# Sequential choice functions and stability problems

Danilov V.I Central Institute of Economics and Mathematics of the RAS.
Nahimovskii prospect 47, 117418, Moscow; email<EMAIL_ADDRESS>
(01.01.2024)

The concept of sequential choice functions is introduced and studied. This
concept applies to the reduction of the problem of stable matchings with
sequential workers to a situation where the workers are linear.

##  1 Introduction

Our interest is based on the concept of stability of contract systems in a
bipartite situation (conditionally, firms and workers). We assume everywhere,
that agent preferences are described by Plott choice functions (Section 2). In
this case the follows fundamental properties are known (see Section 3): the
existence of solutions, the polarization of workers and firms, the lattice
structure of the set of solutions.

In the paper, we additionally assume that the choice functions of workers are
sequential. Sequentiality here means that the choice is made by sequential
application of $q$ linear orders or criteria: the first item is selected from
the list of available firms according to the first criterion, the next item is
selected from the remaining menu according to the second criterion, and so on.
This concept was introduced relatively recently ([5], see also [10]). We show
that in the case of sequential workers, it is possible to modify the problem
in such a way that the solutions do not actually change (so that both problems
are equivalent), but all workers become linear. It is possible that the second
problem (with linear workers) is easier to analyze (for example, for
construction of rotations).

In the next Section, we recall the definition of Plott choice functions and
introduce the operation $*$ of sequential application of choice functions.
Here some statements about this operation are established, which will be
useful in the future. Section 3 recalls the concept of the stability of a
matching or a contract system. In Section 4, a design of the modification of
the problem is proposed and the main theorems on the reduction of workers are
formulated. After some discussion in Section 5, we provide the proofs of the
main theorems.

##  2 Plott and sequential choice functions

Let $X$ be a set. A _choice function_ (hereinafter CF) on $X$ is a mapping
$C:2^{X}\to 2^{X}$, which satisfies the condition $C(A)\subseteq A$ for any
$A\subseteq X$. The subset $A$ is called _menu_ , and $C(A)$ is called
_choice_ from this menu. However, we will not be interested in any CFs, but
only in those that somehow model the human choice of the best from the
available. There is an extensive literature on this topic (see [1]). As
undoubted feature of rational behavior the following property should be
recognized, which is most often called consistency (there are other names:
independence of rejected alternatives, etc.).

_Consistency_. If $C(A)\subseteq B\subseteq A$, then $C(B)=C(A)$.

This axiom is usually supplemented by other conditions. For us, the following
one will be important, which is called substitution (or the Heredity axiom):

_Substitution._ If $B\subseteq A$ then $C(A)\cap B\subseteq C(B)$.

In other words, if an element from $B$ is selected in a larger set $A$, then
it is also selected in the smaller one. In a sense, the items look like as
competitors, and appearance of a new item in the menu reduces the chances of
other items being selected. This property is not as undoubted as the
consistency, but it is very relevant to the stability problems.

We say that a CF is _Plott_ one if it is consistent and substitute. Plott in
[12] introduced a class of choice functions, which he called ”path
independent”. These functions were distinguished by the condition

$C(A\cup B)=C(C(A)\cup B)$

Later it was realized that they are exactly the CFs having the properties of
consistency and substitution. The basic properties of such CFs can be found in
[1, 11, 8]. Therefore, we will be brief and limit ourselves only to those
concepts that will be needed for the future.

A CF $C$ _is cardinally monotonous_ if $A\subseteq B$ implies
$|C(A)|\leq|C(B)|$.

A menu $A$ is called _acceptable_ if $C(A)=A$.

A Plott function $C$ is called _non-empty_ if $C(A)\neq\emptyset$ when $A$ is
non-empty. For this, it is sufficient to require that all singletons be
acceptable.

With each Plott CF $C$ one can associate the _Blair relation_ (or the revealed
preference) $\preceq=\preceq_{C}$ on the set $2^{X}$ by setting

$A\preceq B,\text{ if }C(A\cup B)\subseteq B.$

This relation is reflexive and transitive. In the class of menus equivalent to
$A$, there is the smallest, namely $C(A)$. There is also the largest. On the
set of acceptable menus, the relation $\preceq$ is antisymmetric, that is a
(partial) order. Moreover, this poset is a lattice in which the join (sup) of
two acceptable menus $A$ and $B$ is equal to $C(A\cup B)$.

An element $x$ is called _desirable_ (in relation to a menu $A$, or in a state
$A$) if $x\in C(A\cup x)$. In other words, the agent would like to add $x$ to
the menu. In another way, we can say that $A\prec_{C}A\cup x$, that is, that
the menu $A\cup x$ is strictly better (according to Blair) than $A$.

We can not do without examples. Here and further, we assume that $X$ is finite
set. The simplest example gives maximization of a linear order. Let $\leq$ be
a linear order on the set $X$. Then in each (non-empty) subset $A\subseteq X$
there is a (single) maximal element $a=\max_{\leq}(A)$. The mapping
$A\mapsto\max_{\leq}(A)$ gives a CF which, as it is easy to understand, is
Plott function. We will call such CFs _linear_.

A more interesting and widely used in the literature on stability example of
Plott function gives the choice (using a linear order $\leq$) of not one best,
but the first $q$ best elements, the choice with a _quota_ $q$. This choice is
sometimes called _responsive_. It is also a Plott CF. The sequential CF, which
will be discussed below, is again the choice of the $q$ best, but with respect
to different linear orders. But it is better to start with the more general
operation $*$.

Having several CFs, one can use them to build new CFs. The most popular
operation is that of the union. Below we will introduce two more new
operations. The main for further is the operation of _sequential (or serial)
application_ of two (or more) CFs. When we first chose (and TAKE them away)
from the menu using the first CF, and choice from the rest using the second
one. Formally, CF $F*G$ is defined as:

$(F*G)(A)=F(A)\cup G(A-F(A)).$

This operation, as it is easy to understand, is associative,
($F*(G*H)=(F*G)*H$), and therefore, it makes sense to write $F*G*H*...$
without parentheses. In general case, this the operation is not commutative.

Proposition 1. _If $F$ and $G$ are Plott CFs, then $F*G$ is Plott CF._

Proof. It is necessary to check the consistency and substitutability of $F*G$.

Consistency. By virtue of the definition of $*$, we have $F(A)\cup
G(A-FA)\subseteq B$. In particular, $F(A)\subseteq B$, from where (using the
consistency of $F$) $F(B)=F(A)$. Next, $G(A-FB)=G(A-FA)$ is also contained in
$B$ and does not overlap with $FB$. Therefore, $G(A-FB)\subseteq B-FB\subseteq
A-FB$, and from the consistency of $G$ we obtain $G(B-FB)=G(A-FB)=G(A-FA)$.

Substitute. Let $A\subseteq B$, and let $a\in A$ be chosen from $B$ (by
$F*G$); it must be shown that it is also chosen from $A$. If $a\in F(B)$ then,
from the substitutability of $F$, $a\in F(A)$ and everything is fine.
Therefore, we can assume that $a\notin FA$ (and even more so it does not
belong to $FB$) and therefore belongs to $G(B-FB)$. In addition, $a$ belongs
to $A-FA$, which is contained in $B-FB$. Therefore, from the substitutability
of $G$, $a\in G(A-FA)$. $\Box$

Definition. A CF $C$ is called _sequential_ if it is represented as
$F_{1}*...*F_{q}$, where $F_{1},...,F_{q}$ are linear CFs.

According to Proposition 1, any sequential CF is Plott CF. In addition, it is
$q$-_quotable_ , that is, the choice from $A$ has $q$ elements if $|A|\geq q$,
and coincides with $A$ if $|A|\leq q$. Plott CF with quota 1 is linear. If $F$
and $G$ are quotable with quotas $q$ and $r$, then $F*G$ is quotable with
quota $q+r$.

Sequential CFs provide a lot of examples of quota-based CFs. However, not all
as it shown in [5].

In the follows, we will need another construction. It can be conditionally
called ”integration”. Let $\pi:Y\to X$ be a mapping of sets. We look at this
mapping as a fibration of $Y$ over the base $X$; for each point $x\in X$ the
set $Y_{x}=\pi^{-1}(x)$ is the _fibre_ over $x$. In this sense, $Y$ is a union
of its fibres. Or $Y$ is fibred into fibres $Y_{x}$, where $x$ runs through
$X$.

So, let us assume that for each fibre $Y_{x}$ a CF $C_{x}$ is given, and in
addition, a CF $C$ is given on the base $X$. Then one can form the following
CF $D$ on $Y$ (conditionally $\int C_{x}dC$) by the formula (where $B\subseteq
Y$):

$D(B)=\bigcup_{x\in C(\pi(B))}C_{x}(B_{x}),$

where $B_{x}=B\cap Y_{x}$. In other words, $D(B)$ consists of those $b$
(belonging to $B$) that $b\in C_{x}(B_{x})$, where $x=\pi(b)\in C(\pi(B))$. In
short, it is necessary that the element $\pi(b)$ was selected in the set
$\pi(B)$, and $b$ was selected in $B_{\pi}(b)$.

Proposition 2. _If $C$ is Plott CF, and all CFs $C_{x}$ are non-empty Plott
CFs, then the CF $D$ is also Plott one._

Proof. Let the element $b$ belong to $B$, but not $D(B)$, and
$B^{\prime}=B-b$. We have to check what $D(B^{\prime})=D(B)$; this will give
the consistency of $D$.

Let’s consider two cases: when $b$ is the only element in its fibre, and when
it is not the only one.

