Enhanced E1 transition between weakly-bound excited states in the nucleus 27Ne

TL;DR

This paper models the enhanced E1 transition in 27Ne, explaining it through deformation and weakly-bound neutron effects, particularly the halo structure and shell-structure peculiarities.

Contribution

It provides a theoretical explanation for the strong E1 transition in 27Ne by incorporating deformation and halo neutron effects, advancing understanding of weakly-bound nuclear states.

Findings

E1 transition strength explained by prolately deformed states

Transition between halo components accounts for observed strength

Shell-structure influences weakly-bound neutron behavior

Abstract

Inspired by the recently-reported strong electric-dipole (E1) transition between the weakly-bound first and second excited states, 3/2- at 765 keV and 1/2+ at 885 keV, in the nucleus 27Ne, the E1 transition is estimated in a model by properly taking into account the effect of both deformation and weakly-bound neutrons. In addition to both the spin-parities, 1/2+ and 3/2-, and observed nearly degenerate energies of the two excited states, the observed order of magnitude of the E1 transition strength between the two states is very naturally explained in the case that these two excited states are prolately deformed, in terms of the transitions between the halo components of the wave functions of the weakly-bound odd-neutrons, s1/2 -> p3/2 and s1/2 -> p1/2, in addition to the large probability of the p3/2 component in the weakly-bound neutron [330 1/2] orbit. The large probability is the result of the shell-structure unique in weakly-bound or resonant neutrons.

Tables

Table 1. Table 1: The contributions ⟨ ℓ 1 ​ j 1 ​ Ω 1 π = 1 / 2 − | r ​ Y 10 | ℓ 2 ​ j 2 ​ Ω 2 π = 1 / 2 + ⟩ quantum-operator-product subscript ℓ 1 subscript 𝑗 1 superscript subscript Ω 1 𝜋 1 superscript 2 𝑟 subscript 𝑌 10 subscript ℓ 2 subscript 𝑗 2 superscript subscript Ω 2 𝜋 1 superscript 2 \langle\,\ell_{1}\,j_{1}\,\Omega_{1}^{\pi}=1/2^{-}\,|rY_{10}|\,\ell_{2}\,j_{2}\,\Omega_{2}^{\pi}=1/2^{+}\,\rangle in the unit of fm to the matrix element ⟨ [ 330 1 / 2 ] | r ​ Y 10 | [ 200 1 / 2 ] ⟩ quantum-operator-product delimited-[] 3301 2 𝑟 subscript 𝑌 10 delimited-[] 2001 2 \langle\,[330\,1/2]\,|rY_{10}|\,[200\,1/2]\,\rangle , using the wave functions shown in Figs. 2 and 3. See eq. ( 13 ).

		[330 1/2]
		$p_{1/2}$	$p_{3/2}$	$f_{5/2}$	$f_{7/2}$	$h_{9/2}$	$h_{11/2}$
[200 1/2]	$s_{1/2}$	0.165	0.608
	$d_{3/2}$	$-$0.105	$-$0.038	$-$0.066
	$d_{5/2}$		0.003	$-$0.000	$-$0.061
	$g_{7/2}$			$-$0.013	$-$0.002	$-$0.002
	$g_{9/2}$				$-$0.025	$-$0.000	$-$0.004

Table 2. Table 2: The contributions ⟨ ℓ 1 ​ j 1 ​ Ω 1 π = 1 / 2 − | r ​ Y 10 | ℓ 2 ​ j 2 ​ Ω 2 π = 1 / 2 + ⟩ quantum-operator-product subscript ℓ 1 subscript 𝑗 1 superscript subscript Ω 1 𝜋 1 superscript 2 𝑟 subscript 𝑌 10 subscript ℓ 2 subscript 𝑗 2 superscript subscript Ω 2 𝜋 1 superscript 2 \langle\,\ell_{1}\,j_{1}\,\Omega_{1}^{\pi}=1/2^{-}\,|rY_{10}|\,\ell_{2}\,j_{2}\,\Omega_{2}^{\pi}=1/2^{+}\,\rangle in the unit of fm to the matrix element ⟨ [ 330 1 / 2 ] | r ​ Y 10 | [ 200 1 / 2 ] ⟩ quantum-operator-product delimited-[] 3301 2 𝑟 subscript 𝑌 10 delimited-[] 2001 2 \langle\,[330\,1/2]\,|rY_{10}|\,[200\,1/2]\,\rangle , calculated by using the wave functions shown in Figs. 5 and 6. See eq. ( 13 ).

