# Northcott numbers for the weighted Weil heights

Masao Okazaki, Kaoru Sano  Graduate School of Mathematics, Kyushu University,
Motooka 744, Nishi-ku, Fukuoka 819-0395, Japan<EMAIL_ADDRESS>Faculty of Science and Engineering, Doshisha University, Kyoto, 610-0394,
Japan<EMAIL_ADDRESS>
###### Abstract.

We answer the question of Vidaux and Videla about the distribution of the
Northcott numbers for the Weil height. We solve the same problem for the
weighted Weil heights. These heights generalize both the absolute and relative
Weil height. Our results also refine those of Pazuki, Technau, and Widmer.

###### Key words and phrases:

wighted Weil height, Northcott property, Bogomolov property, Northcott number

###### 2020 Mathematics Subject Classification:

Primary 11G50

##  1\. Introduction

Let $h:\overline{\mathbb{Q}}\rightarrow\mathbb{R}_{\geq 0}$ be the absolute
logarithmic Weil height. For each $\gamma\in\mathbb{R}$ and
$a\in\overline{\mathbb{Q}}$, we set

$h_{\gamma}(a):=\deg(a)^{\gamma}h(a),$

where $\deg(a):=[\mathbb{Q}(a):\mathbb{Q}]$. We call $h_{\gamma}$ the
$\gamma$-weighted Weil height, introduced in [PTW21]. We remark that $h_{0}$
(resp. $h_{1})$ is the absolute (resp. relative) Weil height. For each
$A\subset\overline{\mathbb{Q}}$ and $C>0$, we set

$\displaystyle N_{\gamma}(A,C)$ $\displaystyle:=\\#\left\\{a\in A\mid
h_{\gamma}(a)<C\right\\}\text{ and }$ $\displaystyle\mu_{A}$
$\displaystyle:=\left\\{\text{root of unity in }A\right\\}.$

Here note that the equality $\mu_{A}=\left\\{a\in A\mid
h_{\gamma}(a)=0\right\\}$ holds for all $\gamma\in\mathbb{R}$ (see, e.g.,
[BG06, Theorem 1.5.9]). The following definitions are fundamental to study the
finiteness for the weighted heights.

######  Definition 1.1 ([BZ01] and [PTW21]).

  * (1)

We say that $A$ has the $\gamma$-Northcott property (or $\gamma$-(N) for
short) if $N_{\gamma}(A,C)<\infty$ holds for all $C>0$.

  * (2)

We say that $A$ has the $\gamma$-Bogomolov property (or $\gamma$-(B) for
short) if $N_{\gamma}(A\setminus\mu_{A},C)<\infty$ holds for some $C>0$.

For each subset $A\subset\overline{\mathbb{Q}}$, we set

$\text{Nor}_{\gamma}(A):=\inf\left\\{C>0\mid N_{\gamma}(A,C)=\infty\right\\}$

and call it the $\gamma$-Northcott number of $A$ (see [PTW21] or [VV16]). By
definition, a subset $A$ has $\gamma$-(N) (resp. $\gamma$-(B)) if and only if
$\text{Nor}_{\gamma}(A)=\infty$ (resp.
$\text{Nor}_{\gamma}(A\setminus\mu_{A})>0$). Note that $\gamma$-(B)
immediately follows from $\gamma$-(N). The property $0$-(N) (resp. $0$-(B)) is
usually called the Northcott property (resp. Bogomolov property). There are
several examples of infinite extensions of $\mathbb{Q}$ that has $0$-(N) or
$0$-(B) (see, e.g., [AD00], [BZ01], [DZ08], [Pot16], [Sch73], or [Wid11]). On
the one hand, the Lehmer conjecture asserts that $\overline{\mathbb{Q}}$ has
$1$-(B) and
$\text{Nor}_{1}(\overline{\mathbb{Q}}\setminus\mu_{\overline{\mathbb{Q}}})\geq\log(s)$,
where $s=1.176\ldots$ is the smallest known Salem number (see, e.g., [Sil07,
p.100]). In [PTW21], they gave fields that have $1$-(B) but not $0$-(B). More
strictly, for each $\gamma\leq 1$ and $\epsilon>0$, they constructed a field
$L\subset\overline{\mathbb{Q}}$ that has $\gamma$-(N) but not
$(\gamma-\epsilon)$-(B) (see Theorem 4.3 for the explicit construction of
$L$). As we will see in Lemma 2.1, the sets

$\displaystyle I_{N}(A)$ $\displaystyle:=\left\\{\gamma\in\mathbb{R}\mid
A\text{ has }\gamma\text{-(N)}\right\\}\text{ and }$ $\displaystyle I_{B}(A)$
$\displaystyle:=\left\\{\gamma\in\mathbb{R}\mid A\text{ has
}\gamma\text{-(B)}\right\\}$

are intervals of forms $(\gamma,\infty)$ or $[\gamma,\infty)$ for some
$\gamma\in\mathbb{R}\cup\\{-\infty\\}$. Thus the field $L$ in [PTW21]
satisfies that $(\gamma-\epsilon,\infty)\supset I_{B}(L)\supset
I_{N}(L)\supset[\gamma,\infty)$. However, the values of $\inf I_{B}(L)$ and
$\inf I_{N}(L)$ were not mentioned in [PTW21]:

Here we remark that $\inf I_{N}(L\setminus\mu_{L})=\inf I_{B}(L)$ holds as we
will see in Corollary 2.4. Furthermore, in [PTW21, Theorem 3], they also
constructed a field $L$ such that $c\leq\text{Nor}_{0}(L)\leq 2c$ for a given
$c>0$. Thus the following question naturally arises.

######  Question 1.2.

For a given $\gamma\leq 1$, is there a field $L\subset\overline{\mathbb{Q}}$
satisfying the following conditions (1), (2), or (3), respectively?

  * (1)

$I_{N}(L)=I_{B}(L)=[\gamma,\infty)$.

  * (2)

$I_{N}(L)=(\gamma,\infty)\subsetneq[\gamma,\infty)=I_{B}(L)$ with ${\rm
Nor}_{\gamma}(L)=c$ for a given $c>0$.

  * (3)

$I_{N}(L)=I_{B}(L)=(\gamma,\infty)$.

######  Remark 1.3.

In Remark 2.6, we will see that
$I_{N}(\overline{\mathbb{Q}}\setminus\mu_{\overline{\mathbb{Q}}})=(1,\infty)$.
Thus, in Question 1.2, we restricted our attention to the case $\gamma\leq 1$.

Our main result is a positive answer to Question 1.2 of the case $\gamma<1$:

######  Theorem 1.4.

Let $\gamma<1$ be a real number. We can construct a field
$L\subset\overline{\mathbb{Q}}$ satisfying the condition (1), (2), or (3) in
Question 1.2, respectively.

######  Remark 1.5.

Theorem 1.4 especially answers [VV16, Question 6].

######  Remark 1.6.

We note that Theorem 1.4 covers neither the case $\gamma=1$ nor
$\gamma=-\infty$. We will discuss fields $L$ satisfying each of
$I_{N}(L)=I_{B}(L)=[1,\infty)$,
$I_{N}(L)=(1,\infty)\subsetneq[1,\infty)=I_{B}(L)$, and
$I_{N}(L)=I_{B}(L)=\mathbb{R}$ in Section 5.

######  Remark 1.7.

Another remarkable result related to calculating Northcott numbers is [PTW21,
Theorem 1]. Pazuki, Technau, and Widmer constructed a field such that the
Northcott number for the house on its ring of integers equals a given real
number $c\geq 1$. Due to its simple definition, some stronger results of the
house are known. Indeed, [PTW21, Theorem 1] also succeeded in controlling the
distribution of integers whose value of the house converges in the Northcott
number. In addition, Dimitrov claims to have proved an analogue of the Lehmer
conjecture for the house ([Dim19, Theorem 1]).

The other result concerns the following theorem.

######  Theorem 1.8 (e.g., [DZ08, Theorem 2.1]).

Let $L\subset\overline{\mathbb{Q}}$ be a field with $0$-(N). Then for any
$d>0$, the set

$L^{(d)}:=\left\\{a\in\overline{\mathbb{Q}}\mid[L(a):L]\leq d\right\\}$

also has $0$-(N).

The well-known fact that any number field has $0$-(N) is an immediate
consequence of Theorem 1.8. We give a simple counterexample to the
generalization of Theorem 1.8 to the case of $h_{\gamma}$.

######  Proposition 1.9.

Let $\mathbb{Q}^{\rm tr}$ be the field of all totally real numbers. Then
$\mathbb{Q}^{\rm tr}$ has $\gamma$-(N) for all $\gamma>0$ but $\mathbb{Q}^{\rm
tr}(\sqrt{-1})$ does not have $\gamma$-(N) for all $0<\gamma\leq 1$. More
strictly, the set $\mathbb{Q}^{\rm tr}(\sqrt{-1})\setminus\mu_{\mathbb{Q}^{\rm
tr}(\sqrt{-1})}$ does not have $\gamma$-(N) for all $0<\gamma\leq 1$.

### Outline of this paper

In Section 2, we study properties of $I_{N}(A)$ and $I_{B}(A)$ that we use
later. As a consequence of Proposition 2.3, we prove Proposition 1.9 at the
end of Section 2. In Section 3, we give some lemmata needed to calculate
Northcott numbers. In Section 4, we give a more explicit form of Theorem 1.4
and prove it. We also compare our fields with those constructed in [PTW21]. In
Section 5, we give some supplemental remarks on the cases $\gamma=1$ and
$\gamma=-\infty$ of Question 1.2.

##  2\. Intervals associated with $\gamma$-(N) and $\gamma$-(B)

Let $A\subset\overline{\mathbb{Q}}$. In this section, we give some basic
properties of $I_{N}(A)$ and $I_{B}(A)$ defined in Section 1.

######  Lemma 2.1.

Let $\gamma\in\mathbb{R}$. If $A\subset\overline{\mathbb{Q}}$ has $\gamma$-(N)
(resp. $\gamma$-(B)), the set $A$ also has $\delta$-(N) (resp. $\delta$-(B))
for all $\delta>\gamma$.

###### Proof.

Let $\delta>\gamma$ and $A\subset\overline{\mathbb{Q}}$. Since the inequality
$h_{\gamma}(a)\leq h_{\delta}(a)$ holds for all $a\in\overline{\mathbb{Q}}$,
the property $\delta$-(N) of $A$ immediately follows from $\gamma$-(N) of $A$.
On the one hand, a set $A$ having $\gamma$-(B) means that there exists a
constant $D>0$ such that $h_{\gamma}(a)\geq D$ for all $a\in
A\setminus\mu_{A}$. Thus $\delta$-(B) of $A$ also follows from $\gamma$-(B) of
$A$ since we know that $h_{\delta}(a)\geq h_{\gamma}(a)\geq D$ for all $a\in
A\setminus\mu_{A}$ and $\mu_{A}=\left\\{a\in A\mid h_{\delta}(a)=0\right\\}$.
∎

######  Proposition 2.2.

For each subset $A\subset\overline{\mathbb{Q}}$, the sets $I_{N}(A)$ and
$I_{B}(A)$ are intervals of forms $(\gamma,\infty)$ or $[\gamma,\infty)$ for
some $\gamma\in\mathbb{R}\cup\\{-\infty\\}$.

###### Proof.

This is an immediate consequence of Lemma 2.1. ∎

By the definitions of $\gamma$-(N) and $\gamma$-(B), we know that
$I_{N}(A)\subset I_{B}(A)$, i.e., $\inf I_{N}(A)\geq\inf I_{B}(A)$. We
actually have $\inf I_{N}(A\setminus\mu_{A})=\inf I_{B}(A)$. This is an
immediate consequence of the following proposition.

######  Proposition 2.3.

Let $\gamma\in\mathbb{R}$. If $A\subset\overline{\mathbb{Q}}$ has
$\gamma$-(B), the set $A\setminus\mu_{A}$ has $\delta$-(N) for all
$\delta>\gamma$.

