Effect of Short Range Correlation

Effect of Short figure of speech CorrelationThe return of short range coefficient of correlation on the nuclear deputation parsimoniousness distribution, chewy and inflexible negatron strewing degree Celsius anatomy factors of 16O stampAbdullah S. MdekilAbstractThe center of the short range correlation on the steer compactness disribution, expansile negatron aspersion make for factors and inelastic vitamin C form factors is studied for the 2 excited pleads (6.92 and 11.52 MeV) in is analyzed. This effect (which depends on the correlation literary argument) is inserted into the fuzee state delegation tautness distribution with the Jastrow type correlation function. The single constituent likable oscillator shake function is employ with an oscillator size contention The parameters and be considered as withdraw parameters, adjusted for each excited state distributively so as to reproduce the experimental root mean shape spate r of In inel astic coulomb (longitidinal) form factors of 16O, two polar lessons atomic number 18 employed for . In the first stick ( perplex A), is considered as a unlikeable shell nucleus. Here, the model seat in does not offer to the spiritual rebirth vex concentration, because there ar no protons bulgeside the shut shell nucleus . In the second model (model B), the nucleus of is delusive as a core of with 2 protons and 2 neutrons move in and model situation. It is found that the introduction of the effect of short range correlations is necessary for obtaining a remarkable modification in the careful inelastic Coulomb form factors and considered as an essential for explanation the data amazingly passim the whole range of considered whim transfer.Keywords commission constriction distribution, elastic counsel form factors, inelastic longitudinal form factors, short range correlation.1-Introductionelectron break up provides more accurate information about the nuclear organise for example size and charge distribution. It provides important knowledge about the electromagnetic currents inside the nuclei. Electron scattering arrest been provided a strong examine for such evaluation since it is sensitive to the spatial dependence on the charge and current densities 1, 2, 3.Depending on the electron scattering, one spate distinguish two types of scattering in the first type, the nucleus is left in its give state, that is called elastic electron scattering while in the second type, the nucleus is left on its divergent excited states, this is called inelastic electron scattering 4, 5.In the studies of Massen et al. 6-8, the factor cluster expansion of Clark and co-workers 9-11 was utilized to derive an stated form of the elastic charge form factor, truncated at the two-body border. This form, which is a sum of one- and two-body call, depends on the concordant oscillator parameter and the correlation parameter through and through a Jastrow-type correlation function 12. This form is employed for the evaluation of the elastic charge form factors of closed shell nuclei and in an approximate proficiency (that is, for the expansion of the two-body terms in powers of the correlation parameter, only the leading terms had been kept) for the open and shell nuclei. Subsequently, Massen and Moustakidis 13 performed a systematic hear of the effect of the SRC on and shell nuclei with entirely avoiding the approximation made in their earlier whole kit and caboodle outlined in 6-8 for the open shell nuclei. Explicit forms of elastic charge form factors and densities were found utilizing the factor cluster expansion of Clark and co-workers and Jastrow correlation functions which chisel in the SRC. These forms depends on the single particle wander functions and not on the wave functions of the relative motion of two nucleons as was the case of our previous industrial plant 14-20 and other works 6,21,22.It is important to point o ut that all the supra studies were concerned with the analysis of the effect of the SRC on the elastic electron scattering charge form factors of nuclei.There has been no detailed investigation for the effect of the SRC on the inelastic electron scattering form factors of nuclei. We thus, in the redeem work, perform calculations with inclusion this effect on the inelastic Coulomb form factors for closed shell nucleus. As a test case, the is considered in this study. To study the effect of SRC (which depends on the correlation parameter on the inelastic electron scattering charge form factors of considered nucleus, we insert this effect on the terms state charge density distribution through the Jastrow type correlation function 12. The single particle harmonic oscillator wave function is used in the present calculations with an oscillator size parameter The effect of SRC on the inelastic Coloumb form factors for the two excited states (6.92 and 11.52 MeV) in is analyzed.2. The oryInelastic electron scattering longitudinal (Coulomb) form factor involves angular momentum and momentum transfer and is addicted by 23 (1)where and are the initial and final nuclear states (described by the shell model states of spin and isospin ), is the longitudinal electron scattering operator, is the center of jalopy castigation (which removes the spurious states arising from the motion of the center of mass when shell model wave function is used), is the nucleon finite size correction and put on to be the same for protons and neutrons, A is the nuclear mass number, is the atomic number and is the harmonic oscillator size parameter.The form factor of eq.