		[330 1/2]
		$p_{1/2}$	$p_{3/2}$	$f_{5/2}$	$f_{7/2}$	$h_{9/2}$	$h_{11/2}$
[200 1/2]	$s_{1/2}$	0.044	0.197
	$d_{3/2}$	$-$0.024	$-$0.007	$-$0.044
	$d_{5/2}$		0.021	$-$0.000	$-$0.082
	$g_{7/2}$			$-$0.011	$-$0.003	$-$0.002
	$g_{9/2}$				$-$0.043	$-$0.000	$-$0.009

Equations

$${\Psi }_{\Omega }(\vec{r})=\frac{1}{r}\sum _{\ell j}{R}_{\ell j\Omega }(r){\mathbf{Y}}_{\ell j\Omega }(\hat{r}),$$

$$H{\Psi }_{\Omega }={\epsilon }_{\Omega }{\Psi }_{\Omega }$$

$$V(r)=V_{WS}f(r),$$

$$V_{coupl}(\vec{r})=-\beta k(r)Y_{20}(\hat{r}),$$

$$V_{so}(r)=-V_{WS}v{\left(\frac{\Lambda }{2}\right)}^2\frac{1}{r}\frac{df(r)}{dr}(\vec{\sigma }\cdot \vec{\ell }),$$

$$f(r)=\frac{1}{1+exp\left(\frac{r-R}{a}\right)}$$

$$k(r)=R{V}_{WS}\frac{df(r)}{dr}.$$

$$\left(\frac{d^2}{d{r}^2}-\frac{\ell (\ell +1)}{r^2}+\frac{2m}{{\hslash }^2}({\epsilon }_{\Omega }-V(r)-V_{so}(r))\right)R_{\ell j\Omega }(r)=\frac{2m}{{\hslash }^2}\sum _{{\ell }^{{}^{\prime }}{j}^{{}^{\prime }}}⟨{\mathbf{Y}}_{\ell j\Omega }\mid V_{coupl}\mid {\mathbf{Y}}_{{\ell }^{{}^{\prime }}{j}^{{}^{\prime }}\Omega }⟩R_{{\ell }^{{}^{\prime }}{j}^{{}^{\prime }}\Omega }(r)$$

$$⟨{\mathbf{Y}}_{\ell j\Omega }\mid V_{coupl}\mid {\mathbf{Y}}_{{\ell }^{{}^{\prime }}{j}^{{}^{\prime }}\Omega }⟩$$

$$B(E1$$

$$⟨[3301/2]|r{Y}_{10}|[2001/2]⟩$$

$$⟨[3301/2]|r{Y}_{11}|[\widetilde{2001/2}]⟩$$

$$B(E1;I^{\pi}=K^{\pi}=1/2^{+}\rightarrow I^{\pi}=3/2^{-},K^{\pi}=1/2^{-})\,=\,[e^{n}_{eff}(E1)]^{2}\,(0.145)\quad\mbox{e${}^{2}$ fm${}^{2}$}$$

$$⟨[3301/2]|r{Y}_{10}|[2001/2]⟩=\sum _{{\ell }_1,j_1,{\ell }_2,j_2}⟨{\ell }_1{j}_1{\Omega }_1^{\pi }=1/2^{-}|r{Y}_{10}|{\ell }_2{j}_2{\Omega }_2^{\pi }=1/2^{+}⟩$$

$$⟨[3301/2]|r{Y}_{10}|[2001/2]⟩$$

$$⟨[3301/2]|r{Y}_{11}|[\widetilde{2001/2}]⟩$$

$$B(E1;I^{\pi}=K^{\pi}=1/2^{+}\rightarrow I^{\pi}=3/2^{-},K^{\pi}=1/2^{-})\,=\,[e^{n}_{eff}(E1)]^{2}\,(0.00082)\quad\mbox{e${}^{2}$ fm${}^{2}$}$$

Full text

I INTRODUCTION

The study of nuclei far from the $\beta$ stability line has shown many new interesting phenomena which are unexpected from our common sense about stable nuclei; among others, halo phenomena, the change of the shell structure, $\ell$-$s$ splittings and magic numbers. In the present work I study the unusually strong $E1$ transition between the two excited states of the nucleus ${}^{27}_{10}$Ne17, by properly taking into account the possible deformation and the weakly-bound odd neutron.

