 ( )2 1

sphe defo

l l

Lm l L

F q F q F q

I q l lm l m L l l L I ql =− =

= +

= + + − + ∑ ∑

2 2 2 2. .| ( ) | ( ) ( )DW

C sphe defoF q F q F qξ= +

Taking into considerations of the Coulomb distortion effect:

Finally, the scattering cross sections can be obtained: 2

22m C

qd Fd q

µσ σ η=Ω

20

02

( ) ( ) ( )l

Lp p p L

Lr r Yρ ρ ρ

=

= + ∑r

0.5 1.0 1.5 2.0 2.5 3.010-6

10-5

10-4

10-3

10-2

10-1

100

q (fm-1)

|F(q

)|2

Expt. ξ=0.0 ξ=1.0 ξ=1.6

14N

Nuclear charge form factors of 14N

2 2 2 2. .| ( ) | ( ) ( )DW

C sphe defoF q F q F qξ= +

20

02

( ) ( ) ( )l

Lp p p L

Lr r Yρ ρ ρ

=

= + ∑r

0.5 1.0 1.5 2.0 2.5 3.010-7

10-5

10-3

10-1

27Al

q (fm-1)

Expt. ξ=0.0 ξ=1.0 ξ=0.5

|F(q

)|2

Nuclear charge form factor of 27Al

2 2 2 2. .| ( ) | ( ) ( )DW

C sphe defoF q F q F qξ= +

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.

61

Thanks for your attention