Possible halo structure of ^62,72Ca by forbidden-state-free locally peaked Gaussians

The Hamiltonian of a core$+2n$ system consists of the $n$-core kinetic ($T$) and potential ($U$) and the $n$-$n$ potential ($v$) terms,

where $m_{N}$ and $A$ are the nucleon mass and the mass number of the core nucleus, respectively. We follow the cluster-orbital shell model approach [44], by reducing the three-body problem to a two-body problem using two independent $n$-core relative distance vectors, $\bm{r}_{1}$ and $\bm{r}_{2}$ with their respective momentum conjugates $\bm{p}_{1}$ and $\bm{p}_{2}$. The center-of-mass kinetic energy is subtracted and the corresponding kinetic energies are given by $T_{i}={\bm{p}_{i}^{2}}/{(2\mu)}$ with $\mu=m_{N}A/(A+1)$. Our choices for the $n$-core and $n$-$n$ potentials will be given later.

The total wave function with the angular momentum $J$ and its $z$ component $M$ is expanded in terms of $K$ basis functions $\Phi_{JM}(\alpha_{i})$ $(i=1,\dots,K)$:

where $\alpha_{i}$ denotes a set of variational parameters for the $i$th basis. These basis functions are not restricted to be orthogonal. A set of linear coefficients $\bm{c}=(c_{1},\dots,c_{K})^{t}$ is determined variationally by solving the generalized eigenvalue problem

where $\mathcal{H}$ and $\mathcal{B}$ are the Hamiltonian and overlap matrices with elements defined by

and

An optimal set of $\alpha_{i}$ is selected by the stochastic variational method (SVM) [25, 26]. Its efficiency was demonstrated by a number of examples. See, e.g., Refs. [28, 27, 45]. We increase the basis one by one by selecting the one that gives the lowest energy among randomly generated candidates until the energy convergence is met. This procedure greatly reduces the total number of basis $K$, which helps the description of nucleon systems around a heavy core, where the number of possible configurations is quite large.

The efficiency of a variational calculation strongly depends on a choice of basis functions. Some basic requirements for a “good” basis include the following in the present case: (i) Weakly bound neutron orbits should be well described, because the system may have large amplitude at and beyond the surface of the core nucleus. (ii) The evaluation of the matrix elements in Eqs. (4) and (5) should be easy and extendable to systems with more neutrons bound to the core. (iii) The removal of many forbidden states should be performed numerically easily and stably.

The sp wave function with total (orbital) angular momentum $j$ ($l$), with their corresponding angular and spin wave functions, $Y_{l}(\hat{\bm{r}})$ and $\chi_{1/2}$, is defined by

where the LPG function is given by

Here, $N_{kl}$ is the normalization constant

and $a$ is a parameter related to the width of the LPG function. We assume $k$ to be a non-negative integer for the sake of simplicity. The LPG with $k=0$ is nothing but the ordinary Gaussians. Most of basic matrix elements between the LPG bases can be obtained analytically [31]. Since the LPG reaches a maximum at $r=\sqrt{(2k+l)/a}$, a suitable combination of $a$ and $k$ can describe such wave packets that are centered at different positions and have different widths. Because of this flexibility, the LPG can describe even linear chain structure [31]. Obviously the tail of a weakly bound orbit can be described well by a superposition of the LPG. This flexibility is vital to describe the sp orbits around a heavy core nucleus.

The LPG basis satisfies the requirements (i) and (ii). Concerning the requirement (iii), we introduce a projection operator onto the forbidden ($F$) space,

where the sum extends over all the orbits occupied in the core nucleus. The forbidden states $\psi_{nljm}$ are defined by the bound-state solutions of the one-body Hamiltonian, $T+U$. As will be seen later, they are very well approximated by the harmonic-oscillator (HO) functions

with the principal quantum number ($n$) and an appropriately chosen oscillator parameter $\nu$. We use this approximation in what follows. The FFLPG sp orbit is defined by

Because $\psi_{n^{\prime}l}^{\nu}(r)$ is a combination of LPG’s, $\phi_{k^{\prime}l}^{\nu}(r)\ (k^{\prime}=0,\ldots,n^{\prime})$, $\left<\psi_{n^{\prime}l}^{\nu}\right.\left|\phi_{kl}^{a}\right>$ is readily obtained by using

which leads to easy determination of $\bar{\phi}_{kljm}^{a}$.