Though low-energy $E1$ transitions are often observed in $\gamma$ decays of nuclei, in both spherical and deformed nuclei their strength is usually orders of magnitude weaker than the Weisskopf unit. This is due to several reasons: (a) The isoscalar part of the dipole operator corresponds to the center of mass motion and thus does not contribute to $E1$ transitions; (b) Since the isovector part of the residual dipole-dipole interaction is repulsive, the major part of the unperturbed $E1$ strength lying originally in the low energy region is shifted to the region of higher energy, especially to the giant dipole resonance (GDR); (c) There is no appreciable amount of low-energy $E1$ strength in one-particle excitations of spherical stable nuclei due to the nuclear shell structure. If the nuclear potential is expressed by a harmonic oscillator potential, $E1$ transitions which connect the states with different parities do not occur between the one-particle orbits belonging to the same major shell. In realistic nuclear potentials the parities of one-particle orbits in a given major shell of lighter nuclei with 2$<$N$\leq$8, 8$<$N$\leq$20 and 20$<$N$\leq$28, are the same, while in the major shells of heavier nuclei 28$<$N$\leq$50, 50$<$N$\leq$82, and 82$<$N$\leq$126 …, so-called high-j orbits such as 1g9/2, 1h11/2, and 1i13/2 …, respectively, appear due to the large spin-orbit splitting. Those high-j orbits have the parity different from other (normal) orbits in respective major shells. Nevertheless, since the angular momentum of the high-j orbit differs by more than (or equal to) 2$\hbar$ from normal-parity orbits belonging to the same major shell, $E1$ transitions do not occur between the one-particle orbits belonging to a given major shell of the spherical nuclear potential. Essentially the same mechanism continues to hinder strong low-frequency $E1$ transitions in quadrupole-deformed nuclei. Namely, one can hardly find an energetically close-lying pair of one-particle levels, of which the asymptotic quantum-numbers [$N\,n_{z}\,\Lambda\,\Omega$] allow $E1$ transitions between them.

In contrast, in nuclei with weakly bound neutrons the origin of the hindrance (c) may be removed due to the change of the shell structure or the one-particle wave-functions, while the hindrance (b) may be drastically reduced due to the very weak coupling of weakly-bound neutrons with the well-bound core.

In Ref. CL18 the lower limit of 0.030 e2 fm2 for the $B(E1;\,1/2^{+}\rightarrow 3/2^{-})$ value in 27Ne was reported, which represented one of the strongest $E1$ strengths observed among the bound discrete states in this mass region. It was also reported that the $E1$ transition strength is at least 30 times larger than the one measured for the decay of the 3/2- state to the 3/2+ ground state. In the present paper I show an estimate of the $E1$ strength between the two excited states with weakly-bound neutrons based on the deformed Woods-Saxon potential, taking explicitly into account both the deformation and the feature of the wave functions of weakly-bound neutrons in the deformed finite potential. On the other hand, in Ref. CL18 the result of large-scale shell-model calculations was included, which could reproduce the observed order of magnitude of the $E1$ strength depending on effective interactions by using the harmonic oscillator basis in the calculation of radial integrals.

Based on the measured $B(E2;0^{+}_{gr}\rightarrow 2^{+}_{1})$ values obtained in the analysis of heavy-ion inelastic scattering experiments JG07 ; HI05 it is often stated that even-even nuclei next to 27Ne, 26Ne and 28Ne, are deformed. However, I note that the measured $E(2_{1}^{+})$ value systematically varies as 2.018, 1.304 and 0.792 MeV in 26Ne, 28Ne and 30Ne, respectively, and the measured energy ratio $E(4_{1}^{+})/E(2_{1}^{+})$ is 2.308 and 2.822 in 28Ne and 30Ne, respectively. These data suggest that the transition from spherical to deformed shape takes place in the isotope chain of 26Ne to 30Ne. Therefore, I expect that the nucleus 27Ne lies on the border of deformed and spherical nuclei. Then, it is likely that both deformed and spherical states appear in the region of low excitation energy of 27Ne. I suppose the ground state of 27Ne is basically spherical since the spin-parity of $3/2^{+}$ is naturally expected for the neutron number N=17 in spherical shape. On the other hand, the almost degenerate 1/2+ and 3/2- levels imply that these states are deformed. If I consider one-particle levels in the deformed potential this degeneracy is naturally understood, and as explained later, I can estimate the deformations of these states as $\beta\approx 0.4$ based on the deformed Woods-Saxon potential. This estimate reasonably coincides with the values estimated in Refs. JG07 ; HI05 . Therefore, in the present work I concentrate on the estimate of the $B(E1;\,1/2^{+}\rightarrow 3/2^{-})$ value assuming a prolate deformation for both excited states. I note that since the neutron separation energy $S_{n}$ of the ground state of 27Ne is 1.515 MeV, the $S_{n}$ values of the excited states 1/2+ and 3/2- are only 0.63 and 0.75 MeV, respectively. Then, in the estimate of $E1$ matrix-elements the radial part of relevant one-particle wave-functions has to be carefully prepared when the states have such small $S_{n}$ values.

