The least-squares analysis of the moments of the charge distribution in the mean-field models

Comparing the moments of the charge distribution in the mean-field models with experimental values from electron scattering, each value of the related moments of the point proton and neutron distributions is estimated in $^{40}$Ca, $^{48}$Ca and $^{208}$Pb by the least-squares analysis(LSA).


Introduction
Electron scattering is one of the best probes to investigate the point proton distribution in nuclei among various methods [1].Since the electromagnetic interaction and the reaction mechanism are well understood theoretically [2], the scattering process is completely separated from assumptions on the nuclear structure.Moreover, the relationship between the nuclear charge distribution and the point proton one is well defined [3].The nuclear charge density is composed of the proton and neutron charge densities, but is dominated by the former.The neutron density is less than about 1% of the total density, and its radial dependence oscillates to yield the integrated charge to be zero, as shown in Fig. 1.As a result, electron scattering has been used as a probe to explore the point proton distribution throughout the periodic table [4], and has not been employed so far to investigate the point neutron distribution, which is another fundamental quantity to dominate nuclear structure.The neutron distribution has been studied with other probes mainly through strong interaction which is dealt with rather model-dependently, compared with electromagnetic interaction [1].
Recently, it has been pointed out that the neutron charge density plays an appreciable role in the nth( n ≥ 4) moments of the nuclear charge distribution [5], because at the nuclear surface, contribution from the neutron charge density to the total one becomes non-negligible, as expected from Fig. 1.The nth moment of the nuclear charge distribution is shown to include the contribution from the m(≤ (n − 2))th moment of the point neutron distribution, in addition to the m(≤ n)th moment of the point proton distribution [5].This fact implies that both the point proton and the point neutron distribution are explored together, using the same experimental data through well-known electromagnetic interaction.It is not possible, however, to extract the neutron contribution model-independently, because the nth( n ≥ 4) moment has several components whose values are not determined uniquely by experiment.Strictly speaking, even the second moment of the charge density is not simply given by the point proton distribution [6].ρ cp (r) ρ cn (r)×100 Figure 1: The charge density(ρ c (r)) of 208 Pb as a function of the nuclear coordinate(r) calculated by the relativistic mean-field model(NL3), taking into account the finite size of the nucleon.The proton(ρ cp (r)) and the neutron(ρ cn (r)) charge densities are shown by the black and blue curves, respectively.The former is almost the same as the total charge density, ρ c (r).For details, see the text.
Hence, in order to estimate the value of each component separately, Ref. [6] has employed the least-squares analysis(LSA) for the values of the moments predicted in the mean-field(MF) models, which have been widely used to describe the fundamental properties of nuclei for long time [7].
The LSA used to analyze the moments is as follows.First, many MF models are arbitrarily chosen within the same model-framework like the relativistic MF(RMF)-framework, or the Skyrme-type MF(SMF)-framework in the literature [8,9].In Ref. [6], two sets composed of the 11 RMF-models and the 9 SMF-ones were prepared.Next, the nth moment and its component, for example, the fourth moment(Q 4 c ) of the charge distribution and the second moment of the point neutron distribution(R2 n ) in 48 Ca, are calculated using each model.Third, a pair of the two calculated values are plotted on the (R 2 n − Q 4 c )-plane as an element, (R 2 n,i , Q 4 c,i ), by the model named i in the set.Fourth, the regression equation of the least-squares line(LSL) for the elements is obtained, together with the standard deviation and the correlation coefficient between two quantities.Finally, the value of the component R 2 n is determined by the cross point(LSL-value) of the LSL with the line of the experimental value of Q 4 c observed through electron scattering.Thus, the LSA is not a way to determine the experimental value of each component, but to estimate the value accepted in the used model-framework.Actually, Ref. [6] has estimated the values of R n allowed in the RMF-and SMF-frameworks for 40 Ca, 48 Ca and 208 Pb within 1% error 2 , together with the ones of the root mean-square-radius(msr), R p , of the point proton distribution 3 , using the experimental data available at present [4,10].
The present paper is an extension of Ref. [6], where the components of Q 4 c in the MF models are explored.In this paper, the sixth moments of the charge distribution(H 6 c ) will be investigated.As far as the author knows, the structure of H 6 c will be discussed for the first time in nuclear physics.Among the components of H 6  c , the values of R 2 n and the fourth moment of the 2 The relativistic and non-relativistic expressions of the sixth moment Neglecting the center of mass correction, the sixth moment of the nuclear charge density is described as where r 6 cτ is given by with the proton charge density, ρ cp (r), and the neutron charge density, ρ cn (r), satisfying d 3 rρ cp (r) = Z, the number of the protons in the nucleus, and d 3 rρ cn (r) = 0, respectively.Following Refs.[5,6], it is convenient to use the Fourier component of ρ cτ (r) for calculations of H 6 c , ρcτ (q) = d 3 r exp(iq • r)ρ cτ (r) = d 3 rj 0 (qr)ρ cτ (r).
Then, Eq.( 2) is written as where ρGτ (q) = d 3 rj 0 (qr)G Eτ (q 2 )ρ τ (r) , ρF τ (q) = d 3 rj 0 (qr)F 2τ (q 2 )W τ (r). (5) In the above equations, G Eτ (q 2 ) and F 2τ (q 2 ) denote the Sachs and Pauli form factors of the nucleon, and ρ τ (r) and W τ (r) stand for the relativistic point nucleon distribution and the spinorbit density, respectively.The explicit forms of the nucleon form factors are provided in Ref. [6] as, with r p = 0.877 fm, r 2 ± = (0.830) 2 ∓ 0.058 fm 2 .The relativistic expressions of ρ τ (r) and W τ (r) are written in the MF models as [5,6], where j α denotes the total angular momentum of a single-particle, κ α = (−1) jα−ℓα+1/2 (j α +1/2), and ℓ α the orbital angular momentum.The function G α (r) and F α (r) stand for the radial parts of the large and small components of the single-particle wave function, respectively, with the normalization, The spin-orbit density is a relativistic correction due to the anomalous magnetic moment of the nucleon, µ τ (µ p = 1.793, µ n = −1.913), and its role is enhanced by the effective nucleon mass, M * (r) = M + V σ (r), V σ (r) being the σ meson-exchange potential which behaves in the same way as the nucleon mass, M , in the equation of motion.The value of M * depends on the relativistic MF models, but is nearly equal to 0.6M .Eq.( 7) satisfies d 3 r ρ τ (r) = Z for τ = p, and N , the number of the neutrons in the nucleus, for τ = n, respectively, while Eq.( 8) does d 3 r W τ (r) = 0, as it should.Using Eq.( 6) to (8), Eq.( 4) is calculated in the same ways as in Ref. [5], In the right-hand sides of the above equations, we have used the same notations as in Ref. [5].Finally, the relativistic expression of the sixth moment is described in terms of H 6 cn and H 6 cn of the proton and neutron charge distributions expressed, respectively, as where the components of H 6 cn and H 6 cn are given by The notation r n Wτ is defined as where N τ represents Z and N for τ = p and τ = n, respectively.The non-relativistic expression of the sixth moment which is equivalent to the above relativistic one up to 1/M 2 may be obtained by the Foldy-Wouthuysen(F-W) transformation [5].For the phenomenological non-relativistic models, however, it is not possible to derive the consistent relativistic corrections to them, since their original four-component frameworks are not known.In the present paper, the free Dirac equation is employed as the nuclear part of the relativistic Hamiltonian for the F-W transformation [5,6].Then, the non-relativistic expression of the sixth moment is obtained from Eq.( 10) by replacing the relativistic matrix elements r n τ and r n Wτ with r n nr+r,τ and r n nr,Wτ , respectively, where the subscript nr of the matrix elements in the right-hand sides indicates that they should be calculated in the two-component framework of the wave functions4 .

