Effect of Short Range Correlation

Effect of Short Range CorrelationThe yield of piffling range coefficient of correlation on the atomic beef density distri exactlyion, resilient and in compromising electron scattering coulomb configuration federal agents of 16O lens nucleusAbdullah S. MdekilAbstractThe effect of the short range correlation on the flower density disribution, tractile electron scattering form figures and in rubbery Coulomb form factors is studied for the two excited offers (6.92 and 11.52 MeV) in is analyzed. This effect (which depends on the correlation parameter) is cut ined into the cornerstone state intrust density distribution through the Jastrow type correlation function. The single particle openhearted oscillator vibrate function is used with an oscillator size parameter The parameters and argon considered as free parameters, adjusted for each excited state separately so as to regorge the experimental root mean squ ar post wheel spoke of In nonresilient coulomb (longi tidinal) form factors of 16O, two different mystifys are engaged for . In the first ensample (model A), is considered as a disagreeable slip nucleus. Here, the model billet in does not contribute to the mutation adulterate density, because there are no protons outside the closed jaw nucleus . 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. It is found that the admission of the effect of short range correlations is necessary for obtaining a remarkable modification in the cipher inelastic Coulomb form factors and considered as an essential for invoice the data amazingly throughout the whole range of considered momentum transfer.Keywords charge density distribution, elastic charge form factors, inelastic longitudinal form factors, short range correlation.1-IntroductionElectron scattering provides more accurate information roughly the nu abstemious structure for example size and charge distri bution. It provides important knowledge about the electromagnetic currents inside the nuclei. Electron scattering have been provided a good test 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 can distinguish two types of scattering in the first type, the nucleus is left in its ground state, that is called elastic electron scattering while in the second type, the nucleus is left on its different 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 explicit form of the elastic charge form factor, truncated at the two-body term. This form, which is a sum of one- and two-body terms, depends on the good-hearted oscillator parameter and the correlation parameter through a Jastrow-type correlation function 12. This form is employed for the evaluation o f the elastic charge form factors of closed shell nuclei and in an approximate technique (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 guinea pig of the effect of the SRC on and shell nuclei with entirely avoiding the approximation made in their earlier works 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 introduce the SRC. These forms depends on the single particle jar functions and not on the wave functions of the relative motion of two nucleons as was the case of our previous works 14-20 and other works 6,21,22.It is important to point out that all the above studies were interpose to with the analysis of the effect of the SRC on th e 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 present 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 ground 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. Theory dead electron scattering longitudinal (Coulomb) form factor involves angular momentum an d momentum transfer and is give 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 mass correction (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 assumed 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 hyaloplasm elements reduced 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 given by The reduced matrix elements in spin and isospin space of the longitudinal operator between the final and initial many particles states of the system including configuration 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 label single particle states (isospin included) for the shell model space. The in eq. (4) is calculated 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 modulation charge density, expressed as the sum of the crossway of the times the single particle matrix elements, given by 26. (8)Here, is the radial 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 understandation charge density which depends on the model used for core polarization. To take the core-polarization effects into consideration, the model space rene wal charge density is added to the core-polarization transition charge density that describes the collective modes of nuclei. The total transition charge density becomes (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 hazard 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 associate to the ground state point nucleon densities by 28, 29 (13)in unit of electronic charge per unit volume (e.fm-3).An expression of the correlated density (where the effect of the SRCs is included), consists 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 set 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 occupation 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 13 (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 w here the occupation probability is 0 or 1. To blend in 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 chain nuclei have the same atomic number but this time different 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 discussionThe 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 square (rms) charge radii of nuclei under study. The elastic electron scattering charge form factors which is simply the Fourier transform of the ground state CDD.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). 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 dashed curves are the calculated results without the inclusion of the effect of the SRC obtained with and fm. The solid 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 radius 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 substitution region ( 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 radius calculated with the above 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 inelas tic Coulomb form factors is studied for the two excited states (6.92 and 11.52 MeV) in. Core polarization effects are taken into consideration by means 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 reproduce 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 occupation 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 also 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 enlarge 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) sho uld 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 factors 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 pro babilities, 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)State (fm)(fm-2)(fm) (fm)(fm-2)(fm)302.704