In Sec. II the main points of the model used are briefly summarized, and the results of numerical calculations in the present case of 27Ne are given, while in Sec. III the $E1$ transition between well-bound neutron [200 1/2] and [3301/2] orbits is calculated by using the present model and is compared with the result in Sec. II. Conclusion and discussions are given in Sec. IV.

II MODEL AND NUMERICAL CALCULATIONS

One-particle energy spectra in the quadrupole-deformed potential are known to have a close relation to the observed low-lying energy spectra in odd-A deformed nuclei BM75 . In other words, the one-particle picture in deformed nuclei seems to work much better than that in spherical nuclei. This is certainly because in the phenomenological quadrupole-deformed potential the major part of the residual interaction in the spherical potential, quadrupole-quadrupole interaction, is already included in the one-body potential. Spectroscopic properties of low-energy levels of light deformed odd-N (or odd-Z) nuclei have been successfully analyzed based on deformed one-particle potential. For example, see the successful analysis of the spectroscopic properties of 25Mg and 25Al in Ref. BM75 taking the stable nucleus 24Mg as the deformed core. In Ref. IHS07 the data on the deformed nucleus 11Be were analyzed in terms of the model similar to the one in the present article. However, one may wonder whether it is appropriate to use one-body potential in the description of such a light nucleus as 11Be. In the present work I apply the deformed one-body potential to the nucleus 27Ne with weakly-bound neutrons, knowing the successful work on the well-bound neutrons in 25Mg BM75 .

In the case of 27Ne the ground state (3/2*+*) may be basically a spherical state, though in the 17th neutron wave-function the components other than $2d_{3/2}$ may not be negligible. (The spectroscopic factor with rather large experimental errors obtained from a d(26Ne, 27Ne)p reaction can be found, for example, in Ref. SMB12 .) In contrast, I take the model that both the first (3/2-) and second (1/2+) excited states as the deformed 26Ne core plus one neutron in the quadrupole-deformed potential. This is partly because the coexistence of spherical and deformed shape is expected in the region of low-excitation energy of 27Ne as described in Introduction and partly because in one-particle spectra of the prolately deformed potential in Fig. 1 it is so easily expected that in nuclei with N=17 the deformed $3/2^{-}$ and $1/2^{+}$ states, which are almost degenerate, are energetically close-lying to a spherical $3/2^{+}$ state.

As a spherical one-body potential I take the Woods-Saxon potential described in p.238-240 of Ref. BM69 . Then, I examine the shell structure of one-particle energy spectra in the axially-symmetric quadrupole-deformed potential which is constructed in a standard way. I write the single-particle wave-function in the body-fixed (intrinsic) coordinate system as

[TABLE]

which satisfies

[TABLE]

where $\Omega$ expresses the component of one-particle angular-momentum along the symmetry axis and is a good quantum number. The Hamiltonian $H$ in Eq. (2) consists of the kinetic energy part and the potential energy part. The axially symmetric quadrupole-deformed potential for neutrons consists of the following three parts.

[TABLE]

[TABLE]

[TABLE]

where $\Lambda$ is the reduced Compton wave-length of nucleon $\hbar$/$m_{r}c$,

[TABLE]

and

[TABLE]

The values of the parameters in the potential except the depth $V_{WS}$ are taken from Ref. BM69 . For example, the diffuseness $a$ = 0.67 fm and the radius $R$= $r_{0}A^{1/3}$ with $r_{0}$ = 1.27 fm. Taking the value $v$ = 32 in (5) gives the same spin-orbit potential in Ref. BM69 .

