The n-th order Moment of the Nuclear Charge Density and Contribution from the Neutrons

TL;DR

This paper derives relativistic expressions for the moments of nuclear charge density, highlighting the dependence on neutron density and implications for electron scattering in neutron-rich nuclei.

Contribution

It provides a consistent derivation of relativistic and non-relativistic expressions for nuclear charge density moments up to second order using the Foldy-Wouthuysen transformation.

Findings

Relativistic and non-relativistic msr expressions are equivalent up to 1/M^2.

Higher moments depend on neutron density, especially the 4th moment.

The 4th moment offers insights into neutron distribution in nuclei.

Abstract

The relativistic expression for the $n$-th order moment of the nuclear charge density is presented. For the mean square radius(msr) of the nuclear charge density, the non-relativistic expression, which is equivalent to the relativistic one, is also derived consistently up to $1/M^2$ with use of the Foldy-Wouthuysen transformation. The difference between the relativistic and non-relativistic expressions for the msr of the point proton density is also discussed. The $n(\ge 4)$-th order moment of the nuclear charge density depends on the point neutron density. The 4-th order moment yields a useful information on the msr of the point neutron density, and is expected to play an important role in electron scattering off neutron-rich nuclei.

Tables

Table 1. Table 1: The mean square radius(msr) of the charge distribution of 40 Ca, 48 Ca and 208 Pb in the unit of fm 2 . The calculated values, using parameters of the relativistic nuclear models NL3 [ 2 ] and NL-SH [ 1 ] , and of the non-relativistic one SLy4 [ 3 ] , are listed. The experimental values are those used in the nuclear models to fix their parameters. For details, see the text.

	${⟨r^2⟩}_p+r_p^2$	$\left(r_{+}^2-r_{-}^2\right)N/Z$	${⟨r^2⟩}_{W_p}$	${⟨r^2⟩}_{W_n}N/Z$	${⟨r^2⟩}_c$	Exp.
NL3
40Ca	$12.060$	$-0.1200$	$0.0222$	$-0.0244$	$11.938$	$11.90$
48Ca	$12.073$	$-0.1680$	$0.0263$	$-0.1573$	$11.774$	$11.91$
208Pb	$30.467$	$-0.1844$	$0.1054$	$-0.1460$	$30.242$	$30.28$
NL-SH
40Ca	$11.928$	$-0.1200$	$0.0225$	$-0.0248$	$11.806$	$11.90$
48Ca	$11.998$	$-0.1680$	$0.0268$	$-0.1594$	$11.698$	$11.91$
208Pb	$30.304$	$-0.1844$	$0.1071$	$-0.1482$	$30.304$	$30.28$
SLy4
40Ca	$12.404$	$-0.1200$	$0.0000$	$0.0000$	$12.318$	$12.18$
48Ca	$12.602$	$-0.1680$	$0.0000$	$-0.1014$	$12.366$	$12.11$
208Pb	$30.448$	$-0.1844$	$0.0579$	$-0.0865$	$30.284$	$30.25$

Table 2. Table 2: The 4th order moment of the charge distribution of 40 Ca, 48 Ca and 208 Pb. The value of each term in Eq.( 44 ) is listed in the unit of fm 4 , except for the one of R 4 subscript 𝑅 4 R_{4} which is given by 0.590 0.590 0.590 , 0.396 0.396 0.396 and 0.329 0.329 0.329  fm 4 for 40 Ca, 48 Ca and 208 Pb, respectively. The experimental values are obtained by the Fourier-Bessel analyses of data in ref. [ 36 ] . For details, see the text.

	$R_{4p}$	$R_{2p}$	$R_{2n}$	$R_{2W}$	$R_{4W_p}$	$R_{4W_n}$	${⟨r^4⟩}_c$	Exp.
NL3
40Ca	$183.921$	$24.941$	$-4.432$	$-0.000$	$0.457$	$-0.525$	$204.952$	$199.991$
48Ca	$178.085$	$24.968$	$-7.278$	$-0.283$	$0.738$	$-4.962$	$191.664$	$194.714$
208Pb	$1115.64$	$65.197$	$-20.254$	$-0.067$	$8.269$	$-12.304$	$1156.81$	$1171.58$
NL-SH
40Ca	$178.612$	$24.652$	$-4.382$	$-0.000$	$0.499$	$-0.569$	$199.401$	$199.991$
48Ca	$174.602$	$24.805$	$-7.190$	$-0.286$	$0.780$	$-4.973$	$188.134$	$194.714$
208Pb	$1098.69$	$64.839$	$-20.046$	$-0.068$	$8.381$	$-12.427$	$1139.70$	$1171.58$
SLy4
40Ca	$194.854$	$25.694$	$-4.555$	$-0.000$	$-0.073$	$0.061$	$216.572$	$199.991$
48Ca	$196.171$	$26.126$	$-7.260$	$-0.223$	$-0.014$	$-3.475$	$211.722$	$194.714$
208Pb	$1121.53$	$65.155$	$-19.371$	$-0.051$	$4.716$	$-7.500$	$1164.81$	$1171.58$

