11institutetext: Institut für Physik, Universität Rostock, D-18051 Rostock,
Germany 22institutetext: School of Physics, Nanjing University, Nanjing
210093, China 33institutetext: Key Laboratory of Nuclear Physics and Ion-Beam
Application (MoE), Institute of Modern Physics, Fudan University, 200433,
Shanghai, China 44institutetext: Shanghai Research Center for Theoretical
Nuclear Physics, NSFC and Fudan University, 200438, Shanghai, China
55institutetext: School of Physics Science and Engineering, Tongji University,
Shanghai 200092, China 66institutetext: Laboratory of Physics, Kanto Gakuin
University, Yokohama 236-8501, Japan 77institutetext: Research Center for
Nuclear Physics (RCNP), Osaka University, Osaka 567-0047, Japan
88institutetext: College of Science, Nanjing University of Aeronautics and
Astronautics, Nanjing, China

# Alpha-like correlations in 20Ne, comparison of quartetting wave function and
THSR approaches

G. Röpke 11 C. Xu 22 B. Zhou 3344 Z. Z. Ren 55 Y. Funaki 66 H. Horiuchi 77 M.
Lyu 88 A. Tohsaki 77 T. Yamada 66

(Received: date / Revised version: date)

###### Abstract

20Ne can be considered as a double-magic 16O core nucleus surrounded by four
nucleons, the constituents of an $\alpha$-like quartet. Similar to other
nuclei (212Po, 104Ti, etc.) with a quartet on top of a double-magic core
nucleus, significant $\alpha$-like correlations are expected. Correlations in
the ground state of 20Ne are investigated using different approaches. The
quartetting wave function approach (QWFA) predicts a large $\alpha$-like
cluster contribution near the surface of the nuclei. The Tohsaki-Horiuchi-
Schuck-Röpke (THSR) approach describes $\alpha$-like clustering in nuclear
systems. The results of the QWFA in the Thomas-Fermi and shell-model
approximation are compared with THSR calculations for the container model.
Results for the $\alpha$ formation probability and the rms radii are shown.

###### pacs:

PACS-keydiscribing text of that key and PACS-keydiscribing text of that key

##  1 Introduction

The liquid-drop model of nuclei, which can be considered as a simple version
of a local density approach, reflects important properties of nuclear
structure, for example the famous Bethe-Weizsäcker mass formula. Other
properties such as the occurrence of magic numbers are explained by the shell
model, which considers nucleonic quasiparticle states, and many properties of
nuclei, including pairing, are studied in this approach, see e.g. the
fundamental book by Ring and Schuck RingSchuck . However, the description of
correlations, in particular of $\alpha$-like clusters in nuclei, requires
going beyond the quasiparticle approach.

The nucleus 212Po shows a significant $\alpha$-like correlation in the skin
region Po14 ; Xu16 ; Xu17 . It can be assumed that it consists of a double-
magic, relatively stable 208Pb core nucleus surrounded by an $\alpha$-like
cluster. This $\alpha$-like quartet experiences a potential pocket for the
center-of-mass (c.m.) motion outside a critical radius $r_{\rm cr}$ where it
can exist as a quasi-bound state. Its intrinsic structure is dissolved at
smaller distances when the nucleon density of the core nucleus exceeds a
critical value $n_{\rm cr}=0.03$ fm-3. The reason for this is the Pauli
principle, which applies to the nucleons that form the $\alpha$ particle.
Their mutual interaction is blocked in the dense medium of the nucleons of the
core nucleus that occupy the Fermi sphere in momentum space. This is a
consequence of the antisymmetrization of the full many-fermion wave function
of the entire nucleonic system. The $\alpha$ particle, which is preformed in a
near-surface pocket, can escape from the 212Po nucleus by tunneling. The
calculations were performed using the quartet wave function approach (QWFA).
It was found that the calculated $\alpha$ decay half-life agrees well with the
observed data Xu16 ; Xu17 .

A similar behavior is expected for other nuclei consisting of a double magic
core nucleus and an additional $\alpha$ cluster. Calculations were performed
for 104Te Yang20 . The observed half-life of $\alpha$ decay was successfully
reproduced in QWFA for 104Te as well as for additional $\alpha$-decaying
nuclei Yang21 . Improvements of the quartet model have been made in Refs.
Bai19 ; Wang22 , see also Jin23 ; Li23 . Using QWFA, the influence of
$\alpha$-like clustering in nuclei on the nuclear symmetry energy was analyzed
in Ref. Yang23 .

Another nucleus, consisting of a double-magic core nucleus surrounded by an
$\alpha$-like cluster, is 20Ne. In this work, we present calculations within
QWFA and compare them with other approaches. A main result is the preformation
fraction of $\alpha$-like clusters and the point rms radius which are
determined by the wave function including correlations. We compare the Thomas-
Fermi approximation with shell model calculations.

A consistent description of quartetting ($\alpha$-like correlations) has
recently been developed in the framework of the Tohsaki-Horiuchi-Schuck-Röpke
(THSR) approach THSR . This approach provides an excellent description of low-
density 4$n$ nuclei such as 8Be, the Hoyle state of 12C and excited states of
16O, but has also been applied to more complex nuclei such as 20Ne Bo12 ; Bo13
; Bo14  as well as 4$n$ nuclei with additional nucleons Lyu ; Lyu1 . Recently,
calculations for 20Ne were performed by Bo et al. Bo23  using the two-
parameter container model. A review on microscopic clustering in light nuclei
was presented in Ref. Freer18 .

In this work, we also compare the two approaches. Heavy nuclei with a large
number of nucleons like 212Po are not yet computable with the THSR approach.
The QWFA provides better results for heavier nuclei where a mean-field
approach for the core nucleus is more justified. For 20Ne, both approaches are
feasible. The comparison of the results for the quartetting wave function
approach and THSR calculations allows a better understanding of the
description of correlations in nuclear systems.

We study the c.m. motion of a quartet
$\\{n_{\uparrow},n_{\downarrow},p_{\uparrow},p_{\downarrow}\\}$ as a new
collective degree of freedom and compare the wave functions for both
approaches, the QWFA and the THSR approach. Instead of an uncorrelated Fermi
gas model for the cluster environment, an improvement of the quartet wave
function approach is investigated to achieve a consistent description of
cluster formation in a clustered medium.

After a brief explanation of the QWFA in Sec. 2, we carry out calculations
using the Thomas-Fermi approach in Sec. 3. Calculations with the shell model
are shown in Sec. 4. Comparisons with the THSR approach are discussed in Sec.
5, and concluding remarks are made in Sec. 6.

##  2 The quartet wave equation

The treatment of the interacting many-nucleon system requires some
approximations which can be obtained in a consistent way from a Green’s
function approach. The quartetting wave function approach Po14 ; wir
considers the wave function $\Psi({\bf r}_{1}{\bf r}_{2}{\bf r}_{3}{\bf
r}_{4})$ of the quartet (spin and isospin variables are dropped) which obeys
the in-medium wave equation

$\displaystyle[E_{4}\\!-\\!\hat{h}_{1}\\!-\\!\hat{h}_{2}\\!-\\!\hat{h}_{3}\\!-\\!\hat{h}_{4}]\Psi({\bf
r}_{1},{\bf r}_{2},{\bf r}_{3},{\bf r}_{4})$
$\displaystyle=\int\\!\\!d^{3}{\bf r}_{1}^{\prime}\,d^{3}{\bf
r}_{2}^{\prime}\langle{\bf r}_{1}{\bf
r}_{2}|\hat{B}(1,2)\,\,\hat{V}_{N-N}|{\bf r}_{1}^{\prime}{\bf
r}_{2}^{\prime}\rangle\Psi({\bf r}_{1}^{\prime},{\bf r}_{2}^{\prime},{\bf
r}_{3},{\bf r}_{4})$ $\displaystyle+\int d^{3}{\bf
r}_{1}^{\prime}\,\,d^{3}{\bf r}_{3}^{\prime}\langle{\bf r}_{1}{\bf
r}_{3}|\hat{B}(1,3)\,\,\hat{V}_{N-N}|{\bf r}_{1}^{\prime}{\bf
r}_{3}^{\prime}\rangle\Psi({\bf r}_{1}^{\prime},{\bf r}_{2},{\bf
r}_{3}^{\prime},{\bf r}_{4})$ $\displaystyle+{\rm
four\,\,further\,\,permutations,}$ (1)

with the single-quasiparticle Hamiltonian (single-nucleon shell states
$|n,\nu\rangle$)

$\hat{h}_{i}=\frac{\hbar^{2}\hat{p}_{i}^{2}}{2m}+[1-\hat{f}_{\nu_{i}}]\,V_{\nu_{i}}^{\rm
mf}(\hat{r}),\qquad\hat{f}_{\nu}=\sum_{n}^{{\rm occ.}}|n,\nu\rangle\langle
n,\nu|$ (2)

denotes the phase space which, according to the Pauli principle, cannot be
used for an interaction process of a nucleon with an intrinsic quantum state
$\nu=\sigma,\,\tau$. In addition to the nucleon-nucleon potential
$\hat{V}_{N-N}$, the nucleon-nucleon interaction terms also contain the
blocking operator $\hat{B}(1,2)=[1-\hat{f}_{1}-\hat{f}_{2}]$ for the first
term on the r.h.s. of Eq. (1), and corresponding expressions for the other 5
terms. The mean-field potential $V_{\nu_{i}}^{\rm mf}(\hat{r})$ contains the
strong core-nucleon interaction $V^{\rm ext}(r)$ as well as the Coulomb
potential of the core nucleus. It is considered as an external potential. The
Pauli blocking terms, which are given by the occupation numbers
$\hat{f}_{\nu}$, are not easy to treat as will be explained below. The mean-
field approach treats the motion within the cluster independent of the motion
in the surrounding medium, and neglects any correlations between the two. If
such further correlations exist, clusters with a larger number of nucleons are
formed. This concept is known from the shell model at the one-particle level,
for pairing at the two-particle level. We first discuss here the motion of
four nucleons under the influence of an external potential.

A main aspect of the cluster approach is the introduction of the center-of-
mass (c.m.) motion $\bf R$ as new collective degree of freedom, and ${\bf
s}_{j}=\\{\bf S,s,s^{\prime}\\}$ for the intrinsic motion (Jacobi-Moshinsky
coordinates). As shown in Po14 , the normalized quartet wave function
$\Phi({\bf R},{\bf s}_{j})$,

$\int d^{3}R\,\int d^{9}s_{j}\,|\Phi({\bf R},{\bf s}_{j})|^{2}=1,$ (3)

can be decomposed in a unique way (up to a phase factor),

$\Phi({\bf R},{\bf s}_{j})=\varphi^{{\rm intr}}({\bf s}_{j},{\bf
R})\,\Psi^{\rm c.m.}({\bf R})$ (4)

with the individual normalizations

$\int d^{3}R\,|\Psi^{\rm c.m.}({\bf R})|^{2}=1\,,{\rm and}\int
d^{9}s_{j}|\varphi^{{\rm intr}}({\bf s}_{j},{\bf R})|^{2}=1$ (5)

for arbitrary ${\bf R}$.