With the use of the sp basis defined above, we construct an antisymmetrized two-neutron basis

where $P_{12}$ exchanges the neutron labels 1 and 2, and $[j_{1}j_{2}]_{JM}$ denotes the tensor product. Here, the set of variational parameters of the $i$th basis $\alpha_{i}$ in Eq. (2) stands for $\alpha_{i}=(a_{1,i}k_{1,i}l_{1,i}j_{1,i},a_{2,i}k_{2,i}l_{2,i}j_{2,i})$. This two-neutron basis can easily be extended to core$+$few nucleon systems by successively coupling another nucleon one by one.

Note that the variational parameter $\alpha_{i}$ comprises eight variables: two continuous ones ($a_{1,i}$ and $a_{2,i}$) and six discrete ones. As will be shown later, both of short- and long-ranged LPG’s have to be superposed to properly describe the asymptotics of the FF halo wave functions and also continuum states with high angular momenta ($l_{1,i}$ and $l_{2,i}$) have to be included to reach convergence. Therefore, discretizing each component of $\alpha_{i}$ on certain grids would lead to enormous basis dimension even for the present three-body system. More crucial is that one has to take into account the possibility of producing two bound states with the same spin and parity. A way to overcome these difficult problems is to reduce the basis dimension by the SVM. An interested reader should refer to Chapter 4 of Ref. [26].

The above basis of Eq. (13) is completely FF. In contrast to this approach, a popular way of eliminating the forbidden components is to add a pseudo-potential to the Hamiltonian as in Ref. [24],

and to attempt at reaching a stable eigenvalue by taking $\lambda\to\infty$. In practice, $\lambda\approx 10^{4}$ MeV is taken [28]. This pseudo-potential method has the advantage of its simplicity. We compare both approaches in the next section.

We take the Woods-Saxon (WS) form for the $n$-⁶⁰Ca potential

where $f_{\rm WS}(r)=\left[1+\exp\left(r-R_{\rm WS}\right)/a_{\rm WS}\right]^{-1}$ with $R_{\rm WS}=r_{0}A^{1/3}$ and $r_{0}=1.27$ fm. We use two sets for the diffuseness parameter: One is a standard one, $a_{\rm WS}=0.67$ fm (set A), and the other is a larger one, $a_{\rm WS}=0.80$ fm (set B).

The ⁶⁰Ca core nucleus is assumed to have the $N=40$ closed configuration, that is, the occupied orbits include $0s_{1/2}$, $0p_{3/2}$, $0p_{1/2}$, $1s_{1/2}$, $0d_{5/2}$, $0d_{3/2}$, $1p_{3/2}$, $1p_{1/2}$, $0f_{7/2}$, $0f_{5/2}$. We generate these forbidden states by assuming the WS potential parameters of Eq. (2-182) in Ref. [46]. Those potential parameters are $V_{0}=40.00$ MeV and $V_{1}=17.60$ MeV. Note that the potential makes both of $0g_{9/2}$ and $2s_{1/2}$ orbits unbound. The HO parameter $\nu$ is determined so as to maximize the average of the squared overlaps, $\sum_{nljm\in{F}}\langle\psi_{nljm}|\psi^{\nu}_{nljm}\rangle^{2}/40$ (see Ref. [47]). The maximum average value is 0.988 for set A and the resulting $\nu$ value is 0.212 fm^-2, while in set B case they are 0.989 and 0.203 fm^-2. Approximating the forbidden states with the HO wave functions is quite reasonable.