Components of the sixth moment of the charge density
Before the LSA is performed in the next section, it will be shown in this section how each component in Eq.( 11) and ( 12) contributes to H 6 c by taking a few MF-models.  4Ca, 48 Ca and 208 Pb in units of fm 6 .They are calculated by using parameters of the relativistic nuclear models, NL3 [9] and NL-SH [11], and of the non-relativistic one, SLy4 [12].The values of H  40 Ca, 48 Ca and 208 Pb in units of fm 6 .The values are calculated by using parameters of the relativistic nuclear models, NL3 [9] and NL-SH [11], and of the non-relativistic one, SLy4 [12].The others show the values of the components of H 6 cn .For the details, see the text.) of the charge distributions in 40 Ca, 48 Ca and 208 Pb in units of fm 6 .FB and SOG indicate the Fourier-Bessel-and the sum-of-Gaussiananalyses of the experimental data from electron scattering [4], respectively.In the brackets, the values of H c are listed in units of fm.The values calculated with the MF-models are given by , where H 6 cp and H 6 cn stand for the sixth moments of the proton and neutron charge distributions, respectively, as listed in Table 1 and 2. NL3 [9] and NL-SH [11] are the relativistic MF-models, while SLy4 [12] the non-relativistic one.For the details, see the text.4: The results of the least squares analysis taken from Ref. [6].The numbers in the parentheses for the RMF-and SMF-models denote the error taking account of the experimental one and the standard deviation of the calculated values from the least-squares line.All the numbers are given in units of fm.
2 show the values of the components of H 6 cp and H 6 cn , respectively.They are calculated using the two RMF-models, NL3 [9] and NL-SH [11], and the one SMF-model, SLy4 [12].For the SMF models, Eq.( 14) and (15) are employed in Eq.( 11) and (12).These phenomenological models are constructed so as to reproduce the fundamental quantities of various nuclei like the binding energies, the root msr of the charge distribution, etc., and have widely been used for studying other various nuclear phenomena [8,14].The values of the last terms of Eq.( 11) and (12) which are not listed in the tables are given, respectively, by Table 3 compares the MF-results with the experimental values of H 6 c = H 6 cp − H 6 cn which are evaluated by two ways.As mentioned in the previous section, the one is calculated by using the FB-coefficients(FB), and the other the SOG-parameters(SOG) in Ref. [4].Unfortunately, their experimental errors can not be estimated from the published data [4], where there are not enough information on the experimental and model-dependent errors.Hence, the only four digits as the significant figures are kept, so that the sixth root of H 6 c in units of fm yields up to the three digits after the decimal point, as in the brackets of Table 3.The numbers listed in Ref. [4] for the root msrs of the charge distributions are given in the same way.The calculated values in Table 1 and 2, however, are shown up to the three digits after the decimal point so as to compare the model-predictions with one another in detail.
Let us go into more details of Table 1, 2 and 3 in order.In Table 1 shows that the main component, H It is noted that the difference between the values of H Wp in the relativistic and non-relativistic models is not only on their absolute values, but also on their signs in 40 Ca and 48 Ca, as seen in Table 1.
Tables 1 and 2 show that H  40 Ca is larger than that of 48 Ca.This fact is reproduced by the three MF models in Table 3.The reason why the value of H 6 c in 40 Ca is larger than that in 48 Ca will be discussed in the next section.
For later discussions, the previous results obtained by the LSA in Ref. [6] 5 are summarized in Table 4.The notation, δR, stands for the neutron-skin thickness, δR = R n − R p , and Q cp and Q cn represent the fourth root of the mean fourth moments of the proton and neutron charge distributions, respectively.The experimental values of R c and Q c obtained by FB-method [10] are also listed in Table 4.
The relationship between Q c , Q cp and Q cn is given by [6], where Q 4 cp and Q 4 cn are expressed as with the notations for the protons, and for the neutrons, Comparing Table 4 to Tables 1 and 2, it is seen that the neutron contribution from H 6 cn to H 6 c is increased, compared to that from Q 4 cn to Q 4 c , as expected.Ref. [6] has shown that Q 2n is the main component of Q 4 cn , while in H 6 cn , H Wn is much larger than H 2n in 48 Ca and 208 Pb.The contribution from the spin-orbit density is more important in H 6 cn than in Q 4 cn .Increase of the contribution in the relativistic models, compared with that of the non-relativistic models, is due to the fact that the spin-orbit density is enhanced by the small effective mass of nucleon in the relativistic equation of motion [13].
In contrast to the values of H 6 cp of 40 Ca and 48 Ca in Table 1, those of Q 4 cp in Table 4 are almost the same in the RMF-and SMF-models.Thus, the difference between the models depends on which moment is discussed.

The least-squares analysis
In the present section, the values of the various moments of the point proton and neutron distributions allowed in the MF-framework are estimated in 40 Ca, 48 Ca and 208 Pb, by the LSA using the experimental values of H 6 c with the SOG-and FB-methods.The only SOG-values, however, will be indicated in all the figures below, in order to make them simpler.The results using the FB-values will be mentioned, if necessary.
The way of the LSA is the same as that in Ref. [6] for n and H 6 cn and the line of the quasi-experimental value of H 6 cn .In Ref. [6] the LSL-values of R 2 n , R 2 p , and Q 4 p have been estimated for 40 Ca, 48 Ca and 208 Pb, by using the experimental values of R 2 c and Q 4 c obtained from the FB-analysis in Refs.[4,10].In the present paper, the LSL-values of H 6 p and Q 4 n will be also determined by using the experimental values of H 6 c from the both FG-and SOB-analyses.In the following subsections, the sixth moments of the charge density in 40 Ca, 48 Ca and 208 Pb will be analyzed.The MF-models employed in the present paper are the same as those in Ref. [6].They are 11 kinds of parameterization for nuclear interactions as RMF-models and 9 kinds of parameterization as the SMF-models, which are chosen arbitrarily among more than 100 models accumulated in nuclear physics [8,14].We number them, following Ref.  because there is no reason why they should be taken into account together [6,28].The equations of the LSLs and their standard deviations(σ) are listed in the last of the subsection as Table 5.Both RMF-and SMF-models provide almost the same LSLs with the small values of σ as 0.7718 and 1.5701, respectively, and with the correlation coefficient to be about 1.0000.The two LSLs cross the horizontal line which expresses the SOG experimental value of H 6 c indicated on the right-hand side of the figure.The values of the intersection points are described on the top of the figure for the RMF(rel.)-andSMF(non.)-frameworks,respectively.Those are the LSLvalues of H 6 cp accepted in the RMF-and SMF-ones, according to the LSA.They are 5581.031and 5572.169fm 6 in the RMF-and SMF-frameworks, respectively, which will be used below as a 'quasi-experimental value' in 40 Ca.The way of the designation in Fig. 2 will be used in all the following figures in the present paper.
It is seen in Fig. 2 that the RMF-and SMF-models underestimate the experimental value of H 6 c .As a result, the LSL-value of H 6 cp allowed by the MF-frameworks is much larger than the values calculated with the MF-models used in the present LSA.For latter discussions in §5, the experimental value of H 6 c obtained from the FB-analysis in Table 3 should be remembered.The FB-value is 4.234 × 10 3 fm 6 , which is overestimated by all the MF-models except for SKI(1) in Fig. 2, in contrast to the SOG-value given by 5.390 × 10 3 fm 6 .5 in the last of the present subsection.For details, see the text.
In Fig. 3 is shown H 6 cp as a function of H 6 p .Here, the LSL-values of H 6 cp determined in Fig. 2 are used as the quasi-experimental value, as shown by the solid and dashed horizontal lines for the RMF-and SMF-models, respectively.The calculated values in the RMF-and SMF-models distribute over the similar region of the LSLs with the correlation coefficients, r = 0.9996 and 0.9998, respectively, but underestimate the LSL-values of H 6 p .The order of the MF-models according to the magnitude of H 6 p is almost the same as that of Fig. 2, and will also be seen to be the same as those of Fig. 4 for Q 4  p and of Fig. 5 for R 2 p . Figure 4   Figure 5 shows the correlation of R 2 p with H 6 cp .As seen in Table 1, H 2p with R 2 p contributes to H 6  cp by only about 2%, but they are well correlated with r = 0.9466 and 0.9729 in the RMFand SMF-models, respectively.The LSL-values of R 2 p determined from H 6 cp in Fig. 5 will be compared to those from R 2 c and Q 4 c in Ref. [6] in §5. Figure 6 shows the correlation between Q 4 n and H 6 cn .They are related as in Eq.( 12), where Q 4 n is defined as r 4 n in H 4n .As mentioned before, the quasi-experimental values of H 6 cn are obtained from the experimental value of H 6 c and the LSL-values of H 6 cp by using the definition in Eq. (10).The differences between the quasi-experimental values of H 6 cn and between the LSLs in the two frameworks in Fig. 6 are understood in the same way as in Figs. 8 and 9 of Ref. [6] for the (R 2 n − Q 4 cn )-correlation.Fig. 8 in Ref. [6] shows similar differences to those in Fig. 6, while such differences disappear in neglecting artificially the spin-orbit densities in the both frameworks, as exhibited in Fig. 9 in Ref. [6].The order of the MF-models according to the magnitude of H ) Figure 6: The sixth moment of the neutron charge density(H 6 cn ) as a function of the fourth moment of the point neutron distribution(Q 4 n ) in 40 Ca.The designation in the figure is the same as in Fig. 2. For details, see the text.
Figure 7 shows the correlation of H 6 cn with R 2 n .The LSL-values are underestimated by the calculated values in the same way as in the previous figures.The value of the correlation coefficient of the relativistic models in this figure is the worst among those in 40 Ca, but is r = 0.9393, while that of the non-relativistic models 0.9978.
The distribution of the circles in Fig. 7 is similar to that in Fig. 6.The difference between the LSLs is understood in the same way as in the previous figure, as a result of the different contributions from the spin-orbit densities in the two frameworks.The spin-orbit densities make the difference between the two LSLs clear, but change a little the LSL-values of R 2 n [6].Comparing Fig. 5 to Fig. 7, the LSL-value of R 2 p is larger than that of R 2 n in both the RMFand SMF-models.In the same way, the LSL-value of Q 4 p in Fig. 4 is larger than that of Q 4 n in Fig. 6.This result will be discussed in the following section, together with those in 48 Ca and 208 Pb.