The Hamiltonian of a four-nucleon cluster can be written as

$\displaystyle H$ $\displaystyle=$
$\displaystyle\left(-\frac{\hbar^{2}}{8m}\nabla_{R}^{2}+T[\nabla_{s_{j}}]\right)\delta^{3}({\bf
R}-{\bf R}^{\prime})\delta^{3}({\bf s}_{j}-{\bf s}^{\prime}_{j})$ (6)
$\displaystyle+V({\bf R},{\bf s}_{j};{\bf R}^{\prime},{\bf s}^{\prime}_{j})$

with the kinetic energy of the c.m. motion of the cluster, and the kinetic
energy $T[\nabla_{s_{j}}]$ of the internal motion. The interaction $V({\bf
R},{\bf s}_{j};{\bf R}^{\prime},{\bf s}^{\prime}_{j})$ contains the mutual
interaction $V_{ij}({\bf r}_{i},{\bf r}_{j},{\bf r}^{\prime}_{i},{\bf
r}^{\prime}_{j})$ between the quartet particles as well as the interaction
with an external potential (for instance, the mean-field potential of the core
nucleus), with strict fulfillment of the Pauli principle.

For the c.m. motion we have the wave equation

$\displaystyle-\frac{\hbar^{2}}{8m}\nabla_{R}^{2}\Psi^{\rm c.m.}({\bf
R})-\frac{\hbar^{2}}{4m}\int d^{9}s_{j}$ $\displaystyle\times\varphi^{{\rm
intr},*}({\bf s}_{j},{\bf R})[\nabla_{R}\varphi^{{\rm intr}}({\bf s}_{j},{\bf
R})][\nabla_{R}\Psi^{\rm c.m.}({\bf R})]-$
$\displaystyle-\frac{\hbar^{2}}{8m}\int\\!\\!d^{9}s_{j}\varphi^{{\rm
intr},*}({\bf s}_{j},{\bf R})[\nabla_{R}^{2}\varphi^{{\rm intr}}({\bf
s}_{j},{\bf R})]\Psi^{\rm c.m.}({\bf R})$
$\displaystyle+\\!\\!\int\\!\\!d^{3}R^{\prime}\,W({\bf R},{\bf
R}^{\prime}),\Psi^{\rm c.m.}({\bf R}^{\prime})\\!=\\!E\,\Psi^{\rm c.m.}({\bf
R}),$

with the c.m. potential

$\displaystyle W({\bf R},{\bf R}^{\prime})=\int
d^{9}s_{j}\,d^{9}s^{\prime}_{j}\,\varphi^{{\rm intr},*}({\bf s}_{j},{\bf
R})\left[T[\nabla_{s_{j}}]\right.$ (7)
$\displaystyle\left.\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\times\delta^{3}({\bf
R}-{\bf R}^{\prime})\delta^{9}({\bf s}_{j}-{\bf s}^{\prime}_{j})+V({\bf
R},{\bf s}_{j};{\bf R}^{\prime},{\bf s}^{\prime}_{j})\right]\varphi^{{\rm
intr}}({\bf s}^{\prime}_{j},{\bf R}^{\prime}).$

For the intrinsic motion we find the wave equation

$\displaystyle-\frac{\hbar^{2}}{4m}\Psi^{\rm c.m.*}({\bf
R})[\nabla_{R}\Psi^{\rm c.m.}({\bf R})][\nabla_{R}\varphi^{{\rm intr}}({\bf
s}_{j},{\bf R})]$ $\displaystyle-\frac{\hbar^{2}}{8m}|\Psi^{\rm c.m.}({\bf
R})|^{2}\nabla_{R}^{2}\varphi^{{\rm intr}}({\bf s}_{j},{\bf R})$
$\displaystyle+\int d^{3}R^{\prime}\,d^{9}s^{\prime}_{j}\,\Psi^{\rm
c.m.*}({\bf R})\left[T[\nabla_{s_{j}}]\delta^{3}({\bf R}-{\bf
R}^{\prime})\delta^{9}({\bf s}_{j}-{\bf s}^{\prime}_{j})\right.$
$\displaystyle\left.+V({\bf R},{\bf s}_{j};{\bf R}^{\prime},{\bf
s}^{\prime}_{j})\right]\Psi^{\rm c.m.}({\bf R}^{\prime})\varphi^{{\rm
intr}}({\bf s}^{\prime}_{j},{\bf R}^{\prime})$ $\displaystyle=F({\bf
R})\varphi^{{\rm intr}}({\bf s}_{j},{\bf R})\,.$ (8)

Both the c.m. and intrinsic Schrödinger equations, Eqs. (2) and (2),
respectively, are coupled by contributions containing the expression
$\nabla_{R}\varphi^{{\rm intr}}({\bf s}_{j},{\bf R})$. This expression
vanishes in homogeneous matter, and we recover the in-medium Schrödinger
equation for $\alpha$ clusters in matter without external potential. Then, the
eigenvalue $F({\bf R})$ of Eq. (2) represents the bound state energy of the
$\alpha$ particle which is shifted in dense matter because of Pauli blocking.

The contribution of the gradient terms was recently investigated by Yang et
al. Yang23 . It can be shown that the second term of Eq. (2) vanishes. In the
present work, we neglect the contributions of the gradient terms. This
corresponds to a local density approximation, as is often used in many-body
theories.

##  3 Quartets in nuclei in Thomas-Fermi approximation

###  3.1 Mean field for the c.m. motion

We would like to emphasize that in general non-local interactions are
possible. In particular, the Pauli blocking considered in the following is
non-local. To simplify the calculations, local approximations are often used,

$\displaystyle W({\bf R},{\bf R}^{\prime})\approx W({\bf R})\delta^{3}({\bf
R}-{\bf R}^{\prime}),$ $\displaystyle W({\bf R})=W^{\rm ext}({\bf R})+W^{\rm
intr}({\bf R}).$ (9)

$W^{\rm ext}({\bf R})=W^{\rm mf}({\bf R})$ is the contribution of external
potentials, here the mean field of the core nucleons. The interaction within
the cluster according Eq. (2) gives the contribution $W^{\rm intr}({\bf R})$.
We give a short description, for details see Refs. wir ; R17 ; Ro18 .

If we know the nucleon densities of the core nucleus, the mean fields can be
easily calculated. The mean-field contribution $W^{\rm mf}({\bf R})$ is
obtained by double folding the density distribution of the core nucleus and
the intrinsic density distribution of the quartet at c.m. position $\bf R$
with the interaction potential of the nucleons. An $\alpha$-like Gaussian
density distribution was assumed for the bound quartet.

For the Coulomb interaction we calculate the double-folding potential

$V^{\rm Coul}_{\alpha-{\rm O}}(R)=\int d^{3}r_{1}\int d^{3}r_{2}\rho_{{\rm
O}}({\bf r}_{1})\rho_{\alpha}({\bf r}_{2})\frac{e^{2}}{|{\bf R}-{\bf
r}_{1}+{\bf r}_{2}|}\,.$ (10)

The charge density of the $\alpha$ nucleus according to

$\rho_{\alpha}(r)=0.2114\,\,{\rm fm}^{-3}\,e^{-0.7024\,\,r^{2}/{\rm fm}^{2}}$
(11)

reproduces the measured rms point radius 1.45 fm. For the density distribution
of 16O, the expression Qu2011

$n^{\rm WS}_{B,{\rm O}}(r)=\frac{0.168\,{\rm fm}^{-3}}{1+e^{(r/{\rm
fm}-2.6)/0.45}}$ (12)

was given which reproduces the rms point radius 2.6201 fm, or Gaussians wir .
The convolution integral (10) is easily evaluated in Fourier representation
and gives for the parameter values considered here wir

$\displaystyle V^{\rm Coul}_{\alpha-{\rm O}}(R)=\frac{16\times 1.44}{R}\,{\rm
MeV\,\,fm}$ $\displaystyle\times\left[{\rm Erf}(0.7683\,\,R/{\rm
fm})-0.9097\,\,(R/{\rm fm})\,\,e^{-0.2274\,\,R^{2}/{\rm fm}^{2}}\right]\,.$

For the nucleon-nucleon contribution to the mean field, a parametrized
effective nucleon interaction (distance $s$)

$V_{N-N}(s/{\rm fm})=c\,\exp(-4s)/(4s)-d\,\exp(-2.5s)/(2.5s)$ (14)

can be used which is motivated by the M3Y interaction M3YReview , $s$ denotes
the distance of nucleons. The parameters $c,d$ are adapted to reproduce known
data, see Po14 ; Xu16 ; Xu17  for the case of a lead core nucleus. For the
oxygen core nucleus, parameter values $c,d$ are given below in Eq. (22). As
also known from other mean-field approaches, we fit the mean field parameter
to measured data. The nucleonic contribution $V^{\rm N-N}_{\alpha-{\rm O}}(R)$
to the mean field is calculated in analogy to Eq. (10) replacing the Coulomb
interaction by the nucleon interaction (14). With both contributions, the
mean-field part of the cluster potential is

$W^{\rm ext}({\bf R})=W^{\rm mf}({\bf R})=V^{\rm Coul}_{\alpha-{\rm
O}}(R)+V^{\rm N-N}_{\alpha-{\rm O}}(R).$ (15)

The local approximation $W^{\rm intr}({\bf R})$, Eq. (3.1), for the intrinsic
contribution to the effective c.m. potential is more involved. It contains the
binding energy of the cluster taking into account the Pauli blocking of the
surrounding matter. The local density approximation neglects any gradient
terms so that homogeneous-matter results can be used.

The intrinsic wave equation (2) describes in the zero density limit the
formation of an $\alpha$ cluster with binding energy $B_{\alpha}=28.3$ MeV. In
homogeneous matter, the binding energy is reduced due to Pauli blocking. The
shift of the binding energy is determined by the baryon density
$n_{B}=n_{n}+n_{p}$ and the asymmetry $\delta=2n_{p}/n_{B}-1$. For the c.m.
momentum ${\bf P}=0$, the Pauli blocking term depends on the baryon density
$n_{B}$ Po14 ; wir  as

$\displaystyle W^{\rm Pauli}(n_{B},\delta)$ $\displaystyle\approx$
$\displaystyle 4515.9\,{\rm MeV\,fm}^{3}n_{B}$ (16)
$\displaystyle-100935\,{\rm MeV\,fm}^{6}n_{B}^{2}(1+\delta^{2})$
$\displaystyle+1202538\,{\rm MeV\,fm}^{9}n_{B}^{3}(1+3\delta^{2})\,.$

This approximation formula applies to the density range $n_{B}\leq n_{\rm
crit}=0.02917$ fm-3. In particular, the bound state is dissolved and merges
with the continuum of the scattering states at the critical density $n_{\rm
crit}$ (introduced as Mott density). A more detailed discussion of this ansatz
for the Pauli blocking term will follow below, see section 4. For the
intrinsic wave function of the quartet, we can assume an $\alpha$-like
Gaussian function to describe the bound state. The width parameter of the free
$\alpha$ particle is only weakly changed when it approaches the critical
density, see Ref. Po14 .

Below the critical density, $n_{B}\leq n_{\rm crit}$, the intrinsic potential

$W^{\rm intr}({\bf R})=-B_{\alpha}+W^{\rm Pauli}[n_{B}({\bf R})],\qquad
n_{B}\leq n_{\rm crit}$ (17)

results in a local density approximation. The intrinsic energy of the quartet
for densities above the critical density is a minimum if all four nucleons are
at the Fermi energy (ideal Fermi gas), for symmetric matter and $n_{B}\geq
n_{\rm crit}$

$W^{\rm intr}({\bf R})=4E_{F}[n_{B}({\bf R})],$ (18)

with

$E_{F}(n_{B})=(\hbar^{2}/2m)(3\pi^{2}n_{B}/2)^{2/3}.$ (19)

###  3.2 Thomas-Fermi rule and results for 20Ne in local density approximation

The quartetting wave function approach for 20Ne in local density approximation
has been considered in Ref. wir . We are presenting some results for the
effective potential $W({\bf R})$ and the wave function $\psi({\bf R})$, see
Fig. 1. To this purpose, we use empirical data from the nuclei involved.

