Sizes of the Neutron–Proton Halo in Nucleon-Stable States of the 6Li Nucleus

The matter, neutron, and proton radii of nucleon-stable 1+ and 0+ states of the 6Li nucleus are studied theoretically within the no-core shell model. The results have been comparatively analyzed with the radii of the 0+ state of the 6He nucleus. To increase the accuracy of calculations, we have developed an extrapolation procedure. A new definition of the quantitative measure has been proposed and justified to describe the properties of a halo formed by loosely bound neutrons and protons in the A-nucleon problem. The sizes of the halo in the indicated states of 6Li have been calculated for the first time. It has been demonstrated that the sizes of their halos are close to those of the two-neutron halo in 6He. Thus, additional reliable evidence of the existence of the neutron–proton halo in the discussed states of 6Li has been obtained.


Sizes of the Neutron-Proton Halo in Nucleon-
The matter, neutron, and proton radii of nucleon-stable 1 + and 0 + states of the 6 Li nucleus are studied theoretically within the no-core shell model.The results have been comparatively analyzed with the radii of the 0 + state of the 6 He nucleus.To increase the accuracy of calculations, we have developed an extrapolation procedure.A new definition of the quantitative measure has been proposed and justified to describe the properties of a halo formed by loosely bound neutrons and protons in the A-nucleon problem.The sizes of the halo in the indicated states of 6 Li have been calculated for the first time.It has been demonstrated that the sizes of their halos are close to those of the two-neutron halo in 6 He.Thus, additional reliable evidence of the existence of the neutron-proton halo in the discussed states of 6 Li has been obtained.

DOI: 10.1134/S0021364023602130
The nuclear halo, i.e., a low-density region on the surface of nuclei, is actively studied experimentally and theoretically.This halo consists of one, two, and more loosely bound neutrons and/or protons.It is not a neutron skin, which naturally appears in neutronrich nuclei.The nuclear halo is manifested in experiments in large cross sections for the Coulomb decay and narrow momentum distributions of its products (these features described in [1] underlie the concept of the halo), a high probability of the transfer of nucleons from the halo to another nucleus [2], the specific properties of gamma and beta transitions between states having the halo structure, and transitions between states one of which (initial or final) has the halo structure and the other does not have such a structure [3][4][5][6].
Various theoretical schemes are used to describe this effect.The quantitative characteristics of one-and two-nucleon halos are most simply interpreted in schemes describing the interaction of the core as a whole with a nucleon (nucleons) and the interaction between nucleons.These schemes differ in the corenucleon potential.Two main concepts are used to construct such a potential: (i) a potential with repulsion at small distances [7] and (ii) a potential with forbidden states [8,9].The approaches involving potentials of the former type include semimicroscopic schemes, where the core is considered as a system of nucleons and the antisymmetrization of the outer and inner nucleons is performed after the variational procedure [10,11].
The modern theory of light nuclei step-by-step approaches the description of the properties of these nuclei with completely microscopic methods treating a nucleus as a system of interacting nucleons using realistic NN interactions obtained, in particular, within the chiral effective field theory (see, e.g., [12]).With these ab initio approaches, a high-quality description of the total binding energies of nuclei with the mass numbers , the energies of their low levels, angular momenta, and some electromagnetic transitions in them is achieved.
Compared to the listed characteristics, the possibility of the ab initio description of sizes of nuclei is significantly limited because they are determined strongly by long-range correlations between nucleons, whereas their total binding energies are determined mainly by the interaction between nucleons at short distances.Consequently, it is difficult to achieve the convergence of the calculation of nuclear radii on bases acceptable for modern computers.For this reason, it is necessary to use more complex and less reliable procedures of extrapolation of the results to a large functional basis.Another problem is even more serious: the convergence of nuclear radii is not monotonic unlike the monotonic variation of the binding energy with increasing the size of the basis.Thus, accurate studies of halo effects are currently performed mainly with the canonical 6 He halo nucleus.

