#  Integral dimension of a noetherian ring

Caijun Zhou
Department of Mathematics, Shanghai Normal University,
Shanghai, 200234, China

##  1 Introduction

Throughout this paper we always use $R$ to denote a commutative noetherian
ring with an identity. The main purpose of the paper is to introduce a new
notion, integral dimension, for noetherian rings.

Let $J$ be an ideal of $R$. An element $x\in J$ is said to be integral over
$J$, if there is an equation of the form

$x^{n}+a_{1}x^{n-1}+\cdots+a_{n}=0$

where $a_{i}\in J^{i}$ for $1\leq i\leq n$. It is not difficult to show that
the set of all elements which are integral over $J$ is an ideal of $R$. We
will use $\overline{J}$ to denote this ideal, and call it the integral closure
of the ideal $J$. An ideal $I$ with $I\subseteq\overline{J}$ will be said to
be integral over $J$.

One of the main interests on $\overline{J}$ is when $\overline{J^{n}}$ is
contained in a power of $J$. Let us recall several remarkable results
established in the past years. Rees [Re] showed that if $R$ is an analytically
unramified local ring and $J$ an ideal of $R$, then there is an integer $k$,
depending on $J$, such that $\overline{J^{n}}\subseteq{J^{n-k}}$ for $n\geq
k$.

Let $R$ be the ring of convergent power series in $n$ variables
$Z_{1},\cdots,Z_{n}$ over the field of complex numbers C. Let $f\in R$ be a
non unit element and $J=(\partial f/\partial{Z_{1}},\cdots,\partial
f/\partial{Z_{n}})$ be the Jacobian ideal of $f$. It can be proved [cf. HS]
that $f$ is integral over $J$. Briançon and Skoda [BS] showed by analytic
method that $f^{n}\in J$ , and this is known as the Briançon-Skoda theorem.

Lipman and Sathaye [LS] extended the theorem of Briançon-Skoda to arbitrary
regular rings by purely algebraic method. They proved if $R$ ia a regular ring
of dimension $d>0$, then for all ideals $J$ of $R$,
$\overline{J^{n}}\subseteq{J^{n-d+1}}$ for $n\geq d-1$.

Later on, Huneke [ Hu, Theorem 4.13] proved a far reaching result for a broad
class of noetherian rings. For example, He obtained that, for any reduced
excellent local ring $R$, there exists an integer $k\geq 0$, such that for all
ideals $J$ of $R$, $\overline{J^{n}}\subseteq{J^{n-k}}$ for $n\geq k$. The
result of Huneke is known as the uniform Briançon-Skoda theorem. Recently,
Andersson and Wulcan [AW] presented global effective versions of the Briançon-
Skoda-Huneke theorem by analytic method.

In this paper, instead of considering the integral closure of the power ideal
$\overline{J^{n}}$, we will study the power of the integral closure of an
ideal ${\overline{J}}^{n}$. We prove in the next section for any ideal $J$ of
$R$,

$J^{n}\subseteq{\overline{J}}^{n}\subseteq\overline{J^{n}}.$

Hence the comparison of ${\overline{J}}^{n}$ with the power of $J$ may be
easier than that of $\overline{J^{n}}$. Motivated by this, we introduce the
following notion of integral dimension for noetherian rings.

###### Definition 1.1

Let $R$ be a noetherian ring. If there exists an integer $k\geq 0$ such that
for all ideals $J$ of $R$ and for $n\geq k$

${\overline{J}}^{n}\subseteq{J^{n-k}},$ $None$

then we define the integral dimension of $R$, denoted by $i(R)$, to be the
least integer $k$ such that (1.1) holds for all $J$ and all $n\geq k$. If such
$k$ does not exist, then we set $i(R)=\infty$.

It is obvious that $i(R)=k<\infty$ if and only if $k$ is the least integer
such that for every pair of ideals $J\subseteq I$, where $I$ is integral over
$J$,

$I^{n}\subseteq J^{n-k}$

for $n\geq k$. Moreover, we have $Q^{k+1}=0$ for the nilradical $Q$ of $R$
because it is integral over the zero ideal. Clearly, if $R$ is an artinian
ring, then $i(R)\leq l(R)$, where $l(R)$ denotes the length of $R$.

It follows easily from the Lipman-Sathaye theorem, $i(R)\leq d-1$ for every
regular ring of dimension $d>0$. The above mentioned result of Huneke implies
that if $R$ is a reduced excellent local ring, then $i(R)<\infty$.

In this paper, we will prove some basic facts about the behavior of $i(R)$
under several operations on $R$. The results show that $i(R)$ behaves well
under operations such as localization and completion.

We will give a lower bound for $i(R)$ in terms of the dimension $d$ of $R$.
Explicitly, we will show $i(R)\geq d-1$. In particular, it follows from
Lipman-Sathaye theorem, if $R$ is a regular ring of dimension $d>0$, then
$i(R)=d-1$. We can prove a noetherian ring $R$ with $i(R)=0$ if and only if
$R$ is a regular ring of dimension $d\leq 1$.

The main goal of the paper is to show that $i(R)<\infty$ for a large class of
noetherian rings. Recall that a $S$-algebra $R$ is said to be essentially of
finite type over a ring $S$, if $R$ is a localization of a finitely generated
$S$-algebra. Our main result of the paper states as follows:

Theorem 3.6 Let $R$ be a noetherian ring of finite dimension. If $R$ satisfies
one of the following conditions, then $i(R)<\infty$.

(i) $R$ is essentially of finite type over a local ring $S$.

(ii) $R$ is a ring of characteristic $p$, and $R$ is module finite over
$R^{p}$.

(iii) $R$ is essentially of finite type over the ring of integer numbers Z.

In particular, one can conclude from Theorem 3.6, $i(R)<\infty$ for any
noetherian local ring $R$. It implies that the notion of the integral
dimension becomes a well-defined notion and has concrete meaning in the class
of local rings. We do not know whether $i(R)$ is finite for every noetherian
ring $R$. It seems very difficult to give an answer to the question in
general. However, we can prove (see, Corollary 2.11) that
$\text{dim}(R)<\infty$ if $i(R)<\infty$. We make the following conjecture.

###### Conjecture 1.2

Let $R$ be a noetherian ring of finite dimension. Then $i(R)<\infty.$

Huneke [Hu] observed that the uniform Briançon-Skoda property plays an
important role in the study of the uniform Artin-Rees property of a noetherian
ring. As pointed out in Theorem 3.2, the finiteness property of integral
dimension is also useful in proving a ring with such uniform Artin-Rees
property. The proof of our main result Theorem 3.6 depends heavily on the
uniform Artin-Rees theorem of Huneke [Hu]. The main technique of Huneke came
from the paper of Lipman and Sathaye [LS] in characteristic 0 and the theory
of tight closure of [HH1] [HH2] in characteristic $p$. It can not be applied
directly to proving a result more general than Theorem 3.6.

Even for the local case, it would be very interesting to reveal more about the
relationship between $i(R)$ and $i(R/aR)$ for an element $a$ in $R$ with
$\text{dim}(R/aR)=\text{dim}(R)-1$. We guess that they may be related to each
other by something like the multiplicities, $e(R)$ and $e(R/(a))$ , of the
maximal ideals of $R$ and $R/(a)$. The answer is not known even for $i(R)$ and
$i(R[X])$, where $R[X]$ is the polynomial ring of $R$ in the variable $X$.
Hence there are a lot of questions about integral dimension remained for
further studying.

##  2 Basic properties

In this section, we will recall some basic facts about the integral closure of
an ideal, and then presents some properties of the integral dimension $i(R)$.
Some of the results are known, one can find in [cf. HS], we will prove them
for convenience of readers.

Let $Q$ be the nilradical of $R$ and $R_{red}$ be the ring of $R$ modulo $Q$.
Since there is an integer $n$ such that $Q^{n}=0$, it is easy to see that $Q$
is contained in $\overline{J}$ for every ideal $J$ of $R$, and an element $x$
is integral over $J$ if and only if $\bar{x}$, the image of $x$ in $R_{red}$,
is integral over $JR_{red}$. Thus we have:

###### Proposition 2.1

Let $J$ be an ideal of $R$. Then $\overline{JR_{red}}=\overline{J}R_{red}$.

Integral dependence behaves well under localization:

###### Proposition 2.2

Let $J$ be an ideal of $R$. Then for any multiplicatively closed subset $T$ of
$R$, $T^{-1}\overline{J}=\overline{T^{-1}J}.$

  * Proof.

Clearly, $T^{-1}\overline{J}\subseteq\overline{T^{-1}J}$ by the definition of
integral dependence. Let $a\in\overline{T^{-1}J}$. By the definition of
integral dependence again, there is an equation of the form

$a^{n}+a_{1}a^{n-1}+\cdots+a_{n}=0$

in the ring $T^{-1}R$, where $a_{i}\in T^{-1}J^{i}$ for $1\leq i\leq n$.
Choose an element $b\in T$ such that $ba\in R$ and $ba_{i}\in J^{i}$ for
$1\leq i\leq n$. Hence we have

${(ba)}^{n}+a_{1}b{(ba)}^{n-1}+\cdots+b^{n}a_{n}=0$

is an equation with coefficients in $R$, and there is an element
$b^{\prime}\in T$ such that

$b^{\prime}({(ba)}^{n}+a_{1}b{(ba)}^{n-1}+\cdots+b^{n}a_{n})=0$

in $R$. It shows that

${(b^{\prime}ba)}^{n}+a_{1}b^{\prime}b{(b^{\prime}ba)}^{n-1}+\cdots+{(b^{\prime}b)}^{n}a_{n}=0$

is the desired equation which implies $b^{\prime}ba$ is integral over $J$. So
$a\in T^{-1}\overline{J}$, and $\overline{T^{-1}J}\subseteq
T^{-1}\overline{J}$. This proves the proposition.

###### Corollary 2.3

Let $T$ be any multiplicatively closed subset of $R$. Then $i(T^{-1}R)\leq
i(R)$.

  * Proof.

If $i(R)=\infty$, there is nothing to prove. Now, assume that $i(R)=k$ is
finite. For any ideal $K$ of $T^{-1}R$, there exists an ideal $J$ of $R$ such
that $K=T^{-1}J$. By Proposition 2.2, $\overline{K}=T^{-1}{\overline{J}}$.
Note that ${\overline{J}}^{n}\subseteq J^{n-k}$ for $n\geq k$. Thus, for
$n\geq k$, we have

${\overline{K}}^{n}={(T^{-1}{\overline{J}})}^{n}=T^{-1}{\overline{J}}^{n}\subseteq{T^{-1}{{J}}^{n-k}}={(T^{-1}J)}^{n-k}=K^{n-k}$

and it follows that $i(T^{-1}R)\leq k$.

It is well-known the dimension of $R$ can be computed as follows:

$\text{dim}(R)=sup\\{\ \text{dim}(R_{\texttt{m}})\ |\ \texttt{m}\text{ a
maximal ideal of R}\\},$

where $R_{\texttt{m}}$ denotes the localization of $R$ at the maximal ideal m.
Similarly, we have the following computation for integral dimension.

###### Corollary 2.4

For a noetherian ring $R$, $i(R)=sup\\{\ i(R_{\texttt{m}})\ |\
\texttt{m}\text{ a maximal ideal of R}\\}.$

  * Proof.

Let us set

$t=sup\\{\ i(R_{\texttt{m}})\ |\ \texttt{m}{\text{ a maximal ideal of }}R\\}.$

By Corollary 2.3, we have $i(R)\geq t.$ On the other hand, if $t=\infty$, then
$i(R)=\infty$ and thus $i(R)=t$ holds in this case.

Now, assume that $t<\infty$. For any ideal $I$ of $R$, we have
$i(R_{\texttt{m}})\leq t$ and

$(\overline{IR_{\texttt{m}}})^{n}\subseteq(IR_{\texttt{m}})^{n-t}$

for all for $n\geq t$ and for any maximal ideal m of $R$. By Proposition 2.2,
$\overline{IR_{\texttt{m}}}=\overline{I}R_{\texttt{m}}$. It implies

${\overline{I}}^{n}R_{\texttt{m}}\subseteq I^{n-t}R_{\texttt{m}}$

for all maximal ideals m. Hence ${\overline{I}}^{n}\subseteq I^{n-t}$ for
$n\geq t,$ and consequently $i(R)\leq t$. This proves the corollary.

Now, we recall a useful tool, the reduction of an ideal, for studying the
integral closure of an ideal. An ideal $J\subseteq I$ is said to be a
reduction of the ideal $I$, if there exists an integer $k$ such that
$I^{k+l}=JI^{k}$. One important result about reduction is the following
proposition.

###### Proposition 2.5

Let $R$ be a noetherian ring and $J\subseteq I$ a pair of ideals of $R$. Then
the following conditions are equivalent.

(i) $J$ is a reduction of $I$.

(ii) $I$ is integral over $J$.

  * Proof.

(i) $\Rightarrow$ (ii) By assumption, there is an integer $k$ such that
$I^{k+l}=JI^{k}$. As $R$ is a noetherian ring, the ideal $I^{k}$ is finitely
generated. Set $I^{k}=(a_{1},a_{2},\cdots,a_{n})$. For $a\in I$ and $i(1\leq
i\leq n)$, write $aa_{i}=\sum_{j=1}^{n}a_{ij}a_{j}$ for some $a_{ij}\in J$.
Let $A=(a_{ij})$ be the matrix $(\delta_{ij}a-a_{ij})$, where $\delta_{ij}$ is
the Kronecker delta function. By Cramer’s Rule, $\text{det}(A)I^{k}=0$. In
particular, $\text{det}(A)a^{k}=0$, and an expansion of $\text{det}(A)a^{k}=0$
yields the desired equation of integral dependence of $a$ over $J$. Hence $I$
is integral over $J$.

