TX_1~ABS:AT/TX_2:ABS~AT


56 UHD Journal of Science and Technology | Jul 2020 | Vol 4 | Issue 2

1. INTRODUCTION

There are two essential reasons why electron-nucleus 
scattering is such a successful apparatus used for studying 
nuclear structure. The primary one belongs to the reality 
that the main interaction occurring between the electron 
and the nucleus is well known [1]. The origin of  the second 
reason that makes electron scattering is a valuable method in 
examining the properties of  nuclear structure comes from 
its ability to identify the excited states, spins, and parities, 
through the calculations of  the reduced matrix elements of  
nuclear transitions. Basically, in the electron scattering with 
a relatively weak interaction, the interactions of  the electron 
with charge and the nuclear current density occur where they 
described by the theory of  quantum electrodynamics [2].

One of  the great (standard) effective interactions for light nuclei 
is the Cohen-Kurath [3], for 1p-shell (1p

1/2,
 1p

3/2
) nuclei with 

core 
2
He4. In addition, different macroscopic and microscopic 

theories have been used to analyze excitation states in Be nucleus. 
The form factor calculations were done by utilizing the model 
space (MS) wave functions alone which were not sufficient 
for duplicating the experimental data of  the electron-nucleus 
scattering [4]. Therefore, the electron scattering Coulomb form 
factors in the p-shell nucleus (Be9) have been investigated by 
taking into account higher energy configurations outside the 
p-shell MS which are named core polarization effects [5].

Many research studies have focused their efforts on the 
improvement and development of  the electron scattering. 
Starting with Hofstadter who was the primary to utilize high-
energy electron beams given by the Stanford linear electron 
accelerator to discover electron scattering and an old work 
of  Sir Nevill Mott which was used electrons against point 
nuclei in his experiment as the relativistic scattering of  Dirac 
particles. Then, he established a series formulation for the 
cross-section of  the elastic scattering, also he allowed to 
estimating formula [6], [7].

Elastic and Inelastic Electron-Nucleus 
Scattering Form Factors for Be9 Nucleus
Hawar Muhamad Dlshad, Aziz Hama-Raheem Fatah, Adil Mohammed Hussain
Department of Physics, College of Science, University of Sulaimani, Sulaymaniyah, Kurdistan Region, Iraq

A B S T R A C T
The computations of the elastic and inelastic Coulomb form factors for the electron-nucleus scattering of beryllium nucleus 
Be9 have performed with core polarization (CP) effects including the realistic Michigan sum of three range Yukawa (M3Y) 
interaction and the other residual interaction which is modified surface delta interaction (MSDI). In the calculations, root 
mean square charge density and charge radii include for the ground states. The perturbation theory is adopted to compute 
the CP using the harmonic oscillators potential to calculate single-particle radial wave functions. The comparison has 
been done between the theoretical calculations of Coulomb form factors by MSDI interaction, realistic M3Y interaction, 
and the experimental results that measured by other workers, it noticed that the Coulomb form factors for the (M3Y) 
interaction gave a reasonable depiction of the measured data.

Index Terms: Beryllium Nucleus, Core Polarization Effect, Electron Scattering, Form Factor, Harmonic Oscillator.

Corresponding author’s e-mail: Hawar Muhamad Dlshad, Department of Physics, College of Science, University of Sulaimani, Sulaymaniyah, 
Kurdistan Region, Iraq. E-mail: hawar.muhamaddlshad@gmail.com

Received: 05-04-2020 Accepted: 13-08-2020 Published: 17-08-2020

Access this article online

DOI: 10.21928/uhdjst.v4n2y2020.pp56-62 E-ISSN: 2521-4217

P-ISSN: 2521-4209

Copyright © 2020 Dlshad, et al. This is an open access article 
distributed under the Creative Commons Attribution Non-Commercial 
No Derivatives License 4.0 (CC BY-NC-ND 4.0)