There is no information about ⁶¹Ca. Even its stability is unknown. The valence neutron orbit of ⁶¹Ca belongs to the $2n+l=4$ shell comprising the $0g$, $1d$, and $2s$ orbits. The order of these sp orbits is under debate [42, 38, 39, 19, 43, 40, 41]. The standard shell-model filling with the spin-orbit interaction arranges the sp levels in the order of $0g_{9/2}$, $2s_{1/2}$, and $1d_{5/2}$ at $A\approx 60$ [46], while Ref. [37] predicted the inverted order of $2s_{1/2}$, $1d_{5/2}$, and $0g_{9/2}$. In the latter case, the halo structure would appear in the ground state of ⁶²Ca [18]. Following this inverted situation, we determine $V_{0}$ in Eq. (15) so as to set the $2s_{1/2}$ sp energy to be $-0.01$ MeV, resulting in $V_{0}=44.03$ MeV (set A) and 41.89 MeV (set B), respectively. The value turns out to be slightly stronger than the standard value $V_{0}\approx 40$ MeV [46].

We vary the spin-orbit strength $V_{1}$ in Eq. (15) to simulate different conditions for the ⁶¹Ca structure. Figure 1 plots the sp energies of the $0g_{9/2}$ and $1d_{5/2}$ orbits. Note that the standard value of $V_{1}$ for ⁶⁰Ca is $0.44V_{0}\approx 19$ MeV for set A [46], resulting in the sp levels of $0g_{9/2}$, $2s_{1/2}$, and $1d_{5/2}$ order. In Fig. 1, both sets exhibit similar sp energy dependence as a function of $V_{1}$, though the value of $V_{1}$ for set B tends to be stronger than that for set A to make $0g_{9/2}$ and $2s_{1/2}$ states degenerate due to more diffused nuclear surface. To fix the range of $V_{1}$, the following two extreme cases are considered:

Vanishing spin-orbit limit. In Ref. [37], the $1d_{5/2}$ and $0g_{9/2}$ sp energies of ⁶¹Ca are predicted to be 1.14 and 2.29 MeV. To realize this situation, we need to take a very small $V_{1}$ value, $V_{1}\approx 0$ for set A and $\approx 5$ MeV for set B, respectively. We call this choice of $V_{1}$ the vanishing spin-orbit (so) limit.

Degenerate $sg$ limit. The sp energies of the $2s_{1/2}$ and $0g_{9/2}$ orbits are degenerate at $V_{1}=11.40$ MeV for set A and at $V_{1}=21.09$ MeV for set B. This choice of $V_{1}$ is called the degenerate $sg$ limit. Despite both sp energies being the same, their respective root-mean-square (rms) radii are quite different: 36.0 and 5.04 fm for the $2s_{1/2}$ and $0g_{9/2}$ orbits of set A, and 36.1 and 5.28 fm for these orbits of set B. In both sets, the halo features are noticed in the $s$ states, while the rms radii shrink significantly for the $g$ states due to the $l=4$ centrifugal barrier.

We examine the energy spectrum of ⁶²Ca by varying $V_{1}$ between the vanishing and degenerate limits. As the outcomes of both sets A and B are qualitatively the same, we discuss the results obtained with set A, unless otherwise mentioned.

Before discussing the structure of ⁶²Ca, we evaluate the power of the FFLPG approach. We use the Minnesota (MN) potential [48] for $v_{12}$ of Eq. (1). The MN potential, a soft-core central potential, is designed to reasonably well reproduce the energies and sizes of $s$-shell nuclei [28]. Since the two neutrons should be antisymmetric in the spin-orbital space, it is expected that they gain the attraction mostly in the relative $s$-wave and spin-singlet state. The spin-orbit and tensor components of $v_{12}$ are therefore expected to play an insignificant role in the core+$n$+$n$ model. Refer to Refs. [49, 50, 51] for an example of demonstrating that the MN potential gives similar results as the realistic $n$-$n$ ones.

