Pauli nonlocality and the nucleon effective mass

Over a wide range of the single-particle energies, the nucleon motion in medium is overwhelmingly governed by the nuclear mean field, known as the shell-model potential for bound states and optical potential (OP) for scattering states. The single-particle potential is also a key quantity in the mean-field studies of the equation of state (EOS) of nuclear matter (NM) as well as the structure of finite nuclei [1, 2]. The nucleon OP was widely studied in the Brueckner-Hartree-Fock (BHF) calculations of NM using free nucleon-nucleon (NN) interaction [2, 3, 4, 5], and in the mean-field calculations of NM on the Hartree-Fock (HF) level using different choices of the effective NN interaction [6, 7, 8, 9, 10, 11]. The mean-field description of nucleon OP in the NM limit provide an important physics input for the microscopic models of nucleon OP of finite nuclei, in particular, different versions of the folding model [11, 12, 13, 14, 15].

The microscopic studies of NM have repeatedly shown the impact by the Pauli blocking and increasing strength of higher-order NN correlations at high densities [2]. These in-medium effects are considered as the physics origin of the density dependence introduced into various effective NN interactions used in the nuclear structure and reaction studies. In the present work, we employ the CDM3Y6 density dependent version [16] of the M3Y interaction [20] which was used successfully in the HF studies of NM [6, 7, 21, 22, 23], and in the folding model calculation of the nucleon and nucleus-nucleus OP at low and medium energies [16, 13, 14, 24, 25, 26, 11, 17]. The CDM3Yn density dependence (n=1-6) of the original M3Y interaction [20] was first parametrized in Ref. [16] to reproduce the saturation of symmetric NM on the HF level. The CDM3Y3 and CDM3Y6 versions were later modified to take into account the rearrangement term (RT) of the single-particle potential in NM into the folding calculation of the nucleon-nucleus and nucleus-nucleus OP [11, 17], based on the Hugenholtz and van Hove (HvH) theorem [27, 28] which is exact for all interacting Fermi systems independently of the interaction between fermions.

When the antisymmetrization of the nucleon-nucleus system is taken explicitly into account, the exchange term of the folded nucleon-nucleus potential becomes nonlocal in coordinate space [12, 15], and a nonlocal folding model of nucleon OP using the CDM3Yn interaction was suggested in Ref. [29], with the RT properly taken into account. The calculable $R$-matrix method [30, 31] is used to solve the OM equation with the nonlocal kernel of the folded nucleon OP. This method was well tested in the OM analysis of elastic nucleon scattering at energies up to 40 MeV [32], using the phenomenological nonlocal nucleon OP [33, 34, 35, 36].

The momentum dependence of nucleon OP at the NM saturation density $\rho_{0}$ is due mainly to the exchange term up to $k\approx 2$ fm^-1, and the antisymmetrization of the nucleon-nucleus system is, therefore, the main origin of the momentum dependence of nucleon OP at low momenta. As a result, it becomes possible to determine the nucleon effective mass from the momentum dependence of the folded nucleon OP of finite nuclei, based on the WKB local approximation, in the OM analysis of low-energy elastic nucleon scattering on target nuclei with neutron excess. The present work is our first attempt to determine the radial- and isospin dependence of the nucleon effective mass near the saturation density ($\rho\lesssim\rho_{0}$) from the radial strengths of the real folded nucleon OP. The obtained results for $m^{*}$ allowed us to determine the neutron-proton effective mass splitting $\Delta m^{*}$ at $\rho\approx\rho_{0}$ for ⁴⁸Ca, ⁹⁰Zr, and ²⁰⁸Pb targets, which is complementary to the $\Delta m^{*}$ values obtained in different mean-field calculations of asymmetric NM.

The knowledge about the nucleon effective mass is of wide importance for different nuclear physics and nuclear astrophysics studies [37, 38, 39, 40, 41]. In particular, the direct link of the nucleon effective mass to the density dependence of nuclear symmetry energy [5, 42, 43], the liquid-gas phase transition in the neutron-rich NM, and the temperature profile of hot protoneutron stars [44] and neutrino emission therefrom [45].

An effective density-dependent NN interaction is the essential input for the mean-field calculation of NM, and we recall briefly the CDM3Yn density dependent versions of the M3Y interaction used in our model. Originally, parameters of the CDM3Yn density dependence were parametrized [16] to reproduce the NM saturation properties on the HF level. These parameter sets were extended later to include a realistic isovector part [23, 11, 14] as well as the rearrangement term of the single-particle potential [11]. The HF results for the energy per nucleon $E/A$ of NM obtained with the CDM3Yn interaction are compared in Fig. 1 with results of the ab initio variational calculation using the Argonne V18 interaction [46]. One can see a nice agreement of the HF results with those of the ab initio calculation over a wide range of densities.

