Core Polarization Effects With M3Y-Sigma

Republic of Iraq Ministry of Higher Education And Scientific Research University of Baghdad College of Science

Core Polarization Effects with M3Y-Sigma Meson Interaction for Longitudinal Form

Factors of 48Ca Nucleus.

A Thesis Submitted to the Department of Physics, College of Science, University of Baghdad, in Partial Fulfillment of the Requirements for the Degree of Master of Science in Physics.

By

SADEQ SALTAN MASHAAN

B.Sc. Babylon University (2001)

Prepared under the Supervision of

Dr. Firas Zuhair Majeed

Jan 2014 1st Rabee 1435

Acknowledgements First of all profusely and all thanks are due to Allah lord of the

whole creation who enabled me to achieve this research and peace is upon his messenger Mohammad.

To acknowledge the pleasure of my supervisor Dr. Firas Zuhair

Majeed from suggesting this project, help, guidance, and advice throughout the work.

I would like to thank the Head and the staff of Physics Department

for all the assistance they gave. My thanks are also extended to the Dean of the College of Science for his support to perform this work.

Authors would like to express their thanks to prof. Dr. Raad A. Radhi

for his assistance to provide us the original versions of interaction and form factors algorithm and they are expressing their thanks to prof.B.A.Brown for his international code OXBASH.

Many thanks go to all my colleagues and all the members of the

nuclear group for their support. I am grateful to the staff of the library of the college of science and

staff of the central library of the University of Baghdad for providing references.

With my regards for all

Sadeq

Certificate I certify that the preparation of this thesis, entitled "Core Polarization Effects with M3Y-Sigma Meson Interaction for Longitudinal Form Factors of 48Ca Nucleus'' was prepared under our supervision by Sadeq Saltan Mashaan. At the College of science University of Baghdad in partial fulfillment of the requirements for the degree of Master of Science in physics. Signature:

Name: Dr. Firas Z. Majeed

Title: Assistant Professor

Date: 16 / 01/ 2014

In view of the available recommendations, I forward this thesis for debate by the

examining committee.

Signature:

Name: Dr. Raad M.S AL-Haddad

Title: professor

Head of Physics Department, College of Science

Date: / 01/ 2014

We certify that we have read this thesis, entitled "Core Polarization Effects with M3Y-Sigma Meson Interaction for Longitudinal Form Factors of 48Ca Nucleus'' and as examining committee, examined the student Sadeq Saltan Mashaan on its contents, and that in our opinion it is adequate for the partial fulfillment of the requirements for the degree of Master of Science in Physics.

Signature:

Name: Dr. Zaheda A. Dakhil Title : Professor (Chairman) Date : / 01/ 2014

Signature: Signature:

Name: Dr. Khalid S. Jassim Name: Dr. Gaith .N .Flaiyh Title : Assistant Professor Title : Assistant Professor (Member) (Member) Date : / 01/ 2014 Date : / 01/ 2014

Signature:

Name: Dr. Firas Z. Majeed Title : Assistant Professor

(Supervisor) Date : / 01/ 2014

Approved by the University Committee of Postgraduate Studies.

Signature:

Name: Dr. Saleh Mahdi Ali Title : Professor

Dean of the College of Sciences, University of Baghdad Date : / 01/ 2014

قل الروح عن ونك يسأل و أمر من الروح وتيتم أ وما يرب

لي ق إ م ل الع من

الرحيم الرحمن هللا بسم

العلي هللا صدق العظيم

) 85آية سورة اإلسراء/ (

TABLE OF CONTENTS

No. Subjects Page No.

English Cover

Arabic Cover

Acknowledgements

Certificate

اآلية الكريمة TABLE of CONTENTS I TABLE of Tables III TABLE of Figures IV Abstract VI CHAPTER ONE INTRODUCTION

1 Introduction. 1 1.1 Electron scattering. 1 1.2 Aim of the Present Work. 9 CHAPTER TWO ELECTRON SCATTERING 2.1 General Theory. 10 2.2 The Reduced Single-Particle Matrix Elements. 12 2.3 Many-Particle Matrix Elements. 15

2.4 The Reduced Single-Particle Matrix Elements of the Longitudinal Operator )( L=η . 16

2.5 Correction to the Form Factor. 18 CHAPTER THREE CORE-POLARIZATION EFFECTS 3.1 Introduction. 21

3.2 Effective and Residual Interaction(exchange M3Y by sigma meson). 23

3.3 Relative and center of mass transformation coefficient. 30 3.4 Some Properties of Brody-Moshinisky coefficient. 33 3.5 Energy Matrix Element with Oscillator Function. 33

3.6 The realistic Sigma meson (S=0 T=0) effective Nucleon-Nucleon interactions. 36

CHAPTER FOUR RESULTS, DISCUSSION AND CONCLUSIONS

4.1 Introduction. 41

4.2 The 48Ca Nucleus. 42

4.2.1 The Charge Form Factor for Jπ T= 0+ 4 State. 43

4.2.2 The C2 Charge Form Factor For Jπ T= 2+ 4 state. 46 4.2.2.1 The first Jπ T= 21

+ 4 state with Ex=3.658 MeV. 46 4.2.2.2 The second Jπ T= 2+ 4 state with Ex=5.949MeV. 48 4.2.2.3 The third Jπ T= 2+ 4 state with Ex=6.557 MeV. 50 4.2.2.4 The fourth Jπ T= 2+ 4 state with Ex=7.275 MeV. 52 4.2.2.5 The fifth Jπ T= 2+ 4 state with Ex=7.363 MeV. 54 4.2.3 The Inelastic Longitudinal C4 form factors (4 P

+P) State. 56

4.2.3.1 The first J P

πP T= 4 P

+P 4 state with Ex=4.134 MeV. 56

4.2.3.2 The second J P

πP T= 4 P

+ P4 state with Ex=6.387 MeV. 58

4.2.3.3 The third J P

πP T= 4 P


4.2.3.4 The fourth J P

πP T= 4 P


4.2.3.5 The fifth J P

πP T= 4 P


4.3 Conclusions. 66 4.4 Futurisms Research. 67 References 68

Table of Tables


(3-1) The values of the best fit to the potential parameters. 40

(4-1) The OBDM elements for the C0 transition with FPD6 interaction for ground state (Ex=0 MeV). 44

(4-2) The interaction OBDM elements for the C2 transition with FPD6 for 21+ state (Ex=3.658 MeV). 46

(4-3) The interaction OBDM elements for the C2 transition with FPD6 for 22+ state (Ex=5.949 MeV). 48

(4-4) The interaction OBDM elements for the C2 transition with FPD6 for 23+ state (Ex=6.557 MeV).

50


52


54


56


58


60


62


64

Table of Figures


1 Elastic charge form factor (C0) for the ground state 00+. 45

2 Charge form factor for the C2 state, with Ex=3.658 MeV (21+). 47




6 Charge form factor for the C2 state, with Ex=7.363 MeV (25+).

55

7 Charge form factor for theC4 state, with Ex=4.134 MeV ( 41+).

57


59


61


63


65

Abstract

Inelastic longitudinal electron scattering form factors of Ca48 has been investigated in the framework of Microscopic theory. The investigations have been performed in terms of the configuration mixing shell model with limiting number of orbital in the model space outside the inert core. The discarded space has been included, which is called core-polarization effects, which considers particle-hole excitations from the core orbits and from the model space orbits into the higher orbits with 2ħω excitation energy.

The sigma meson of two body Michigan sum of three range Yukawa potential

(M3Y) interaction has been utilized as a residual interaction. The FPD6 model space

was used as an effective interaction to generate the model space wave functions. The

simple harmonic oscillator potential is used as a single particle wave functions where

an analytical solution is possible.

The effective fitting interaction parameter was produced using the sigma(δ)

mesons exchange between two nucleons by using inelastic electron scattering from

the multiple transition (C0,C2 and C4) to five excitation energy levels for C2 and C4,

in fp-model space of P

48PCa by seven(E,P0,P1,P2,P3,P4 and P5) Yukawa potential

interaction (M3Y-type).

The C0, C2 and C4 form factors are enhanced by including core-polarization

effects, where previous studies give effective charges greater than those of free space,

to account for the core polarization effects.

Introduction

1.1Electron scattering

Inelastic electron scattering has been confirmed to be a powerful tool for studying properties of excited states of nuclei, in particular, their spins, parities and the strength, and structure of the transition operators connecting the ground and the excited states. Although some of this information is available from experiments, the electron scattering method has certain unique features that indicate its continued use. The information existing from experiments is certain to increase in quantity and quality[1].

Born approximation calculations, is based on realistic shell model wave functions, which should be carried out for those light elements available as suitable targets for experiments. Efforts to make realistic theories for heavy nuclei have already made large progress and should continue to do so [1]. It is clear that the enhanced theories, need extensive computer capabilities. It would be necessary that specialist modifying the theories and calculations to be in attachment so that an efficient program, without extensive duplication of effort may be carried out.

There are two fundamental reasons why electron scattering is such an efficient tool for studying nuclear structure. The first is that the basic interaction between the electron and the target nucleus is well defined. The electron interacts with the electromagnetic charge and current density of the nucleus. Since the interaction is relatively weak one can make calculations on the target nucleus without greatly affecting its structure. This is in difference to the situation with strongly interacting projectiles where the scattering mechanism cannot be clearly known and separated from structure effects in the target. By using electron scattering, one can directly join the cross section to the transition matrix elements of the local charge and current density operators, and thus directly to the structure of the target itself. Of course, the same concepts take an action involving real photons, but electrons have the second great advantage that for a fixed energy loss of the electron. One can vary the three-momentum transferred to the nucleus (q) [1].

Electron scattering is an efficient tool for spectroscopy of bound and unbound states in nuclei. Since transitional form factors have different q dependence with different multi-polarity by varying the momentum transfer, specific multi-poles can be excited and thus selectivity to spins is done.

Modern inelastic electron scattering experiments have produced well defined data as well as multi-pole transitions in some of doubly even 2p-lf shell nuclei within a measured momentum-transfer range up to q=3.0 fmP

-1P [1].

Theoretical solutions to the nuclear many-body problem are partly phenomenological, and thus theory and experiment are closely related to each other. Theory takes its inspiration from experiment in guiding the frame work of the models and their criteria; the nuclear shell model is the primary example [2,3,4]. Nuclear experiment takes its inspiration from theory in helping to choose which experiments are most important to coincide or disprove model assumptions. The nuclear shell model codes like OXBASH[5], MSHELL[6], ANTOINE[7], RETSSCHIL[8], VECSSE[9], and NATHAN[10] are globally used.

Besides the single particle wave function, the two body matrix element (TBME) has a most important action in the calculation of the nuclear properties named above, so that different shell model space effective interactions have been introduced depending on two body interaction strength, where fitting parameters are obtained from real nuclear interactions and fitting process [1].

After we have chosen the model space and searching for the most suitable model space effective interaction depending on literature, moreover through microscopic theory, the discarded space has been included as a first order correction, that is particle hole state (p-h), and using the mixing interaction in order to calculate these effects, as a residual interaction. Some of the most widely mixing interactions used to calculate this effect are; (modified-surface delta function interaction (MSDI)[11,12], density dependence Michigan sum of three-range Yukawa potential (M3Y-DD)[13, 14,…, 24], Skyrme-type Hamiltonian ( SKX) [25, 12, 26, 27], modified Skyrme-type Hamiltonian (MSK7) [28,12], Dirac-Hartree Hamiltonian (NL3)[29], Gogny interactions D1S[30, 26], ……….etc.) and the process is called (Core polarization).

For Nuclei of A>40, Z, N≥ 20, the fp-shell model space is the suitable space [12], where the nucleus Ca40 is considered as a core.