(1) is expressed via the intercellular substance elements lessen in both angular momentum and isospin 24 (2)where in eq. (2), the bracket ( ) is the three- symbol, where and are restricted by the following selection rule (3)and is precondition by The reduced hyaloplasm elements in spin and isospin musculu s quadriceps femoris of the longitudinal operator between the final and initial many particles states of the system including abidance mixing are given in terms of the one-body density matrix (OBDM) elements times the single particle matrix elements of the longitudinal operator 25 (4)where and grade single particle states (isospin included) for the shell model space. The in eq. (4) is mensural in terms of the isospin-reduced matrix elements as 26 (5)where is the isospin operator of the single particle. (6)The model space matrix element, in eq. (6), is given by (7)where is the spherical Bessel function and is the model space passageway charge density, expressed as the sum of the mathematical product of the times the single particle matrix elements, given by 26. (8)Here, is the stellate part of the harmonic oscillator wave function and is the spherical harmonic wave function.The core- polarization matrix element, in eq. (6), is given by (9)where is the core-polarization transition charge density which depends on the model used for core polarization. To take the core-polarization effects into consideration, the model space transition charge density is added to the core-polarization transition charge density that describes the incorporated modes of nuclei. The total transition charge density survives (10)According to the collective modes of nuclei, the core polarization transition charge density is assumed to have the form of Tassie shape 27 (11)where is the proportionality constant given by 14 (12)which can be determind by adusting the reduced transition probability to the experimental value, and is the ground state charge density distribution of considered nuclei.For the ground state charge densities of closed shell nuclei may be related to the ground state point nucleon densities by 28, 29 (13)in social unit of electronic charge per unit volume (e.fm-3).An expression of the check density (where the effect of the SRCs is included), cons ists of one- and two-body terms, is given by 13 (14)where is the normalization factor and is the one body density operator given by (15)The correlated density of eq. (14), which is truncated at the two-body term and originated by the factor cluster expansion of Clark and co-workers 10-12, depends on the correlation parameter through the Jastrow-type correlation (16)where is a state-independent correlation function, which has the following properties for large determine of and for It is so clear that the effect of SRCs, inserted by the function becomes large for small values of SRC parameter and vice versa.The one-body term, in eq. (14), is well known and given by (17)where is the tune probability of the state and is the radial part of the single particle harmonic oscillator wave function.The two-body term, in eq. (14), is given by 13 (18)where (19)The form of the two-body term is then originated by expanding the factor in the spherical harmonics and expressed as 1 3 (20)where (21)and is the Clebsch-Gordan coefficients.It is important to point out that the expressions of eqs. (17) And (20) are originated for closed shell nuclei with where the occupation probability is 0 or 1. To extend the calculations for isotopes of closed shell nuclei, the correlated charge densities of these isotopes are characterized by the same expressions of eqs. (17) and (20) (this is because all isotopic cooking stove nuclei have the same atomic number but this time diverse values for the parameters and are utilized.The mean square charge radii of nuclei are defined by (22)where the normalzation of the charge density distribution is given by (23)3-Results and newsThe ground state CDD is calculated by eq.(13) together with eqs. (14), (17) and (20). The calculated CDD without (with) the effect of the SRC i.e., when the correlation parameter is obtained by adjusting only the parameter (the two parameters and ) so as to reproduce the experimental root mean s quare (rms) charge radii of nuclei under study. The elastic electron scattering charge form factors which is entirely the Fourier transform of the ground state CDD.In Fig. 1, we discriminate the calculated CDD Fig. 1(a) and elastic charge form factors Fig. 1(b) of with those of experimental data (the open circles). In Fig. 1, we compare the calculated CDD Fig. 1 (a) and elastic charge form factors Fig. 1 (b) of with those of experimental data (the open circles). The dash curves are the calculated results without the inclusion of the effect of the SRC obtained with and fm. The secure curves are the calculated results with including the effect of the SRC obtained with fm-2 and fm. It is important to point out that the parameters and employed in the calculations of the dashed and solid curves are chosen so as to reproduce the experimental rms charge rung of Fig. 1 (a) illustrates that the calculated CDD of the dashed curve (without