We note that the functional CDM3Yn density dependence of the original M3Y interaction [20] was parametrized [16] to reproduce on the HF level the saturation of symmetric NM at $\rho_{0}$, as well as to give a realistic (real) nucleus-nucleus OP within the double-folding model that properly describes the nuclear rainbow pattern observed in elastic $\alpha$-nucleus and light heavy-ion refractive scattering. The rainbow scattering data were proven to be sensitive to the real OP at small distances where the dinuclear overlap density is reaching up to twice the saturation density $\rho_{0}$ [24], so that the use of the CDM3Yn interaction is well validated for the low-energy domain of nuclear EOS (see Fig. 1).

According to Landau theory for an infinite system of interacting fermions [47], the single-particle energy is determined [11] as derivative of the energy per nucleon $\varepsilon\equiv E/A$ of NM with respect to the nucleon momentum distribution $n_{\tau}(k)$

$E_{\tau}(\rho,k)$ is, thus, the change of the NM energy at density $\rho$ caused by the removal or addition of a nucleon with the momentum $k$. The single-particle potential $U_{\tau}(\rho,k)$ consists of both the HF and rearrangement terms

with the explicit expressions of $U_{\tau}(k)$ given in Ref. [11]. At the Fermi momentum $(k\to k_{F\tau}),\ E_{\tau}(k_{F\tau})$ determined from Eqs. (1)-(2) is exactly the Fermi energy given by the Hugenholtz-van Hove (HvH) theorem [27], which is satisfied on the HF level only when the effective NN interaction is density independent, with the RT equal zero [11, 48]. In the mean-field calculation (1)-(2), the RT originates naturally from the density dependence of the in-medium NN interaction that implicitly accounts for the higher-order NN correlations as well as three-body force [1, 4, 5].

For the spin-saturated NM, the (spin-independent) direct (D) and exchange (EX) parts of the CDM3Yn interaction [16, 14] are used in the HF calculation of NM

The radial dependence of the isoscalar (IS) and isovector (IV) terms $v^{\rm D(EX)}_{00(01)}(s)$ is kept unchanged as that used for the original M3Y interaction [20]. The IS density dependence $F_{0}(\rho)$ was determined [16] to reproduce the saturation of NM on the HF level. The IV density dependence $F_{1}(\rho)$ was determined [11] based on the isospin dependence of nucleon OP given by the BHF calculation of NM by Jeukenne, Lejeune and Mahaux (JLM) [49], and and fine tuned by the folding model description of the charge exchange reaction to the isobar analog states in medium-mass nuclei [26].

Using the exact expression of the RT given by the HvH theorem, a compact method was suggested [11] to account for the RT on the HF level by adding a correction term to the CDM3Yn density dependence, $F_{0(1)}(\rho)\to F_{0(1)}(\rho)+\Delta F_{0(1)}(\rho)$, used in the HF-type calculation of the single-particle potential

where $|\mbox{\boldmath$k$}\sigma\tau\rangle$ are the ordinary plane waves. Treating explicitly the isospin dependence, the single-particle potential (4) is expressed [11] in terms of the IS and IV parts as

where (-) sign pertains to proton and (+) sign to neutron. $U^{\rm(M3Y)}_{0}$ and $U^{\rm(M3Y)}_{1}$ are, respectively, the IS and IV parts of the single-particle potential given by the density independent M3Y interaction

Because the original M3Y interaction is momentum independent, the momentum dependence of the single-particle potential (5) is due to the exchange term of $U^{\rm(M3Y)}_{0(1)}(\rho,k)$, with the nucleon momentum $k$ determined self-consistently as

The extended HF approach (4)-(8) provides a consistent description of both the single-particle potential for bound nucleons, $U_{\tau}(\rho,k)$ with $k<k_{F\tau}$, and nucleon optical potential, $U_{\tau}(\rho,k)$ with $k>k_{F\tau}$. Such an approach is the well-known continuous approximation for the single-particle potential [1, 50], where nucleon OP in NM is determined as the mean-field potential felt by nucleon being scattered on NM at energy $E>0$. The momentum of scattered nucleon is determined by the same relation (8) but with $E_{\tau}$ replaced by the nucleon incident energy $E$. In this way, the energy- and momentum dependence of nucleon OP are treated on the same footing as illustrated in Fig. 2. Therefore, an important constraint for the present approach is that at $E>0$ the energy dependence of the potential (5) should agree reasonably with the observed energy dependence of nucleon OP. The single-particle potential (5) evaluated for symmetric NM at the saturation density $\rho_{0}$ using the CDM3Y6 interaction is compared with the empirical data [51, 52, 53] for nucleon OP at $E>0$ in Fig. 2. One can see that the inclusion of the RT significantly improves the agreement with the empirical data at energies $E<50$ MeV. Moreover, the momentum dependence of nucleon OP in this low-energy region is found nearly linear and due mainly to the Pauli exchange term in Eq. (4).