Figure 1: Effective potential $W^{\rm TF}(R)$ for the center of mass motion of
the quartet on top of 16O. The Thomas-Fermi model has been used. The formation
of a pocket is shown.

The mean-field contribution $W^{\rm ext}(R)$ (15) is given by the double-
folding Coulomb and $N-N$ potentials. Empirical values for the densities of
the $\alpha$ particle (11) and the 16O core nucleus (12) are known from
scattering experiments, such as the rms radii, so that the Coulomb interaction
$V^{\rm Coul}_{\alpha-{\rm O}}(R)$ (3.1) as well as the nucleon-nucleon
interaction $V^{\rm N-N}_{\alpha-{\rm O}}(R)$ can be calculated.

With respect to $W^{\rm intr}({\bf R})$, the local density approximation is
also very simple: There are two regions separated by the critical radius
$r_{\rm crit}=3.302$ fm in which the density of the 16O core nucleus (12) has
the critical value $n_{B}(r_{\rm crit})=n_{\rm crit}=0.02917$ fm-3. We obtain
$-B_{\alpha}+W^{\rm Pauli}[n_{B}(r_{\rm crit})]=4E_{F}[n_{B}(r_{\rm crit})]$,
and the bound state merges with the continuum of scattering states.

For $R>r_{\rm crit}$, the intrinsic part $W^{\rm intr}(R)$ contains the bound
state energy -28.3 MeV of the free $\alpha$ particle, which is shifted due to
Pauli blocking. At $r_{\rm crit}$ the bound state merges with the continuum,
so that we have the condition (symmetric matter)

$W(r_{\rm crit})=W^{\rm ext}(r_{\rm crit})+4E_{F}(n_{\rm crit})=\mu_{4},$ (20)

the intrinsic wave function changes from a bound state to four uncorrelated
quasiparticles on top of the Fermi sphere (the states below the Fermi energy
are already occupied).

For $R<r_{\rm crit}$, the Fermi energy $4E_{F}[n(R)]$ appears in addition to
the mean-field contribution $W^{\rm ext}(R)$. In the Thomas-Fermi model, for a
given potential $W^{\rm ext}(R)$ the density is determined by the condition
that $W^{\rm ext}(R)+4E_{F}[n_{B}(R)]$ remains a constant, here $\mu_{4}$. We
find the effective potential $W^{\rm TF}(R)$, which is continuous but has a
kink at $r_{\rm crit}$. It is an advantage of the Thomas-Fermi model that the
condition $W^{\rm TF}(R)=\mu_{4}=$ const applies to the entire range $R<r_{\rm
crit}$, independently of the shape of the mean-field potential $W^{\rm
ext}(R)$ and the corresponding density distribution. We analyze this property
in the following section.

While the Coulomb part of the external potential as well as the intrinsic part
of the effective potential $W^{\rm TF}(R)$ are fixed, the two parameters $c,d$
in Eq. (14) for the $N-N$ part of the external potential can be adjusted such
that measured data are reproduced. In the case of heavy nuclei that are
$\alpha$ emitters, such as 212Po Po14 , two conditions can be formulated:
i) For $\alpha$ emitters, the normalized solution of the c.m. wave equation
(neglecting the decay) gives the energy eigenvalue $E_{\alpha}=E_{\rm
tunnel}$. This eigenvalue should correspond to the measured energy after
decay, which is given by the $Q$ value.
ii) This value $E_{\rm tunnel}$ should coincide with the value $\mu_{4}$. In
the context of the local density approach, this is the value that the four
nucleons must have in order to implement them into the core nucleus. We denote
this condition

$E_{\alpha}=\mu_{4}$ (21)

as the Thomas-Fermi rule wir . With both conditions, the parameter $c,d$ for
the double folding $N-N$ interaction potential are found, and values for the
preformation factor and the half-life of the $\alpha$ decay were determined
for heavy nuclei, see Ref. Po14 ; Xu16 ; Xu17 , where further discussions were
made.

In contrast to the $\alpha$ decay of 212Po where the $Q$ value can be used to
estimate the chemical potential $\mu_{4}$ Po14 , the nucleus 20Ne is stable.
However, we can use the additional bonding in the transition from 16O
($B_{{}^{16}{\rm O}}=127.66$ MeV) to 20Ne ($B_{{}^{20}{\rm Ne}}=160.645$ MeV)
by adding the four nucleons. The difference fixes the position of the in-core
effective potential $\mu_{4}=B_{{}^{16}{\rm O}}-B_{{}^{20}{\rm Ne}}=-33.0$
MeV.

Another condition is that the solution of the Schrödinger equation for the
four-nucleon c.m. motion in the effective potential $W(R)$ gives the energy
eigenvalue $E_{\alpha,{\rm bound}}$ at this value -33 MeV, so that the energy
eigenvalue of the $\alpha$-like cluster coincides with the Fermi energy
$\mu_{4}$ (the Thomas-Fermi rule, see also the discussion in Ref. Xu16 ). Both
conditions are used to fix the parameters $c,d$. The values

$c=4650\,\,{\rm MeV\,\,\,\,and}\,\,\,d=1900\,\,{\rm MeV}$ (22)

have been found wir .

The resulting effective potential $W^{\rm TF}(R)$ (17) for the center of mass
motion of the quartet is shown in Fig. 1. One can see the formation of a
pocket near the surface caused by the formation of an $\alpha$-like cluster.
The sharp kink at the critical radius $r_{\rm crit}=3.302$ fm is a consequence
of the local approximation for the Pauli blocking term. A smooth behavior is
expected if the finite extension of the $\alpha$-like cluster is taken into
account so that the kink generated by the local density approximation is
smeared out.

The wave function for the quartet center-of-mass motion $\psi^{\rm TF}_{\rm
c.m.}(R)$ is found as a solution of the Schrödinger equation, mass $4m$, with
the potential $W^{\rm TF}(R)$. The energy eigenvalue is -33.0 MeV. A graph of
$(4\pi)^{1/2}R\,\psi^{\rm TF}_{\rm c.m.}(R)$ is shown in Fig. 2. As a result,
in Ref. wir  the rms point radius 2.864 fm for 20Ne was calculated which is in
good agreement with the experimental rms point radius of 2.87 fm. The
normalization is $4\pi\int_{0}^{\infty}r^{2}\psi^{2}_{\rm c.m.}(r)dr=1$.
Integrating from 0 to $r_{\rm crit}=3.302$ fm, the part of the quartet where
the internal structure is the product of free states, comes out at 0.3612. The
remaining part where the internal structure is given by an $\alpha$-like bound
state is 0.6388.

Figure 2: Wave function for the c.m. motion of the quartet. A prefactor
$(4\pi)^{1/2}R$ is introduced so that the integral over $R$ of the squared
quantity is normalized to 1. The solution for the Thomas-Fermi model
$\psi^{\rm TF}_{\rm c.m.}(R)$ (red, dashed) is compared with the non-
interacting shell-model calculation $\psi_{2s^{4}}(R)$ (blue). The shift of
the maximum is caused by the formation of a pocket, see Fig. 1. Dotted line:
$r_{\rm crit}=3.302$ fm.

A further discussion of optical model description and double-folding potential
is given in App. A. Note that the standard approaches of optical model
potentials have a diverging repulsive potential below $r_{\rm crit}$.

###  3.3 Discussion of the Thomas-Fermi rule $E_{\alpha}=\mu_{4}$

The condition $E_{\alpha}=\mu_{4}$ (21) is a consequence of the Thomas-Fermi
model, which applies to infinite matter: an additional nucleon with given spin
and isospin can be introduced at the corresponding chemical potential
$\mu_{\sigma,\tau}$. At zero temperature, this coincides with the
corresponding Fermi energy (plus the potential energy). For finite system such
as nuclei, the energy levels of the single-nucleon states are discrete. If we
add a nucleon to the core nucleus in which all the single-nucleon states below
a certain energy are occupied, the next free single-nucleon state that is free
has a distance to the chemical potential. This means, under these
considerations, the quartet cannot be introduced at $\mu_{4}$ but at a higher
value $E_{\alpha}>\mu_{4}$ which is now a new parameter. This aspect has been
worked out already in Xu17 . We do the same here for 20Ne.

We compare our calculations with values for 212Po. The $\alpha$ decay energy
$Q_{\alpha}$ was introduced as the difference between the binding energy of
the mother nucleus (212Po) and the binding energies of the daughter nuclei
(208Pb and $\alpha$). Similarly, we have -4.73 MeV, so that the energy
eigenvalue of the Schrödingier equation comes out as
$E^{0}_{\alpha}-Q_{\alpha}=-28.3-4.73$ MeV=-33.03 MeV. As a second condition,
we used the results for 212Po. If $d=3415.56$ remains the same, the given
energy eigenvalue is of the Schrödingier equation is reproduced with
$c=10623$. This results in the value $\mu_{4}=-32.388$ MeV and
$P_{\alpha}=0.72$ follow. If we take $c=11032$ from Po, we get $d=3513.46$ as
well as $\mu_{4}=-32.12$ MeV and $P_{\alpha}=0.74$.

We reproduce a large preformation factor $P_{\alpha}$ in both cases. In
contrast to the Thomas-Fermi model, the condition $E_{\alpha}=\mu_{4}$ is not
valid. The value of $\mu_{4}$ is not below $E_{\alpha}$ as expected from the
shell model consideration, but $E_{\alpha}<\mu_{4}$. This means that it is
energetically more favorable for the nucleus to form correlated quartets
instead of remaining in uncorrelated single-nucleon (shell model) states. This
will be seen from the THSR calculations, in which the core nucleus 16O also
shows $\alpha$-like correlations.

##  4 Shell model calculations

###  4.1 Comparison with the harmonic oscillator model

The local density approximation (Thomas-Fermi model) is not able to describe
the nuclear structure of the core nucleus. In particular, the Thomas-Fermi
rule must be replaced by a more microscopic approach, see Po14 ; Xu16 ; Xu17 .
However, the behavior of the effective c.m. potential which remains almost
constant within the core nucleus, is also interesting in the case that shell
model states are used. A first attempt was made in Ref. wir  with harmonic
oscillator states. The results of the simple Thomas-Fermi model, in particular
the approximate constancy of the c.m. quartetting potential in the core
nucleus and the Thomas-Fermi rule, can be verified. However, the harmonic
oscillator potential is not realistic for nuclei, especially near the surface
of the core nucleus where $\alpha$-like quartets are formed. We present here
calculations with more realistic potentials (units MeV, fm), see also Mirea .
The intrinsic nucleon-nucleon interaction $W^{\rm intr}(R)$, which is
suppressed due to Pauli blocking, is not considered in this section 4.1.