Let $L$ denote the relative orbital angular momentum between the two neutrons. The MN potential in the spin-singlet and even $L$ channel reads $200e^{-1.487r^{2}}-91.85e^{-0.405r^{2}}$ in MeV, where $r$ is the two-nucleon distance in fm. For the spin-triplet and odd $L$ channel it is given by $(200e^{-1.487r^{2}}-178e^{-0.639r^{2}})(u-1)$, where $u$ is a parameter and usually taken to be around 1. In what follows, $u$ is set to 1 and no interaction acts between the two neutrons in the spin-triplet and odd $L$ channel.

To demonstrate the power of the FFLPG approach, we focus on such a state that is dominated by the weakly bound $2s_{1/2}$ orbit because the approach is expected to have the advantage in describing nodal orbits properly. We use the set A potential with $V_{1}=0$ and include the LPG bases restricted to $l_{1}=l_{2}=0$. Figure 2 plots the energy as a function of $K$, for different choices of $k_{\rm max}$. The candidates for the basis states are generated randomly in the interval $[0.1,40]$ fm for $b=1/\sqrt{a}$ and in the interval $[0,k_{\rm max}]$ for $k$, and the best one is selected by the SVM algorithm. The results with the pseudo-potential method are also shown for comparison. The FFLPG approach significantly improves the energy convergence. We confirm that $k_{\rm max}=4$ truncation virtually gives the same curve as the one with $k_{\rm max}=3$. The FFLPG calculation leads to convergence within a few tens of basis functions, while the pseudo-potential method needs more than a hundred bases to reach convergence. As for the pseudo-potential method, it appears that the convergence is fastest when the ordinary Gaussians with $k_{\rm max}=0$ are used, and there is no significant advantage in using the LPG bases with $k_{\rm max}>0$. Probably this occurs because the $k=0$ bases have the largest overlap with the $0s_{1/2}$ and $1s_{1/2}$ forbidden states, with the orthogonality requirement being most efficiently met by a superposition of those ordinary Gaussians. However, the energy obtained with $K=150$ is $-0.0537$ for $k_{\rm max}=0$ and $-0.0533$ MeV for $k_{\rm max}=1$, respectively, which still misses the FFLPG energy of $-0.0539$ MeV. What is more serious in the pseudo-potential method is its numerical instability. Because the $k=0,1$ bases have in general large overlap with the forbidden states, the calculation becomes numerically unstable due to the large prefactor $\lambda$ in Eq. (14). It is very hard to extend the basis size without breaking the linear independence of the selected bases. With $k_{\rm max}=2,3$ this instability problem is recovered and we get the energy of $-0.0539$ MeV by the pseudo-potential method with $K=200$.

Here, we should make a comment on the role of $k$ in the LPG basis. As shown above, the FFLPG basis with $k_{\rm max}>0$ accelerates the energy convergence compared to the ordinary Gaussians with $k=0$. To understand the reason, we plot in Fig. 3 the $s$-wave radial functions with different $k$ values for some choices of $a$. The thick curves are for $\bar{\phi}^{a}_{k0}$ in Eq. (11), while the thin curves are for $\phi^{a}_{k0}$ that in general contains forbidden states. Note that the forbidden states are $\psi^{\nu}_{00}$ and $\psi^{\nu}_{10}$. In the case of $b=1/\sqrt{\nu}\equiv b_{0}$ the FFLPG function thus vanishes for $k=0$ and 1, whereas it has two nodes at short distances for $k=2$ and 3 because it is orthogonal to the forbidden states. With increasing $b$, the amplitude of the FFLPG function becomes smaller in the inner region and finally has no node for $k=3$ at $b=2.5b_{0}$ due to small overlap with the forbidden states. The FFLPG basis with $k=0$ has small amplitude beyond the nuclear surface. With increasing $k$, however, it has large amplitude beyond the nuclear surface and damped amplitude at short distances. This property meets the requirement needed for the neutron orbits around the core nucleus and offers the possibility of efficiently describing neutron orbits with large radial extension. The FFLPG basis is also advantageous to gain the energy from the two-neutron interaction because it can have large relative $s$-wave components around the nuclear surface.