48 Ca
Figure 8 shows the relationship between H 6 cp and H 6 c in 48 Ca.The correlation coefficients are 0.9987 and 0.9992 for the RMF-and SMF-models, respectively, and the equations of LSLs are listed, together with their standard deviations, in Table 6 at the end of this subsection.
The LSL-values of H 6 cp accepted in the RMF-and SMF-models are 4805.335and 4721.394fm 6 in the RMF-and SMF-schemes, respectively.In contrast to the case of 40 Ca, the calculated values of H 6 c and H 6 cp in the RMF-models other than NL-Z(7) underestimate the ones of their experimental and LSL-values, while those of the SMF-models, except for SKI(1), overestimate them.The mean values of H 6 c and H 6 cp in the RMF models are smaller than those in the SMF ones, whereas the LSL-value of H 6 cp in the RMF-framework is larger than that in the SMF-one.The experimental value, 4298.520fm 6 , of H 6 c in 48 Ca is smaller than that, 5389.740fm 6 , in 40 Ca.Its main reason is due to the increasing negative contribution from the neutron charge density.Moreover, it should be noticed that the LSL-values of H 6 cp are also smaller in 48 Ca than those in 40 Ca in Fig. 2.These results will be understood from Fig. 9, which shows that the LSL-values of H 6 p is smaller in 48 Ca than those in 40 Ca in Fig. 3.In Fig. 9 is shown H 6 cp as a function of H 6 p in 48 Ca.The calculated values of the RMF-and SMF-models are almost on the same line, but most of the former underestimate the LSL-value, cp ) in 48 Ca.The designation in the figure is the same as in Fig. 2. The values of the slope(a), the intercept(b) and the standard deviation(σ) of the LSLs are listed in Table 6.For details, see the text.
whereas the latter overestimate their LSL-value.As mentioned before, both LSL-values of H 6 p are smaller than those in 40 Ca in Fig. 3.The three examples in Table 1 shows also such a tendency, in contrast to H 2p with R 2 p .Thus, higher moments reveal not only the contribution of the neutron charge density, but also the detailed change of the point proton distributions in the isotopes.p and H 6 cp in 48 Ca.The RMF-models underestimate again the LSL-value, while the SMF-ones except for SKI(1) overestimate their LSL-value in the same way as in Fig. 9.The LSL-values of Q 4 p are given as 183.914 and 182.500 fm 4 in the RMF-and SMF-frameworks, respectively.They are smaller than those in 40 Ca in Fig. 4 showing 198.613 and 199.877 fm 4 .It should be noted that in contrary to the SOG-value, the FB-one in Table 3 yields the smaller LSL-values in 40 Ca as 167.897 and 168.880 fm 4 in the RMF-and SMF-schemes, respectively than 172.420 and 171.436 fm 4 in 48 Ca.As for H 6 p , both the SOG-and FB-values yield the larger LSL-values in 40 Ca than those in 48 Ca.
Figure 11 shows H  n with H 6 cn in 48 Ca is shown in Fig. 12.The values of r are obtained as 0.9669 and 0.9902 for the RMF-and SMF-frameworks, respectively.The quasi-experimental values of H 6 cn for the two frameworks are different, since they are obtained by Eq.( 10) using the values of H 6 cp given to each framework by Fig. 8.The difference between the two LSLs in Fig. 12 is owing mainly to the different contributions from H Wn to H 6 cn in the RMF-and SMF-models, as seen in Table 2.
On the one hand, in Fig. 10, the calculated values of Q 4 p in the RMF-models are smaller than the values of the SMF-ones, except for SKI(1), but their LSLs yield the almost same LSL-values of Q 4 p .On the other hand, in Fig. 12, most of the calculated values of Q 4 n are distributed over the same region between 220 and 250 fm 4 , but the LSL-values of the RMF-and SMF-frameworks are different from each other, as shown on the top of the figure as 244.1112 and 217.569 fm 4 , respectively.These values together with the LSL-values in Fig. 10 provide δQ = Q n − Q p = 0.2701 and 0.1651 fm in the RMF-and SMF-frameworks, respectively, which show the value of δQ in the RMF-one is larger by about 0.105 fm than that in the SMF-one, in a similar way as for the neutron skin thickness defined by δR = R n − R p in Refs.[6,29].n ) in 48 Ca.The designation in the figure is the same as in Fig. 2. For details, see the text.
Figure 13 shows the correlation of R 2 n with H 6 cn in 48 Ca.The values of the correlation coefficients are 0.9267 for the RMF-models, while 0.9958 for the SMF-ones.The former value is the worst among those for 48 Ca.The distribution of the calculated circles is similar to that in Fig. 12.The line-spacing between the two LSLs is also almost the same as in Fig. 12.Most of the circles by both the RMF-and SMF-models are in the same region of R 2 n around 13 fm 2 , in contrast to those for the protons in Fig. 11 where all of the closed circles distribute over the lower region of R 2 p than the open circles, except for that of SKI(1).However, whereas the LSL-values of R 2 p are almost the same in the RMF-and SMF-schemes in Fig. 11, the LSL-values of R 2 n are different from each other in Fig. 13.The LSL-value of R 2 n in the RMF-framework is 13.282 fm 2 which is underestimated by the most of the solid circles, while that in the SHF-one 12.622 fm 2 which is overestimated by the open ones, except for SKI (1).The two LSL-values yield the difference between R n -values between the RMF-and SMF-schemes to be about 0.092 fm, which is almost equal to 0.105 fm of the difference between δQ mentioned before.The least-squares line y(x) = ax + b and the standard deviation σ depicted in Figure 8 to 13 for the relativistic(RMF) and the non-relativistic(SMF) models.