If $b$ is the only one, then $\pi(B^{\prime})=\pi(B)-\pi(b)$. In addition,
$\pi(b)$ is not selected from $\pi(B)$. In fact, if $\pi b\in C(B)$, then $b$
as the only element of the fibre $B_{\pi}b$ is selected in this fibre and
belongs to $D(B)$. So (due to the consistency of $C$) $C(\pi B^{\prime})=C(\pi
B)$. And, for the points $y$ of $C(\pi B)$, the fibres $B\cap X_{y}$ and
$B^{\prime}\cap X_{y}$ are the same, as the choices from them. Therefore
$D(B^{\prime})=D(B)$.

If $b$ is not the only one in the fibre $B\cap X_{\pi}b$, then $\pi
B^{\prime}=\pi B$. In all other fibres the choice does not change, so it
remains to look at the choice from the fibre above $\pi b$, which we denote as
$B_{0}$ (and the CF $C_{\pi b}$ as $C_{0}$). Since $B^{\prime}_{0}=B_{0}-b$,
and $b$ is not selected by the function $C_{0}$, then (due to the consistency
of $C_{0}$) $C_{0}(B^{\prime}_{0})=C_{0}(B_{0})$. This proves the consistency
of $D$.

In the same style, the substitution is checked. Here we assume that $b\in
D(B)$ and must check that the remaining elements of $D(B)$ survive when
selected from $B^{\prime}=B-b$. Again, if $\pi B^{\prime}=\pi B$, then the
other fibres do not change and we only need to consider the fibre containing
$b$, that is, $Y_{\pi b}$. And in it (due to the substitutability of $C_{\pi
b}$), deleting $b$ leads to the fact that everything that was selected
continues to be selected.

If $\pi B^{\prime}$ is less than $\pi B$, then this can only happen if $b$ is
the only element of the fibre $B_{\pi}b$. In this case $\pi B^{\prime}=\pi
B-\pi b$. Also, $\pi b\in C(\pi B)$. From the substitutability of $C$, we get
that all other points from $C(\pi B)$ remain in $C(\pi B^{\prime})$, as well
as the choices from the fibres above them. As for the fibre above $\pi b$,
this fibre consistes only of $b$, which was removed. $\Box$

Especially important is the case when all CFs $C_{x}$ are linear.

Corollary. _Let (under the assumptions of Proposition 2) all CFs $C_{x}$ be
linear. Then, for the set $B\subseteq Y$, the following statements are
equivalent:_

1) $D(B)=B$;

2) $B$ _is bijectively mapped to $\pi(B)$ and_ $C(\pi(B))=\pi(B)$.

In particular, the size of $D(B)$ is the same as the size of $C(\pi(B))$.
Therefore, if CF$C$ is cardinally monotonous, then $D$ is cardinally
monotonous.

##  3 Stability problem

Plott CFs have found the greatest (the best) use in problems of stable
matchings or contract systems. We will not formulate this problem in a general
way, referring to [7], but will confine ourselves to a bipartite situation.
There are two groups of agents, $M$ (conditionally, _firms_) and $W$
(conditionally, _workers_), and a set $E$ of potential _contracts_ between
them. In fact, we tell about a bipartite (multi)graph, with the shares $M$ and
$W$ and the set $E$ of edges between vertices from different groups. Each
contract $e\in E$ is concluded between some firm $m$ and some worker $w$. A
firm can conclude several contracts with the same worker at the same time, not
to say about contracts with several workers. The same is true for workers.

The set of contracts available to a firm $m$ is denoted by $E(m)$; similar to
$E(w)$ for a worker $w$. The preference of firm $m$ is given by a CF $C_{m}$
on the set $E(m)$; preferences of worker $w$ is given by CF $C_{w}$ on $E(w)$.
We assume everywhere else that $C_{m}$ and $C_{w}$ are Plott CFs. A _contract
system_ is an arbitrary subset $S\subseteq E$. $S(m)=S\cap E(m)$, and $S(w)$
is understood similarly.

Definition. The set (or system) of contracts $S$ is called _stable_ if

1\. For any firm $m\in M$ the set $S(m)$ is acceptable; the set $S(w)$ is
acceptable for every worker $w$.

2\. There are no _blocking_ contracts, that is, contracts not from $S$ which
are desirable for both parties to the agreement. In other words, if a contract
$e$ (between $m$ and $w$) belongs both $C_{m}(S(m)\cup e)$ and $C_{w}(S(w)\cup
e)$, then $e\in S$.

This is the standard definition of stability. Sometimes we call a stable set
by a _solution_ of the problem. The following fundamental facts are known ([9,
1]):

1\. The set of solutions St is nonempty.

2\. The set of solutions St equipped with the Blair order $\preceq_{M}$ is a
lattice. In particular, it contains the best solution for all firms.

3\. The Blair order $\preceq_{W}$ on St is opposite to the order
$\preceq_{M}$.

If CFs of all agents are cardinally monotonous, then two additional statements
are true [2]:

4\. The size of $S(a)$ for any agent $a$ (firm or worker) does not depend on
the choice of a stable set $S$.

5\. The joint $S\vee T$ of two stable sets $S$ and $T$ is arranged as follows:
$(S\vee T)(m)=C_{m}(S(m)\cup T(m))$ for any firm $m\in M$.

The reader will lose little if will considers that there is only one company
and one worker. The fact is that all firms can be aggregated into one agent
$M$ (if you like, the ”Union of Entrepreneurs”), considering the set of its
contracts equal to $E=\coprod_{m}E(m)$, and its CF $C_{M}$ equal to the direct
sum of CFs $C_{m}$. The same for workers; they can be aggregated into one
agent $W$ (”Trade Union”). The solutions to these two problems are completely
equivalent. However, to avoid doubts, we will work with the original
(disaggregated) setup. Moreover, our main goal is to disaggregate agents (in
our case, the workers) to such an extent that they have become ”linear” in the
sense that their CFs are linear. Such a linear worker will conclude only one
contract (or even will remain without contracts). This situation belongs to
the type ”many-to-one”, which simplifies the problem somewhat.

Let us spend some time to convince the reader that the situation ”many-to-one”
is really a bit simpler than the general one. We will assume that all workers
$w$ are ”linear”; as for firms, we will assume that they have Plott and
cardinally monotonous CFs. As we mentioned above, the existence of stable
systems takes place in the general situation, however, in our situation there
is a simple way to build stable systems inductively. More precisely, let us
assume that we have a solution $S_{0}$ in the problem in which a certain
worker $w_{0}$ has been deleted. Then it is possible to canonically rebuild
$S_{0}$ into the solution $S$ of the original problem, already with the worker
$w_{0}$. Below we describe in detail one step of this procedure, inspired by
Gale-Shapley algorithm.

Let us assume. that we have a problem $I$ with agents $M$ and $W$, a set of
contracts $E$, and choice functions that are linear for workers and cardinally
monotonous for firms. Let $I(-w_{0})$ denote the problem in which the worker
$w_{0}$ is deleted, and let $S_{0}$ be some stable set in the problem
$I(-w_{0})$.

We return the worker $w_{0}$ together with the bouquet $E(w_{0})$ of his
contracts. Some of these contracts may be of interest to firms. Say that a
contract $e$ from $E(w_{0})$ is _interested_ for a firm $m$ if $e\in E(m)$ and
$e\in C_{m}(S_{0}(m)\cup e)$. The set of such contracts is denoted by $D(m)$,
and let $D=\cup_{m}D(m)$.

It may turn out that $D$ is empty (let’s call this _situation 0_), that is no
firm has shown interest in contracts with $w_{0}$. Obviously, in this case
$S=S_{0}$ is stable in the problem $I$.

Otherwise, $D$ is nonempty, and let $d$ be the best (for $w_{0}$) contract
from this set, the contract with some firm $m$.

_Situation 1_. $C_{m}(S_{0}(m)\cup d)=S_{0}(m)\cup d$. Let $S=S_{0}\cup d$.
Again, it is quite obvious that $S$ is stable in the problem $I$. The main
thing here is that the set $S(m)=S_{0}(m)\cup d$ is acceptable for $m$.

_Situation 2._ $C_{m}(S_{0}(m)\cup d)$ is strictly less than $S_{0}(m)\cup d$.
Due to the assumption of the cardinal monotonicity of CF $C_{m}$, the size
$C_{m}(S_{0}(m)\cup d)$ cannot be less than the size of
$C_{m}(S_{0}(m))=S_{0}(m)$. Therefore, $C_{m}(S_{0}(m)\cup d)$ is obtained
from $S_{0}(m)$ by throwing out exactly one old contract $s$ and the addition
of the contract $d$ instead. This rejected contract $s$ was a contract between
$m$ and some worker $w_{1}$ (different, of course, from $w_{0}$). And if we
now delete the worker $w_{1}$ along with the set $E(w_{1})$ of his contracts,
that is, consider the problem $I(-w_{1})$, then the system
$S_{1}=(S_{0}-s)\cup d$ will be stable. The fact is that a new desirable
contract $d^{\prime}$ could appear only at the firm $m$. But if $d^{\prime}$
is desirable in the state $(S_{0}(m)-s)\cup d$, it is all the more desirable
in the state $S_{0}$ (let’s recall the transitivity of the relation
$\preceq_{m}$).

This completes the description of the first step of the procedure. In the
situations 0 and 1, we already got a stable $S$. If the situation 2 is
realized, it is necessary to repeat the operation, returning the worker
$w_{1}$, and so on. Sooner or later, this process it will end up in the
situation 0 or 1. Indeed, if at the first step we are in the situation 2, the
position of the firm $m$ is strictly improving (in the sense of
$\preceq_{m}$), and nothing changes for other companies. In the second step
(if we gets in the situation 2 again), there is an improvement for another
firm and so on. But firms cannot be improved infinitely, and sooner or later
the process gets into situation 0 or 1 and gives a stable state $S$ for the
original problem $I$.