Equations

$${\rho }_c(r)=\int \frac{d^{3}q}{(2\pi {)}^3}\exp (-i\mathbf{q}\cdot \mathbf{r})\tilde{\rho }(q).$$

$$\tilde{\rho }(q)=⟨0|\sum _{k=1}^A\exp (i\mathbf{q}\cdot {\mathbf{r}}_{\mathbf{k}})\left(F_{1k}(q^2)+\frac{{\mu }_k}{2M}{F}_{2k}(q^2)\mathbf{q}\cdot {\mathbf{\gamma }}_k\right)|0⟩,$$

$$\tilde{\rho}(q)=\int d^{3}x\exp(i\boldsymbol{q}\!\cdot\!\boldsymbol{x})\sum_{\tau}\Bigl{(}G_{E\tau}(q^{2})\rho_{\tau}(x)+F_{2\tau}(q^{2})W_{\tau}(x)\Bigr{)},$$

$$G_{E\tau }(q^2)=F_{1\tau }(q^2)-{\mu }_{\tau }{q}^2{F}_{2\tau }(q^2)/(4M^2).$$

$${\rho }_{\tau }(r)$$

$$W_{\tau }(r)$$

$${\rho }_{\tau }(r)$$

$$W_{\tau }(r)$$

$${\int }_0^{\infty }dr\left(G_{\alpha }^2+F_{\alpha }^2\right)=1.$$

$$\rho_{c}(r)=\sum_{\tau}\Bigl{(}\rho_{c\tau}(r)+W_{c\tau}(r)\Bigr{)}$$

$${\rho }_{c\tau }(r)$$

$$W_{c\tau }(r)$$

$$g_{\tau }(x)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }dq{e}^{iqx}{G}_{E\tau }(q^2),f_{2\tau }(x)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }dq{e}^{iqx}{F}_{2\tau }(q^2).$$

$$G_{Ep}(q^2)$$

$$G_{En}(q^2)$$

$$r_p=0.81\text{fm},r_{\pm }^2=(0.9{)}^2\mp 0.06{\text{fm}}^2.$$

$$⟨r^{2n}{⟩}_c=\sum _{\tau }⟨r^{2n}{⟩}_{c\tau },Z⟨r^{2n}{⟩}_{c\tau }=\int d^{3}r{r}^{2n}{\rho }_{c\tau }(r)=(-{\nabla }_q^2{)}^n{\tilde{\rho }}_{\tau }(q){|}_{q=0}.$$

$${\tilde{\rho }}_{\tau }(q)=\int d^{3}r{e}^{i\mathbf{q}\cdot \mathbf{r}}{C}_{\tau }(q^2,r)=\int d^{3}r{j}_0(qr)C_{\tau }(q^2,r),$$

$$C_{\tau }(q^2,r)=G_{E\tau }(q^2){\rho }_{\tau }(r)+F_{2\tau }(q^2)W_{\tau }(r).$$

$$\tilde{F}(q)=\int d^{3}r{j}_0(qr)G(q^2)F(r),G(x)=\frac{1}{(1+\Lambda x{)}^{\alpha +1}},(\alpha =0,1,\cdots ),$$

$$F_{2p}$$

$$F_{2n}$$

$$(-{\nabla }_q^2{)}^n\tilde{F}(q)=\sum _{k=0}^{2n}{}_{2n}{\mathrm{C}}_k\int d^{3}rF(r)J_k,$$

$$J_{2k}$$

$$J_{2k-1}$$

$$G^{(n)}(x)=\frac{(\alpha +n)!}{\alpha !}\frac{(-\Lambda {)}^n}{(1+\Lambda x{)}^{\alpha +1+n}}.$$

$$J_{2k},J_{2k-1}⟶(2k)!{}_{\alpha +k}{\mathrm{C}}_{\alpha }{r}^{2n-2k}{\Lambda }^k,$$

$$(-{\nabla }_q^2{)}^n\tilde{F}(q){|}_{q=0}=\int d^{3}r{r}^{2n}F(r)+\sum _{k=1}^n\frac{(2n+1)!}{(2n+1-2k)!}{}_{\alpha +k}{\mathrm{C}}_{\alpha }{\Lambda }^k\int d^{3}r{r}^{2n-2k}F(r).$$

$$(-{\nabla }_q^2)\tilde{F}(q){|}_{q=0}$$

$$(-{\nabla }_q^2{)}^2\tilde{F}(q){|}_{q=0}$$

$$+60(\alpha +1)(\alpha +2){\Lambda }^2\int d^{3}rF(r).$$

$$⟨r^2{⟩}_c=⟨r^2{⟩}_p+r_p^2+(r_{+}^2-r_{-}^2)\frac{N}{Z}+⟨r^2{⟩}_{W_p}+⟨r^2{⟩}_{W_n}\frac{N}{Z},$$

$$⟨r^k{⟩}_{\tau }=\frac{1}{N_{\tau }}\int d^{3}r{r}^k{\rho }_{\tau }(r),⟨r^k{⟩}_{W_{\tau }}=\frac{1}{N_{\tau }}\int d^{3}r{r}^k{W}_{\tau }(r).$$

$$⟨r^2{⟩}_{c,\mathrm{nonrel}}=⟨r^2{⟩}_{p,\mathrm{nonrel}}+r_p^2+(r_{+}^2-r_{-}^2)\frac{N}{Z}-\frac{1}{Z}\sum _{i=1}^A\frac{{\mu }_i}{M^2}({\kappa }_i+1)+\frac{3}{4M^2}.$$