FIELDS, PARTICLES, AND NUCLEI
diverse ab initio schemes including various theoretical approaches based on the Monte Carlo method [13,14,18,24], hyperspherical harmonic method [17], coupled cluster method [26], and other methods.The most noticeable schemes are various versions of the no-core shell model (NCSM).The NCSM calculations and similar no-core configuration interaction (NCCI) calculations [22,23] of the matter and charge radii, as well as calculations of the total binding energies, involve realistic NN [19] and NN + 3N [15,21,24] interactions.In addition, the NCSM version adapted to the SU(3) symmetry [20] and the NCSM with the ( ) continuum (NCSMC) [25] are also used.In [27], we proposed a new promising scheme of the two-dimensional extrapolation of the sizes of the 6 He nucleus.
The ab initio study of the halo (i.e., region with a low nucleon density) of the other nucleon-stable sixnucleon system 6 Li seems interesting.Indirect indications of the neutron-proton halo in 6 Li are presented, e.g., in [3][4][5][6].Numerous ab initio studies are devoted to this system.Its matter, charge, and neutron radii were also studied [16,[28][29][30][31][32][33][34][35][36].At the same time, the problem of determining the size of the neutron-proton halo in the light system with , which is very relevant for the interpretation of the experimental results mentioned above, is still unsolved because of the identity of nucleons in the core and the halo.Experimental measurements also do not clarify the situation.The difference between the matter radii of an exotic nucleus and the corresponding core nucleus can be used as a quantitative measure of the size of the halo in a relatively heavy system.At the same time, the motion of the center of mass of the core plays a significant role for the halo whose mass is only a factor of 2-4 smaller than the mass of the core, and the volume where nucleons of the core move is noticeably larger than the measured and calculated volumes of the core nucleus.This volume for a neutron-rich nucleus, e.g., 6 He can be estimated within a high accuracy as the volume corresponding to the charge radius, but this method is inapplicable to the neutron-proton halo.
The aims of this work are to quantitatively study the size parameters of loosely bound nucleons in the nucleon-stable states of the 6 Li nucleus and to compare them to the characteristics of the halo in the ground state of the 6 He nucleus, which were described in [27].For these calculations, we used the same methods and techniques as in [27].
We briefly describe these methods.The М-scheme of the NCSM as one of the most reliable and justified ab initio approaches is used together with the Dae-jeon16 potential [37], which was derived in the N3LO approximation of the chiral effective field theory [38] smoothed using the similarity renormalization group (SRG) transformation [39].This potential is "global"; i.e., it is constructed to calculate various characteristics of nuclei with mass numbers and is tested in extensive calculations of the total binding energies, the binding energies of nucleons and clusters, excitation energies, radii, the spins of nuclear states, the decay widths of resonances, and the reduced probabilities of electromagnetic transitions.These tests showed that these characteristics are generally well reproduced.The NCSM calculations were performed with the Bigstick code [40].The oscillatory bases in those calculations were limited by the cutoff parameter .To improve the calculated radii, the two-dimensional extrapolation procedure mentioned above was applied to the shape of the surface of each radii on the ( ) plane; both these coordinates enter the extrapolated formula.
We begin the description of the experimental situation and the developed approach with terms.As in most of the modern works, the distributions of all nucleons (nuclear matter), neutrons, and protons for point nucleons are characterized by the matter ( ) , n e u t r o n ( r n ), and proton (r p ) rms radii, respectively.The last radius is obtained from the measured charge radius r ch using the expression [41] (1) Here, and are the mean squared charge radii of the proton and neutron, respectively; is the Darwin-Foldy relativistic correction, and is the spin-orbit correction to the nuclear charge density.The values fm 2 , = 0.033 fm 2 , fm 2 are usually used.According to the Particle Data Group [42], = 0.877 (7) fm.This value is used to process the measurement results to obtain nuclear radii for point nucleons.However, this radius determined exactly from the spectroscopic data for the muon hydrogen atom [43] is 0.84184(67) fm.Consequently, two different values can be obtained using these two values.The matter radius r m is extracted directly from the differential cross sections for the elastic scattering of protons at high momentum transfers using the Glauber multiple scattering theory.The radius cannot be measured and is calculated using the relation (2) A universal approach used in NCSM calculations allows the calculation of all three radii.After the calculation of the binding energy and wavefunction of the state, the matter, neutron, and proton radii of the nucleus for point nucleons are calculated.The square of the radius of the corresponding system in the shell model is determined as Here, the OBTD and TBTD are the one-and twobody transition densities expressed in terms of the matrix elements of the products of the secondary quantization fermion operators as (7) (8) where is the set of the quantum numbers of the ith nucleon, stands for the tensor product, and and λ are the parameters determining the rank of the tensor product of the corresponding operators.
The NCSM calculations of the radii are supplemented by the two-dimensional extrapolation procedure proposed and described in detail in our work [27].This procedure was implemented using the conventional χ 2 method in the free TMinuit minimization package, which is included in the ROOT CERN data analysis environment with the open initial code.We used three values = 10, 12, and 14.The inclusion of smaller values in the input database of the extrapolation, for which radius values are unstable, can only reduce the quality of the results.As known from [19,22,44] and confirmed by our calculations, the radii with decreasing increase at low parameters much faster and they decrease at high parameters .To take into account this circumstance, we N max * ω ω used several nonuniform meshes in the parameter as in [27].The result corresponding to the minimum χ 2 value was taken as optimal.
This method was used to calculate the size characteristics of the 1 + ground and 0 + excited states of the 6 Li nucleus.The latter state is nucleon-stable because the decay into the α + d open channel is parity forbidden.The theoretical calculations show that the spatial distributions of neutrons and protons are almost completely symmetric: the difference is ~0.02-0.04fm.The radius of the 0 + excited state is insignificantly larger than the radius of the 1 + state.The size characteristics of the discussed states obtained in the extrapolation procedure, as well as the total binding energies, are summarized in Table 1.