The test example presented in the previous subsection confirms that the FFLPG offer much faster and numerically stabler results than the pseudo-potential method. In this subsection, we study the structure of ⁶²Ca in the ${}^{60}{\rm Ca}+n+n$ model using the FFLPG expansion. We take $k_{\rm max}=3$ and $l_{\rm max}=10$ and generate the Gaussian fall-off parameter $b=1/\sqrt{a}$ in the interval [0.1, 40] fm. All possible different combinations of $(l_{1},j_{1})$ and $(l_{2},j_{2})$ are taken into account. For example, the number of combinations are respectively 21, 55, and 69 for $J^{\pi}=0^{+}$, $2^{+}$, and $4^{+}$ states.

We exhibit in Fig. 4 the energies of the states with $J^{\pi}=0^{+},2^{+}$, and $4^{+}$ as a function of $K$. The spin-orbit strength is $V_{1}=11.40$ MeV, set A of the degenerate $sg$ limit. Note that the energy converges rapidly on a few hundred bases for all the states. This is because all the bases are made free from the forbidden states. The fourth digit of the energy does not change on $K=1500$ for these lowest states. The second bound states are found for the $0^{+}$, $2^{+}$, and $4^{+}$ states. We further increase the number of basis to lower the second bound states and confirm that they converge very well at $K=2000$.

The truncation of $l_{\rm max}=10$ appears considerably large. One might question why so large a value is needed. Most of the sp levels are unbound in the present case, that is, they are nonresonant continuum states. The large value of $l_{\rm max}$ therefore suggests the need to account of the continuum effect to bind the two neutrons. Table 1 lists the $l_{\rm max}$ dependence of the basic properties of the $0_{1}^{+}$ and $0_{2}^{+}$ states. As the table indicates clearly, $\epsilon_{2n}$ increases, while $\langle v_{12}\rangle$ gets more attractive as a function of $l_{\rm max}$. The $l_{\rm max}$ dependence of the two-neutron rms distance, $r_{12}$, shows an apparent correlation with $\langle v_{12}\rangle$. Although we confirm that the truncation with $l_{\rm max}=10$ takes into account most of the continuum effect, it appears that there may be still some room to improve the convergence by increasing $l_{\rm max}$ further.

In addition to the six states drawn in Fig. 4, we obtain two bound states with $6^{+}$ and $8^{+}$. Table 2 summarizes the energies, the rms neutron radii, and the occupation probabilities for those bound states. Note that the $0g_{9/2}$ and $2s_{1/2}$ sp states are set to be degenerate. The ground state exhibits nonhalo structure, occupying the $(g_{9/2})^{2}$ configuration by 94%. The configuration also produces the $2^{+}_{1},4^{+}_{1},6^{+}$, and $8^{+}$ bound states that have almost the same structure as the ground state. Halo structure is realized as the $0^{+}_{2}$ and $4^{+}_{2}$ states. Both states have the rms neutron radius, $r_{2n}$, larger than 10 fm. As shown by the occupation probability, the $0^{+}_{2}$ state may be called an $s$-wave two-neutron halo. It should be noted, however, that its $r_{2n}$ value of about 13 fm is by far smaller than the rms radius of the $2s_{1/2}$ sp orbit, which is about 36 fm. This dramatic reduction is of course due to the correlated motion of the two neutrons. The one-neutron halo character of the $4^{+}_{2}$ state is brought about by the coupling of the $g_{9/2}$ neutron with the $s_{1/2}$ neutron, as revealed by the occupation probabilities. This suggests that both states of two- and one-neutron halo structure can coexist within a narrow energy spacing. Despite the fact that it is barely bound, the $2^{+}_{2}$ state shows no characteristics of halo structure. Its $r_{2n}$ value is considerably large, 6.77 fm, but it is by far smaller than those of the $0^{+}_{2}$ and $4^{+}_{2}$ states. Note that the recoil kinetic energy is not negligible as the binding energy is small. This is because the main configuration of the $2^{+}_{2}$ state is $(g_{9/2}d_{5/2})$.