The coupled differential equations for the radial wave-functions are written as

[TABLE]

where

[TABLE]

The details of the model can be found, for example, in Ref. IH19 and the quoted papers therein. The important point in the present work is that the coupled differential equations are integrated in coordinate space with the correct asymptotic behavior of radial wave functions in respective $(\ell,j)$ channels for $r\rightarrow\infty$. In particular, the r-dependence of a given $(\ell,j)$ component thus obtained is generally different from that of eigenfunctions in the spherically symmetric potential due to the presence of the coupling term with the deformed field on the right-hand-side of Eq. (8), which is proportional to $\beta$ . Such a unique r-dependence of the solution of Eq. (8) is clearly seen, for example, in the d5/2 component of the [200 1/2] orbit shown in Fig. 5. Some detailed discussion related to this point is given in the last paragraph of Sec. III. In this way, the values of the radial matrix-element of $E1$ operator, $\langle r\rangle$, between weakly-bound one-particle wave-functions in the deformed potential are carefully calculated including the possible halo effect.

For axially symmetric quadrupole-deformed shape an infinite number of channels with ($\ell,j,\Omega$) are coupled for a given $\Omega$. In practice, the number of coupled channels or the maximum value of ($\ell,j$) pairs included for a given $\Omega$ is determined so that the resulting eigenvalues and wave functions do not change in a meaningful way by taking a larger number of coupled channels. In the present work s1/2, d3/2, d5/2, g7/2, and g9/2 are included in the calculation of the [200 1/2] orbit, while in the calculation of the [330 1/2] orbit p1/2, p3/2, f5/2, f7/2, h9/2, and h11/2 are included. It is noted that the ($\ell,j,\Omega$) component obtained by integrating the coupled differential equations (8) in the present model contains properly the contributions from ($n,\ell,j,\Omega$) orbits with various radial nodes $n$.

In Fig. 1 I show the calculated one-neutron energy eigenvalues in the deformed Woods-Saxon potential produced by the 26Ne core as a function of quadrupole deformation parameter $\beta$. All parameters of the potential are taken from those given by Ref. BM69 except the depth, $V_{WS}$ = $-$41.0 MeV, which was used so that the crossing energy of the levels [220 1/2] and [330 1/2] is around $\varepsilon_{\Omega}=-$650 keV. The value of 650 keV is chosen to be approximately equal to the observed binding energies of the neutron in the 1/2*+* and 3/2*-* states. The two levels cross at $\beta=0.44$ in the case of Fig. 1. If $V_{WS}$ = $-$43 MeV is used the two levels cross at $\beta$ = 0.42. In both cases the energies of two levels at the crossing point are always close to the energy of 1d3/2 at $\beta$ = 0. This is consistent with the assumption of the spherical ground state and deformed excited states, which I made in this study. The band-head states generated by the [200 1/2] and [330 1/2] configurations should be 1/2*+* and 3/2*-, respectively. The latter can be understood from the fact that the main components of the [330 1/2] state are $1f_{7/2}$ and $2p_{3/2}$, therefore, the decoupling parameter of the [330 1/2] band is between $-$4 (for f7/2*) and $-$2 (for p3/2).  Consequently, the angular momentum of the band-head state of the [330 1/2] configuration is not equal to $I=1/2$, but 3/2.

The formula for $B(E1)$ in the present case is written as (see eq. (4-91) in Ref. BM75 for the reduced matrix-elements)

[TABLE]

where $[\widetilde{200\,1/2}]$ expresses the time-reversed intrinsic configuration of [200 1/2]. By using one-particle wave functions of the [330 1/2] and [200 1/2] states calculated for $V_{WS}=-41.0$ MeV and $\beta$ = 0.44 I obtain

[TABLE]

where the smallness of the $rY_{11}$ matrix elements is the result of cancellation between major contributions. Using the values of (11) in (10), I obtain

[TABLE]

In Figs. 2 and 3 the components of the radial wave functions, $R_{\ell j\Omega}(r)$ of the [200 1/2] and [330 1/2] states, respectively, are shown as a function of radial coordinate. It is seen that due to the small binding energies the major part of the wave functions especially with smaller orbital angular-momentum lies outside the Woods-Saxon potential. Namely, the large halo phenomena, in particular in the orbits $s_{1/2}$ in Fig. 2 and $p_{3/2}$ and $p_{1/2}$ in Fig. 3, are recognized.