###### Proof.

Take any $C>0$ and $a\in A\setminus\mu_{A}$ with $h_{\delta}(a)<C$. Since $A$
has $\gamma$-(B), there exists a constant $D>0$ independent of $a\in
A\setminus\mu_{A}$ such that $h_{\gamma}(a)\geq D$. Thus we have

$\deg(a)=\left(\frac{h_{\delta}(a)}{h_{\gamma}(a)}\right)^{\frac{1}{\delta-\gamma}}<\left(\frac{C}{D}\right)^{\frac{1}{\delta-\gamma}}.$

If $\delta\geq 0$, then we have

$h(a)\leq\deg(a)^{\delta}h(a)=h_{\delta}(a)<C.$

By Theorem 1.8, the conditions (2) and (2) conclude that
$N_{\delta}(A\setminus\mu_{A},C)<\infty$ for the case $\delta\geq 0$.

On the one hand, if $\delta<0$, then (2) implies that
$\deg(a)^{-\delta}<(C/D)^{\frac{-\delta}{\delta-\gamma}}$. Therefore we have

$h(a)=\deg(a)^{-\delta}h_{\delta}(a)<\left(\frac{C}{D}\right)^{\frac{-\delta}{\delta-\gamma}}C.$

Again by Theorem 1.8, the conditions (2) and (2) also conclude that
$N_{\delta}(A\setminus\mu_{A},C)<\infty$ for the case $\delta<0$. ∎

######  Corollary 2.4.

For any subset $A\subset\overline{\mathbb{Q}}$, the equality $\inf
I_{N}(A\setminus\mu_{A})=\inf I_{B}(A)$ holds. In addition, if $A$ contains
only finitely many roots of unity, we have $\inf I_{N}(A)=\inf I_{B}(A)$.

######  Remark 2.5.

We can not remove the assumption “excluding $\mu_{A}$” in Proposition 2.3.
This is because $\mathbb{Q}^{\rm ab}$, the maximal abelian extension of
$\mathbb{Q}$, has $0$-(B) (see, e.g., [AD00], [AZ00], or [AZ10]) but not
$\gamma$-(N) for all $\gamma\in\mathbb{R}$ since $\mathbb{Q}^{\rm ab}$
contains all roots of unity.

######  Remark 2.6.

In [Dob79], Dobrowolski proved that there exists $D>0$ such that $f(a)\geq D$
for all $a\in\overline{\mathbb{Q}}^{\times}$, where

$f(a):=\left(\frac{\log^{+}(\deg(a))}{\log^{+}\log(\deg(a))}\right)^{3}h_{1}(a)$

and $\log^{+}(\cdot):=\max\\{1,\log(\cdot)\\}$. It is clear that for any
$\varepsilon>0$, if $\deg(a)$ is sufficiently large, the inequality

$\deg(a)^{\varepsilon}>\left(\frac{\log^{+}(\deg(a))}{\log^{+}\log(\deg(a))}\right)^{3}$

holds. Thus $\overline{\mathbb{Q}}$ has $(1+\varepsilon)$-(B) for any
$\varepsilon>0$ by Theorem 1.8. Therefore, by Proposition 2.3, the set
$\overline{\mathbb{Q}}\setminus\mu_{\overline{\mathbb{Q}}}$ has $\gamma$-(N)
for all $\gamma>1$. Note that this immediately follows from Proposition 2.3 if
we assume the Lehmer conjecture.

As a consequence of Proposition 2.3, we prove Proposition 1.9.

###### Proof of Proposition 1.9.

It is known that $\mathbb{Q}^{\rm tr}$ has $0$-(B) (see [Pot16] or [Sch73]).
Thus $\mathbb{Q}^{\textrm{tr}}$ has $\gamma$-(N) for each $\gamma>0$ by the
equality $\mu_{\mathbb{Q}^{\textrm{tr}}}=\\{\pm 1\\}$ and Proposition 2.3. On
the one hand, the field $\mathbb{Q}^{\textrm{tr}}(\sqrt{-1})$ does not have
$\gamma$-(N) for all $0<\gamma\leq 1$. Indeed, the algebraic number
$a_{k}:=((2-\sqrt{-1})/(2+\sqrt{-1}))^{1/k}$ is an element of
$\mathbb{Q}^{\textrm{tr}}(\sqrt{-1})$ for each $k\in\mathbb{Z}_{>0}$ (see,
e.g., [ASZ14, Section 5]). Since we have the inequalities

$h_{\gamma}(a_{k})=\deg(a_{k})^{\gamma}h(a_{1}^{1/k})\leq\frac{2^{\gamma}}{k^{1-\gamma}}h(a_{1})\leq
2h(a_{1}),$

the field $\mathbb{Q}^{\textrm{tr}}(\sqrt{-1})$ has infinitely many elements
with bounded value of $h_{\gamma}$. ∎

##  3\. Some remarks on height lower bounds

Throughout the rest of the paper, we denote the set of positive integers by
$\mathbb{N}$. This section is devoted to giving some technical lemmata to
calculate the $\gamma$-Northcott numbers. Theorem 3.1 and Lemma 3.2 allow us
to get a lower bound for the Northcott number of our fields.

######  Theorem 3.1.

Let $K$ be a number field. Assume that $a\in\overline{\mathbb{Q}}$ satisfies
that $[K(a):K]>1$. We set $M:=K(a)$ and $m:=[M:K]$. Then we have the
inequality

$h(a)\geq\frac{1}{2(m-1)}\left(\frac{\log(N_{K/\mathbb{Q}}(D_{M/K}))}{m[K:\mathbb{Q}]}-\log(m)\right),$

where $N_{K/\mathbb{Q}}$ is the usual norm and $D_{M/K}$ is the relative
discriminant ideal of the extension $M/K$.

###### Proof.

See [Sil84, Theorem 2]. ∎

######  Lemma 3.2.

Let $\gamma\in\mathbb{R}$ and $A\subset\overline{\mathbb{Q}}$. We set

$\delta_{\gamma}(B):=\inf h_{\gamma}(B)$

for each non-empty subset $B\subset A$. Let $A_{0}\subsetneq A_{1}\subsetneq
A_{2}\subsetneq\cdots$ be an ascending chain of non-empty subsets of $A$
satisfying that

  * (1)

$A_{i}$ has $\gamma$-(N) for all $i\in\mathbb{Z}_{\geq 0}$ and

  * (2)

$A=\bigcup_{i\in\mathbb{Z}_{\geq 0}}A_{i}$.

Then we have

${\rm
Nor}_{\gamma}(A)=\liminf_{i\rightarrow\infty}\delta_{\gamma}(A_{i}\setminus
A_{i-1}).$

###### Proof.

See [PTW21, Lemma 2]. ∎

######  Proposition 3.3.

Let $\gamma\leq 1$ be a real number. We also let $(p_{i})_{i\in\mathbb{N}}$,
$(q_{i})_{i\in\mathbb{N}}$, and $(d_{i})_{i\in\mathbb{N}}$ be strictly
increasing sequences of prime numbers. Assume that there exists
$i_{0}\in\mathbb{N}$ such that for all $i>i_{0}$, the inequality $p_{i}<q_{i}$
and

$p_{i},q_{i}\notin\\{d_{1},p_{1},q_{1},\ldots,d_{i-1},p_{i-1},q_{i-1}\\}$

hold. We set

$V(i,\gamma):=\begin{cases}\displaystyle\log(p_{i})-\frac{\log(d_{i})}{2}&(\gamma=1),\\\
\displaystyle\frac{\log(p_{i})}{d_{i}^{1-\gamma}}&(0\leq\gamma<1),\\\
\displaystyle\frac{\log(p_{i})}{(d_{1}\cdots
d_{i-1})^{-\gamma}d_{i}^{1-\gamma}}&(\gamma<0).\end{cases}$

Then the field $L:=\mathbb{Q}((p_{i}/q_{i})^{1/d_{i}}\mid i\in\mathbb{N})$
satisfies that

${\rm Nor}_{\gamma}(L)\geq\liminf_{i\rightarrow\infty}V(i,\gamma).$

Especially, we have the followings.

  * (1)

$L$ has $\gamma$-(N) if $\liminf_{i\rightarrow\infty}V(i,\gamma)=\infty$ and

  * (2)

$L$ has $\gamma$-(B) if $\liminf_{i\rightarrow\infty}V(i,\gamma)>0$.

###### Proof.

We set

$K_{0}:=\mathbb{Q},\ K_{i}:=K_{i-1}((p_{i}/q_{i})^{1/d_{i}}),\ \text{and}\
F_{i}:=\mathbb{Q}((p_{i}/q_{i})^{1/d_{i}})$

for each $i\in\mathbb{N}$. By Lemma 3.2, it is enugh to show that the
inequality

$\liminf_{i\rightarrow\infty}\delta_{\gamma}(K_{i}\setminus
K_{i-1})\geq\liminf_{i\rightarrow\infty}V(i,\gamma)$

holds. Take any $i>i_{0}$ and $a\in K_{i}\setminus K_{i-1}$. Note that the
equality $K_{i-1}(a)=K_{i}$ holds since $[K_{i}:K_{i-1}]=d_{i}$ is a prime
number and $a\notin K_{i-1}$. Thus we have the inequalities

$d_{i}\leq\deg(a)\leq d_{1}\cdots d_{i}$

and

$h_{\gamma}(a)\geq\frac{\deg(a)^{\gamma}}{2(d_{i}-1)}\left(\frac{\log(N_{K_{i-1}/\mathbb{Q}}(D_{K_{i}/K_{i-1}}))}{d_{i}[K_{i-1}:\mathbb{Q}]}-\log(d_{i})\right)$

by Theorem 3.1. To estimate $h_{\gamma}(a)$ from below, we give a lower bound
for $N_{K_{i-1}/\mathbb{Q}}(D_{K_{i}/K_{i-1}})$. Since
$F_{i}=\mathbb{Q}((p_{i}q_{i}^{d_{i}-1})^{1/d_{i}})$ and
$(p_{i}q_{i}^{d_{i}-1})^{1/d_{i}}$ is a root of the $p_{i}$-Eisenstein
polynomial $X^{d_{i}}-p_{i}d_{i}^{d_{i}-1}\in\mathbb{Q}[X]$, we know that
$p_{i}$ ramifies totally in $F_{i}$ (see, e.g., [FT93, Theorem 24 (a)]). Thus
it holds that $p_{i}^{d_{i}-1}\mid D_{F_{i}/\mathbb{Q}}$ (see, e.g., [Neu99,
p.199, (2.6)] and [Neu99, p.201, (2.9)]). Now we note that the equalities

$N_{K_{i-1}/\mathbb{Q}}(D_{K_{i}/K_{i-1}})D_{K_{i-1}/\mathbb{Q}}^{[K_{i}:K_{i-1}]}=D_{K_{i}/\mathbb{Q}}=N_{F_{i}/\mathbb{Q}}(D_{K_{i}/F_{i}})D_{F_{i}/\mathbb{Q}}^{[K_{i}:F_{i}]}$

hold (see, e.g., [Neu99, p.202, (2.10)]). Since $K_{i-1}$ is the compositum of
$F_{1}$, $\ldots$ , $F_{i-1}$ and
$p_{i}\notin\\{d_{1},p_{1},q_{1},\ldots,d_{i-1},p_{i-1},q_{i-1}\\}$, we know
that $p_{i}$ does not ramify in $K_{i-1}$ (see, e.g., [Hil98, Theorem 85] and
[Viv04, Lemma 4.1]). Hence $p_{i}\nmid D_{K_{i-1}/\mathbb{Q}}$ holds.
Therefore (3) yields that

$p_{i}^{[K_{i}:F_{i}](d_{i}-1)}\mid
N_{K_{i-1}/\mathbb{Q}}(D_{K_{i}/K_{i-1}}).$

Replacing $p_{i}$ with $q_{i}$ in the above discussion, we also get

$q_{i}^{[K_{i}:F_{i}](d_{i}-1)}\mid
N_{K_{i-1}/\mathbb{Q}}(D_{K_{i}/K_{i-1}}).$

Since the equality $[K_{i}:F_{i}]=d_{1}\cdots d_{i-1}=[K_{i-1}:\mathbb{Q}]$
holds, the conditions (3) and (3) yield that

$2\log(p_{i})\leq\log(p_{i}q_{i})\leq\frac{\log(N_{K_{i-1}/\mathbb{Q}}(D_{K_{i}/K_{i-1}}))}{[K_{i-1}:\mathbb{Q}](d_{i}-1)}.$