O R I G I N A L  R E S E A R C H  A R T I C L E UHD JOURNAL OF SCIENCE AND TECHNOLOGY



Hawar Muhamad Dlshad, et al.: Electron Scattering Form Factors for Be9

UHD Journal of Science and Technology | Jul 2020 | Vol 4 | Issue 2 57

Elastic and inelastic electron scattering for the light nuclei 
using Born approximation had performed by Uberall 
and Ugincius [8]. In the last decades, the single-particle 
quadrupole transitions of  Coulomb electron-nucleus 
scattering form factors studied in the B10 which is the 
p-shell nucleus by Majeed [9], whereas the studies included 
a microscopic theory in the core polarization (CP) effects for 
the excitation states up to 2ℏω by employing the modified 
surface delta interaction (MSDI).

The charge density distributions and charge radii of  the 
nucleus were distinguished from the investigation of  elastic 
electron scattering data [10]. However, Sharrad et al. [11] 
have used the charge density distributions of  the ground 
state for determining the Coulomb form factors using 
the approximation rule which is the plane wave Born 
approximation with the two-body short range correlation. 
Consequently, Radhi et al. [12] presented inelastic Coulomb 
and electromagnetic form factors for F19 in each positive 
parity and negative parity states by applying the single-particle 
states shell model and Hartree–Fock method.

At present, Raheem et al. [13] have been calculated the elastic 
Coulomb C

0
 form factors for a few sd-shell nuclei using 

nucleon-nucleon effective interaction, which is two-body 
(Michigan sum of  three range Yukawa [M3Y]) as residual 
interactions with considering the CP matrix elements.

This work is devoted to calculate the theoretical Coulomb 
electron scattering form factors for Be9 by considering the 
role of  the MS besides the CP effects using MSDI and the 
realistic interaction named M3Y including root mean square 
charge density along with charge radii for the ground states. 
The harmonic oscillator (HO) wave function will be adopted 
as a single particle wave function. To do this, first needed to 
use shell model code (OXBASH) to calculate the one-body 
density matrix (OBDM) elements [14], [15].

Finally, the theoretical calculations of  Coulomb form 
factors by MSDI, M3Y interactions are compared with the 
experimental results.

2. THEORY

2.1. Coulomb Form Factor 
The Coulomb electron scattering form factors of  a given 
multipolarity (J) is a function of  transfer momentum (q) and 
it can be described in term of  reduced matrix elements (in 
spin state) of  the transition operator [16]:

 ( ) ( ) ( )2
4

| |  
2 1J f J ii

F q J T q J
Z J


=

+
 (1)

Where, Ji and Jf are, respectively, the initial and final total 
angular momentum, while Z is the number of  proton (atomic 
number), TJ(q) is the multipole operator of  electron scattering, 
and J T q Jf J i| |( )  is the reduced many body matrix element.
The best description of  the experimental form factors requires 
to correct the form factor in Equation (1) corresponding to 
the center of  mass correction and the finite size correction 
of  the nucleon [17]:

 ( ) ( ) ( )
( )2 2 0.43

4
2

4
| |    

2 1

q b A

A
J f J i

i

F q J T q J e
Z J


−

=
+

 (2)

Here, b is the HO size parameter that obtained from the 
experiment, A is the nuclear mass number, and the final 
term in the above equation is the correction coefficient. The 
reduced matrix element in Equation (1) can be expressed 
in two terms, the first one is MS term and the other is CP 
term [18].

J T q J J T q J J T q Jf J i f J i
MS

f J i
CPz z z

| | | | | |τ τ τδ( ) = ( ) + ( )  

 (3)

The MS reduced matrix element in the spin and isospin 
spaces of  the transition operator TJ is performed as the 
sum of  the product of  the (OBDM) elements which are in 
neutron-proton formalism OBDM(β, α ,J, τ

z
, i, f) multiplied 

by the single-particle reduced matrix elements as follow [18]:

J T q J OBDM J i f Tf J i
MS

z Jz zτ
β α

τβ α τ β α( ) = ( )∑
� ,

, , , , , � | |  

 (4)

In addition, the CP reduced matrix element can be 
represented as:

 