(ii) $\Rightarrow$ (i) Suppose that $I$ is integral over $J$. Write
$I=(b_{1},b_{2},\cdots,b_{s})$. For each $i,1\leq i\leq s$, there is an
equation of the form

$b_{i}^{n_{i}}+c_{i1}b_{i}^{n_{i}-1}+\cdots+c_{in_{i}}=0$

where $c_{ij}\in I^{j}$ for $1\leq j\leq n_{i}$. Thus it implies
$b_{i}^{n_{i}}\in JI^{n_{i}-1}$. Put $k=\sum_{j=1}^{s}n_{j}+1$. It is clear
that if $r_{1}+r_{2}+\cdots+r_{s}=k$, every element of the form
$b_{1}^{r_{1}}b_{2}^{r_{2}}\cdots b_{s}^{r_{s}}$ lies in $JI^{k-1}$, and this
proves that $I^{k}=JI^{k-1}$.

As an easy consequence of the proposition, we have the following result
mentioned in the last section.

###### Corollary 2.6

Let $J$ be an ideal of $R$. Then for any integer $n$,
$J^{n}\subseteq{\overline{J}}^{n}\subseteq\overline{J^{n}}.$

  * Proof.

It follows from Proposition 2.5, $J$ is a reduction of ${\overline{J}}$. Thus
there is an integer $k$ such that ${\overline{J}}^{k+1}=J{\overline{J}}^{k}$.
For an integer $n$, we have
${({\overline{J}}^{n})}^{k+1}=J^{n}{({\overline{J}}^{n})}^{k}.$ By Proposition
2.5 again, ${\overline{J}}^{n}$ is integral over $J^{n}$, and consequently

$J^{n}\subseteq{\overline{J}}^{n}\subseteq\overline{J^{n}}.$

Let $\varphi:R\rightarrow S$ be a morphism of noetherian rings. The property
of persistence of integral closure states that
$\varphi(\overline{J})\subseteq\overline{\varphi(J)}$ for any ideal $J$ of
$R$. This follows as by applying $\varphi$ to an equation of integral
dependence of an element $a$ over $J$ to obtain an equation of integral
dependence of $\varphi(a)$ over the ideal $\varphi(J)S$.

###### Proposition 2.7

Let $\varphi:R\rightarrow S$ be a faithfully flat morphism of noetherian
rings. Then $i(R)\leq i(S).$

  * Proof.

If $i(S)=\infty$, there is nothing to prove. Now, we assume $i(S)=k$ is
finite. Let $J$ be an ideal of $R$. Since $i(S)=k$, it shows
${\overline{\varphi(J)}}^{n}\subseteq{\varphi(J)}^{n-k}$ for $n\geq k$. By
persistence of integral closure, we have
$\varphi(\overline{J})\subseteq\overline{\varphi(J)}$. So
${{{\varphi(\overline{J})}}}^{n}\subseteq{\varphi(J)}^{n-k}.$ It implies
${{{\varphi{(\overline{J}}^{n})}}}\subseteq{\varphi(J^{n-k})}.$ Note that
$\varphi$ is a faithfully flat morphism of noetherian rings, we conclude
${\overline{J}}^{n}\subseteq J^{n-k}$ for $n\geq k$, and this proves $i(R)\leq
i(S)$.

If $R$ is a local ring with the unique maximal ideal m, we use $\hat{R}$ to
denote the m-adic completion of $R$. Since $\hat{R}$ is faithfully flat over
$R$, $i(R)\leq i(\hat{R})$ by Proposition 2.7. We prove that this inequality
is indeed an equality.

###### Proposition 2.8

Let $(R,\texttt{m})$ be a noetherian local ring. Then $i(R)=i(\hat{R}).$

  * Proof.

It suffices to prove $i(\hat{R})\leq i(R)$ by Proposition 2.7. If
$i(R)=\infty$, there is nothing to prove. In fact, such case cannot happen, we
will prove in the next section, $i(R)$ is always a finite number. Now, put
$i(R)=k$, and naturally regard $R$ as a sub-ring of $\hat{R}$. Let
$K=(b_{1},b_{2},\cdots,b_{r})$ be an arbitrary ideal of $\hat{R}$.

If $K$ is a $\texttt{m}\hat{R}$-primary ideal, then there exists an integer
$t$ such that $\texttt{m}^{t}\hat{R}\subseteq K$. By the definition of
completion of a ring, we can choose elements $a_{i}\in R$ such that
$b_{i}-a_{i}$ lies in $m^{t+1}\hat{R}$ for $1\leq i\leq r$. Let $J$ be the
ideal of $R$ which is generated by $a_{1},a_{2},\cdots,a_{r}$. Clearly,
$K\subseteq J\hat{R}+\texttt{m}K$. So it follows that $K=J\hat{R}$ by Nakayama
lemma. Since the integral closure $\overline{K}$ of $K$ is also a
$\texttt{m}\hat{R}$-primary ideal, there is an ideal $I$ of $R$, replacing $I$
by $I+J$, we may assume $J\subseteq I$, such that $\overline{K}=I\hat{R}$. By
Proposition 2.5, $J\hat{R}$ is a reduction of $I\hat{R}$, and there is an
integer $n_{0}$ such that $I^{n_{0}+1}\hat{R}=JI^{n_{0}}\hat{R}$. As $\hat{R}$
is faithfully flat over $R$, we obtain $I^{n_{0}+1}=JI^{n_{0}}$. Hence for
$n\geq k$, $I^{n}\subseteq J^{n-k}$, and consequently

${\overline{K}}^{n}=I^{n}\hat{R}\subseteq
J^{n-k}\hat{R}=(J\hat{R})^{n-k}=K^{n-k}.$

For an arbitrary ideal $K$ of $\hat{R}$, we have for $n\geq k$ and all
positive integer $i$

${\overline{K}}^{n}\subseteq{\overline{K+\texttt{m}^{i}\hat{R}}}^{n}\subseteq({K+\texttt{m}^{i}\hat{R}})^{n-k}\subseteq
K^{n-k}+\texttt{m}^{i}\hat{R}$

by the case we have just proved. So ${\overline{K}}^{n}\subseteq K^{n-k}$ for
$n\geq k$ by Nakayama lemma again. This proves $i(\hat{R})\leq i(R)$, and the
proof of the proposition is complete.

Now, we turn to giving a lower bound for $i(R)$. We will show that if $R$ is
of finite dimension $d>0$, then $i(R)$ is at least $d-1$. Let us begin with
recalling Monomial Conjecture of Hochster which asserts that given any system
of parameters $x_{1},x_{2},\cdots,x_{d}$ of a $d$-dimensional local ring $R$,
then for all $n>1$

$(x_{1}x_{2}\cdots
x_{d})^{n}\notin(x_{1}^{n+1},x_{2}^{n+1},\cdots,x_{d}^{n+1}).$

Hochster proved the conjecture in the equicharacteristic case [Ho1] [Ho2]. In
mixed cases, the conjecture remains undetermined, despite much effort.
However, Hochster [Ho1] pointed out, for any system of parameters
$x_{1},x_{2},\cdots,x_{d}$ of $R$, there exists an integer $t_{0}$, such that
for all $t\geq t_{0}$, $x_{1}^{t},x_{2}^{t},\cdots,x_{d}^{t}$ satisfying
Monomial Conjecture. In particular, we have:

###### Lemma 2.9

If $R$ is a noetherian local ring of dimension $d>0$, then there exists a
system of parameters $x_{1},x_{2},\cdots,x_{d}$ of $R$ such that for all $n>0$

$(x_{1}x_{2}\cdots
x_{d})^{n}\notin(x_{1}^{n+1},x_{2}^{n+1},\cdots,x_{d}^{n+1}).$

By means of Lemma 2.9, we can prove the following interesting result.

###### Proposition 2.10

Let $R$ be a noetherian ring dimension $d>0$. Then $i(R)\geq d-1$.

  * Proof.

Choose a maximal ideal m of $R$ such that $\text{ht}(\texttt{m})=d$. It is
clear, $\text{dim}(R_{\texttt{m}})=d$. By Corollary 2.3, $i(R)\geq
i(R_{\texttt{m}})$. Replacing $R$ by $R_{\texttt{m}}$, we may assume $R$ is a
local ring with the maximal ideal m.

If $d=1$, it follows from the definition of integral dimension that $i(R)\geq
0$, and thus the conclusion is true in this case.

Now we assume $d\geq 2$. By Lemma 2.9, there is a system of parameters
$x_{1},x_{2},\cdots,x_{d}$ of $R$ such that the element $(x_{1}x_{2}\cdots
x_{d})^{n}$ is not contained in the ideal
$(x_{1}^{n+1},x_{2}^{n+1},\cdots,x_{d}^{n+1})$ for any $n>0$.

Let us consider the two ideals
$I=(x_{1}^{d},x_{1}^{d-1}x_{d},x_{2}^{d},x_{2}^{d-1}x_{d},\cdots,x_{d-1}^{d},x_{d-1}^{d-1}x_{d},x_{d}^{d})$
and $J=(x_{1}^{d},x_{2}^{d},\cdots,x_{d}^{d})$ of $R$. Since for each $i\
(1\leq i\leq{d-1})$

$(x_{i}^{d-1}x_{d})^{d}=(x_{i}^{d})^{d-1}x_{d}^{d}\in(x_{i}^{d},x_{d}^{d})(x_{i}^{d},x_{i}^{d-1}x_{d},x_{d}^{d})^{d-1},$

it yields that

$(x_{i}^{d},x_{i}^{d-1}x_{d},x_{d}^{d})^{d}=(x_{i}^{d},x_{d}^{d})(x_{i}^{d},x_{i}^{d-1}x_{d},x_{d}^{d})^{d-1}.$

By Proposition 2.5, $x_{i}^{d-1}x_{d}$ is integral over the ideal
$(x_{i}^{d},x_{d}^{d})$, and thus $x_{i}^{d-1}x_{d}$ is integral over $J$ for
$i\ (1\leq i\leq d-1)$. It shows $I$ is generated by elements which are
integral over $J$, and thus $I$ is integral over $J$.

Suppose that $i(R)<d-1$. We must have $I^{d-1}\subseteq J$ by the definition
of integral dimension. In particular, it implies the multiplication of the
following $d-1$ elements

$x_{1}^{d-1}x_{d},\cdots,x_{d-1}^{d-1}x_{d}$

lies in $J$, i.e.

$(x_{1}x_{2}\cdots x_{d})^{d-1}\in(x_{1}^{d},x_{2}^{d},\cdots,x_{d}^{d}),$

and this contradicts the choices of $x_{1},x_{2},\cdots,x_{d}$. Therefore
$i(R)\geq d-1$, and this proves the conclusion.

As an immediate consequence of Corollary 2.4 and Proposition 2.10, we have:

###### Corollary 2.11

Let $R$ be a noetherian ring. If $i(R)<\infty$, then $\text{dim}(R)<\infty.$

Motivated by Proposition 2.10, we will say a noetherian local ring $R$ of
dimension $d$ is of minimal integral dimension if $i(R)=d-1$. It follows from
the Lipman-Sathaye theorem [LS], $i(R)\leq d-1$ for every regular ring of
dimension $d>0$. So by Proposition 2.10 , we have:

###### Corollary 2.12

Let $R$ be a regular local ring of dimension $d>0$. Then $R$ is of minimal
integral dimension, i.e. $i(R)=d-1$.

In the rest of the section, we consider the local ring $R$ with $i(R)=0$.
Clearly every ideal of $R$ is integrally closed for such rings. By Proposition
2.10, we have $\text{dim}R\leq 1$. If $R$ is an artinian local ring, then
every ideal $I$ of $R$ with $I\neq R$ is integral over the zero ideal $0$. So
$I=0$, and $R$ must be a field. For $\text{dim}R=1$, we have

###### Proposition 2.13

Let $(R,\texttt{m})$ be a noetherian local ring with $i(R)=0$. If $R$ is not a
field, then $R$ is a one dimensional regular local ring.

  * Proof.

As we point out as above, under the assumption of $R$, $R$ must be a local
ring of dimension 1. Since every ideal of $R$ is integrally closed, it follows
that $R$ is reduced because the nilradical of $R$ is integral over the zero
ideal.

Now, we show that $R$ is a domain. It suffices to prove that $R$ has only one
minimal prime ideal. Let $P_{1},P_{2},\cdots,P_{r}$ are all the minimal prime
ideals of $R$.

If $r\geq 2$, we can choose elements $a,b$ such that

$a\in P_{1},a\notin\cap_{i=2}^{r}P_{i}\ \text{and}\
b\in\cap_{i=2}^{r}P_{i},b\notin P_{1}.$

It is obvious $ab=0,(a+b)\notin P_{1}$ and $a^{2}-(a+b)a=0.$ It shows $a$ is
integral over the ideal $(a+b)R$, and thus $a\in(a+b)R$ because of
$\overline{(a+b)R}=(a+b)R$.

Write $a=(a+b)c$ for some $c\in R$. If $c\notin\texttt{m}$, then $c$ is a unit
of $R$. So

$a+b=c^{-1}a\in P_{1},$

and this contradicts the choice of $a+b$. Hence $c\in\texttt{m}$, and it
implies $a(1-c)=bc$. Thus

$a\in\cap_{i=2}^{r}P_{i}$

because $1-c$ is a unit of $R$, which also contradicts the choice of $a$.
Hence $r=1$ and $R$ is a domain.

Let $Q(R)$ be the field of the fractions of $R$. In order to prove that $R$ is
a regular local ring, it suffices to show that $R$ is integrally closed in
$Q(R)$ [Ma]. Let $\frac{a}{b}$ be an element of $Q(R)$ such that $\frac{a}{b}$
is integral over $R$, where $a,b\in R$. Hence there exists an equation of the
form

$(\frac{a}{b})^{n}+a_{1}(\frac{a}{b})^{n-1}+\cdots+a_{n}=0$

where $a_{i}\in R$ for $(1\leq i\leq n)$. It yields that

$a^{n}+a_{1}ba^{n-1}+\cdots+b^{n}a_{n}=0.$

It implies $a$ is integral over the ideal $bR$. Since $bR$ is integrally
closed, it shows $a\in bR$, and thus $\frac{a}{b}\in R$. Therefore $R$ is a
integrally closed domain, and the proof of the proposition is complete.

