Effect of Short Range Correlation
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The effect of short range correlation on the nuclear charge density distribution, elastic and inelastic electron scattering coulomb form factors of 16O nucleus
Abdullah S. Mdekil
Abstract
The effect of the short range correlation on the charge density disribution, elastic electron scattering form factors and inelastic Coulomb form factors is studied for the two excited states (6.92 and 11.52 MeV) in is analyzed. This effect (which depends on the correlation parameter) is inserted into the ground state charge density distribution through the Jastrow type correlation function. The single particle harmonic oscillator wave function is used with an oscillator size parameter The parameters and are considered as free parameters, adjusted for each excited state separately so as to reproduce the experimental root mean square charge radius of In inelastic coulomb (longitidinal) form factors of 16O, two different models are employed for . In the first model (model A), is considered as a closed shell nucleus. Here, the model space in does not contribute to the transition charge density, because there are no protons outside the closed shell 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 introduction of the effect of short range correlations is necessary for obtaining a remarkable modification in the calculated inelastic Coulomb form factors and considered as an essential for explanation 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.
1Introduction
Electron scattering provides more accurate information about the nuclear structure for example size and charge distribution. 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. [68], the factor cluster expansion of Clark and coworkers [911] was utilized to derive an explicit form of the elastic charge form factor, truncated at the twobody term. This form, which is a sum of one and twobody terms, depends on the harmonic oscillator parameter and the correlation parameter through a Jastrowtype correlation function [12]. This form is employed for the evaluation of the elastic charge form factors of closed shell nuclei and in an approximate technique (that is, for the expansion of the twobody 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 study of the effect of the SRC on and shell nuclei with entirely avoiding the approximation made in their earlier works outlined in [68] for the open shell nuclei. Explicit forms of elastic charge form factors and densities were found utilizing the factor cluster expansion of Clark and coworkers and Jastrow correlation functions which introduce the SRC. These forms depends on the single particle wave functions and not on the wave functions of the relative motion of two nucleons as was the case of our previous works [1420] and other works [6,21,22].
It is important to point out that all the above 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 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
Inelastic electron scattering longitudinal (Coulomb) form factor involves angular momentum and momentum transfer and is given 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 matrix 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 onebody 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 isospinreduced 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 transition charge density, expressed as the sum of the product 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 corepolarization matrix element, in eq. (6), is given by
(9)
where is the corepolarization transition charge density which depends on the model used for core polarization. To take the corepolarization effects into consideration, the model space transition charge density is added to the corepolarization 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 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 unit of electronic charge per unit volume (e.fm3).
An expression of the correlated density (where the effect of the SRC's is included), consists of one and twobody 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 twobody term and originated by the factor cluster expansion of Clark and coworkers [1012], depends on the correlation parameter through the Jastrowtype correlation
(16)
where is a stateindependent correlation function, which has the following properties: for large values of and for It is so clear that the effect of SRC's, inserted by the function becomes large for small values of SRC parameter and vice versa.
The onebody 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 twobody term, in eq. (14), is given by [13]
(18)
where
(19)
The form of the twobody term is then originated by expanding the factor in the spherical harmonics and expressed as [13]
(20)
where
(21)
and is the ClebschGordan 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 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)
3Results and discussion
The 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 fm2 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 central 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 fm1. 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 fm2 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. 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) 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 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 probabilities, the chosen values for and as well as the rms charge radius calculated with the effect of the SRC of 16O.
(fm) 
Model B 
Model A 
fm2L) 
(MeV) 
State 

(fm) 
(fm2) 
(fm) 
(fm) 
(fm2) 
(fm) 

[30] 
2.704 
2.6 
1.75 
2.732 
2.35 
1.76 
[34] 
6.92 [36] 

2.668 
2.5 
1.72 
2.678 
1.79 
1.67 
[37] 
11.52 [37] 
In Fig. 2, the inelastic Coulomb form factors for the transition ( MeV and fm4 [36]) are demonstrated. In Fig. 2 (a), the calculated form factors (the solid curve) are obtained via utilizing the values of fm and fm2. It is obvious from this figure that the calculated form factors without the effect of the SRC (the dashed curve) does not describe correctly the experimental data (the open circle symbols) [36], where the diffraction minimum of this curve is not reproduced in the correct place of momentum transfer as well as the experimntal data are slightly over predicted for the second maximum. Inclusion the effect of SRC leads to give a notable improvement in the calculated form factors (the solid curve) and then leads to describe the data amazingly for all range of momentum transfer. It is clear that the magnitude of the experimental form factors along the diffraction minimum as well as along the first and second maxima is well reproduced by the solid curve.
In Fig. 2 (b), the calculated inelastic form factors revealed by the solid curve (with the effect of the SRC) are obtained with applying the values of fm and fm2. The calculated C2 form factors of the dashed and solid curves are in very good agreement with those of experimental data along the first maximum but the dashed curve does not agree well with the data at the region q>1.9 fm1. The inclusion of the effect of SRC leads to overestimate slightly these data in the region of q>1.9 fm1. However, the calculated C2 form factors by the solid curve agrees quite well with the experimental data in the model A [Fig. 2 (a)].
Fig. 2. Inelastic Coulomb form factors for the transition to the (6.92 MeV) state in. The dashed and solid curves are the calculated form factors without and with the inclusion of the effect of the SRC, respectively. The open circle symbols are those of the experimental data taken from ref. [34].
In Fig. 3, the inelastic Coulomb form factors for the transition ( MeV and fm4 [37]) are presented. In Fig. 3 (a), the calculated form factors (the solid curve) are obtained via utilizing the values of fm and fm2. It is clear from this figure that the calculated result displayed by the dashed curve (without the effect of the SRC) does not predict correctly the experimental data at qâ‰¥1.7 fm1where the diffraction minimum of this curve is not reproduced in the correct place of momentum transfer. Inserting the effect of the SRC leads to improve the calculated form factors by the solid curve and then makes the calculations to be more closer to the data at q<1.7 fm1. The rms charge radius estimated by using the above values of and is 2.678 fm, which is less than the experimental value by 0.04 fm, which corresponds to a decrease of nearly 1.4%of the experimental value.
In Fig. 3 (b), the calculated form factors with the effect of SRC are obtained using the values of fm and fm2. This figure illustrates that the calculated result displayed by the dashed curve (without the effect of the SRC) does not predict correctly the experimental data, where the diffraction minimum of this curve is not reproduced in the correct place of momentum transfer. Introducing the effect of the SRC leads to provide a notable modification in the calculated form factors displayed by the solid curve and subsequently predicts the data amazingly throughout the whole range of considered momentum transfer. Fig. 4.6 shows that the C2 form factors with inclusion the effect of the SRC calculated by model B [Fig. 3(b)] is better describing the data than that of model A [Fig. 3 (a)].
Fig. 3. Inelastic Coulomb form factors for the transition to the (11.52 MeV) state in . The dashed and solid curves are the calculated form factors without and with the inclusion of the effect of the SRC, respectively. The open circle symbols are those of the experimental data taken from ref. [35].
4. Conclusions
The effect of the SRC on the CDD, elastic electron scattering form factors and inelastic Coulomb form factors for the two excited states in is analyzed. This effect is included in the present calculations through the Jastrow type correlation function. It is concluded that the introduction of the effect of the SRC is necessary for obtaining a notable modification in the calculated inelastic Coulomb form factors and also essential for explanation the data astonishingly throughout the whole range of considered momentum transfer.
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