� , , , , ,

�

� ,
f J i CP z

J

T q J J OBDM J i f

T

z

z

δ β α τ

β δ α

τ
β α

τ

( ) = ( )∑
 (5)

Where, α and β are, respectively, the initial and final single-
particle states for the MS when isospin included, the index 



Hawar Muhamad Dlshad, et al.: Electron Scattering Form Factors for Be9

58 UHD Journal of Science and Technology | Jul 2020 | Vol 4 | Issue 2

τz is the third component of  nucleons Pauli isospin which 
used to identify the nucleons with τz = 1, −1 for protons 
and neutrons, respectively, and the OBDM determined 
macroscopically for the elastic scattering by the initial and 
final nuclear wave functions, while it obtained from OXBASH 
code for inelastic scattering. The single-particle matrix 
element is determined from:

β ατ β τ α β β α α| | | |T j j n l j qr n lJ J Jz z= ( )Υ , ,  (6)

Where, 〈nβ, lβ│jJ (qr)│nα lα〉 is the radial part matrix element 
of  the spherical Bessel function jJ (qr) which is calculated in 
[19, Equation (23)] of  our published article; and j jJ zβ τ αΥ  

represents the reduced matrix element of  the spherical 
Harmonics Υ J z .

The single-particle matrix element can represent according 
to the first-order perturbation theory as [20]: 

 

β δ α β α

β α

| | | |

| |

T V
Q

E H
T

T
Q

E H
V

CP
a o

b o

Λ Λ

Λ

=
−

+
−

�

 (7)

The single-particle matrix element β δ ατ| |TJ z  in the 

above equation obtained from the particle hole excitation 
with the first-order perturbation including residual interaction 
(V) for the MSDI and M3Y interaction [20].

β α

βα αα α α

α α α β α α

α

| |

| | | |

V
Q

E H
O

V O
e e e e

b o−
=

− + −
−( )∑

Λ

Γ

Γ Λ

, ,1 2 1 2

2 1 1 2
1 ++ + +( )

× +( ) +( )






+

α

δ δ
β α
α α

2 2 1

1 1
1 2

Γ Γ

Λ
Γbp ah

�terms�with�exchaanged� �and� �by�an�overall�minus�sign�α α1 2

 

 (8)

Here, Ho is the unperturbed Hamiltonian, Ea, Eb are the initial 
and final states of  energy, the Q operator projects the outside 
space of  the MS, both the indices α1 and α2, are, respectively, 

run over particle and hole states, 
β α
α α

Λ
Γ1 2








�is the six-j 

symbol and e is the single-particle energy. Every matrix 
element in the Equation (7) is obtained in iso-scalar (T = 0) 
and isovector (T = 1) formalism with Λ = JT and Γ=J’ T’.

The single-particle energies are calculated by [21]:

e n l
l f r for j l

l f r
nlj

nl
= + −





+
− +( ) ( ) = −

(
2

1
2

1
2

1
1
2

1
2


� � � � �,

� � )) = +








 nl

for j l� � �,
1
2

 

 (9)
Where f r A A A

nl
( ) ≈ − = −

− − −20 45 25
2
3 1 3 2 3� �, � � � �/ /� � .

2.2. Ground-State Form Factor and the Charge Density
It is clear that the electron scattering is one of  the most 
powerful tools for analyzing the charge density distributions 
of  the nucleus. Since the charge density is a measurable 
quantity, subsequently, it is another way of  calculating the 
form factor. Moreover, the elastic form factor is occurring 
when J = 0 (zero spin) and is obtainable from the simple 
form of  the Fourier transform as [22]:

 F q
Z

r j qr r dr0
0
0 0

24( ) = ( ) ( )
∞

∫
π

ρ� � � � � (10)

Where, F
0
 (q) is the ground-state form factor, r is the radius 

of  the nucleus, and ρ
0
 (r) is the charge density.

The entirety of  all protons point charge is the representation 
of  operator of  transition charge density ( )ˆ  JM r

 of  a 
nucleus [23].