A more systematic way to find a suitable simple basis of single-particle
states is to use the Woods-Saxon potential WS  for $Z=N$, see Ref. Ro18 ,

$V_{\rm WS}(r)=\frac{V_{0}(1+3\kappa/A)}{1+\exp[(r-R_{0}A^{1/3})/a]}$ (23)

with $V_{0}=-52.06$ MeV, $\kappa=0.639$, $R_{0}=1.26$ fm, $a=0.662$ fm. The
normalized solution $\psi_{2s}(r)$ for the 2$s$ state is shown in Fig. 3,
eigenvalue $E_{2s}=-9.162$ MeV. For comparison, the harmonic oscillator wave
function

$\psi_{2s}^{\rm HO}(r)=-\left(\frac{a^{\rm HO}}{\pi}\right)^{3/4}e^{-a^{\rm
HO}r^{2}/2}\left(a^{\rm
HO}r^{2}-\frac{3}{2}\right)\left(\frac{2}{3}\right)^{1/2},$ (24)

is also shown, where the parameter $a^{\rm HO}=0.31047$ fm is chosen so that
the values coincide at $r=0$. A scaling of the $r$-axis is considered to make
both coincide, $\psi_{2s}^{\rm HO}(r^{\prime})=\psi_{2s}(r)/(1+0.0024719\,r)$.
(The amplitude correction is necessary to reproduce the correct value of the
minimum). This defines the relationship $r^{\prime}=f_{\rm scal}(r)$ shown in
Fig. 3.

Figure 3: Normalized wave function $\psi_{2s}(r)$ for the Woods-Saxon
potential (23). For comparison, the harmonic oscillator wave function
$\psi_{2s}^{\rm HO}(r)$ is also given, where the potential parameter $a^{\rm
HO}$ is chosen so that $\psi_{2s}(0)$ coincides. The scaling function $f_{\rm
scal}(r)$ give full coincidence of both wave functions.

Neglecting any intrinsic interaction, the 2$s$ wave functions can be used to
construct the quartet wave function

$\Phi_{2s^{4}}({\bf R,S,s,s}^{\prime})=\psi_{2s}({\bf
r}_{n,\uparrow})\,\psi_{2s}({\bf r}_{n,\downarrow})\,\psi_{2s}({\bf
r}_{p,\uparrow})\,\psi_{2s}({\bf r}_{p,\downarrow}).$ (25)

The wave function for the c.o.m. motion follows as (Jacobi-Moshinsky
coordinates ${\bf R,S,s,s}^{\prime}$ wir )

$\psi_{2s^{4}}({\bf R})=\left[\int
d^{3}Sd^{3}sd^{3}s^{\prime}|\Phi_{2s^{4}}({\bf
R,S,s,s}^{\prime})|^{2}\right]^{1/2}\,.$ (26)

The evaluation of the 9-fold integral in (26) is very time-consuming. An
approximation can be given comparing with the solution for the harmonic
oscillator wir

$\displaystyle\varrho_{2s^{4}}^{\rm cm,HO}(a,R)=|\psi^{\rm
HO}_{2s^{4}}(R)|^{2}=\left(\frac{a}{\pi}\right)^{3/2}e^{-4aR^{2}}$
$\displaystyle\times\frac{1}{10616832}(24695649+14905152\,aR^{2}+354818304\,a^{2}R^{4}$
$\displaystyle-876834816\,a^{3}R^{6}+1503289344\,a^{4}R^{8}-1261699072\,a^{5}R^{10}$
$\displaystyle+613416960\,a^{6}R^{12}-150994944\,a^{7}R^{14}+16777216\,a^{8}R^{16}).$

The parameter $a^{\prime\prime}=0.287038$ fm can be chosen to reproduce the
value at $R=0$ (three-fold integral). The scaling $R^{\prime\prime}=f_{\rm
scal}(R)+0.174\,(e^{R/2.924}-1)$ fulfills normalization and improves the
asymptotic behavior for large $R$, so that $\varrho_{2s^{4}}^{\rm
cm}(R)\approx\varrho_{2s^{4}}^{\rm cm,HO}(a^{\prime\prime},R")$. A plot of
$(4\pi R^{2})^{1/2}\psi_{2s^{4}}(R)$ is shown in Fig. 2. The normalization
$\int_{0}^{\infty}4\pi R^{2}\psi^{2}_{2s^{4}}(R)dR=1$ holds.

We reconstruct the effective potential from the wave function
$\psi_{2s^{4}}(R)=(\varrho_{2s^{4}}^{\rm cm}(R))^{1/2}$ wir . If we restrict
us to $s$ states ($l=0$) and introduce
$u_{2s^{4}}(R)=(4\pi)^{1/2}R\psi_{2s^{4}}(R)$, we have

$W_{2s^{4}}(R)-E_{2s^{4}}=\frac{\hbar^{2}}{8m}\frac{1}{u_{2s^{4}}(R)}\frac{d^{2}}{dR^{2}}u_{2s^{4}}(R).$
(28)

The result is shown in Fig. 4.

Figure 4: The c.m. potential $W_{2s^{4}}(R)$, Eq. (28), compared with the
Woods-Saxon potential of the quartet.

We conclude from this: The effective c.m. potential $W(R)$ remains almost
constant within the core as expected from the Thomas-Fermi model. The value
$E_{2s^{4}}=-36.65$ MeV is near to the estimate $\mu_{4}=-33$ MeV from the
Thomas-Fermi rule. It is slightly increasing near the surface, possibly
because the quartet is not localized at a point, but smeared out, so that it
”feels” the weakening of the potential near the surface. Another reason could
be the gradient terms in Eq. (2), which are neglected here. A similar behavior
was also observed for the harmonic oscillator potential in wir . In contrast
to the harmonic oscillator, where the effective potential increases with $R$,
the behavior near the surface is now more realistic. The weakening of the
Thomas-Fermi rule has been shown in Refs. Po14 ; Xu16 ; Xu17 ; wir .

###  4.2 Intrinsic interaction and Pauli blocking

We have introduced an effective c.m. potential $W(R)$, which describes the
influence of the environment (here the core nucleus) on the c.m. motion of the
quartet in mean-field approximation. Specifically, we have simulated a quartet
of 4 uncorrelated nucleons in $2s$ states moving under the influence of the
core nucleus 16O. The corresponding potential $W_{2s^{4}}(R)$ shows
approximately the constancy of the chemical potential required within the
Thomas-Fermi model.

To describe the formation of an $\alpha$-like cluster, we need to consider the
interaction within the quartet. To estimate the intrinsic interaction of the
quartet, we add for $R>r_{\rm crit}$ the energy shift $W^{\rm intr}({\bf R})$,
Eq. (17), which describes the formation of the cluster and the dissolution due
to Pauli blocking, see fig. 5. The Coulomb potential is added, and the free
effective potential of the shell model $W_{2s^{4}}(R)$ is used instead of
$W^{\rm ext}$. A harmonic oscillator base was essentially used here Ro18 . We
denote this approximation for the potential for the c.m. motion as $W_{\rm
appr}(R)$.

Obviously, this c.m. potential $W_{\rm appr}(R)$ is only a rough
approximation. In particular, the sharp peak due to the sudden switching off
of the intrinsic interaction at $r_{\rm crit}=3.302$ fm does not seem
realistic. A similar peak at $r_{\rm crit}$ was also obtained for the heavy
isotopes Xu16 ; Xu17 , but it was less pronounced than for the light isotope
20Ne.

The behavior for large $R$ is correctly reproduced, the asymptote
$\lim_{R\to\infty}W(R)=-28.3$ MeV is the binding energy of the $\alpha$
particle, and the Coulomb repulsion is well represented. The attractive $N-N$
interaction is also visible, as in other approaches using an optical
potential, see App. A. As the density of the core increases, the binding
energy of the $\alpha$ cluster is weakened due to Pauli blocking, and a pocket
is formed. The behavior for small $R\leq 2$ fm is also well reproduced. The
fluctuations around the Thomas-Fermi value are due to the shell structure.

An improvement of the effective quartet potential is particularly necessary in
the vicinity of the critical density. Instead of a sharp switchover, in which
all correlations above the critical density are omitted, these decrease
continuously. Quartet correlations are also present for $R\leq r_{\rm crit}$.
They can provide a contribution as resonances in the continuum, which
decreases steadily with increasing density. Furthermore, Pauli-blocking is
calculated for uncorrelated nucleons in the environment, which is expressed in
the use of the Fermi function. Correlations in the surrounding matter would
also reduce the Pauli blocking. Taking into account the c.m. movement of the
$\alpha$-cluster, the Pauli blocking is also reduced.

Furthermore, we are dealing with an inhomogeneous system, so that gradient
terms can become important. As an extended system, the $\alpha$-like cluster
is determined not only by the properties of the surrounding matter at the
position of the center of mass, but by the properties within the extension of
the cluster. Finally, the Pauli principle is a non-local effect, which is
treated as local only after some approximations. We have collected several
arguments which show that the effect of Pauli blocking should be treated as a
continuous function of density. This can help to reduce the peak at the
critical radius.

Figure 5: Quartet c.m. potentials $W(R)$. The Thomas-Fermi approximation
$W^{\rm TF}(R)$ is compared with the calculation $W_{\rm appr}(R)$ using
harmonic oscillator shell model states. Note the peak at $r_{\rm crit}=3.302$
fm.

###  4.3 Shell-model calculations

First results to use shell-model calculations for 20Ne to perform calculations
within the QWFA have been presented in Refs. Yang21 ; Bai19 . We use the
widely-used Woods-Saxon potential

$V_{\rm WS}\left(r\right)=\frac{V_{0}}{1+\textrm{exp}(\frac{r-R_{0}}{a})},$
(29)

together with the spin-orbit coupling interaction

$V_{\rm so}\left(r\right)=\frac{1}{2\mu^{2}r}\left(\frac{\partial}{\partial
r}\frac{\lambda V_{0}}{1+\textrm{exp}(\frac{r-R_{\rm so}}{a_{\rm
so}})}\right)\bf l\cdot\bf s$ (30)

to determine the shell model wave functions of quartet nucleons in 20Ne. The
strength of the Woods-Saxon potential is parameterized as

$V_{0}=-46\left[1\pm 0.97\left(\frac{N-Z}{A}\right)\right]$ (31)

(“$+$” for protons and “$-$” for neutrons). The parameter $R_{0}$ is
$1.43\,A^{1/3}$ fm for both protons and neutrons while the parameter $R_{\rm
so}$ is $1.37\,A^{1/3}$ fm. The diffusivity parameter $a$ and $a_{\rm so}$ are
chosen to be the same value 0.7 fm. $\mu$ is the reduced mass of the
$\alpha$-core system and the normalization factor of the $ls$ coupling
strength $\lambda$ is 37.5 for neutrons and 31 for protons, respectively. The
Coulomb potential we adopt is

$\displaystyle V_{C}(r)$ $\displaystyle=$ $\displaystyle(Z-1)e^{2}(3R_{\rm
Coul}^{2}-r^{2})/2R_{\rm Coul}^{3},\quad r\leq R_{\rm Coul},$ (32)
$\displaystyle=$ $\displaystyle(Z-1)e^{2}/r,\quad r>R_{\rm Coul}.$

with the radius $R_{\rm Coul}=1.25\,A^{1/3}$ fm. The effective c.m. potential
constructed from the shell model quartet state for 20Ne is shown in Fig. 6.