Thus we conclude that

$\displaystyle~{}h_{\gamma}(a)$ $\displaystyle\geq$
$\displaystyle~{}\deg(a)^{\gamma}\left(\frac{\log(p_{i})}{d_{i}}-\frac{\log(d_{i})}{2(d_{i}-1)}\right)$
$\displaystyle\text{by }(\ref{3.3}),(\ref{3.7})$ $\displaystyle=$
$\displaystyle~{}\left(\frac{\deg(a)}{d_{i}}\right)^{\gamma}\left(\frac{\log(p_{i})}{d_{i}^{1-\gamma}}-\frac{d_{i}^{\gamma}\log(d_{i})}{2(d_{i}-1)}\right)$
$\displaystyle\geq$
$\displaystyle~{}\begin{cases}\displaystyle\frac{\log(p_{i})}{d_{i}^{1-\gamma}}-\frac{d_{i}^{\gamma}\log(d_{i})}{2(d_{i}-1)}&(0\leq\gamma\leq
1)\\\ \displaystyle(d_{1}\cdots
d_{i-1})^{\gamma}\left(\frac{\log(p_{i})}{d_{i}^{1-\gamma}}-\frac{d_{i}^{\gamma}\log(d_{i})}{2(d_{i}-1)}\right)&(\gamma<0)\end{cases}$
$\displaystyle\text{by }(\ref{3.2})$ $\displaystyle=$
$\displaystyle~{}\begin{cases}\displaystyle
V(i,1)-\frac{\log(d_{i})}{2(d_{i}-1)}&(\gamma=1),\\\ \displaystyle
V(i,\gamma)-\frac{d_{i}^{\gamma}\log(d_{i})}{2(d_{i}-1)}&(0\leq\gamma<1),\\\
\displaystyle V(i,\gamma)-\frac{\log(d_{i})}{2(d_{1}\cdots
d_{i})^{-\gamma}(d_{i}-1)}&(\gamma<0).\end{cases}$

Since we have the equalities

$\displaystyle\lim_{i\rightarrow\infty}\frac{\log(d_{i})}{2(d_{i}-1)}=0,$
$\displaystyle\lim_{i\rightarrow\infty}\frac{d_{i}^{\gamma}\log(d_{i})}{2(d_{i}-1)}=0$
$\displaystyle(0\leq\gamma<1),\text{ and}$
$\displaystyle\lim_{i\rightarrow\infty}\frac{\log(d_{i})}{2(d_{1}\cdots
d_{i})^{-\gamma}(d_{i}-1)}=0$ $\displaystyle(\gamma<0),$

we get the inequality (3) by letting $i\rightarrow\infty$. ∎

##  4\. On Theorem 1.4

###  4.1. Proof of Theorem 1.4

In this section, we prove Theorem 1.4. More precisely, we prove the following
theorem.

######  Theorem 4.1.

Let $\gamma<1$ and $c>0$ be real numbers. We also let
$(p_{i})_{i\in\mathbb{N}}$, $(q_{i})_{i\in\mathbb{N}}$, and
$(d_{i})_{i\in\mathbb{N}}$ be strictly increasing sequences of prime numbers
with $q_{i}<p_{i+1}$ and $p_{i}<q_{i}<2p_{i}$ for all $i\in\mathbb{N}$.
Furthermore, we let $f(x)$ be $\log(x)$, $c$, or $1/\log(x)$. If $\gamma\geq
0$, then we assume that the inequalities

$\exp(f(d_{i})d_{i}^{1-\gamma})\leq p_{i}\leq 2\exp(f(d_{i})d_{i}^{1-\gamma})$

hold for all $i\in\mathbb{N}$. On the one hand, if $\gamma<0$, then we assume
that the inequalities

$\exp(f(d_{i})(d_{1}\cdots d_{i-1})^{-\gamma}d_{i}^{1-\gamma})\leq p_{i}\leq
2\exp(f(d_{i})(d_{1}\cdots d_{i-1})^{-\gamma}d_{i}^{1-\gamma})$

hold for all $i\in\mathbb{N}$ and the equality

$\lim_{i\rightarrow\infty}\frac{i\log(d_{i})}{d_{i}^{-\gamma}}=0$

holds. We set $L:=\mathbb{Q}((p_{i}/q_{i})^{1/d_{i}}\mid i\in\mathbb{N})$.

  * (1)

If $f(x)=\log(x)$, the field $L$ satisfies that
$I_{N}(L)=I_{B}(L)=[\gamma,\infty)$.

  * (2)

If $f(x)=c$, the field $L$ satisfies that
$I_{N}(L)=(\gamma,\infty)\subsetneq[\gamma,\infty)=I_{B}(L)$ with ${\rm
Nor}_{\gamma}(L)=c$.

  * (3)

If $f(x)=1/\log(x)$, the field $L$ satisfies that
$I_{N}(L)=I_{B}(L)=(\gamma,\infty)$.

######  Remark 4.2.

By the theorem of Bertrand-Chebyshev, we can take $(p_{i})_{i\in\mathbb{N}}$
and $(q_{i})_{i\in\mathbb{N}}$ in Theorem 4.1. Furthermore, for each
$\gamma<0$, the sequence of prime numbers $(d_{i})_{i\in\mathbb{N}}$ satisfies
the equality

$\lim_{i\rightarrow\infty}\frac{i\log(d_{i})}{d_{i}^{-\gamma}}=0$

if, for example, the inequality $d_{i}^{-\gamma}\geq i^{2}$ holds for each
$i\in\mathbb{N}$.

###### Proof of Theorem 4.1.

Let $V(i,\gamma)$ be the quantity defined in Proposition 3.3. First, we prove
the assertion in the case $\gamma\geq 0$. We prepare some notation. By the
assumptions $\exp(f(d_{i})d_{i}^{1-\gamma})\leq p_{i}$ and $\gamma<1$, there
exists $i_{0}\in\mathbb{N}$ such that $d_{i}<p_{i}$ for all $i>i_{0}$.
Combining the assumption $q_{i}<p_{i+1}$, we know that

$p_{i},q_{i}\notin\\{d_{1},p_{1},q_{1},\ldots,d_{i-1},p_{i-1},q_{i-1}\\}$

for all $i>i_{0}$. Thus we can apply Proposition 3.3 to our setting. For each
$\delta\geq\gamma$ with $\delta<1$ and $\varepsilon\leq\gamma$, we have the
inequalities

$V(i,\delta)\geq\frac{f(d_{i})d_{i}^{1-\gamma}}{d_{i}^{1-\delta}}=d_{i}^{\delta-\gamma}f(d_{i})=:L_{1}(i,\delta)$

and

$0<h_{\varepsilon}((p_{i}/q_{i})^{1/d_{i}})=d_{i}^{\varepsilon-1}\log(q_{i})<\frac{\log(4)}{d_{i}^{1-\varepsilon}}+\frac{f(d_{i})}{d_{i}^{\gamma-\varepsilon}}=:U_{1}(i,\varepsilon)$

by the assumption $q_{i}<2p_{i}\leq 4\exp(f(d_{i})d_{i}^{1-\gamma})$.

  * (1)

For $\delta\geq\gamma$, since we have

$V(i,\delta)\geq
L_{1}(i,\delta)=d_{i}^{\delta-\gamma}\log(d_{i})\rightarrow\infty$

as $i\rightarrow\infty$, the field $L$ has $\delta$-(N) by Proposition 3.3.

For $\varepsilon<\gamma$, because we observe that

$0<h_{\varepsilon}((p_{i}/q_{i})^{1/d_{i}})<U_{1}(i,\varepsilon)=\frac{\log(4)}{d_{i}^{1-\varepsilon}}+\frac{\log(d_{i})}{d_{i}^{\gamma-\varepsilon}}\rightarrow
0$

as $i\rightarrow\infty$, the field $L$ does not have $\varepsilon$-(B).

  * (2)

Proposition 3.3 and the inequality

$V(i,\gamma)\geq L_{1}(i,\gamma)=c$

imply that $\text{Nor}_{\gamma}(L)\geq c$ holds. On the one hand, due to we
notice that

$h_{\gamma}((p_{i}/q_{i})^{1/d_{i}})<U_{1}(i,\gamma)=\frac{\log(4)}{d_{i}^{1-\gamma}}+c\rightarrow
c$

as $i\rightarrow\infty$, the inequality $\text{Nor}_{\gamma}(L)\leq c$ also
holds. Hence we get the equality $\text{Nor}_{\gamma}(L)=c$. It also concludes
that $I_{N}(L)=(\gamma,\infty)\subsetneq[\gamma,\infty)=I_{B}(L)$ by
Proposition 2.3.

  * (3)

For $\delta>\gamma$, since we have

$V(i,\delta)\geq
L_{1}(i,\delta)=\frac{d_{i}^{\delta-\gamma}}{\log(d_{i})}\rightarrow\infty$

as $i\rightarrow\infty$, the field $L$ has $\delta$-(N) by Proposition 3.3.

For $\varepsilon\leq\gamma$, because we observe that

$0<h_{\varepsilon}((p_{i}/q_{i})^{1/d_{i}})<U_{1}(i,\varepsilon)=\frac{\log(4)}{d_{i}^{1-\varepsilon}}+\frac{1}{d_{i}^{\gamma-\varepsilon}\log(d_{i})}\rightarrow
0$

as $i\rightarrow\infty$, the field $L$ does not have $\varepsilon$-(B).

Next, we prove the assertion in the case $\gamma<0$. As in the case
$\gamma\geq 0$, we can apply Proposition 3.3 to our setting. For each negative
real numbers $\delta\geq\gamma$ and $\varepsilon\leq\gamma$, we have the
inequalities

$V(i,\delta)\geq\frac{f(d_{i})(d_{1}\cdots
d_{i-1})^{-\gamma}d_{i}^{1-\gamma}}{(d_{1}\cdots
d_{i-1})^{-\delta}d_{i}^{1-\delta}}=(d_{1}\cdots
d_{i})^{\delta-\gamma}f(d_{i})=:L_{2}(i,\delta)$

and

$\displaystyle 0$
$\displaystyle<h_{\varepsilon}((p_{1}/q_{1})^{1/d_{1}}\cdots(p_{i}/q_{i})^{1/d_{i}})$
$\displaystyle=(d_{1}\cdots
d_{i})^{\varepsilon}h((p_{1}/q_{1})^{1/d_{1}}\cdots(p_{i}/q_{i})^{1/d_{i}})$
$\displaystyle\leq(d_{1}\cdots
d_{i})^{\varepsilon}\left(h((p_{1}/q_{1})^{1/d_{1}})+\cdots+h((p_{i}/q_{i})^{1/d_{i}})\right)$
$\displaystyle=(d_{1}\cdots
d_{i})^{\varepsilon}\left(\frac{\log(q_{1})}{d_{1}}+\cdots+\frac{\log(q_{i})}{d_{i}}\right)$
$\displaystyle\leq(d_{1}\cdots
d_{i})^{\varepsilon}\left(\left(\frac{1}{d_{1}}+\cdots+\frac{1}{d_{i}}\right)\log(4)+\left(\frac{f(d_{1})}{d_{1}^{\gamma}}+\cdots+\frac{f(d_{i})}{(d_{1}\cdots
d_{i})^{\gamma}}\right)\right)$
$\displaystyle<\frac{i\log(4)}{d_{i}^{-\varepsilon}}+\frac{f(d_{1})+\cdots+f(d_{i-1})}{d_{i}^{-\varepsilon}}+\frac{f(d_{i})}{(d_{1}\cdots
d_{i})^{\gamma-\varepsilon}}$ $\displaystyle=:U_{2}(i,\varepsilon)$

by the assumption $q_{i}<2p_{i}\leq 4\exp(f(d_{i})(d_{1}\cdots
d_{i-1})^{-\gamma}d_{i}^{1-\gamma})$.

  * (1)

For $\delta\geq\gamma$, since we have

$V(i,\delta)\geq L_{2}(i,\delta)=(d_{1}\cdots
d_{i})^{\delta-\gamma}\log(d_{i})\rightarrow\infty$

as $i\rightarrow\infty$, the field $L$ has $\delta$-(N) by Proposition 3.3.