 ( ) ( ) ( )2
1

   ˆ Õ Ù  
Z

k
JM JM k

kk

r r
r

r



=

−
=∑

 



 (11)

Where, J is the multipolarity of  the operator, M is the 
projection quantum number takes 2J + 1 values, −J ≤ M ≤ 
J, ΥJM (Ωk) represents the spherical Harmonic, and 

 

r rk−( ) 
is a Dirac delta function.

The matrix element in the reduced form of  the operator 
( )ˆ JM r
  is gotten when the transition happens from initial 

nuclear spin Ji to the final nuclear spin Jf and complying the 
inequality Ji ≤ J ≤ Jf , from Equation (4) where ( )ˆ ,zJ JT r ≡



then it is given by:

( ) ( ) 
  ,

  ( , , , , , ) ˆ   ˆf J i z JJ r J OBDM J i f r
 

      = ∑  

 (12)



Hawar Muhamad Dlshad, et al.: Electron Scattering Form Factors for Be9

UHD Journal of Science and Technology | Jul 2020 | Vol 4 | Issue 2 59

For the ground state (J = 0) as mentioned before. Moreover, 
Ji= Jf, and the charge density J

p r( )  define of  the nucleus 
is getting from this matrix element [23].

( ) ( )

( )





,

1
   

4 ( 2 1)

1
  ( , , , , , )   

( 2 )

ˆ

ˆ
4 1

p
J f J i

i

z J
i

r J r J
J

OBDM J i f r
J  

 


     


=
+

=
+ ∑

 (13)

The single-particle matrix element in Equation (4) can be 
re presented by the radial wave functions of  HO 
 n l n lr rα α β β( ) ( )    and 〈lβ jβ║ΥJ (Ωr) ║lα jα〉 which is the 

spherical Harmonic reduced matrix element.

( ) ( ) ( ) ( )| |    ˆ | |J n l n l J rr r r j j        = Υ Ω


  (14)

After putting Equation (14) in Equation (13), the nuclear 
charge density becomes [24]:

 

ρ
π

α β τ
α β

α α β β

J
p

i
z

n l n l

r
J

OBDM J i f

r r j

( ) =
+

( ) ( )

∑� ( ) ( , , , , , )

�

,

1
4 2 1

  ββ α| |Υ ΩJ r j( )
 (15)

For the ground-state nucleus, (τz= 1, −1), it makes J = 0 and 

j j jJ r j jβ α αδ πα β| |Υ Ω( ) = +( ), /2 1 4 . After utilizing 
the Delta-Kronecker in Equation (15) and putting τz= 1 for 
protons, the equation is rewritten as:

 

ρ
π

α α
α

α α α

0

2

1
4 2 1

0 1

2 1

p

i

n l

r
J

OBDM i f

j R r

( ) =
+( )

( )

+( ) ( )

∑
�

, , , , ,

�
 (16)

the index α≡nα lα jα used for all closed shells for the ground 
state.

The ground-state charge radii and charge distribution 
considered as two great determinable quantities experimentally, 
meanwhile, they can be calculated theoretically. The mean 
square radius for the nucleus gets from the charge density 
integration in Equation (16) [22].

 r r r dr
ch

p2
0

2= ( )∫
 

� � � (17)

Under the effect of  the point-proton folded charge density 
distribution, in Equation (10), the charge density needs to 
be corrected by the folding factor [25].

 ρ
π

fo

r r

ar r
a
e a fm

 

 

−( ) = =−
−( )

' �
�
� �,� . � �

'

1
0 6532

3 6

2

2
 (18)

Now, for the normalized charge density with the target 
nucleus atomic number Z, the root-mean-square in 
Equation (17) gives [24]:

 r
Z

r r r r drch
p

fo
2

0
21= ( ) −( )∫ 

   

� � �'  (19)

3. RESULTS AND DISCUSSION

The Beryllium nucleus has 12 known isotopes, but only one 
of  these isotopes (Be9) is stable and a primordial nuclide. The 
microscopic structure of  the stable nucleus Be9 imagined as 
being composed of  a tightly bound core He4 plus five loosely 
bound nucleons outside the core divided over the p-shell (1 
p3/2, 1 p1/2). On the other way, it consists of  four protons 
and five neutrons.