In all $\Omega^{\pi}$ = 1/2*+* orbits the probability of the $s_{1/2}$ component becomes unity in the limit of $\varepsilon_{\Omega}\rightarrow 0$ TM97 ; IH04 . However, at the binding energy of 622 keV in Fig. 2 I do not find the tendency that the probability of the s1/2 component in the [200 1/2] orbit is particularly large. The probability of the s1/2 component in Fig. 2 is 0.518 compared with 0.492 in the well-bound case of Fig. 5. The binding energy at which the probability of s1/2 component starts to steeply increase depends very much on individual orbits. See Fig. 3 of Ref. IH04 . In the case of the [200 1/2] level, which is the energetically highest $\Omega^{\pi}=1/2^{+}$ level for prolate shape that comes from the sd-shell, the steep increase of the s1/2 probability starts first for |\varepsilon_{\Omega}|\;\raisebox{-1.72218pt}{\tiny\stackrel{{\scriptstyle{\textstyle<}}}{{\sim}}}\;500 keV.

In Table I the respective contributions $\langle\ell_{1}\,j_{1}\,\Omega_{1}^{\pi}=1/2^{-}\,|rY_{10}|\,\ell_{2}\,j_{2}\,\Omega_{2}^{\pi}=1/2^{+}\rangle$ to the matrix element in Eq. (11)

[TABLE]

are shown. It is seen that the large positive contribution comes from the two transition matrix-elements between particular halo components, $s_{1/2}\rightarrow p_{1/2}$ and $s_{1/2}\rightarrow p_{3/2}$, while almost all other contributions that are much smaller have different (negative) sign and, thus, cancel partially the halo contribution. Halo components hardly polarize the well-bound core, while the latter (negative) contributions may be considerably affected by the polarization of the core. Consequently, if the dependence of the $e_{eff}^{n}(E1)$ value on one-particle wave-functions is taken into account, the value $|e_{eff}^{n}(E1)|$ = (Z/A)$e$ may be the first approximation for halo contributions, while other (negative) contributions may be associated with the $|e_{eff}^{n}(E1)|$ value which is considerably smaller than (Z/A)$e$. Then, the magnitude of the cancellation of the $E1$ matrix-element coming from the halo components by non-halo components may become smaller. However, since the detailed dependence of $e_{eff}^{n}$ on one-particle wave-functions lies outside the scope of the present paper, here I use $|e_{eff}^{n}(E1)|$ = (Z/A)$e$ for all contributions, as an estimate of the $B(E1)$ value. Then, I obtain $B(E1;I^{\pi}=K^{\pi}=1/2^{+}\rightarrow I^{\pi}=3/2^{-},K^{\pi}=1/2^{-})$ = 0.020 e2 fm2, which is the order of magnitude of the measured $B(E1)$ value. Since there are several adjustable parameters in the present model such as the radius and diffuseness parameters of the deformed Woods-Saxon potential and the dependence of $e_{eff}^{n}(E1)$ value on one-particle orbits, here I am satisfied with the agreement of the order of magnitude. However, one has to keep in mind that the value of $e_{eff}(E1)$ is the important parameter which really decides calculated $B(E1)$ values, though it is often carelessly chosen.

From Table I it is seen that, for example, the $f_{7/2}$ component has 40 percent probability in the one-particle wave-function of [330 1/2], nevertheless, it has only a negligible contribution to the present $E1$ transition matrix-element.

III E1 TRANSITION BETWEEN WELL-BOUND NEUTRON ORBITS, [200 1/2]

AND [330 1/2]

In order to understand the characteristic feature of the enhanced $E1$ transition between the weakly-bound neutron orbits, [200 1/2] and [330 1/2], which is obtained in the previous section, in this section the $E1$ matrix-element is calculated in the case that both orbits are well-bound, say, bound by about 8 MeV, using the same parameters of the Woods-Saxon potential except the depth.