208 Pb
As shown in Ref. [6], the experimental values of R 2 c and Q 4 c of 208 Pb are well reproduced by both RMF-and SMF-models, unlike those of 40 Ca and 48 Ca.Fig. 14 shows the correlation between H 6 cp and H 6 c in 208 Pb.The circles of the calculated values are distributed over both the left-and right-hand sides of the LSL-values of H 6 cp .This fact implies that the mean value of H 6 c , H 6 c , in the MF-models well reproduce the SOG-value and that the value of H 6 cp is nearly equal to its LSL-value.The difference between the LSL-values of the RMF-and SMF-models in Fig. 14 is about 1% of their values.For 208 Pb, the FB-and SOG-analyses provide almost the same experimental value as shown in Table 3, although Appendix indicates unphysical r-dependences in the charge distributions.In contrast to Fig. 14 7.For details, see the text.

The correlation of H 6
cp with H 6 p is depicted in Fig. 15, which shows that most of the circles are near the LSL-values of H 6 p .The difference between the LSL-values of H 6 p in the RMF-and SMF-frameworks is less than 1% of their values, as in the previous figure .
Figure 16 shows the correlation between Q  In Fig. 17 is shown the correlation of R 2 p with H 6 cp in a similar way as for previous figures.Although the contribution of H 2p with R 2 p to H 6 cp is small as in Table 1, the values of r is obtained as 0.9878 and 0.9559, for the RMF-and SMF-models, respectively.The LSL-value of R 2 p are 29.935 in the RMF-framework, while 29.810 fm 2 in the SMF-one.Ref. [6] provides 29.843(0.252)and 29.738(0.295)fm 2 obtained from Q 4 cp , and 29.733(0.155)and 29.671(0.154)fm 2 from R 2 c in the RMF-and SMF-frameworks, respectively.The present values are slightly larger than the previous ones.
Figure 18 shows the correlation between Q 4 n and H 6 cn .The values of r are obtained as 0.9442 and 0.9982 in the RMF-and SMF-models, respectively.The quasi-experimental value of H 6 cn is obtained by Eq.( 10) using the LSL-values of H 6 cp in Fig. 14.Hence, the difference between the quasi-experimental values of H 6 cn is the same as that between the LSL-values of H 6 cp in Fig. 14 for the RMF-and SMF-frameworks.In the RMF-framework, the ratio of the quasi-experimental value of H 6  cn to the experimental one of H 6 c is 0.053, whereas that of Q 4 cn to Q 4 c is 0.028 in Ref. [6].Thus, the neutron contribution is increased in H 6 c , compared to that in Q 4 c .The spacing between the two LSLs comes from the difference between the values of H Wn in the RMF-and SMF-models, as in Table 2.In contrast to the case of 48 Ca in Fig. 12, the calculated values of Q 4 n in the RMF-and SMF-models are distributed over regions near their LSL-values.
On the one hand, Fig. 16 yields almost the same LSL-values of Q 4 p in the RMF-and SMFframeworks.The difference between them is less than 1%.On the other hand, those of Q 4 n in Fig. 18 are different from each other by about 9%.These values provide δQ = Q n − Q p = 0.3345 and 0.2032 fm in the RMF-and SMF-frameworks, respectively, which show the value of δQ in the RMF-one is larger by about 0.131 fm than that in SMF-one, in a similar way as in 48 Ca. n with H 6 cn in 208 Pb is show in Fig. 19.It shows that, although the main component of H 6  cn is H 4n with Q 4 n , as seen in Table 2, the circles for R 2 n in the small component H 2n are distributed in a similar way as in Fig. 18.The values of r are 0.9252 and 0.9980 for the RMF-and SMF-models, respectively.The former is the worst value among those in figures for 208 Pb, while the best one 1.0000 for the RMF-models in Fig. 15.The LSL-values of R 2 n are determined to be 33.070 and 31.611fm 2 for the RMF-and SMF-frameworks, respectively, while they are obtained to be 32.943(2.934)and 31.516(0.657)fm 2 from Q 4 cn in Ref. [6].Although the The least-squares line y(x) = ax + b and the standard deviation σ depicted in Figure 14 to 19 for the relativistic(RMF) and the non-relativistic(SMF) models.
value of Q 4 cn in Ref. [6] are estimated on the basis of the FB-analysis, the values of R 2 n contain the present results within their errors.
In the previous papers [6,29], it has been pointed out that whereas the LSL-values of R p are almost the same in the RMF-and SMF-frameworks, the LSL-values of R n are larger by about 0.1 fm in the RMF-framework than those in the SMF-one, in both 48 Ca and 208 Pb.As a result, their δRs differ from each other by about 0.1 fm.In the same way, the present LSA in Fig. 19 provides the LSL-values of R n to be larger by about 0.128 fm in the RMF-framework than in the SMF-one.Together with the LSL-values of R p in Fig. 17, the difference between δRs in the two frameworks is obtained as 0.117 fm.This 0.1 fm difference between δRs in the RMF-and SMF-frameworks is understood, according to Refs.[28,29].The difference between δQs in 48 Ca and 208 Pb mentioned before may be explained in a similar way as for δRs.