Note that for firms, the state $S$ is not worse than the initial one $S_{0}$.
As for the workers, their position fluctuates in different directions during
the process, sometimes improving (for $w_{0}$), then deteriorating (for
$w_{1}$). As a kind of curiosity, we note that it may well turn out that in
the final stable state $S$ the initiator of the process $w_{0}$ may turn out
to be a loner.

One can see it in the following simple example.

$m$$w$$w_{0}$$l$$r$$e$

The firm ranks the contracts as follows: $l>e>r$, for the worker $w$ on the
contrary $r>l$. When $w_{0}$ is absent, the contract $r$ is stable for the
pair $m,w$. But here $w_{0}$ appears, the firm switches to $e$, abandoning
$r$. The worker $w$ temporarily drops out, then returns, the firm offers him
the contract $l$, to which $w$ agrees. The contract $e$ is broken, and we get
a stable set $\\{l\\}$.

In principle, this can be used to improve (for firms) a stable system of
contracts by introducing a ”fictitious” worker who disappears at the end of
the process. However, I have not yet been able to accurately describe the
conditions…

Another benefit of the hypothesis that all workers have linear CFs is that we
can compare two systems $S$ and $T$, considering their symmetric difference
$S\Delta T$ and analyzing the resulting rotations. However, so far this is
only a direction for further work.

##  4 Disaggregation of sequential workers

A similar task of disaggregation was considered in [7]. But there The
disaggregated agents turned out to be not quite linear, their FVS were set in
weak orders. In addition, the solutions in the tasks were not completely
equivalent, although ”almost equivalent”.

To achieve the linearity of at least some workers, we assume that FW $C_{w}$
each worker’s $w$ is sequential, that is, it has the form
$C_{w}=C_{w_{1}}*...*C_{w_{q}(w)}$, where $q(w)\geq 1$ is the ”quota” $w$, and
the components FB $C_{w_{i}}$ are linear with linear orders $\leq_{w_{i}}$. In
this situation , it is quite natural to understand the worker $w$ as a queue
of sub-agents (clones) $w_{1},...,w_{q}$, which take turns selecting available
contracts, guided by each time by its ranking $\leq_{w_{i}}$. Therefore, the
idea arises to transform the task in such a way that these sub-agents with
their ”simple” linear preferences become new workers in it. In the hope that
the modified task (let’s denote it $\widetilde{I}$ to distinguish it from the
original $I$) in some sense, it was equivalent to the original one.

In fact, we will work in a slightly more general situation, since this does
not complicate the presentation too much. Namely, we will assume that the
”intermediate” CFs $C_{w_{i}}$ are not linear, but only Plott CFs.

To set the modified problem $\widetilde{I}$ we need to define ”successfully”
the sets $\widetilde{M}$ and $\widetilde{W}$ of new agents, the set
$\widetilde{E}$ of new contracts, as well as CFs $\widetilde{C}$ for new
agents.

_The construction of $\widetilde{M}$ and $\widetilde{W}$._ In principle, we
have already described it. The firms remain old, although we will write
$\widetilde{m}$ instead $m$ to not be confused. The fact is that the firms
change a lot of contracts (see below). They need to negotiate not with the old
worker $w$, but with $q(w)$ his ”clones”. As for the workers, each worker $w$
is replaced by a group $\\{w_{1},...,w_{q(w)}\\}$ of their ”clones” (a typical
clone is written as $\widetilde{w}=(w,i)$). So the new set $\widetilde{W}$ has
the form $\coprod_{w\in W}\\{w\\}\times\\{1,...,q(w)\\}$ and has a natural
projection $\pi:\widetilde{W}\to W$.

_The construction of_ $\widetilde{E}$ is not much more complicated. Every old
contract $e$ between $m$ and $w$ ”multiplies” in $q(w)$ copies of the form
$(e;i)$, where $1\leq i\leq q(w)$. The parties of such a contract are the firm
$\widetilde{m}$ and the new worker $w_{i}$. Again, there is a natural mapping
$\widetilde{E}\to E$, which we will again denote as $\pi$.

The set $\widetilde{E}(w_{i})$ is identified with $E(w)$. The set
$\widetilde{E}(\widetilde{m})$ is represented as a ‘multi-storey building’
over $E(m)$. In Fig. 1 we have depicted the change of the graphs when we start
with two agents and three contracts and then split the worker into two
subagents.

${m}$

$w$$C_{m}$$C_{w}$$\widetilde{C}_{\widetilde{m}}$${C}_{1}$${C}_{2}$$\widetilde{m}$$w_{1}$$w_{2}$$\pi$

Fig. 1

_The construction of CFs_ $\widetilde{C}$. This is the most delicate part. New
choice functions are defined differently for firms and workers. It is clear
with workers: the set $\widetilde{E}(w_{i})$ is canonically identified with
$E(w)$, and its CF $\widetilde{C}_{w_{i}}$ is actually equal to $C_{w_{i}}$.
It is a little more difficult with firms, because the set on which we need to
set the CF $\widetilde{C}_{\widetilde{m}}$ is changed. Roughly speaking, for
the firm $\widetilde{m}$, all contracts with clones of the worker $w$ are the
same (the firm considers them as contracts with $w$), however preference is
given to a clone with a smaller number. Formally, it looks like this. The set
$\widetilde{E}(\widetilde{m})$ is displayed (by mapping $\pi$) over $E(m)$; a
fibre over contract $e$ (that is, $\pi^{-1}(e)$) between $m$ and $w$ consists
of $\widetilde{e}=(e,i)$, where $i$ runs through $\\{1,...,q(w)\\}$. Suppose
now that $\widetilde{A}$ is a subset of $\widetilde{E}(\widetilde{m})$. Then
the CF $\widetilde{C}_{\widetilde{m}}(\widetilde{A})$ works like this: you
need to take the set of $C_{m}(\pi(\widetilde{A}))$ in $E(m)$ and then the
”minimally” rise this set in $\widetilde{A}$. The minimality of the rise means
that the contract $e$ from $\pi(\widetilde{A})$ rises to the contract $(e,i)$
from $\widetilde{A}$ with the minimum possible number $i$. In other words,
this is the construction of an ”integral” CF from Section 2. Proposition 2
gives that $\widetilde{C}_{\widetilde{m}}$ are Plott CFs (and cardinally
monotonous if $C_{m}$ were such).If $\widetilde{S}$ is a set of contracts in
the task $\widetilde{I}$, we can form the set $S=\pi(\widetilde{S})$ contracts
in the original problem. Theorem 1._If $\widetilde{S}$ is stable set in the
problem $\widetilde{I}$, then $S=\pi(\widetilde{S})$ is stable in the problem
$I$._ This gives the mapping (denoted by $\pi_{*}$) from ${\bf
St}(\widetilde{I})$ to ${\bf St}(I)$.Theorem 2. _The mapping $\pi_{*}:{\bf
St}(\widetilde{I})\to{\bf St}(I)$ is bijective._The proofs (somewhat tedious,
or involved) of both theorems are postponed to Section 5. These theorems say
that the problems $I$ and $\widetilde{I}$ are equivalent at least as the sets
of their solutions. However, as we have already said, both sets ${\bf
St}(\widetilde{I})$ and ${\bf St}(I)$ are provided with partial order
structures. Let us be more precise. Let $S$ and $T$ be two stable sets in the
problem $I$. Say that $S\preceq T$ (or, more precisely, $S\preceq_{M}T$) if
$S(m)\preceq_{m}T(m)$ for all $m\in M$ (where $\preceq_{m}$ are Blair’s orders
with respect to CFs $C_{m}$). Similarly the relation $\widetilde{\preceq}$ for
the task $\widetilde{I}$ is defined. We assert that the mapping $\pi_{*}$ not
just mapping of sets, but mapping of posets.Theorem 3. 1) _If $\widetilde{S}\
\widetilde{\preceq}\ \widetilde{T}$, then $\pi(\widetilde{S})\ {\preceq}\
\pi(\widetilde{T})$._ 2) _If all workers are sequential, and the firms are
cardinally monotonous, then the opposite is true, that ts $\pi_{*}:{\bf
St}(\widetilde{I})\to{\bf St}(I)$ is isomorphism of posets._Perhaps? the
statement 2) is true even without the assumption about cardinal monotonicity,
but I was unable to obtain a proof in the general case.The proof of point 1)
is done separately for each company. Therefore, we fix $m$ and do not mention
it in further notation. The relation $\widetilde{S}\ \widetilde{\preceq}\
\widetilde{T}$ means that
$\widetilde{C}(\widetilde{S}\cup\widetilde{T})\subseteq\widetilde{T}$. Let’s
recall the definition of $\widetilde{C}$. We have to construct
$\widetilde{S}\cup\widetilde{T}$ in $E$, that is form $S\cup T$, take $C(S\cup
T)$ and minimally raise this set to $\widetilde{S}\cup\widetilde{T}$. In
particular, $\pi(\widetilde{C}(\widetilde{S}\cup\widetilde{T}))=C(S\cup T)$.
And since
$\widetilde{C}(\widetilde{S}\cup\widetilde{T})\subseteq\widetilde{T}$, we get
that $C(S\cup T)\subseteq T$. That is, $S\preceq T$. Proof of 2). Here we
assume that the CFs of workers are sequential, and the CFs of firms are
cardinally monotonous. Then in the modified problem $\widetilde{I}$ CFs of new
workers are linear, and CFs of new firms is also cardinally monotonous (see
Corollary from Section 2). In this situation, join
$\widetilde{S}\vee\widetilde{T}$ of stable systems $\widetilde{S}$ and
$\widetilde{T}$ is given by a simple explicit formula: for any worker
$\widetilde{m}$