It should be noted here that the Pauli exchange term in Eq. (4) gives a good agreement of the energy (or momentum) dependence of nucleon OP with the empirical data at low energies only. At higher energies ($E>50$ MeV or $k>2$ fm^-1) the agreement becomes worse, and nucleon OP (5) is more attractive compared to the empirical data (see, e.g., Fig. 3 in Ref. [17]). Such a behavior of nucleon OP given by the HF calculation is well expected in light of the effective G-matrix interaction derived from the solution of Brueckner–Bethe–Goldstone equation [12], where both the direct and exchange parts of the G matrix interaction are energy dependent. This is also the reason why a slight linear energy dependence was added to the CDM3Y6 interaction, in terms of $g(E)$ factor [16], for the double-folding calculation of nucleus-nucleus OP at medium energies. Consequently, the nucleon effective mass $m^{*}$ at high momenta originates not only from the Pauli nonlocality, but also from the higher-order NN correlations in nuclear medium. The present work is focused, however, on the determination of $m^{*}$ at low energies ($E<50$ MeV) and corresponding momenta $k<2$ fm^-1 from the momentum dependence of nucleon OP, which is due entirely to the Pauli nonlocality as shown above.

As discussed in Sec. II, the momentum dependence of the single-particle potential in NM at low momenta is determined mainly by the exchange term which results on a nonlocal single-nucleon potential in the coordinate space. At positive energy ($E>0$) the single-nucleon potential (4)-(5) can be treated as nucleon OP in NM, with the associated nucleon effective mass $m^{*}_{\tau}$ determined [1, 37] as

where $m$ is the free nucleon mass. In general, the nucleon effective mass arises from both the momentum- and energy dependence of the single-particle potential [1], known as the $k$- and $E$-effective masses, which characterize the spacetime nonlocality. Within the time independent HF formalism [11], the total nucleon effective mass is determined by the same relation (35), and it is associated with the spatial nonlocality of the nucleon mean-field potential. At the Fermi momentum ($k\to k_{F\tau}$) , the nucleon effective mass (35) is obtained naturally from the Fermi energy (1) by the Hugenholtz-van Hove theorem [27]. The knowledge about $m^{*}_{\tau}$ at the saturation density $\rho_{0}$ and different neutron-proton asymmetries $\delta=(\rho_{n}-\rho_{p})/\rho$ is essential for the determination of the nuclear symmetry energy $E_{\rm sym}$ and its slope parameter $L$, the two key ingredients of the EOS of neutron rich NM [41, 42].

The neutron and proton effective masses (35) obtained from the single-particle potential (4)-(5) in asymmetric NM at the saturation density $\rho_{0}$ with $k\to k_{F\tau}$ are shown in Fig. 7 as dashed and dotted lines, respectively. Although the $m^{*}_{\tau}$ values are still poorly known at high NM densities and/or large $\delta$ values, the empirical nucleon effective mass in symmetric NM ($\delta=0$) at the saturation density $\rho_{0}$ is well established to be around $m^{*}/m\approx 0.73$ [37]. Our extended HF calculation of NM using the CDM3Yn interaction gives $m^{*}/m\approx 0.737$ at $\rho\approx\rho_{0}$ and $\delta=0$ (see Fig. 9), in a good agreement with the empirical data. The isospin dependence of the nucleon effective mass is governed by the neutron-proton effective mass splitting

which is associated directly with the nuclear symmetry energy and its slope parameter. The knowledge of $m^{*}_{n-p}$ is also of importance for the determination of the neutron/proton ratio during stellar evolution or cooling of protoneutron star [38, 40, 41]. Depending on the isospin dependence of the in-medium NN interaction, one obtains very different results for the neutron-proton effective mass splitting. A survey of different mean-field studies in Ref. [41] shows that $m^{*}_{n-p}$ values depend linearly on the neutron-proton asymmetry parameter $\delta$, and are ranging widely from negative- to positive values, up to $m^{*}_{n-p}\approx 0.637~{}\delta$. An analysis of the empirical constraints for the density dependence of nuclear symmetry energy [38] from both the nuclear physics experiments and astrophysical observations has led to the empirical constraint $m^{*}_{n-p}(\rho_{0},\delta)\approx(0.27\pm 0.25)\delta$. Our extended HF calculation of asymmetric NM using the CDM3Y6 interaction gives $m^{*}_{n-p}(\rho_{0},\delta)\approx(0.20\pm 0.02)\delta$ at the Fermi momentum, which is well inside the empirical boundary.