Discussions
The LSA yields for the set of the models the LSLs peculiar to its framework, like the RMFor the SMF-approximation.The LSLs are expected to be inherent to the framework, and not to depend strongly on the choice of the sample-models for the set.Indeed, there is a following example.The previous authors [30] has explored the correlation between the slope τ,2 are given in units of fm 2 , those of Q 4 in units of fm 4 , and the values of a of the asymmetry-energy density(L) and the neutron skin thickness(δR), by the LSA using 47 models in the RMF-and SMF-frameworks.They have analyzed all results in the two framework together, and found the LSL-equation as δR = 0.147 × 10 −2 L + 0.101 with r = 0.979.In the present LSA, 11 and 9 models are chosen arbitrarily among the RMF-and SMF-models, respectively [6].If these 20 models are used in the same purpose as in Ref. [30], one finds the LSL-equation as δR = 0.151 × 10 −2 L + 0.102 with r = 0.991 [28], which is almost the same result as that by 47models [30].Thus, it is reasonable to assume that the LSL does not depend on how the models are chosen for the sample-set in the specified framework.
In the present paper, the moments of the charge distribution have been analyzed.The nth moment(R n ) in the nucleus [5].Then, the LSA is performed between R τ,m denote the slope and the intercept of the LSL-equation, respectively.By the definition of LSL, the above equation provides the relationship between the mean values of the moments calculated by the models in the set as, When such equations for the various moments under consideration are solved simultaneously, one obtains the average value of each calculated moment.For example, if one is interested in the five moments, p and R 2 n , then the same number of the LSLs are selected as Employing the LSL-coefficients listed in Table 4 and 5 in Ref. [6], the mean values in 48 Ca and 208 Pb are obtained as in Table 8.
There is a kind of the sum rule with respect to the coefficients of the LSL-equations.Eqs.( 17) and (18) provide which is nothing but the definition of a c )-plane.According to the definition of the moments as in Eqs.( 10) and ( 16), R (n) c is assumed to be described in the form: τ,m stands for the coefficient coming from the nucleon-size, W  12) and (16).For example, α In calculating Eq.( 21) by sample-models, the mean values satisfy Eqs.( 21) and ( 22) provide Finally, the above equation and Eq.( 20) yield the sum rule on the slopes of the LSL-equations as Here, the slope of the LSL-equation with respect to the moment of the spin-orbit density, a Wτ ,m , has been defined in the same way as Eq.( 17) for the moment, R (m) τ , as which gives a similar equation to Eq.( 20) as The study of the LSLs on W (n,m) τ is out of the present purpose, but an example is found for 48 Ca in Fig. 20 of Ref. [6].It is possible to study the LSL by expanding W (n,m) τ in terms of the moments, r Wτ , of the spin-orbit density as in Eqs.(10) and (16).
When the moment of the charge distribution is written for the protons and the neutrons separately as in Eqs.( 10) and ( 16), and the LSL-equations are obtained as then the LSL-sum rule is described as Eq.( 24) and ( 29) are convenient for examining how each moment of the point proton and neutron distributions contributes to the R c , and whether or not the calculations of the LSLs are performed correctly, as for the energy-weighted sum rule in the random phase approximation(RPA) [31,32,33,34].
The LSL-sum rule, Eq.( 24), makes clear the relationship between the experimental value of R (n) c and the LSL-values of the moments.The LSL-value is given by Eq.( 17) as where R c,exp , respectively.Eqs.( 18) and (30) give the relationship between the LSL-value and the mean value as By inserting the above equation and similar equation of the moment for the spin-orbit density into the right-hand side of Eq.( 22), and using the LSL-sum rule, Eq.( 24), one obtains where the LSL-value of W c,exp ) in the same way as in Eq.( 30) for R (m) τ .Thus, in spite of the fact that the nth moment of the charge density is a function of several moments of the point proton and neutron distributions, the value of each moment is provided uniquely by the LSL-value, R c,exp ).For example. the value of R 2 p,LSL (Q 4 c,exp ) is not necessary to be equal to that of R 2 p,LSL (R 2 c,exp ) as It is natural to expect that there is an ideal set A of the MF-models, whose mean value R 2 c A reproduces the experimental value of R 2 c,exp , because R 2 c,exp is usually used as one of the inputquantities to fix the free interaction parameters.Then, Eq.( 33) provides the LSL-value as In order to obtain R 2 p,LSL (Q 4 c.exp ) = R 2 p,LSL (R 2 c.exp ), Eq. (34) implies that the set A should reproduce the value of Q 4 c,exp also, as Q 4 c A = Q 4 c,exp .An arbitrary set A in the same framework as for A does not necessarily satisfy Eq.( 35), but is expected to have the same LSL as In order fot the set Similar discussions can be made for other higher moments R (n) c and their components R (m) τ .The MF-framework does not assure that the set A reproduces the experimental values of the higher moments(n ≥ 4) by the mean values, and that the set A have the relationship like Eq.( 37) for higher moments(n ≥ 4) also.When the equation like Eq.( 37) is not satisfied for R  The results of the SMF-models for 48 Ca in Table 8 seem to show approximately the relationship in Eq. (37).The values of R 2 p,LSL (R 2 c,exp ) and R 2 p,LSL (Q 4 c,exp ) are almost the same, as 11.372 and 11.336 fm 2 , while the mean values of R 2 p = 11.772fm 2 are rather different from those LSL-values.Table 8 gives the values of the right-hand side in Eq.( 37) to be about 0.435 fm 2 , while that of the left-hand side to be about 0.400 fm 2 .These values compensate the differences between R 2 p and R  3, it is seen that the predicted values are fairly different from the FB-and SOG-values.The predicted values for 40 Ca and 48 Ca are between those by FB-and SOG-method, while the ones for 208 Pb are smaller than those of FB-and SOG-analysis.These facts may be expected from Figs.21, 23 and 25 in Appendix, and affect the LSL-values as in the following Table 10.The last column in Table 9 shows the contribution of the neutrons to the charge distributions.In 48 Ca, both the RMF-and SMF-frameworks predict the ratio of H 6 cn to H 6 cp to be about 10%, while Ref. [6] has obtained the value of Q 4 cn /Q 4 cp to be about 5%.), the proton charge(H 6 cp ), and the neutron charge(H 6 cn ) distributions calculated in the relativistic(RMF) and the nonrelativistic(SMF) framework in 40 Ca, 48 Ca and 208 Pb.They are related as H 6 c = H 6 cp − H 6 cn .The mean square radii of the proton distributions in those nuclei are determined by the leastsquares method with the experimental values obtained through Fourier-Bessel analyses of the electron scattering data [6].The numbers are given in units of fm 6 .The last row shows the ratio of H are employed in the Fourier-Bessel-analyses of the electronscattering cross sections.The numbers in the parentheses denote the error taking account of the experimental one and the standard deviation of the calculated values from the least-squares line [6].All the numbers are given in units of fm.For details, see the text.FB-ones for R 2 c and Q 4 cτ , according to the experimental values in Ref. [10].Since the value of root msr R p is frequently used in the literature [4], all the values in Table 10 are given in units of fm.The numbers in the parentheses stand for the error taking account of the experimental one and the standard deviation of the calculated values from the least square line [6].It is seen that the LSL-values in 40 Ca and 48 Ca decrease from R 2 c to H 6 c in the RMF-and SMF-framework, respectively, while those in 208 Pb increase inversely.This different behavior between Ca and Pb is due to fact that the predicted values for Pb in Table 9 is smaller than the FB-values, while those for Ca larger, as mentioned on Table 9  and Q 4 cτ to be compared to.Table 10 shows that the values of R p are larger than the those of R n in 40 Ca, yielding the negative values of the neutron-skin thickness, δR = R n − R p , in contrast to the positive ones in 48 Ca and 208 Pb.Figs. 5 and 7 for the SOG-values also show the same relationship in 40 Ca.This fact may be understood qualitatively, according to the constraint of the Hugenholtz-Van Hove(HVH) theorem in the MF-approximation [35,36,37], as below.
Ref. [28] has derived the following equations for R n and R p of asymmetric semi-infinite nuclear matter on the basis of the HVH theorem, where, in addition to he Fermi momentum(k F ) and the asymmetry parameter(I = (N − Z)/A), V c (> 0), J, L and K denote the Coulomb energy, the asymmetry-energy coefficient, the slope of the asymmetry energy, and the incompressibility coefficient.respectively.All the values of the right-hand sides of the above equations are given in each RMF-and SMF-model [28].Eqs.(38) and (39) provide the neutron skin thickness of the matter, δR M = R n − R p , as The contribution of the third term in the parenthesis to δR M is less than 10%.The above equation is examined in detail for 208 Pb in Ref. [28], but may be useful for rough discussions on medium-heavy nuclei also.
In 40 Ca, I = 0 implies δR M < 0, so that R p > R n , whereas in nuclei with N > Z like 48 Ca and 208 Pb, we have R n > R p , because of I > V c /4J [28].This interpretation may also be related to the result that Q 4 n < Q 4 p in 40 Ca, as mentioned before with respect to Figs. 4 and 6.The correlation between R 2 c and R 2 n is not required in the definition of R 2 c , but Ref.
[6] has shown their well-defined LSL in the used models.This fact implies that the correlation between R 2 p and R 2 n is constrained by other requirements in the MF-models.One of them may be Eq.( 40) which is dominated by the Coulomb-and asymmetry-energy.Hence, when the mean values of R 2 c and Q 4 c are almost equal to their experimental values, respectively, the LSLvalue of R 2 n from R 2 c may be nearly equal to that from Q 4 c .Indeed, for example, in 208 Pb, the (R .964(0.802) fm 2 in the RMF-models [6].In the SMF-models, As shown in Appendix, the experimental values of H 6 c are not determined by the FB-and SOG-analyses in Ref. [4].Ref. [10] employed the FB-method also did not provide those values from their experiment.If the FB-and SOG-coefficients in Ref. [4] are used, H 6 c -values are calculated as in Table 3. Table 11 lists the LSL-values of the components of H 6 c , in order to show how their values are sensitive to the ways of the analyses.The values of the components in the rows of SOG are taken from the corresponding figures in §4.Those in the FB-rows are obtained by replacing the horizontal lines in the figures by the FB-ones.It is noticeable that the LSL-values of Q 4 p and R 2 p in 40 Ca are smaller than those in 48 Ca in the FB-analysis, while the former are larger than the latter in the SOG-analysis.This result is due to the big difference between the FB-and SOG-values of H 6 c in 40 Ca, as 4.234 × 10 3 and 5.390 × 10 3 fm 6 , respectively.Before closing this subsection, there is a comment on the relationship between the LAS in the present paper and that in Refs.[38,39,40] by JLab.They have analyzed the cross sections for the parity-violating electron scattering(PVES) observed at q = 0.8733 fm −1 in 48 Ca and at 0.476 and 0.398 fm −1 in 208 Pb, q being the momentum transfer from the electron to the nucleus.On the one hand, Table 4 lists the values of δR = R n −R p in the present method, as 0.220(0.062) in the RMF-framework and 0.121(0.036)fm in the SMF-one for 48 Ca, while 0.275(0.070) in the RMF-framework and 0.162(0.068)fm in the SMF-one for 208 Pb.On the other hand, Ref. [40] has obtained δR = 0.121(0.050)for 48 Ca, and Ref. [39] 0.283(0.071)fm for 208 Pb, in the set including the both RMF-and SMF-models.These values are all within the errors in each nucleus, in spite tive mass is determined by the zero-range interactions with the momentum-dependence [12].Their parameters are related to the enhancement factor κ of the sum of the dipole excitation strengths [46].The value of κ observed in photo-excitations are mainly due to charge-exchange interactions with finite range [7,32].It is not clear, however, how many % of the value of κ should be explained by the momentum-dependent zero-range forces [7,32].Moreover, the momentum-dependent zero-range forces may include the finite range effects without charge exchange operators [32].Thus, there seems to have possibility to require additional parameters in both the RMF-and SMF-models for improvement.In constructing a new MF-model, the LSL and LSL-values obtained in this paper may offer a guide for reproducing the moments, where it is enough to take care of the LSL-values of R  It is well known that for analyzing experimental cross section in electron scattering, there are methods which are called "model-independent" ways.The one is Fourier-Bessel(FB)-method [47], and the other sum-of-Gaussians(SOG)-method [48].In these cases, however, "model-independent" means the analysis to provide charge distribution which fits the experimental cross section without regard to its special functional form like Fermitype [49].In fact, the FB-method can not use the infinite series for the analysis, while the SOG-one assumes a finite number of the Gaussian distribution of the fractional charge in the nucleus.Strictly speaking, these analyses imply that if experimental data to be analyzed by FB-method are given by different momentum-transfer regions from those by the SOG-one, then the obtained charge distributions may be different from each other.In principle, in order to determine uniquely the density distribution from experiment, an infinite number of data may be required.Hence, it is necessary to know whether or not the experimental values obtained from these "model-independent" analyses are appropriate for the present purpose.
In this Appendix, the problem of the FB-and SOG-methods is pointed out, which may be avoidable at present for the study of the moments higher than the fourth.Fig. 20 shows the charge density in 40 Ca, using the FB(red curve)-and SOG(blue curve)parameters listed in Ref. [4].As a reference, the charge density calculated with NL3 [9] is depicted with the black curve.According to Ref. [4], the cutoff radius(R) of the FB-series is taken to be 8 fm, while the number of the terms N = 14.The numbers, R 1 and R 12 , indicate examples of the Gaussian-parameters specified in Ref. [4].Fig. 21 shows the charge density of Fig. 20 multiplied by r 2+n (n = 2, 6), where the curves are normalized by the maximum value(Max) of each function.
With respect to the values of R 2 c and Q 4 c of the charge distribution, there is not a large difference between the two analyses, as listed in Table 12.The SOG-values are obtained with the upper limit of the integral for the moments to be 20 fm.Ref. [4] listed the same values of R c for 40 Ca, as 3.450(10) and 3.480(3) fm in FB-and SOG-analyses, respectively.The values in the NL3 model [9] and those of r 8 c are also listed in Table 12, as a reference.
Figures 22 and 23 show the charge density and the one multiplied by r 2+n in 48 Ca, respectively, in the same way as the previous figures for 40 Ca.The dashed parts of FB(red)-and NL3(black)-curves in Fig. 22 indicate the negative densities.The obtained values of R c for 48 Ca are listed in Table 12, which are the same as those in Ref. [4], as  (2).The latter has 13 ones and provides R c = 5.499 (1).
In the FB-analysis, the oscillation of the charge density is observed, whereas in the SOG-one, the density decreases monotonically with r up to 20 fm.As a results, the ratio of the FB-value to the SOG-one is 1.0000 for R c , Q c and H c in 208 Pb, as expected from Fig. 25.The obtained values are Q c = 5.851 and H c = 6.128 fm, and the value of R c is the same as 5.503 fm listed in Ref. [4].Calculations of R c in an accuracy within 1% [50] should take care of the problems of the density-tails in the FB-and SOG-analyses, but a few % difference is not essential for the present discussions.In the discussions on H 6 c in the present paper, however, the difference between the FB-and SOG-values are not disregarded, because of the two reasons.The one is that the cutoff radius R = 8 fm for 40 Ca and 48 Ca in the FB-analyses seems not to be enough for covering most of the region where the function r 8 ρ c (r) is finite, as seen in Figs.21 and 23.In the FB-analysis, the value of the nth moment is proportional to R n .In SOG-calculations, the cut-off radius is required to be at least 12 fm for the enough convergence of the integral on H 6 c .The other reason is that the unphysical negative density in FB-distribution and the bump in SOG-one contribute to the values of H 6 c , as seen in Figs.21, 23 and 25.Since there are not enough experimental data to improve