Full text

**The n-th order Moment of the Nuclear Charge Density

and Contribution from the Neutrons **

Haruki Kurasawa111kurasawa@faculty.chiba-u.jp1 and Toshio Suzuki2

1 

Department of Physics, Graduate School of Science, Chiba University,

Chiba 263-8522, Japan

2 

Research Center for Electron Photon Science, Tohoku University,

Sendai 982-0826, Japan

The relativistic expression for the $n$-th order moment of the nuclear charge density is presented. For the mean square radius(msr) of the nuclear charge density, the non-relativistic expression, which is equivalent to the relativistic one, is also derived consistently up to $1/M^{2}$ with use of the Foldy-Wouthuysen transformation. The difference between the relativistic and non-relativistic expressions for the msr of the point proton density is also discussed. The $n(\,\geq 4)$-th order moment of the nuclear charge density depends on the point neutron density. The 4-th order moment yields a useful information on the msr of the point neutron density, and is expected to play an important role in electron scattering off neutron-rich nuclei.

1 Introduction

At present it is unavoidable for nuclear models to include unknown parameters. In order to fix their values, some experimental values have to be used as inputs. One of them is the mean square radius(msr) of nuclei which is a fundamental quantity in nuclear physics. In most of the previous papers[1, 2, 3], the msr is calculated using their point proton density, or the one convoluted with a single proton density, and is compared with the values obtained from the charge distribution assumed so as to reproduce experimental cross section of electron scattering.

Electron scattering is an unambiguous tool to examine the nuclear charge distribution, since the electromagnetic interaction is well understood theoretically[4, 5]. Indeed, its msr is determined with high accuracy throughout the periodic table. For example, the root msr of 208Pb is reported to be 5.5012(0.0013) fm [6, 7].

Unfortunately, however, the point proton density obtained in nuclear models is not determined uniquely from the experiment. The nuclear charge density deduced from electron scattering is composed of several elements. At least in the charge density responsible for a small momentum transfer $q<1$ fm*-1*[8], they are the point neutron density, the proton and neutron spin-orbit densities and a single-proton and -neutron charge density, in addition to the point proton density. Their contributions are not distinguished from each other experimentally or model-independently.

Nuclear models so far have not taken particular care to the elements other than the point proton density, since their contributions are believed to be small, and it was not a main purpose of phenomenological models to reproduce the msr within a few % accuracy. Recently, however, more ambitious attempt called ab initio calculations has been carried out, trying to explain accurately the experimental values of the msr[9, 10]. Moreover, there are plans to perform experiments to compare the cross section to the model ones with high accuracy, aiming to determine a small difference 0.03 $\sim$ 0.05 fm between the root mean square radii of point-proton and -neutron distributions[11]. In order to respond to these challenges, it is necessary for nuclear models to make clear a role of each element in the nuclear charge density. Although main contribution to the charge density comes from the point proton density, it is not obvious whether or not other elements are always negligible. One of the purposes of the present paper is to explore the contribution of each element in the nuclear charge density to the msr.

Another purpose is to propose a complementary method to the previous ones[11] for investigating the neutron density of nuclei. It is one of the fundamental problems in nuclear physics how the protons and neutrons are distributed in nuclei. In contrast to the proton distribution, however, the neutron distribution is not well known yet, since there is no simple way to explore it experimentally. For example, in elastic electron scattering, the cross section is strongly dominated by the proton charge distribution, and the contribution from the neutrons is hidden behind the one from protons. As a result, it is hard to extract information on the neutron density by the analysis of the charge density profile. In hadron scattering[12], there is a different kind of difficulties. Although contributions to the cross section from neutrons and protons are comparable, they are not distinguishable from each other, because in the strong interaction, the reaction mechanism is not well understood, and the physical meaning of the parameters employed in the analyses is not obvious[11]. As a unique experiment to observe directly the neutron weak charge density, the measurement of the parity-violating asymmetry in the polarized-electron scattering has recently been performed[13, 14]. It is a promising, but very difficult and time-consuming experiment. At present, the value of the form factor is available only for 208Pb at a single value of the momentum transfer, $q=0.475\,\textrm{fm}^{-1}$, with the error of about 10%. Thus, it does not seem that the neutron density profiles are extracted soon from experiment without the help of nuclear models. In the present paper, it will be shown that instead of discussing the charge density profiles themselves, the analyses of their moments provide us with the useful information on the neutron distribution in nuclei.