The fp-shell model space consists of the 1fR7/2R,2pR3/2R,1fR5/2R and 2pR1/2R valence orbits. The potential model, modified-surface one-boson exchange potential (MSOBEP), developed for the sd-shell [31, 32, 33] was successfully applied to obtain FPD6 ( six parameter density-dependent interaction for the lower part of the fp shell) [34,…, 41]. The effective SPE obtained for FPD6 for Ca48 , however can be improved by the inclusion of monopole interactions (beyond that already contained in FPD6) [42].

The full Effective two body matrix elements (ETBME) approach to the fp shell interaction has been feasible by the (the group matrix fitted to fp shell nuclei with density dependence parameters) GXPF1 and has been applied to all fp shell nuclei (A= 47-65 )[43, 44, 45, 46].

The theoretical calculation of electron scattering cross section multipole form factors, multi-pole transition probability, ….etc. are calculated by the use of nuclear shell model wave functions, which produced from the solution of Schrodinger differential equation and consisted of varies suggested potential energy function such as Harmonic Oscillator (HO)[11].

So, the electron scattering form factor had an interest during the past century and one can review them as related to the present work as follows:

In (1929) Mott [47] studied the electron scattering where he derived an expression of the cross-section for the scattering of accelerated electrons by a point nucleus. Collines and Waldman (1940) [48] established the first experiment on electron excitation of nuclei to discrete levels.

Inclusion of the so-called form factor is done first by Lyman et al (1951) [49]; they have made the first experiment, which is sensitive to nuclear size. The nuclear size can be found by multiplying the Mott’s cross-section by the nuclear form factor which depends on the charge, current and magnetization distribution in the target nucleus, where this factor can be found experimentally as a function of the momentum transfer (q), by knowing the energies of the incident and scattered electrons, and the scattering angle. Electron scattering may be classified into two kinds; Elastic and inelastic electron scattering, the elastic is the electrons are scattered from the nucleus, leaving the nucleus in its ground state and study the ground-states properties such as static distributions of charge and magnetization which are feasible. The latter kind which determines transition densities corresponding to the initial and final nuclear state in question, for the three quantities in the nucleus that interact with the passing electron, namely the distribution of charge, current, and magnetization had studied by Uberall and Ugincius (1969)[50]. The electron scattering process can be explained according to the first Born-approximation as an exchange of a virtual photon carrying a momentum (q) between the electron and the nucleus, by Deforest and Walecka (1966) [51]. The first Born-approximation is being valid only if Zα <<1, where Z is the atomic number and α is the fine structure constant (α = 1/137). According to this approximation, the interaction of the electron with the charge distribution of the nucleus is considered as an exchange of a virtual photon with zero

angular momentum along the direction of the momentum-transfer (q). This is called Coulomb or longitudinal scattering.

The possibility of studying single-particle nuclear transition charge densities with inelastic electron scattering have discussed by Walecka and Willey(1963) [52], and they have pointed out that such experiments lead to important background about the nature of the “effective charge”. A few simple calculations of cross sections are carried out for various models of the effective charge and it is shown that the momentum transfer dependence of the cross section distinguishes between them.

Bellicard et al (1967) [53] measured elastic electron scattering form factor for Ca48,40 using 750 MeV-electrons.

The low lying states of the calcium isotopes Ca42 through Ca50 were discussed within the framework of the conventional shell model by Mc Grory et al (1970) [54] and they considered Ca40 as an inert core, and best results had been obtained in the range of 0 ≤ q ≤ 2.5 fm-1 .

Differential cross sections for inelastic electron scattering from Ca44,42,40 and Ti50 had been measured by Heisenberg et al (1971) [55].

Vautherin and Brink (1972) [56] verified that many of the nuclear ground-state properties are well reproduced by the Hartree-Fock calculations using the Skyrme effective forces (SHF).

De Jager (1974) [57] calculated the radii of Ca44,42 theoretically and compared the results with the experimental values of the foregoing nuclei.

Relative sizes of Ca48,40 from the scattering of 79 MeV electrons had been calculated by Hiebert (1975) [58].

Backward-angle high resolution inelastic electron scattering on Ca48,44.42,40 had been carried out by Steffen et al (1978) [59], they observed of a very strong magnetic dipole ground state transition in Ca isotopes.

Monopole transitions from the 1+0 ground states to 2+0 excited states for Ca40 (3.353 MeV), Ca42 (1.837 MeV) , Ca44 (1.884 MeV) and Ca48 (4.272 MeV) had been investigated by Gräf et al (1978) [60] with high resolution inelastic electron

scattering (FWHM ≈ 30 keV) at low momentum transfer (0.29fmP

-1P<q<0.53fmP

-1P).

These results are used together with known ground state charge radii and the average number of holes in the sd shell in the ground state to estimate the number of particle-hole excitations in the wave functions of the excited 0 P

+P states .

Bellicard et al (1980) [61] studied the elastic electron scattering cross section for Ca48 at 757.5 MeV, the results are compared with experimental data specially in the

range of q ≥ 2 and they obtained a good estimated results.

Itoh et al (1983) [62] studied the electro excitation of giant electric dipole and electric quadruple cross section of Ca,44.42 by electron scattering, good agreement had been produced.

Coulomb form factors for the 0 P

+P to 2 P

+P transitions in the even-even 1fR7/2R-shell

nuclei were studied by Iwamoto et al (1982) [63] in terms of the shell model within the f n1 2/7

, pf n

2/3

1

2/721 − configurations and with the effective interactions, it was proven

that the characteristics of the C2 form factors in the higher-momentum-transfer region, have not shown by the simple model, but can be explained by the mixing of the one-particle excitations into the 2pR3/2 Rorbit in the shell-model wave functions. E2 transition strengths and q moments were also discussed in connection with the C2 form factors. Ca48 had been widely studied experimentally by Wise et al (1985) [64] utilizing inelastic electron scattering, they completed the measurements over a momentum transfer range q=(0.6 to 3)fmP

-1P for a wide variety of multiple transitions

for normal and abnormal transitions.

The modeling of the mean square charge radii of nuclei in the Calcium region was determined by Emrich et al (1983) [65] for several proton and neutron shell closures, measurements on the six stable Calcium isotopes (Z = 20),where there isotopes covered the 1fR7/2 Rshell between the doubly magic nuclei Ca40 (N = 20) and Ca48 (N = 28). The charge radii of these Calcium nuclei have been studied by electron scattering.

Heisenberg (1983) [66] through his work on charge and current transition densities reviewed a few examples to explain how electron scattering modified our picture of nuclear physics, and he said “The measurement of electron scattering from nuclei has been a major contribution to the advancement of the field of nuclear structure”.

Also, a review for inelastic electron scattering from nuclei was published by Heisenberg and Block (1983) [67].

Inelastic electron scattering form factors for the excitation of 2+ states in some 2p-1f shell nuclei has been carried out by Mukherjee and Sharma (1984) [68]. The quadruple moments of the first excited states as well as the reduced transition probabilities for E2 transitions are also discussed in connection with the C2 form factors. The overall agreement between the calculated and the experimental results is quite good.

Sick (1984) [69] introduced a brief study of electron scattering form factor for wide range of mass number (A).

Brown et al (1985) [70] concluded that the inelastic electron scattering on Ca48 is one of the key experiments in making the association between experimental and theoretical energy levels. They had used the shell-model multi-particle transition densities together with the SKX HF single-particle radial wave functions to calculate the longitudinal and transverse electron scattering form factors.

Horie and Yokoyama (1987) [71] carried out a study on inelastic electron scattering form factor in some closed shell nuclei within the framework of the first-order perturbation theory, which is included core polarization (particle-hole) effect, for studying inelastic electron scattering form factor in some closed shell nuclei. It was shown that the suppression of transition strength, which had been observed in electron scattering and harmonic reaction, can be obtained theoretically in all the closed shell nuclei.

Platchkov et al (1988) [72] measured the radius of (1f7/2) neutron orbit by the use of elastic electron scattering experiment, and he obtained r=3.96±0.05.

Form factors and transition charge densities for the quadruple and hexadecupole electro excitation of some lf-2p shell nuclei have been carried out by Raina and Sharma (1988) [73]. It turns out that the available form factor data out to about 2.5fm-1 could be reproduced in most of the eases in a fairly satisfactory manner in terms of reasonable values of effective charges.

Zheng and Zamick(1989) [74] studied the relations between polarized-proton-nucleus and un polarized-transverse-electron-nucleus scattering and their applications in Ca42 .

http://prola.aps.org/search/field/author/Horie_H

The electron scattering form factors has been measured by Itoh et al (1989) [75], for 2+ ,3+ and 5+ states up to 7 MeV excitation in Ca42 and Ca44 , the range of the incident electron energy were (62.5–250) MeV.

Frois and Sick (1991) [76], at World Scientific introduced general review on electron scattering form factor for nuclei with the mass region from A=3 →209.

Effects of the neutron spin-orbit density on the nuclear charge density, in relativistic models, have been investigated by Kurasawa and Suzuki (2000) [77]. They explained the difference between the cross sections of elastic electron scattering of Ca40 and Ca48

, which is not reproduced in the non-relativistic models.

Results on ongoing research of elementary electric and magnetic nuclear excitations at the superconducting Darmstadt electron linear accelerator (S-Dalinac) were presented by Richter (2000) [78].

Precise measurements of charge form factor for some fp shell nuclei have been carried out by Denyak et al (2004) [79] for 0 ≤ q ≤ 2.5 fm-1.

The electron scattering form factor in 48Ca with the same theoretical framework and use M3Y of Bertsch as a residual interaction had studied by Majeed (2009) [80].

The electron scattering form factor in 48Ca with the same theoretical framework and used the shell model codes M3Y interaction had studied by Jassim (2012) [81].

The electron scattering form factor in 48Ca with the same theoretical framework and use M3Y of Nakada and MSDI interaction had studied by Kasim (2012) [82].

Al-lamy (2013) [83] used separate (N-N)channel to study the electron scattering form factor in 48Ca and proved that S=0 T=0 channel has the major effect.

The theoretical part of the present work includes the formulations of the inelastic electron scattering and will be performed in chapter two.

The derivation of Core Polarization (CP) effects with higher configuration in the first order perturbation theory and the two-body matrix elements of three parts of the realistic interaction: central, spin orbit and strong tensor force will be introduced in

chapter three, where the two-body matrix elements calculation in harmonic oscillator single-particle basis using Moshinisky transformation.

The results, discussion and conclusions will be demonstrated in chapter four.

A computer program is written in FORTRAN 90 language to include realistic interaction M3Y in the original code which calculates the model space form factors (zero-order) and the first-order core polarization effects. This code is written by prof. R.A.Radhi.

1.2 Aim of the Present Work

The aim of present work is to use electron scattering formalism (theoretical) to study the difference between sigma(δ) meson exchange interaction belongs to M3Y-DD with different sets of fitting parameters as function of momentum transfer, with use of 48Ca as a testing nucleus, with C0, C2 and C4 longitudinal form factors as a transitional multipolarties, with five exited states for C2 and C4, the reason behind the selection of this nucleus and in the same time the transitional multipolarities are to give the model space part no contribution and to make the core part dominant so reflecting to the effects of residual interactions and its sets of fitting parameters.

Sigma(δ) meson of Michigan sum of three-range Yukawa potential (M3Y) interaction of Nakada et al (2008) [84] will be adopted as a residual interaction for the core polarization matrix elements.