4.1 40 CaFigure 2
Figure 2 shows the relationship between H 6 cp and H 6 c in 40 Ca.The calculated values with the RMF-models are indicated by the closed circles, while those with the SMF-ones are by the open circles.Each circle in the figure bears the number to specify the corresponding nuclear model mentioned in the above.The LSLs are shown for the RMF-and SMF-models, separately,

6 )Figure 2 :
Figure 2: The sixth moment of the charge density(H 6 c ) as a function of the sixth moment of the proton charge density(H 6 cp ) in 40 Ca.The closed(open) circles are calculated in various relativistic(non-relativistic) mean-field models.The number assigned to each circle represents a model used in the calculations as explained in the text.The lines of the least-square fitting are shown by the solid ones for the relativistic and non-relativistic models, respectively.Those lines cross the horizontal line which expresses the experimental value indicated on the righthand side of the figure.Their intersection points are described on the top of the figure for the relativistic(rel.)and non-relativistic(non.) frameworks, respectively.The values of the slope(a), the intercept(b) and the standard deviation(σ) of the least-squares lines are listed in Table5in the last of the present subsection.For details, see the text.

6 )Figure 4 :
Figure 4: The sixth moment of the proton charge density(H 6 cp ) as a function of the fourth moment of the point proton distribution(Q 4 p ) in 40 Ca.The designation in the figure is the same as in Fig.2.For details, see the text.

6 )Figure 5 :
Figure 5: The sixth moment of the proton charge density(H 6 cp ) as a function of the mean square rarius of the point proton distribution(R 2 p ) in 40 Ca.The designation in the figure is the same as in Fig.2.For details, see the text.

6 )Figure 7 :
Figure 7: The sixth moment of the neutron charge density(H 6 cn ) as a function of the mean square radius of the point neutron distribution(R 2 n ) in 40 Ca.The designation in the figure is the same as in Fig.2.The values of the slope(a), the intercept(b) and the standard deviation(σ) of the LSL are listed in Table5.For details, see the text.