For $\varepsilon<\gamma$, because we observe that

$\displaystyle 0$
$\displaystyle<h_{\varepsilon}((p_{1}/q_{1})^{1/d_{1}}\cdots(p_{i}/q_{i})^{1/d_{i}})$
$\displaystyle<U_{2}(i,\varepsilon)$
$\displaystyle<\frac{i\log(4)}{d_{i}^{-\varepsilon}}+\frac{i\log(d_{i})}{d_{i}^{-\varepsilon}}+\frac{\log(d_{i})}{(d_{1}\cdots
d_{i})^{\gamma-\varepsilon}}$ $\displaystyle\rightarrow 0$

as $i\rightarrow\infty$, the field $L$ does not have $\varepsilon$-(B).

  * (2)

Proposition 3.3 and the inequality

$V(i,\gamma)\geq L_{2}(i,\gamma)=c,$

imply that $\text{Nor}_{\gamma}(L)\geq c$ holds. On the one hand, due to we
notice that

$\displaystyle
h_{\gamma}((p_{1}/q_{1})^{1/d_{1}}\cdots(p_{i}/q_{i})^{1/d_{i}})<$
$\displaystyle~{}U_{2}(i,\gamma)$ $\displaystyle<$
$\displaystyle~{}\frac{i\log(4)}{d_{i}^{-\gamma}}+\frac{ic}{d_{i}^{-\gamma}}+c$
$\displaystyle\rightarrow$ $\displaystyle~{}c$

as $i\rightarrow\infty$, the inequality $\text{Nor}_{\gamma}(L)\leq c$ also
holds. Hence we get the equality $\text{Nor}_{\gamma}(L)=c$.

  * (3)

For $\delta>\gamma$, since we have

$V(i,\delta)\geq L_{2}(i,\delta)=\frac{(d_{1}\cdots
d_{i})^{\delta-\gamma}}{\log(d_{i})}\rightarrow\infty$

as $i\rightarrow\infty$, the field $L$ has $\delta$-(N) by Proposition 3.3.

For $\varepsilon\leq\gamma$, because we observe that

$\displaystyle 0$
$\displaystyle<h_{\varepsilon}((p_{1}/q_{1})^{1/d_{1}}\cdots(p_{i}/q_{i})^{1/d_{i}})$
$\displaystyle<U_{2}(i,\varepsilon)$
$\displaystyle<\frac{i\log(4)}{d_{i}^{-\varepsilon}}+\frac{i}{d_{i}^{-\varepsilon}\log(d_{1})}+\frac{1}{(d_{1}\cdots
d_{i})^{\gamma-\varepsilon}\log(d_{i})}$ $\displaystyle\rightarrow 0$

as $i\rightarrow\infty$, the field $L$ does not have $\varepsilon$-(B).

∎

In summary, the fields $L=\mathbb{Q}((p_{i}/q_{i})^{1/d_{i}}\mid
i\in\mathbb{N})$ in Theorem 4.1 satisfy the following:

Table 1. Stratification of $L$ $f(x)$ | $I_{N}(L)\text{ and }I_{B}(L)$
---|---
$\log(x)$ | $I_{N}(L)=I_{B}(L)=[\gamma,\infty)$
$c>0$ |  $I_{N}(L)=(\gamma,\infty)\subsetneq[\gamma,\infty)=I_{B}(L)$ with $\text{Nor}_{\gamma}(L)=c$
$1/\log(x)$ | $I_{N}(L)=I_{B}(L)=(\gamma,\infty)$

###  4.2. Comparison with previous work

Now we compare our result with that in [PTW21]. The following is a variant of
[PTW21, Theorem 3] and [PTW21, Theorem 4].

######  Theorem 4.3.

Let $\gamma<1$ and $c>0$ be non-negative real numbers. We also let
$(p_{i})_{i\in\mathbb{N}}$ and $(d_{i})_{i\in\mathbb{N}}$ be strictly
increasing sequences of prime numbers. Furthermore, we let $f(x)$ be
$\log(x)$, $c$, or $1/\log(x)$. Assume that the inequalities

$\exp(f(d_{i})d_{i}^{1-\gamma})\leq p_{i}\leq 2\exp(f(d_{i})d_{i}^{1-\gamma})$

hold for all $i\in\mathbb{N}$. We set $L:=\mathbb{Q}(p_{i}^{1/d_{i}}\mid
i\in\mathbb{N})$.

  * (1)

If $f(x)=\log(x)$ the field $L$ satisfies that
$I_{N}(L)=I_{B}(L)=[\gamma,\infty)$.

  * (2)

If $f(x)=c$ the field $L$ satisfies that
$I_{N}(L)=(\gamma,\infty)\subsetneq[\gamma,\infty)=I_{B}(L)$ with $c/2\leq{\rm
Nor}_{\gamma}(L)\leq c$.

  * (3)

If $f(x)=1/\log(x)$ the field $L$ satisfies that
$I_{N}(L)=I_{B}(L)=(\gamma,\infty)$.

We can similarly prove Theorem 4.3 as Theorem 4.1. In fact, the field
constructed in [PTW21, Theorem 4] is that in Theorem 4.3 (2) replaced $\gamma$
with $\gamma-\epsilon/2$ for given positive real numbers $\gamma\leq 1$ and
$\epsilon\leq 2\gamma$. Indeed, the field has $\gamma$-(N) but not
$(\gamma-\epsilon)$-(B). Theorem 4.3 (1) (resp. (3)) actually gives a positive
answer to Question 1.2 (1) (resp. (3)) of the case $\gamma\geq 0$. However, to
calculate the Northcott number, we employed one more sequence
$(q_{i})_{i\in\mathbb{N}}$. This idea is based on the argument in [VW15,
Section 1]. We also note that a similar field not having $0$-(N) was dealt
with in [GR17, Example 4.6].

##  5\. Remarks on the cases $\gamma=1$ and $\gamma=-\infty$

In this section, we partially deal with the cases $\gamma=1$ and
$\gamma=-\infty$ of Question 1.2. Before dealing with the case $\gamma=1$, we
note that constructing a field $L$ satisfying the condition (2) or (3) in
Question 1.2 will disprove the Lehmer conjecture. Thus we deal with neither
the case $\gamma=1$ of Question 1.2 (2) nor (3).

Now we give a field $L\subset\overline{\mathbb{Q}}$ such that
$I_{N}(L)=I_{B}(L)=[1,\infty)$.

######  Proposition 5.1.

Let $(p_{i})_{i\in\mathbb{N}}$ and $(q_{i})_{i\in\mathbb{N}}$ be strictly
increasing sequences of prime numbers with $q_{i}<p_{i+1}$ and
$p_{i}<q_{i}<2p_{i}$ for all $i\in\mathbb{N}$. Then the field
$L:=\mathbb{Q}((p_{i}/q_{i})^{1/p_{i}}\mid i\in\mathbb{N})$ satisfies that
$I_{N}(L)=[1,\infty)$.

###### Proof.

Let $(d_{i})_{i\in\mathbb{N}}:=(p_{i})_{i\in\mathbb{N}}$ and $V(i,\gamma)$ be
the quantity defined in Proposition 3.3. Since we have

$V(i,1)=\frac{1}{2}\log(p_{i})\rightarrow\infty$

as $i\rightarrow\infty$, the field $L$ has $1$-(N) by Proposition 3.3.

For $\varepsilon<1$, because we observe that

$0<h_{\varepsilon}((p_{i}/q_{i})^{1/p_{i}})=\frac{\log(q_{i})}{p_{i}^{1-\varepsilon}}<\frac{\log(2p_{i})}{p_{i}^{1-\varepsilon}}\rightarrow
0$

as $i\rightarrow\infty$, the field $L$ does not have $\varepsilon$-(B). ∎

Next, we deal with fields $L\subset\overline{\mathbb{Q}}$ such that
$I_{N}(L)=(1,\infty)\subsetneq[1,\infty)=I_{B}(L)$. In [Sch73], Schinzel
proved that $\mathbb{Q}^{\textrm{tr}}(\sqrt{-1})$ has $1$-(B). On the one
hand, Proposition 1.9 and 2.3 imply that $\mathbb{Q}^{\textrm{tr}}(\sqrt{-1})$
does not have $\varepsilon$-(B) for all $\varepsilon<1$. Thus
$\mathbb{Q}^{\textrm{tr}}(\sqrt{-1})$ is an example of such the fields. Here
note that $\mathbb{Q}^{\textrm{tr}}(\sqrt{-1})$ contains infinitely many roots
of unity and thus $N_{1}(\mathbb{Q}^{\textrm{tr}}(\sqrt{-1}))=0$. By using the
result in [Amo16], we can also construct a field
$L\subset\overline{\mathbb{Q}}$ satisfying that
$c\leq\text{Nor}_{1}(L)<\infty$ for a given $c>0$. For each integer $b\geq 2$
and prime number $p\geq 3$, we set

$\displaystyle L_{b,p}$
$\displaystyle:=\mathbb{Q}(\zeta_{p^{i}},b^{1/p^{i}}\mid i\in\mathbb{N}),$
$\displaystyle\langle b\rangle$ $\displaystyle:=(\textrm{the subgroup of
}L_{b,p}^{\times}\textrm{ generated by }b),\text{ and}$
$\displaystyle\sqrt{\langle b\rangle}$ $\displaystyle:=\left\\{a\in
L_{b,p}\mid a^{n}\in\langle b\rangle\text{ for some }n\in\mathbb{N}\right\\},$

where $\zeta_{m}$ is a primitive $m$-th root of unity for each
$m\in\mathbb{N}$. Amoroso gave the following theorem.

######  Theorem 5.2 ([Amo16, Theorem 3.3]).

If $p\nmid b$ and $p^{2}\nmid b^{p-1}-1$, the set
$L_{b,p}\setminus\sqrt{\langle b\rangle}$ has $0$-(B).

######  Proposition 5.3.

Let $c>0$ be a real number. We also let a prime number $b\in\mathbb{N}$
satisfy that $b\geq\exp(c)$ and $b\in\left\\{9n+2\mid
n\in\mathbb{Z}\right\\}$. Then the field
$L^{\prime}_{b,3}:=\mathbb{Q}(b^{1/3^{i}}\mid i\in\mathbb{N})$ satisfies that
$I_{B}(L^{\prime}_{b,3})=[1,\infty)$ and $c\leq{\rm
Nor}_{1}(L^{\prime}_{b,3})<\infty$.

######  Remark 5.4.

By the theorem of Dirichlet on arithmetic progression, we can take a prime
number $b$ in Proposition 5.3.

###### Proof of Proposition 5.3.

Note that $3\nmid b=9n+2$ and $3^{2}\nmid b^{3-1}-1=(9n+3)(9n+1)$ hold. Thus,
by Proposition 2.3 and Theorem 5.2, the set
$L^{\prime}_{b,3}\setminus\sqrt{\langle b\rangle}$ has $1$-(N). Thus there are
only finitely many $a\in L^{\prime}_{b,3}\setminus\sqrt{\langle b\rangle}$
such that $h_{1}(a)\leq\log(b)$. On the one hand, we know that
$h_{1}(a)\geq\log(b)$ for all $a\in\sqrt{\langle b\rangle}\setminus\\{\pm
1\\}$ and that $h_{1}(b^{1/3^{i}})=\log(b)$ for all $i\in\mathbb{N}$. Hence we
have $\text{Nor}_{1}(L^{\prime}_{b,3})=\log(b)\geq c$. ∎

Finally, we deal with fields $L\subset\overline{\mathbb{Q}}$ such that
$I_{N}(L)=I_{B}(L)=\mathbb{R}$. We remark that any number field satisfies the
condition by Theorem 1.8. Here we give such a field of infinite extensions of
$\mathbb{Q}$.

######  Proposition 5.5.

Let $(d_{i})_{i\in\mathbb{N}}$, $(p_{i})_{i\in\mathbb{N}}$, and
$(q_{i})_{i\in\mathbb{N}}$ be strictly increasing sequences of prime numbers.
Assume that the inequalities

$\exp(d_{i}^{1+i^{2}})\leq p_{i}<q_{i}<p_{i+1}$

hold for all $i\in\mathbb{N}$. Then the field
$L:=\mathbb{Q}((p_{i}/q_{i})^{1/d_{i}}\mid i\in\mathbb{N})$ satisfies that
$I_{N}(L)=\mathbb{R}$.