Electron Scattering

2.1 General Theory The differential scattering cross-section for the scattering of an electron into a solid angle (dΩ) from a nucleus of charge (Ze) and mass (M) in the plane-wave Born approximation (PWBA) is given by [51,85,86]:

Ωd

dσ = R

Mottdd

Ωσ

R

ƒRrecR 2),(∑

JJ qF θ (2-1)

where R

Mottdd

Ωσ

R

is the Mott cross-section for high-energy scattering electron from a

point spineless nucleus, which is given by:

R Mottd

d

Ωσ

R

= 2

)2/(2siniE2

)2/cos(Z

θ

θα (2-2)

where ( )137/12 == ce α is the fine structure constant [51], Z is the atomic

number of the target nucleus, θ is the scattering angle and ERi R is the energy of

incident electron.

The recoil factor of the nucleus is given by:

ƒRrecR =

1

)2/(sin2

1 2−

+ θ

MiE

(2-3)

where M is the mass of the target.

the total nuclear form factor FRJR(q,θ) which is containing two parts the longitudinal (Coulomb) part L

JF and the transverse (electric and magnetic) part TJF with multi-

polarity J as a function of momentum transfer q is given by:

++= )2/(tan

2)()( 2

2

22

42

θµµ

qq

qFqq

qF LJJ

2)(qFT

J (2-4)

The four-momentum transfer µq is given by, (with ћ=c=one for abbreviation):

222 )( fi EEqq −−=µ (2-5)

with 222 )()2/(sin4 fifi EEEEq −+= θ

22 )2/(sin4 ωθ += fiEE (2-6)

where iE and fE are the initial and final total energy of the electron scattering

respectively and ω is the energy transfer from the electron to the nucleus. The squared transverse form factor is given as the sum of the squared electric and

magnetic form factors as follows:

2

)(qFTJ =

2

)(qF ElJ +

2

)(qF magJ (2-7)

The form factor of the given multipolarity J as a function of momentum transfer (

q ) is given as:

2

)(qFJη =

)12(4

2 +JZ i

π

2

)( JqTJ iJf

η∧ (2-8)

The subscript η represents longitudinal (L) or transverse (T) (electric( El) or

magnetic (mag)), Ji and Jf are the total angular momentum of the initial and the final

states, respectively, and η

JT∧

)(q is the electron scattering multipole operator.

By the use of Wigner-Eckart theorem in isospin space, the form factor can be

written in terms of the matrix element reduced in both angular momentum (J) and

isospin (T) (triple-bar matrix elements) and is given as[87]:

2

1,0,2

2)1(

)12(4)( ii

TTJff

i

ZiT

f

Zf

zfTfT

iJ TJTTJ

T

T

T

M

T

TJZqF ∑

−−

+=

=

∧

−η

η π (2-9) where:

2NZTz

−=

The bracket ( )..................... denotes the 3j-symbol. Since Tzf=Tzi for electron scattering, then

MT=zero.

2.2 The Reduced Single-Particle Matrix Elements:

The single-particle wave functions α are specified by:

tt zmjln ααααα = (2-10)

Where αn is the principal quantum number, αl is the orbital angular momentum

quantum number and αj is the total single-particle angular momentum quantum

number. The state ztt is the single-particle isospin state, with zt =21 for proton

and 21

− for neutron.

The radial part )(rRln nl=αα is normalized as:

1)( 22

0=∫

∞drrrRnl (2-11)

The single particle transition operator depends on the single nucleon, which is a

proton or a neutron and it can be written as:

−++= TTT nzpzztˆ)1(

21ˆ)1(

21

ˆ ηηη ττ (2-12)

where:

η

npT ,ˆ is the single particle operator for proton/neutron.

By rearranging equation (2-12), the transition operator can be written as:

ηηηηηηη τ 10ˆˆ)ˆˆ(

21)ˆˆ(

21ˆ

== +=−++= TTnpznpztTTTTTTT (2-13)

where η0

ˆ=TT is the isoscalar part of the operator and η

1ˆ=TT is the isovector part of the

operator.

The reduced single-particle matrix element of the isoscalar and isovector parts

between two single-particle states (α) and (β) are [11]:

βαβαβα ηηη10

ˆˆˆ=+== JTJTJT TTT

ttjlnTTjlntjlnTtjln JnJpJT 1ˆˆ21ˆ βββ

ηηαααβββ

ηααα +=

ttjlnTTjln zJnJp τβββηη

ααα ˆˆˆ21 −+ (2-14)

Since:

⟨𝑡𝑡𝑧|1|𝑡𝑡𝑧⟩ = (−1)𝑡−𝑡𝑧 𝑡 0 𝑡−𝑡𝑧 0 𝑡𝑧

⟨𝑡||1||𝑡⟩ , (2-15)

and

𝑡 0 𝑡−𝑡𝑧 0 𝑡𝑧

= 1√2𝑡+1

(2-16)

then:

=1 2112

112

1+t

(2-17)

hence:

2122112

1 =+= t (2-18)

and:

⟨𝑡𝑡𝑧|𝜏𝑧|𝑡𝑡𝑧⟩ = (−1)𝑡−𝑡𝑧 𝑡 1 𝑡−𝑡𝑧 0 𝑡𝑧

⟨𝑡||𝜏𝑧||𝑡⟩ (2-19)

where:

𝑡 1 𝑡−𝑡𝑧 0 𝑡𝑧

= (−1)𝑡−𝑡𝑧 𝑡𝑧𝑡(2𝑡+1)(𝑡+1)

(2-20)

Then:

621

21

21

21

)1)(12(1

=∴

++=

z

zz

tttt

τ

τ

The reduced single particle matrix element becomes:

βββη

αααη βα jlnTjlnt

tITT

zJtz

z

TJTˆ)(

212ˆ ∑+

= (2-21)

where:

0

1

1

)1(

)(

21

=

=

−

=−

T

T

for

fortI

ztzT

(2-22)

The reduced single-particle matrix elements in proton-neutron formalism could be

rewritten as:

≡βα ηzJtT βββ

ηααα jlnTjln

zJtˆ (2-23)

2.3 UMany Particle Matrix Elements

Many particle matrix elements of the electron scattering operator ηΛT are

expressed as the sum of the product of the one-body density matrix elements (OBDM) times the single-particle transition matrix elements [88]: βαβα η

βα

ηΛ

Λ ∑ ΓΓ=ΓΓ TOBDMT fiifˆ,,,ˆ

,

(2-24)

where JT=Λ is the multi-polarity in spin and isospin in respectively and the states

iii TJ≡Γ and fff TJ≡Γ are the initial and final states of the nucleus, while α and β

denote the final and initial single-particle states, respectively (isospin is included).

In isospin representation, the value of JTOBDM can be written as a sum of isoscalar

and isovector parts of the ztJOBDM , as [89]:

200

2)1(0,

,=−

−

−=TJ

iTfTztJ OBDM

TTTT

OBDMzizf

if

201

6)2(1, =

−

+TJOBDM

TTiTT

ztzizf

f (2-25)

where 12

][ ~

,',,

+

⊗=

+

Jttt JaaJ

OBDMi

J

jjfJ zzz , )12)(12(

][ ~

',

++

⊗=

ΓΓΛ+

TJijjfTJ aa

OBDM

Where; a + , a ~ and ⊗ are making, destructing and coupling the particle in initial

and final states respectively.

The elements of the OBDM for longitudinal C0 (J=zero) form factor are given by:

( )0=∆TOBDM ( )npn j +=1212

212

+++

jJT ii (2-26)

( )1=∆TOBDM ( )npn j −= 1212

6)1)(12(

++++

jJ

TTTT i

z

iii (2-27)

where ( )npnj + and ( )npnj − are the occupation numbers .

2.4 UThe Reduced Single-Particle Matrix Elements of The Longitudinal

UOperator U ( L=η ):

The longitudinal form factor describes the spatial distribution of the charge (the

transition charge densities), so the longitudinal scattering can be considered as a

result of the interactions of the incident electrons with the charge distribution of the

nucleus [51].

The longitudinal form factor operator is defined as [90, 51]

),(ˆ)()()(ˆzrJMJ

LzJMt trYqrjrdqT ρΩ∫= (2-28)

where )(qrjJ is the spherical Bessel function, )( rJMY Ω is the spherical harmonic

function and ),(ˆ ztr

ρ is the nucleon charge density operator, which is given by :

)()(),(ˆ izz rrtetr

−= δρ (2-29) with

2)(1)( ite z

zτ+

= , zz t2=τ

and

)( irr

−δ is Dirac delta function

From equations (2-29) and (2-30), one obtains:

)()()()(ˆrJMJz

LzJMt YqrjteqT Ω= (2-30)

The reduced single-particle matrix element of the longitudinal operator between the

initial β and final α states can be written as[51]

ββααββααβα jlYjllnqrjlnteT rJJzL

zJt 21)(

21)()(ˆ Ω= (2-31)

For the spherical harmonic, the reduced matrix element, is given by [11]

( ) 21

121

21

21 +

−=Ω α

ββααj

rJ jlYjl

++−+ Jll βα)1(1

××

−

+++ βαβα

π

jJjJjj

210

214

)12)(12)(12( (2-32)

Relation (2-32) could be expressed as:

ββααβαβαβα lnqrjlnjjCllelPteT JJJzL

zJt )(),(),()(ˆ = (2-33)

where JP and

JC represent the coefficients of the electric parity-selection rules given

by,

++−+= JllJ llP βα

βα )1(121),( (2-34)

−

+++−=

+ βαβαα

βα π

jJjjJjjjC jJ

210

214

)12)(12)(12()1(),(21

21 (2-35)

while

)()()()( 20

rRrRqrjdrrlnqrjln lnlnJJ ββααββαα

∞

∫= (2-36)

The solution of Schrodinger equation using harmonic oscillator will generate the

radial component of the single particle wave function and the size parameter b, which

is set to the value brms that reproduces the root-mean square (rms) charge radii.

Therefore, with these wave functions, the radial matrix elements of Bessel function

can be treated analytically as:

21

2 )!1()!1()exp(!)!12(

2)(

−−−

+= αβββαα nnyy

Jlnqrjln

JJJ

21

)21()2

1(

++Γ++Γ× ααββ lnln

)!1()!1(

1!!

)1(1

0

1

0 −−−−∑

−∑×

−

=

+−

= ααββ

α

α βα

α ββ

β mnmnmm

n

m

mmn

m

)2

3()23(

))322(21(

++Γ++Γ

+++++Γ×

ββαα

βααβ

lmlm

Jmmll

);23);22(

21( yJmmllJF +−−−−× αβαβ

(2-37)

where: 2

2

= bqy ,

Γ is gamma function

and ( ).......F is the confluent hypergeometric function, which can be evaluated

using [51]:

.......!2)1(

)1(1,,2+

++++=

yyyF

χχκκ

χκχκ (2-38)

2.5 Corrections to The Form Factor: Nucleon- Nucleon experiments show that the nucleon is not a point but has a finite

size that must be corrected for, so the nuclear form factor of equation (2-9) must be

multiply by this correction, which is found to be as[91],

443.0

.

2

)(q

sf eqF−

= (2-39)

On the other hand, there is a non-physical excited states called spurious states, due

to the fact that the interaction potential represents an average potential with respect to

a fixed origin. The problems of removing spurious states have been investigated

previously [92]. It has been shown that the calculated shell-model form factors for the

non- spurious ground state should be multiplied by the center-of-mass(𝑐.𝑚)

correction [91]:

Abq

mc eF 4.

22

= (2-40)

Where; b is the harmonic-oscillator size parameter, and A is the nuclear mass

number.