The result of Proposition 2.13 means that a one dimensional noetherian local
ring with minimal integral dimension is a regular local ring. It would be very
interesting to study the properties of local rings with nonzero minimal
integral dimension. We conjecture:

###### Conjecture 2.14

Let $(R,m)$ be a reduced local ring with minimal integral dimension $i(R)>0$.
Then $R$ is a regular local ring.

##  3 Main result

In this section, we will present several sufficient conditions for a ring $R$
having finite integral dimension. Let us begin with recalling the notion of
uniform Artin-Rees property studied by Huneke [Hu].

Let $R$ be a noetherian ring, we say that $R$ has the uniform Artin-Rees
property, if for every pair $N\subseteq M$ of finitely generated $R$-modules,
there exists a number $k$, depending on $N,M$, such that for all ideal $I$ of
$R$ and all $n\geq k$

$I^{n}M\cap N\subseteq I^{n-k}N.$

First of all, we observe that if $R$ has the uniform Artin-Rees property, then
we can reduce the question when $i(R)$ is finite to the the reduced case when
$i(R_{red})<\infty$.

###### Proposition 3.1

Let $R$ be a noetherian ring. Then

(i) $i(R_{red})\leq i(R)$.

(ii) If $i(R_{red})<\infty$, then $i(R)<\infty,$ provided that $R$ has the
uniform Artin-Rees property.

  * Proof.

(i) If $i(R)=\infty$, we must have $i(R_{red})\leq\infty$ by (i), and the
conclusion holds in this case.

Now we assume $i(R)$ is a finite number and put $i(R)=k$. For any ideal $K$ of
$R_{red}$, there is an ideal $I$ with $Q\subseteq I$ such that $K=IR_{red}$.
Note that ${\overline{I}}^{n}\subseteq I^{n-k}$ for $n\geq k$. By Proposition
2.1, $\overline{K}={\overline{I}}R_{red}$. So

${\overline{K}}^{n}={\overline{I}}^{n}R_{red}\subseteq I^{n-k}R_{red}=K^{n-k}$

for $n\geq k$, this shows $i(R_{red})\leq k$, and the proof of (i) is
complete.

(ii) Let $Q$ be the nilradical of $R$. If $Q=0$, then $R=R_{red}$, and there
is nothing to prove. Now, we assume $Q\neq 0$. Let $n_{0}$ be the positive
integer such that $Q^{n_{0}}\neq 0$ and $Q^{n_{0}+1}=0.$ Since $R$ has the
uniform Artin-Rees property, for each $i(1\leq i\leq n_{0})$, there exists
integer $k_{i}>0$ such that for all ideals $I$ of $R$ and all $n\geq k_{i}$

$I^{n}\cap Q^{i}\subseteq I^{n-k_{i}}Q^{i}.$ $None$

Put $k=i(R_{red})$. By Proposition 2.1, for any ideal $I$ of $R$,
$\overline{IR_{red}}=\overline{I}R_{red}$. It yields

${\overline{I}}^{n}R_{red}\subseteq I^{n-k}R_{red}$

for $n\geq k$. Equivalently, we have for $n\geq k$

${\overline{I}}^{n}\subseteq I^{n-k}+Q.$ $None$

Since $I^{n-k}\subseteq{\overline{I}}^{n-k}$, it shows

${\overline{I}}^{n}\subseteq I^{n-k}+{\overline{I}}^{n-k}\cap Q.$

Hence if $n\geq k+k_{1}$, it follows from (3.1)

${\overline{I}}^{n}\subseteq I^{n-k}+{\overline{I}}^{n-k-k_{1}}Q,$

and then by (3.2), we have for $n\geq n-2k-k_{1}$

${\overline{I}}^{n}\subseteq I^{n-k}+({{I}}^{n-2k-k_{1}}+Q)Q\subseteq
I^{n-2k-k_{1}}+Q^{2}.$ $None$

In the following, we will use induction on $i$ to prove

${\overline{I}}^{n}\subseteq I^{n-ik-k_{1}-\cdots-k_{i-1}}+Q^{i}.$ $None$

for $i\ (2\leq i\leq n_{0}+1)$ and for all $n\geq ik+k_{1}+\cdots+k_{i-1}$.

Clearly, the conclusion holds for $i=2$ by (3.3). Suppose that

${\overline{I}}^{n}\subseteq I^{n-jk-k_{1}-\cdots-k_{j-1}}+Q^{j}.$

for some $j$ with $(2\leq j<n_{0})$ and for all $n\geq
jk+k_{1}+\cdots+k_{j-1}$. Then we have

${\overline{I}}^{n}\subseteq I^{n-jk-k_{1}-\cdots-
k_{j-1}}+{\overline{I}}^{n-jk-k_{1}-\cdots-k_{j-1}}\cap Q^{j}.$

Hence by (3.1), we have for $n\geq jk+k_{1}+\cdots+k_{j-1}+k_{j}$

${\overline{I}}^{n}\subseteq I^{n-jk-k_{1}-\cdots-
k_{j-1}}+{\overline{I}}^{n-jk-k_{1}-\cdots-k_{j-1}-k_{j}}Q^{j}.$

It follows from (3.2) that for $n\geq(j+1)k+k_{1}+\cdots+k_{j-1}+k_{j}$

${\overline{I}}^{n}\subseteq I^{n-(j+1)k-k_{1}-\cdots-k_{j}}+Q^{j+1}.$

Hence by induction, (3.4) holds for all $i(2\leq i\leq n_{0}+1)$. Note that
$Q^{n_{0}+1}=0$. Thus we have shown

${\overline{I}}^{n}\subseteq I^{n-(n_{0}+1)k-k_{1}-\cdots-k_{n_{0}}}.$

for $n\geq(n_{0}+1)k+k_{1}+\cdots+k_{n_{0}}$ and for all ideals $I$ of $R$.
Therefore $i(R)<\infty$, and this ends the proof of (ii).

To give a sufficient condition for a ring $R$ having the uniform Artin-Rees
property, Huneke [Hu] studied two ideals $T(R)$ and $CM(R)$ of $R$.

For an integer $k$, set
$T_{k}(R)=\underset{I,n}{\bigcap}(I^{n-k}:\overline{I^{n}})$, where the
intersection is taken over all $n$ and all ideals $I$ of $R$. We define
$T(R)=\underset{k}{\bigcup}T_{k}(R)$. It is clear if $k$ increases, the ideals
$T_{k}$ also increase. Hence $T(R)$ is an ideal of $R$. If $T(R)=R$, we say
that $R$ has the uniform Briançon-Skoda property. Clearly, $R$ has the uniform
Briançon-Skoda property if and only if there exists an integer $k\geq 0$ such
that for $n\geq k$ and all ideals $I$ of $R$

$\overline{I^{n}}\subseteq I^{n-k}.$

By Corollary 2.6, one can see easily $i(R)<\infty$ if $R$ has the uniform
Briançon-Skoda property.

We need the following notion of standard complex of free modules to describe
the definition of the ideal $CM(R)$. Recalling that a complex of finitely
generated free $R$-modules $F\textbf{.}$:

$0\rightarrow
F_{n}\overset{f_{n}}{\to}F_{n-1}\rightarrow\cdots\rightarrow{F_{1}}\overset{f_{1}}{\to}\
{F_{0}}$

is said to satisfy the standard condition on rank if
$\text{rank}(f_{n})=\text{rank}(F_{n})$, and
$\text{rank}(f_{i+1})+\text{rank}(f_{i})=\text{rank}(F_{i})$ for $1\leq i<n$.
Here we think $f_{i}$ as given by a matrix $A_{i}$, and the rank of $f_{i}$ is
the determinantal rank of $A_{i}$. Let $I(f_{i})$ be the ideal generated by
the rank-size minors of $A_{i}$. We say that $F\textbf{.}$ satisfies the
standard condition on height if $\text{ht}(I(f_{i})\geq i$ for all $i$. Note
that the height an ideal $I$ is $\infty$ if $I=R$. We define $CM(R)$ to be the
ideal generated by all elements $a\in R$ such that for all complexes
$F\textbf{.}$ of finitely generated free $R$-modules satisfying the standard
conditions on height and rank, $x\text{H}_{i}(F\textbf{.})=0$ for all $i\geq
1$.

As an easy consequence of Buchsbaum-Eisenbud criterion theorem for exactness
of a free complex [BE], it follows that $CM(R)=R$ if and only if $R$ is a
Cohen-Macaulay ring. One of the important questions concerning $CM(R)$ is when
$CM(R)$ is not contained in any minimal prime ideal of $R$. This question is
closely related to the existence of uniform local cohomological annihilators
of $R$ which was introduced by the author [Zh1].

Recall that an element $a\in R$ is said to be a uniform local cohomological
annihilator of $R$, if

(i) $a$ is not contained in any minimal prime ideal of $R$,

(ii) For every maximal ideal m, $a$ kills the $i$-th local cohomology module
$\text{H}_{\texttt{m}}^{i}(R)$ for $i<\text{ht}(\texttt{m}).$

It is known [cf. Zh2] that $CM(R)$ is not contained in any minimal prime ideal
of $R$ if and only if $R$ has a uniform local cohomological annihilator.
Moreover, Zhou proved a noetherian domain of finite dimension has a uniform
local cohomological annihilator, provided $R$ is a homomorphic image of a
Cohen-Macaulay ring [Zh1] or $R$ is an excellent ring [Zh2].

By means of $T(R/P)$ and $CM(R/P)$ for every prime ideal $P$ of $R$, Huneke
[Hu] proved an important criterion for rings having the uniform Artin-Rees
property. The same proof of Huneke works for the following criterion, which
replaces the condition $T(R/P)\neq 0$ of [Hu, Theorem 3.4] by $i(R/P)<\infty$.

###### Theorem 3.2

Let $R$ be a noetherian ring with infinite residue fields. Assume that for all
nonzero prime ideals $P$ of $R$ the following conditions hold:

(i) $i(R/P)<\infty.$

(ii) $CM(R/P)\neq 0$.

Then $R$ has the uniform Artin-Rees property.

Combining Theorem 3.2 with the mentioned results of the author [Zh1] [Zh2] ,
we have:

###### Corollary 3.3

Let $R$ be a finite dimensional noetherian ring with $i(R/P)<\infty$ for any
prime ideal of $R$. If $R$ satisfies one of the following conditions:

(i) $R$ is an excellent ring.

(ii) $R$ is a quotient ring of a Cohen-Macaulay ring of finite dimension.

Then $R$ has the uniform Artin-Rees property.

The following remarkable result is proved by Huneke [Hu, Theorem 4.12], which
gives three sufficient conditions for rings having the uniform Artin-Rees
property. In particular, the theorem asserts that every local ring has the
uniform Artin-Rees property.

###### Theorem 3.4

Let $R$ be a noetherian ring of finite dimension. $R$ has the uniform Artin-
Rees property if $R$ satisfies one of the following conditions:

(i) $R$ is essentially of finite type over a local ring.

(ii) $R$ is a ring of characteristic $p>0$, and $R$ is module finite over
$R^{p}$.

(iii) $R$ is essentially of finite type over the ring of integer numbers Z.

The another remarkable result of Huneke [Hu] is the following uniform
Briançon-Skoda theorem [Hu, Theorem 4.13].

###### Theorem 3.5

Let $R$ be a reduced noetherian ring of finite dimension. $R$ has the uniform
Briançon-Skoda property if $R$ satisfies one of the following conditions:

(i) $R$ is essentially of finite type over an excellent local ring.

(ii) $R$ is a ring of characteristic $p>0$, and $R$ is module finite over
$R^{p}$.

(iii) $R$ is essentially of finite type over the ring of integer numbers Z.

Now, we turn to the main result of the paper. By means of Proposition 3.1,
Theorem 3.4 and Theorem 3.5, we can prove the following main result, which is
a weaker generalization of Theorem 3.5.

###### Theorem 3.6

Let $R$ be a noetherian ring of finite dimension. Then $i(R)<\infty$ if $R$
satisfies one of the following conditions:

(i) $R$ is essentially of finite type over a noetherian local ring.

(ii) $R$ is a ring of characteristic $p$, and $R$ is module finite over
$R^{p}$.

(iii) $R$ is essentially of finite type over the ring of integer numbers Z.

  * Proof.

(i) By Corollary 2.3, we may assume $R$ is of finite type over a local ring
$(S,\texttt{m})$. Let $\hat{S}$ be the completion of $S$ in the m-adic
topology. It is known $\hat{S}$ is faithfully flat over $S$, and thus
$B=R\otimes_{S}\hat{S}$ is of finite type and faithfully flat over $\hat{S}$.
From Proposition 2.7, it suffices to prove $i(B)<\infty$. Replacing $R$ by
$B$, and $S$ by $\hat{S}$, we may assume $R$ is of finite type over an
excellent ring $S$.

Write $R=S[X_{1},X_{2},\cdots,X_{r}]/I$, where $I$ is an ideal of the
polynomial ring $S[X_{1},X_{2},\cdots,X_{r}]$. Let $P_{1},P_{2},\cdots,P_{t}$
be all the minimal prime ideals of $I$. Set $p_{i}=P_{i}\cap S$ for $1\leq
i\leq t$. It is not difficult to see that

$R_{red}=S[X_{1},X_{2},\cdots,X_{r}]/P_{1}\cap P_{2}\cap\cdots\cap P_{t}$

is of finite type over the reduced excellent ring $S/p_{1}\cap
p_{2}\cap\cdots\cap p_{t}$. By Theorem 3.5, $R_{red}$ has the uniform
Briançon-Skoda property. In particular $i(R_{red})<\infty$. Now from Theorem
3.4 and Proposition 3.1, we conclude $i(R)<\infty$, and this proves (i).