###### Proof.

Let $V(i,\gamma)$ be the quantity defined in Proposition 3.3. For $\gamma<0$,
since we have

$V(i,\gamma)\geq\frac{d_{i}^{1+i^{2}}}{(d_{1}\cdots
d_{i-1})^{-\gamma}d_{i}^{1-\gamma}}\geq\frac{d_{i}^{1+i^{2}}}{d_{i}^{1-i\gamma}}=d_{i}^{i(i+\gamma)}\rightarrow\infty$

as $i\rightarrow\infty$, the field $L$ has $\gamma$-(N) by Proposition 3.3. ∎

######  Acknowledgments.

The authors would like to thank professor Pazuki, Technau, and Widmer for
their favorable comments. Professor Pazuki informed the authors of [GR17,
Example 4.6]. The valuable comments of professor Technau made many sentences
in the draft better. The suggestion of professor Widmer highly improved the
proof of Proposition 3.3. The first author was supported by JST SPRING, Grant
Number JPMJSP2136. The second author is supported by JSPS KAKENHI Grant Number
JP20K14300.

## References

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††thanks: Email<EMAIL_ADDRESS>
#  Influence of the Pauli exclusion principle on the $\alpha$ decay

M. M. Amiri  O. N. Ghodsi  Department of Physics, Faculty of Science,
University of Mazandaran, P.O. Box 47415-416, Babolsar, Iran.

###### Abstract

In this study, the effects of repulsive nucleon-nucleon interactions arising
from the Pauli exclusion principle were examined regarding the half-lives of
heavy even-even nuclei with $84\leq Z\leq 92$. The Pauli exclusion principle
is applied to our investigations by renormalizing the nucleon-nucleon
interactions according to the Bohr-Sommerfeld quantization condition. It is
also applied to the double-folding ($DF$) formalism as a modification term by
investigating the kinetic energy variation at the overlapping regions between
the densities of the alpha and daughter nuclei. The standard deviation for the
calculated half-lives from their corresponded experimental data is 0.340 for
the renormalized DF formalism. Its value reduces to 0.309 for the DF formalism
modified with the kinetic energy contribution estimated by the extended
Thomas-Fermi approach. The calculated $\alpha$ decay half-lives are more
consistent with experimental data if the kinetic energy contribution is being
associated with the DF formalism.

##  I  Introduction

Over the past decades, the $\alpha$ decay process has attracted much interest
toward investigating the decay and fusion mechanisms of heavy and superheavy
nuclei [1, 2, 3, 4, 5, 6, 7, 13, 8, 9, 10, 11, 12]. Various approaches have
been adopted to investigate the properties of the $\alpha$ decay processes
[14, 15, 16, 17, 18, 19]. These approaches are based on nucleon-nucleon
interactions or density functionals of the emitted cluster and daughter nuclei
[20, 21, 22].

The properties of cluster and daughter nuclei play a significant role in
calculating the interaction potential between two interacting nuclei. The
nucleon density distribution is one of the most critical factors in assessing
nuclear properties that can affect the interaction potential [23, 24, 25, 26,
27] and thereby confront theoretical calculations with experimental data.
Meanwhile, investigations of cluster decay and nuclear fusion are
predominantly relevant for the interaction potentials in the partial and full
overlap density regions [28, 29, 30]. Although the semi-microscopic DF
potentials based on the effective M3Y nucleon-nucleon ($NN$) interaction can
well reproduce most of the scattering data, they fail in describing many
reactions that are strongly affected by the characteristics of the potential
below the barrier in the internal region [31]. This deficiency can be due to
the non-consideration of the repulsive core in the DF formalism [32]. However,
the Pauli exclusion principle ($PEP$) satisfactory and antisymmetrization
impact would be important ones compensating such insufficiencies [33]. To this
goal, the contribution of an increase in the intrinsic kinetic energy of the
nuclear densities at small separation distances can be employed due to the
antisymmetrization of both the interaction matrix element and the distortion
[34].

As a result of the repulsive force would be remarkable at minimal distances,
the internal kinetic energy reaches a maximum value when the densities of
participating nuclei have a complete overlap. On the other hand, the PEP
antisymmetrization and nuclear incompressibility would be an obstacle to the
occurrence of a complete overlap in the total system that guarantees
saturation properties of nuclear matter [35].

In some formalism like DF, the sudden approximation is being used. The
densities of the colliding nuclei are assumed to be unchanged at all distances
during the overlapping process. The densities of two interacting nuclei
overlap so that an increasing repulsive force would be expected due to this
approximation and PEP. In some theoretical studies, this increase in energy of
the dinuclear system is being simulated within a repulsive force in the NN
interaction, and the nucleus-nucleus potential is being modified in
consequence. Such modifications have been made mostly for fusion reactions,
which led to the better reproduction of experimental data. For instance, it is
shown that the incorporation of Pauli blocking properly with the DF model can
achieve by taking into account the redefinition of the density matrices of the
free isolated nuclei [33]. Microscopically, from the standard M3Y potential
developed with the density-dependent Pauli blocking potential of the density
overlap of two colliding nuclei and consequent appearance some shallow pockets
in $\alpha$-core potentials of $\alpha$+208Pb, $\alpha$+197Au, $\alpha$+209Bi,
and $\alpha$+238U, their originated fusion cross-sections were consistent with
the experimental data [36]. Also, the simultaneous cross-section studies based
on the experimental evidence for the reactions 12C + 12C, 12C + 16O, and 16O +
16O proposed a soft Gaussian repulsive core to the Woods-Saxon potential [37].
Moreover, the unexpected steep falloff of fusion cross sections at energies
far below the Coulomb barrier for the reactions 64Ni + 64Ni, 58Ni + 58Ni, and
64Ni + 100Mo were successfully explained by combination using the DF potential
based on the M3Y interactions supplemented with a simulated repulsive core
[30, 35].

On the other hand, such an additional repulsive force increases the kinetic
energy of the total interaction [38, 36]. Therefore, the desired modifications
can be investigated by estimating the variation of the kinetic energy in the
dinuclear system. An increase in kinetic energy acts as a repulsive force in
the dinuclear system that prevents the unexpected increase of the density
overlapping at interior regions. Despite the well-illustrated overlapping
effect in the density functional theory ($DFT$) [39, 40], the effect of Pauli
blocking is not well embedded in the DF formalism. For achieving such
prevention, a repulsive force can be simulated in the DF formalism due to the
PEP, which plays a similar role as like as the kinetic energy contribution in
the DFT. Therefore, incorporating the PEP with the DF would be associated with
a repulsive core simulation, which is one of the possible solutions to
conserve the dinuclear system around the saturation density. To this aim, we
intend to examine the influences of the PEP on the $\alpha$ decay within the
DF formalism.

Furthermore, the $\alpha$ decay process is also a low-energy phenomenon, and
it can not actually cause a sensible variation in the ground-state properties
of an alpha emitter[15, 41]. On the other hand, it is shown that an $\alpha$
cluster is mostly formed in the pre-surface region of the nucleus. Some
sensible density overlapping between alpha and daughter nuclei in more dense
regions due to the considering the effect of Pauli blocking from the saturated
core density [42]. Therefore, the interior regions of the interaction
potentials between alpha and daughter nuclei would be affected by the PEP,
where the tunnelling process occurs. One of the possible mechanism that is
extensively being used for simultaneously applying the PEP and clusterization
state in the dinuclear system is the renormalizing the strength of effective
NN interaction due to the Bohr-Sommerfeld (BS) quantization condition [43, 39,
44, 45, 46, 2, 3]. This procedure compensates for the deficiency of the PEP in
the DF formalism to some extent.

Since the effect of the Pauli blocking behaves as a repulsive force in the
dinuclear system[38], an increase in kinetic energy in the dinuclear system
would be expected due to densities overlap. Due to the fact that the effect of
Pauli blocking is not embedded in the DF formalism, some promotions would be
required to properly incorporate the DF formalism and the effect of Pauli
blocking for the $\alpha$ decay studies. On the other hand, the Hartree-Fock
model based on Skyrme forces is a successful approach for examining the
ground-state properties of nuclei. With a similar approach adopted in the
Hartree-Fock model, and different from what the BS condition suggests, we
intend to estimate an increase in the kinetic energy due to the Pauli blocking
within the extended Thomas-Fermi approach. Consequently, the $\alpha$ decay
half-lives of even-even heavy nuclei will be calculated that are being
affected by embedding kinetic energy causing by folding the densities of alpha
and daughter nuclei into the nucleon-nucleon M3Y interactions in the DF
formalism. A comparison between these results with their corresponding values
obtaining through the renormalizing the strength of effective NN interaction
of total potential due to the BS condition will be presented.

This paper is organized as follows: The formalism of potential, half-life
calculations in Sec. II, and our result and discussion are given in Sec. III.
This paper ends with the main results and conclusions presented in Sec. IV.

##  II Theoretical framework

###  II.1 Double-Folding formalism and the $\alpha$ decay half-life

The total potential $V(R)$ is written by

$V(R)=V_{C}(R)+V_{N}(R),$ (1)

where $V_{C}$, $V_{N}$ are the coulomb and nuclear parts of the total
potential, respectively. Also, $R$ denotes the vector joining the center of
masses of the two nuclei. In this study, we investigate the $\alpha$ decay of
even-even nuclei that the transferred angular momentum for these decay
processes are zero. The nuclear part is obtained by the double-folding model
within folding the densities of the alpha and the daughter nuclei with the
effective M3Y interaction,

$\displaystyle V_{N}(R)$ $\displaystyle=\lambda_{0}V_{F}(R)$
$\displaystyle=\lambda_{0}\iint{\rho_{1}(\textbf{r}_{1})\,V_{eff}(\textbf{s})\,\rho_{2}(\textbf{r}_{2})\,d^{3}\textbf{r}_{1}\,d^{3}\textbf{r}_{2}},$
(2)

where, $\textbf{s}=\textbf{R}+\textbf{r}_{1}-\textbf{r}_{2}$ corresponds to
the distance between two specified interacting points of the interacting
nuclei, whose radius vectors are $\textbf{r}_{1}$ and $\textbf{r}_{2}$,
respectively. The $V_{eff}(\textbf{s})$ is an effective nucleon-nucleon
interaction [32, 47]. The energy-dependent M3Y Reid-NN forces with zero range
approximation that used in our calculations have following explicit forms
[48]:

$\displaystyle V_{eff}$
$\displaystyle(\textbf{s})=7999\frac{\exp(-4s)}{4s}-2134\frac{\exp(-2.5s)}{2.5s}+J_{00}\delta(\textbf{s})$
$\displaystyle J_{00}=-276(1-0.005E/A),$ (3)

where $E$ and $A$ are the incident energy in the center-of-mass frame and the
mass number of the alpha particle, respectively. In Eq.(II.1), $\rho_{1}$ is
taken for the density distribution function of the spherical $\alpha$ particle
in its Gaussian form used in this study [49] that is given as

$\rho(r)=0.4229\,e^{-0.7024\,r^{2}},$ (4)

and $\rho_{2}$ is the density of the daughter nucleus that is determined by
Hartree-Fock-Bogoliubov calculations based on the set of Skyrme SLy4
parametrization [50] as result of its capability for well reproducing the
$\alpha$ decay energies of heavy and superheavy nuclei, in this study [51].

The parameter $\lambda_{0}$ in Eq.(II.1) changes the folded potential strength
that is known as the strength parameter. It can be determined by using the BS
quantization condition [52, 53, 54]:

$\int_{R_{1}}^{R_{2}}\sqrt{\frac{2\mu}{\hbar^{2}}|V(R)-Q|}\,dR=(2n+1)\frac{\pi}{2}=(G-\ell+1)\frac{\pi}{2},$
(5)

where $R_{2}$, $R_{3}$ are classical turning points are obtained by $V(R)=Q$
(the $\alpha$ decay energy), and for $0^{+}\rightarrow 0^{+}$ $s$-wave decay
the inner turning point is at $R_{1}=0$.