$(\widetilde{S}\vee\widetilde{T})(\widetilde{m})=\widetilde{C}_{\widetilde{m}}(\widetilde{S}(\widetilde{m})\cup\widetilde{T}(\widetilde{m})).$

Now let $S$ and $T$ be stable sets in the problem $I$, and $\widetilde{S}$ and
$\widetilde{T}$ be their ”prototypes” in ${\bf St}(\widetilde{I})$ (so that
$T=\pi(\widetilde{T})$). Suppose that $S\preceq_{M}T$. Form the join
$\widetilde{S}\vee\widetilde{T}$. Further we fix (and do not write)
$\widetilde{m}$. This join is equal to
$\widetilde{C}(\widetilde{S}\cup\widetilde{T}$) and is obtained in three
steps: take the projection $S\cup T$, select from it $C(S\cup T)$, and rise
$\widetilde{S}\cup\widetilde{T}$. From where
$\pi(\widetilde{S}\vee\widetilde{T})=C(S\cup T)$. According to our assumption,
$C(S\cup T)=T$. So that $\pi$ of the stable system
$\widetilde{S}\vee\widetilde{T}$ is equal to $T$. But due to Theorem 2,
$\widetilde{T}$ is the only stable system in $\widetilde{I}$, which can be
constructed for $T$. Therefore,
$\widetilde{S}\vee\widetilde{T}=\widetilde{T}$, and
$\widetilde{S}\preceq_{M}\widetilde{T}$. $\Box$

##  5 Discussion

Thus, we have shown that if all the workers are sequential, then one can
replace them with workers possessing linear CFs. This makes a little easier
task analysis, because now each worker concludes no more than one contract.
Workers left without a contract in one stable system will be the same in any
stable system (provided that firms are radically monotonous). We cannot delete
them, because the removal may lead to the expansion of the set St. However,
even in this case one can delete them, and then see which stable systems
disappear after their return. So, in principle, we can assume that the workers
conclude exactly one contract. Of course, it is possible to ”reduce” firms
instead of workers, if they are sequential. But it is worth emphasizing that
the ”reduction” to linearity remains ”one-sided”: we reduce either workers or
firms. To do this simultaneous for both, if possible, it requires some other
approach. In the work [7] a simultaneous reduction of all agents was
performed, but the agents there were reduced to such that their CFs were weak
orders (rather than linear ones). And although it would seem that the weak and
linear orders are very close, in the stability problems the difference becomes
quite significant. This can be seen on the following simple example.Example.
The skeleton of the problem looks like this:
$m$$m_{0}$$m^{\prime}$$w$$w_{0}$$w^{\prime}$$a$$a^{\prime}$$b$$b^{\prime}$$c$$c^{\prime}$$d$Three
firms: $m,m^{\prime}$ and $m_{0}$, three workers $w,w^{\prime}$ and $w_{0}$,
and seven contracts $a,b,c,a^{\prime},b^{\prime},c^{\prime}$ and $d$. The
preferences of the workers are as follows: $w$ likes $b$ more than $a$,
$w^{\prime}$ likes $b^{\prime}$ more than $a^{\prime}$, $w_{0}$ wants $d$ the
least, and $c$ and $c^{\prime}$ are equivalent for him. For $m$ the contract
$a$ is better than $b$ and $c$, which are equivalent for it. Similarly for
$m^{\prime}$: $a^{\prime}$ is better than $b^{\prime}$ and $c^{\prime}$, which
are equivalent. There are four stable systems in this problem. 1\. The best
for firms: $\overline{S}=\\{a,a^{\prime},d\\}$. 2\. The best for workers:
$\underline{S}=\\{b,c,c^{\prime},b^{\prime}\\}$. 3\. The left:
$S_{l}=\\{a,b^{\prime},c^{\prime}\\}$, and 4\. The right:
$S_{r}=\\{b,c,a^{\prime}\\}$. From the point of view of firms, $\overline{S}$
is the join of $S_{l}$ and $S_{r}$. However, if we take the union of $S_{l}$
and $S_{r}$, we get a set that does not contain $d$. So their join
$\overline{S}$ is not contained in the union. And $\overline{S}$ is not a
choice of the firms from $S_{l}\cup S_{r}$. Blair[4] gives a more complex
example where the lattice of stable sets is not distributive. But all this is
a little aside. As for inapplicability of the above construction for reducing
both firms and workers to linear ones, then here is another simple example.
With two participants and two contracts. One of which (say, the left one) is
better for both agents. But they are allowed a quota of 2. So they use both
contracts. But if we literally apply the proposed above construction, we get
two $M$ agents and two $W$ agents. For all these clones, the left contract is
better than the right one, and everyone will choose the left one. So the
projection of the solution of modified problem is not a solution in the
original one.It would be possible to do so. First we ”linearize” the workers,
and then (assuming firms are sequential) repeat the construction, but this
time in relation to firms. The failure here is such. Even if the firm’s CF
$C_{m}$ was sequential, the CF $\widetilde{C}_{\widetilde{m}}$ of the modified
firm is not longer sequential and even quotable. However, if the firms were
initially linear, then after the proposed construction modification they
remain linear. Therefore, if the firms were linear, and the workers are
sequential (it is better, of course, to consider the opposite), then the
problem can be reduced to the classical matching case when all agents are
linear.

##  6 Proofs of Theorems 1 and 2

Before proceeding to the proofs of Theorems 1 and 2, it is worth making a few
remarks. Firstly, it is possible to ”split” workers separately for every
worker. The rest remain unchanged. Therefore, we can assume that we are
dealing with a single worker, whom we will designate as $w_{0}$. Secondly, we
can assume that it splits into only two subagents, which are designated as 1
and 2 instead of $w_{1}$ and $w_{2}$. Thirdly, we can aggregate all the firms
into one and assume that we have only one firm $m$. With these simplifying
assumptions, the problem is shown in Figure 1.Proof of the Theorem 1. Here we
start with a stable set $\widetilde{S}$ and have to check the stability of its
projection $S=\pi(\widetilde{S})$. Stability means acceptability and non-
blocking. What does the acceptability of $\widetilde{S}$ mean for the firm
$\widetilde{m}$ and the workers $i=1,2$?Lemma 1. _$\widetilde{S}$ is
acceptable for $\widetilde{m}$ if and only if $\pi:\widetilde{S}\to S$ is a
bijection and $S$ is acceptable for $m$._This is seen from Corollary of
Proposition 2. The part $\widetilde{S}$ located on the first floor (denote it
$\widetilde{S}(1)$) consists of contracts with the worker 1. The other part,
$\widetilde{S}(2)$, represents contracts with the worker 2. Denote