In the following section, the relativistic charge density will be defined. In §3, the expression of the $n$-th order moment will be derived. In §3.1 and §3.2, the 2nd(msr) and 4th order moments will be discussed in detail, respectively. For the msr, the non-relativistic expression, which is equivalent to the relativistic one up to order $1/M^{2}$, will be derived consistently, according to the Foldy-Wouthuysen(F-W) transformation. The new term which has not been discussed so far is obtained as a relativistic correction. The difference between the mean square radii of the relativistic and non-relativistic point proton densities will also be shown in an analytic way. Unlike the msr, the $n(\,\geq 4)$-th order moment depends on the point neutron density. In the 4th order moment, the msr of the point neutron density and the 4th order moments of the neutron spin-orbit density play a crucial role, as corrections to the main term from the point proton density.

The present results may be useful for reducing ambiguity of nuclear model parameters, and hence, for recent detailed discussions of the nuclear proton and neutron density profiles[11]. Moreover, there is a possibility that not only the change of the proton distribution, but also the one of the neutron distribution will be explored throughout neutron-rich and proton-rich nuclei with the conventional and well-established electron scattering experiment[15, 16].

The final section is devoted to a brief summary of the present paper.

2 Relativistic nuclear charge density

The relativistic charge density of the nuclear ground state $|\,0\,\rangle$ is given by[17]

[TABLE]

Its Fourier component is described as

[TABLE]

where $A$ denotes the nucleon number of the system, $F_{1}$ and $F_{2}$ the Dirac and Pauli form factor of the nucleon, respectively, $\mu_{k}$ the anomalous magnetic moment, and $M$ the nucleon mass[18]. Throughout the present paper, the following values will be used, $\mu_{k}=1.793$ and $-1.913$ for the proton(p) and neutron(n), respectively and $M=939$ MeV. The center-of-mass corrections will not be taken into account.

The above matrix element is rewritten in terms of Sachs form factor,

[TABLE]

where $\tau$ represents p and n. The Sachs form factor is related to the Dirac and Pauli form factor as[18]

[TABLE]

The point nucleon density $\rho_{\tau}$ and the spin-orbit density $W_{\tau}$ are given by[17]

[TABLE]

The first equation satisfies $\int d^{3}r\,\rho_{\tau}(r)=Z$ for $\tau=p$ and $N$ for $\tau=n$, respectively, while the last equation $\int d^{3}r\,W_{\tau}(r)=0$, as should be. Their explicit forms in relativistic nuclear mean field models are written as[17]

[TABLE]

In the above equations, $j_{\alpha}$ denotes the total angular momentum of a single-particle, $\kappa_{\alpha}=(-1)^{j_{\alpha}-\ell_{\alpha}+1/2}(j_{\alpha}+1/2)$, $\ell_{\alpha}$ being the orbital angular momentum, and $G_{\alpha}(r)$ and $F_{\alpha}(r)$ stand for the radial parts of the large and small component of the single-particle wave function, respectively, with the normalization,

[TABLE]

In Eq.(8), the effective nucleon mass is defined by $M^{*}(r)=M+V_{\sigma}(r)$, $V_{\sigma}(r)$ being the $\sigma$ meson-exchange potential which behaves in the same way as the nucleon mass in the equation of motion. The spin-orbit density is a relativistic correction due to the anomalous magnetic moment of the nucleon, and its role is enhanced by the effective mass in relativistic nuclear models[17]. The reason why Eq.(8) is called the spin-orbit density will be seen later in discussing its non-relativistic reduction.

The relativistic nuclear charge density Eq.(1) is finally written as,

[TABLE]

by convoluting a single-proton and -neutron density,

[TABLE]

with the functions,

[TABLE]

The momentum-transfer dependence of the nucleon form factors is not well known yet, and are still under discussions theoretically[19, 20, 21]. Experimentally also there are various versions to fit the electron scattering data at present[22, 23]. In this paper, the following Sachs and Pauli form factors will be employed[17, 24, 25, 26, 27],

[TABLE]

with

[TABLE]

These form factors have been determined by fitting electron scattering data within a relativistic framework, but we note that there are still discussions on the values of $r_{p}$ and $r_{\pm}^{2}$ themselves[28].

3 Relativistic expression of the n-th order moment

The relativistic charge density Eq.(1) satisfies $\int d^{3}r\,\rho_{c}(r)=Z$. The mean $2n$-th order moment $\langle\,r^{2n}\,\rangle_{c}$ of the nuclear charge distribution is given by

[TABLE]

Here, according to Eq.(3), we have defined the notations

[TABLE]

with

[TABLE]

In order to calculate Eq.(15), it is convenient to define the integral,

[TABLE]

since the Pauli form factor is written in the form,

[TABLE]

with use of the notations $a=r_{p}^{2}/12$ and $b=1/4M^{2}$. Then, we have

[TABLE]

where ${}_{n}\mathrm{C}_{k}$ denotes the binomial coefficient and

[TABLE]

with

[TABLE]

In the limit $q\rightarrow 0$, $J_{2k}$ and $J_{2k-1}$ become of

[TABLE]

which yields

[TABLE]

In the present paper, we will discuss in detail the case $n=1$, and $2$, which are given explicitly as,

[TABLE]

3.1 The 2nd order moment of the charge density

Eq.(15) and Eq.(19) provide us with the relativistic expression for the msr of nuclear charge distribution[29],

[TABLE]

where we have defined with $N_{p}=Z$ and $N_{n}=N$,

[TABLE]

The msr depends on a single-proton and a single-neutron size through the second and the third term of the above equation. They are not in a negligible order, but unfortunately, it seems that their experimental values are still under discussion [28]. We note, however, that their contributions to the msr is almost eliminated in taking the difference between the values for two nuclei, for example, in discussing isotope and isotone shift, or mirror nuclei. Then, the msr is approximately given by the point proton radius and the proton and neutron spin-orbit densities. Among them, if protons fill the shells of spin-orbit partners, contributions from their spin-orbit term is negligible, as will be shown later. In that case, the msr of the charge density is given by the ones of the point proton density and the neutron spin-orbit density in relativistic models.