A general discussion of the Pauli blocking term is necessary to avoid the peak
in Figs. 5 and 6. Various approximations were made when calculating the
effective potential. We mention the neglect of the gradient terms and the non-
local property of the potential $W({\bf R},{\bf R}^{\prime})$, Eq. (3.1), in
particular due to the Pauli blocking term. We emphasize that Eq. (16) was
derived for $\alpha$-particles in an uncorrelated medium. At zero temperature,
the medium can be strongly correlated and form $\alpha$ matter. A correlated
medium was considered in Ref. Tak04 , and the merging with the continuum was
observed at a slightly higher critical density. If we use this calculation to
construct the Pauli blocking shift, this could possibly lead to a smoother
transition and reduce the peak.

Figure 6: Quartet c.m. potential $W(R)$ for 20Ne using shell model states.

For 20Ne, the probability to find the $\alpha$-particle in the localized shell
model states can be defined as

$\displaystyle{\cal F}_{\alpha}=\int dR\,4\pi R^{2}\rho_{\rm quartet}^{\rm
c.m.}(R)\left|\left\langle\varphi_{\alpha}^{\rm intr}|\varphi_{\rm
quartet}^{\rm intr}\right\rangle(R)\right|^{2},$ (33)

where $\left\langle\varphi_{\alpha}^{\rm intr}|\varphi_{\rm quartet}^{\rm
intr}\right\rangle(R)$ is the overlap between the intrinsic wave functions of
a quartet $\varphi_{\rm quartet}^{\rm intr}$ and a free $\alpha$-particle as a
function of c.m. variable $R$. The density at the c.m. position $\bf R$ is
$\rho_{\rm quartet}^{\rm c.m.}(R)=\mid{\Psi_{\rm quartet}^{\rm c.m.}(\bf
R)}\mid^{2}$. As expected, the probability ${\cal F}_{\alpha}=2.004\times
10^{-3}$ is quite small for 20Ne as the wave function of the quartet is
approximated by a product of shell model states. However, the probability
${\cal F}_{\alpha}$ is significantly enhanced for the $\alpha$ \+ doubly magic
core system 20Ne as compared to those of their neighboring isotopes We show in
Fig. 7 the overlap between the wave functions of the quartet and the
$\alpha$-particle as a function of c.m. coordinate $R$ for 20Ne. It is clearly
demonstrated that there exists a peak in the region beyond the critical radius
(i.e. the surface region of the core). Inside the core, the probability to
find the $\alpha$-like state is quite low for 20Ne.

Figure 7: The overlap between the intrinsic wave functions of the quartet and
the $\alpha$-particle as a function of c.o.m. coordinate $R$ for the
$\alpha$+doubly magic core system 20Ne.

##  5 Comparison with the THSR model and other approaches

###  5.1 Calculations for 20Ne

Figure 8:  Variational calculations for the energy of 16O with respect to the
harmonical osciallator parameter $b$ and size parameter $\beta_{0}$ using the
THSR wave function THSR .

The THSR ansatz adeptly describes the low-density regime of $\alpha$ matter as
well as the shell model states, particularly when the c.m. wave function
coincides with the intrinsic wave function. Notably, when four $\alpha$
clusters merge into a ${}^{16}\mathrm{O}$-like configuration, the
antisymmetrization process gives rise to nucleonic $s$ and $p$ orbitals,
especially as the inter-cluster distance approaches zero. Deviations in the
Gaussian width parameters signal the presence of correlations. The $N\alpha$
THSR wave function THSR  can be written as,

$\Phi_{n\alpha}^{\rm THSR}\\!\\!\propto{\cal A}\
\Big{\\{}\prod_{i=1}^{n}\exp\Big{[}-\frac{2(\vec{X}_{i}-\vec{X}_{G})^{2}}{b^{2}+2\beta_{0}^{2}}\Big{]}\phi(\alpha_{i})\Big{\\}},$
(34)

where $\vec{X}_{i}$ and $\vec{X}_{G}$ are the c.m. coordinate of the $\alpha$
cluster and the total c.m. coordinate of $N\alpha$ cluster, respectively.
Figure 8 presents a THSR calculation for ${}^{16}\mathrm{O}$, where the
parameter $\beta_{0}$ reflects deviations from shell model behavior. The
observed energy minimum at a finite $\beta_{0}$ (size parameter of the THSR
wave function) signifies the existence of $\alpha$-like correlations even in
the ground state.

The uncorrelated mean-field approximation, often invoked to compute Pauli
blocking effects, may not be universally valid. In particular, $\alpha$ matter
exemplifies a scenario where the medium undergoes a transformation into a
correlated state. Analogous reconfigurations are evident in pairing phenomena
at temperatures descending below the critical value. The Tohsaki-Horiuchi-
Schuck-Röpke (THSR) formalism was conceived to elucidate $\alpha$ clustering
within such tenuous nuclear environments, exemplified by the Hoyle state of
${}^{12}\mathrm{C}$. Here, the environment of an $\alpha$ cluster is composed
of other $\alpha$ clusters, leading to a pronouncedly clustered structure.
This method has been successfully employed to investigate various $4n$ nuclei,
including ${}^{20}\mathrm{Ne}$.

Figure 9: Energy curve of 20Ne with the increase of the size parameter $\beta$
using the intrinsic THSR wave function. The asymptotic -154.16 MeV for the
binding energy of separated ${}^{16}\mathrm{O}$ and $\alpha$ clusters is also
shown.

The microscopic THSR wave function for the nucleus ${}^{20}\mathrm{Ne}$ can be
written as

$\displaystyle{\widehat{\Phi}}_{{\rm THSR}}(\beta)={\cal
A}[\exp(-\frac{8r^{2}}{5(b^{2}+2\beta^{2})}\phi(\alpha)\phi({{}^{16}{\rm
O}})],$ (35)

where ${\bf r}={\bf X}_{1}-{\bf X}_{2}$. ${\bf X}_{1}$ and ${\bf X}_{2}$
represent the center-of-mass coordinates of the $\alpha$ cluster and the
${}^{16}\mathrm{O}$ cluster, respectively. It should noted that the
${}^{16}\mathrm{O}$ cluster is described as the shell model wave function.

Fig. 9 shows the energy curve with the increase of the size parameter $\beta$.
This can be transformed to the energy curve as a function of the inter-cluster
distance. The extracted effective $\alpha$-O potential would be of interest.
It should be noted, however, that the inter-cluster distance between clusters
cannot be precisely defined, especially when clusters are in close proximity,
owing to the effects of antisymmetrization.

It is not directly possible to define the inter-cluster distance $D$ in THSR
approach. According to Matsuse Mat75  one can introduce the distance $D$
according the relation for the rms radii

$20\langle r^{2}\rangle_{\rm Ne}=16\langle r^{2}\rangle_{\rm O}+4\langle
r^{2}\rangle_{\alpha}+\frac{16}{5}\langle D^{2}\rangle$ (36)

so that

$\langle D^{2}\rangle=\frac{25}{4}\langle r^{2}\rangle_{\rm
Ne}-\frac{195}{16}b^{2}.$ (37)

follows. We used this quantity $D$ for the distance $r$ in Fig. 11.

Very recently, the $5\alpha$ clustering structure of ${}^{20}\mathrm{Ne}$ was
scrutinized by Bo et al. Bo23  utilizing the THSR framework, which adopts the
container model. In this model, the intrinsic $\alpha$ cluster width parameter
$b$ is complemented by two additional parameters: $\beta_{1}$ (denoting the
width of the ${}^{16}\mathrm{O}$ core nucleus) and $\beta_{2}$ (representing
the center-of-mass motion of the residual $\alpha$ cluster). As illustrated in
Fig. 10, the energy minimum is observed at $\beta_{1}=1.5$ fm and
$\beta_{2}=3.0$ fm, corresponding to an energy of approximately $-155.3$ MeV.
The GCM calculations yield an energy of $-156.4$ MeV. Additionally, the
calculated rms radius is $2.96$ fm. A notable aspect of the THSR wave function
is its inclusion of the shell model limit, thereby ensuring an accurate
representation of the ground state of the ${}^{16}\mathrm{O}$ core nucleus.
The orthogonality between the additional fifth $\alpha$ particle and the core
states is rigorously preserved. Theoretical calculations yield favorable
comparisons with empirical data for both the binding energy and the root-mean-
square (rms) radius of the ground state. The disparity between the values of
$\beta_{1}$ and $\beta_{2}$ suggests the presence of an $\alpha$ particle atop
the doubly-magic ${}^{16}\mathrm{O}$ core. These results from the $5\alpha$
calculations set the stage for future studies to develop a more realistic
${}^{16}\mathrm{O}$-$\alpha$ effective interaction.

Figure 10: Contour plot for the ground state of ${}^{20}\mathrm{Ne}$ in the
spherical $\beta_{1}$ and $\beta_{2}$ parameter space.

Figure 11: 16O - $\alpha$ effective interaction potential as function of the
center-of-mass distance $R$. The THSR calculations (blue full line) are
compared with the Thomas-Fermi approximation of the quartetting wave function
(red full line). The total potential (TF, THSR) is shown as well as the
Coulomb contribution (dashed lines). In addition, the Coulomb interaction for
the harmonic oscillator density of the O-core is also shown.

###  5.2 $\alpha$ matter

The equilibrium composition of homogeneous nuclear matter at low densities and
temperatures is a complex problem, since below the saturation density of
symmetric matter $\rho_{\rm sat}=0.15$ fm-3 a thermodynamic instability occurs
and clusters are formed. The highest binding energy per nucleon is found for
the nucleus 56Fe. Here we only consider the formation of $\alpha$-clusters
from the nucleons.

At a fixed baryon density, the mass fraction of the $\alpha$ clusters
increases with decreasing temperature. At a critical temperature, a quantum
condensate can be formed. In analogy to pairing, the $\alpha$-like quartets
are the bosonic components of the condensate. However, they are modified by
the medium Fun08 . As known from pairing, where the Bogoliubov transformation
allows to describe the nuclear matter below the critical temperature, below
the critical temperature for quartetting we have to consider a correlated
medium, the so-called $\alpha$ matter.

In analogy to the THSR approach for low-density nuclei such as 8Be or the
Hoyle state of 12C, calculations for periodic $\alpha$-like structures were
performed in Tak04 . Orthonormal Bloch states were introduced so that Pauli
blocking by nucleons bound in $\alpha$-clusters is strictly realized. One
problem is the separation of the c.m. contribution to the kinetic energy,
which is solved by a simple ansatz based on the energy gap at zero momentum.
As a result, in Ref. Tak04  it was shown that the bound state merges with the
continuum at about $0.2\rho_{\rm sat}$.

We have also performed exploratory calculations with a separable potential
adapted to reproduce the free-$\alpha$ properties mass and rms radius, see
Appendix B. The difference between the energy per nucleon in the uncorrelated
free-nucleon state and the $\alpha$-matter state is shown in Fig. 12. A value
$\rho_{\rm Mott}=0.04$ fm-3 was found for the dissolution of the bound state.

For comparison, in Fig. 12 also shown is the shift of the binding energy for
uncorrelated matter where the surrounding nuleons occupy free single-particle
states,

$E_{\rm bound}^{\rm uncorr}(n_{B})=-7.07\,{\rm MeV}+W^{\rm
Pauli}(n_{B})-E_{F}(n_{B}).$ (38)

Compared to the Pauli blocking by free nucleons considered in Eq. (16), the
blocking in $\alpha$-matter is smaller because the distribution in momentum
space is spreed out, and the blocking is less efficient. As a result, the
critical density where bound states are dissolved, comes out to be larger if
cluster formation in the surrounding is taken into account. We expect that
this modification makes the peak in the figures 5 and 6 smoother. Further
investigations are necessary to find a better treatment of the dissolution of
clusters due to Pauli blocking.