(ii) First, we show that $R_{red}$ is module finite over $(R_{red})^{p}$. In
fact, let $P_{1},P_{2},\cdots,P_{r}$ are all the minimal prime ideal of $R$.
It is easy to see $R^{p}/(P_{i}\cap R^{p})$ is naturally isomorphic to
$(R/P_{i})^{p}$ and $P_{i}\cap R^{p}={P_{i}}^{p}$ is a prime ideal of $R^{p}$.
Hence the nilpotent ideal of $R^{p}$

$I=P_{1}\cap P_{2}\cap\cdots\cap P_{r}\cap
R^{p}={P_{1}}^{p}\cap{P_{2}}^{p}\cap\cdots\cap{P_{r}}^{p}$

is an intersection of prime ideals of $R^{p}$. Conversely the nilradical ideal
J is clearly contained in $I$, so $I$ is the nilradical ideal of $R^{p}$. It
is easy to see $P_{i}^{p}$ are all the distinct minimal prime ideals of
$R^{p}$. Moreover, there is a natural isomorphism between the rings $R^{p}/I$
and $({R_{red}})^{p}$. So if $R$ is module finite over $R^{p}$, then $R_{red}$
is module finite over $R^{p}/I$, and thus is module finite over
$({R_{red}})^{p}$.

Now from Theorem 3.5, $R_{red}$ has the uniform Briançon-Skoda property. In
particular $i(R_{red})<\infty$. Now from Theorem 3.4 and Proposition 3.1, we
conclude $i(R)<\infty$, and this ends the proof of (ii).

(iii) Clearly, $R_{red}$ is also essentially of finite type over the ring of
integer numbers Z. Hence from Theorem 3.5 $i(R_{red})<\infty$. The conclusion
follows from Theorem 3.4 and Proposition 3.1, and this ends the proof of the
theorem.

An immediate consequence of Theorem 3.5 is the following finiteness property
of $i(R)$ for any noetherian local ring $R$.

###### Corollary 3.7

Let $R$ be a noetherian local ring. Then $i(R)<\infty$.

References

  * [AW]

Andersson, M., Wulcan, E.: Global effective versions of the Briançon-Skoda-
Huneke theorem. Invent. Math.  200, 607-651(2015)

  * [BE]

Buchsbaum, D., Eisenbud, D.: What makes a complexes exact. J. Algebra 25,
259-268(1973)

  * [BS]

Briançon, J., Skoda, H.: Sur la clóture intégrale d’ un idéal de germes de
fonctions holomorphes en un point de ${\textbf{C}}^{n}$. C.R. Acad. Sci. Paris
S$\acute{e}$r. A. 278, 949-951(1974)

  * [Ei]

Eisenbud, D.: Commutative algebra. With a view toward algebraic geometry.
Graduate Texts in Mathematics 160, Springer, New York (1995)

  * [Gr]

Grothendieck, A.: Local Cohomology. Lect. Notes Math. 41, Springer-Verlag, New
York (1963)

  * [HK]

Herzog, J., Kunz, E.: Der Kanonische Module eines Cohen-Macaulay Rings.
Lecture Notes in Mathematics 238, Springer-Verlag, New York(1971)

  * [Ho1]

Hochster, M.: Contracted ideals from integral extensions of regular rings.
Nagoya Math. J. 51 25-43(1973)

  * [Ho2]

Hochster, M.: Canonical elements in local cohomology modules and the direct
summand conjecture. J. Algebra 84, 503-553(1983)

  * [HH1]

Hochster, M., Huneke C.: Tight closure, invariant theory, and the Briançon-
Skoda theorem. J. Amer. Math. Soc. 3, 31-116(1990)

  * [HH2]

Hochster, M., Huneke C.: Infinite integral extensions and big Cohen-Macaulay
algebras. Annals of math. 3, 53-89(1992)

  * [HH3]

Hochster, M., Huneke C.: Comparison of symbolic and ordinary powers of ideals.
Invent. Math. 147, 349-369(2002)

  * [HS]

Huneke, C., Swanson, I.: Integral Closure of Ideals, Rings and Modules. London
Math. Soc. Lect. Note Ser. 336, Camb. Univ. Press 2006

  * [Hu]

Huneke, C.: Uniform bounds in noetherian rings. Invent. Math.  107, 203-223
(1992)

  * [H$\rm{\ddot{u}}$]

H$\rm{\ddot{u}}$bl, R.: Derivations and the integral closure of ideals, with
an appendix Zeros of differentials along ideals. by I. Swanson. Proc. Amer.
Math. Soc. 127, 3503-3511 (1999)

  * [It]

Itoh, S. Integral closures of ideals of the principal class. Hiroshima Math.
J. 17 373-375 (1987)

  * [Ku]

Kunz, E.: On noetherian rings of characteristic p. Am. J. Math. 98, 999-1013
(1976).

  * [LS]

Lipman, J., Sathaye, A.: Jacobian ideals and a theorem of Briançon-Skoda.
Mich. Math. J. 28, 199-222 (1981)

  * [Ly]

Lyubeznik, G.: A property of ideals in polynomial rings. Proc. Am. Math. Soc.
98, 399-400 (1986)

  * [Ma]

Matsumura, H.: Commutative Ring Theory, Cambrige Univ. Press, New York, (1986)

  * [NR]

Northcott, D., G., Rees, D.: Reductions of ideals in local rings. Proc.
Cambridge Phil. Soc. 50, 145-158 (1954)

  * [Re]

Rees, D.: A note on analytically unramified local rings. J. London Math. Soc.
36, 24-28 (1961)

  * [Sc]

Schenzel, P.: Cohomological annihilators. Math. Proc. Camb. Phil. Soc. 91,
345-350 (1982)

  * [Sh1]

Sharp, R.: Local cohomology theory in commutative algebra. Q. J. Math. Oxford
21, 425-434 (1970)

  * [Sw1]

Swanson, I.: Joint reductions, tight closure and the Briançcon-Skoda Theorem.
J. Algebra 147, 128-136 (1992)

  * [Sw2]

Swanson, I.: Joint reductions, tight closure and the Briançcon-Skoda Theorem.
J. Algebra 170, 567-583 (1994)

  * [Zh1]

Zhou, C.: Uniform annihilators of local cohomology. J. Algebra 305, 585-602
(2006)

  * [Zh2]

Zhou, C.: Uniform annihilators of local cohomology of excellent rings. J.
Algebra 315, 286-300 (2007)


# Further evidence for shape coexistence in 79Znm near doubly-magic 78Ni

L. Nies<EMAIL_ADDRESS>European Organization for Nuclear Research (CERN),
Meyrin, 1211 Geneva, Switzerland  Institut für Physik, Universität Greifswald,
17487 Greifswald, Germany  L. Canete  University of Jyvaskyla, Department of
Physics, Accelerator laboratory, P.O. Box 35(YFL) FI-40014 University of
Jyvaskyla, Finland  Department of Physics, University of Surrey, Guildford,
GU2 7X5, United Kingdom  D. D. Dao  Université de Strasbourg, CNRS, IPHC UMR
7178, F-67000 Strasbourg, France  S. Giraud  Present address: Facility for
Rare Isotope Beams, Michigan State University, East Lansing, Michigan 48824,
USA  GANIL, Bd Henri Becquerel, BP 55027, F-14076 Caen Cedex 5, France  A.
Kankainen<EMAIL_ADDRESS>University of Jyvaskyla, Department of
Physics, Accelerator laboratory, P.O. Box 35(YFL) FI-40014 University of
Jyvaskyla, Finland  D. Lunney  Université Paris-Saclay, CNRS/IN2P3, IJCLab,
91405 Orsay, France  F. Nowacki  Université de Strasbourg, CNRS, IPHC UMR
7178, F-67000 Strasbourg, France  B. Bastin  GANIL, Bd Henri Becquerel, BP
55027, F-14076 Caen Cedex 5, France  M. Stryjczyk  University of Jyvaskyla,
Department of Physics, Accelerator laboratory, P.O. Box 35(YFL) FI-40014
University of Jyvaskyla, Finland  P. Ascher  Université de Bordeaux,
CNRS/IN2P3—Université, CNRS/IN2P3, LP2I Bordeaux, UMR 5797, F-33170 Gradignan,
France  K. Blaum  Max-Planck-Institut für Kernphysik, 69117 Heidelberg,
Germany  R. B. Cakirli  Department of Physics, Istanbul University, Istanbul
34134, Turkey  T. Eronen  University of Jyvaskyla, Department of Physics,
Accelerator laboratory, P.O. Box 35(YFL) FI-40014 University of Jyvaskyla,
Finland  P. Fischer  Institut für Physik, Universität Greifswald, 17487
Greifswald, Germany  M. Flayol  Université de Bordeaux, CNRS/IN2P3—Université,
CNRS/IN2P3, LP2I Bordeaux, UMR 5797, F-33170 Gradignan, France  V. Girard
Alcindor  GANIL, Bd Henri Becquerel, BP 55027, F-14076 Caen Cedex 5, France
A. Herlert  FAIR GmbH, Planckstraße 1, 64291 Darmstadt, Germany  A. Jokinen
University of Jyvaskyla, Department of Physics, Accelerator laboratory, P.O.
Box 35(YFL) FI-40014 University of Jyvaskyla, Finland  A. Khanam  University
of Jyvaskyla, Department of Physics, Accelerator laboratory, P.O. Box 35(YFL)
FI-40014 University of Jyvaskyla, Finland  Department of Applied Physics,
Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland  Department of
Physics, University of Helsinki, P.O. Box 43, FI-00014 Helsinki, Finland  U.
Köster  European Organization for Nuclear Research (CERN), Meyrin, 1211
Geneva, Switzerland  Institut Laue-Langevin, 38000 Grenoble, France  D. Lange
Max-Planck-Institut für Kernphysik, 69117 Heidelberg, Germany  I. D. Moore
University of Jyvaskyla, Department of Physics, Accelerator laboratory, P.O.
Box 35(YFL) FI-40014 University of Jyvaskyla, Finland  M. Müller  Max-Planck-
Institut für Kernphysik, 69117 Heidelberg, Germany  M. Mougeot  University of
Jyvaskyla, Department of Physics, Accelerator laboratory, P.O. Box 35(YFL)
FI-40014 University of Jyvaskyla, Finland  Max-Planck-Institut für Kernphysik,
69117 Heidelberg, Germany  D.A. Nesterenko  University of Jyvaskyla,
Department of Physics, Accelerator laboratory, P.O. Box 35(YFL) FI-40014
University of Jyvaskyla, Finland  H. Penttilä  University of Jyvaskyla,
Department of Physics, Accelerator laboratory, P.O. Box 35(YFL) FI-40014
University of Jyvaskyla, Finland  C. Petrone  IFIN-HH, P.O. Box MG-6, 077125
Bucharest-Magurele, Romania  I. Pohjalainen  University of Jyvaskyla,
Department of Physics, Accelerator laboratory, P.O. Box 35(YFL) FI-40014
University of Jyvaskyla, Finland  A. de Roubin  Present address: KU Leuven,
Instituut voor Kern- en Stralingsfysica, B-3001 Leuven, Belgium  University of
Jyvaskyla, Department of Physics, Accelerator laboratory, P.O. Box 35(YFL)
FI-40014 University of Jyvaskyla, Finland  V. Rubchenya  Deceased.  University
of Jyvaskyla, Department of Physics, Accelerator laboratory, P.O. Box 35(YFL)
FI-40014 University of Jyvaskyla, Finland  Ch. Schweiger  Max-Planck-Institut
für Kernphysik, 69117 Heidelberg, Germany  L. Schweikhard  Institut für
Physik, Universität Greifswald, 17487 Greifswald, Germany  M. Vilen
University of Jyvaskyla, Department of Physics, Accelerator laboratory, P.O.
Box 35(YFL) FI-40014 University of Jyvaskyla, Finland  J. Äystö  University of
Jyvaskyla, Department of Physics, Accelerator laboratory, P.O. Box 35(YFL)
FI-40014 University of Jyvaskyla, Finland

###### Abstract

Isomers close to doubly-magic ${}^{78}_{28}$Ni50 provide essential information
on the shell evolution and shape coexistence near the ${Z=28}$ and ${N=50}$
double shell closure. We report the excitation energy measurement of the
$1/2^{+}$ isomer in ${}^{79}_{30}$Zn49 through independent high-precision mass
measurements with the JYFLTRAP double Penning trap and with the ISOLTRAP
Multi-Reflection Time-of-Flight Mass Spectrometer. We unambiguously place the
$1/2^{+}$ isomer at $942(10)\text{\,}\mathrm{keV}$, slightly below the
$5/2^{+}$ state at $983(3)\text{\,}\mathrm{keV}$. With the use of state-of-
the-art shell-model diagonalizations, complemented with Discrete Non
Orthogonal shell-model calculations which are used here the first time to
interpret shape coexistence, we find low-lying deformed intruder states,
similar to other ${N=49}$ isotones. The $1/2^{+}$ isomer is interpreted as the
band-head of a low-lying deformed structure akin to a predicted low-lying
deformed band in 80Zn, and points to shape coexistence in 79,80Zn similar to
the one observed in 78Ni. The results make a strong case for confirming the
claim of shape coexistence in this key region of the nuclear chart.

The atomic nucleus, a conglomerate of protons and neutrons, is a complex many-
body system with unique features. The nuclear shell model has successfully
described various nuclear properties, including the emergence of shell
closures [1] and magic numbers [2].