6 )Figure 8 :
Figure 8: The sixth moment of the charge density(H 6 c ) as a function of the sixth moment of the proton charge density(H 6cp ) in48 Ca.The designation in the figure is the same as in Fig.2.The values of the slope(a), the intercept(b) and the standard deviation(σ) of the LSLs are listed in Table6.For details, see the text.

6 )Figure 9 :
Figure 9: The sixth moment of the proton charge density(H 6 cp ) as a function of the sixth moment of the point proton distribution(H 6 p ) in 48 Ca.The designation in the figure is the same as in Fig.2.For details, see the text.

Figure 10
Figure10shows the correlation between Q4  p and H 6 cp in48 Ca.The RMF-models underestimate again the LSL-value, while the SMF-ones except for SKI(1) overestimate their LSL-value in the same way as in Fig.9.The LSL-values of Q 4 p are given as 183.914 and 182.500 fm 4 in the RMF-and SMF-frameworks, respectively.They are smaller than those in40 Ca in Fig.4showing 198.613 and 199.877 fm 4 .It should be noted that in contrary to the SOG-value, the FB-one in Table3yields the smaller LSL-values in40 Ca as 167.897 and 168.880 fm 4 in the RMF-and SMF-schemes, respectively than 172.420 and 171.436 fm 4 in 48 Ca.As for H 6 p , both the SOG-and FB-values yield the larger LSL-values in40 Ca than those in48 Ca.Figure11shows H 6 cp as a function of R 2 p in48 Ca.The contribution of R 2 p through H 2p to H 6

6 )Figure 11 :
Figure 11:  The sixth moment of the proton charge density(H 6 cp ) as a function of the mean square radius of the point proton distribution(R 2 p ) in48 Ca.The designation in the figure is the same as in Fig.2.For details, see the text.The correlation of Q 4n with H 6 cn in48 Ca is shown in Fig.12.The values of r are obtained as 0.9669 and 0.9902 for the RMF-and SMF-frameworks, respectively.The quasi-experimental values of H 6 cn for the two frameworks are different, since they are obtained by Eq.(10) using the values of H 6 cp given to each framework by Fig.8.The difference between the two LSLs in Fig.12is owing mainly to the different contributions from H Wn to H 6 cn in the RMF-and SMF-models, as seen in Table2.On the one hand, in Fig.10, the calculated values of Q 4 p in the RMF-models are smaller than the values of the SMF-ones, except for SKI(1), but their LSLs yield the almost same LSL-values of Q 4 p .On the other hand, in Fig.12, most of the calculated values of Q 4 n are distributed over the same region between 220 and 250 fm4 , but the LSL-values of the RMF-and SMF-frameworks are different from each other, as shown on the top of the figure as 244.1112 and 217.569 fm4 , respectively.These values together with the LSL-values in Fig.10provide δQ = Q n − Q p = 0.2701 and 0.1651 fm in the RMF-and SMF-frameworks, respectively, which show the value of δQ in the RMF-one is larger by about 0.105 fm than that in the SMF-one, in

6 )Figure 12 :
Figure 12: The sixth moment of the neutron charge density(H 6 cn ) as a function of the fourth moment of the point neutron distribution(Q 4n ) in48 Ca.The designation in the figure is the same as in Fig.2.For details, see the text.

6 )Figure 13 :
Figure 13: The sixth moment of the neutron charge density(H 6 cn ) as a function of the mean square radius of the point neutron distribution(R 2 n ) in 48 Ca.The designation in the figure is the same as in Fig.2.For details, see the text.

6 )Figure 14 :
Figure 14: The sixth moment of the charge density(H 6 c ) as a function of the sixth moment of the point proton charge density(H 6 cp ) in 208 Pb.The designation in the figure is the same as in Fig.2.The values of the slope(a), the intercept(b) and the standard deviation(σ) of the LSLs are listed in Table7.For details, see the text.

6 )Figure 16 :
Figure 16: The sixth moment of the proton charge density(H 6 cp ) as a function of the fourth moment of the point proton distribution(Q 4 p ) in 208 Pb.The designation in the figure is the same as in Fig.2.For details, see the text.

6 )Figure 17 :
Figure 17: The sixth moment of the proton charge density(H 6 cp ) as a function of the mean square radius of the point proton distribution(R 2 p ) in 208 Pb.The designation in the figure is the same as in Fig.2.For details, see the text.

6 )Figure 18 :
Figure 18: The sixth moment of the neutron charge density(H 6 cn ) as a function of the fourth moment of the neutron distribution(Q 4 n ) in 208 Pb.The designation in the figure is the same as in Fig.2.For details, see the text.

6 )Figure 19 :
Figure 19: The sixth moment of the neutron charge density(H 6 cn ) as a function of the mean square radius of the neutron distribution(R 2 n ) in 208 Pb.The designation in the figure is the same as in Fig.2.For details, see the text.
have no unit.For details, see the text.

c
) is expressed in terms of the m(≤ n)th moments of the point proton distribution(R (m) p ) and the m(< (n−2))th ones of the point neutron distribution(R (m) to the spin-orbit density, and C (n) term including the nucleon-size only.The examples of α (n) τ,m and W (n,m) τ are given below Eqs.( ), determined through the single experimental value, R(n) c,exp .It should be noticed that the LSL-values in Eq.(32) depend on n of R (n) c,exp , as explicitly indicated in Eq.(30) as R

c
(n ≥ 4), then the MF-framework with R 2 p,LSL (R 2 c.exp ) fails to reproduce its experimental value, although Eq.(32) for R (n) c,exp still holds.It is explored numerically how the LSL-values of the higher moments are related to that of the lowest moment, R 2 p,LSL (R 2 c,exp ), used as an input-quantity.
the point neutron and proton distributions in each framework, instead of the experimental values of R (n) c s with complicated corrections.

Figure 21 :
Figure 21: The charge density multiplied r 2+n in 40 Ca.The charge density is obtained by the Fourier-Bessel(FB)-and sum-of Gaussians(SOG)-analyses of electron scattering data, as in Fig 20.The curves are normalized by the maximum value(Max) of each function.For details, see the text.

Figures 24 and 25
Figures 24 and 25 are the same as Figs.20 and 21, but for 208 Pb.Ref.[4] listed in the FB-method the two different values of R c in 208 Pb for the two corresponding sets of experimental data.The one was obtained with R = 12.0 fm, while the other R = 11.0 fm.The former, which is used in the present paper, has 17 coefficients of FB-series, yielding

208Figure 24 :Figure 25 :
Figure24:The charge density of 208 Pb.The figure is depicted in the same way as Fig.20, but the dashed parts of red and black curves indicate the negative densities.The FB-and SOG-coefficients are taken from Ref.[4].For details, see the text.