For nuclei in which Ζα«1, the Plane Wave Born Approximation (PWBA) is

expected to describe the electron scattering data very well, except in the region of the

diffraction minima, where the PWBA goes to zero. The Coulomb distortion of the

electrons increases the q, and an effective momentum transfer can be used to include

these effects. The effective momentum transfer( effq )is given by [93]:

+=ci

eff REZeqq

2

231 , rmsc rR

35

= (2-41)

In the realistic calculation of the form factor, it is necessary to take into account the

effects of finite proton size, center of mass motion and Coulomb distortion of

electron waves. Thus, the form factor for a given multipolarity J can be written in

terms of the matrix elements reduced both in angular momentum and in isospin

spaces as [94]: 2

1,0,2

2)()1(

)12(4)( i

TTJf

i

ZiT

f

Zf

zfTfT

iJ qT

T

T

T

M

T

TJZqF Γ∑ Γ

−−

+=

=

∧

−η

η π

2.

2. )()( qFqF sfmc ×× (2-42)

Core-Polarization Effects

3.1 UIntroduction

The shell-model theory has shown that the true space may be divided into three

separated spaces which are: model space, inert core and higher configurations. Higher

orbits might be included or excluded according to the choice of the researcher and the

model that uses, but core orbits as have been proved [95], has an active contribution

in the calculation of form factors. The main problem is that the inclusion of core

orbits makes the space (Hilbert Space) very vast so, it must be separate between the

two spaces ( a core part and a valence part) to express the interaction between the

core and the valence particles, and that among the valence particles.

Through microscopic theory, the core polarization effect on the form factor

combines shell-model wave functions and configurations with higher energy as first

order perturbations; these are called "core-polarization effects".

In the shell model, it is unable to solve Schrodinger equation in the full Hilbert

space, because of the huge number of configurations, so one must trnucate it to a

smaller part of the configurations with finite dimenstions Hilbert space. This is

called the shell model-space or simply the model space. For this reason, it must use

effective interactions and operators. The effective interaction is used to measure the

nuclear properties microscopically, starting with realastic N-N interaction using

quantum mechanical many-body theory. For light nuclei, there are different effective

interactions such as the FPD6 [42], GXPF1[43], interactions for fp-shell. Some

special types of model space effective interactions are tabulated in the code

OXBASH, V.2005 [5].

The reduced matrix elements of the electron scattering operator ( ηΛT

) consist of

two parts, one is for the "Model space" matrix elements, and the other is for the

"Core-polarization" matrix elements,

.

ˆˆˆCP

ifMS

ifif TTT ΓΓ+ΓΓ=ΓΓ ΛΛΛηηη δ (3-1)

Where, MS

if T ΓΓ Ληˆ are the model-space matrix elements ,

and, CP

if T ΓΓ Ληδ are the discarded space matrix elements.

iΓ and fΓ are described by the model-space wave functions.

The model-space in the light fp-shell nuclei is defined by the following

configuration: (1f7/2,2p3/2,1f5/2,2p1/2 ).

The fp-shell model-space matrix elements are written as the sum of the product of

the one-body density matrix elements (OBDM) multiplied by the single-particle

matrix elements, which is expressed by:

MSfi

MSif TOBDMT βαβα η

βα

ηΛΛ ∑ ΓΓ=ΓΓ ˆ),,,(ˆ

,

(3-2)

where α and β denote the final and initial single particle orbits, respectively

(isospin is included) for the 1f 2p-shell model-space.

The core-polarization matrix elements in equation (3-1) can be written as [95]:

cpfi

cpif TOBDMT βδαβαδ η

βα

ηΛΛ ∑ ΓΓ=ΓΓ ˆ),,,(ˆ

, (3-3)

3.2 Effective and Residual Interaction(exchange M3Y by sigma meson):

The so-called Michigan 3-range Yukawa (M3Y) interaction [14] has been derived

from the bare N-N interaction, by fitting the Yukawa functions to the G-matrix. It is

represented by the sum of the Yukawa functions. The M3Y-type interactions will be

tractable in various models. It has been shown that the M3Y interaction gives matrix

elements similar to reliable shell model interactions [31]. Moreover, with a certain

modification, M3Y-type interactions have successfully been applied to nuclear reactions

[84], including electron scattering. A class of the M3Y-type interactions have been

applied to electron scattering form factor calculations, with Paris and Ried fitting[86]. In

this work, we shall use M3Y- Paris and Ried type interaction has been used as a

Residual interaction and investigate the modification of the calculations. Once a reliable

interaction is obtained, one can describe various nuclear properties accurately and

systematically, which helps us to understand the underlying structure, and to make

predictions for unobserved properties. The fp shell is a region where the shell model can

play an indispensable role, and is at the frontier of our computational abilities. In the fp

shell one can find the interplay of collective and single-particle properties, both of

which the shell model can describe within a unified framework. Since the protons and

neutrons occupy the same major shell, the proton-neutron interaction is relatively strong

and one can study the relative collective effects such as T = 0 pairing. For all these

applications, a suitable effective interaction for fp-shell nuclei is required. Because of

the spin-orbit splitting, there is a sizable energy gap between the 1f7/2 orbit and the other

three orbits (1f7/2,2p3/2,1f5/2,2p1/2). Thus, there exists an N or Z = 28 “magic” number

inside the major shell. For shell-model calculations around this magic number, Ni56 has

often been assumed as an “inert core”. However, it has been shown that this core is

rather soft [97] and the closed shell model for the magic number 28 provides a very

limited description, especially for nuclei near N or Z=28 semi-magic. It is necessary to

assume essentially the full set of fp shell configurations and the associated unified

interaction in order to describe the complete set of data and to have some predictive

power. The effective interaction can, in principle, be derived from the free nucleon-

nucleon interaction. In fact, such microscopic interactions have been suggested for the

fp shell.

Quantum mechanics predicts that the inclusion of effective interaction might be

carried out through the perturbation theory and correcting the model space wave

functions and excitation energies. The first step is to define the effective interaction

between the two particles and adding it to the total Hamiltonian as follows [98,11].

VHH )( += 0 (3-4)

where )0(H represents the unperturbed Hamiltonian and V is the effective

nucleon-nucleon interaction.

The )0(H might be described by a complete orthonormal basis which are

unperturbed wave functions Φ, which satisfy Schrödinger equation:

iii EH Φ=Φ)0( i = 1,2,…. (3-5)

In Hilbert space the true wave function can be written in terms of the unperturbed

basis wave function as:

∑ Φ=Ψ∞

=1iiia (3-6)

Where; ia the amplitude of eignfunction of Ф𝑖,however, the summation runs over a

complete orthonormal set of eigenfunctions Ф𝑖.

The model space wave function Ψ′ can be obtained from the expansion,

∑ Φ=Ψ∈Mi

iia' (3-7)

The effective Hamiltonian effH is required to reproduce the true eigenvalue E for the

model wave function 'Ψ , and the effective interaction is defined by [87]:

µµ Ψ+=Ψ PVHPH effeff ˆ)(ˆ )0( (3-8)

where the eigenvalues are a subset of the eigenvalues of the original Hamiltonian in

the full space, µ = 1, 2, ……. d, where d denoting the dimension of the model space. P is a projection operator.

The division of the Hilbert space into a model-space (MS) and a discarded part can be

accomplished by the construction of the so called projection operators (P) that is

optionally built which is a small part of the Hilbert space, where the remaining part is

the discarded space (core + higher configurations) obtained by the effect of operator Q

and they were expressed as:

∑ ΦΦ=∈Mi

iiP (3-9)

i

MiiQ Φ∑ Φ=

∉

ˆ (3-10)

where P projects onto the model space and Q projects off the model space, and they have the properties:

1=+QP , PP =2 , QQ =2 , 0== PQQP (3-11)

P and Q are expressed in terms of the eigenfunctions of the unperturbed

Hamiltonian H (0), they satisfy the commutation relations:

00 =

)(H,P , 00 =

)(H,Q (3-12)

So the application of these operators does not change the properties of the unperturbed hamiltonian.

The true wave function Ψ can be written as:

Ψ+Ψ=Ψ+=Ψ QQP ˆ')ˆˆ( (3-13) where: Ψ=Ψ P' (3-14) For a particular state the true wave function Ψ satisfies Schrödinger equation:

Ψ=Ψ+=Ψ EVHH rms)ˆ( )0( (3-15)

where V is the free nucleon-nucleon interaction and E represents the true energy of

the system.

Making simplifications in order to get the most familiar sort of the effective

interaction [11]:

.....VHEQV

HEQVV

HEQVVV

)()()(eff +−−

+−

+=000

(3-16)

One can derive the expression for the true wave function Ψ in terms of model

wave function 'Ψ [87,3,11]:

Ψ−

+Ψ=Ψ VHEQ

rms)0(' (3-17)

and it can be iterated to give:

.....''')0()0()0(

+Ψ−−

+Ψ−

+Ψ=Ψ VHEQV

HEQ

VHEQ

rmsrms (3-18)

Since the operator Q projects off the model-space, i.e.:

0ˆ =Ψ′Q (3-19)

Therefore, the true wave function Ψ is normalized to the relation:

1=Ψ′Ψ′=ΨΨ (3-20)

By using equation (3-13) and equation (3-15) one can verify the relation:

Ψ=Ψ VV rmseff' (3-21)

Equation (3-21) shows that the action of the effective interaction on the model-

space wave function yields the same result as the action of realistic nucleon-nucleon

interaction on the true wave function.

The first order perturbation theory says that the single-particle matrix element for

the higher-energy configurations can be expressed as [11]:

βαβαβδα ηηηresJJresJ V

HEQTT

HEQVT

)0()0(ˆˆˆ

−+

−= (3-22)

where resV represents a residual nucleon-nucleon interaction, and might be

simplified as fallows [11]:

hVpeeee

pThVHEQT res

hpJ

hpresJ βαβα

αβ

ηη ×+−−

×∑=−

1ˆˆ,)0(

(3-23)

pVheeee

hTpTHEQV res

hpJ

hpJres βαβα

βα

ηη ×+−−

×∑=−

1ˆˆ,)0(

(3-24)

the summation includes all possible particle-hole states and ke is the single-particle

energy (k ≡ α, β, p, h).

In equations (3-23) and (3-24), the energy denominators are the excitation

energies of the intermediate states with respect to the unperturbed valance states α

and β .

In order to reduce the single-particle matrix elements, Wigner-Eckart theorem [88]

is used, with taking care of the proper normalization of the angular momentum

coupled two-particle states one can obtain from equation (2-23) the expression:

∑+

=−

MJm

jmjres

h

hpp h

hhpp

J,M j

mjJMmjmjV

HE

QηTmj

,,

,,,12)0( ββαα

JMmjmjeeee

jTjppph

hpJ ααη

αβ

×+−−

××1ˆ

)1)(1(hphresphh J

jjVjjJMmjmjβαβαββ

δδ ++××× (3-25)

where the summation over the three Clebsch-Gorden coefficients can be performed

to yield a 3j-symbol and 6j-symbol, which is given as (from the relations those couple

the Wigner –symbols)[11]:

−

−−=

−

β

β

α

α

ααββ

ηαα

j

m

J

M

j

m

mjmjresV

HE

QMJTmj )1(

)0(,ˆ

++−

+−−+

×∑×J

J

j

j

j

j

Jjj

eeeeJjTj

Jjjph

h

hppJh

hp '

')1()1'2(ˆ

',,

βα

αβ

η β

)1)(1(' hphresp J

jjVjj βαβα δδ ++×× (3-26)

where

...

...is the 6j-symbol.