Experimental data on the spatial characteristics of the studied states of the 6 Li nucleus are very scarce.The charge radius of the ground state is well measured, whereas data on the sizes of the 0 + excited state are absent.The various experimental data for the charge radius of the ground state of the 6 Li nucleus, as well as the radius of the proton distribution extracted from these data using Eq. ( 1) with the value 0.877 (7) fm, are summarized in Table 2. Agreement between these results is good.The radius calculated with the Daejeon16 potential is close to the corresponding experimental data.The experimental matter radius of the ground state of the nucleus presented in [48] is r m = (2.09± 0.02) fm.Taking into account that the ab initio approach used in this work reproduces well the charge radius of the nucleus 6 Li and the radius of the system of point protons, as well as the closeness of the sizes of the proton and neutron systems demonstrated above, this result is doubtful.This conclusion is additionally confirmed indirectly by the fact that the method used in this work gives the matter radius of the 6 He nucleus in good agreement with the experimental data [27].
The experimental data and calculation results reported above demonstrate that the sizes of the discussed states are large for such light nuclei.Since the core of this system is the very compact hard nucleus 4 He, these data can be considered as additional (indi-  rect, qualitative) indications of the neutron-proton halo in these states.
It is senseless to define the size of the neutron-proton halo as the difference between the neutron and proton radii, as in 6 He.Nevertheless, to quantitatively estimate the sizes of the halo in 6 Li states, we use the results obtained in [27], where the size parameters of the 6 He nucleus were calculated by the same method.Table 3 presents these parameters obtained using the extrapolation procedure.We note that the matter radii of all three discussed six-nucleon states are approximately the same.This is a reason to assume that the motion characteristics of the center of mass of the 4 He core and, the more so, its polarization by outer nucleons are very close for all systems under consideration.
The analysis of the main virtual fragmentation channels of these states: 6 Li(1 + ) → 4 He (F1), 6 Li(0 + ) → 4 He (F2), and 6 He(0 + ) → 4 He (F3) theoretically confirms this assumption.Here, the subscripts 1 and 2 indicate the ground and first excited states of the 4 He core, respectively, and and are the "singlet deuteron" and its two-neutron isobaric analog with the isospin , respectively, which are not bound in the free state but are localized in the field of the core nucleons.
For this analysis, we recall the main characteristics of the cluster channels.These are the projection of the wavefunction of the system on a channel, which is characterized by the masses of the fragments and , their relative orbital angular momentum l, channel spin S, the total angular momentum of the system J, the cluster form factor expressed in terms of the wavefunctions of the fragments as and the spectroscopic factor, i.e., the norm of the cluster form factor: The spectroscopic factors of all six listed channels, which are their statistical weights, are given in Table 4.This table demonstrates that the degree of polarization of the core by the outer nucleons, i.e., the ratio of the spectroscopic factors of the virtual decay into the first excited and ground states of the α particle, is approximately 1/5 for all three cases.
The main evidence for the close similarity of the properties of the core in all three systems is the closeness of the shapes of the functions characterizing the relative motion of the centers of masses of the fourand two-nucleon fragments.This is illustrated in Figs. 1 and 2. These functions for the channels with 4 He in both the ground and first excited states coincide with each other within a high accuracy.The contributions to the wavefunctions from numerous other channels with small spectroscopic factors are small; moreover, a similar behavior can also reasonably be expected for them.
Thus, the analysis of the presented data shows that the sizes of the neutron-proton halo can be estimated with a high accuracy under the assumption that the radius of the nucleon distribution in the core coincides with the radius of the proton subsystem in 6 He: . As a result, the size of the neutron-proton halo in the states of the 6 Li nucleus can be determined by the formula (9) and the size of the halo in 6 He can be redetermined similarly.These quantities are also presented in Table 3.All radii are close to each other, which reliably indicates the existence of the neutron-proton halo in the discussed states of the 6 Li nucleus.It is noteworthy that the difference between the sizes of the halo and the matter radii of the states, which is at first glance not too large, really indicates a drastic difference between the properties of nuclear matter in the region corresponding to the radius of the core and the peripheral region.The nucleon densities in these regions differ by approximately an order of magnitude.A more detailed quantitative comparison of the sizes of the halo in three studied states interestingly shows that the difference between the sizes of the two-neutron halo in 6 He and the neutron-proton halo in the state 1 + of the 6 Li nucleus is noticeably smaller than the difference between the sizes of 0 + isobaric analog states.This property is particularly surprising because the halo in the 1 + state of the 6 Li nucleus and the halos of two other discussed states belong to different types.The first halo is a "tango halo," i.e., a bound state of a three-boy system, where one of three pair subsystems in the free state is bound, whereas two other subsystems are not bound.Two 0 + states belong to a Borromean type; i.e., they do not have bound pair subsystems.We repeat that the 0 + state of the 6 Li nucleus with the isospin T = 1 is a bound state of the alpha particle and a neutron-proton pair, which is in a singlet state, and its decay into the channel is forbidden by the selection rules in the isospin and parity.
To conclude, we list the main results of this work.
(i) The size characteristics of nucleon-stable states of the 6 Li nucleus have been calculated ab initio with the Daejeon16 NN potential, which is the most successively used to describe the properties of light nuclei.The results have been analyzed in comparison with the respective characteristics of the ground state of the 6 He nucleus.