The global quantum number $G$ of a cluster state can be obtained by the
Wildermuth condition [43]:

$G=2N+\ell=\sum_{i=1}^{4}g_{i},$ (6)

where $N$ is the number of nodes of the $\alpha$-core wave function; $\ell$ is
the relative angular momentum of the cluster motion, and, $g_{i}$; is the
oscillator quantum number of a cluster nucleon. For the $\alpha$ decay, we can
take $G$ as

$G=2N+\ell=\begin{cases}18,&\text{\qquad$N\leq 82$},\\\
20,&\text{\qquad$82<N\leq 126$},\\\ 22.&\text{\qquad$N>126$.}\end{cases}$ (7)

The half-life of the $\alpha$ decay is $T_{1/2}=\hbar\ln 2/\Gamma_{\alpha}$.
In this relation, $\Gamma_{\alpha}$ is the $\alpha$ decay width of the cluster
state within the Gurvitz and Kälbermann method, determined as [55]

$\Gamma_{\alpha}=FP_{\alpha}\frac{\hbar^{2}}{4\mu}\exp\left(-2\int_{R_{2}}^{R_{3}}k(R)\,dR\right),$
(8)

where $F$ is normalization factor can be defined as below

$F\int_{R_{1}}^{R_{2}}\frac{dR}{2k(R)}\,=1,$ (9)

The $k(R)=(2\mu/\hbar^{2}[V(R)-Q])^{1/2}$ is the wave number. Also,
$P_{\alpha}$ and $\mu$ are the alpha formation probability and reduced mass,
respectively.

###  II.2  Alpha decay half-life within Wentzel-Kramers-Brillouin
approximation

In the Preformed Cluster Model (PCM) viewpoint, a formation probability is
being attributed to the alpha cluster before the tunneling process.
Consequently, the $\alpha$ decay half-lives can be calculated by $T_{1/2}={\ln
2}/{\lambda}$ with the PCM viewpoint, where $\lambda$ is decay constant that
is a multiplication of the barrier penetrability P, the assault frequency
$\nu_{0}$, and cluster formation probability $P_{\alpha}$. Moreover, many
researchers adopted $P_{\alpha}=1$, fission-like model, to calculate the
$\alpha$ decay half-lives. However, in this study, the calculation of the
$\alpha$ decay half-lives will be done the same way for both conditions. It
should be noted that the strength parameter is assumed to be $\lambda_{0}=1$
for calculating the $\alpha$ decay half-lives through the WKB approximation.

The barrier penetrability $P$ can be calculated by semi-classical Wentzel-
Kramers-Brillouin (WKB) approximation

$P=\exp\left(-\frac{2}{\hbar}\int_{R_{4}}^{R_{5}}\sqrt{\frac{2\mu}{\hbar^{2}}|V(R)-Q_{\alpha}|}dR\right)\,.$
(10)

where $R_{4}$, $R_{5}$ are classical turning points. In this study, the
experimental $Q_{\alpha}$-values are taken from refs. [56, 57]. By assuming
that the $\alpha$ particle vibrates in a harmonic oscillator potential with
oscillation frequency $\omega$, the assault frequency $\nu_{0}$ can be
determined as illustrated in ref.[58].

###  II.3 Cluster Formation Model

In the cluster formation model (CFM), it is assumed that the parent nucleus is
a compilation of different cluster states [15]. For each preformation, there
is a different wave function and a different Hamiltonian. Therefore, we assume
that for each preformation or clusterization, there is a clusterization state
represented by a wave function. If the parent nucleus has $N$ different
clusterization states with total energy $E$, the Hamiltonian $H_{i}$ belongs
to the $i$th clusterization defined with an $i$th wave function, therefore

$H_{i}\Psi_{i}=E\Psi_{i}\qquad i=1,2,\dots,N.$ (11)

Therefore, this nucleus is described by a total time-independent wave function
that is a linear combination of these clusterization orthonormalized wave
functions

$\Psi=\sum^{N}_{i=1}a_{i}\Psi_{i},$ (12)

where $a_{i}$ are the amplitudes for the clusterization states of the complete
set and within the orthogonality condition,

$\sum_{i}^{N}|a_{i}|^{2}=1.$ (13)

Each cluster has specific formation energy $E_{fi}$ that

$E_{fi}=|a_{i}|^{2}E.$ (14)

The probability of the alpha clusterization state $P_{\alpha}$ is equivalent
to $a_{\alpha}^{2}$. It can be calculated as

$P_{\alpha}=|a_{\alpha}|^{2}=\frac{E_{f\alpha}}{E}.$ (15)

here, $a_{\alpha}$ and $E_{f\alpha}$ denote the coefficient of the $\alpha$
clusterization and the formation energy of an $\alpha$ cluster. $E$ is
composed of the $E_{f\alpha}$ and the interaction energy between $\alpha$
cluster and daughter nuclei. The detailed illustrations are provided in
ref.[15]. In the framework of CFM, the $\alpha$ cluster formation energy
$E_{f\alpha}$ and total energy $E$ of a considered system can be expressed as

$\displaystyle E_{f\alpha}=$ $\displaystyle 3B(A,Z)+B(A-4,Z-2)$
$\displaystyle-2B(A-1,Z-1)-2B(A-1,Z),$ (16) $E=B(A,Z)-B(A-4,Z-2),$ (17)

$B(A,Z)$ is the binding energy of the nucleus with mass number $A$ and proton
number $Z$. The defined energies in Eq.II.3 and Eq.17 belong to even-even
nuclei, and for an odd atomic number or odd neutron number, the formation
energies can be found in refs. [60, 59].

Moreover, the formation probability of each cluster state calculates by the
CFM can well reproduce a more realistic formation probability, which follows
the calculation of Varga et al. [61, 62].

##  III Results and discussions

Generally, the order of magnitude of alpha tunneling is about $10^{-21}$ s
[14, 58, 63] that proceed with the $\alpha$ decay studies are being done under
sudden approximation. Also, this approximation is being used in DF formalism,
in which the densities of the interacting nuclei are being assumed to be
frozen at all distances during the interaction. One can expect that the
densities of the nucleons in the compound system become increased as the
folding density distributions of alpha and daughter nucleus begins at the
nuclear surface and rise to total overlap. Meanwhile, the PEP would be more
apparent in following this accumulation of nucleons. Hence, a variation in the
kinetic energy at constant volume would be expected [33, 38]. In this case,
such kinetic energy is not attributed to the kinetic energy of the emitted
alpha particle. Hence, the $\alpha$-core interaction potential that is being
calculated by DF formalism can be modified by adding the mentioned kinetic
energy term that causing by overlapping densities of the alpha and daughter
nuclei.

For estimating the kinetic energy well illustrated in the DFT [22, 64], the
self-consistent Hartree-Fock calculations are being performed comprising SLy4
Skyrme interaction. The variation of the kinetic energy of the density overlap
of two colliding nuclei based on the DFT is being obtained by

$\displaystyle\Delta K(R)=$
$\displaystyle\frac{\hbar^{2}}{2m}\iint\bigg{\\{}\tau[\rho_{1p}(\mathbf{r})+\rho_{2p}(\mathbf{r}-\mathbf{R}),\rho_{1n}(\mathbf{r})$
$\displaystyle+\rho_{2n}(\mathbf{r}-\mathbf{R})]-\tau[\rho_{1p}(\mathbf{r})+\rho_{1n}(\mathbf{r})]$
$\displaystyle-\tau[\rho_{2p}(\mathbf{r})+\rho_{2n}(\mathbf{r})]\bigg{\\}}d\mathbf{r}.$
(18)

where $\tau$ denotes the kinetic energy density. The two nuclei are
overlapping at $R$ and completely separated at infinity $R=\infty$. The
contribution of the kinetic energy density for the dinuclear system coincide
with overlapping the densities of the alpha and daughter nuclei would be
clarified by the extended Thomas-Fermi approach (ETF) and considering the
semi-classical correction of the second-order $\hbar$ [65] proposed as

$\displaystyle\tau_{q}(\textbf{r})=$
$\displaystyle\frac{3}{5}(3\pi^{2})^{\frac{2}{3}}\rho_{q}^{\frac{5}{3}}+\frac{1}{36}\frac{(\mathbf{\nabla}\rho_{q})^{2}}{\rho_{q}}+\frac{1}{3}\Delta\rho_{q}+\frac{1}{6}\frac{\mathbf{\nabla}\rho_{q}.\mathbf{\nabla}f_{q}}{f_{q}}$
$\displaystyle+\frac{1}{6}\rho_{q}\frac{\Delta
f_{q}}{f_{q}}-\frac{1}{12}\rho_{q}\left(\frac{\mathbf{\nabla}f_{q}}{f_{q}}\right)^{2}$
$\displaystyle+\frac{1}{2}\rho_{q}\left(\frac{2m}{\hbar^{2}}\right)^{2}\left(\frac{W_{0}}{2}\frac{\mathbf{\nabla}(\rho+\rho_{q})}{f_{q}}\right)^{2},$
(19)

where $q$ denotes proton and neutron and $f_{q}(\textbf{r})$ is the effective
mass form factor that is given as

$\displaystyle f_{q}(\textbf{r})=$ $\displaystyle
1+\frac{2m}{\hbar^{2}}\frac{1}{4}\left[t_{1}\left(1+\frac{x_{1}}{2}\right)+t_{2}\left(1+\frac{x_{2}}{2}\right)\right]\rho(\textbf{r})$
$\displaystyle-\frac{2m}{\hbar^{2}}\frac{1}{4}\left[t_{1}\left(x_{1}+\frac{1}{2}\right)-t_{2}\left(x_{2}+\frac{1}{2}\right)\right]\rho_{q}(\textbf{r}).$
(20)

the parameters $x_{1}$, $x_{2}$, $t_{1}$, $t_{2}$, and $W_{0}$ are obtained by
fitting different properties of nuclei, $m$ and $\rho=\rho_{1}+\rho_{2}$ are
the nucleon mass and nuclear densities, respectively.

The calculated kinetic energy typically for $\alpha$ \+ 208Pb is presented by
red dotted line in Fig.1(a). This figure indicates that the kinetic energy is
impressive in the interior region, where two nuclei have sensible overlap.
Also, its value is confronted by a gradual depression concerning the low
overlapping densities at nuclear surfaces.

The estimated kinetic energy by the ETF approach for the dinuclear system can
be accompanied as a corrective term with the DF model that explicitly
presented in Fig.1(a). The presented results in Fig.1(b) indicate that the
influence of the considered kinetic energy on the interior regions of the
Coulomb barrier is quite evident. This kinetic energy due to the PEP acts as a
repulsive force that hinders a large density overlap in the dinuclear system.
Consequently, the half-life calculations would be affected by this
modification. For seeking such influences, by employing the M3Y potentials and
modified potentials with added kinetic energy terms, the half-lives of the
even-even nuclei with $84\leq Z\leq 92$ are calculated through WKB
approximation that are listed as ${T^{(2)}_{1/2}}$ and ${T^{(3)}_{1/2}}$ in
Tab.1. The standard deviation (SD) of the 48 nuclei,
$\sigma=\sqrt{\frac{1}{N}\sum_{i=1}^{N=48}[\log(T^{Calc}_{i}/T^{Exp}_{i})]^{2}}$,
of the calculated half-lives from their corresponded experimental data is
employed for having a better insight into this modification. The obtained SD
value by use of WKB approximation for the standard M3Y potentials is $1.245$,
which varies to $0.787$ for the half-lives obtained by modified potentials
within kinetic energy considerations. This SD promotion indicates how the
kinetic energy application to the DF model would be productive on the
calculation of the $\alpha$ decay half-lives.

On the other hand, for considering the PEP in the total system, we
renormalized the NN interactions according to the BS quantization condition.
Consequently, the $\alpha$ decay half-lives are being calculated within the
Gurvitz method, and their logarithms are expressed as ${T^{(1)}_{1/2}}$ in
Tab.1. The SD value 0.994 is obtained for this case. The obtained results
express that the $\alpha$ decay half-lives that are being calculated by
kinetic energy application to the DF model are more consistent with the
experimental data in comparison with their corresponding values obtained by
the renormalizing and Gurvitz method.

It is noticeable that all the $\alpha$ decay half-lives are calculated with
$P_{\alpha}=1$ assumption. Extensively, it has shown that the alpha formation
probability has a remarkable role in the $\alpha$ decay studies [66, 67, 68].
However, if the calculations of the $\alpha$ decay half-lives within WKB
approximation, in the PCM viewpoint, associates with the cluster preformation
factors estimating by the CFM [60, 69, 70] can result in a good agreement with
the experimental data. The preformation factors calculated by CFM are
presented as $P^{CFM}_{\alpha}$ in Tab.1. The SD values for half-lives
obtained by M3Y potentials change to $0.561$, and it reduces to $0.309$ for
those obtained by modified potentials as a consequence of the application of
the preformation factors estimated by the CFM. Also, the logarithms of
calculated half-lives with $P_{\alpha}$ applications for M3Y and modified
potentials with kinetic energies are presented as ${T^{(4)}_{1/2}}$ and
${T^{(5)}_{1/2}}$ in Tab.1. Furthermore, if the alpha cluster preformation
factors estimating by the CFM are being applied to half-lives calculated
within the Gurvitz method, the SD value changes from 0.994 to 0.340.