In space and isospace coordinates, equation (3-26) becomes:

'

,,

,,)0(, )1()1'2)(1'2(ˆ Jhjpj

hpThtpt

JhjpjresTJ eeee

TJtjVHEQTtj ++−×

+−−++

∑=− αβ

ββααη

'',

'

'')1(

TJhresp

phph

Thtpt jjVjjT

T

t

t

t

t

J

J

j

j

j

jβα

βαβα×××−×

++

)1)(1(ˆ, hpppTJhh tjTtj βαη δδ ++×× (3-27)

The Greek-symbols is used to denote quantum numbers in space and isospace

coordinates, Equation (3-27) becomes:

∑+−−

−=

− Γ

Γ++

Λ,

2,

1 21

2

)0()1(ˆ

αα αααβ

αβη βα

eeeeV

HEQT res

Λ

Γ×+Γ

β

α

α

α 12

)12(

)1)(1(ˆ2112

21 βαααη δδααβααα ++××× Λ

ΓTVres

(3-28)

where the index pptj≡1α runs over particle states and hhtj≡2α runs over hole

states and ke is the single-particle energy, ααα tj≡ and βββ tj≡ are the final and

initial single-particle states, ''TJ≡Γ correspond to the total spin in both spaces,

respectively, ηΛT denotes the single-particle transition operator of rank J in space

coordinate and rank T in isospace )( JT≡Λ , =T 0 or 1 are denoting the isoscalar or

isovector contribution [99].

Similarly one can find the contribution of the second term of equation (3-24)

as:

Γ++

ΓΛ

Λ

Γ×+Γ

+−−−

∑=−

β

α

α

αβα

αααβ

αβ

αα

η

2121

2

,2,1)0( )12()1(ˆ

eeeeT

HEQVres

)1)(1(ˆ122112 βααα

η δδααβααα ++××× ΛΓ

TVres (3-29)

The core-polarization form factor can be evaluated by replacing the single-particle

matrix elements in equation (3-3) with the two matrix- elements which are given in

equations (3-28) and (3-29).

The single-particle energies are calculated according to [11]:

21

21

121

212

12−=

+=

+−+ω−+=

lj

lj

for

for

nl)r(f)l(

nl)r(fl)ln(nlje

(3-30)

with:

3/2253/145

3/220)(

−−−=

−−≈

AA

MeVAnl

rf

ω

(3-31)

For the two-body matrix elements of the residual interactionΓ21 βααα resV and

Γ12 βααα resV , which appear in equations (3-28) and (3-29), the Michigan sum of

three range Yukawa potential (M3Y) interaction of Berstch et. al [14] is adopted. A

transformation of the wave function from jj to LS coupling must be done to get the

relation between the two-body shell model matrix elements and the relative and

center of mass coordinates, using the harmonic oscillator radial wave functions with

Brody-Moshinsky transformation.

3.3 Relative and center of mass transformation coefficient: For a two-particle system in a harmonic-oscillator potential, it shall characterize by

their coordinates and quantum numbers . For the purpose of our discussion, two

systems of coordinates are introduced:

1.Laboratory frame, where the two particles are described by their intrinsic

coordinates with respect to the center of the potential well, →

r1 and →

r2 , and

corresponding radial quantum numbers, n1, n2 and 1l , 2l , respectively.

2.Center of mass frame, where the system is characterized by the relative

coordinate r and the coordinate R of the center of mass of the two-particles,

defined as [100]:

)( 21 rrr −= , )(2

121 rrR += (3-32)

The wave functions in the two coordinate systems can be written as follows

[100,101]:

• Laboratory frame:

λµ

λµ φφψ

×=

→→

)()()2(),1( 212211 2211rrlnln Ho

lnHoln

Ho

∑=21

2211mm

mlml λµ )()( 21 2211

→→

rr lnln φφ (3-33)

The notation )2(),1( 2211 lnln means particle number 1 is in the orbit 11ln and particle

number 2 is in the 22ln orbit, Holn 11

φ and Holn 22

φ are the wave functions for particle 1 and 2,

respectively.

• Center of mass frame:-

λµ

λµ φφψ

×=

→→

)()( RrlL HoNL

Honl

Ho =∑mM

lmLM λµ )()(→→

Rr NLnl φφ (3-34)

where HoNL

Honl φφ , are the wave functions of relative and center of mass, respectively.

The principal and the radial quantum numbers n, are corresponding to the relative

motion, and N, L to that of the center of mass motion.

Eq (3-33) can be wrirren in terms of eq (3-34) as:

λµ

φφ

×

→→

)()( 212211

rr Holn

Holn

λµλ φφ

×=

→→

∑ )()(2211 RrlnlnnlNLM HoNL

Honl

nlNL

(3-35)

The coefficient 2211 lnlnnlNLM λ that appears in equation (3-35) is called the

transformation braketts (Brody-Moshinisky brackets).

The square of λM gives the propability that the two particle systems will be found

in a state of relative motion characterized by the oscillator function ),()( φθlmnl YrR and

center of mass motion ),()( ΦΘLMNL YRR

The transformation braket vanishes for all combinations of its parametres which

do not satisfy the total angular momentum

Llll +=+= 21λ

and the degenerated eigen value (energy) in this case is:

ωω )232()

232(),( 22112211 +++++= lnlnlnlnE

ωω )232()

232(),( +++++= LNlnNLnlE (3-36)

it follows from the conservation law of energy that [101]:

),( 2211 lnlnE = ),( NLnlE

LNlnlnln +++=+++ 2222 2211 (3-37)

Therefore, the transformation of Moshinsky bracket vanishes for all combinations

of its parameters which do not satisfy the energy condition of equation (3-37), and

any summations over λ will be restricted to [101,100],

2121 llll +≤≤− λ

LlLl +≤λ≤− (3-38)

The wave functions )(rnlφ describe states with energy ω)232( ++ ln and have the

form:

),()()( ϕθφ lmnlnl YrRr = (3-39)

Where,

( )[ ]( )[ ] l

lnl

nl rrlnlrR 22

2/1

2

322

exp!!12

!)!122(2)( απ

α−

+++

=++−

( )∑

= ++−+−

×n

k

kkk

klknkrln

0

22

!)!122()!(!)(!)!12(!21 α

is the radial wave function

),( ϕθ ′′′′mlY is the angular wave function and

ωα m=2 .

Thus the (nl,NL) sum in equation (3-35) is finite.

3.4 Some properties of Brody-Moshinisky coefficient: The Brody-Moshinisky transformation coefficient has symmetry properties in a

certain calculation, one reviewd them as [101]:

1- ( ) ( ) ( )11222211 ;1; lnlnnlNLMlnlnnlNLM Lλ

λλ

−−= (3-40)

2- ( ) ( ) ( )22112211 ;1; 1 lnlnNLnlMlnlnnlNLM lλ

λλ

−−= (3-41)

3- ( ) ( ) ( )11222211 ;1; 2 lnlnNLnlMlnlnnlNLM lLλλ

−−= (3-42)

The Moshinisky transformation coefficients are real and independent of µ .

3.5 Energy matrix element with oscillator function: TheHamiltonian of the harmonic oscillator potential is the only potential that can

be separated into relative and center of mass coordinate system, this property also

important in order to get the matrix element of the residual interaction as it is a

function of relative coordinate ( r ).

In order to make the two body wave functions compatable with the LS-scheme of

the residual interaction , because of the strong two-body spin-orbit force in nuclei,

which leads to the required shell structure, and the orginal two body wave function is

represented with jj scheme, it is necessary to transform the two body wave function

from jj scheme to LS scheme as follows [101]:

=×

JMljlj ][2211

φφ

∑ ++++JSjljl

SSjj

λλλ 22/12

12/11)12)(12)(122)(112(

[ ]JMslnln )2,1()2(),1( 2211 χψ λ ×× (3-43)

Where –is 9j-symbol.

Thus:

[ ]JMs

s

Js

JMljljJM

lnlnljlj

lnjlnj

)2,1()2(),1();(

)]2()1([)2(),1(

22112211)(

222111 2211

χψγ

φφ

λλ

λ ××=

×=Φ

∑

where

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

1

2

1

υυυυ

χχυυχ′

′′

′∑ ′= MSMS

(3-45)

and λµψ is the spatial part of the two particle orbital wave function given by equation

(3-44).

The coefficient );( 2211)( ljljI

sλγ is the j-j to L-S transformation coefficient[11].

λ++λ++=λγ

JSjljl

SjjljljIs 22/12

12/11)12)(12)(122)(112()22;11()( (3-46)

and when squared it gives the probability that the state [ ]JMSχψ λ × will be realized

in the original j-j coupled eigenfunction.

Thus instead of equation (3-44) we use:

∑+

=ΦS

JS

nnlljjJM ljljTlnjlnj

λλγδδδ

),()1(2

1);,( 2211)(

222111

212121

))],()(),([)()],()(),(([

JMsT

JMs

lnlnlnln

122112121

2211

2211

χ×ψ−+

χ×ψ×

λ

λ (3-47)

As a consequence of the defining equation (3-45):

)2,1()1()1,2( 1MS

sMS ′

+′ −= χχ (3-48)

and from equation (3-45) it follows that [107] :

)2(),1()1()1(),2( 2211221121 lnlnlnln ll

λµλ

λµ ψψ −+−= (3-49)

Furthermore, from the properties of the Moshinisky coefficient [101]:

( ) ( )11222211 ;)1(; lnlnnlNLMlnlnnlNLM Lλ

λλ

−−=

(3-44)

If one combine these results together with the fact that [92]: lLll )1()1( 21 −=− −+ ,

which follows from equation (3-37), one see that JMΦ can be written as:

)1(2(1);,(

212121

222111nnlljj

JM Tlnjlnjδδδ+

=Φ

∑∑ λ++

λλ

×−−γ×nlNL

lTSJS

SlnlnnlNLMljlj );()1(1),( 22112211

)(

JMslnln )]2,1()2(),1(([ 2211 χ×ψ× λ (3-50)

In this equation, the terms involving the phase factors show that even l wave

function, which are symmetric under the interchange of the two particles, must go

with S=1,T=0 and S=0,T=1, where as antisymmetric spatial states (l odd) are

associated with S=1,T=1 and S=0,T=0. Thus, if we merely replace )( JSλγ by

);()1(2(

)1(1),( 2211)(

2211)(

212121

ljljljlj JS

nnlljj

lTSJ

S λλ γδδδ

γ+

−−=′

++

(3-51)

Matrix elements of general energy operators V between basis states given by

equation (3-50) have the form [101,102]:

( ) ( )TnljlnjVTnljlnj JMJM ;,;, 444333222111 ΦΦ

),(),( 4433)(

2211)( ljljljlj J

SLNln nlNL S S

JS λ

′′′′ ′λ′ λλ γ′γ′= ∑ ∑∑∑

);();( 44332211 lnlnnlNLMlnlnLNlnM λλ′ ′′′′×

JMSLNlnJMSLNln RrVRr ])]()([[])]()([[ ′′′′′′′′′′ χ×φ×φχ×φ×φ× (3-52)

One can abbreviate the two-body matrix element of a general operator as:

( ) ( )TnljlnjVTnljlnj JMJM ;,;, 444333222111 ΦΦ = Γ4321 jjVjj

3.6 UThe realistic Sigma meson (S=0 T=0) effective Nucleon-Nucleon

interactions:

The two-body matrix elements which are used in the present work to calculate the

core-polarization effect are not obtained from a fit to spectroscopic data such as

modified surface delta interaction (MSDI), but they are calculated from free nucleon-

nucleon interaction. It can be reasonably well-understood in terms of the meson-

exchange current models, so that the usual procedure is to start with theoretical forms

of the N-N interaction, based on varying degrees of meson-exchange theory, such as

Paris, Bonn and Ried potentials [103]. Basically, one wants analytical form of the N-

N potential, which is in general obtained by fitting some set of potential parameters,

i.e., strength and ranges to the ground state (g.s) properties of deuteron and the low-

energy N-N scattering phase-shift (up to≈300 MeV), so that one can then

extrapolated to the off-energy shell behavior that required by many-body systems.