Figure 12: Shift of the binding energy per nucleon for an $\alpha$-cluster as
function of the nucleon density $n_{B}$. The difference of the energy per
nucleon in $\alpha$-matter and in momentum eigenstates (red) is compared with
the shift (blue) in uncorrelated matter, Eq. (38).

Another approach to show that nuclear matter dissolves into clusters at low
density was presented in Ref. Ebran20 . Restricted Hartree-Fock calculations
were performed that allow the formation of separate cluster structures. Even
with this approach, the strict separation of the kinetic energy of the c.m.
remains open. An unresolved question is whether the disappearance of the
cluster structures and the appearance of a homogeneous phase is a first-order
transition.

###  5.3 Other approaches to $\alpha$-clustering in nuclei

Based on a local density approach with composition and energy shifts derived
from R1 ; R11 , Typel T14  has considered the formation of $\alpha$-particle
correlations at the surface of heavy nuclei to study the neutron skin
thickness of heavy nuclei and the slope of the nuclear symmetry energy. The
$\alpha$ particle density was considered as a function of radius for the tin
isotopes 108Sn to 132Sn, and it was shown that as the neutron density at the
skin increases, the abundance of $\alpha$-particles is suppressed as a result
of Pauli blocking. The experimental evidence for the $\alpha$ cluster
formation in the surface of neutron-rich tin isotopes was given using quasi-
free $\alpha$ cluster-knockout reactions T20 ; T21 . Note that the occurrence
of $\alpha$-cluster at the surface of 48Ca and 208Pb and and its impact on the
extraction of symmetry energy from skin thickness is also investigated by
using QWFA Yang23 . Strong closed shell structure effects and complex
derivative terms of the intrinsic wave function are properly taken into
account in QWFA Yang23 .

The question of $\alpha$ formation in the ground state of heavy nuclei has
been investigated using the AMD approach AMD ; AMD04  in several recent
publications. The AMD approach also describes the suppression of clusters
using the Pauli blocking effect. The manifestation of clustering at the
surface region where the density is low has induced many investigations on
$\alpha$-break-up reactions. We do not give a comprehensive account of various
investigations of specific isotopes Freer18 ; Yos19 ; Chi17 ; Tan21 ; Yos22 ;
Qi10 ; Nak23 ; Kimura22 ; PG11  in this paper. We would like to emphasize that
the approach of the quartet wave function presented here is also of interest
for these examples.

##  6 Conclusions

We investigated the c.m. motion of an $\alpha$-like quartet, which moves under
the influence of a core nucleus, here the 16O nucleus. In local density
approximation, an effective potential $W(R)$ for the quartet c.m. motion is
obtained, which shows a pocket structure near the surface of the nucleus. This
is important for the preformation of $\alpha$ particles near the surface. A
new aspect is the behavior of $W(R)$ inside the core nucleus, i.e. for $R\leq
r_{\rm crit}$, where the quartet bound state is dissolved due to Pauli
blocking. In contrast to earlier studies, which assume an increase in the
effective $\alpha-^{16}$O-potential with decreasing $R$, in a Thomas-Fermi
approach $W^{\rm TF}(R)=\mu_{4}$ remains constant in this range $R\leq r_{\rm
crit}$ Po14 ; Xu16 ; Xu17 ; wir . In the present work, we also show for the
shell model approach that the effective potential $W(R)$ remains almost
constant in the core nucleus. The reason for this is the exchange interaction
or Pauli blocking between the quartet nucleons and the core nucleus.

For large distances, the empirically determined M3Y potential used for $W(R)$
agrees with the optical potentials derived from scattering experiments. Near
the surface of the nucleus the Pauli blocking becomes relevant. A pocket that
is formed for the effective potential $W^{\rm TF}(R)$ is also retained after
the introduction of single-particle shell model states for the core nucleus.
However, the local density approximation for the Pauli blocking should be
improved, and it is expected that sharp peak structures observed for $W(R)$ in
shell model calculations will be smeared out.

Of interest is the comparison with the THSR approach THSR ; Toh17 , which
treats the quartets self-consistently. If a mean-field description for the
surrounding medium based on uncorrelated single-particle states is no longer
possible, correlations in the medium, especially quartetting, should be taken
into account. The full antisymmetrization of the many-body wave function is a
great challenge. The THSR approach offers us such a self-consistent,
antisymmetrized treatment of quartetting of all nucleons. A variational
principle with Gaussian wave functions was used, and nuclei with $A\leq 20$
were treated in this way. Interesting results were obtained for 20Ne Bo ; Bo2
; Bo3  considering the full antisymmetrization of the $\alpha$\- and 16O-wave
functions . We have tried to find appropriate observables in the THSR approach
to derive an effective potential $W(R)$ and a wave function $\psi(R)$ for the
quartet c.m. motion, to compare them with the quartetting wave function
approach.

Our general vision is to treat quartetting in the nuclear matter self-
consistently, as is the case for pairing. The approaches described in this
paper provide only partial answers to this project. The THSR approach comes
closest to this goal, but it contains some restrictions, so that it is not
generally applicable. Although the quartet wave function approach is generally
applicable, it contains several approximations that still need to be improved.
One main problem is the treatment of Pauli blocking. The local approximation
with a cut-off of $\alpha$-like clusters at the critical density needs to be
improved in future work.

### Acknowledgement

We would like to dedicate this work to the memory of our esteemed colleague
and friend, Peter Schuck, with whom we have had the privilege of collaborating
for many years. Peter’s broad interests and profound insights in nuclear
physics have been an inspiration to us all. We are deeply grateful for his
companionship and contributions throughout the years. He will be dearly
missed. His spirit and dedication to the pursuit of knowledge continue to
guide us, and in his memory, we commit to advancing the work he so
passionately embraced.

C. Xu is supported by the National Natural Science Foundation of China (Grant
No. 12275129). This work was supported in part by the National Natural Science
Foundation of China under contract Nos. 12175042,12147101. Zhongzhou REN
thanks the support of National Natural Science Foundation of China with grant
number 12035011. The work of G.R. received support via a joint stipend from
the Alexander von Humboldt Foundation and the Foundation for Polish Science.

##  Appendix A Optical model description and double-folding potential

We discuss the effective c.m. potential $W^{\rm TF}(R)$ and compare with other
approaches, see also wir . In particular, we check whether the choice (22) for
the double-folding potential parameters $c,\,\,d$ are realistic. Several
approaches to the optical potential are shown in Fig. 13.

The elastic scattering of $\alpha$ particles on the 16O nucleus was
investigated, and the corresponding optical potentials were inferred. There is
a large uncertainty for small values of $R$. A first expression for the real
part of the optical potential is Fadden66

$-\frac{V_{0}}{1+e^{(r-r_{0}A^{1/3})/a}}$ (39)

with $V_{0}=43.9$ MeV, $r_{0}=1.912$ fm and $a=0.451$ fm. Improvements were
madein Ref. Fukui16  considering the 16O (6Li,d) 20Ne transfer reaction, where
the model potential (39) with $r_{0}=1.25$ fm and $a=0.76$ fm was used,
$V_{0}$ was adjusted to reproduce the value 4.73 MeV of the binding energy.
Kumar and Kailas Kumar  give the parameter values $V_{0}=142.5$ MeV,
$r_{0}=1.18$ fm and $a_{0}=0.76$ fm.

Another approach Michel  was compared with experiments Oertzen ; Oertzen1 .
They used the expression

$-V_{0}\frac{1+\alpha e^{-(r/\rho)^{2}}}{[1+e^{(r-R_{R})/(2a_{R})}]^{2}}$ (40)

with $V_{0}=38$ MeV, $\rho=4.5$ fm, $R_{R}=4.3$ fm, $a_{R}=0.6$ fm, and the
energy-dependent $\alpha=3.625$. More recently, in Ref. Ohkubo  a density
dependent effective M3Y interaction (DDM3Y) was used, and a double-folding
potential was derived (Fig. 3 in Ohkubo ) which ranges at $R=0$ to -110 MeV.

Figure 13: Optical model potentials from $\alpha$ \- 16O scattering: The
double-folding Coulomb plus nucleon-nucleon interaction and the Thomas-Fermi
approach $W^{\rm TF}$ plus the medium-dependent $\alpha$-binding energy in
comparison with empirical expressions by Michel Michel , McFadden Fadden66 ,
and Kumar Kumar .

Note that $V_{\rm eff}(R)=W(R)+B_{\alpha}\approx V^{\rm Coul}_{\alpha-{\rm
O}}(R)+V^{\rm N-N}_{\alpha-{\rm O}}(R)$ is the mean field relative to the free
$\alpha$ particle. Below $R=5$ fm, Pauli blocking terms occur, see Eq. (17).
The agreement with Michel et al. Michel  is quite good. We conclude that the
choice (22) is reasonable. The standard approaches of the optical model
potentials have a diverging repulsive potential below $r_{\rm crit}$.

##  Appendix B $\alpha$-shifts

This Section is not yet completed, in preparation

In order to obtain a simple model to reproduce the essential properties of the
$\alpha$-particle, we consider a microscopic model to describe correlations.
With the separable interaction R1

$V(p_{1},p_{2};p^{\prime}_{1},p^{\prime}_{2})=-\frac{\lambda}{\Omega}e^{\frac{(p_{2}-p_{1})^{2}}{4\gamma^{2}}}e^{\frac{(p^{\prime}_{2}-p^{\prime}_{1})^{2}}{4\gamma^{2}}}\delta_{p_{1}+p_{2},p^{\prime}_{1}+p^{\prime}_{2}}\delta_{\sigma\tau,\sigma^{\prime}\tau^{\prime}}$
(41)

with $\Omega$ the normalization volume, $\lambda=1449.6$ MeV fm3,
$\gamma=1.152$ fm-1, we solve the $\alpha$ cluster within a variational ansatz

$\Phi_{\alpha}^{\rm Gauss}(p_{1},p_{2},p_{3},p_{4})=\frac{1}{\rm
norm^{2}}e^{-(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+p_{4}^{2})b^{2}/4}$ (42)

with the c.m. momentum $P=p_{1}+p_{2}+p_{3}+p_{4}$. The norm follows from

${\rm
norm}=\sum_{p}e^{-b^{2}p^{2}/2}=\int\frac{d^{3}p\Omega}{(2\pi)^{3}}e^{-b^{2}p^{2}/2}=\frac{\Omega}{(2\pi
b^{2})^{3/2}}$ (43)

We calculate the kinetic energy

$\displaystyle T=\frac{\hbar^{2}}{2m}\frac{1}{\rm
norm^{4}}\sum_{p}e^{-(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+p_{4}^{2})b^{2}/2}(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+p_{4}^{2})$
$\displaystyle=4\frac{\hbar^{2}}{2m}\frac{1}{\rm
norm}\int\frac{d^{3}p\Omega}{(2\pi)^{3}}e^{-b^{2}p^{2}/2}p^{2}=12\frac{\hbar^{2}}{2mb^{2}}$
(44)

From this, 1/4 is connected with the c.m. motion (introducing Jacobian
coordinates,
$p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+p_{4}^{2}=2q_{1}^{2}+\frac{3}{2}q_{2}^{2}+\frac{4}{3}q_{3}^{2}+\frac{1}{4}P^{2}$).
The intrinsic kinetic energy is $9\hbar^{2}/(2mb^{2})$.

