Elastic and Inelastic Electron-Nucleus Scattering Form Factors for Be⁹ Nucleus

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]:

Where, J_i and J_f are, respectively, the initial and final total angular momentum, while Z is the number of proton (atomic number), T_J(q) is the multipole operator of electron scattering, and 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]:

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].

The MS reduced matrix element in the spin and isospin spaces of the transition operator T_J 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]:

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

Where, α and β are, respectively, the initial and final single-particle states for the MS when isospin included, the index τ_z is the third component of nucleons Pauli isospin which used to identify the nucleons with τ_z = 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:

Where, is the radial part matrix element of the spherical Bessel function j_J (qr) which is calculated in [19, Equation (23)] of our published article; and represents the reduced matrix element of the spherical Harmonics Y_Jτz.

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

The single-particle matrix element 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].

Here, H_o is the unperturbed Hamiltonian, E_a, E_b are the initial and final states of energy, the Q operator projects the outside space of the MS, both the indices a₁ and a₂, are, respectively, run over particle and hole states, 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]:

Where .

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]:

Where, F₀ (q) is the ground-state form factor, r is the radius of the nucleus, and ρ₀ (r) is the charge density.

The entirety of all protons point charge is the representation of operator of transition charge density of a nucleus [23].

Where, J is the multipolarity of the operator, M is the projection quantum number takes 2J + 1 values, −J ≤ M ≤ J, Y_JM (Ω_k) represents the spherical Harmonic, and is a Dirac delta function.

The matrix element in the reduced form of the operator is gotten when the transition happens from initial nuclear spin J_i to the final nuclear spin J_f and complying the inequality J_i ≤ J ≤ J_f , from Equation (4) where then it is given by:

For the ground state (J = 0) as mentioned before. Moreover, J_i= J_f, and the charge density define of the nucleus is getting from this matrix element [23].

The single-particle matrix element in Equation (4) can be represented by the radial wave functions of HO and which is the spherical Harmonic reduced matrix element.

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

For the ground-state nucleus, (τ_z=1, -1), it makes J = 0 and . After utilizing the Delta-Kronecker in Equation (15) and putting τ_z= 1 for protons, the equation is rewritten as:

the index α≡n_α l_α j_α 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].

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].

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

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 (J^π 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 C₀ and C₂, respectively. The calculated form factor of sum C₀ + C₂ is shown in Fig. 1.

Fig. 1. Elastic Coulomb C₀+C₂ form factors for Be9. The experimental data were taken from reference [28].

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.

TABLE 1 The calculated OBDM elements for the Coulomb C₀+C₂ transition of Be9

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

The C₂ transition for Coulomb scattering is taking place between the ground state of (J^π T = 3/2 ⁻1/2) and the first excited state (J^π 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.

Fig. 2. Inelastic Coulomb C₂ form factors for 5/2 ⁻1/2 state of Be⁹ with E = 2.43 MeV. The experimental data were taken from reference [28].

TABLE 2 The calculated OBDM elements for the Coulomb C₂ transition of Be9

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 = C₂, it occurs between the ground state (J^π T= 3/2 ⁻1/2) and the second excited state (J^π 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 (J^π T = 7/2 ⁻1/2) are not substantially different from that of the first excited state (J^π 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).

Fig. 3. Inelastic Coulomb C₂ form factors for 7/2 ⁻1/2 state of Be⁹ with E = 6.38 MeV. The experimental data were taken from reference [28].

TABLE 3 The calculated OBDM elements for the Coulomb C₂ transition of Be9