In non-relativistic models, the following expression of the msr has been widely used[30],

[TABLE]

In the r.h.s. of the above equation, the first term stands for the msr of the point proton density calculated with non-relativistic wave functions. The values of the second and the third term are taken from the Sachs form factors determined experimentally within a relativistic framework, as used in Eq.(21). The 4th term is added as a non-relativistic reduction of the spin-orbit terms in Eq.(21)[24, 30], and the 5th one is explained as coming from the Darwin-Foldy(D-F) term which is a relativistic correction to the nuclear charge density[31]. In fact, however, the consistency of these terms is doubtful, since they should be calculated consistently, according to the Foldy-Wouthuysen(F-W) unitary transformation of the four-component framework to the two-component one[5, 18, 32].

The F-W transformation for Dirac equation with electromagnetic field has been performed by various authors[24, 33, 34]. For example, Nishizaki and the present authors[34] have obtained a single-particle charge operator $\hat{\rho}(q)$ up to order $1/M^{2}$,

[TABLE]

where $D_{1}$ and $D_{2}$ are defined as

[TABLE]

The relativistic corrections are those proportional to $1/M^{2}$. The second term of $D_{1}$ is called Darwin-Foldy term, while $D_{2}$ the spin-orbit term in ref.[33]. The first term and the $F_{2}$ part of the Darwin-Foldy term in $D_{1}$ are replaced by the Sachs form factor, according to Eq.(4),

[TABLE]

Then the second derivative of Eq.(23) yields

[TABLE]

where $D_{1}^{\prime}$ denotes the first derivative with respect to $q^{2}$. Finally, Eq.(27) provides us with the expression of the msr up to $1/M^{2}$,

[TABLE]

where $R$ represents the contributions of order $1/M^{2}$ except for the one from the $F_{2}$ part included in the Sachs form factor of Eq.(26),

[TABLE]

The terms of the r.h.s. in Eq.(28) are formally consistent with each other, but in the present framework the values of $G^{\prime}_{E_{\tau}}(0)$ up to order $1/M^{2}$ are unknown. In the relativistic expression Eq.(21), they are taken from Eq.(14) determined by experiment with use of the relationship,

[TABLE]

If the same values are employed as in Eq.(22) , the consistency in Eq.(28) becomes obscure. Since the structure of $G_{E_{\tau}}$ is not well known theoretically at present[19, 20, 21], this ambiguity remains in the expression of Eq.(28).

In comparing Eq.(28) with Eq.(22), the former equation has an additional term in $R$, which is the last one in Eq.(29). It stems from the $F_{1p}$ part of $D_{2}$ in Eq.(25), while the first term in $R$ from its $F_{2}$ part. The last term contributes additively to the proton part of the first one, cancelling more its neutron part with the negative sign of $\mu_{k}$.

We note that the second term $3/4M^{2}=0.0331\,\textrm{fm}^{2}$ in $R$ comes from the $F_{1p}$ part of the Darwin-Foldy term in Eq.(24), whose $F_{2}$ part is taken in $G_{E_{\tau}}$ of Eq.(26).

When the nuclear part of the hamiltonian is different from the Dirac equation, the F-W transformation yields different relativistic corrections. For the mean field hamiltonian in the $\sigma$-$\omega$ model, Nishizaki et al.[34] have obtained, instead of Eqs.(24) and (25),

[TABLE]

In this case, Eq.(29) is replaced by

[TABLE]

in the approximation $M^{*}(r)\approx M^{*}$. The second term is proportional to $1/M^{*2}$, while the first term to $1/MM^{*}$, because of the definition of $F_{2\tau}$. Most of the relativistic models[2] has $M^{*}\approx 0.6M$ which enhances the relativistic corrections. In particular, the D-F correction $3/4M^{2}$ is enhanced by $(M/M^{*})^{2}$, yielding $3/4M^{*2}\approx 0.0920\,\textrm{fm}^{2}$. The enhancement of the $F_{2}$ part in $D_{1}$, Eq.(31), is absorbed into the Sachs form factor, if it is taken from experiment.

It may be instructive to show in a more naive way where the last two terms in $R$ of Eq.(29) stem from in the non-relativistic reduction, and why the D-F corrections are enhanced by the effective mass as in Eq.(33) . Since they are independent of $\mu_{k}$, it is expected that they are contained in the first term of the relativistic expression Eq.(21).