Table 2 also lists the decomposition of the energy $E$ into the sp energy, the recoil kinetic energy, the two-neutron interaction energy, and the rms two-neutron distance. The positive value of $\epsilon_{2n}$ indicates that the two neutrons move mostly in the continuum states. The recoil kinetic energies are small due to the factor $1/60m_{N}$. Clearly, the two-neutron interaction $v_{12}$ plays a decisive role to make the two neutrons bound. At a closer look, the $2_{2}^{+}$ state shows the largest sp energy and a relatively large energy gain from the two-neutron interaction energy. The state is realized by the coupling of the continuum states such as the $d_{5/2}$ and other high $l$ orbits due to the two-neutron correlation.

The $(g_{9/2})^{2}$ dominance in the ground state can be explained by the pairing antihalo effect pointed out in Ref. [16]. A combination of the higher angular momentum states is more advantageous to gain energy from the pairing. In fact, $\left<v_{12}\right>$ values are $-1.55$ and $-0.48$ MeV for $0^{+}_{1}$ and $0_{2}^{+}$, respectively. To make the ground state a halo, the energy difference between the $2s_{1/2}$ and $0g_{9/2}$ orbits should be sufficiently larger than the energy gain of the two-neutron interaction with the $(g_{9/2})^{2}$ configuration.

As mentioned above, the $4^{+}_{2}$ state is found to have one-neutron halo structure constructed dominantly from the $(s_{1/2}g_{9/2})$ configuration. That configuration would suggest a doublet state with $5^{+}$. However, its dominant configuration is in the spin-triplet and odd $L$ channel. Possible existence of this unnatural-parity state crucially depends on the choice of $u$ parameter. Since we set $u=1$, $v_{12}$ vanishes and no doublet state appears. To be more definitive about its existence, we have to test other realistic $n$-$n$ potentials.

As discussed above, when the spin-orbit strength $V_{1}$ of the $n$-⁶⁰Ca potential is taken to be the degenerate $sg$ limit, the $J^{\pi}=0^{+}$ halo structure appears as the excited state. It is interesting to examine how the $0^{+}$ state changes if the spin-orbit strength is weakened towards the vanishing so limit. We introduce a multiplicative factor $f$ and set the spin-orbit strength as $V_{1}=fV_{1}^{\prime}\,(0\leq f\leq 1)$ with $V_{1}^{\prime}=11.40$ MeV for set A. The limit of $f\approx 0$ is the case used in Ref. [37], where the energy difference between the $2s_{1/2}$ and $0g_{9/2}$ orbits is large (see Fig. 1) and only one bound state of two-neutron halo structure is predicted.