${S}(i)=\pi(\widetilde{S}(i)),\ \ i=1,2.$

These sets do not intersect and together give $S$, $S=S(1)\sqcup S(2)$. In
turn, $\widetilde{S}(i)=S(i)\times\\{i\\}$, that is, $S(i)$, raised by $i$-th
floor of the building $\widetilde{E}=E\times\\{1,2\\}$. $\Box$ Lemma 2.
_$\widetilde{S}$ is acceptable for worker $i$ if and only if
$S(i)=C_{i}(S(i))$._ $\Box$ Already from Lemma 1 we see that ${S}$ is
acceptable for $m$. And it remains unsure that ${S}$ is acceptable for $w$ as
well. That is, $S$ coincides with $C_{w}(S)=C_{1}(S)\cap C_{2}(S-C_{1}(S))$.
This can be seen from the followingLemma 3. $S(1)=C_{1}(S)$,
$S(2)=C_{2}(S-S(1))$.Proof. It is enough to show the first one, because then
$S-S(1)=S(2)$ and then everything follows from Lemma 2. Let $e\in SbeS(1)$; we
assert that $e\notin C_{1}(S(1)\cup e)$. In fact, suppose that $e\in
C_{1}(S(1)\cup e)$, and let $e_{1}=(e,1)$ be the lifting of $e$ to the first
floor. This $e_{1}$ does not belong to $\widetilde{S}(1)$ because otherwise,
$\pi(e_{1})=e$ would belong to $S(1)$. But it doesn’t belong and
$\widetilde{S}(2)$, as well as $\widetilde{S}(2)$, is located on the second
floor. So $e_{1}$ does not belong to $\widetilde{S}$. Let’s show that $e_{1}$
blocks $\widetilde{S}$ in the problem $\widetilde{I}$. Indeed,
$e\in\widetilde{C}_{\widetilde{m}}({\widetilde{S}}\cup e_{1})$. To see this,
check that $e=\pi(e_{1})\in C_{m}(S\cup e)$, but $e\in S$, and $S=C_{m}(S)$
(see Lemma 1). So the contract $e_{1}$ is desirable for the firm
$\widetilde{m}$. But $e_{1}$ is also desirable for worker 1, because $e\in
C_{1}(S(1)\cup e)$ according to the assumption made above. Thus, $e_{1}$
blocks $\widetilde{S}$, which contradicts the stability of the system
$\widetilde{S}$. So, we have shown that $e\notin C_{1}(S(1)\cup e$. Due to the
consistency of the CF $C_{1}$ we have $S(1)=C_{1}(S(1))=C_{1}(S(1)\cup e)$.
$\Box$ Thus, the acceptability of $S$ for all agents has been verified._Non-
blockability._ Now let us check the $S$ is non-blockable. Suppose that a
contract $e$ between $m$ and $w$ does not lie in $S$ and is desirable for both
$m$ and $w$. That is 1) $e\in C_{m}(S(m)\cup e)$ and 2) $e\in C_{w}(S(w)\cup
e)$. Recall that $C_{w}(S\cup e)$ is obtained by application of CF $C_{1}$ to
menu $S\cup e$ and then by application of $C_{2}$ to what is left. Let’s
assume that we chosen $e$ already in the first step, that is, $e\in
C_{1}(S\cup e)$. Due to the substitutability of $C_{1}$, the contract $e$ is
selected in a smaller set $S(1)\cup e$, $e\in C_{1}(S(1)\cup e)$. But this
means that $e_{1}=(e,1)$ is desirable for the worker 1. We argue that $e_{1}$
is desirable for the firm $\widetilde{m}$ as well. In fact, let’s turn to
$\widetilde{C}_{\widetilde{m}}(\widetilde{S}\cup e_{1})$. To get this set, we
should to descend $\widetilde{S}\cup e_{1}$ (using $\pi$), apply CF $C_{m}$ to
the image, and then ”minimally” raise the resulting set. Projection of
$\pi(\widetilde{S}\cup e_{1})$ is equal to $S\cup e$, from where, using 1), we
get that $e$ belongs to the choice from the projection. But the ”minimal” rise
of $e$ into the set of $\widetilde{S}\cup e_{1}$ is equal to $e_{1}$.
Therefore, $e_{1}$ belongs to $\widetilde{C}_{\widetilde{m}}(S\cup e_{1})$ and
is desirable for $\widetilde{m}$. Thus, $e_{1}$ blocks $\widetilde{S}$,
contrary to the assumption of stability. Suppose now that $e$ is selected in
the second step, that is, $e\notin C_{1}(S\cup e)$, but $e\in C_{2}(S\cup
e-C_{1}(S\cup e))$. Since $C_{1}(S\cup e)$ does not contain $e$, then from the
consistency $C_{1}(S\cup e)=C_{1}(S)=S(1)$. So $e$ is also selected from the
smaller set of $S(2)\cup e$, $e\in C_{2}(S(2)\cup e)$. This means that $e_{2}$
is desirable for the worker $2$. But $e_{2}$ is also desirable for the firm
$\widetilde{m}$ (in the state $\widetilde{S}$). So $e_{2}$ blocks
$\widetilde{S}$, contrary to the assumption of stability. This proves that $S$
is not blocked. $\Box$Proof of Theorem 2. _Injectivity_ of $\pi_{*}$ is almost
obvious. Indeed, the set $S=\pi(\widetilde{S})$, according to Lemma 3, is
canonically decomposed into $S(1)=C_{1}(S)$ and $S(2)=C_{2}(S-S(1)$. Therefore
$\widetilde{S}$ turns out by lifting $S(1)$ to the first floor and $S(2)$ to
the second one._Surjectivity._ To prove the surjectivity of $\pi_{*}$ we have
to build, for each stable $S$ (in the problem $I$), a stable $\widetilde{S}$
(in the problem $\widetilde{I}$) such that $\pi(\widetilde{S})=S$. As a matter
of fact, we have the only candidate for the role of $\widetilde{S}$, namely,
the canonical rise of $S$. More precisely: $S$ is split into
$S(1)=C_{1}(S(1))$ and $S(2)=C_{2}(S-S(1))$ (Lemma 3). So, to build
$\widetilde{S}$, we raise $S(1)$ to the level 1, and rise $S(2)$ to the level
2. As a result, $\widetilde{S}$ is covered (without intersections) by the sets
$\widetilde{S}(i)$ equal to $S(i)\times\\{i\\}$. In other words,
$\widetilde{S}$ consists of contracts $(s,i)$, where $s\in S_{i}$, $i=1,2$.
Obviously, $\widetilde{S}$ is projected (bijectively) onto $S$. And it remains
to check that $\widetilde{S}$ is stable in the problem $\widetilde{I}$. That
is check the acceptability and non-blockability of $\widetilde{S}$.
_Acceptability._ Let’s start with the acceptability for $\widetilde{m}$. As
mentioned above (Lemma 1), this means that the projection of $\widetilde{S}$
onto $S$ is a bijection, and $S=C_{m}(S)$. The first is obvious from the
construction of $\widetilde{S}$, and the second follows from the acceptability
of $S$ for $m$. Acceptability for $i=1,2$. The set $\widetilde{S}(i)$ is set
$S(i)$ raised to the $i$-th floor. And we need to check that
$C_{i}(S(i))=S(i)$. But by the definition, $S(1)$ is equal to $C_{1}(S)$, and
$S(2)=C_{2}(S-C_{1}(S))$. And if the choice from $S$ is equal to $S(1)$, then
even more (due to the substitutability of $C_{1}$) the choice from $S_{1}$ is
equal to $S_{1}$. So, the acceptability of $\widetilde{S}$ is established.
Let’s show the non-blocking. _Non-blocking._ Suppose that some contract
$\widetilde{e}$ is desirable for $\widetilde{m}$ and for some of the workers.
Desirability for $\widetilde{m}$ means that
$\widetilde{e}\in\widetilde{C}_{\widetilde{m}}(\widetilde{S}\cup\widetilde{e})$.
That is, that $e\in C_{m}(S\cup e)$ and, therefore, $e$ is desirable for $m$
in the state $S$. Let’s first assume that $\widetilde{e}$ is desirable for the
worker 1. This means, firstly, that $\widetilde{e}=e_{1}$ is located on the
first floor and, secondly, that
${e_{1}}\in\widetilde{C}_{1}(\widetilde{S}(1)\cup{e_{1}})$, that is $e\in
C_{1}(S(1)\cup e)$. We claim that in fact $e$ belongs to $C_{1}(S\cup e)$ and
even more so to $C_{w}(S\cup e)$, that is, $e$ is desirable for $w$ in the
state of $S$. Indeed, by virtue of parh independence

$C_{1}(S\cup e)=C_{1}(C_{1}(S)\cup e)=C_{1}(S(1)\cup e)$

and therefore contains $e$. So, $e$ is desirable for both $m$ and $w$. Due to
the stability of $S$, this means that $e\in S$ and even $S(1)$. But in this
case, $\widetilde{e}$ belongs to $\widetilde{S}$ and there is no blocking. We
get that $\widetilde{e}$ cannot be desirable for 1. Now let’s assume that
$\widetilde{e}$ is desirable for both the firm $\widetilde{m}$ and the worker
$2$. The first implies, as before, that $e$ is desirable for $m$. The second
means that $\widetilde{e}=e_{2}$ is located on the second floor and that
${e_{2}}\in\widetilde{C}_{2}(\widetilde{S}(2)\cup{e_{2}})$, that is, that
$e\in C_{2}(S(2)\cup e)$. We again assert that $e\in C_{w}(S\cup e)$, that it
is desirable for $w$. Really,

$C_{w}(S\cup e)=C_{1}(S\cup e)\cup C_{2}(S\cup e-C_{1}(S\cup e)).$

Since $\widetilde{e}$ is undesirable for worker 1, $e\notin C_{1}(S\cup e)$
and (due to the consistency of $C_{1}$) $C_{1}(S\cup e)=C_{1}(S)=S_{1}$.
Therefore

$C_{2}(S\cup e-C_{1}(S\cup e))=C_{2}(S\cup e-S(1))=C_{2}(C_{2}(S-S(1))\cup
e)=C_{2}(S(2)\cup e),$

and the last term contains $e$. So $C_{w}(S\cup e)$ contains $e$. We get that
$e$ is desirable for $m$ and $w$, and therefore belongs to $S$. Moreover, $e$
belongs to $S(2)$. But then $e_{2}$ belongs to $\widetilde{S}$, and again it
does not block. Thus, the set $\widetilde{S}$ is stable, which proves Theorem
2. $\Box$

## References

  * [1] Aizerman M.A., Aleskerov F.T. _Theory of Choice._ North-Holland, 1995
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  * [3] Alkan A., Gale D. Stable schedule matching under revealed preference // J. Economic Theory. 112 (2) (2003) 289-306.
  * [4] Blair C. The lattice structure of the set of stable matchings with multiple partners // Math. Oper. Res. 13 (4) (1988) 619-628.
  * [5] Chambers, C. P. and B. Yenmez. 2018. On lexicographic choice. // Economics Letters, 171 (1018) 222–224.
  * [6] Danilov V.I. Stable system of schedule contracts. // Journal of the New Economic Association. n.3 (2021) 32–49 (in Russian)
  * [7] Danilov V. I., Karzanov A. V. Stable and meta-stable contract networks // Journal of Mathematical Economics, Volume 108, October 2023, 102888
  * [8] Danilov V.I. and G.A. Koshevoy, Mathematics of Plott choice functions. // Mathematical Social Sciences, 49 (2005), 245-272.
  * [9] Fleiner T. A fixed-point approach to stable matchings and some applications // Math. Oper. Res. 28(1) (2003) 103-126.
  * [10] Horan S., Manjunath V. Lexicographic Composition of Choice Functions // arXiv:2209.09293v1 [econ.TH] 19 Sep 2022
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LA-UR-21-31648

]https://www.matthewmumpower.com

#  $\beta^{-}$-delayed fission in the coupled Quasi-particle Random Phase
Approximation plus Hauser-Feshbach approach

M. R. Mumpower<EMAIL_ADDRESS>[ Theoretical Division, Los Alamos National
Laboratory, Los Alamos, NM 87545, USA  T. Kawano  Theoretical Division, Los
Alamos National Laboratory, Los Alamos, NM 87545, USA  T. M. Sprouse
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545,
USA

###### Abstract

Beta-delayed neutron emission and $\beta$-delayed fission ($\beta$df)
probabilities were calculated for heavy, neutron-rich nuclei using the Los
Alamos coupled Quasi-Particle Random Phase Approximation plus Hauser-Feshbach
(QRPA+HF) approach. In this model, the compound nucleus is initially populated
by $\beta$-decay and is followed through subsequent statistical decays taking
into account competition between neutrons, $\gamma$-rays and fission. The
primary output of these calculations includes branching ratios along with
neutron and $\gamma$-ray spectra. We find a relatively large region of heavy
nuclides where the probability of $\beta$df is near 100%. For a subset of
nuclei near the neutron dripline, delayed neutron emission and the probability
to fission are both large which leads to the possibility of multi-chance
$\beta$df (mc-$\beta$df). We comment on prospective neutron-rich nuclei that
could be probed by future experimental campaigns and provide a full table of
branching ratios in ASCII format in the supplemental material for use in
various applications.