For many years, the nucleon effective mass in finite nuclei has been a focus of different nonrelativistic and relativistic nuclear structure studies (see, e.g., Refs. [68, 69, 70]) which provided accurate estimates of the $m^{*}_{\tau}$ values for the single-particle states of bound nucleons (with $k\lesssim k_{F\tau}$). Although the neutron- and proton effective masses, and the corresponding single-particle energies and spectroscopic factors are well described by the structure studies, it remains difficult to deduce therefrom an explicit isospin dependence of $m^{*}_{\tau}$ or the neutron-proton effective mass splitting (36). The method often used so far for this purpose is to deduce the isospin dependence of the nucleon effective mass from the isospin dependence of the phenomenological nucleon OP given by the best OM fit to elastic nucleon scattering data at different energies [39]. In the present work, we aim to determine the nucleon effective mass from the semi-microscopic nucleon OP predicted by the local folding model presented in Sec. III.2. Given a good OM description of elastic nucleon scattering at low energies by the local folded nucleon OP shown Sec. III.3, it is reasonable to determine the effective mass of nucleon scattered by the mean field of target from the momentum dependence of the real folded OP as

Here $U^{(\tau)}_{\rm D(EX)}$ are the direct and exchange parts of the local neutron (or proton) folded OP determined from Eqs. (14) and (33), respectively. The energy or momentum dependence of the folded nucleon OP (38) is embedded in the exchange term only, which was shown in Sec. III.2 to be resulted from the Pauli nonlocality of the folded nucleon OP. The in-medium momentum $k$ of scattered nucleon is determined self-consistently from the real folded nucleon OP by Eq. (34).

It is obvious from Eq. (34) that the momentum $k$ of scattered nucleon depends explicitly on the nucleon-nucleus distance $R$, and the nucleon effective mass (38) is, therefore, also radial dependent. The in-medium momentum of scattered neutron determined from the real folded $n+^{208}$Pb OP at energies of $E=1\sim 45$ MeV is shown in panel (a) of Fig. 8, and one can see that at each energy the neutron momentum $k$ changes gradually from its maximum of about $1.6\sim 2$ fm^-1 in the center to $0.2\sim 1.5$ fm^-1 at the surface, lying above the corresponding Fermi momentum. Over the same radial range, the neutron effective mass $m^{*}_{n}/m$ is changing from about $0.75\sim 0.78$ to unity at the surface. Such a radial dependence of $m^{*}$ is similar to that found in the nuclear structure studies [68, 69, 70]. However, the latter is usually enhanced to above unity at the surface for the bound single-particle states lying close to the Fermi level.

From the radial dependence of the neutron effective mass and the neutron- and total ground-state densities of ²⁰⁸Pb shown in panel (a) of Fig. 9, it is straightforward to infer the density dependence of the effective mass of neutron scattered off ²⁰⁸Pb target at different energies, over the density range $0\lesssim\rho\lesssim\rho_{0}$. One can see in panel (a) of Fig. 9 that the average total density in the center of ²⁰⁸Pb target is $\bar{\rho}\approx\rho_{0}$, and the neutron- and proton effective masses determined at distances $0\lesssim R\lesssim 3$ fm (shown in panels (b) of Figs. 9 and 10) can represent, therefore, the corresponding $m^{*}_{\tau}/m$ values in asymmetric NM at $\rho\approx\rho_{0}$, $k\gtrsim k_{F\tau}$ and neutron-proton asymmetry $\bar{\delta}=(\bar{\rho_{n}}-\bar{\rho_{p}})/\bar{\rho}\approx 0.185$. Because of the target valence neutrons, the deduced $\bar{\delta}$ value in the target center is smaller than the total difference in neutron- and proton numbers $(N-Z)/A$. Since our local folding model (14) and (33) predicts the real nucleon OP at $E>0$ or $k\gtrsim k_{F\tau}$, approaching the Fermi momentum from above, we have computed the real folded nucleon OP for ⁴⁸Ca, ⁹⁰Zr, and ²⁰⁸Pb targets at $E=0.05$ MeV (i.e., 50 keV above the Fermi level), and deduced the neutron- and proton effective masses using Eq. (38). The obtained results are presented in Table 1 and shown in Figs. 7 as rhombuses and circles. One can see that the $m^{*}_{\tau}/m$ values obtained at $\bar{\rho}\approx\rho_{0}$ and $k\gtrsim k_{F\tau}$ for these targets follow approximately the trend given by the extended HF calculation of the single-particle potential in NM.