Similar to the atomic shell, nuclei can be excited, resulting in a dense level
structure. Ground and excited states can show different shapes resulting from
the microscopic wave function [3]. Deformed excited states often emerge near
closed shells, where the excitation of multiple nucleons across the shell gap
can be energetically favorable, leading to deformation through the increased
number of particles found in the valence space [4]. While typically the
coexistence of ground states and deformed excited states at low energies are
observed, shape inversion can appear when the ground state becomes deformed in
coexistence with a spherical excited state [4, 5, 6]

Research on shape coexistence close to the doubly-magic nucleus
${}^{78}_{28}$Ni50 has gained momentum only recently [7]. Low-lying intruder
states, often indicators of shape coexistence, have been studied in ${N=49}$
isotones through transfer-reaction experiments [8, 9, 10, 11]. First evidence
supporting shape coexistence, such as the claimed discovery of a $0^{+}_{2}$
intruder state in ${}^{80}_{32}$Ge48 from Ref. [12], could, however, not be
confirmed in subsequent experiments [13, 14].

More recently, the doubly-magic nature of 78Ni was supported through the
measurement of its E($2_{1}^{+}$) value [15], as well as $\gamma$-ray
spectroscopy [16] and mass spectrometry of ${}^{79}_{29}$Cu50 [17],
reinforcing the persistence of the ${Z=28}$ gap. The potential appearance of
shape coexistence in 78Ni is furthermore theorized to be a pathway into a new
island of inversion at ${N=50}$ [18].

The isomer in ${}^{79}_{30}$Zn49, a long-lived nuclear state, provides a
unique opportunity to further study the interplay between the single-particle
and collective degrees of freedom in the close vicinity of 78Ni. The first
spectroscopy of 79Zn from (d,p) transfer reactions found evidence for the
presence of intruder states and tentatively assigned a spin-parity of
$1/2^{+}$ to the isomeric state with an excitation energy of
$1100(150)\text{\,}\mathrm{keV}$, as well as a close-lying $5/2^{+}$ state
with $983(3)\text{\,}\mathrm{keV}$ [19], leaving the exact state ordering
uncertain. The spin-parity was confirmed for the $9/2^{+}$ ground and the
$1/2^{+}$ isomeric states through magnetic moment measurements from collinear
laser spectroscopy [20, 21]. More significantly, these works found a large
isomer shift in the charge radius but could not connect this increase to a
deformation unambiguously.

Recent studies have shown that shape changes may be linked to multiple
particle-hole excitations of both protons and neutrons [22, 23, 24, 25, 26].
However, a precise and accurate excitation energy measurement of the $1/2^{+}$
isomer to confirm claims of shape coexistence in 79Zn is missing from the
above-mentioned work. Such measurement will unambiguously determine the state
ordering, validate the particle-hole excitation character, and benchmark the
binding energy predictions of the employed shell-model interactions near 78Ni.

While the ground-state mass of 79Zn is precisely known [27, 28, 29] via mass
measurements, the uncertainty on the excitation energy from the transfer-
reaction experiment is rather large. Given the importance of shape coexistence
in the immediate vicinity of 78Ni, we present two independent high-precision
mass spectrometry experiments of the isomer 79Znm using the JYFLTRAP double
Penning trap [30] at the Ion Guide Isotope Separator On-Line (IGISOL) facility
[31] in Jyväskylä (Finland), and the Multi-Reflection Time-of-Flight Mass
Spectrometer (MR-ToF MS) of ISOLTRAP [32] at ISOLDE/CERN [33] (Switzerland).
The experiments were performed at different facilities and with different
techniques to ensure the isomer’s production and the accuracy of its
excitation energy. The results are interpreted by large-scale shell model
calculations, utilizing the valence space of interactions used in Refs. [20,
21, 17, 18]. While offering a more detailed and accurate picture of the
nuclear structure in this critical region, the calculations highlight the
relative fragility of the doubly-magic shell strength.

Table 1:  Frequency ratio $r$ or time-of-flight difference $\Delta t$, mass-excess values ME, and excitation energy of the isomer $E$ determined in this work. The values for $J^{\pi}$ and $T_{1/2}$ are from Ref. [20], ME${}_{\text{lit}}$ is deduced from the ground state mass (weighted average of the measurements from Refs. [28, 27, 29]) in combination with the excitation energy reported in [19]. 84Kr+ from [34] was used as a reference for the ToF-ICR measurements, while 79Zn+ in its ground state from [28, 27, 29] was used for the MR-ToF MS measurements.  Nuclide | $J^{\pi}$ | $T_{1/2}$ | Method |  $r$ or $\Delta t$ | ME $($\mathrm{keV}$)$ | ME${}_{\text{lit}}$ $($\mathrm{keV}$)$ | Diff. $($\mathrm{keV}$)$ |  $E$ $($\mathrm{keV}$)$
---|---|---|---|---|---|---|---|---
79Znm | $1/2^{+}$ |  $>200$ ms |  ToF-ICR MR-ToF MS |  $0.940796186(144)$ $274.2(40)\text{\,}\mathrm{ns}$ |  $-52490(12)$ $-52489(14)$ | $-52332(150)$ |  $-158(150)$ $-157(150)$ |  $942(12)$ $943(14)$

For the Penning-trap measurements, the ions of interest were produced via
proton-induced fission using $35$-MeV protons from the K130 cyclotron
impinging onto a $15\text{\,}\mathrm{mg}\text{\,}{\mathrm{cm}}^{-2}$ thick
natU target at IGISOL. The reaction products were stopped and thermalized in
the helium gas cell of the fission ion guide [35], leaving a large fraction of
the products singly charged. The ions were extracted from the chamber with a
sextupole ion guide [36] and accelerated to 30 keV. A 55∘ dipole magnet was
used to mass-separate the ions based on their mass-to-charge ratio $m/q$. The
mass-separated beam was then stopped in a radiofrequency quadrupole cooler and
buncher (RFQ-cb) [37] and released as ion bunches into the double Penning trap
JYFLTRAP [30]. In the first trap, either the ground- or isomeric-state ions of
79Zn were selected using mass-selective buffer-gas cooling [38]. The selected
ions were transferred to the second trap, where the high-precision mass
measurements were performed using the Time-of-Flight Ion Cyclotron Resonance
(ToF-ICR) method [39]. The cyclotron resonance frequency ${\nu_{c}=qB/(2\pi
m)}$ of the $1/2^{+}$ state in 79Zn+ was measured using a
$100\text{\,}\mathrm{ms}$ pulse of quadrupolar rf excitation. Altogether four
ToF-ICR spectra were measured for the $1/2^{+}$ state in 79Zn+. An example of
such a ToF-ICR spectrum is shown in Fig. 1(a).

Figure 1: (a) A typical ToF-ICR spectrum for the $1/2^{+}$ state in 79Zn+.
Colored bins indicate the number of detected ions. Darker shades correspond to
more ions and lighter shades to fewer ions. The solid red line represents the
fit to the data points (black) using the model from Ref. [39]. The cyclotron
frequencies are indicated with a vertical black dashed line for the ground
state (not present in this spectrum) and a vertical blue dash-dotted line for
the isomer. (b) Time-of-Flight spectrum for the MR-ToF MS data. The ToF of the
ground state is indicated by a vertical black dashed line, and the ToF of the
isomer by a vertical blue dash-dotted line. The solid red line represents the
fit to the data using the model from Ref. [40].

The magnetic field strength $B$ was determined using 84Kr+ as a mass
reference. The mass of the $1/2^{+}$ state in 79Zn was obtained from the
measured frequency ratio $r=\nu_{c,ref.}/\nu_{c}$ between the 84Kr+ reference
ions and the isomeric-state ions of 79Zn+ as $m=r(m_{ref}-m_{e})+m_{e}$, where
$m_{ref}$ is the mass of the reference ion and $m_{e}$ is the electron mass.
Two sources of systematic uncertainties were taken into account in the
analysis, the fluctuation of the magnetic field being $8.18\times
10^{-12}\times\Delta t\mathrm{~{}min}^{-1}$ [41], where $\Delta t$ represents
the time between two reference measurements, and mass-dependent uncertainties
being $2.2\times 10^{-10}\times(m-m_{ref})/\mathrm{u}$ [42].

For the MR-ToF MS measurements, neutron-rich zinc isotopes were produced at
the isotope separation online facility ISOLDE at CERN [33] by impinging a
$1.4$-GeV proton beam onto a solid tungsten block to generate an intense
spallation neutron flux [43]. The zinc isotopes were then produced through
neutron-induced fission processes in an adjacent thick uranium carbide target.
Using the tungsten converter resulted in reduced production of isobaric
neutron-deficient nuclides. The radioactive fission products then diffused
through the target material into a cold quartz transfer line [44, 45] which
further eliminated contamination of surface-ionized elements, e.g. Ga, Rb, and
Sr [46]. The remaining radioactive species were then ionized by the Resonance
Ionization Laser Ion Source (RILIS) [47], using an element-selective three-
step ionization scheme for zinc. The ion beam was then mass-separated using
the general-purpose mass-separator dipole magnet, removing non-isobaric
contamination, before being sent to the ISOLTRAP mass spectrometer [32].

The quasi-continuous ion beam was cooled and bunched in a linear RFQ-cb [48]
with a storage time of $10\text{\,}\mathrm{ms}$ before being captured in the
MR-ToF MS [49] using the in-trap lift method [50]. After trapping times of up
to $43\text{\,}\mathrm{ms}$, the ion bunch was ejected onto a single-ion
counting detector using the same in-trap lift.

The excitation energy of the isomeric state ${E=\left[(\Delta
t/t_{0})^{2}+2\Delta t/t_{0}\right]m_{0}c^{2}}$ is related to the time-of-
flight difference $\Delta t$ between the ground state with mass $m_{0}$ and
the isomeric state, the absolute flight time of the ground state $t_{0}$, and
the speed of light in vacuum $c$. For long flight times, where ${\Delta t\ll
t_{0}}$, this reduces to ${E\approx 2\Delta t/t_{0}\times m_{0}c^{2}}$. Figure
1 (b) shows the ToF spectrum for the MR-ToF MS data. Zinc was delivered from
ISOLDE as a pure beam, thus only the ground state (black dashed line) and the
isomeric state (blue dash-dotted line) of 79Zn were present in the spectrum. A
mass resolving power $R=t_{0}/2\Delta t_{\text{FWHM}}=300\,000$ was reached,
sufficient to resolve the two states. The asymmetric peak shape was fitted
with a multi-component exponentially modified Gaussian (“hyper-EMG”) [40] to
extract the absolute ToF $t_{0}=$4\,277\,871(2)\text{\,}\mathrm{ns}$$ of the
ground state and the ToF difference $\Delta t=$274(4)\text{\,}\mathrm{ns}$$
between the two states. The excitation energy was then calculated using the
ground state mass with
ME${}_{\text{lit}}=$-53\,432.1(18)\text{\,}\mathrm{keV}$$, which we calculated
as the weighted mean of the results from Refs. [27, 28, 29]. The excitation
energy was measured for different ion loads to account for ion-ion
interactions during the storage time in the MR-ToF MS [51, 52].

The results of the two independent measurements are summarized in Tab. 1. The
extracted excitation energies of the isomer agree very well, resulting in a
weighted mean of $942(10)\text{\,}\mathrm{keV}$. The isomer energy is lower
than $1100(150)\text{\,}\mathrm{keV}$ as given in the NUBASE 2020 evaluation
[53], which is based on the transfer reaction experiment from Ref. [19]. Our
value is significantly more precise and unambiguously sets the isomeric state
below the $5/2^{+}$ state located at $983(3)\text{\,}\mathrm{keV}$. We note
that the result agrees with the value $943(3)\text{\,}\mathrm{keV}$, obtained
from the beta-decay spectroscopy of 79Cu [54]. Here, we confirm the existence
of the isomer and provide a direct measure of its excitation energy.

Table 2:  Occupation of orbitals in the full proton $pf$ and neutron $sdg$ valence space for low-lying states in 79,80Zn and 78Ni (the latter taken from Ref. [18]). Eexp and Etheo (in $\mathrm{MeV}$) are the experimental and theoretical excitation energies. Ecorr and E${}^{*}_{\rm{corr}}$ (in MeV) are the total correlation energy and the correlation energy difference of an excited state with respect to its ground state. $n^{*}_{\nu}$ and $n^{*}_{\pi}$ are the total numbers of protons and neutrons above Z=28 and N=50, respectively.  Nuclide | $J^{\pi}$ | Eexp | Etheo | Ecorr | E${}^{*}_{\rm{corr}}$ | $n^{*}_{\nu}$ | $\nu_{g9/2}$ | $\nu_{d5/2}$ | $\nu_{s1/2}$ | $\nu_{g7/2}$ | $\nu_{d3/2}$ | $n^{*}_{\pi}$ | $\pi_{f7/2}$ | $\pi_{f5/2}$ | $\pi_{p3/2}$ | $\pi_{p1/2}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
79Zn | $9/2^{+}$ | 0.0 | 0.0 | -11.72 | - | 0.53 | 8.47 | 0.27 | 0.04 | 0.18 | 0.04 | 2.49 | 7.51 | 1.79 | 0.50 | 0.20
| $1/2^{+}$ | 0.94 | 0.83 | -18.59 | -6.87 | 1.84 | 7.17 | 0.81 | 0.54 | 0.34 | 0.15 | 2.82 | 7.18 | 1.45 | 0.95 | 0.42
| $5/2^{+}$ | 0.98 | 0.94 | -18.23 | -6.51 | 1.82 | 7.18 | 1.06 | 0.31 | 0.33 | 0.12 | 2.79 | 7.20 | 1.51 | 0.87 | 0.41
80Zn | $0^{+}_{1}$ | 0.0 | 0.0 | -10.80 | - | 0.49 | 9.50 | 0.23 | 0.03 | 0.19 | 0.04 | 2.48 | 7.52 | 1.90 | 0.44 | 0.14
| $0^{+}_{2}$ | - | 2.16 | -17.12 | -6.32 | 2.74 | 7.26 | 1.20 | 0.71 | 0.52 | 0.31 | 3.08 | 6.92 | 1.33 | 1.28 | 0.47
78Ni | $0^{+}_{1}$ | 0.0 | 0.0 | -8.00 | - | 0.38 | 9.62 | 0.12 | 0.02 | 0.20 | 0.04 | 0.57 | 7.44 | 0.38 | 0.15 | 0.04
| $0^{+}_{2}$ | - | 2.65 | -24.09 | -16.09 | 2.70 | 7.30 | 1.11 | 0.81 | 0.43 | 0.35 | 2.35 | 5.65 | 0.98 | 0.94 | 0.43

To interpret the present experimental data, shell-model calculations with the
PFSDG-U interaction [18, 15] were performed for 79,80Zn. The valence space is
spanned across the full $pf$ shell for protons and full $sdg$ shell for
neutrons, with 60Ca as an inert core. This interaction has been successfully
used in the 78Ni region to describe, among others, the two-neutron separation
energies $S_{2n}$ along the zinc isotopic chain [17], as well as the magnetic
g-factor in 79Zn [21].