For the depth $V_{WS}$ = $-$ 55.81 MeV the 1d3/2 orbit in the spherical potential ($\beta$=0) is bound by $-$7.99 MeV, which is roughly the energy of the least-bound neutron in $\beta$-stable nuclei. In Fig. 4 calculated neutron one-particle energies in the potential are plotted as a function of quadrupole deformation parameter $\beta$. It is seen that the [200 1/2] and [330 1/2] orbits cross with each other again at the energy which is almost equal to that of the 1d3/2 level at $\beta$ = 0. The deformation parameter at the level crossing is $\beta$ = 0.30 for this potential depth. However, I am interested in the $E1$ matrix element between the [200 1/2] and [330 1/2] orbits at $\beta$ = 0.44 that is the deformation, for which the enhanced $E1$ matrix element was obtained in Sec. II. For $\beta$ = 0.44 I obtain

[TABLE]

Then, using the values of (14) in (10), I obtain

[TABLE]

When I use $|e_{eff}^{n}(E1)|$ = (Z/A)$e$ for all contributions, I obtain $B(E1;I^{\pi}=K^{\pi}=1/2^{+}\rightarrow I^{\pi}=3/2^{-},K^{\pi}=1/2^{-})$ = 0.00011 e2 fm2, which is more than two orders of magnitude smaller than the $B(E1)$ value, 0.020 e2 fm2, obtained for weakly-bound neutrons in Sec. II.

In Table II the respective contributions $\langle\ell_{1}\,j_{1}\,\Omega_{1}^{\pi}=1/2^{-}\,|rY_{10}|\,\ell_{2}\,j_{2}\,\Omega_{2}^{\pi}=1/2^{+}\rangle$ to the matrix element in Eq. (14) are shown. It is seen that in Table II ; (i) the positive contributions from $s_{1/2}\rightarrow p_{3/2}$ and $s_{1/2}\rightarrow p_{1/2}$ to the matrix element of $rY_{10}$ are a factor of 3 to 4 smaller than the corresponding contributions in Table I and (ii) the cancellation of the positive contributions by other negative contributions is efficient. The result (i) comes partly from the halo effect of the radial wave functions of the weakly-bound s and p neutrons in the case of Table I and partly from the larger probability (0.512) of the p3/2 component in the weakly-bound [330 1/2] orbit in the case of Table I compared with the probability (0.299) of the p3/2 component in the well-bound [330 1/2] orbit in Table II. The much larger probability of the p3/2 component in the weakly-bound [330 1/2] orbit is mainly the result of the unique shell-structure that when both 2p3/2 and 1f7/2 neutron orbits are weakly bound or one-particle resonant the 2p3/2 level is almost degenerate with or slightly lower than the 1f7/2 level IH07 ; IH10 ; IH12 . In contrast, in the case of Fig. 4 the 1f7/2 level lies lower than the 2p3/2 level by 2.4 MeV, therefore, for moderately-deformed prolate shape f7/2 is the main component of the lowest $\Omega^{\pi}$ = 1/2*-* level in the fp-shell denoted by [330 1/2]. The strong cancellation (ii) is expected, since the $rY_{10}$ matrix element vanishes if the wave functions of the [200 1/2] and [330 1/2] orbits are exactly those specified by the respective asymptotic quantum numbers and since the wave functions of well-bound neutrons are more similar to those specified by asymptotic quantum numbers than those of weakly-bound neutrons.

From the analysis of numerical results which is described above and in the previous section it is understood that the enhanced $E1$ transition between weakly-bound excited states of 27Ne obtained in the present work comes from the halo behavior of s and p components of the wave functions of the weakly-bound neutron [200 1/2] and [330 1/2] orbits, in addition to the shell-structure unique in weakly-bound neutrons.