Moreover, we employed some various phenomenological formulas based on the
Geiger Nuttall law [71] that have been proposed to estimate the $\alpha$ decay
half-lives for comparing with the estimated half-lives. The Viola-Seaborg-
Sobiczewski ($VSS$) semi-empirical relationship was determined as

$\log(T_{1/2})=\frac{(aZ+b)}{\sqrt{Q_{\alpha}}}+cZ+d+e$ (21)

where Z is the atomic number of the parent nucleus and the constants a, b, c,
and d are $1.66175$, $-8.5166$, $-0.20228$, and $-33.9069$, respectively [72].
For the even-even nuclei the parameter $e$ is assumed to be zero [73].

Also, the Royer analytic formula ($RF$) for the $\alpha$ decay half-lives was
proposed as [74],

$\log(T_{1/2})=a+bA^{\frac{1}{6}}\sqrt{Z}+\frac{cZ}{\sqrt{Q_{\alpha}}}$ (22)

where A and Z denote the mass and atomic numbers of the parent nuclei. For the
even-even nuclei the constants a, b, and c are $-25.31$, $-1.1629$, and
$1.5864$, respectively.

Furthermore, the universal decay law ($UDL$) in charged-particle emission and
exotic cluster radioactivity was presented by Qi et al.[75]

$\log(T_{1/2})=aZ_{\alpha}Z_{d}\sqrt{\frac{A}{Q_{\alpha}}}+b\sqrt{AZ_{\alpha}Z_{d}(A_{\alpha}^{\frac{1}{3}}+A_{d}^{\frac{1}{3}})}+c$
(23)

here, $A=(A_{\alpha}A_{d})/(A_{\alpha}+A_{d})$ and constants $a=0.4314$,
$b=-0.4087$, and $c=-25.7725$ were obtained by fitting to experimental of both
alpha and cluster decays [75]. The logarithms of the calculated half-lives
obtained by VSS, RF, and UDL formulas are presented as ${T^{VSS}_{1/2}}$,
${T^{RF}_{1/2}}$, and ${T^{UDL}_{1/2}}$ in Tab.1, respectively. The SD values
of the calculated half-lives obtained by VSS, RF, and UDL formulas are
$0.704$, $0.385$, and $0.484$, respectively. Also, by comparison on SD values
of the calculated half-lives obtained by these mentioned phenomenological
formulas and those obtained by M3Y and modified DF potentials, one would be
clarified that the calculated half-lives within simultaneous kinetic energy
modification and alpha preformation factor applications obtained by CFM along
with the RF formula are more consistent with the experimental data. Therefore,
the role of this modification in the $\alpha$ decay studies would be more
impressive.

In order to simplify the estimation of kinetic energy through the
sophisticated Hartree-Fock and ETF approaches, a double-folded integral with a
delta function $V_{0}\delta(\textbf{s})$ is simulated as

$V_{Rep}(R)=V_{0}\iint\rho_{1}(\textbf{r}_{1})\delta(\textbf{s})\rho_{2}(\textbf{r}_{2})\,d\textbf{r}_{1}d\textbf{r}_{2}.$
(24)

here, $V_{0}$ is the strength of this simulated repulsive force obtaining at
where the alpha and daughter have a total overlap $(R=0)$. Its value is being
adjusted so that to reproduce the estimated kinetic energy by the ETF
approach, especially where the PEP becomes more apparent in the total system.
For instance, the simulated repulsive force for the dinuclear system ($\alpha$
\+ 208Pb) is displayed in Fig.2. The result in this figure indicates that the
overlapping of the $\alpha$ and daughter nuclei would be associated with some
energy variations at nuclear surfaces. Furthermore, the presented results in
this figure indicate that the treatment of our simulation is close to the
treatment of kinetic energy with fair approximation. Also, the treatments and
the magnitudes of such simulation and estimated kinetic energy are exciting
the same from the first turning point and thereafter. Hence, this similarity
at $R=0$ and from the first turning point is supporting the choice of such
simulated double-folded integral with a delta function.

Concerning to the calculated kinetic energy for selected isotopic groups,
their values at origin are being determined and presented in Fig.3. The
results in Fig.3 indicate the quantized shifts for examined kinetic energies
according to an increase in atomic numbers. Also, for the specific isotope
group, the kinetic energy has a peak for the parent nucleus that its daughter
has a magic neutron closed-shell, which can be as a result of the shell-
closure effect at magic neutron number $N=126$. One can expect that the PEP
causes a nucleon interacting space at full strength. It more increases the
kinetic energy during the density overlap of alpha and the nuclei with the
full occupied layers, especially magic nuclei. This treatment can be justified
as the fully occupied valence layer in the magic nuclei, causing a higher
nucleon accumulation in constant volume in comparison with the nuclei that
have a non-full valence layer.

It is noticeable that each nucleus has its characteristic $f_{q}(r)$ that
relates to the density distributions of the nucleons calculated by the SLy4,
in this paper. For examining the $f_{q}(r)$ contributions to the shell-closure
treatment displayed in Fig.3, their values and the amount of the nucleon
densities for the 206-210Pb are being estimated at $R=0$, where to nuclei have
complete overlap. The estimated effective mass form factor $f_{p}(r)$ for
206Pb, 208Pb, and 210Pb are 1.4344, 1.4366, and 1.4328, respectively. Also,
the estimated effective mass form factor $f_{n}(r)$ for 206Pb, 208Pb, and
210Pb are 1.4595, 1.4628, and 1.4620, respectively. On the other hand, by
considering the dependency of $f_{q}(r)$ to the variation of the total energy
$E$ with respect to the kinetic energy density,
$f_{q}(r)=(2M^{q}/\hbar^{2})(\delta E/\delta\tau$), one can expect that the
higher effective mass form factor would result in a higher kinetic energy.
According to the estimated $f_{p}(r)$ and $f_{n}(r)$, the double closed-shell
nucleus like 208Pb has more kinetic energy than the isotopes in its vicinity,
as indicated in Fig.3. Hence one can expect that the shell-closure treatment
in Fig.3 can be due to the effective mass form factors that are being obtained
by density distributions calculated by SLy4 force.

The all adjusted $V_{0}$ values at $R=0$ with respect to the daughter neutron
numbers for the investigated heavy nuclei are presented in Fig.4. The results
in this figure indicate an obvious shell-closure effect for a characterized
isotope group. As shown in Fig.4, quadratic dependencies with respect to the
daughter neutron number are obtained for the nuclei with valence holes and
linear dependencies for the isotopes with the valence particles.

Instead of discrete calculation for the strength of the such simulation, the
mentioned two different dependencies made us drawn to fit these obtained
$V_{0}$ values with respect to the atomic and mass numbers of daughter nuclei.
These fitted relations are illustrated as below:

$V_{0}=\begin{cases}a+bZ_{d}+(cZ_{d}+d)A_{d}^{1/3}\sqrt{N_{0}-N_{d}},&\text{$N_{d}<N_{0}$},\\\
a+b(A_{d}-N_{0}).&\text{$N_{d}\geq N_{0}$.}\end{cases}$ (25)

where the coefficients $a=145$, $b=0.3125$, $c=0.0255$, and $d=1.8575$ are
obtained and $N_{0}$ is equivalent to the neutron magic number $N=126$.
$A_{d}$, $Z_{d}$, and $N_{d}$ are the mass, atomic, and neutron numbers of
daughter nuclei, respectively. Also, our simulation can compensate for the
non-consideration of the kinetic energy in the DF potential as like as the
estimated kinetic energy through the extended Thomas-Fermi approach. The
$\alpha$ decay half-lives are being calculated upon implementing the simulated
double-folded integral with a delta function and $P_{\alpha}$ application
obtained by CFM. Their logarithms are presented as ${T^{(6)}_{1/2}}$ in Tab.1.
The obtained SD for the calculated half-lives through the modified DF
potentials with this simulation is $0.298$, which indicates a good consistency
with SD values obtained by RF formula and those obtained by the DF potential
with the kinetic energy modification and $P_{\alpha}$ application. Although
both kinetic energy consideration and the NN renormalization due to the BS
quantization condition in the DF model can reproduce the effect of the PEP in
the total system, the calculated $\alpha$ decay half-lives within the kinetic
energy consideration are more consistent with the experimental data. The
detailed properties of all considered states are presented in Tab.2.

##  IV SUMMARY AND CONCLUSION

In this study, the effects of the repulsive forces arising from the PEP are
investigated on the calculations of the $\alpha$ decay half-life for even-even
nuclei with $84\leq Z\leq 92$. Such repulsive force between NN interactions is
investigated by considering an increase in the kinetic energy in the dinuclear
systems. The densities of the alpha and daughter nuclei have an overlap. Also,
the effect of the PEP is investigated on the $\alpha$ decay by the
renormalizing the NN interactions by the BS quantization condition. To this
intention, the ETF approach is used to estimate such kinetic energies that
have arisen at the overlapping regions. Subsequently, the results expose some
energy variation at nuclear surfaces causes by the applied interior
modifications to the DF formalism, which can affect the $\alpha$ decay half-
life calculations.

The SD value for the half-lives obtained by the standard M3Y potentials is
1.245, which reduces to 0.787 for modified M3Y potentials by the PEP inclusion
by increasing kinetic energy consideration at overlapping regions. These
values varied to 0.561 and 0.309 by the $P_{\alpha}$ application on the half-
lives obtained by M3Y and modified M3Y potentials, respectively. On the other
hand, the SD value 0.340 is being obtained for the normalized potentials
within the BS condition for embedding the PEP. These obtained results reveal
that the $\alpha$ decay half-lives would be more affected by the repulsive
force considerations on the DF formalism, which proceed to the more consistent
half-lives with the experimental data. Moreover, we proposed a pocket formula
instead of the kinetic energy estimations through the sophisticated Hartree-
Fock and ETF approaches, associated with DF formalism.

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Figure 1:  (a) The calculated $V_{C}(R)$, $V_{N}(R)$, and kinetic energy,
$K_{E}(R)$, typically for $\alpha$ and 208Pb; (b) the dashed line is M3Y total
potential and dotted line is the total potential with added kinetic energy.
The solid horizontal line indicates the $Q_{\alpha}$-value for the
${}^{212}Po$.

Figure 2:  Solid line is the kinetic energy estimated by ETF approach and
dashed line is the simulated double-folded integral with a delta function,
typically for the overlapping of $\alpha$ and 208Pb.

Figure 3:  The kinetic energy values for the overleaping $\alpha$ and daughter
nuclei examined at $R=0$ with respect to daughter neutron number.

Figure 4:  The strength of the simulated double-folded integral with a delta
function for the overleaping $\alpha$ and daughter nuclei adjusted at $R=0$
due to the kinetic energy.