The calculated excitation energies for 79Zn (Tab. 2) are in good agreement
with the experimental results with $1/2^{+}$ and $5/2^{+}$ at
$0.83\text{\,}\mathrm{MeV}$ and $0.94\text{\,}\mathrm{MeV}$, respectively. We
find that the two low-lying excited states in 79Zn show a one-particle-two-
hole configuration, consistent with other ${N=49}$ isotones [8, 9, 10, 11].
While the s1/2 and d5/2 orbitals lie together at the neutron Fermi surface in
the vicinity of 78Ni, the correlated ${N=50}$ neutron gap
${S_{N}(^{81}\text{Zn})-S_{N}(^{80}\text{Zn})}$, calculated from the present
theoretical values, remains sizeable at about $4.0\text{\,}\mathrm{MeV}$ and
is in agreement with the effective single-particle energies of Ref. [6] and
with the experimental value provided in Ref. [34]. These states usually
recover enough correlation energy (total energy minus the monopole part) to
compensate for their energy gap loss. This is observed in our calculations
shown in Tab. 2, where the total correlation energy is extracted. The excited
$1/2^{+}$ and $5/2^{+}$ states recover correlation energy on the order of
$\sim$ $$6.5\text{\,}\mathrm{MeV}$-$6.9\text{\,}\mathrm{MeV}$$ compared to the
$9/2^{+}$ the ground state.

The low excitation energy of these two states can be understood as the balance
between an average of 1.3 neutrons excited across the shell gap with respect
to the ground state ($\sim$5.5\text{\,}\mathrm{MeV}$$) and the correlation
energies ($\sim$6.5\text{\,}\mathrm{MeV}$-$6.9\text{\,}\mathrm{MeV}$$) which
compensate and result in very low-lying excitation energies for these one-
particle-two-holes states.

An inspection of the partial occupancies of the first two excited states (see
Tab. 2) reveals strong neutron mixing for the orbitals above ${N=50}$, as well
as different proton occupancies with respect to the ground state, invalidating
the spherical single particle-hole nature of these states and rather
suggesting a deformed shape. This neutron mixing and proton reshuffling is
visualized in Fig. 2, where the occupancy differences of the excited states
with respect to their ground states are plotted.

The structure of the excited states can be interpreted within the Discrete
Non-Orthogonal shell-model (DNO-SM) method, newly developed in Ref. [55] and
applied recently in Refs. [56, 57]. This approach expands the shell-model wave
functions in the deformed Hartree-Fock basis rather than the usual spherical
m-scheme basis. This allows the extraction of the corresponding deformation
amplitudes of a given state in the $(\beta,\gamma)$ plane.

Figure 3 (top panel) depicts such expansion for the ground state and the first
two excited states of 79Zn. Two clear patterns emerge: in the ground state,
the main components of the wave function tend towards the low-deformation
region with a small $\beta$ value, while for the excited states, the wave
functions are more fragmented and have, on average, a larger deformation.
Also, we find the clustering of the wave-function components consistent with
the deformation parameters $\beta\approx 0.15$ (ground state) and
$\beta\approx 0.22$ (isomeric state), deduced from the nuclear charge radius
in Ref. [20]. Moreover, the DNO-SM calculations reveal that both $1/2^{+}$ and
$5/2^{+}$ states belong to the same rotational structure, which is
characterized by the $K=1/2$ components of 100% and 96%, respectively (see
K-component extraction in Refs. [55, 57]).

Figure 2: Occupancy differences between excited states in 79,80Zn, and 78Ni
with their respective ground states. The data for 78Ni is taken from [18].
Figure 3: DNO-SM expansions in the ($\beta,\gamma$) plane (using the same
energy scale) for low-lying states in 79Zn ($9/2^{+}_{1}$, $1/2^{+}_{1}$ and
$5/2^{+}_{1}$ in upper panel) and 80Zn ($0^{+}_{1}$ and $0^{+}_{2}$ in lower
panel). The radius of circles represents the normalized probability of finding
a deformation of ($\beta,\gamma$) in the corresponding state.

To probe further into the deformed character of the low-lying states in 79Zn,
we have complemented the calculations with the investigation of 80Zn. We find
two low-lying $0^{+}$ states: the spherical ground-state and an excited
deformed state of two-particle-two-hole nature at $2.16\text{\,}\mathrm{MeV}$
(see Tab. 2). Again, a larger correlation energy ($\sim$
$6.3\text{\,}\mathrm{MeV}$) is observed for the excited 0+ state on the same
order of magnitude as those of the deformed intruder states in 79Zn,
indicating the same deformation nature of these states. Figure 3 shows the
wave function expansions for these two 0+ states (bottom panel). There is a
clear similarity between the spherical ground states for
79Zn($9/2^{+}$)/80Zn($0^{+}_{1}$) and the deformed excited states for
79Zn($1/2^{+}$, $5/2^{+}$)/80Zn($0^{+}_{2}$), advocating for the deformed
nature of the observed isomer in 79Zn, as well as its $5/2^{+}$ companion.

Finally, the present shape coexistence discussed for 79,80Zn can be put in
perspective with the shape coexistence recently observed and discussed for
78Ni [18, 15]: the deformed intruder 0${}^{+}_{2}$ of 78Ni has 2.7 neutron p-h
excitations on average, in remarkable agreement with present values quoted in
Tab. 2 for 0${}^{+}_{2}$ of 80Zn. Both 0${}^{+}_{2}$ states have $\sim 2.4-3$
protons on average in the ${f_{5/2},p_{3/2},p_{1/2}}$ shells, leading to close
collective structures. Therefore, the shape coexistence in 78Ni and in the
presented 79-80Zn reveal striking similarities.

To summarize, we have established the level ordering and determined the
excitation energy of the isomer in 79Zn by means of high-precision mass
spectrometry. Two measurements were performed independently, using different
production methods and measurement techniques at different radioactive ion
beam facilities. We show unambiguously that the $1/2^{+}$ isomeric state with
$942(10)\text{\,}\mathrm{keV}$ lies below the $5/2^{+}$ state with
$983(3)\text{\,}\mathrm{keV}$. The new DNO-SM calculations tool provides the
theoretical analysis, predicting the occurrence of low-lying deformed intruder
states. The $1/2^{+}$ isomer is interpreted as the band-head of a low-lying
deformed structure of the same nature as a predicted low-lying deformed band
in 80Zn. These findings provide an additional indication for shape coexistence
in 79,80Zn, similar to the one suggested for 78Ni.

We thank the ISOLDE technical group and the ISOLDE Collaboration for their
support. We acknowledge the support of the German Max Planck Society, the
French Institut National de Physique Nucléaire et de Physique des Particules
(IN2P3), the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme (Grant Agreements No. 682841
‘ASTRUm’, 654002 ‘ENSAR2’, and 101020842 ‘EUSTRONG’), as well as the German
Federal Ministry of Education and Research (BMBF; Grants No. 05P18HGCIA,
05P21HGCI1, and 05P21RDFNB). L.N. acknowledges support from the Wolfgang
Gentner Programme of the German Federal Ministry of Education and Research
(Grant No. 13E18CHA). This work has been supported by the Academy of Finland
under the Finnish Centre of Excellence Program (Nuclear and Accelerator Based
Physics Research at JYFL 2012-2017), and under Academy of Finland grants No.
275389, 284516, 312544, 295207, and 306980. We acknowledge the support of the
European Union’s Horizon 2020 research and innovation program grant agreement
No 654002 (ENSAR2) and No. 771036 (ERC CoG MAIDEN). We thank for the bilateral
mobility grant from the Institut Français in Finland, the Embassy of France in
Finland, the French Ministry of Higher Education and Research, and the Finnish
Society of Science and Letters. We are grateful for the mobility support from
PICS MITICANS (Manipulation of Ions in Traps and Ion sourCes for Atomic and
Nuclear Spectroscopy). S.G. thanks for the mobility grant from the EDPSIME. F.
N and D. D. D. acknowledge the financial support of CNRS/IN2P3, France, via
ABI-CONFI master projet.

The JYFLTRAP experiment was conducted by L.C, S.G, A.K, B.B, P.A, T.E, V.G.A.,
A.J., A.K, I.D.M, D.A.N, F.D.O., H.P., C.P., I.P., A.D.R., V.R., M.V., and
J.Ä. The ISOLTRAP experiment was conducted by L.N., R.B.C., P.F., M.F., A.H.,
D.La., M.Mü., M.M., Ch.S., and was conceived by U.K. The theoretical
calculations were performed by D.D.D. and F.N. Funding and supervision were
provided, in parts, by K.B. and L.S. The Manuscript was prepared by L.N.,
D.D.D., A.K., D.Lu., F.N., and M.S. All authors contributed to the editing of
the manuscript.