In this paper, the CP effects are calculated according to 
Equation (7) which include M3Y and MSDI interactions.

The potential parameters of  M3Y which known as three 
range potential contain spin orbit, central, and tensor 
interactions are obtained from Bertsch et al. [26]. Besides, the 
MSDI strength parameters that used in the calculations of  the 
CP effects are A

T
, B, and C. Where T is defined as the isospin 

(1, 0). They have taken the values as A
0 
=A

1
=B=25/A and 

C=0 [20], where A is the mass number of  Beryllium nucleus, 
B and C are the correction parameters. It has the HO length 
parameter b = 1.791 fm [27].

FORTRAN 2008 used as a computer program for calculating 
CPM3Y and MSDI in the elastic and inelastic form factors. 
In addition, the OBDM elements calculated with the shell 
model code OXBASH for excitation states but is obtained 
from the occupation numbers for the closed-shell orbits 
(ground state).

Beryllium nucleus has a ground state, whereas its value is 
( �J Ti i

  = 3/2 −1/2) E = 0.0 MeV. Here, two transitions are 
under investigation representing C

2
 with E = 2.43 MeV where 

the transition occurs to the excited state (Jf Tf = 5/2 
−1/2) 

and the other excited state is (Jf Tf  = 7/2 
−1/2) E = 6.38 



Hawar Muhamad Dlshad, et al.: Electron Scattering Form Factors for Be9

60 UHD Journal of Science and Technology | Jul 2020 | Vol 4 | Issue 2

MeV o[28]. In all graphs, the form factors with MS and CP 
effects including the realistic (M3Y) interactions representing 
as the red lines, the form factors with MSDI interaction 
performs as blue lines and the small filled circles represent 
the experimental values for the electron scattering form 
factors.

3.1. Elastic Coulomb form Factor for 3/2 −1/2 State
For elastic electron scattering, the scattered electron leaves 
the nucleus in the ground-state configuration. The ground 
state has (Jπ T= 3/2 −1/2) with E = 0.0 MeV. The 
multipoles entering the elastic scattering are J = 0, 2 with the 
corresponding Coulomb transition C

0
 and C

2
, respectively. 

The calculated form factor of  sum C
0
 + C

2
 is shown in Fig. 1.

The obtained OBDM elements are shown in Table 1. The 
calculated root-mean-square (charge radii) for the ground 
state without folding is 2.629 fm but with folding is 2.505 fm, 
while experimentally is equal to 2.519 fm [29]. The results 
with M3Y interaction have a great agreement with measured 
data in the transfer momentum domain of  1.1 ≤ q ≤2.5 fm−1. 
On the contrary, the calculations with MSDI interaction have 
a bad deal with the experimental data excepting the area of  
1.5 ≤ q ≤ 2 fm−1 where they have similarities with each other.

3.2. Inelastic Coulomb form Factor for 5/2 −1/2 State
The C

2
 transition for Coulomb scattering is taking place 

between the ground state of  (Jπ T = 3/2 −1/2) and the first 
excited state (Jπ T = 5/2 −1/2) with excitation energy of  E 
= 2.43 MeV. The computed and measured Coulomb form 
factors of  inelastic electron scattering for the Be9 nucleus are 
shown in Fig. 2. The OBDM elements which calculated with 
OXBASH code are listed in Table 2. In this transition, the 
calculations with MSDI are not able to denote an adequate 
description of  the experimental data for the region of  transfer 
momenta (q = 0.8 fm−1) and (q = 1.8 fm−1), but once the CP 
effect with M3Y interaction is applied, making the results of  
the total theoretical form factors fitting the experimental data 
along with all regions of  transfer momenta.