For higher energies, description in terms of a potential looses its meaning due to

meson exchange.

Potentials derived from the description of free nucleon-nucleon scattering are

customarily referred to as a realistic N-N effective interaction. One of the interactions

of this type, which is frequently used in shell-model calculations, is the one obtained

by Hamada-Johnston [104]. In another approach one-derived two-body matrix

elements in a particular single-particle basis directly from the scattering phase-shift.

In this case, interaction of the explicit form of the realistic interaction is avoided by

Elliott et al [105].

The direct application of a realistic interaction in a shell-model calculation does

not lead to an acceptable agreement with spectroscopic data. This is due to the fact

that a shell-model calculation is necessarily restricted to a finite, truncated model

space. For a reasonably complete description in terms of a realistic interaction, one

should take into account the scattering of nucleons into many more single-particle

states. In particular, also the presence of the core assumed to be inert in the model

calculation has a strong modifying influence of the interaction between the nucleons

outside the core. This means that for a model calculation one needs the so-called

“effective interaction”. For a given configuration space this effective interaction can,

in principle, be constructed from the nucleon-nucleon interaction when all processes

that take place outside the chosen configuration space is accounted for in term of

perturbation theory.

The realistic M3Y effective N-N interaction, which is used in electron scattering

(Vres =v12) is expressed as the central potential part v c)(

12 [103]:

vv c)(

1212 = (3-53)

The four potentials are expressed as: [100]

)(( 12

)()()(

12 rfPtvC

nSEn

SE

n

c ∑= (3-54)

The relative coordinate is denoted by rrr 2112 −= and rr 1212 =

Correspondingly,

the relative momentum is defined by2

)(21

12PPP

−= , L

12 is the relative orbital

angular momentum, PL r 121212 ×= as[103,101]:

(3-55)

where )(SEnt is the strength parameter in central part for (singlet-even) nucleon-

nucleon interaction, ( nµ =1/Rn) [84].

where: 𝑅𝑛 = ℏ𝑚δ 𝑐

is the exchange meson wavelength.

The two-body matrix elements of the realistic M3Y effective N-N interaction

consist of four parts: the central matrix element, the spin-orbit matrix element, the

tensor matrix element and the density dependance matrix elements, therefore, one

have:

Γ4321 jjVjj res = Γ4321 jjVjj C (3-56)

Where:

Γ4321 jjVjj C are the two-body matrix elements of the centeral potential.

rerf

nn

rn

1212

12

)( µµ−

=

The two-body matrix elements of the centeral potential are given as [100,101]:

Γ4321 jjVjj C ),(),( 4433)(

2211)( ljljljlj J

SnlNLn S

JS λ

λλ γγ∑ ∑

′

′=

);( 2211 lnlnLNlnM ′′′′× ′λ )()2()();( 4433 rRrVrRlnlnnlNLM nlSln βλ ′′×

(3-57)

According to this fitting, the radial integral is:

)()2()( rRrVrR nlln β′ ∑ ∫=

∞−

′=3

1 0

2)()(2i

rnlln

i

i drrerRrRViβ

β

),,(2

),,(2

),,(2 3

3

32

2

21

1

1 ββ

ββ

ββ

nllnRadVnllnRadVnllnRadV ′+′+′= (3-58)

where ),,( 1β′ nllnRad is the radial part at 1β

),,( 2β′ nllnRad is the radial part at 2β

),,( 3β′ nllnRad is the radial part at 3β

where: β =µ =1/R, and R is the range parameter of the nucleon-nucleon

interaction.

The first range parameters of the interaction (R1) between two nucleons in centeral

and spin-orbit force is 0.25 fm, the second range (R2) is 0.4 fm, and the longest range

(R3) is 1.414fm. But the longest range of the interaction (R3=0.7 fm) is only used in

r2Y(x) fit to the tensor force.

There are four different ways in which pairs of protons and/or neutrons can be

combined, and the nuclear force is different in each case.

• The "even" part of N-N interaction, i. e., is applied when the relative wave

function is even (symmetric) under exchange of two particles. Notice that it is

attractive as described above with a short range repulsion.

• The "odd" part of the N-N interaction is applied when the relative wave

function is odd (anti-symmetric) under exchange of the radial wave function.

Notice that it basically consists of only the short-range repulsion.

The SE shows the Singlet-Even (S=0, T=0) part applied to an L=1 (p state) of the n-

p system with anti-parallel spins.

The values of the best fit to the potential parameters ( )(SEnt ) are shown in table (3-

1) in [84].

Table (3-1). Shows the values of the best fit to the potential parameters [28].

Parameters Unit M3Y-E M3Y-P0 M3Y-P1 M3Y-P2 M3Y-P3 M3Y-P4 M3Y-P5

RR1RP

(c) Fm 0.25 0.25 0.25 0.25 0.25 0.25 0.25

tR1RP

(SE) Mev 9958 11466 8599.5 8027 8027 8027 8027

RR2RP

(c) Fm 0.40 0.40 0.40 0.40 0.40 0.40 0.40

tR2RP

(SE) Mev -3556 -3556 -3556 -2880 −2637 −2637 −2650

RR3RP

(c) fm 1.414 1.414 1.414 1.414 1.414 1.414 1.414

tR3RP

(SE) Mev -10.463 -10.463 -10.463 -10.463 −10.463 −10.463 −10.463

Results, Discussion and Conclusions

4.1 Introduction

The calculation of electron scattering form factor needs a lot of matters to be included in order to make this calculation feasible and fast in time because of large amount of terms represent the problem under study. These matters are mathematics, quantum mechanical theories, nuclear shell model theories and formulas. These different matters must be transformed into a computer program to be used by a computer having the capability to run these huge calculations.

The abnormal phenomena inherent with the calculated form factor need an experience and “know how” that’s related to the fp-shell nuclei and their strange, mystery, abnormal, and unexpected results. It make the calculations deviate from the experimental one and many efforts were exhausted in order to give a possible analysis and justifications about these cases.

The calcium nucleus( Ca48 ) can be considered as dual behavior nucleus under the effect of two nucleon interaction in fp-shell model space which consist of four orbitals which is 1f7/2, 2p3/2, 1f5/2 and 2p1/2 with single particle energies (calculated by use equations (3-30), (3-31) and (3-36)) equal to 48.2440 MeV,49.9261 MeV, 54.1312 MeV and 59.1879 MeV respectively. The two nucleons interaction in of Ca48

can be affected by the really and the nature of Ca48

which can be considered as an inert core for some isotopes with mass number larger than Ca48 like Cr isotopes, Fe isotopes, Ti isotopes, Co isotopes and Ni isotopes beside Ca isotopes with mass number larger than 49. But sometimes Ca48 will be considered as an active nucleus with eight nucleons distributed in fp-shell. The Ca40 as an inert core at sometimes this behavior might be extended to involve the eight nucleons in the orbit (1f 2/7 ) in the interaction process and makes S32 as an inert core for an active nucleus like Ca48 . It dual behavior of Ca48 is reproduced from the excitation energy of incident particles of the nuclear reaction and the response energy function of the nucleus.

The discarded space effects have been included in order to account the contribution of configurations from outside of the model space in the transition calculation.

The model space effective interaction (FPD6) [34, 35] has been used to give the (1fR7/2R,2pR3/2R,1fR5/2R,2pR1/2 R) shell model wave functions for P

48PCa because it is derived from

real N-N model space interaction in fp shell nuclei.

The core polarization effects are calculated by the realistic effective interaction (exchange M3Y by sigma meson) as a residual interaction. In this interaction, the Paris and Ried fitting [96] have been used to calculate the radial integral in equation (3-66). In this approach one-derived two-body matrix elements in a particular single-particle basis directly from the scattering phase shift. The HO potential is used to calculate the radial part of the single-particle wave functions, with size parameter (b) fitted to get the root mean square radius (rms) of each nuclei.

4.2 UThe UPU

48UPUCa Nucleus

The nucleus Ca48 is the lightest doubly magic nucleus with a neutron excess. It is known to be a good shell-model nucleus and thus provides an excellent testing

ground of nuclear models. In fact, the nucleus Ca48 is more inert than Ca40 , Ni48 and Ni56

because of the closed sub shell neutron1fR7/2R so that it is an interesting one in fp shell nuclei. The single particle wave functions of the harmonic oscillator (HO) with size parameter b= 1.988 fm are used [25] .

The linking processes of the sigma meson interaction and linking of the interaction intensity with fitting parameters of nucleon interaction have been swing in the middle of q values and qualitative constant of q < 2fmP

-1P. These matter will be certain, it have

quantum tails of the N-N matrix elements interaction. It will be causing that there is interference region between high mass sigma(δ) meson that have low wavelength (𝑅 = ℏ

𝑚δ 𝑐 ) with middle mass sigma(δ) meson that have high wavelength.

It is known that the high mass sigma meson is a repulsive type, however, the middle mass sigma meson is attractive type, according to the table (3-1) of fitting parameters [25].

The sigma (δ) meson interaction is a suitable channel to study fitting parameters of core polarization effect as a residual interaction.. It just through previous technology from two nucleon matrix elements, adding to it might be swing from orbit to another, and negative to positive behavior value which make it difficult behavior predictive of the interaction, especially. If we know that the total number of exchange nucleon matrix elements interaction from the fp shell has up to 12,000 elements which makes it difficult to predict using the extrapolation process of matrix elements in its normal form.

One can use the energy levels or multiple of gamma emission as a method to study the fitting parameters effective to the meson exchange process, but also, it is known

that the properties mention (energy levels and multiple gamma emission) are dependent on the very small point (→ 0) in a momentum space, so there is no study provided of accurately fitting parameters effective, finally, the inelastic electron scattering can be using to create the possibility to build study of meson exchange.

The OBDM for the model space orbits are obtained from the OXBASH code [5], using the FPD6 [34, 35] interaction where the OBDM for the core orbits are obtained from the occupation numbers according to equations (2-27), (2-28).

4.2.1 UThe Charge Form Factor for J UP

πPU T= 0 UP

+PU 4 State

Elastic coulomb form factors (C0) are displayed in figure (1) in comparison with the experimental data from ref.[106].

In Fig (1), the form factors are calculated by using M3Y(for E,P0,P1,P2,P3,P4 and P5) as a residual interaction.

There is no difference between reaction and another because they do not depend on the interaction as a transition, than :

E=P0=P1=P2=P3=P4=P5

The OBDM elements are given in table (4-1).

Table (4-1): The OBDM elements for the C0 transition with FPD6 interaction for ground state (Ex=0 MeV).

OBDM (∆T=1) OBDM (∆T=0) JRf JRi NRlj

0.0 6.0 1/2 1/2 1SR1/2

0.0 8.4853 3/2 3/2 1PR3/2

0.0 6.0 1/2 1/2 1PR1/2

0.0 10.3923 5/2 5/2 1dR5/2

0.0 8.4853 3/2 3/2 1dR3/2

0.0 6.0 1/2 1/2 2SR1/2

3.72442 5.76985 7/2 7/2 1fR7/2

0.11016 0.17066 3/2 3/2 2PR3/2

0.06561 0.10164 5/2 5/2 1fR5/2

0.02769 0.04290 1/2 1/2 2PR1/2

Fig.(1) Elastic charge form factor for (C0) for the ground state 𝟎𝟎+. The experimental data are taken from ref. [28].

4.2.2 The C2 Charge Form Factor For Jπ T= 2+ 4 state

0.00 1.00 2.00 3.001E-6

1E-5

1E-4

1E-3

1E-2

1E-1

Ca48

EX=0.000 MevC0 charge form factor

IF(q

)I2

q(fm )-1

___E___P0___P1___P2___P3___P4___P5

+

Exp.