In using the Dirac equation[35], the $n$-th order moment of the density distribution as to the small component $F(r)$ is expressed in terms of the large component $G(r)$,

[TABLE]

which provides us with

[TABLE]

In the first term of the r.h.s., we have used the approximation

[TABLE]

corresponding to the approximation for the small component,

[TABLE]

Hence, the msr of the single-particle is written as

[TABLE]

where we have used the fact that

[TABLE]

The second term of the r.h.s in Eq.(35) is nothing but the last two terms in $R$. In this way, they are understood as a contribution of the small component to the msr.

In relativistic mean field models, the nuclear part of the hamiltonian contains one-body potentials. In the $\sigma-\omega$ model, the equations for the radial parts of the wave function are written as[35]

[TABLE]

where $V_{0}(r)$ denotes the potential due to the $\omega$-meson exchange. In this case, we obtain

[TABLE]

which gives

[TABLE]

This is not exactly the same as, but similar to the corresponding term of Eq.(33), because of $V_{\sigma}(r)\approx-V_{0}(r)$ in the relativistic models. Thus, it is understood why the last two terms of Eq.(33) are enhanced by the nucleon effective mass.

The above derivation of the relativistic corrections implies that, when we compare the relativistic and non-relativistic results with each other, the difference between $\langle\,r^{2}\,\rangle_{p}$ in Eq.(21) and $\langle\,r^{2}\,\rangle_{p,\mathrm{nonrel}}$ in Eq.(28) should be taken into account. If $\langle\,r^{2}\,\rangle_{p,\mathrm{nonrel}}$ is fixed so as to reproduce experimental values of the msr, $\langle\,r^{2}\,\rangle_{p,\mathrm{nonrel}}$ may contain a part of the relativistic correction in Eq.(39).

The 4th term of Eq.(22) is the same as the first term in Eq.(29). In fact, it is derived by various ways[24, 30] without concerning the consistency with other terms. For example, if we neglect simply the small component $F_{\alpha}$ in the relativistic spin-orbit density Eq.(8), we have

[TABLE]

which yields

[TABLE]

In the case of $k=2$, together with Eq.(9) neglecting $F_{\alpha}$, the above equation gives the spin-orbit correction of Eq.(22) and Eq.(29), but does not the one in Eq.(33) from the F-W transformation.

We note that Eq.(40) is written as[17],

[TABLE]

The above is the reason why $W_{\tau}(r)$ is called the spin-orbit density[17]. The previous authors[24] have pointed out that there is the exact cancellation of the spin-orbit terms between the fully occupied spin-orbit partners in their non-relativistic limit. This fact is easily understood from the above expression, and also valid in the last term of Eq.(29).

Now, we investigate the msr of the nuclear charge distribution given by Eq.(21) in the relativistic models and by Eq.(28) in the non-relativistic framework. Table 1 shows how each term contributes to the msr in the cases of 40Ca, 48Ca, and 208Pb. Among the experimental data available at present[7, 36], those for the above three nuclei are suitable as examples for the present purpose, since they have been well investigated using a mean field approximation, and used to fix the parameters of the model hamiltonian. In this approximation, 48Ca is described as the $f_{7/2}$ closed shell nucleus, while 40Ca is the double closed shell one, and in 208Pb, protons occupy up to $h_{11/2}$ shell and neutrons up to $i_{13/2}$ shell.

In Table 1, the sum of the first two terms and each of the rest in Eq.(21) are listed separately, in the relativistic cases, NL3[2] and NL-SH[1]. In non-relativistic calculations of Eq.(28) with SLy4, its second and the third term are taken from Eq.(30), as in the relativistic models. The values of the first term in the relativistic correction $R$ given by Eq.(29) are listed as $\langle\,r^{2}\,\rangle_{W_{p}}$ and $\langle\,r^{2}\,\rangle_{W_{p}}N/Z$, while the second term, $3/4M^{2}=0.0331\,\textrm{fm}^{2}$, is included in $\langle\,r^{2}\,\rangle_{c}$ in the Table 1. The last term of $R$ in Eq.(29) does not contribute to $\langle\,r^{2}\,\rangle_{c}$ in Ca, but does in 208Pb. Its value, 0.0162 fm2, is added to $\langle\,r^{2}\,\rangle_{c}$ of 208Pb.

The experimental values of the msr employed as inputs for fixing the parameters of NL3 and SLy4 are also listed in Table 1, according to the refs.[2, 3]. For NL-SH, the same values as the ones of NL3 are putted, which are taken from ref.[36]. Unfortunately, however, it is not clear for the authors which corrections to the msr of the calculated point proton densities are taken into account in the refs.[1, 2, 3], in reproducing the quoted experimental data.

As seen in the Table 1, the corrections from the last three terms to the first two terms in Eq.(21) are less than $3\%$ in relativistic models, in spite of the fact that the spin-orbit densities are enhanced by the effective mass[17]. The sum of the first two terms for 48Ca, however, is slightly larger than for 40Ca, while the total sum value of the former is smaller than the latter.

