Effect of short-range correlations on the single proton 3s1/2 wave function in 206Pb

We consider the experimental data for difference, Δρc(r), between the charge density distributions of the isotones 206Pb - 205Tl, deduced by analysis of elastic electron scattering measurements and corresponds to the shell model 3s1/2 proton orbit. We investigate the effects of two-body short-range correlations. This is done by: (a) Determining the corresponding single particle potential (mean-field), employing a novel method, directly from the single particle proton density and its first and second derivatives. We also carried out least-square fits to parametrized single particle potentials; (b) Determining the short-range correlations effect by employing the Jastrow correlated many-body wave function to derive a correlation factor for the single particle density distribution. The 3s1/2 wave functions of the determined potentials reproduce fairly well the experimental data within the quoted errors. The calculated charge density difference, Δρc(r), obtained with the inclusion of the short-range correlation effect does not reproduce the experimental data.


Introduction
The shell model has been very successful in explaining many features of nuclei [1,2]. The relation between the shell model wave functions and the real nuclear ones is rather complicated. Important information about it may be deduced from the measured charge density difference, # ( ), between charge density distributions of the isotones 206 Pb -205 Tl [3]. The experimental data of # ( ) show a clear maximum at the center of 206 Pb with two additional maxima. This seems to be the shape obtained from a 3s1/2 single proton wave-function, in agreement with the simple shell model. It was pointed out in the literature that commonly used central potentials, such as the Woods-Saxon potential, lead to a 3s1/2 charge density in disagreement with experimental data. In particular, the central density obtained from the Woods-Saxon potential is too large by about 30%. This difference between data and the Woods-Saxon results was considered earlier in the literature resulting with the statement that it is accounted for by the effect of two-body short range correlations (SRC) on the shell model wave functions [3,4]. However, it was also stated in Ref, [5] that "there is no significant discrepancy between the experimental data and the Hartree-Fock calculations using reasonable effective interactions." In this work we take a closer look at this problem and ask: (i) Is there a single particle potential V(r) whose 3s1/2 wave function reproduces the experimental data? To answer this question we have developed a novel method, using the single particle Schrodinger equation for a wave function ( ⃗) with eigenenergy E, to determine the central potential ( ⃗) directly from the measured single particle proton density of 3s1/2 wave function and its first and second derivatives, assuming known for all ⃗. We apply the method to the experimental data of the charge distribution of the proton 3s1/2 orbit given by the charge density difference, # ( ) [3]. We have also carried out fits to parametrized single particle potentials. (ii) Is the effect of short-range correlations (SRC) on the nuclear density explains the experimental data? To assess the effect of SRC we have used the Jastrow many-body correlated wave function, with a two-body correlation factor, and derived a simple and a good approximate method for calculating the effect of SRC on the single particle density distribution. We point out that the resulting single particle potential, if found, will provide a stringent limit on the effects of short correlation on the expected values of long-range operators, an important test for the shell model. The potential can also be used as an additional experimental constraint in determining a modern energy density functional (EDF) for providing more reliable predictions of properties of nuclei and nuclear matter [6,7]. In the next section we provide a short presentation of the novel methods used in the calculations and in section 3 we present some results with the conclusion given in section 4. Some results were presented in Refs. [8,9,10].

Formalism
2.1 Deducing the potential from the single-particle density From the single particle Schrodinger equation, where ( ⃗) is a real local and non-singular potential, follows that for a given single particle wave function ( ⃗), known for all ⃗, and given eigenvalue E, the corresponding single particle potential V is uniquely determined by ( ⃗) = + (3) Here, ;<= ( ) is the radial wave function for the orbit with principal number n, orbital angular momentum l and total angular momentum j and <= is the eigenfunction of the angular momenta l and j. In the following we limit the discussion to the proton 3s1/2 orbit, in which the angular momentum l = 0 and therefore no centrifugal and spin-orbit potentials appear in ( ⃗). Substituting (3) in (1) we obtain the Schrodinger equation for the radial wave function ;<= ( ). Using the Schrodinger Equation for ;<= ( ) we find that, where #D; ( ) and where ;<= ( ) is obtained by integrating the single particle density distribution over the angles and spin variables, it is possible to extract the wave function ;<= ( ) from the experimental data for the charge density of the 3s1/2 proton orbit and deduce the corresponding single particle potential using Eq. (4). This leads to having some numerical complications. Therefore, we develop below a method to determine the potential directly from ;<= ( ) and its first and second derivatives. Using Eq. (4) for the relation between ( ) and ;<= ( ), we obtain the simple relation between ( ) and ;<= -, When ;<= -= 0, ;<= -⁄ = 0. From Eq. (4), for non-singular potential, the term on the right hand side (r.h.s) of Eq. (6) in the large square brackets also vanishes. From Eqs. (5) and (6) we obtain the relation, When ;<= = 0, ;<= ⁄ = 0. From Eq. (4), for non-singular potential, the term in the square brackets of (7) vanishes and K , R ?@A K6 , also vanish. We add that a commonly used central nuclear potential is the Woods Saxon (WS) potential, where, U , E and a U are the depth, half radius and diffuseness parameters, respectively. For the Coulomb potential we adopt the form obtained from a uniform charge distribution of radius #Z , where #Z is deduced from the experimental value of the charge root mean square radius.We emphasize that in elastic electron-nucleus scattering measurements the charge density distribution, #Z ( ⃗) is deduced but in theoretical models the point proton density distribution, b ( ⃗) is calculated. They are related by where -= E E-bfg with bfg = 0.85 fm being the corresponding charge root mean square (rms) radius [11]. The Fourier transform of the charge density #Z ( ⃗), determined by the convolution relation of Eq. (10) is given by