3.3. Inelastic Coulomb form Factor for 7/2 −1/2 State
The squared inelastic scattering of  Coulomb form factors 
for Be9 is displayed in Fig. 3. The symbol of  this transition 
(Coulomb transition) C

J
 = C

2
, it occurs between the ground 

state (Jπ T= 3/2 −1/2) and the second excited state (Jπ T= 7/2 
−1/2) with transition energy E = 6.38 MeV. The OBDM 
elements are tabulated in Table 3. Fig. 3 shows the plot 
of  measured and calculated data for the squared inelastic 
Coulomb scattering form factors. The ratio of  agreement 
between the results of  both M3Y and MSDI interactions 

for the Be9 form factors and the measured data are quite 
strong between (q = 1 fm−1) and (q = 2.5 fm−1). Taking into 
consideration that the form factors for the second excited 
state (Jπ T = 7/2 −1/2) are not substantially different from 

Fig. 1. Elastic Coulomb C
0
+C

2
 form factors for Be9. The experimental 

data were taken from reference [28].

TABLE 1: The calculated OBDM elements for the 
Coulomb C0+C2 transition of Be

9

j1 j2 OBDM (n) OBDM (p)
C0 1s1/2 1s1/2 2.8284 2.8284

1p1/2 1p1/2 0.5305 0.6072
1p3/2 1p3/2 1.6249 2.5707

C2 1p1/2 1p3/2 0.2502 0.1623
1p3/2 1p1/2 −0.2502 −0.1623
1p3/2 1p3/2 −0.4610 −0.2982

TABLE 2: The calculated OBDM elements for the 
Coulomb C2 transition of Be

9

j1 j2 OBDM (n) OBDM (p)
1p1/2 1p3/2 −0.4820 −0.8767
1p3/2 1p1/2 0.4187 0.5365
1p3/2 1p3/2 0.6372 0.1167

TABLE 3: The calculated OBDM elements for the 
Coulomb C2 transition of Be

9

j1 j2 OBDM (n) OBDM (p)
1p1/2 1p3/2 0.2724 0.2442
1p3/2 1p1/2 −0.1264 −0.1242
1p3/2 1p3/2 −0.7953 −0.1735



Hawar Muhamad Dlshad, et al.: Electron Scattering Form Factors for Be9

UHD Journal of Science and Technology | Jul 2020 | Vol 4 | Issue 2 61

that of  the first excited state (Jπ T = 5/2 −1/2), whence a 
noticeable change is observed in the computation of  form 
factors by M3Y and MSDI interactions. The MSDI decreased 
faster than the M3Y during the increase of  momentum 
transfer, especially at the point of  (q = 3 fm−1).

4. CONCLUSION

In the present work, it is possible to consider the following 
conclusions:
• The basic calculations include the Coulomb form factors 

for the ground state and other excitation states.
• The ground-state Coulomb form factors (C

0
 transitions) 

for the M3Y interaction and the ground-state charge radii 
with folding effect give the best fit with the experimental 
data for beryllium (Be9) nucleus.

• For Be9 nucleus which is under consideration, the quality 
of  similarity between the computed Coulomb form 
factors F

C
(q) and those of  the measured data become 

even better in the using of  the CP effects with including 
M3Y residual interaction for all Coulomb C

2
 transitions 

because M3Y interaction is more realistic nucleon-
nucleon interaction that adopted for the CP calculation.

• The calculation of  Coulomb form factors FC(q) with 
(MSDI) interaction is dealing with the surface nucleons 
only. Therefore, it has a limited agreement with the 
experimental results.

• The HO succeeded to describe the wave functions 
completely.

5. CONFLICTS OF INTEREST

There are no conflicts of  interest.

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[9] F. A. Majeed. “The effect of core polarization on longitudinal form 

Fig. 2. Inelastic Coulomb C
2
 form factors for 5/2 −1/2 state of Be9 with 

E = 2.43 MeV. The experimental data were taken from reference [28].

Fig. 3. Inelastic Coulomb C
2
 form factors for 7/2 −1/2 state of Be9 with 

E = 6.38 MeV. The experimental data were taken from reference [28].



Hawar Muhamad Dlshad, et al.: Electron Scattering Form Factors for Be9

62 UHD Journal of Science and Technology | Jul 2020 | Vol 4 | Issue 2

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