α + d
(ii) To improve the calculated results, the matter, proton, and neutron radii of the nuclei, which have been obtained within the no-core shell model with various oscillatory bases with different parameters and , have been extrapolated using our new two-dimensional procedure to the infinite functional basis.
(iii) The calculated size of the proton system in the ground state of the 6 Li nucleus is in agreement with experimental data.The calculated matter radius of this nucleus together with the result formulated in the preceding sentence makes the known measurements of this parameter doubtful.
(iv) A new definition for the quantitative measure of the nuclear halo has been proposed to characterize the halo formed by loosely bound neutron and proton.The sizes of the halo in nucleon-stable states of the 6 Li and 6 He nuclei have been calculated with this definition.
(v) It has been demonstrated that the sizes of the two-neutron halo in 6 He are close to those of the halo in the nucleon-stable states of the 6 Li nucleus.Thus, additional reliable evidence of the existence of the neutron-proton halo in the discussed states of the 6 Li nucleus has been obtained.It has been shown that the type of the halo slightly affects its sizes.He , and 4 He channels for the 6 He(0 + ), 6 Li(0 + ), and 6 Li(1 + ) states, respectively.  He , 4 He , and 4 He channels for the 6 He(0 + ), 6 Li(0 + ), and 6 Li(1 + ) states, respectively.

Table 1 .
Calculated total binding energies E tot and the matter (r m ), neutron (r n ), and proton (r p ) radii of the ground 1 + and excited 0 + states of the6Li nucleus

Table 2 .
[42]rimental charge (r ch ) and proton (r p ) radii of the ground state of the6Li nucleus.The proton radius is obtained with the radius of the proton from[42] p R

Table 3 .
Calculated matter (r m ), neutron (r n ), and proton (r p ) radii of the 0 + ground state of the6Не nucleus and the ground 1 + and excited 0 + states of the6Li nucleus, as well as the radius of the core r core used to calculate the radius of the halo r halo in the new definition

Table 4 .
Spectroscopic factors F1, F2, and F3 (see the main text) of the virtual decay channels of the6He and6Li states + 1