Effect of short-range correlations on the single particle density
Recently [12], we have derived and employed a simple and a good approximate method for determining the effect of short-range correlations (SRC) on the single particle densities of nucleons in nuclei. The method is obtained assuming: (i) the Jastrow ground state many body wave-function in the independent pair approximation [13,14], and (ii) the relation between the exchange terms and the direct terms of the two-body matrix element of the correlation function is similar to that of the contact interaction U u ⃗ v − ⃗ = w. The Jastrow ground state wave-function has the form, where A is the number of nucleons, ~I66 is the normalization constant, Φ( ⃑ E , ⋯ ⃑ | ) is the independent particle model (IPM) wave-function for the ground state, given by the Slater determinant,  where the parameter is deduced from the experimental data on the charge form factor. The effect of SRC on the proton single particle density distribution is determined by evaluating the matrix element of the density operator using the expansion where, g = 2h -h 2 . We adopt in our calculations the approximation of Ref. [14] with the ratio U '~ 0.5, where d being the "healing distance" and U h is the average volume per particle. In the calculations we use the shell model harmonic oscillator wave functions.

Results
We present in Figure 1a assuming an increase by 0.005 fm, similar to the change between nuclei in this region [15]. We obtain the scaling parameter α = 5.4792/(5.4792 + 0.005) = 0.9990, the ratio between the charge rms radius of 205 Tl to that of the 81 core protons in 206 Pb. The results for ># ( ) is shown in Figure1a (dashed line).   Figure 3 for # -( ). the charge density of the proton 3s1/2 orbit using the Jastrow correlated wave function.
Using the Jastrow wave function, Eq. (13), we determined the effect of SRC on the charge density distributions of 206 Pb and 205 Tl. In the calculations we adopted the shell model harmonic oscillator wave function with the size parameter = 1/5.9 fm -2 and the parameter = 1.96 fm -2 . In Figs. 3 and 4 we show the results for the 3s1/2 proton charge density and the square of the radial wave-function, respectively, and compare with experimental data. The dotted lines show the experimental results for the difference between 206 Pb and 205 Tl charge distributions and the solid line is the corresponding calculated results. The dashed and dashed dotted lines are for the harmonic oscillator 3s1/2 proton orbit in 206 Pb with and without the effect of correlations, respectively. It is seen from Figures 3 and 4 that although the effect of SRC is to reduce the 3s1/2 single proton charge density at r = 0 by about 30%, the calculated density disagrees with the experimental data by more than a factor of 2, particularly in the region of r = 2 -4 fm.

Conclusions
We have considered the experimental data for the charge density difference between the isotones 206 Pb -205 Tl, deduced by the analysis of elastic electron scattering measurements and corresponds to the shell model 3s1/2 proton orbit. We have investigated the effects of two-body short-range correlations. This was done by: (a) Determining the corresponding single particle potential (mean-field), employing a novel method, directly from the single particle proton density distribution and its first and second derivatives. We also carried out least-square fits to parametrized single particle potentials. The 3s1/2 wave functions of the determined potentials reproduce fairly well the experimental data within the quoted errors.; (b) Determining the effect of short-range effect correlations by employing the Jastrow correlated many-body wave function to derive a correlation factor for the density distributions of single particle orbits. The calculated 3s1/2 density obtained with the inclusion of the effect of short-range correlations does not reproduce the experimental data. Our conclusion does not contradict the existence of SRC effects in the real nuclear wave function, as demonstrated by the high momentum tail in the experimental data for the charge form factor. Our results indicate that the shell-model wave function may yield charge density distributions which are in agreement with experimental data. More accurate experimental data, with uncertainty smaller by a factor of two or more, may answer the question how well can the data be reproduced by a 3s1/2 single particle wave function obtain from mean-field calculations. It will also help to determine more exactly the deviation between data and our SRC calculations. Of course, more accurate calculations of the effect of SRC are needed.