Table 1 :
The sixth moment of the nuclear charge distribution(H 6 c ) in

Table 2 :
6 cp show the contributions from the proton charge distribution to H 6 c , and other numbers the values of the components of H 6 cp .The neutron charge distribution(H 6 cn ), which contributes to H 6 c as H 6 c = H 6 cp − H 6 cn , are listed in Table 2.For the details, see the text.The sixth moment of the neutron charge distribution(H 6 cn ) in

Table 3 :
The values of the sixth moments(H 6 c 6cn contributes to H 6 c by 9 ∼ 12% in48Ca, and by 4 ∼ 5% in 208 Pb.The neutron contribution is more remarkable in48Ca, rather than in 208 Pb.Almost a half of the value of H 4p is cancelled by H 6 cn in both nuclei.The term, H 4n , accounts for more than 50% of H 6 cn .The contribution of H 2n depending on R 2 n is about 10% of H 6 cn in 48 Ca, and 5% in 208 Pb.In 208 Pb, the value of H 6 cn in SLy4 is smaller by more than 15%, compared to that in the RMF.Table 3 shows that most of the calculated values of H 6 c are between the FB-and SOG-values.Exceptions are those of SLy4 for 48 Ca and 208 Pb which overestimate the both FB-and SOGexperimental values.The difference between those of the SLy4 and RMF models stems mainly from their values of H 6 p , as seen in Table 1.It is noticeable in Table 3 that the experimental value of H 6 c in the values of H 6 cp and H 6 c calculated in various MF models are plotted in the (H 6 cp − H 6 c )-plane, and the LSL between H 6 cp and H 6 c is obtained.The value of H 6 cp is determined by the cross point of the LSL and the line of the experimental value of H 6 c .The determined value is called the LSL-value, as mentioned in §1.Since the LSL-value of H 6 cp is obtained with small standard deviation(σ) and the correlation coefficient(r) nearly equal to 1, it will be used as the 'quasi-experimental value' of H 6 cp for other LSA in the following steps.Second, the values of H 6 p and H 6 cp calculated using the same MF models are plotted in the (H 6 p − H 6 cp )-plane to obtain the LSL for them.The LSL-value of H 6 p is determined by the cross point of the LSL and the line of the quasi-experimental value of H 6 cp determined in the first step.Third, the values of Q 4 p are calculated, and are plotted with the calculated values of H 6 cp in the (Q 4 p − H 6 cp )-plane.The cross point of the LSL for (Q 4 p − H 6 cp ) and the line of the quasi-experimental value of H 6 cp give the value of Q 4 p in the MF-framework.Fourth, the same procedure is repeated in order to estimate the values of R 2 p .Fifth, the values of Q 4 n and H 6 cn are calculated and are plotted in the (Q 4 n − H 6 cn )-plane to obtain their LSL.In order to obtain the quasi-experimental value of H 6 cn , Eq.(10) is employed with the quasi-experimental one of H 6 cp .The value of Q 4 n is determined by the cross point of the LSL between Q 4 n and H 6 cn and the line of the quasi-experimental value of H 6 cn .Finally, the values of R 2 n are calculated, and are plotted with H 6 cn .The value of R 2 n in the used model-framework is determined by the cross point of the LSL between R 2 The sixth moment of the proton charge density(H 6 cp ) as a function of the sixth moment of the point proton distribution(H 6 p ) in40Ca.The solid and dashed horizontal lines indicate the quasi-experimental values of H 6 cp for the RMF-and SKM-models, respectively.The designation in the figure is the same as in Fig.2.For details, see the text.SMF-models underestimate the LSL-values of Q 4p .The standard deviations of the LSLs are a little large, compared with those in Figs2 and 3, but their correlation coefficients are still close to 1.0000, as r = 0.9830 and 0.9945 in RMF-and SMF-frameworks, respectively.
shows H 6 cp as a function of Q 4 p in 40 Ca.As in Figs. 2 and 3, both RMF-and ) Figure 3: 6cn is almost the same as that of H 6 cp in Fig.2.All the calculated values of Q 4 n indicated by the circles underestimate their LSL-values obtained as 185.130 and 187.254 fm 4 for the RMF-and SMF-scheme, respectively.

Table 5 .
For details, see the text.

Table 5 :
The least-squares line y(x) = ax + b and the standard deviation σ depicted in Figure2to 7 for the relativistic(RMF) and the non-relativistic(SMF) models.
The sixth moment of the proton charge density(H 6 cp ) as a function of the fourth moment of the point proton distribution(Q 4 p ) in48Ca.The designation in the figure is the same as in Fig.2.For details, see the text.forRMF-and SMF-models, respectively.The distribution of the circles is almost the same as those for H 6 cp , H 6 p and Q 4 p in Figs. 8 to 10.
6 cp as a function of R 2 p in 48 Ca.The contribution of R 2 p through H 2p to H 6 cp is small, as seen in Table 1, they are well correlated with the values of r to be 0.9613 and 0.9572 ) Figure 10:

Table 6 :
The sixth moment of the proton charge density(H 6 cp ) as a function of the sixth moment of the point proton distribution(H 6 p ) in208Pb.The designation in the figure is the same as in Fig.2.For details, see the text.by the mean values calculated with the models in both (Q 4 p − Q 4 cp )-and (Q 4 p − H 6 cp )-correlations for 208 Pb, as will be discussed in the next section.
4 p and H 6 cp .The ratio of the LSL-value of Q 4 p from H 6 cp to that from Q 4 cp in Ref.[6] is, for example, about 1.007 in the RMF-framework.Such a small difference in 208 Pb is owing to the fact that the LSL-values of Q 4 p are almost reproduced ) Figure 15:

Table 8 :
The LSL-values of R 2 τ in48Ca and 208 Pb listed as R 2 τ,LSL (a Table 8 lists the values needed for above arguments for Q 4 c in 48 Ca and 208 Pb.The mean values of R 2 c and Q 4 c are nearly equal to the corresponding experimental values, with one exception.The exception is in the SMF-models for 48 Ca which provide 12.310 fm 2 for R 2 c against 11.910 fm 2 of the experimental value, and 210.706 fm 4 for Q 4 c against 194.734fm 4 of the experimental one.As a result, apart from the exception, all the models yield each nucleus almost the same values for R 2 In these calculations, the samplemodels seem to play a role of the set A which reproduces the experimental values of Q 4 Once the experimental values of H 6 c is determined, then it would be possible to discuss in more detail the n-dependence of the LSL-values, together with the relationship of the LSL-value, R 2 τ,LSL (H 6 c,exp ), to the mean values.Since there are no reliable data on H 6 c yet, the examples of the predicted values in the present LSA are listed in Table 9.If R 2 p,LSL (H 6 c,exp ) is assumed to be equal to R 2 p,LSL (R 2 c,exp ), then the values of H 6 c,exp is calculated with Eq.(30) and the coefficients of the LSL-equations, H 6 c = aH 6 cp + b, in Table 5 to 7. Comparing those values with the ones in Table

Table 9 :
The values of the sixth moments of the nuclear charge(H 6 c

Table 10 :
6cn to H 6 cp .For the details, see the text.The results of the least-squares analysis for the moments of the point neutron and proton distributions in40Ca,48Ca and 208 Pb.The notations for the components of the moments in the second row and the first column are defined in the text, where τ denotes the protons(τ = p) and neutrons(τ = n).The values of R p is obtained from the analyses of H 6 before .Thus if the experimental values of H 6 c are observed with small experimental errors and those of Q 4 c are determined more precisely, understanding the MF-models would be advanced.The LSL-values of H 6 p and Q 4 n estimated from H 6 c in the previous section are not listed in Table 10, because there are no values from R 2

Table 11 :
[6] LSL-values of the components of the sixth moment H 6 c in40Ca,48Ca and 208 Pb.The experimental values of H 6 c (Exp) is obtained by the two ways, which are the Fourier-Bessel(FB)-and the sum-of-Gaussians(SOG)-analysis of the electron-scattering cross section[4].The nuclear models used to estimate the LSL-values are the relativistic (SMF) and non-relativistic(SMF) mean-field ones.The numbers of H 6 c (Exp), H 6 cp and H 6 p are listed in units of 10 3 × fm 6 , while those of H 6 cn in 10 2 × fm 6 .The numbers of Q 4 p and Q 4 n are given in units of 10 2 × fm 4 , while those of R 2 p and R 2 n in 10 × fm 2 .For etails, see the text.thyare31.382(0.698)fmagainst31.507(716)fm2[.They  withi the errors as written in the parentheses[6].These results shows that the values of R 2 n determined by other physical quantities like Eq.(40) are assured to be consistent with the one in Q 4 c in the MF-framework.It is no doubt to have further information on R 2 n after determining the experimental values of H 6 c in future.