##  I Introduction

Beta-delayed fission ($\beta$df) is a two step nuclear decay process in which
fission follows electron capture (EC), $\beta^{+}$ decay or $\beta^{-}$ decay.
Discovered by experiments in the 1960’s, this process starts with a parent
nucleus which undergoes EC or $\beta^{+/-}$ decay to an excited state in the
daughter nucleus. The populated state may fission if the excitation energy is
near or greater than the fission barrier of the daughter nucleus and is in
competition with particle emission and $\gamma$ de-excitation. This rare decay
mode in near-stable isotopes is limited by the relatively small $\beta$-decay
Q-value of the parent nucleus making it a unique probe of low energy structure
of atomic nuclei. In nuclei with extreme neutron-excess this decay mode may be
more prevalent, playing a role in the cosmos by influencing the formation of
the elements found on the periodic table in astrophysical events Kodama+75 .

Difficulty in the production of exotic nuclei which may undergo fission
coupled with small branching ratios relative to other processes (such as
$\alpha$-decay) makes the investigation of $\beta$df very challenging in a
laboratory environment Andreyev+13 . Just under thirty cases of $\beta$df on
both the neutron-deficient and neutron-rich side of stability have been
studied experimentally with most of this progress coming in recent years at
radioactive beam facilities. The bulk of current data resides on the neutron-
deficient side of stability, e.g. most recently the two isotopes of thallium
($Z=81$), 178Tl and 180Tl have been shown to be the lighest measured $\beta$df
precursors Andreyev+10 ; Liberati+13 ; Elseviers+13 . While the $\beta$df
branching ratio of the neutron-deficient isotope 242Es at $0.6$(2) $\times$
$10^{-2}$ remains one of the largest measured to date, see Table 1 of Ref.
Andreyev+13  for full list. The partial $\beta$df half-lives of neutron-
deficient nuclei have been found to follow a systematic trend Ghys+15 ,
however further measurements are required to determine if the systematics can
be extended to neutron-rich nuclei. Measurements have been made on only six
neutron-rich nuclei, all of which exhibit extremely small branchings relative
to measurements of neutron-deficient nuclei. The reason for this observation
comes from the fact that experiments cannot yet reach nuclei where the fission
barrier height ($B_{\textrm{f}}$) is sufficiently smaller than
$Q_{\beta^{-}}$. Such low branching ratios are beyond the fidelity of any
current nuclear model prediction.

Excluding the paucity of experimental data, the description of $\beta$df
persists as an open challenge to theory. This stems from the requirement of an
all encompassing description of the complexity of the atomic nucleus. To list
a few examples: the $\beta$-strength function, excited states, single particle
and collective effects as well as a fission properties that depend on the
nuclear potential-energy landscape, and traversal through it (resulting in the
production of fission fragments) must all be modeled. Of particular interest
to the description of $\beta$df are those neutron-rich nuclei that may
participate in the astrophysical rapid neutron capture or $r$-process of
nucleosynthesis Mumpower+16r . While these nuclei remain out of reach to
experimental facilities, heavy $r$-process nuclei have relatively large
$Q_{\beta^{-}}$ compared to $B_{\textrm{f}}$ which is a favorable condition
for large $\beta$df branching ratios. The work of Thielemann et al.
Thielemann+83  laid the groundwork for the theoretical description of
$\beta$df and its application to the rapid neutron capture or $r$ process of
nucleosynthesis. Statistical approaches, as studied here, offer further
exploration of delayed fission phenomenon Mumpower+18a ; Minato+21 .

In this work, we study the $\beta$df of neutron-rich nuclei using the recently
developed Quasi-particle Random Phase Approximation plus Hauser-Feshbach
(QRPA+HF) framework. This microscopic approach starts with a compound nucleus
initially populated by $\beta$-decay and follows the subsequent statistical
decay taking into account competition between neutrons, $\gamma$-rays and
fission. We find a relatively large region of the chart of nuclides where the
probability of $\beta$df is near 100% that prevents the production of
superheavy elements in nature Mumpower+18a . The decay chains of very neutron-
rich nuclei near the neutron dripline exhibit large neutron multiplicity and
large probability to fission leading to the possibility of multi-chance
$\beta$df or mc-$\beta$df for short. This decay mode results in multiple
fission chances after $\beta$-decay analogous to multi-chance neutron induced
fission. We discuss the theoretical basis of our model and provide tabulated
values in the supplemental material.

##  II Model

The Quasi-particle Random Phase Approximation plus Hauser-Feshbach (QRPA+HF)
approach was introduced in Refs. Kawano+08 ; Mumpower+16a  using version 3.3.3
of the Los Alamos Hauser-Feshbach statisitical decay code. Here, we apply this
framework to the calculation of delayed-neutron emission in the presence of
$\beta$-delayed fission and now use version 3.5.0. Among other modifications,
the change in version number represents the modifications to the code to
include the description of fission during the statistical decay. The
calculation of $\beta$df in this approach is a two step process starting with
the $\beta$-decay of the precursor nucleus ($Z$,$A$) where $Z$ represents the
atomic number, and $A$ is the total number of nucleons, followed by the
statistical decay of the subsequent daughter generations, ($Z+1$,$A-j$) where
$j$ is the number of neutrons emitted, ranging from $0$ to $10$. Ground state
properties are taken from the 2012 version of the Finite Range Droplet Model
FRDM2012  unless otherwise noted. The procedure for the QRPA+HF calculation is
shown schematically in Fig 1 and outlined in detail below.

The $\beta$-decay of the precursor nucleus provides the initial population of
the daughter nucleus ($Z+1$,$A$), and is given by the QRPA formalism of Refs.
Krumlinde+84 ; Moller+90 ; Moller+97 ; Moller+03 . QRPA allows for the
calculation of excited state properties by using a small perturbing force. We
use the latest QRPA calculations from Ref. Moller+18a  which provides both the
Gamow-Teller (GT) and First-Forbidden (FF) contributions, a notable upgrade in
this work compared to our existing global model calculations Mumpower+16a ;
Mumpower+18a . In this framework, based on a folded-Yukawa interaction, the
$\beta$-decay half-life is calculated via

$\frac{1}{T_{1/2}}=\sum_{0\leq E_{x}\leq
Q_{\beta}}S_{\beta}(E_{x})f(Z,Q_{\beta}-E_{x})\ ,$ (1)

where $S_{\beta}(E_{x})$ is the $\beta$-strength function at excitation energy
$E_{x}$ in the daughter nucleus and $f$ is the Fermi function which takes into
account the phase space contribution. The energy factor can be approximated as
$f\sim(Q_{\beta}-E_{x})^{5}$ so that the low-energy excitations or near
ground-state transitions dominate the calculation of the half-life. Thus, the
output of the QRPA (the $\beta$-strength function) at low energies is of
critical importance to understanding this phenomenon Mustonen+16 . Where data
exists, we may supplement the theoretical calculation with the experimental
data as in Ref. Mumpower+16a , however, we note that in the case of neutron-
rich nuclei which undergo $\beta$df there is no data with which to compare or
include.

The QRPA solutions for $\beta$-decay strongly depend on the level structure in
the participating nuclei. Further, these solutions are strongly peaked
depending on the unknown level structure for the nuclei of interest. We
therefore apply a smoothing procedure to the $\beta$-strength distribution by
a Gaussian,

$\omega(E_{x})=C\sum_{k}b^{(k)}{1\over{\sqrt{2\pi}\Gamma}}\exp\left\\{-{{[E^{(k)}-E_{x}]^{2}}\over{2\Gamma^{2}}}\right\\},$
(2)

where $b^{(k)}$ are the branching ratios from the parent state to the daughter
states $E^{(k)}$, $E_{x}$ is the excitation energy of the daughter nucleus,
$C$ is a normalization constant, and the Gaussian width is taken to scale
proportional to $A^{-0.57}$ for excitations above 2 MeV, as used in Ref.
Moller+97 ; Moller+03 . The choice in this work results in a spreading of
roughly $200$ keV for $A=200$ nuclei, and is roughly equal to the error in the
mass model. Other Gaussian widths have been used in our previous works, for
instance a constant value of $\Gamma=100$ keV in Refs. Kawano+08 ;
Mumpower+16a . The choice of the Gaussian width represents an intrinsic
uncertainty associated with modern QRPA methods.

With the population of the first daughter nucleus defined by the
$\beta$-strength function from the QRPA calculation, the statistical decay can
then be followed where the competition between neutron emission, $\gamma$-rays
and fission occurs. In this second stage of the calculation, the daughter
nucleus is assumed to be in a compound state, which means the nucleus is
governed by its overall properties rather than the details of the formation
process Weisskopf+56 . This assumption leads to the independent factorization
of exit channel probabilities; for our purposes it is a good approximation.
Independent factorization of exit channel probabilities may be modified in
certain situations by taking into account the memory of the entrance channel
via correction factors Bertsch+17 .