In non-relativistic models[3], the ambiguity coming from the relativistic corrections in Eq.(28) may be within $2\%$. We note, however, that if non-relativistic model fixes parameters, so as to reproduce the experimental values with the msr of the point proton density, it may contain relativistic corrections implicitly. In fact, Table 1 shows that the msr value of the point proton density in SLy4 is similar to the ones in the relativistic models which include a non-negligible relativistic correction enhanced by the effective mass. When one employs experimental values of the msr as inputs in phenomenological models[1, 2, 3], or when one compares phenomenological models with each others for precise discussions[11], these small corrections should be taken into account carefully.

Table 1 shows also that the cancellation of the spin-orbit terms between the spin-orbit partners does not hold exactly in the relativistic models, as was already pointed out in ref.[37].

3.2 The 4th order moment of the charge density

The function $r^{6}\rho_{c}(r)$ has a peak around the nuclear surface in a similar way as $r^{4}\rho_{c}(r)$. As a result, it is expected that the 4th order moment also reflects well the structure of the nuclear surface, as the msr does. This fact implies that it is useful to investigate the 4th order moment for understanding more details of the nuclear surface structure. Compared with the msr, however, the 4th order moment has not been explored in detail so far.

Fig.1 shows the contribution of the $n$-th order moment of the charge density to the form factor squared for elastic electron scattering off 48Ca and 208Pb [38], which is defined as

[TABLE]

where the $2n$-th order moment is calculated using the Fourier-Bessel analyses of the experimental data[36]. It is seen that the msr dominates the form factor in 48Ca up to $q\approx 0.3\,\textrm{fm}^{-1}$, and in 208Pb up to $q\approx 0.2\,\textrm{fm}^{-1}$. Above these momentum transfer regions, the 4th order moment also begins to contribute to the form factors. Up to about $q\approx 0.5\,\textrm{fm}^{-1}$ in 48Ca, and $q\approx 0.35\,\textrm{fm}^{-1}$ in 208Pb, it may be possible to explore well the 4th order moment, together with the msr. Nevertheless, the previous papers have not focused on the 4th order moment, as far as the authors know. As will be shown bellow, the 4th order moments provide us with rich information about the nuclear surface. In particular, it is noticeable that the point neutron density, in addition to the neutron spin-orbit density, contributes appreciably to it.

Eq.(15) and Eq.(20) give the 4th order moment of the nuclear charge distribution,

[TABLE]

where the following abbreviations are employed

[TABLE]

While the msr is independent of the point neutron density, the 4th order moment depends on it through its 2nd order moment in $R_{2n}$. Generally, the $2n$-th order moment of the charge distribution depends on the $2(n-k)$-th ( $k=1,2\cdots,n-1$ ) order moment of the point neutron density.

Table 2 shows the 4th order moments of Eq.(44) calculated for 40Ca, 48Ca and 208Pb by using the relativistic model NL3[2] and NL-SH[1]. As was seen in the detailed discussions on the msr, Eq.(44) of the 4th order moment should be used in relativistic framework, but not in non-relativistic calculations. For comparison, however, we will show in Table 2 the results of the non-relativistic calculations with SLy4[3], where $R_{4p}$, $R_{2p}$ and $R_{2n}$ are calculated with non-relativistic single-particle wave functions, and the contributions of the spin-orbit densities are obtained with use of Eq.(41), replacing its $G_{\alpha}$ by non-relativistic wave functions. The non-relativistic expression of the 4th order moment, which is equivalent to Eq.(44), may be obtained by the expansion of the nuclear hamiltonian up to order $1/M^{4}$, according to the F-W transformation [39].

Table 2 shows that the main contribution to the value of $\langle\,r^{4}\,\rangle_{c}$ comes from $R_{4p}$, as expected. In the present calculations with the relativistic models, more than $10\%$ correction to that stems from the msr of the proton distribution $R_{2p}$ in 40Ca and 48Ca, and about 6% in 208Pb. The sum of two terms $R_{4p}$ and $R_{2p}$, however, overestimates the experimental values, except for the case of NL-SH for 208Pb . In the case of SLy4 for 48Ca, the value of $R_{4p}$ itself exceeds the experimental one. Hence, it is necessary to have a negative contribution from the neutron density.

In 48Ca, $R_{2n}$ reduces the value of $R_{2p}$ by about 29.1$\%$ in the case of NL3. The sum of $|R_{2n}|$ and $|R_{4W_{n}}|$ amounts to 6.87$\%$ of $R_{4p}$, and to 47.6$\%$ of the sum of $|R_{2p}|$ and $|R_{4W_{p}}|$. Since the value of $R_{2p}$ is almost fixed by the experimental value of the msr, we may compare $|R_{2n}+R_{4W_{n}}|$ with $R_{4p}-R_{2p}$ in order to see the contribution of the neutrons to $\langle r^{4}\rangle_{c}$. Then we have their ratio 7.99%. In NL-SH, those values are similar to, and in SLy4 a little smaller than the ones of NL3.

In 208Pb, the sum of $|R_{2n}|$ and $|R_{4W_{n}}|$ is 44.3% of the sum of $|R_{2p}|$ and $|R_{4W_{p}}|$ in NL3. The ratio of the sum, $|R_{2n}|$ and $|R_{4W_{n}}|$, to $R_{4p}$ is decreased to be 0.0292, compared with 0.0687 in 48Ca. This is due to the constraint on the $A$-dependence of the msr in the stable nuclei. In more neutron-rich nuclei which are free from the constrain, the contribution of the neutron density to the 4th order moment is expected to be increased appreciably.