4.2.2.1 The first Jπ T= 21+ 4 state with Ex=3.658 MeV

Inelastic (C2) longitudinal form factors have been calculated using sigma meson as

a residual interaction, it is obvious that the amplitudes of different scattering patterns

are in phase with slightly different a moment in especially for q≤1.4 fm-1, but

they were inclined between each authors when q increased and the amplitudes are

arranged from bigger to smaller as follow:

P0> P1> E> P3> P5> P4> P2 (the first peak)

P0> E> P4> P2> P1> P5 = P3 (second )

The OBDM elements are given in table (4-2).

Table (4-2): The interaction OBDM elements for the C2 transition with FPD6

for 𝟐𝟏+ state (3.658 MeV).

Ji Jf OBDM (∆T=0)) OBDM (∆T=1))

7/2 7/2 0.07960 0.05138 7/2 3/2 0.30835 0.19904 7/2 5/2 0.11637 0.07512 3/2 7/2 1.95301 1.26066 3/2 3/2 0.04680 0.03021 3/2 5/2 0.00256 0.00165 3/2 1/2 0.01778 0.01148 5/2 7/2 -0.27266 -0.17611 5/2 3/2 -0.01412 -0.00911 5/2 5/2 0.03051 0.01970 5/2 1/2 0.01765 0.01139 1/2 3/2 -0.04228 -0.02729 1/2 5/2 0.01848 0.01193

Fig.(2) Charge form factor for the C2 state with Ex=3.658 MeV (𝟐𝟏+).

4.2.2.2 The second Jπ T= 2+ 4 state with Ex=5.949MeV

0.00 1.00 2.00 3.001E-6

1E-5

1E-4

1E-3

1E-2

1E-1

1E+0

Ca48

EX= 3.658 MeV

IF(q

)I2

q(fm )-1

C2 charge form factor

___ E___P0___P1___P2___P3___P4___P5

+1

Fig(3) presents C2 form factors that the relation between q=(0.1 - 3)fmP

-1P with

|𝐹(𝑞)|2=(1×10P

-9P – 1×10P

-3P).

At range of q=(0 – 0.5)fmP

-1P there are lowing different between scattering patterns ,

but they were inclined between each other’s when q increased (q=(0.5 – 1.5)fmP

-1P),

and the amplitudes are arranged from bigger to smaller as follows: P0> P1> P3 =

P5> E> P4> P2

One have maximum values of the amplitudes(|𝐹(𝑞)|2=1×10P

-3P) at q=1.5 fmP

-1P.

At range of q=(1.6 – 3)fmP

-1P, the amplitudes are devolved slightly with q increased,

and the amplitudes are arranged from bigger to smaller as follows: P0> E> P2 = P4>

P1> P3 = P5


for 𝟐𝟐+ state (5.949 MeV).

JRi JRf OBDM (∆T=0)) OBDM (∆T=1))

7/2 7/2 -0.11819 -0.07629 7/2 3/2 0.04547 0.02935 7/2 5/2 -0.00837 -0.00540 3/2 7/2 0.29438 0.19002 3/2 3/2 -0.26976 - 0.17413 3/2 5/2 -0.01156 -0.00746 3/2 1/2 -0.01532 -0.00989 5/2 7/2 -0.02310 -0.01491 5/2 3/2 0.01530 0.00988 5/2 5/2 -0.00743 -0.00480 5/2 1/2 -0.00094 -0.00060 1/2 3/2 0.06802 0.04391 1/2 5/2 -0.00374 -0.00242

Fig.(3) Charge form factor for the C2 state with Ex=5.949 MeV (𝟐𝟐+).

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.501E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

1E-2

1E-1

1E+0

1E+1


48CaIF

(q)I2

q(fm )-1

___E___P1___P2___P0

___P3___P4___P5

2+

EX=5.949 Mev

4.2.2.3 The third Jπ T= 2+ 4 state with Ex=6.557 MeV

In fig(4), one can be represent the relation between q=(0 - 3)fmP

-1P with

|𝐹(𝑞)|2=(1×10 P

-9P – 1×10 P

-4P).

At the range of q=(0 – 0.2)fmP

-1P the amplitudes of the seven sets of fitting parameters

are increased slightly. But at the range of q=(0.5 – 1.4)fmP

-1P with|𝐹(𝑞)|2=(1×10 P

-5P –

1×10 P

-4P), there are splitting between each others of the seven sets of fitting

parameters, with maximum amplitudes of |𝐹(𝑞)|2=1×10 P

-4P at q=0.9 fmP

-1P.the

amplitudes are arranged from bigger to smaller as follows:

P0> E> P1> P3 = P5> P4> P2

At q=1.5 fmP

-1P the diffraction minimum value with |𝐹(𝑞)|2=1×10 P

-7P is lower value.

at the range of q=(1.7 – 3)fmP

-1P there are splitting between each others of the seven

sets of fitting parameters, with .the amplitudes are arranged from bigger to smaller as follows:

P0> E> P4> P2> P1> P3 = P5


for 𝟐𝟑+ state (6.557 MeV).

JRi JRf OBDM (∆T=0)) OBDM (∆T=1)) 7/2 7/2 0.23143 0.14939 7/2 3/2 0.01477 0.00953 7/2 5/2 0.02670 0.01723 3/2 7/2 0.06763 0.04365 3/2 3/2 -0.39544 -0.25526 3/2 5/2 0.00317 0.00204 3/2 1/2 -0.04782 -0.03087 5/2 7/2 -0.20087 -0.12966 5/2 3/2 0.00749 0.00484 5/2 5/2 -0.01038 -0.00670 5/2 1/2 -0.01217 -0.00785 1/2 3/2 0.02408 0.01554 1/2 5/2 -0.00453 -0.00292

Fig.(4) Charge form factor for the C2 state with Ex=6.557 MeV (𝟐𝟑+).

0.00 1.00 2.00 3.001E-10

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

1E-2

1E-1

1E+0

1E+1


48CaIF

(q)I2

q(fm )-1

___E___P1___P2___P0

___P3___P4___P5

3+

EX=6.557 Mev

4.2.2.4 The fourth Jπ T= 2+ 4 state with Ex=7.275 MeV

Fig(5) represents C2 form factors for the fourth excited state

In the first peak: At range of q=(0 – 0.2)fm-1 there are small difference between scattering patterns, but they are inclined between each other’s when q increased (q=(0.4 – 1.6)fm-1 ) , and the amplitudes are arranged from bigger to smaller as follows: P0> E> P1> P2 = P3 = P4 = P5

One have maximum values of the amplitudes (|𝐹(𝑞)|2=1×10 P

-2P) at q=1 fmP

-1P .

At q=(2, 2.8) fmP

-1P the diffraction minimum values with |𝐹(𝑞)|2=(1×10 P

-10P, 1×10 P

-9P)

are lower values.

At the second peak: At the range of q=(2.2 – 2.8)fmP

-1P with|𝐹(𝑞)|2=(1×10 P

-7P – 1×10 P

-

5P),one can show the amplitudes of its are lowering with first peak , but at same shape.

.the amplitudes are arranged from bigger to smaller as follows:

P0> P1> E> P3 = P5> P4> P2


for 𝟐𝟐+ state (7.275 MeV).

JRi JRf OBDM (∆T=0)) OBDM (∆T=1)) 7/2 7/2 0.06266 0.04045 7/2 3/2 -0.05720 -0.03693 7/2 5/2 -0.08568 -0.05531 3/2 7/2 0.24400 0.15750 3/2 3/2 -0.02814 -0.01817 3/2 5/2 -0.03463 -0.02236 3/2 1/2 -0.03560 -0.02298 5/2 7/2 1.91335 1.23506 5/2 3/2 0.03926 0.02534 5/2 5/2 -0.03004 -0.01939 5/2 1/2 -0.00592 -0.00382 1/2 3/2 0.05899 0.03808 1/2 5/2 -0.00137 -0.00088

Fig.(5) Charge form factor for the C2 state with Ex=7.275 MeV (𝟐𝟒+).

0.00 1.00 2.00 3.001E-10

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

1E-2

1E-1

1E+0

1E+1


48Ca

IF(q

)I2

q(fm )-1

___E___P1___P2___P0

___P3___P4___P5

4+

EX=7.275 Mev

4.2.2.5 The fifth Jπ T= 2+ 4 state with Ex=7.363 MeV

In fig(6), one can be represented(C2) the relation between q=(0 - 3)fmP

-1P with

|𝐹(𝑞)|2=(1×10 P

-10P – 1×10 P

-3P).

In the first peak: At range of q=(0 – 0.2)fmP

-1P with |𝐹(𝑞)|2=(1×10 P

-9P – 1×10 P

-6P)there are

different between scattering patterns, but they are inclined between each other’s when q=(0.4 – 1.8)fmP

-1P increased, and the amplitudes are arranged from bigger to smaller

as follows:

P0> E > P1> P3 = P5> P4 = P2

One have maximum values of the amplitudes (|𝐹(𝑞)|2=1×10 P

-3P) at q=1 fmP

-1P .

At q=(2.3, 2.6) fmP

-1P diffraction minimum values with |𝐹(𝑞)|2=1×10 P

-10P are lower

values.

At the second peak: At the range of q=(2.3 – 3)fmP

-1P with|𝐹(𝑞)|2=(1×10 P

-10P – 1×10 P

-

6P),one can show different between scattering patterns (splitting, rippling and swing).

The amplitudes are arranged from bigger to smaller as follows:

P0> E> P4 = P2> P1> P3 = P5


for 𝟐𝟓+ state (7.363 MeV).

JRi JRf OBDM (∆T=0)) OBDM (∆T=1)) 7/2 7/2 -0.34331 -0.22160 7/2 3/2 0.02567 0.01657 7/2 5/2 -0.01734 -0.01119 3/2 7/2 0.39721 0.25640 3/2 3/2 0.07856 0.05071 3/2 5/2 0.07061 0.04558 3/2 1/2 0.00427 0.00276 5/2 7/2 0.15140 0.09773 5/2 3/2 0.00767 0.00495 5/2 5/2 -0.01432 -0.00925 5/2 1/2 0.01000 0.0064 1/2 3/2 -0.03347 -0.02160 1/2 5/2 0.01138 0.00735

Fig.(6) Charge form factor for the C2 state with Ex=7.363 MeV (𝟐𝟓+).

4.2.3 The Inelastic Longitudinal C4 form factors (4+) State

0.00 1.00 2.00 3.001E-10

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

1E-2

1E-1

1E+0

1E+1


48CaIF

(q)I2

q(fm )-1

___E___P1___P2___P0

___P3___P4___P5

5+

EX=7.363 Mev

4.2.3.1 The first Jπ T= 4+ 4 state with Ex=4.134 MeV

The fig(7) can be represents C4 form factors for the first excited state with excitation energy Ex=4.134 MeV. The relation between q=(0.2 - 3)fmP

-1P with |𝐹(𝑞)|2=(1×10 P

-6P –

1×10 P

-4P).

In the first peak: At range of q= ( 0.2-0.4)fmP

-1P with |𝐹(𝑞)|2=(1×10 P

-6P–1×10 P

-4P) there is

low different between scattering patterns, but they are inclined clearly between each other’s when q=(0.4 – 1.8)fmP

-1P increased, and the amplitudes are arranged from

bigger to smaller as follows:

P0> E> P4 = P2> P1> P3 = P5

One have maximum values of the amplitudes at q=1.4 fmP

-1P .

The diffraction minimum values are located at 2≤ q ≤2.4 fmP

-1P.

At the second peak: At the range of q=(2.2 – 3)fmP

-1P with|𝐹(𝑞)|2=(1×10 P

-6P – 1×10 P

-

4P),one can show difference between scattering patterns (splitting, rippling and swing).