Figure 1:  (Color online) Schematic of the combined QRPA+HF approach when
applied to $\beta$df. Initial population of the daughter nucleus ($Z+1$,$A$)
is determined by the $\beta$-strength function ($S_{\beta}$) using QRPA. The
competition between neutron emission, $\gamma$ de-excitation and fission are
then handled in the HF framework for which the transmission coefficients
($T_{\textrm{n}}$, $T_{\gamma}$, $T_{\textrm{f}}$) are calculated respectively
at each stage of the statistical decay. In some cases, the fission barrier
heights ($E_{A}$, $E_{B}$) may contain only a single hump ($E_{B}=0$), and
could possibly be much larger than denoted by this schematic. Multi-chance
$\beta$df occurs for the daughter nucleus ($Z+1$,$A-1$) and beyond. The
trailing dots denotes the statistical decay continues until the total
available excitation energy ($Q_{\beta}$) is exhausted.

The excited state transitions show in Fig. 1 can be discrete or in the
continuum. Since there is no known level data for the extremely neutron-rich
nuclei studied here, we rely on the Gilbert-Cameron level density Gilbert+65 .
This level density formula connects a Fermi gas model to a constant
temperature model at a matching energy with parameters taken from systematics
at stability Kawano+06 . Shell corrections are applied to the level density
using the common Ignatyuk et al. prescription Ignatyuk+75 .

To easily reference the compound nucleus in the context of $\beta$-delayed
calculations, we define the compound state as $c^{(j)}_{k}$ where $c^{(j)}$
represents the $j$-th compound nucleus after $j$ neutron emissions and $k$ the
index of the excited state in the same nucleus. Using this shorthand,
$c^{(0)}$ is the daughter nucleus ($Z+1$,$A$), $c^{(1)}$ is the granddaughter
nucleus ($Z+1$,$A-1$) and so on.

The transmission coefficient for $\gamma$ de-excitation in the compound
nucleus, $c^{(j)}$, is denoted by vertical dashed lines in Fig. 1. This
quantity is calculated using the definition,

$T^{(j)}_{\gamma}(E_{\gamma})=2\pi
E_{\gamma}^{(2L+1)}f^{(j)}_{XL}(E_{\gamma})\ ,$ (3)

where the transition occurs with $\gamma$-ray energy $E_{\gamma}=E_{i}-E_{k}$,
and $f^{(j)}_{XL}$ is the $\gamma$-ray strength function of multipole type
$XL$ for compound nucleus $c^{(j)}$. The $\gamma$-ray transmission coefficient
is often written $T_{XL}$ as shorthand for the multipolarities (E1, M2, E2,
M2, etc.). In this work, we use the generalized Lorentzian for the E1
$\gamma$-strength function ($\gamma$SF) Kopecky+87 . Additional low energy
enhancements, such as the M1 scissors mode Larsen+10 ; Mumpower+17  are not
considered here.

The transmission coefficient for neutron emission between two compound states
is represented by diagonal solid lines connecting adjacent compound nuclei in
Fig. 1. The calculation of this quantity occurs between two excited states
$E_{i}$ and $E_{k^{\prime}}$ in neighboring nuclei and can be written down as

$T^{(j+1)}_{n}(E_{i},E_{k^{\prime}})=\sum_{s}\sum_{l}T_{n_{l_{s}}}(E_{i}-S^{(j)}_{n}-E_{k^{\prime}})\
,$ (4)

where $S^{(j)}_{n}$ is the one neutron separation energy of $c^{(j)}$ and the
summation is over all possible partial waves. The difference in energy
$E_{n}=E_{i}-E_{k^{\prime}}$ is the energy of the delayed neutron. The
$T_{n_{l_{s}}}$ values are given using the Koning-Delaroche global optical
potential that is optimized for neutron-rich nuclei Koning+03 ; Koning+05 ;
Goriely+08 . The prime on the second ($k$) index is a reminder that the energy
level is in a different compound nucleus.

The transmission coefficient for fission is approximated using the Hill-
Wheeler formula assuming transmission through a parabolic barrier Hill+53 ,

$T^{(j)}_{\textrm{f}}(E)=\frac{1.0}{1.0+\exp(2\pi\frac{B_{\textrm{f}}-E}{C})}\
,$ (5)

where $B_{\textrm{f}}$ is the fission barrier height, $C$ is the curvature and
$E$ is the relative excitation energy of the $j$-th compound nucleus. The
fission transmission coefficient is represented as a horizontal solid line to
the right of each compound nucleus in Fig. 1. Fission curvatures are defined
separately for even-even, odd-$A$ and odd-odd nuclei as in Ref. Thielemann+83
. The effective transmission coefficient for fission depends on the number of
fission barriers. For instance, with two barriers,

$T^{eff}_{\textrm{f}}=\frac{T_{\textrm{A}}T_{\textrm{B}}}{T_{\textrm{A}}+T_{\textrm{B}}}\
,$ (6)

where $T_{\textrm{A}}$ and $T_{\textrm{B}}$ are the first and second fission
transmission coefficients respectively. We have considered calculations with a
second barrier that is wider and shorter than the first and note that it does
not qualitatively change our results. Therefore, in the remainder of this
work, we only consider the case of a single (maximal) barrier,
$T^{(j)}_{\textrm{f}}=T_{\textrm{A}}$. Note that for all of the transmission
equations above, we have quoted for convinence the ‘lumped’ versions which
remove the spin and parity dependence Mumpower+17 .

We use the Finite-Range Liquid-Drop Model (FRLDM barrier height predictions
from Ref. Moller+15  for our primary results. Later in section IV we study the
variation of our predictions based on different barrier heights. The FRLDM
barrier height predictions have been extensively benchmarked against many
nuclear observables including electron-capture delayed fission data, fission-
fragment charge yields and a handful of prompt neutron-capture data from
weapons tests. The predictions of ground state masses relative to these
barrier heights shows that there are prominent regions of both (n,f) and
$\beta$df for the heavy neutron-rich nuclei near the end of the chart of
nuclides with $\beta$df most influential just beyond the $N=184$ closed shell
towards the neutron dripline Moller+15 .

With the transmission coefficients defined, we can calculate transmission
probabilities for these three channels. In what follows we simplify the
equations by excluding indices of quantum numbers and implicitly take all
transitions to obey spin-parity selection rules which changes our transmission
coefficients into the so-called ‘lumped’ versions. The $\gamma$-emission
transition probability in the $j$-th compound nucleus is taken to be

$p_{j}(E_{i},E_{k})=\frac{1}{N_{j}(E_{i})}T^{(j)}_{\gamma}(E_{i}-E_{k})\rho_{j}(E_{k})$
(7)

where the transition is from a level with high excitation energy $E_{i}$ to a
level of energy $E_{k}$, $T^{(j)}_{\gamma}$ is the $\gamma$-ray transmission
coefficient for $c^{(j)}$, $\rho_{j}(E_{k})$ is the level density in $c^{(j)}$
at energy $E_{k}$ and $N_{j}(E_{i})$ is a normalization factor that we define
shortly. The neutron-emission transition probability from $c^{(j)}$ to
$c^{(j+1)}$ is defined as

$q_{j}(E_{i},E_{k^{\prime}})=\frac{1}{N_{j}(E_{i})}T^{(j+1)}_{n}(E_{i},E_{k^{\prime}})\rho_{j+1}(E_{k^{\prime}})$
(8)

where the transition is from an energy level $E_{i}$ in $c^{(j)}$ to an energy
level $E_{k^{\prime}}$ in $c^{(j+1)}$, $T^{(j+1)}_{n}$ is the neutron
transmission coefficient from $c^{(j+1)}$ to $c^{(j)}$ and
$\rho_{j+1}(E_{k^{\prime}})$ is the level density in $c^{(j+1)}$ evaluated at
$E_{k^{\prime}}$. For fission, the transmission probability from an excitation
energy $E_{i}$ below the fission barrier for compound nucleus $c^{(j)}$ is
calculated as

$r_{j}(E_{i})=\frac{1}{N_{j}(E_{i})}T^{(j)}_{f}(E_{i})\rho^{f}_{j}(E_{i})\ ,$
(9)

where $T^{(j)}_{f}(E_{i})$ is the fission transmission coefficient from Eqn.
(5) and $\rho^{f}_{j}(E_{i})$ is the fission level density in $c^{(j)}$. The
normalization factor, $N_{j}$ is given by the sum over all possible exit
channels

$N_{j}(E_{i})=\int_{0}^{E_{i}}T^{(j)}_{\gamma}(E_{i}-E_{k})\rho_{j}(E_{k})\textrm{d}E_{k}\\\
+\int_{0}^{{E_{i}}-S^{(j)}_{n}}T^{(j+1)}_{n}(E_{i},E_{k^{\prime}})\rho_{j+1}(E_{k^{\prime}})\textrm{d}E_{k^{\prime}}\\\
+\int_{0}^{E_{i}}T^{(j)}_{f}(E_{k})\rho_{j}(E_{k})\textrm{d}E_{k}\ ,$ (10)

where the integration in each case runs over the appropriate energy window.
The units of the $q$, $p$ and $r$ quantities are the same as the level density
and the transition probabilities do not depend on how the initial state,
$E_{i}$ was populated due to the Bohr independence hypothesis of compound
nucleus formation.

The level population in the compound nucleus, $c^{(j+1)}$, at energy, $E_{k}$,
is

$\displaystyle\mathscr{P}_{j+1}(E_{k})=\sum_{i}\mathscr{P}_{j+1}(E_{i})p_{j+1}(E_{i},E_{k})$
$\displaystyle+\sum_{{k^{\prime}}}\mathscr{P}_{j}(E_{k^{\prime}})q_{j}(E_{k^{\prime}},E_{k})$
$\displaystyle+\sum_{i}\mathscr{P}_{j+1}(E_{i})r_{j+1}(E_{i})\ ,$ (11)

with the summation running over over all levels which may feed the compound
state $c^{(j+1)}_{k}$. Each of the terms takes into account the possible
pathways of $\gamma$-ray emission, neutron emission and fission to the excited
state $E_{k}$. The initial population of this recursive function comes from
the $\beta$-decay strength function.