It is interesting that the neutron-proton effective mass splitting (36) obtained from the $m^{*}_{\tau}/m$ values given in Table 1 also depends linearly on the asymmetry parameter, $m^{*}_{n-p}(\rho_{0},\delta)\approx(0.167\pm 0.018)\delta$, which is within the empirical boundary of $m^{*}_{n-p}(\rho_{0},\delta)\approx(0.27\pm 0.25)\delta$ deduced from the terrestrial nuclear physics experiments and astrophysical observations [38]. In neutron-rich NM, nucleons at $\rho_{0}$ are not bound by the in-medium NN interaction (see, e.g., Fig 1 in Ref. [11]), and the results obtained above for $m^{*}_{\tau}$ at $E>0$ should be appropriate for nucleons in the outer core of neutron star which are bound by gravitation only. We note further that the $m^{*}_{n-p}$ value estimated from the folded nucleon OP of finite nuclei is slightly lower than that given by nucleon OP in NM using the same interaction, $m^{*}_{n-p}\approx(0.20\pm 0.02)\delta$, and the value $m^{*}_{n-p}\approx(0.41\pm 0.15)\delta$ estimated from the phenomenological nucleon OP [40], based on the extensive OM analysis of elastic nucleon scattering.

We note finally that the nucleon effective mass is given entirely by the momentum dependence of the exchange term of the folded nucleon OP, so that the nucleon effective masses presented in Table 1 and shown in Fig. 7 originated solely from the spacial Pauli nonlocality of nucleon OP at low energies. We have considered the energy region $0<E<50$ MeV, and found that the nucleon effective mass (38) depends weakly on the energy. For example, $m^{*}_{n}/m\approx 0.7553+0.0004E$ and $m^{*}_{p}/m\approx 0.7241+0.0005E$ for ²⁰⁸Pb target, and this result agrees fairly with the empirical energy dependence of about $0.0007E$ established for the isoscalar $k$ effective mass of nucleons lying above the Fermi level [37].

The generalized folding model of the nonlocal nucleon OP, with the exchange potential calculated exactly in the HF manner and rearrangement term properly included, has been used for the OM analysis of elastic neutron and proton scattering on ^40,48Ca, ⁹⁰Zr, and ²⁰⁸Pb targets at different energies, where the WKB local approximation for the exchange term of the nonlocal folded nucleon OP is validated by a consistently good OM description of the considered elastic data.

Given the accurate local approximation for the nonlocal folded nucleon OP, a compact method is proposed to determine the nucleon effective mass at low momenta from the in-medium momentum dependence of the local folded nucleon OP, which originates mainly from the Pauli nonlocality. The results obtained for the effective mass of nucleon being scattered by the target mean-field potential at $0\lesssim E\lesssim 45$ MeV and $k_{F\tau}\lesssim k\lesssim 2$ fm^-1 seem to follow closely the isospin dependence of $m^{*}_{\tau}$ predicted by the extended HF calculation of single-particle potential in asymmetric NM, using the same density dependent CDM3Y6 interaction.

The neutron-proton effective mass splitting (36) given by the $m^{*}_{\tau}/m$ values obtained from the real folded nucleon OP of finite nuclei at low energies was found to depend also linearly on the asymmetry parameter $\delta$. At positive energies lying slightly above the Fermi level, the $m^{*}_{n-p}$ value determined at the center of finite nuclei (with the average density $\bar{\rho}\approx\rho_{0}$) is slightly smaller than that obtained from the extended HF calculation of the single-particle potential of nucleons in asymmetric NM at $\rho_{0}$, where the Pauli exchange has been shown as the main origin of nucleon effective mass.

Because nucleons at density $\rho\approx\rho_{0}$ in the outer core of neutron star are mainly bound by gravitation, not by the (in-medium) density dependent NN interaction, the $m^{*}_{\tau}$ values obtained in this work from the folded nucleon OP of finite nuclei with neutron excess could be of complementary interest for the mean-field studies of the EOS of neutron star matter.

The authors are indebted to Pierre Descouvemont for his helpful discussions and comments on the calculable $R$-matrix method [30, 31]. We also thank Bao-An Li for his communication and suggestion on the nucleon effective mass. The present research has been supported, in part, by the National Foundation for Scientific and Technological Development of Vietnam (NAFOSTED Project No. 103.04-2021.74).