The r-dependence of the d5/2 component in Fig. 5 is qualitatively different from that of eigenfunctions of the spherically symmetric potential such as 1d5/2 and 2d5/2. In the following I consider the case of moderate deformation. If the coupling term on the right-hand-side of Eq. (8) is weak enough to be treated by perturbation in the ($\ell j\Omega$) channel, the r-dependence of $R_{\ell j\Omega}(r)$ is, roughly speaking, similar to that of the energetically closest-lying (namely, $\varepsilon_{\Omega}\approx\varepsilon_{n\ell j}$) eigenfunction with ($n,\ell,j$) of the spherically symmetric potential. This is usually the case that the $(\ell j\Omega)$ component has a considerable probability in the wave function of the one-particle orbit with $\varepsilon_{\Omega}$. In contrast, if the energy $\varepsilon_{\Omega}$ is far away from any $\varepsilon_{n\ell j}$ for a given ($\ell,j$) the probability of the ($\ell j\Omega$) component in the one-particle orbit with $\varepsilon_{\Omega}$ is usually small. Then, when the radial wave function $R_{\ell j\Omega}(r)$ is expanded in terms of eigenfunctions with ($n,\ell,j$) in the spherically symmetric potential, namely in terms of the solutions of Eq. (8) in the absence of the coupling term on the right-hand-side, the major components of $R_{\ell j\Omega}(r)$ will be generally those with ($n,\ell,j,\Omega$), of which the energies $\varepsilon(n\ell j)$ are closer to $\varepsilon_{\Omega}$. In the case of Fig. 5 the probability of $d_{5/2}$ component in the [200 1/2] orbit is small (0.039), and the energy of the [200 1/2] orbit, $\varepsilon$([200 1/2]), is between $\varepsilon$(1d5/2) and $\varepsilon$(2d5/2), but closer to $\varepsilon$(1d5/2). Consequently, the r-dependence of the d5/2 component in the [200 1/2] orbit shows something between that of 1d5/2 and 2d5/2 orbits and, consequently, has a node at r = 5.2 fm that is outside the nuclear radius. In the case that $\varepsilon$([200 1/2]) is closer to $\varepsilon$(2d${}_{5/2})$, the node moves towards the inside of nuclear radius, and the radial shape of the d5/2 component in the [200 1/2] orbit will become qualitatively similar to that of 2d5/2 orbit. Though the unique r-dependence of some solutions of Eq. (8) is interesting, it is not playing a crucial role in the enhanced E1 transition presently calculated. However, the one-particle wave functions, which have the r-dependence very different from that of any eigenfunctions of spherically symmetric potential, may produce, for example, cross sections of one-particle transfer reactions, which are unexpected from the traditional analysis of the reactions using one-particle wave functions with ($n,\ell,j$) quantum numbers IH74 .

IV CONCLUSION AND DISCUSSIONS

Interpreting that the first and second excited states of 27Ne, 3/2- at Ex=0.765 MeV and 1/2+ at Ex=0.885 MeV, are prolately deformed around $\beta\approx 0.4$, the order of magnitude of the observed enhanced $B(E1;1/2^{+}\rightarrow 3/2^{-})$ value is obtained by the present model. The large $B(E1)$ value comes from the halo behavior of the $s_{1/2}$ component in the [200 1/2] orbit as well as the $p_{1/2}$ and $p_{3/2}$ components in the [330 1/2] orbit, in addition to the large probability (0.512) of the p3/2 component in the [330 1/2] orbit which is even larger than the probability (0.399) of the f7/2 component. The large probability of the p3/2 component is originated from the near degeneracy of the 2p3/2 and 1f7/2 levels at $\beta$ = 0, which is the shell structure unique in the very-weakly bound or resonant one-neutron levels. In this respect, it should be noted that the $E1$ matrix-element between the [200 1/2] and [330 1/2] orbits is in fact vanishing if the wave functions of the two orbits are those exactly specified by the respective asymptotic quantum-numbers, [$N\,n_{z}\,\Lambda\,\Omega$], since the $rY_{1\mu}$ operator cannot change $n_{z}$ by 3.

I also note that the occurrence of almost degenerate low-lying states with different parity, 1/2*+* and 3/2-, is simply the result of deformation. Furthermore, as seen from Fig. 1, those two excited states are expected to be found very close to the basically spherical $3/2^{+}$ ground-state, in agreement with the observed spectra.

The spin-parity 3/2+ of the ground state of 27Ne is easily expected for an almost spherical nucleus with the neutron number N=17. On the other hand, since the shape coexistence is likely in the low-energy region of 27Ne, the lowest-lying levels of 3/2*-* from the [330 1/2] orbit and 1/2*+* from the [200 1/2] orbit are naturally expected in the region of low-excitation energy for prolate deformation, which is certainly favored by the proton number Z=10. It is eagerly hoped to obtain any experimental information on the shape of the 1/2+ state at Ex=0.885 MeV and the 3/2*-* state at Ex=0.765 MeV of 27Ne.

The interpretation that the ground state of 27Ne is basically spherical while the first and second excited states, 3/2*-* and 1/2*+* at 0.765 and 0.885 MeV, respectively, are deformed goes well with the experimental observation CL18 that the present $1/2^{+}\rightarrow 3/2^{-}$ $E1$ transition rate is much larger (at least 30 times) than that measured for the $3/2^{-}$ decay to the 3/2+ ground state.