Table 1:  A comparison between the logarithms of the calculated $\alpha$ decay half-lives and $T_{1/2}$(VSS), $T_{1/2}$(RF), and $T_{1/2}$(UDL). The half-lives are being calculated in the unit of sec. ${Parent}$ |  ${Q_{\alpha}}$[MeV] | ${T^{Exp}_{1/2}}$ | $P_{\alpha}^{CFM}$ | ${T^{(1)}_{1/2}}$ | ${T^{(2)}_{1/2}}$ | ${T^{(3)}_{1/2}}$ | ${T^{(4)}_{1/2}}$ | ${T^{(5)}_{1/2}}$ | ${T^{(6)}_{1/2}}$ | ${T^{VSS}_{1/2}}$ | ${T^{RF}_{1/2}}$ | ${T^{UDL}_{1/2}}$
---|---|---|---|---|---|---|---|---|---|---|---|---
${}^{198}_{84}$Po | 6.3096 | 2.2678 | 0.206 | 1.2479 | 0.9071 | 1.3917 | 1.5932 | 2.0778 | 2.0677 | 1.2815 | 2.0095 | 1.8459
${}^{200}_{84}$Po | 5.9814 | 3.7939 | 0.188 | 2.6637 | 2.3218 | 2.8150 | 3.0476 | 3.5408 | 3.5264 | 2.6939 | 3.4024 | 3.3204
${}^{202}_{84}$Po | 5.7010 | 5.1442 | 0.174 | 3.9741 | 3.6300 | 4.1316 | 4.3895 | 4.8911 | 4.8715 | 3.9961 | 4.6834 | 4.6776
${}^{204}_{84}$Po | 5.4848 | 6.2768 | 0.158 | 5.0466 | 4.7021 | 5.2120 | 5.5034 | 6.0133 | 5.9863 | 5.0675 | 5.7304 | 5.7887
${}^{206}_{84}$Po | 5.3269 | 7.1446 | 0.146 | 5.8648 | 5.5172 | 6.0368 | 6.3528 | 6.8724 | 6.8332 | 5.8909 | 6.5254 | 6.6352
${}^{208}_{84}$Po | 5.2153 | 7.9612 | 0.135 | 6.3954 | 6.0737 | 6.6333 | 6.9434 | 7.5030 | 7.4340 | 6.4953 | 7.0982 | 7.2478
${}^{210}_{84}$Po | 5.4074 | 7.0776 | 0.104 | 5.3241 | 4.9776 | 5.4980 | 5.9606 | 6.4810 | 6.4409 | 5.4666 | 6.0109 | 6.1146
${}^{212}_{84}$Po | 8.9541 | -6.5243 | 0.220 | -7.6543 | -7.9059 | -7.4644 | -7.2483 | -6.8068 | -6.8842 | -7.0964 | -6.8029 | -7.3526
${}^{214}_{84}$Po | 7.8334 | -3.7844 | 0.213 | -4.6055 | -4.9360 | -4.4335 | -4.2644 | -3.7619 | -3.8385 | -4.0678 | -3.7645 | -4.1478
${}^{216}_{84}$Po | 6.9063 | -0.8386 | 0.206 | -1.6597 | -1.9110 | -1.3640 | -1.2249 | -0.6779 | -0.7697 | -1.0235 | -0.7098 | -0.9255
${}^{218}_{84}$Po | 6.1146 | 2.2696 | 0.196 | 1.5048 | 1.2416 | 1.8188 | 1.9493 | 2.5265 | 2.4273 | 2.1070 | 2.4328 | 2.3898
${}^{200}_{86}$Rn | 7.0433 | 0.0783 | 0.229 | -0.7716 | -1.0720 | -0.6157 | -0.4318 | 0.0245 | 0.0343 | -0.6632 | 0.0177 | -0.2277
${}^{202}_{86}$Rn | 6.7737 | 1.0947 | 0.212 | 0.2141 | -0.0878 | 0.3878 | 0.5859 | 1.0615 | 1.0712 | 0.3346 | 0.9875 | 0.8025
${}^{204}_{86}$Rn | 6.5464 | 2.0118 | 0.195 | 1.0671 | 0.7828 | 1.2500 | 1.4928 | 1.9600 | 1.9612 | 1.2235 | 1.8468 | 1.7166
${}^{206}_{86}$Rn | 6.3838 | 2.7393 | 0.178 | 1.6531 | 1.3878 | 1.9021 | 2.1374 | 2.6517 | 2.6223 | 1.8882 | 2.4791 | 2.3919
${}^{208}_{86}$Rn | 6.2606 | 3.3723 | 0.162 | 2.2573 | 1.8872 | 2.3763 | 2.6777 | 3.1668 | 3.1638 | 2.4090 | 2.9655 | 2.9138
${}^{210}_{86}$Rn | 6.1589 | 3.9542 | 0.152 | 2.3159 | 2.3063 | 2.8031 | 3.1245 | 3.6213 | 3.6075 | 2.8507 | 3.3720 | 3.3515
${}^{212}_{86}$Rn | 6.3850 | 3.1565 | 0.120 | 1.6317 | 1.2782 | 1.7801 | 2.1990 | 2.7009 | 2.6409 | 1.8832 | 2.3483 | 2.2841
${}^{214}_{86}$Rn | 9.2084 | -6.5686 | 0.228 | -7.6792 | -7.8166 | -7.3759 | -7.1745 | -6.7338 | -6.8006 | -7.0148 | -6.7258 | -7.2564
${}^{216}_{86}$Rn | 8.1973 | -4.3468 | 0.236 | -4.9573 | -5.2271 | -4.7531 | -4.6000 | -4.1260 | -4.1687 | -4.3628 | -4.0746 | -4.4570
${}^{218}_{86}$Rn | 7.2625 | -1.4559 | 0.234 | -1.9789 | -2.3232 | -1.7862 | -1.6924 | -1.1554 | -1.2314 | -1.4333 | -1.1412 | -1.3603
${}^{220}_{86}$Rn | 6.4046 | 1.7451 | 0.220 | 1.1712 | 0.9214 | 1.4912 | 1.5790 | 2.1488 | 2.0643 | 1.8017 | 2.1025 | 2.0636
${}^{222}_{86}$Rn | 5.5903 | 5.5186 | 0.221 | 4.8612 | 4.7161 | 5.3124 | 5.3717 | 5.9680 | 5.8786 | 5.5380 | 5.8554 | 6.0240
${}^{206}_{88}$Ra | 7.4151 | -0.6198 | 0.222 | -1.3163 | -1.6664 | -1.2171 | -1.0128 | -0.5635 | -0.5391 | -1.1331 | -0.5542 | -0.7904
${}^{208}_{88}$Ra | 7.2730 | 0.1361 | 0.201 | -0.8534 | -1.1977 | -0.7436 | -0.5009 | -0.0468 | -0.0216 | -0.6415 | -0.0985 | -0.3008
${}^{210}_{88}$Ra | 7.1508 | 0.5859 | 0.184 | -0.4438 | -0.7897 | -0.3332 | -0.0545 | 0.4020 | 0.4164 | -0.2070 | 0.2994 | 0.1280
${}^{212}_{88}$Ra | 7.0316 | 1.1845 | 0.170 | -0.0321 | -0.3890 | 0.0858 | 0.3806 | 0.8554 | 0.8251 | 0.2276 | 0.6980 | 0.5574
${}^{214}_{88}$Ra | 7.2725 | 0.3912 | 0.139 | -0.9566 | -1.3345 | -0.8564 | -0.4775 | 0.0006 | -0.0078 | -0.6397 | -0.2229 | -0.4027
${}^{216}_{88}$Ra | 9.5257 | -6.7399 | 0.239 | -7.7321 | -7.8831 | -7.5032 | -7.2615 | -6.8816 | -6.8251 | -7.0864 | -6.7992 | -7.3190
${}^{218}_{88}$Ra | 8.5459 | -4.5986 | 0.241 | -5.4512 | -5.4652 | -4.9964 | -4.8472 | -4.3784 | -4.4146 | -4.5978 | -4.3177 | -4.6967
${}^{220}_{88}$Ra | 7.5924 | -1.7447 | 0.239 | -2.3569 | -2.6258 | -2.1293 | -2.0042 | -1.5077 | -1.5406 | -1.7271 | -1.4485 | -1.6656
${}^{222}_{88}$Ra | 6.6788 | 1.5798 | 0.199 | 0.9979 | 0.7055 | 1.2530 | 1.4066 | 1.9541 | 1.8845 | 1.5817 | 1.8652 | 1.8341
${}^{224}_{88}$Ra | 5.7888 | 5.4964 | 0.183 | 4.9381 | 4.6929 | 5.2814 | 5.4304 | 6.0189 | 5.9432 | 5.5317 | 5.8292 | 6.0194
${}^{226}_{88}$Ra | 4.8706 | 10.7029 | 0.181 | 9.5268 | 9.9942 | 10.6077 | 10.7365 | 11.3500 | 11.3099 | 10.6943 | 11.0226 | 11.5005
${}^{212}_{90}$Th | 7.9579 | -1.4989 | 0.207 | -2.4234 | -2.7755 | -2.3261 | -2.0915 | -1.6421 | -1.6094 | -2.1148 | -1.6370 | -1.8932
${}^{214}_{90}$Th | 7.8271 | -1.0605 | 0.193 | -1.9449 | -2.3820 | -1.9234 | -1.6676 | -1.2090 | -1.1947 | -1.6988 | -1.2580 | -1.4844
${}^{216}_{90}$Th | 8.0723 | -1.5850 | 0.158 | -2.7989 | -3.2102 | -2.7406 | -2.4089 | -1.9393 | -1.9431 | -2.4704 | -2.0809 | -2.3421
${}^{218}_{90}$Th | 9.8490 | -6.9318 | 0.252 | -7.9102 | -7.9604 | -7.5285 | -7.3618 | -6.9299 | -6.9776 | -7.1704 | -6.8803 | -7.3902
${}^{220}_{90}$Th | 8.9530 | -5.0132 | 0.247 | -5.6070 | -5.8344 | -5.3725 | -5.2271 | -4.7652 | -4.8237 | -4.9752 | -4.6993 | -5.0830
${}^{222}_{90}$Th | 8.1269 | -2.6498 | 0.231 | -3.3062 | -3.5433 | -3.0574 | -2.9069 | -2.4210 | -2.4496 | -2.6374 | -2.3737 | -2.6231
${}^{224}_{90}$Th | 7.2985 | 0.1461 | 0.198 | -0.5775 | -0.8251 | -0.3186 | -0.1218 | 0.3847 | 0.3589 | 0.0948 | 0.3515 | 0.2584
${}^{226}_{90}$Th | 6.4508 | 3.2634 | 0.182 | 2.5613 | 2.5199 | 3.0792 | 3.2598 | 3.8191 | 3.7646 | 3.4192 | 3.6765 | 3.7724
${}^{228}_{90}$Th | 5.5200 | 7.7798 | 0.183 | 7.1956 | 7.1081 | 7.6960 | 7.8456 | 8.4335 | 8.3742 | 7.9188 | 8.1915 | 8.5414
${}^{218}_{92}$U | 8.7747 | -3.2924 | 0.186 | -4.2692 | -4.4752 | -4.0493 | -3.7447 | -3.3188 | -3.3451 | -3.7813 | -3.4037 | -3.7199
${}^{222}_{92}$U | 9.4302 | -5.3279 | 0.248 | -6.1426 | -6.3565 | -5.9026 | -5.7510 | -5.2971 | -5.3416 | -5.5056 | -5.2300 | -5.6274
${}^{224}_{92}$U | 8.6198 | -3.0757 | 0.229 | -4.0262 | -4.2472 | -3.7702 | -3.6070 | -3.1300 | -3.1537 | -3.3453 | -3.0871 | -3.3588
${}^{226}_{92}$U | 7.7009 | -0.5719 | 0.206 | -1.1909 | -1.4206 | -0.9212 | -0.7345 | -0.2351 | -0.2506 | -0.4943 | -0.2456 | -0.3531
${}^{228}_{92}$U | 6.8035 | 2.7595 | 0.188 | 2.1752 | 1.9202 | 2.4387 | 2.6460 | 3.1645 | 3.1530 | 2.8303 | 3.0751 | 3.1583
Table 2:  The detailed information of each states that are being investigated. Potential | Half-life calculation method | $\lambda$ | $P_{\alpha}$ | SD | Top name in Tab.1.
---|---|---|---|---|---
M3Y | Gurvitz & Wildermuth condition | $\lambda\neq 1$ | $P_{\alpha}=1$ | 0.994 | ${T^{(1)}_{1/2}}$
M3Y | Gurvitz & Wildermuth condition & $P_{\alpha}$ | $\lambda\neq 1$ | $P_{\alpha}=CFM$ | 0.340 | —
M3Y | WKB | $\lambda=1$ | $P_{\alpha}=1$ | 1.245 | ${T^{(2)}_{1/2}}$
M3Y+Kinetic energy | WKB | $\lambda=1$ | $P_{\alpha}=1$ | 0.787 | ${T^{(3)}_{1/2}}$
M3Y | WKB & $P_{\alpha}$ | $\lambda=1$ | $P_{\alpha}=CFM$ | 0.561 | ${T^{(4)}_{1/2}}$
M3Y+Kinetic energy | WKB & $P_{\alpha}$ | $\lambda=1$ | $P_{\alpha}=CFM$ | 0.309 | ${T^{(5)}_{1/2}}$
M3Y+Simulated term | WKB & $P_{\alpha}$ | $\lambda=1$ | $P_{\alpha}=CFM$ | 0.298 | ${T^{(6)}_{1/2}}$