P0> E> P4> P2> P1> P3 = P5


for 𝟒𝟏+ state (4.134 MeV).

JRi JRf OBDM (∆T=0)) OBDM (∆T=1)) 7/2 7/2 0.04377 0.02825 7/2 3/2 0.08460 0.05461 7/2 5/2 0.05917 0.03820 7/2 1/2 0.04564 0.02946 3/2 7/2 1.93714 1.25042 3/2 5/2 0.01240 0.00801 5/2 7/2 -0.22510 -0.14530 5/2 3/2 -0.03311 -0.02137 5/2 5/2 0.01130 0.00729 1/2 7/2 -0.10962 -0.07076

Fig.(7) Charge form factor for theC4 state with Ex=4.134 MeV ( 𝟒𝟏+).

4.2.3.2 The second Jπ T= 4+ 4 state with Ex=6.387 MeV

0.00 1.00 2.00 3.00

1E-6

1E-5

1E-4

1E-3

1E-2

1E-1

Ca48

EX=4.134 MeV

IF(q

)I2

q(fm )-1

___ E___ P0___

P1___ P2___ P3___ P4___ P5

C4 charge form factor1+

The fig(8) has the same behavior of the fig(7),but the diffraction minimum points at q=(2.2, 2.4)fm-1 are different. The amplitudes of peaks greater than that of fig(7), and swing scattering patterns values. At the region q = (0.2-2) fm-1, the amplitudes are arranged from bigger to smaller as follows: P0> E> P4 = P2> P1> P3 = P5

In the region q = (2-3) fm-1 , the amplitudes are arranged from bigger to smaller as follows: P0> E> P4> P2> P1> P3 = P5


for 𝟒𝟐+ state (5.387 MeV).

Ji Jf OBDM (∆T=0)) OBDM (∆T=1))

7/2 7/2 -0.01533 -0.00990

7/2 3/2 -0.03712 -0.02396

7/2 5/2 -0.11524 -0.07439

7/2 1/2 -0.11657 -0.07525

3/2 7/2 0.18408 0.11882

3/2 5/2 -0.02367 -0.01528

5/2 7/2 0.83569 0.53943

5/2 3/2 0.05093 0.03288

5/2 5/2 -0.01981 -0.01279

1/2 7/2 1.78712 1.15358

Fig.(8) Charge form factor for the C4 state with Ex=6.387 MeV (𝟒𝟐+).

4.2.3.3 The third Jπ T= 4+ 4 state with Ex=6.838 MeV

0.00 1.00 2.00 3.001E-6

1E-5

1E-4

1E-3

1E-2

1E-1

Ca48


IF(q

)I2

q(fm )-1

___E___P0___P1___P2___P3___P4___P5

+2

The fig(9) presents C4 form factors with the relation between q=(0.3 - 3)fmP

-1P with

|𝐹(𝑞)|2=(1×10 P

-6P – 1×10 P

-2P).

One can show there is low difference between scattering patterns at low values of q=(0.3 – 0.5)fmP

-1P with|𝐹(𝑞)|2=(1×10 P

-6P – 1×10 P

-4P). but they are inclined between each

other’s when q increased slightly, and have max. value at q=1.5 fmP

-1P, the peak is

decreased.


P0 > E > P4 = P2 > P1 > P3 = P5


for 𝟒𝟑+ state (6.838 MeV).


7/2 7/2 -0.01989 -0.01284

7/2 3/2 -0.01058 -0.00683

7/2 5/2 -0.10051 -0.06488

7/2 1/2 -0.01122 -0.00724

3/2 7/2 0.19484 0.12577

3/2 5/2 -0.00247 -0.00160

5/2 7/2 1.73102 1.11737

5/2 3/2 0.01277 0.00824

5/2 5/2 -0.00477 -0.00308

1/2 7/2 -0.83072 -0.53623

Fig.(9) Charge form factor for the C4 state with Ex=6.838 MeV (𝟒𝟑+).

4.2.3.4 The fourth Jπ T= 4+ 4 state with Ex=7.283 MeV

0.00 1.00 2.00 3.00 4.001E-6

1E-5

1E-4

1E-3

1E-2

1E-1

Ca48


IF(q

)I2

q(fm )-1

___E___P0___P1___P2___P3___P4___P5

3+

The fig(10) shows that there is low difference between scattering patterns at low values of q=(0.4 – 0.7)fmP

-1P with|𝐹(𝑞)|2=(1×10 P

-6P – 1×10 P

-5P). but they are inclined

between each other’s when q increased slightly, and have max. value at q=1.4 fmP

-1P,

the peak is decreased.


P0 > E > P1 > P2 = P4 > P3 = P5


for 𝟒𝟒+ state (7.283 MeV).


7/2 7/2 -0.31344 -0.20232

7/2 3/2 0.00648 0.00418

7/2 5/2 -0.00024 -0.00016

7/2 1/2 0.00103 0.00067

3/2 7/2 0.06681 0.04313

3/2 5/2 0.00865 0.00559

5/2 7/2 -0.06003 -0.03875

5/2 3/2 -0.02035 -0.01314

5/2 5/2 0.00201 0.00130

1/2 7/2 0.09233 0.05960

Fig.(10) Charge form factor for the C4 state with Ex=7.283 MeV (𝟒𝟒+).

0.00 1.00 2.00 3.001E-6

1E-5

1E-4

1E-3

1E-2

1E-1

Ca48


IF(q

)I2

q(fm )-1

___E___P0___P1___P2___P3___P4___P5

+4

4.2.3.5 The fifth Jπ T= 4+ 4 state with Ex=7.435 MeV

The fig(11) shows that there is low difference between scattering patterns at low values of q=(0.4 – 0.6)fmP

-1P with|𝐹(𝑞)|2=(1×10 P

-6P – 1×10 P

-5P). but they are inclined

between each other’s when q increased slightly, and have max. value at q=1.4 fmP

-1P,

the peak is decreased.


P0 > E > P4 = P2 = P1 > P3 = P5

One can show that the max. value of scattering patterns at q=(1 – 1.5)fmP

-1P, because of

it nucleus range center.


for 𝟒𝟓+ state (7.435 MeV).


7/2 7/2 0.02983 0.01926

7/2 3/2 -0.02639 -0.01703

7/2 5/2 -0.01538 -0.00993

7/2 1/2 -0.00915 -0.00591

3/2 7/2 -0.13709 -0.08849

3/2 5/2 -0.04067 -0.02625

5/2 7/2 0.36736 0.23713

5/2 3/2 0.02302 0.01486

5/2 5/2 0.00077 0.00050

1/2 7/2 0.07519 0.04853

Fig.(11) Charge form factor for the C4 state with Ex=7.435 MeV (𝟒𝟓+).

0.00 1.00 2.00 3.001E-6

1E-5

1E-4

1E-3

1E-2

1E-1

Ca48


IF(q

)I2

q(fm )-1

___E___P0___P1___P2___P3___P4___P5

+5

The interaction equation for sigma(δ) meson when T=0, S=0, which is

expressed as 𝑡(𝑆𝐸),by using the table(3-1)[28]which represents fitting parameter for

nucleon interaction in this channel can be written as follows:

𝑉12(δ) = 𝑡𝑛(𝑆𝐸)

3

𝑛=1

𝑃𝑆𝐸𝑒−( 𝑟12𝑅𝑛12

)

𝑟12𝑅𝑛12

𝑉12(δ) = 𝑡1(𝑆𝐸)𝑃𝑆𝐸

𝑒−( 𝑟12𝑅112

)

𝑟12𝑅112

+ 𝑡2(𝑆𝐸)𝑃𝑆𝐸

𝑒−( 𝑟12𝑅212

)

𝑟12𝑅212

+ 𝑡3(𝑆𝐸)𝑃𝑆𝐸

𝑒−( 𝑟12𝑅312

)

𝑟12𝑅312

Note: One can use table (3-1), and sum process of the fitting parameter (𝑡1(𝑆𝐸) +

𝑡2(𝑆𝐸)) and negligence of 𝑡3(𝑆𝐸) to their small value and equal to the value of all the

interactions we find at the time of the collection process (𝑡1(𝑆𝐸) + 𝑡2(𝑆𝐸)) devolve

Descending and digital output as follows:

P0=7910Mev>E=6853Mev>P3=P4=5390Mev>

P5=5377Mev>P2=5147Mev>P1=5043.5Mev

4.3 UConclusions

We can conclude that the concentration of N-N interaction correspond with the fitting parameters, where q<2fmP

-1P in the inelastic electron scattering form factors are

sweeping with a reaction and other with f(q) in region f(q)=(2-2.5), that means it have quantum tails for short range reflective mesons with the middle attractive mesons according to the relation(𝑅 = ℏ

𝑚δ 𝑐)

One gets the fitting parameter values from the table(3-1).

P0=7910Mev, E=6853Mev, P3=P4=5390Mev, P5=5377Mev, P2=5147Mev, P1=5043.5Mev

4.4 Future Research:

1. Study of inelastic electron scattering form factors from 42,44Ca and how to known the relation effective between a neutron numbers of model space with meson exchange matrix element and fitting parameters have it to all interactions.

2. Using inelastic electron scattering form factors of multiple form factor to study meson’s behavior (like π-meson, ρ-meson and ω-meson).

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U الخالصة

استخدمت النظرية المجهرية في التحقق من عوامل التشكل الستطارة االلكترون الطولية غير المرنة Pلنواة

48PCa والتحقيقات المنجزة ألنموذج خلط التشكيل لعدد محدد من المدارات ألنموذج الفضاء والذي يكون ،

خارج القلب الخامل. الفضاء المستثنى يكون مضمنا و يسمى بتأثيرات استقطاب القلب الذي يفترض تهيج .2ℏwفجوة) لمدارات القلب وانموذج الفضاء التي تكون بطاقة تهيج مقدارها -(جسيمة

)بين النيوكليونين باستخدام δتمت عملية استقراء تأثيرات عوامل الضبط للتبادل الميزوني من نوع ()وبواقع خمس مستويات طاقة C0,C2,C4تقنية االستطارة االليكترونية غير المرنة لالنتقاالت متعددة االقطاب (

P)لنواة الكالسيوم (fpفي انموذج الفضاء ( C2,C4تهيج لكل من

48PCaبواقع سبع تفاعالت )و

(E,P0,P1,P2,P3,P4,P5)من نوع جهد يوكاوا(M3Y).

P)لغرض توليد الدوال الموجية في نواة FPD6تم استخدام التفاعل الفعال من نوع (

48PCa ولالنتقاالت

C0,C2,C4 والتي تتضمن تأثيرات استقطاب القلب، والدراسات السابقة اثبتت تأثير شحني في الفضاء ،اكبر مما في الفضاء الحر عند حساب تأثير استقطاب القلب، كما استخدمت الدوال الموجية من األنموذجي

) باستخدام الحلول التحليلية الممكنة.single particle)كدالة جسيم منفرد(H.Oالنوع المتذبذب التوافقي (

الستقطاب القلب (δ)لميزون سكما M3Y المؤثر التفاعل استخدام

Pلنواة الطولية التشكل عوامل لدراسة بقية كتفاعل

48PCa

مقدمة رسالة إلى كلية العلوم / جامعة بغداد

علوم فيماجستير وهي جزء من متطلبات نيل درجة الفيزياء

:من قبل

صادق سلطان مشعان الشمري) 2001بكالوريوس كلية العلوم جامعة بابل (

:بأشراف

أ.م.د. فراس زهير مجيد

1-ربيع 1435 1-كانون 2014

جمهورية العراق العلميوزارة التعليم العالي والبحث

جامعة بغداد كلية العلوم

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