## References

  * Mayer [1949] M. G. Mayer, On Closed Shells in Nuclei. II, Phys. Rev. 75, 1969 (1949).
  * Haxel  _et al._ [1949] O. Haxel, J. H. D. Jensen, and H. E. Suess, On the "Magic Numbers" in Nuclear Structure, Phys. Rev. 75, 1766 (1949).
  * Jain  _et al._ [2021] A. Jain, B. Maheshwari, and A. Goel,  _Nuclear Isomers: A Primer_ (Springer International Publishing, 2021).
  * Heyde and Wood [2011] K. Heyde and J. L. Wood, Shape coexistence in atomic nuclei, Rev. Mod. Phys. 83, 1467 (2011).
  * Gray  _et al._ [2023] T. J. Gray, J. M. Allmond, Z. Xu, T. T. King, R. S. Lubna, H. L. Crawford, V. Tripathi, B. P. Crider, R. Grzywacz, S. N. Liddick, A. O. Macchiavelli, T. Miyagi, A. Poves, A. Andalib, E. Argo, C. Benetti, S. Bhattacharya, C. M. Campbell, M. P. Carpenter, J. Chan, A. Chester, J. Christie, B. R. Clark, I. Cox, A. A. Doetsch, J. Dopfer, J. G. Duarte, P. Fallon, A. Frotscher, T. Gaballah, J. T. Harke, J. Heideman, H. Huegen, J. D. Holt, R. Jain, N. Kitamura, K. Kolos, F. G. Kondev, A. Laminack, B. Longfellow, S. Luitel, M. Madurga, R. Mahajan, M. J. Mogannam, C. Morse, S. Neupane, A. Nowicki, T. H. Ogunbeku, W.-J. Ong, C. Porzio, C. J. Prokop, B. C. Rasco, E. K. Ronning, E. Rubino, T. J. Ruland, K. P. Rykaczewski, L. Schaedig, D. Seweryniak, K. Siegl, M. Singh, A. E. Stuchbery, S. L. Tabor, T. L. Tang, T. Wheeler, J. A. Winger, and J. L. Wood, Microsecond Isomer at the N=20 Island of Shape Inversion Observed at FRIB, Phys. Rev. Lett. 130, 242501 (2023).
  * Nowacki  _et al._ [2021] F. Nowacki, A. Obertelli, and A. Poves, The neutron-rich edge of the nuclear landscape: Experiment and theory., Prog. Part. Nucl. Phys. 120, 103866 (2021), arXiv:2104.06238 [nucl-th] .
  * Garrett  _et al._ [2022] P. E. Garrett, M. Zielińska, and E. Clément, An experimental view on shape coexistence in nuclei, Prog. in Part. and Nucl. Phys. 124, 103931 (2022).
  * Ahn  _et al._ [2019] S. Ahn, D. W. Bardayan, K. L. Jones, A. S. Adekola, G. Arbanas, J. C. Blackmon, K. Y. Chae, K. A. Chipps, J. A. Cizewski, S. Hardy, M. E. Howard, R. L. Kozub, B. Manning, M. Matoš, C. D. Nesaraja, P. D. O’Malley, S. D. Pain, W. A. Peters, S. T. Pittman, B. C. Rasco, M. S. Smith, and I. Spassova, Direct neutron capture cross section on ${}^{80}\mathrm{Ge}$ and probing shape coexistence in neutron-rich nuclei, Phys. Rev. C 100, 044613 (2019).
  * Montestruque  _et al._ [1978] L. A. Montestruque, M. Cobian-Rozak, G. Szaloky, J. Zumbro, and S. Darden, Study of the 77,79,81,83Se level structure with the 76,78,80,82Se(d,p) reaction, Nucl. Phys. A 305, 29 (1978).
  * Detorie  _et al._ [1978] N. A. Detorie, P. L. Jolivette, C. P. Browne, and A. A. Rollefson, Nuclear structure information for states in ${}^{85}\mathrm{Kr}$ from the ${}^{84}\mathrm{Kr}(d,p)$ and ${}^{84}\mathrm{Kr}(\stackrel{{\scriptstyle\rightarrow}}{{d}},p)$ reactions, Phys. Rev. C 18, 991 (1978).
  * Burks  _et al._ [1986] B. Burks, R. Anderson, T. Clegg, E. Ludwig, B. Karp, and Y. Aoki, A DWBA analysis of the 86Sr(d,p)87Sr reaction and implications for a 87Rb solar neutrino detector, Nucl. Phys. A 457, 337 (1986).
  * Gottardo  _et al._ [2016] A. Gottardo, D. Verney, C. Delafosse, F. Ibrahim, B. Roussière, C. Sotty, S. Roccia, C. Andreoiu, C. Costache, M.-C. Delattre, I. Deloncle, A. Etilé, S. Franchoo, C. Gaulard, J. Guillot, M. Lebois, M. MacCormick, N. Marginean, R. Marginean, I. Matea, C. Mihai, I. Mitu, L. Olivier, C. Portail, L. Qi, L. Stan, D. Testov, J. Wilson, and D. T. Yordanov, First Evidence of Shape Coexistence in the ${}^{78}\mathrm{Ni}$ Region: Intruder ${0}_{2}^{+}$ State in ${}^{80}\mathrm{Ge}$, Phys. Rev. Lett. 116, 182501 (2016).
  * Garcia  _et al._ [2020] F. H. Garcia, C. Andreoiu, G. C. Ball, A. Bell, A. B. Garnsworthy, F. Nowacki, C. M. Petrache, A. Poves, K. Whitmore, F. A. Ali, N. Bernier, S. S. Bhattacharjee, M. Bowry, R. J. Coleman, I. Dillmann, I. Djianto, A. M. Forney, M. Gascoine, G. Hackman, K. G. Leach, A. N. Murphy, C. R. Natzke, B. Olaizola, K. Ortner, E. E. Peters, M. M. Rajabali, K. Raymond, C. E. Svensson, R. Umashankar, J. Williams, and D. Yates, Absence of Low-Energy Shape Coexistence in ${}^{80}\mathrm{Ge}$: The Nonobservation of a Proposed Excited ${0}_{2}^{+}$ Level at 639 keV, Phys. Rev. Lett. 125, 172501 (2020).
  * Sekal  _et al._ [2021] S. Sekal, L. M. Fraile, R. Lică, M. J. G. Borge, W. B. Walters, A. Aprahamian, C. Benchouk, C. Bernards, J. A. Briz, B. Bucher, C. J. Chiara, Z. Dlouhý, I. Gheorghe, D. G. Ghiţă, P. Hoff, J. Jolie, U. Köster, W. Kurcewicz, H. Mach, N. Mărginean, R. Mărginean, Z. Meliani, B. Olaizola, V. Paziy, J. M. Régis, M. Rudigier, T. Sava, G. S. Simpson, M. Stănoiu, and L. Stroe, Low-spin states in ${}^{80}\mathrm{Ge}$ populated in the $\beta$ decay of the ${}^{80}\mathrm{Ga}$ ${3}^{-}$ isomer, Phys. Rev. C 104, 024317 (2021).
  * Taniuchi  _et al._ [2019] R. Taniuchi, C. Santamaria, P. Doornenbal, A. Obertelli, K. Yoneda, G. Authelet, H. Baba, D. Calvet, F. Château, A. Corsi, A. Delbart, J.-M. Gheller, A. Gillibert, J. D. Holt, T. Isobe, V. Lapoux, M. Matsushita, J. Menéndez, S. Momiyama, T. Motobayashi, M. Niikura, F. Nowacki, K. Ogata, H. Otsu, T. Otsuka, C. Péron, S. Péru, A. Peyaud, E. C. Pollacco, A. Poves, J.-Y. Roussé, H. Sakurai, A. Schwenk, Y. Shiga, J. Simonis, S. R. Stroberg, S. Takeuchi, Y. Tsunoda, T. Uesaka, H. Wang, F. Browne, L. X. Chung, Z. Dombradi, S. Franchoo, F. Giacoppo, A. Gottardo, K. Hadyńska-Klęk, Z. Korkulu, S. Koyama, Y. Kubota, J. Lee, M. Lettmann, C. Louchart, R. Lozeva, K. Matsui, T. Miyazaki, S. Nishimura, L. Olivier, S. Ota, Z. Patel, E. Şahin, C. Shand, P.-A. Söderström, I. Stefan, D. Steppenbeck, T. Sumikama, D. Suzuki, Z. Vajta, V. Werner, J. Wu, and Z. Y. Xu, 78Ni revealed as a doubly magic stronghold against nuclear deformation, Nature 569, 53 (2019).
  * Olivier  _et al._ [2017] L. Olivier, S. Franchoo, M. Niikura, Z. Vajta, D. Sohler, P. Doornenbal, A. Obertelli, Y. Tsunoda, T. Otsuka, G. Authelet, H. Baba, D. Calvet, F. Château, A. Corsi, A. Delbart, J.-M. Gheller, A. Gillibert, T. Isobe, V. Lapoux, M. Matsushita, S. Momiyama, T. Motobayashi, H. Otsu, C. Péron, A. Peyaud, E. C. Pollacco, J.-Y. Roussé, H. Sakurai, C. Santamaria, M. Sasano, Y. Shiga, S. Takeuchi, R. Taniuchi, T. Uesaka, H. Wang, K. Yoneda, F. Browne, L. X. Chung, Z. Dombradi, F. Flavigny, F. Giacoppo, A. Gottardo, K. Hadyńska-Klęk, Z. Korkulu, S. Koyama, Y. Kubota, J. Lee, M. Lettmann, C. Louchart, R. Lozeva, K. Matsui, T. Miyazaki, S. Nishimura, K. Ogata, S. Ota, Z. Patel, E. Sahin, C. Shand, P.-A. Söderström, I. Stefan, D. Steppenbeck, T. Sumikama, D. Suzuki, V. Werner, J. Wu, and Z. Xu, Persistence of the $Z=28$ Shell Gap Around ${}^{78}\mathrm{Ni}$: First Spectroscopy of ${}^{79}\mathrm{Cu}$, Phys. Rev. Lett. 119, 192501 (2017).
  * Welker  _et al._ [2017] A. Welker, N. A. S. Althubiti, D. Atanasov, K. Blaum, T. E. Cocolios, F. Herfurth, S. Kreim, D. Lunney, V. Manea, M. Mougeot, D. Neidherr, F. Nowacki, A. Poves, M. Rosenbusch, L. Schweikhard, F. Wienholtz, R. N. Wolf, and K. Zuber, Binding Energy of ${}^{79}\mathrm{Cu}$: Probing the Structure of the Doubly Magic ${}^{78}\mathrm{Ni}$ from Only One Proton Away, Phys. Rev. Lett. 119, 192502 (2017).
  * Nowacki  _et al._ [2016] F. Nowacki, A. Poves, E. Caurier, and B. Bounthong, Shape Coexistence in ${}^{78}\mathrm{Ni}$ as the Portal to the Fifth Island of Inversion, Phys. Rev. Lett. 117, 272501 (2016).
  * Orlandi  _et al._ [2015] R. Orlandi, D. Mücher, R. Raabe, A. Jungclaus, S. D. Pain, V. Bildstein, R. Chapman, G. de Angelis, J. G. Johansen, P. Van Duppen, A. N. Andreyev, S. Bottoni, T. E. Cocolios, H. De Witte, J. Diriken, J. Elseviers, F. Flavigny, L. P. Gaffney, R. Gernhäuser, A. Gottardo, M. Huyse, A. Illana, J. Konki, T. Kröll, R. Krücken, J. F. Lane, V. Liberati, B. Marsh, K. Nowak, F. Nowacki, J. Pakarinen, E. Rapisarda, F. Recchia, P. Reiter, T. Roger, E. Sahin, M. Seidlitz, K. Sieja, J. F. Smith, J. J. Valiente Dobón, M. von Schmid, D. Voulot, N. Warr, F. K. Wenander, and K. Wimmer, Single-neutron orbits near 78Ni: Spectroscopy of the N=49 isotope 79Zn, Phys. Lett. B 740, 298 (2015).
  * Yang  _et al._ [2016] X. F. Yang, C. Wraith, L. Xie, C. Babcock, J. Billowes, M. L. Bissell, K. Blaum, B. Cheal, K. T. Flanagan, R. F. Garcia Ruiz, W. Gins, C. Gorges, L. K. Grob, H. Heylen, S. Kaufmann, M. Kowalska, J. Kraemer, S. Malbrunot-Ettenauer, R. Neugart, G. Neyens, W. Nörtershäuser, J. Papuga, R. Sánchez, and D. T. Yordanov, Isomer Shift and Magnetic Moment of the Long-Lived $1/{2}^{+}$ Isomer in ${}_{30}^{79}{\mathrm{Zn}}_{49}$: Signature of Shape Coexistence near ${}^{78}\mathrm{Ni}$, Phys. Rev. Lett. 116, 182502 (2016).
  * Wraith  _et al._ [2017] C. Wraith, X. F. Yang, L. Xie, C. Babcock, J. Bieroń, J. Billowes, M. L. Bissell, K. Blaum, B. Cheal, L. Filippin, R. F. Garcia Ruiz, W. Gins, L. K. Grob, G. Gaigalas, M. Godefroid, C. Gorges, H. Heylen, M. Honma, P. Jönsson, S. Kaufmann, M. Kowalska, J. Krämer, S. Malbrunot-Ettenauer, R. Neugart, G. Neyens, W. Nörtershäuser, F. Nowacki, T. Otsuka, J. Papuga, R. Sánchez, Y. Tsunoda, and D. T. Yordanov, Evolution of nuclear structure in neutron-rich odd-Zn isotopes and isomers, Phys. Lett. B 771, 385 (2017).
  * Morales  _et al._ [2017] A. Morales, G. Benzoni, H. Watanabe, Y. Tsunoda, T. Otsuka, S. Nishimura, F. Browne, R. Daido, P. Doornenbal, Y. Fang, G. Lorusso, Z. Patel, S. Rice, L. Sinclair, P.-A. Söderström, T. Sumikama, J. Wu, Z. Xu, A. Yagi, R. Yokoyama, H. Baba, R. Avigo, F. Bello Garrote, N. Blasi, A. Bracco, F. Camera, S. Ceruti, F. Crespi, G. de Angelis, M.-C. Delattre, Z. Dombradi, A. Gottardo, T. Isobe, I. Kojouharov, N. Kurz, I. Kuti, K. Matsui, B. Melon, D. Mengoni, T. Miyazaki, V. Modamio-Hoybjor, S. Momiyama, D. Napoli, M. Niikura, R. Orlandi, H. Sakurai, E. Sahin, D. Sohler, H. Schaffner, R. Taniuchi, J. Taprogge, Z. Vajta, J. Valiente-Dobón, O. Wieland, and M. Yalcinkaya, Type II shell evolution in A=70 isobars from the $N\geq 40$ island of inversion, Phys. Lett. B 765, 328 (2017).
  * Leoni  _et al._ [2017] S. Leoni, B. Fornal, N. Mărginean, M. Sferrazza, Y. Tsunoda, T. Otsuka, G. Bocchi, F. C. L. Crespi, A. Bracco, S. Aydin, M. Boromiza, D. Bucurescu, N. Cieplicka-Oryǹczak, C. Costache, S. Călinescu, N. Florea, D. G. Ghiţă, T. Glodariu, A. Ionescu, L. Iskra, M. Krzysiek, R. Mărginean, C. Mihai, R. E. Mihai, A. Mitu, A. Negreţ, C. R. Niţă, A. Olăcel, A. Oprea, S. Pascu, P. Petkov, C. Petrone, G. Porzio, A. Şerban, C. Sotty, L. Stan, I. Ştiru, L. Stroe, R. Şuvăilă, S. Toma, A. Turturică, S. Ujeniuc, and C. A. Ur, Multifaceted quadruplet of low-lying spin-zero states in ${}^{66}\mathrm{Ni}$: Emergence of shape isomerism in light nuclei, Phys. Rev. Lett. 118, 162502 (2017).
  * Stryjczyk  _et al._ [2018] M. Stryjczyk, Y. Tsunoda, I. G. Darby, H. De Witte, J. Diriken, D. V. Fedorov, V. N. Fedosseev, L. M. Fraile, M. Huyse, U. Köster, B. A. Marsh, T. Otsuka, D. Pauwels, L. Popescu, D. Radulov, M. D. Seliverstov, A. M. Sjödin, P. Van den Bergh, P. Van Duppen, M. Venhart, W. B. Walters, and K. Wimmer, ${\beta}^{-}$ decay study of the ${}^{66}\mathrm{Mn}\text{$-$}^{66}\mathrm{Fe}\text{$-$}^{66}\mathrm{Co}\text{$-$}^{66}\mathrm{Ni}$ decay chain, Phys. Rev. C 98, 064326 (2018).