The potential energy results as

$\displaystyle
4^{2}\frac{3}{4}\frac{1}{2}\sum_{12,1^{\prime}2^{\prime}}\phi(p_{1})\phi(p_{2})V(12,1^{\prime}2^{\prime})\phi(p^{\prime}_{1})\phi(p^{\prime}_{2})$
$\displaystyle=-6\lambda\frac{\gamma^{6}b^{3}}{\pi^{3/2}(\gamma^{2}b^{2}+2)^{3}}$
(45)

For the total energy the minimum -28.3087 MeV at $b=1.93354$ occurs. The
energy pro nucleon is -7.04 MeV. The empirical rms point radius is reproduced.

In a next step, we calculate the energy per nucleon $E^{\rm free}(n_{B})$ of
the symmetric matter, baryon density $n_{B}$, in a cubic box of length $La$
with periodic boundary conditions. The volume is $\Omega=(La)^{3}$. We have in
the average one nucleon with given spin and isospin in the elementary box
$a^{3}$, so that $n_{B}=4/a^{3}$. The total number of $\alpha$-particles is
$N_{\alpha}=L^{3}$, the total number of nucleons is $4N_{\alpha}$.

Free nucleon states with ${\bf k}=2\pi/(La)\\{n_{x},n_{y},n_{z}\\}$ are
introduced, which are occupied within the Fermi cube with $k_{F}=\pi/a$ in all
three directions $x,y,z$. Kinetic energy is

$\displaystyle T_{\rm cub}=4\times
3\frac{\hbar^{2}}{2m}\int_{-\pi/a}^{\pi/a}\frac{dk_{x}La}{2\pi}k_{x}^{2}\int_{-\pi/a}^{\pi/a}\frac{dk_{y}La}{2\pi}$
$\displaystyle\times\int_{-\pi/a}^{\pi/a}\frac{dk_{z}La}{2\pi}=\frac{\hbar^{2}}{m}2N_{\alpha}\frac{\pi^{2}}{a^{2}}.$
(46)

The KE per nucleon is $\frac{\hbar^{2}}{m}\frac{\pi^{2}}{2\times
4^{2/3}}n_{B}^{2/3}$. This value $\frac{\pi^{2}}{2\times 4^{2/3}}=1.958$ is a
little bit larger than $3/10(3\pi^{2}/2)^{2/3}=1.808$ for the Fermi sphere
instead of the Fermi cube.

The potential energy is

$\displaystyle V_{\rm cub}=4\times
3/2\sum_{12,1^{\prime}2^{\prime}}V(12,1^{\prime}2^{\prime})=-6\frac{\lambda}{\Omega}\int_{-\pi/a}^{\pi/a}\frac{dk_{1}^{x}La}{2\pi}$
$\displaystyle\dots\int_{-\pi/a}^{\pi/a}\frac{dk_{2}^{z}La}{2\pi}e^{-(k_{2}^{x}-k_{1}^{x})^{2}/2\gamma^{2}}\dots
e^{-(k_{2}^{z}-k_{1}^{z})^{2}/2\gamma^{2}}.$ (47)

With

$\displaystyle
I=\int_{-\pi/a}^{\pi/a}dk_{1}^{x}\int_{-\pi/a}^{\pi/a}dk_{2}^{x}e^{-(k_{2}^{x}-k_{1}^{x})^{2}/2\gamma^{2}}$
$\displaystyle=2\gamma\left[\left(e^{-2\pi^{2}/a^{2}\gamma^{2}}-1\right)\gamma+\frac{1}{a}2^{1/2}\pi^{3/2}{\rm
erf}(2^{1/2}\pi/a\gamma)\right]$

so that

$V_{\rm cub}=-6\frac{\lambda\Omega}{(2\pi)^{6}}I^{3},\qquad
a=(4/n_{B})^{1/3}.$ (49)

and the potential energy per nucleon $-6\frac{\lambda}{(2\pi)^{6}n_{B}}I^{3}$.
The energy per nucleon comes out as

$E_{\rm cub}(n_{B})=\frac{\hbar^{2}}{m}\frac{\pi^{2}}{2\times
4^{2/3}}n_{B}^{2/3}-6\frac{\lambda}{(2\pi)^{6}n_{B}}I^{3}.$ (50)

We compare this result with standard expressions. The chemical potential
contains the kinetic energy degeneracy 4, Fermi wave number
$k_{F}=(3\pi^{2}n/2)^{1/3}$)

$E_{\rm
kin,Fermi}(n)=\frac{\hbar^{2}}{2m}k_{F}^{2}=\frac{\hbar^{2}}{2m}\left(\frac{3\pi^{2}}{2}\right)^{2/3}n^{2/3}$
(51)

so that $\mu(n)=E_{\rm kin,Fermi}(n)+\Delta E^{\rm SE}(n)$. The self-energy
shift of the single-nucleon states can be estimated by the Skyrme model,

$\Delta E^{\rm SE}(n)=-\frac{3}{4}1057.3n+\frac{3}{16}14463.5n^{2}$ (52)

The energy per nucleon follows as

$\displaystyle E/N=\frac{1}{n}\int_{0}^{n}\mu(n^{\prime})dn^{\prime}$ (53)
$\displaystyle=\frac{3\hbar^{2}}{10m}\left(\frac{3\pi^{2}}{2}\right)^{2/3}n^{2/3}-\frac{3}{8}1057.3n+\frac{1}{16}14463.5n^{2}.$

A better parametrization is given by the RMF approach, the DD2 version gives

$\mu(n)=((mc^{2}-s(n))^{2}+(\hbar ck_{F})^{2})^{1/2}-mc^{2}+v(n)$ (54)

(for a parametrization of $s$ and $v$ see R .) The minimum occurs at
$E(0.148327{\rm fm}^{-3})=-16.2784$ MeV. In the subsaturated range of density,
the three approaches are in reasonable agreement, see Fig. 14.

Figure 14: Energy per nucleon as function of the nucleon density $n_{B}$.
Results for the separable potential (50) are compared with RMF-DD2 and Skyrme
calculations. The energy per nucleon for $\alpha$-matter (magenta full line)
takes at zero density the value -28.3/4 MeV.

Between both limits, the free $\alpha$-cluster gas at low densities and the
free-nucleon quasiparticle gas at high densities, we consider a Bloch ansatz
Tak04 .

$\phi_{k}(p)=\frac{1}{N_{k}^{1/2}N_{\alpha}^{1/2}}(2\pi
b^{2})^{1/4}\sum_{m=-N_{\alpha}/2}^{N_{\alpha}/2}e^{imka+impa-b^{2}p^{2}/4}$
(55)

The kinetic energy follows as (4 for spin/isospin, 3 for the components
$x,y,z$)

$\displaystyle T_{\rm Bloch}=4\times 3\frac{\hbar^{2}}{2m}\sum_{k}$
$\displaystyle\times\frac{\sum_{m_{1}m_{2}}\sum_{p}e^{i(m_{1}-m_{2})(k^{x}+p)a-b^{2}p^{2}/2}p^{2}}{\sum_{m_{1}m_{2}}\sum_{p^{\prime}}e^{i(m_{1}-m_{2})(k+{p^{\prime}})a-b^{2}{p^{\prime}}^{2}/2}}.$
(56)

After performing the $p,p^{\prime}$ integrals we have

$\displaystyle T_{\rm
Bloch}/N_{B}=3\frac{\hbar^{2}}{2mb^{2}}\int_{-\pi/a}^{\pi/a}\frac{dk^{x}a}{2\pi}$
$\displaystyle\times\frac{\sum_{m=-L/2}^{L/2}e^{imk^{x}a-a^{2}m^{2}/(2b^{2})}\left(1-\frac{m^{2}a^{2}}{b^{2}}\right)}{\sum_{m^{\prime}=-L/2}^{L/2}e^{im^{\prime}ka-a^{2}m^{\prime
2}/(2b^{2})}}.$ (57)

This expression contains the c.m. kinetic energy. To separate the c.m. energy,
it was proposes in Tak04  to consider the ratio $(x=a/b)$

$N_{c}(x)=1-\frac{1}{4}\left[1-\frac{\sum_{m=-L/2}^{L/2}e^{-x^{2}m^{2}/2}(m^{2}x^{2})}{\sum_{m^{\prime}=-L/2}^{L/2}e^{-x^{2}m^{\prime
2}/2}}\right]$ (58)

as a factor for the kinetic energy to exclude the c.m. kinetic energy in the
low-density region. The evaluation of the potential energy is somewhat lengthy
so that we give only the final result

$\displaystyle V_{\rm Bloch}=-\frac{3}{2\pi^{3/2}}\frac{\lambda
b^{3}}{(b^{2}+2/\gamma^{2})^{3}}\left(\int_{-\pi/a}^{\pi/a}\frac{dk_{1}a}{2\pi}\int_{-\pi/a}^{\pi/a}\frac{dk_{2}a}{2\pi}\right.$
$\displaystyle\left.\times\frac{\Sigma_{1}^{e}[\Sigma_{2}^{e}\Sigma_{3}^{e}+\Sigma_{2}^{o}\Sigma_{3}^{o}]+\Sigma_{1}^{o}[\Sigma_{2}^{e}\Sigma_{3}^{o}+\Sigma_{2}^{o}\Sigma_{3}^{e}]}{\Sigma_{4}(k_{1})\Sigma_{4}(k_{2})}\right)^{3}$
(59)

where

$\displaystyle\Sigma_{1}^{e}=\sum_{m}e^{i2m(k_{1}+k_{2})a/2-m^{2}x^{2}}$
$\displaystyle\Sigma_{1}^{o}=\sum_{m}e^{i(2m+1)(k_{1}+k_{2})a/2-(2m+1)^{2}x^{2}/4}$
$\displaystyle\Sigma_{2}^{e}=\sum_{m}e^{i2m(k_{2}-k_{1})a/2-2m^{2}a^{2}/(b^{2}+2/\gamma^{2})}$
$\displaystyle\Sigma_{2}^{o}=\sum_{m}e^{i(2m+1)(k_{2}-k_{1})a/2-(2m+1)^{2}a^{2}/(2b^{2}+4/\gamma^{2})}$
$\displaystyle\Sigma_{3}^{e}=\sum_{m}e^{i2m(k_{1}-k_{2})a/2-2m^{2}a^{2}/(b^{2}+2/\gamma^{2})}$
$\displaystyle\Sigma_{3}^{o}=\sum_{m}e^{i(2m+1)(k_{1}-k_{2})a/2-(2m+1)^{2}a^{2}/(2b^{2}+4/\gamma^{2})}$
$\displaystyle\Sigma_{4}(k)=\sum_{m}e^{imka-m^{2}a^{2}/(2b^{2})}$

Within our approach one has to search for the minimum of the energy as
function of the width parameter $b$, see Fig. 15. At low densities, the
minimum occurs at $b=1.934$ fm. This value is slightly increasing with
increasing density. At the nucleon density $n_{B}=0.0387$ fm-3 it jumps to the
free nucleon value, see Fig. 15. This is a sharp transition. It is not clear
whether this phase transition is due to the approximations such as the
separation of the c.m. kinetic energy or the Gauss ansatz for the wave
function, or is a real sharp quantum phase transition to a correlated state.
To understand the dissolution of the bound state in the case of a sharp
quantum phase transition, we can use the equilibrium condition of equal
chemical potential $\mu$ in both phases. With the energy per nucleon
$e(n_{B})=E/N_{B}$ we have

$\mu(n_{B})=e(n_{B})+n_{B}\frac{\partial e(n_{B})}{\partial n_{B}}.$ (61)

The disappearance of the $\alpha$-matter phase occurs if the chemical
potential coincides with the free momentum quasiparticle phase.