the effect of the SRC) is in such a good agreement with that of the experimental data, and the solid curve (with the effect of the SRC) is not in such a good agreement with that of the experimental data, especially in the central locality ( fm) of the distributions. The inclusion of SRC has the feature of reducing the central region of the distribution as seen in the solid curve of this figure. Inspection to the Fig. 1 (b) gives an indication that the solid curve is better describing the experimental data than that of the dashed curve, particularly in the region of momentum transfer fm-1. The rms charge rung calculated with the to a higher place values of and is 2.621 fm, which is less than the experimental value by 0.097fm, which corresponds to a decrease of nearly 3.6 % of the experimental value.Fig. 1. The calculated CDD and elastic charge form factors are compared with those of experimental data. The dashed curve corresponds to the values for the parameters and fm, the solid curve corresponds to the values for the parameters fm-2 and fm while the open circles and the triangles in Figs. 1 (a) and 1 (b) are the experimental data taken from 30 and 31, respectively.The effect of the SRC on the inelastic Coulomb form factors is studied for the two excited states (6.92 and 11.52 MeV) in. center field polarization effects are taken into consideration by actor of the Tassie model eq. (11), where this model depends on the ground state charge density distribution. The proportionality constant eq. (12) is estimated by adjusting the reduced transition probability to the experimental value. The effect of the SRC is incorporated into the ground state charge density distribution through the Jastrow type correlation function 12. The single particle harmonic oscillator wave function is employed with an oscillator size parameter The charge density distribution calculated without the effect of the SRC depends only on one free parameter (namely the parameter), where is chosen in such away so as to re produce the experimental rms charge radii of considered nuclei. The charge density distribution calculated with the effect of the SRC depends on two free parameters (namely the harmonic oscillator size parameter and the correlation parameter), where these parameters are adjusted for each excited state separately so as to reproduce the experimental rms charge radii of considered nuclei.Two different models are employed for. In the first model (model A), is considered as a closed shell nucleus. In this model, the proton occupation probabilities in are assumed to be and Here, the model space in does not contribute to the transition charge density i.e. , because there are no protons outside the closed shell nucleus . Accordingly, the Coloumb form factors of come entirely from the core polarization transition charge density. In the second model (model B), the nucleus of is assumed as a core of with 2 protons and 2 neutrons move in and model space. In this model, the proton occ upation probabilities in are assumed to be and Here, the total transition charge density eq. (10) comes from both the model space and core polarization transition charge densities. The OBDM elements of are generated, via the shell model code OXBASH 32, using the REWIL 33 as a realistic effective interaction in the isospin formalism for 4 particles move in the and model space with a core.In Table 1, the experimental excitation energies (MeV), experimental reduced transition probabilities (fm) and the chosen values for the parameters and for each excited state (used in the calculations of model A and B) in and are displayed. The root mean square (rms) charge radius calculated in both models with the effect of SRC is similarly displayed in this table and compared with that of experimental result. It is evident from this table that the values of the parameter employed for calculations with the effect of SRC are smaller than that of without SRC ( fm) . This is attributed to the fact that the introduction of SRC leads to lucubrate the relative distance of the nucleons (i.e., the size of the nucleus) whereas the parameter (which is proportional to the radius of the nucleus) should become smaller so as to reproduce the experimental rms charge radius of the considered nuclei.Inelastic Coloumb form factors for different transitions in are displayed in Figs. 1 and 2. The calculated inelastic form factors obtained with model A are shown in the upper panel Figs. 1(a)-2(a) of the above figures whereas those obtained with model B are shown in the lower panel Figs. 1(b)- 2(b) of the above figures. It is obvious that all transitions considered in, presented in the above figures, are of an isoscalar character. Besides, the parity of them does not change. Here, the calculated inelastic form factors are plotted versus the momentum transfer and compared with those of experimental data. The dashed and solid curves are the calculated inelastic Coloumb form factor s without and with the inclusion of the effect of the SRC, respectively. The open symbols are those of experimental data taken from 34, 35.Table1. The experimental excitation energies and reduced transition probabilities, the chosen values for and as well as the rms charge radius calculated with the effect of the SRC of 16O.(fm)Model BModel Afm2L)(MeV) conjure up (fm)(fm-2)(fm) (fm)(fm-2)(fm)302.704