  * Sels  _et al._ [2019] S. Sels, T. Day Goodacre, B. A. Marsh, A. Pastore, W. Ryssens, Y. Tsunoda, N. Althubiti, B. Andel, A. N. Andreyev, D. Atanasov, A. E. Barzakh, M. Bender, J. Billowes, K. Blaum, T. E. Cocolios, J. G. Cubiss, J. Dobaczewski, G. J. Farooq-Smith, D. V. Fedorov, V. N. Fedosseev, K. T. Flanagan, L. P. Gaffney, L. Ghys, P.-H. Heenen, M. Huyse, S. Kreim, D. Lunney, K. M. Lynch, V. Manea, Y. Martinez Palenzuela, T. M. Medonca, P. L. Molkanov, T. Otsuka, J. P. Ramos, R. E. Rossel, S. Rothe, L. Schweikhard, M. D. Seliverstov, P. Spagnoletti, C. Van Beveren, P. Van Duppen, M. Veinhard, E. Verstraelen, A. Welker, K. Wendt, F. Wienholtz, R. N. Wolf, and A. Zadvornaya, Shape staggering of midshell mercury isotopes from in-source laser spectroscopy compared with density-functional-theory and Monte Carlo shell-model calculations, Phys. Rev. C 99, 044306 (2019).
  * Olaizola  _et al._ [2019] B. Olaizola, L. M. Fraile, H. Mach, F. Nowacki, A. Poves, A. Aprahamian, J. A. Briz, J. Cal-González, D. Ghiţa, U. Köster, W. Kurcewicz, S. R. Lesher, D. Pauwels, E. Picado, D. Radulov, G. S. Simpson, and J. M. Udías, Properties of low-lying states in ${}^{65}\mathrm{Co}$ from lifetime measurements, Phys. Rev. C 99, 024321 (2019).
  * Hakala  _et al._ [2008] J. Hakala, S. Rahaman, V.-V. Elomaa, T. Eronen, U. Hager, A. Jokinen, A. Kankainen, I. D. Moore, H. Penttilä, S. Rinta-Antila, J. Rissanen, A. Saastamoinen, T. Sonoda, C. Weber, and J. Äystö, Evolution of the $N=50$ Shell Gap Energy towards ${}^{78}\mathrm{Ni}$, Phys. Rev. Lett. 101, 052502 (2008).
  * Baruah  _et al._ [2008] S. Baruah, G. Audi, K. Blaum, M. Dworschak, S. George, C. Guénaut, U. Hager, F. Herfurth, A. Herlert, A. Kellerbauer, H.-J. Kluge, D. Lunney, H. Schatz, L. Schweikhard, and C. Yazidjian, Mass Measurements beyond the Major $r$-Process Waiting Point ${}^{80}\mathrm{Zn}$, Phys. Rev. Lett. 101, 262501 (2008).
  * Giraud  _et al._ [2022] S. Giraud, L. Canete, B. Bastin, A. Kankainen, A. Fantina, F. Gulminelli, P. Ascher, T. Eronen, V. Girard-Alcindor, A. Jokinen, A. Khanam, I. Moore, D. Nesterenko, F. de Oliveira Santos, H. Penttilä, C. Petrone, I. Pohjalainen, A. De Roubin, V. Rubchenya, M. Vilen, and J. Äystö, Mass measurements towards doubly magic 78Ni: Hydrodynamics versus nuclear mass contribution in core-collapse supernovae, Phys. Lett. B 833, 137309 (2022).
  * Eronen  _et al._ [2012] T. Eronen, V. Kolhinen, V.-V. Elomaa, D. Gorelov, U. Hager, J. Hakala, A. Jokinen, A. Kankainen, P. Karvonen, S. Kopecky, I. D. Moore, H. Penttilä, S. Rahaman, S. Rinta-Antila, J. Rissanen, A. Saastamoinen, J. Szerypo, C. Weber, and J. Äystö, JYFLTRAP: a Penning trap for precision mass spectroscopy and isobaric purification, Eur. Phys. J. A 48, 46 (2012).
  * Moore  _et al._ [2013] I. Moore, T. Eronen, D. Gorelov, J. Hakala, A. Jokinen, A. Kankainen, V. Kolhinen, J. Koponen, H. Penttilä, I. Pohjalainen, M. Reponen, J. Rissanen, A. Saastamoinen, S. Rinta-Antila, V. Sonnenschein, and J. Äystö, Towards commissioning the new IGISOL-4 facility, Nucl. Instrum. Methods. Phys. Res. B 317, 208 (2013), xVIth International Conference on ElectroMagnetic Isotope Separators and Techniques Related to their Applications, December 2–7, 2012 at Matsue, Japan.
  * Mukherjee  _et al._ [2008] M. Mukherjee, D. Beck, K. Blaum, G. Bollen, J. Dilling, S. George, F. Herfurth, A. Herlert, A. Kellerbauer, H.-J. Kluge, S. Schwarz, L. Schweikhard, and C. Yazidjian, ISOLTRAP: An on-line Penning trap for mass spectrometry on short-lived nuclides, Eur. Phys. J. A 35, 1 (2008).
  * Catherall  _et al._ [2017] R. Catherall, W. Andreazza, M. Breitenfeldt, A. Dorsival, G. J. Focker, T. P. Gharsa, G. T. J, J.-L. Grenard, F. Locci, P. Martins, S. Marzari, J. Schipper, A. Shornikov, and T. Stora, The ISOLDE facility, J. Phys. G Nucl. Part. Phys. 44, 094002 (2017).
  * Wang  _et al._ [2021] M. Wang, W. Huang, F. Kondev, G. Audi, and S. Naimi, The AME 2020 atomic mass evaluation (II). Tables, graphs and references*, Chin. Phys. C 45, 030003 (2021).
  * Al-Adili, A.  _et al._ [2015] Al-Adili, A., Jansson, K., Lantz, M., Solders, A., Gorelov, D., Gustavsson, C., Mattera, A., Moore, I., Prokofiev, A. V., Rakopoulos, V., Penttilä, H., Tarrío, D., Wiberg, S., Österlund, M., and Pomp, S., Simulations of the fission-product stopping efficiency in IGISOL, Eur. Phys. J. A 51, 59 (2015).
  * Karvonen  _et al._ [2008] P. Karvonen, I. Moore, T. Sonoda, T. Kessler, H. Penttilä, K. Peräjärvi, P. Ronkanen, and J. Äystö, A sextupole ion beam guide to improve the efficiency and beam quality at IGISOL, Nucl. Instrum. Methods Phys. Res. B 266, 4794 (2008).
  * Nieminen  _et al._ [2001] A. Nieminen, J. Huikari, A. Jokinen, J. Äystö, P. Campbell, and E. Cochrane, Beam cooler for low-energy radioactive ions, Nucl. Instrum. Methods Phys. Res. A 469, 244 (2001).
  * Savard  _et al._ [1991] G. Savard, S. Becker, G. Bollen, H.-J. Kluge, R. Moore, T. Otto, L. Schweikhard, H. Stolzenberg, and U. Wiess, A new cooling technique for heavy ions in a Penning trap, Phys. Lett. A 158, 247 (1991).
  * König  _et al._ [1995] M. König, G. Bollen, H.-J. Kluge, T. Otto, and J. Szerypo, Quadrupole excitation of stored ion motion at the true cyclotron frequency, Int. J. Mass Spectrom. 142, 95 (1995).
  * Purushothaman  _et al._ [2017] S. Purushothaman, S. Ayet San Andrés, J. Bergmann, T. Dickel, J. Ebert, H. Geissel, C. Hornung, W. Plaß, C. Rappold, C. Scheidenberger, Y. Tanaka, and M. Yavor, Hyper-EMG: A new probability distribution function composed of Exponentially Modified Gaussian distributions to analyze asymmetric peak shapes in high-resolution time-of-flight mass spectrometry, Int. J. Mass. Spectrom. 421, 245 (2017).
  * Canete  _et al._ [2016] L. Canete, A. Kankainen, T. Eronen, D. Gorelov, J. Hakala, A. Jokinen, V. S. Kolhinen, J. Koponen, I. D. Moore, J. Reinikainen, and S. Rinta-Antila, High-precision mass measurements of 25Al and 30P at JYFLTRAP, Eur. Phys. J. A 52, 124 (2016).
  * Canete [2019] L. Canete,  _High precision mass measurements for nuclear astrophysics_ , Ph.D. thesis, University of Jyväskylä (2019).
  * Luis  _et al._ [2012] R. Luis, J. G. Marques, T. Stora, P. Vaz, and L. Zanini, Optimization studies of the CERN-ISOLDE neutron converter and fission target system, Eur. Phys. J. A 48, 90 (2012).
  * Bouquerel  _et al._ [2007] E. Bouquerel, R. Catherall, M. Eller, J. Lettry, S. Marzari, T. Stora, and ISOLDE Collaboration, Purification of a Zn radioactive ion beam by alkali suppression in a quartz line target prototype, Eur. Phys. J.: Spec. Top. 150, 277 (2007).
  * Bouquerel  _et al._ [2008] E. Bouquerel, R. Catherall, M. Eller, J. Lettry, S. Marzari, and T. Stora, Beam purification by selective trapping in the transfer line of an ISOL target unit, Nucl. Instrum. Methods Phys. Res. B 266, 4298 (2008), proceedings of the XVth International Conference on Electromagnetic Isotope Separators and Techniques Related to their Applications.
  * Köster  _et al._ [2008] U. Köster, O. Arndt, E. Bouquerel, V. Fedoseyev, H. Frånberg, A. Joinet, C. Jost, I. Kerkines, R. Kirchner, and T. T. Collaboration, Progress in ISOL target–ion source systems, Nucl. Instrum. Methods Phys. Res. B 266, 4229 (2008), proceedings of the XVth International Conference on Electromagnetic Isotope Separators and Techniques Related to their Applications.
  * Fedosseev  _et al._ [2017] V. Fedosseev, K. Chrysalidis, T. D. Goodacre, B. Marsh, S. Rothe, C. Seiffert, and K. Wendt, Ion beam production and study of radioactive isotopes with the laser ion source at ISOLDE, J. Phys. G Nucl. Part. Phys. 44, 084006 (2017).
  * Herfurth  _et al._ [2001] F. Herfurth _et al._ , A linear radiofrequency ion trap for accumulation, bunching, and emittance improvement of radioactive ion beams, Nucl. Instrum. Methods Phys. Res. A 469, 254 (2001).
  * Wolf  _et al._ [2013] R. Wolf, F. Wienholtz, D. Atanasov, D. Beck, K. Blaum, C. Borgmann, F. Herfurth, M. Kowalska, S. Kreim, Y. A. Litvinov, D. Lunney, V. Manea, D. Neidherr, M. Rosenbusch, L. Schweikhard, J. Stanja, and K. Zuber, ISOLTRAP’s multi-reflection time-of-flight mass separator/spectrometer, Int. J. Mass. Spectrom. 349-350, 123 (2013), 100 years of Mass Spectrometry.
  * Wolf  _et al._ [2012] R. N. Wolf, G. Marx, M. Rosenbusch, and L. Schweikhard, Static-mirror ion capture and time focusing for electrostatic ion-beam traps and multi-reflection time-of-flight mass analyzers by use of an in-trap potential lift, Int. J. Mass. Spectrom. 313, 8 (2012).
  * Rosenbusch  _et al._ [2013] M. Rosenbusch, S. Kemnitz, R. Schneider, L. Schweikhard, R. Tschiersch, and R. N. Wolf, Towards systematic investigations of space-charge phenomena in multi-reflection ion traps, AIP Conf. Proc. 1521, 53 (2013).
  * Maier  _et al._ [2023] F. Maier, F. Buchinger, L. Croquette, P. Fischer, H. Heylen, F. Hummer, C. Kanitz, A. Kwiatkowski, V. Lagaki, S. Lechner, E. Leistenschneider, G. Neyens, P. Plattner, A. Roitman, M. Rosenbusch, L. Schweikhard, S. Sels, M. Vilen, F. Wienholtz, and S. Malbrunot-Ettenauer, Increased beam energy as a pathway towards a highly selective and high-flux MR-ToF mass separator, Nucl. Instrum. Methods Phys. Res. A 1056, 168545 (2023).
  * Kondev  _et al._ [2021] F. Kondev, M. Wang, W. Huang, S. Naimi, and G. Audi, The NUBASE2020 evaluation of nuclear physics properties, Chin. Phys. C 45, 030001 (2021).
  * Delattre [2016] M.-C. Delattre,  _Étude de la structure à basse énergie de 79Zn par décroissance $\beta$ et $\beta$-n de 79Cu et 80Cu_, Ph.D. thesis, Université Paris-Saclay (2016).
  * Dao and Nowacki [2022] D. D. Dao and F. Nowacki, Nuclear structure within a discrete nonorthogonal shell model approach: New frontiers, Phys. Rev. C 105, 054314 (2022).
  * Rocchini  _et al._ [2023] M. Rocchini, P. E. Garrett, M. Zielińska, S. M. Lenzi, D. D. Dao, F. Nowacki, V. Bildstein, A. D. MacLean, B. Olaizola, Z. T. Ahmed, C. Andreoiu, A. Babu, G. C. Ball, S. S. Bhattacharjee, H. Bidaman, C. Cheng, R. Coleman, I. Dillmann, A. B. Garnsworthy, S. Gillespie, C. J. Griffin, G. F. Grinyer, G. Hackman, M. Hanley, A. Illana, S. Jones, A. T. Laffoley, K. G. Leach, R. S. Lubna, J. McAfee, C. Natzke, S. Pannu, C. Paxman, C. Porzio, A. J. Radich, M. M. Rajabali, F. Sarazin, K. Schwarz, S. Shadrick, S. Sharma, J. Suh, C. E. Svensson, D. Yates, and T. Zidar, First Evidence of Axial Shape Asymmetry and Configuration Coexistence in ${}^{74}\mathrm{Zn}$: Suggestion for a Northern Extension of the $N=40$ Island of Inversion, Phys. Rev. Lett. 130, 122502 (2023).
  * Rezynkina  _et al._ [2022] K. Rezynkina, D. D. Dao, G. Duchêne, J. Dudouet, F. Nowacki, E. Clément, A. Lemasson, C. Andreoiu, A. Astier, G. de Angelis, G. de France, C. Delafosse, I. Deloncle, F. Didierjean, Z. Dombradi, C. Ducoin, A. Gadea, A. Gottardo, D. Guinet, B. Jacquot, P. Jones, T. Konstantinopoulos, I. Kuti, A. Korichi, S. M. Lenzi, G. Li, F. Le Blanc, C. Lizarazo, R. Lozeva, G. Maquart, B. Million, C. Michelagnoli, D. R. Napoli, A. Navin, R. M. Pérez-Vidal, C. M. Petrache, N. Pietralla, D. Ralet, M. Ramdhane, M. Rejmund, O. Stezowski, C. Schmitt, D. Sohler, and D. Verney, Structure of ${}^{83}\mathrm{As},$ ${}^{85}\mathrm{As},$ and ${}^{87}\mathrm{As}$: From semimagicity to $\gamma$ softness, Phys. Rev. C 106, 014320 (2022).