The parameters of the present phenomenological models[1, 2, 3] are fixed so as for the point proton distribution to reproduce almost the experimental values of the msr. As a result, relativistic and non-relativistic models yield a similar value of $R_{2p}$, as seen in Table 2. Although, as was discussed in the previous subsection on the msr, $R_{4p}$ calculated in the non-relativistic model is not equivalent to the relativistic ones, we note the following three points. First, the values of $R_{4p}$ are different from each other by 5 to 10$\%$. Second, in the relativistic models, $R_{4p}$ of 40Ca is larger than the one of 48Ca, while in SLy4 the relationship is opposite. Third, on the one hand, the values of the 4th order moment of the charge density by NL3 and NL-SH are in a better agreement with experiment, compared with the ones by SLy4 in Ca, mainly owing to the difference between their $R_{4p}$ values. On the other hand, in 208Pb, the relativistic models fairly underestimate the experimental value. These facts imply that the 4th order moment yields valuable information as to the nuclear surface, in addition to the one from the msr.

4 Summary

The purpose of the present paper is twofold.

The first one is to make clear a role of each component of the nuclear charge density in the mean square radius(msr). The obtained results will be useful for refining or constructing phenomenological nuclear models which employ the experimental values of the msr in order to fix their parameters. Those are also helpful for detailed analyses of experimental data by making use of the nuclear models.

In relativistic models, the msr is dominated by the point proton density with a few % correction from a single-proton and -neutron size and the neutron spin-orbit density in stable neutron-rich nuclei.

For the non-relativistic models, the expression of the msr equivalent to the relativistic one up to order $1/M^{2}$ has been derived, according to the Foldy-Wouthuysen unitary transformation. The terms in the expression are formally consistent with each other, but, in practical use, may not be consistent, since non-relativistic expressions of the nucleon form factors are not known and are usually replaced with the form factors determined by experiment. Moreover, if parameters of non-relativistic models are fixed so as to reproduce the experimental values of the msr without taking into account relativistic corrections, the msr of their point proton density may contain contribution from relativistic effects implicitly.

The second purpose is to propose a complementary method for exploring the neutron density to the previous analyses of experiment[11]. It is to study the $n$-th order moment, instead of the charge density profile itself deduced from electron scattering data. In contrast to the msr, the $n(\,\geq 4)$-th order moment depends not only on the neutron spin-orbit density, but also on the point neutron density.

In the 4th order moment, a large part is explained by the point proton density, but it is apparent that the contribution from the point neutron density and from the neutron spin-orbit density are necessary for reproducing the experimental values of the 4th order moment of the charge density.

For example, in 48Ca, the main components of the 4th order moment of the charge density are, in addition to the 4th and 2nd order moments of the point proton density, the msr of the point neutron density and the 4th order moments of the neutron spin-orbit density. Among them, the 4th order moment of the point proton density dominates the 4th order moment of the charge density, but when 2nd moment of the point proton density is added to it, the sum of them overestimates the experimental value. It is decreased by the negative contributions from the msr of the point neutron density and the 4th order moment of the neutron spin-orbit density. Their negative contributions to the 4th order moment of the point proton density is about 7% in 48Ca, and they almost eliminate a half of the contribution from the 2nd order moment of the point proton density, in using the relativistic models.

In 48Ca and 208Pb, the neutron contribution to the 4th order moment of the charge density is limited, according to the constraint of the $A$-dependence on the msr in stable nuclei. In unstable neutron-rich and proton-rich nuclei, there is not such a constraint on the point neutron density and the neutron spin-orbit density. They are expected to play a more important role in the 4th order moment of the charge density. In electron scattering, the msr of the charge density reflects mainly the point proton distribution selectively, while its 4th moment does the point neutron density additionally. Future experiment on unstable nuclei[15, 16] would show not only change of the point proton distribution, but also of the point neutron distribution as a function of $N$ or $Z$.

A role of the neutron spin-orbit density in electron scattering off neutron rich nuclei has been investigated in more detail in the previous paper[17].

The present study on the 4th order moment of the charge density may also be helpful for the analyses of the experimental data from parity-violating electron scattering[13, 14]. As was shown in Fig.1, in a region of the momentum transfer $q=0.475\,\textrm{fm}^{-1}$, where the experiment on 208Pb has been performed, the 4th order moment considerably contributes to the form factor, together with the msr. In discussing the msr at this region, nuclear models used for the analysis should also reproduce the 4th order moment. In fact, the relativistic and non-relativistic models used in this paper yield the 4th order moments which are fairly different from each other, although they reproduce the msr in a similar way. The components of the 4th order moment have a clear physical meaning separately, and hence, their detailed investigation may reduce the ambiguity of the nuclear models.

Finally, we note that a precise determination of the proton and the neutron size[28] is necessary for more detailed discussions of the moment as to the nuclear charge density.

Acknowledgment

The authors would like to thank Professor T. Suda and Professor T. Tamae for useful discussions.