Figure 5 (a) displays the energy of the $0^{+}$ state as a function of $f$. Only one $0^{+}$ state appears for $f<0.7$ and the existence of the second $0^{+}$ state is possible for $f\geq 0.7$. Figure 5 (b) shows the corresponding rms neutron radii. The halo structure having a radius larger than 10 fm emerges in the ground state for $f<0.7$. For $f>0.7$ the $0_{2}^{+}$ state exhibits the two-neutron halo structure, whereas the ground state turns out to be a compact state. At $f=0.7$ the ground state shows intermediate structure between halo and compact states. The rms radius of the $0_{2}^{+}$ state is extremely large due to the small binding energy of $-$0.03 MeV. This behavior can be understood by showing the occupation probabilities. Figure 5 (c) shows the occupation probabilities of finding $(s_{1/2})^{2}$ and $(g_{9/2})^{2}$ components in the $0^{+}$ states. The contributions of $(d_{5/2})^{2}$ component for the $0^{+}_{1}$ state are at most 0.09 at $f=0.7$ and other contributions are less than 0.05 in total. For the $0_{2}^{+}$ state, those contributions are less about 0.01. As expected, the $(s_{1/2})^{2}$ component dominates in the ground state at $f=0$, resulting in a large rms radius of 12 fm, almost the same structure as the $0_{2}^{+}$ state at $f=1$. By increasing $f$, i.e., reducing the energy difference between the $2s_{1/2}$ and $0g_{9/2}$ orbits, the occupation probability of the $0g_{9/2}$ orbit increases gradually and rises suddenly for $f>0.6$. The rms radius decreases simultaneously with the growing occupation of the $0g_{9/2}$ orbit that has a much smaller radius. An almost equal mixing of the $(s_{1/2})^{2}$ and $(g_{9/2})^{2}$ configurations occurs at $f=0.7$. Here the energy difference between the $2s_{1/2}$ and $0g_{9/2}$ orbit is approximately 0.5 MeV, which is comparable to the difference of $\left<v_{12}\right>/2$ for the two $0^{+}$ states as shown in Table 2. Finally, the ground state becomes $g_{9/2}$-dominant at $f=1$, where the sp energies of the $2s_{1/2}$ and $0g_{9/2}$ orbits are degenerate, while the $0_{2}^{+}$ state exhibits two-neutron halo structure consisting of the $(s_{1/2})^{2}$ configuration.

An extension to the ${}^{70}{\rm Ca}+n+n$ model is straightforward. The reader is referred to Ref. [19] for the structure of ⁷²Ca by the hyperspherical method. We follow the case of ⁶²Ca starting from discussing some constraints on the phenomenological WS potential parameters.

The $n$-${}^{70}{\rm Ca}$ potential should bind the $0g_{9/2}$ orbit due to the assumption of the $N=50$ core. However, no bound $0g_{9/2}$ orbit is generated if we assume the parametrization of Ref. [46]. Instead of searching for such a potential that binds the $0g_{9/2}$ orbit, we simply assume that all the occupied neutron orbits including the $0g_{9/2}$ orbit are described by the HO functions with the oscillator parameter $\nu^{\prime}$ that is scaled from the ⁶⁰Ca parameter $\nu$ by $\nu^{\prime}=(60/70)^{1/3}\nu$. The sp energy spectrum of ⁷¹Ca with respect to $V_{1}$ is virtually the same as Fig. 1 except for the absence of the $0g_{9/2}$ level.

To bind the $2s_{1/2}$ orbit at $-0.01$ MeV, the strength $V_{0}$ turns out to be 40.26 MeV for set A and 38.32 MeV for set B. Since $V_{0}$ gets weaker than the ⁶⁰Ca-$n$ case, we have only one bound $0^{+}$ state that has two-neutron halo structure if $V_{1}$ is taken to be the same as that of ⁶¹Ca: The energy, rms neutron radius, and $(s_{1/2})^{2}$ and $(d_{5/2})^{2}$ occupation probabilities are $-0.14$ MeV, 12.7 fm, and 0.89 and 0.08 for set A, whereas for set B they are $-0.13$ MeV, 13.3 fm, and 0.87 and 0.11, respectively. The $n$-⁷⁰Ca potential used here makes the energy gap between the $2s_{1/2}$ and $1d_{5/2}$ orbits too large to mix those sp states. To make the $1d_{5/2}$ orbit bound is very unlikely because $V_{1}$ has to be taken more than two times larger than the standard value. Within the present phenomenological $n$-${}^{70}{\rm Ca}$ potential, the halo structure emerges if the energy of the $2s_{1/2}$ state is close to zero, which is the same conclusion drawn in Ref. [19].