Figure 15: Energy per nucleon as function of the width parameter $b$ at
different nucleon density $n_{B}$. The result (50) for the uncorrelated
nuclear matter is shown as dotted line.

## References

  * (1) P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer, Berlin 1980).
  * (2) G. Röpke, P. Schuck, Y. Funaki, H. Horiuchi, Zhongzhou Ren, A. Tohsaki, Chang Xu, T. Yamada, and Bo Zhou, Phys. Rev. C 90, 034304 (2014).
  * (3) Chang Xu, Zhongzhou Ren, G. Röpke, P. Schuck, Y. Funaki, H. Horiuchi, A. Tohsaki, T. Yamada, and Bo Zhou, Phys. Rev. C 93, 011306(R) (2016).
  * (4) Chang Xu, G. Röpke, P. Schuck, Zhongzhou Ren, Y. Funaki, H. Horiuchi, A. Tohsaki, T. Yamada, and Bo Zhou, Phys. Rev. C 95, 061306(R) (2017).
  * (5) Shuo Yang, Chang Xu, Gerd Röpke, Peter Schuck, Zhongzhou Ren, Yasuro Funaki, Hisashi Horiuchi, Akihiro Tohsaki, Taiichi Yamada, and Bo Zhou, Phys. Rev. C 101, 024316 (2020).
  * (6) Shuo Yang, Chang Xu, and Gerd Röpke, Phys. Rev. C 104, 034302 (2021).
  * (7) Dong Bai, Zhongzhou Ren, and Gerd Röpke, Phys. Rev. C 99, 034305 (2019).
  * (8) Zhen Wang, Dong Bai, and Zhongzhou Ren, Phys. Rev. C 105, 024327 (2022).
  * (9) Zisheng Jin, Mingshuai Yan, Hao Zhou, An Cheng, Zhongzhou Ren, and Jian Liu, Phys. Rev. C 108, 014326 (2023).
  * (10) Ruijia Li and Chang Xu, Phys. Rev. C 107, 064301 (2023).
  * (11) Shuo Yang, Ruijia Li, and Chang Xu, Phys. Rev. C 108, L021303 (2023).
  * (12) A. Tohsaki, H. Horiuchi, P. Schuck, and G. Röpke, Phys. Rev. Lett. 87, 192501 (2001).
  * (13) Bo Zhou, Zhongzhou Ren, Chang Xu, Y. Funaki, T. Yamada, A. Tohsaki, H. Horiuchi, P. Schuck, and G. Röpke, Phys. Rev. C 86, 014301 (2012).
  * (14) Bo Zhou, Y. Funaki, H. Horiuchi, Zhongzhou Ren, G. Röpke, P. Schuck, A. Tohsaki, Chang Xu, and T. Yamada, Phys. Rev. Lett. 110, 262501 (2013).
  * (15) Bo Zhou, Y. Funaki, H. Horiuchi, Zhongzhou Ren, G. Röpke, P. Schuck, A. Tohsaki, Chang Xu, and T. Yamada, Phys. Rev. C 89, 034319 (2014).
  * (16) Mengjiao Lyu, Zhongzhou Ren, Bo Zhou, Yasuro Funaki, Hisashi Horiuchi, Gerd Röpke, Peter Schuck, Akihiro Tohsaki, Chang Xu, and Taiichi Yamada, Phys. Rev. C 91, 014313 (2015).
  * (17) Mengjiao Lyu, Zhongzhou Ren, Bo Zhou, Yasuro Funaki, Hisashi Horiuchi, Gerd Röpke, Peter Schuck, Akihiro Tohsaki, Chang Xu, and Taiichi Yamada, Phys. Rev. C 93, 054308 (2016).
  * (18) Bo Zhou, Yasuro Funaki, Hisashi Horiuchi, Yu-Gang Ma, Gerd Röpke, Peter Schuck, Akihiro Tohsaki, and Taiichi Yamada, Nature Comm. 14 (2023) 8206.
  * (19) M. Freer, H. Horiuchi, Y. Kanada-En’yo, D. Lee, and U.-G. Meißner Rev. Mod. Phys. 90, 035004 (2018).
  * (20) G. Röpke, P. Schuck, Chang Xu, Zhongzhou Ren, M. Lyu, Bo Zhou, Y. Funaki, H. Horiuchi, A. Tohsaki, and T. Yamada, J. Low. Temp. Phys. 189, 383 (2017) [arXiv:1707.04517 [nucl-th]].
  * (21) G. Röpke, P. Schuck, Y. Funaki, H. Horiuchi, Zhongzhou Ren, A. Tohsaki, Chang Xu, T. Yamada, and Bo Zhou, J. Phys. Conf. Ser. 863, 012006 (2017).
  * (22) G. Röpke,, in Proceedings of the 4th International Workshop on ”State of the Art in Nuclear Cluster Physics” (SOTANCP4), AIP Conf. Proc. 2038, 020008-1-020008-10 (AIP, New York, 2018). arXiv:1810.01274 [nucl-th].
  * (23) W.W. Qu, G.L. Zhang, X.Y. Le, Nucl. Phys. A 868 1 (2011).
  * (24) G.R. Satchler and W.G. Love, Phys. Rep. 55, 183 (1979).
  * (25) M. Mirea, Phys. Rev. C 96, 064607 (2017).
  * (26) N. Schwierz, I. Wiedenhover, and A. Volya, Parameterization of the Woods-Saxon Potential for Shell-Model Calculations. arXiv:0709.3525 (2007).
  * (27) Hiroki Takemoto, Masahiro Fukushima, Satoshi Chiba, Hisashi Horiuchi, Yoshinori Akaishi, and Akihiro Tohsaki, Phys. Rev. C 69, 035802 (2004).
  * (28) T. Matsuse, M. Kamimura, and Y. Fukushima, Prog. Theor. Phys. 53, 706 (1975).
  * (29) Y. Funaki, H. Horiuchi, G. Röpke, P. Schuck, A. Tohsaki, and T. Yamada, Phys. Rev. C 77, 064312 (2008).
  * (30) J.-P. Ebran, M. Girod, E. Khan, R. D. Lasseri, and P. Schuck, Phys. Rev. C 102, 014305 (2020).
  * (31) G. Röpke, Phys. Rev. C 79, 014002 (2009).
  * (32) G. Röpke, Nucl. Phys. A 867, 66 (2011).
  * (33) S. Typel, Phys. Rev. C 89, 064321 (2014).
  * (34) Z. Yang, J. Tanaka, S. Typel, T. Aumann, J. Zenihiro, S. Adachi, S. Bai, P. van Beek, D. Beaume, Y. Fujikawa, J. Han, S. Heil, S. Huang, A. Inoue, Y. Jiang, M. Knösel, N. Kobayashi, Y. Kubota, W. Liu, J. Lou, Y. Maeda, Y. Matsuda, K. Miki, S. Nakamura, K. Ogata, V. Panin, H. Scheit, F. Schindler, P. Schrock, D. Symochko, A. Tamii, T. Uesaka, V. Wagner, K. Yoshida, JPS Conf. Proc. 31, 011019 (2020).
  * (35) J. Tanaka, Z.H. Yang, S. Typel, S. Adachi, S. Bai, P. Van Beek, D. Beaumel, Y. Fujikawa, J. Han, S. Heil, S. Huang, A. Inoue, Y. Jiang, M. Knösel, N. Kobayashi, Y. Kubota, W. Liu, J. Lou, Y. Maeda, Y. Matsuda, K. Miki, S. Nakamura, K. Ogata, V. Panin, H. Scheit, F. Schindler, P. Schrock, D. Symochko, A. Tamii, T. Uesaka, V. Wagner, K. Yoshida, J. Zenihiro, T. Aumann, Science 371, 200 (2021).
  * (36) Y. Kanada-En’yo, Prog.. Theor. Phys. 117, 655 (2007).
  * (37) A. Ono, H. Horiuchi, Prog. Part. Nucl. Phys. 53 (2004) 501.
  * (38) K. Yoshida, Y. Chiba, M. Kimura, Y. Taniguchi, Y. Kanada-En’yo, and K. Ogata, Phys. Rev. C 100, 044601 (2019).
  * (39) Y. Chiba and M. Kimura, Prog. Theor. Exp. Phys. 2017, 053D01 (2017).
  * (40) Y. Taniguchi, K. Yoshida, Y. Chiba, Y. Kanada-En’yo, M. Kimura, and K. Ogata, PRC 103, L031305 (2021).
  * (41) K. Yoshida and J. Tanaka, Phys. Rev. C 106, 014621 (2022).
  * (42) C. Qi, A.N. Andreyev, M. Huyse, R.J. Liotta, P. Van Duppen, and R.A. Wyss, Phys. Rev. C 81, 064319 (2010).
  * (43) T. Nakatsukasa, N. Hinohara, Phys. Rev. C 108, 014318 (2023).
  * (44) Q. Zhao, M. Kimura, B. Zhou, and S. Shin, Phys. Rev. C 106, 054313 (2022).
  * (45) P.-G. Reinhard, J. A. Maruhn, A. S. Umar, and V. E. Oberacker, Phys. Rev. C 83, 034312 (2011).
  * (46) A. Tohsaki, H. Horiuchi, P. Schuck, and G. Röpke, Rev. Mod. Phys. 89, 011002 (2017).
  * (47) Bo Zhou, Zhongzhou Ren, Chang Xu, Y. Funaki, T. Yamada, A. Tohsaki, H. Horiuchi, P. Schuck, and G. Röpke, Phys. Rev. C 86, 014301 (2012).
  * (48) Bo Zhou, Y. Funaki, H. Horiuchi, Zhongzhou Ren, G. Röpke, P. Schuck, A. Tohsaki, Chang Xu, and T. Yamada, Phys. Rev. C 89, 034319 (2014).
  * (49) Bo Zhou, Y. Funaki, H. Horiuchi, Zhongzhou Ren, G. Röpke, P. Schuck, A. Tohsaki, Chang Xu, and T. Yamada, Phys. Rev. Lett. 110, 262501 (2013).
  * (50) L. McFadden and G.R. Satchler, Nucl. Phys. 84, 177 (1966).
  * (51) F. Michel, J. Albinski, P. Belery, T. Delbar, G. Grégoire, B. Tasiaux, and G. Reidemeister, Phys. Rev. C 28, 1904 (1983).
  * (52) W. von Oertzen, M. Freer, and Y. Kanada-Enyo, Phys. Reports 432, 43(2006);
  * (53) W. von Oertzen, in: C. Beck (Ed.), Clusters in Nuclei, vol. 1, p.109 (Springer, Berlin 2010).
  * (54) Y. Hirabayashi and S. Ohkubo, Phys. Rev. C 88, 014314 (2013).
  * (55) T. Fukui, Y. Taniguchi, T. Suhara, Y. Kanada-En’yo, and K. Ogata, Phys. Rev. C 93, 034606 (2016).
  * (56) Ashok Kumar and S. Kailas, Nucl. Phys. A 776, 105 (2006).
  * (57) G. Röpke, Phys. Rev. C 92, 054001 (2015).