Following from Eqn. (11), the total production probability (total branching
ratio) to emit $j$-neutrons or for the $j$-th compound nucleus to fission are
given by,

$\displaystyle P_{j\textrm{n}}=\mathscr{P}_{j}(E_{k}=E_{\text{gs}})\text{
where }r_{j}=0$ (12a) $\displaystyle
P_{j\textrm{f}}=\sum_{k}\mathscr{P}_{j}(E_{k})\text{ where }r_{j}\neq 0.\ $
(12b)

Thus, for $\beta$-delayed neutron emission, we merely need to calculate the
production of the ground state, $E_{\text{gs}}$ for each compound system,
$c^{(j)}$. Conversely, for fission, we must sum over all the excited states
that end with fission in the particular compound system.

Equations 12a and 12b tell us that the statistical decay must end in either
the population of one of the daughter generation’s ground state or by fission.
Ergo, the cumulative probability to emit either neutrons or fission must sum
to unity

$1=\sum_{j=0}^{10}P_{j\textrm{\small n}}+\sum_{j=0}^{10}P_{j\textrm{\small
f}}=P_{\textrm{\small n}}+P_{\textrm{\small f}}$ (13)

where the summation index, $j$, represents the number of neutrons emitted and
runs over all possible daughter nuclei. The value of $P_{j\textrm{n}}$ is the
probability to emit $j$-neutrons without fissioning and $P_{j\textrm{\small
f}}$ is the probability to fission after $j$-neutrons have been emitted. For
very neutron-rich heavy nuclei that may participate in the $r$ process, we
designate a maximum value of $j=10$. The cumulative sum of these two
quantities are denoted by $P_{\textrm{\small n}}$ and $P_{\textrm{\small f}}$
respectively. Some values of $P_{j\textrm{\small n}}$ or $P_{j\textrm{\small
f}}$ may be zero due to selection rules or an exhaustion of the initial
excitation energy. We explicitly assume that other channels such as proton or
alpha emission are very small.

In the context of the approach outlined above, regular or first-chance
$\beta$df is defined as the fission that occurs during the population of the
first compound nucleus formation after $\beta$-decay. Explicitly, if ($Z$,$A$)
is the precursor nucleus that $\beta$-decays, fission occurring in the first
generation daughter nucleus, ($Z+1$,$A$), is first-chance $\beta$df. For heavy
neutron-rich nuclei there are additional chances to fission after
$\beta$-decay stemming from the fact that these nuclei have relatively large
$Q_{\beta}$ values as compared to the fission barriers, $B_{\textrm{f}}$, or
neutron separation energies, $S_{\textrm{n}}$. Thus, neutron-rich nuclei
populated in extreme astrophysical conditions may open the possibility for
each of the populated daughter generations to fission after $\beta$-decay. We
use the term multi-chance $\beta$-delayed fission (mc-$\beta$df for short) to
describe this phenomenon, which is analogous to multi-chance neutron-induced
fission that arises at high neutron incident energies. In Fig. 1 this decay
mode is represented by fission occuring in the second, or higher, daughter
generations. If the sum of the fission probabilities after the first daughter
generation is greater than 10 %, $\sum\limits_{j>0}P_{j\textrm{\small
f}}>10\%$, then we consider this nucleus to undergo mc-$\beta$df.

Figure 2:  (Color Online) Competition between the neutron, gamma and fission
channels in the case of beta-delayed fission of precursor nucleus $Z=97$,
$A=282$, shown as a ratio of the respective transmission coefficient to the
total sum. The total transmission coefficient sum (dashed light line), which
spans many orders of magnitude, can be read from the right Y-axis.

Lastly, it is useful in analysis to compute the average neutron multiplicity
obtained after $\beta$-decay but pre-scission,

$\langle n\rangle=\sum_{j=0}^{10}j\left(P_{j\textrm{\small
n}}+P_{j\textrm{\small f}}\right)$ (14)

where the summation index again represents the number of neutrons emitted and
$P_{j\textrm{\small n}}$, $P_{j\textrm{\small f}}$ are defined in Eqns. (12).

##  III Results

Our results include $\beta$-delayed neutron emission and $\beta$-delayed
fission probabilities for all neutron-rich nuclei from stability to extreme
neutron excess with an upper limit of the mass number at $A=330$ which
represents the extent of the FRDM2012 model. We first discuss individual cases
before going into the global results of applying the QRPA+HF framework to all
nuclei.

As a typical example, we first explore the $\beta$df of $r$-process nucleus
^282_97Bk . This nucleus has $Q_{\beta}\sim 11$ MeV resulting in a population
of high-lying excited states in subsequent daughter generations. A competition
ensues between the neutron, $\gamma$-ray and fission channels during the
statistical decay as shown in Fig. 2. In each successive panel going from
bottom to top, the energies are shifted relative to the ground state of the
first daughter nucleus. For reference, the first daughter generation ^282_98Cf
has $S_{\textrm{n}}=3.33$ MeV and $B_{\textrm{f}}=3.82$ MeV while the second
and third generations have $S_{\textrm{n}}=2.36$ MeV, $B_{\textrm{f}}=4.47$
MeV and $S_{\textrm{n}}=3.65$ MeV, $B_{\textrm{f}}=4.54$ MeV respectively. The
neutron multiplicity is $\langle n\rangle=1.07$ for this decay chain with
large fission chances above $j=0$, thereby satisfying the definition of a
nucleus which undergoes mc-$\beta$df.

Fission dominates the low lying excitations in the first daughter ($j=0$;
bottom panel). Generally one would expect the $\gamma$-channel to have the
largest transmission well below the barrier, however, in some cases such as
this one, the channel may be blocked due to selection rules. Nevertheless, the
total transmission coefficient (right Y-axis) is very small in this energy
regime so neither channel contributes to the resultant probabilities. Above
the neutron separation energy neutron emission is energetically possible,
cutting the fission transmission in half since the barrier height is on the
order of the separation energy in this nucleus. We find that the fission
transmission coefficient again increases above the neutron transmission
coefficient after about 4 MeV due to the higher fission level density relative
to the ground state level density.

In the second daughter ($j=1$; middle panel), the $\gamma$ transmission
coefficient contributes the most to the low excitation energy regime. After
crossing the neutron separation energy neutrons immediately activates with
fission transmission slowly increasing. One of the main reasons for the
difference between this nucleus and the previous is that the barrier heigh in
this nucleus is several MeV higher than the separation energy resulting in a
slower onset of fission taking over. The third nucleus in the decay chain
shows a similar behavior as the first.

Figure 2 hints at a subtle interplay between the $\beta$-strength function,
mass surface and fission barrier heights in predicting $\beta$df properties.
To explore these relationships further we highlight the $\beta$df of 295Fm and
290Am.

Figure 3:  (Color Online) Individual probabilities for delayed neutron and
delayed fission as a function of $j$ neutron emitted from the daughter
nucleus. 295Fm is dominated by fission while 290Am shows more competition
between neutron emission and fission.

Figure 3 shows the individual $P_{j\textrm{\small n}}$ and $P_{j\textrm{\small
f}}$ probabilities after successive neutrons are emitted along the decay
chain. In the case of $\beta$df of 295Fm, fission in the first and second
daughter generations completely govern the decay. The reason for this is that
in the first daughter ^295_101Md has $S_{\textrm{n}}=3.15$ MeV while
$B_{\textrm{f}}=2.69$ MeV, thus fission operates at low excitation energy and
neutron emission is strongly hampered, only reaching a maximum ratio of $25$%
of the total transmission right near the threshold energy. The same scenario
plays out in the second daughter, thus limiting a longer decay chain with
$j>1$.

Conversely, in the $\beta$df of 290Am, we see from Fig. 3 that a longer decay
chain (up to $j=4$) ensues with significant probabilities for both neutron
emission and fission. Here the separation energies of the daughter generations
are all consistently lower than the respective barrier heights, thus providing
ample competition between the neutron and fission channels throughout each
isotope participating in the statistical decay.

Figure 4:  (Color Online) (a) The cumulative probabilities for emitting
neutrons (blue) or fission (red) after $\beta$-decay for neutron-rich
plutonium ($Z=94$) isotopes. The summation of these two terms yields 100%. The
average neutron multiplicity after $\beta$-decay, $\langle n\rangle$, is also
displayed by a dashed line and read from the right Y-axis. (b) The same
quantities shown for the berkelium ($Z=97$) isotopic chain. Solid grey
vertical line indicates the one-neutron dripline in FRDM2012.

Both the cumulative probability to emit a neutron and to fission along an
isotopic chain tend to oscillate as shown in Fig. 4 for neutron-rich isotopes
of plutonium (a) and berkelium (b). The overall trend of the cumulative
probabilities is dependent on the fission barrier heights of the nuclei
involved in the decay. The cumulative $\beta$df probability is small near
stability since fission barriers are relatively large here, in agreement with
experiments. Where fission barriers are relatively low, e.g. near $N=180$ for
plutonium and $N=190$ for berkelium, $\beta$df probabilities may reach near
100%. The odd-even staggering seen in the cumulative curves is sensitive to
nuclear masses because the difference in nuclear masses sets the threshold
energy for neutron emission. Starting from stability, the first nucleus in the
isotopic chain to have $S_{n}<0$ is denoted by the solid grey line in both
panels. This line denotes the extent of the neutron dripline in FRDM2012 that
may be accessible in astrophysical applications.

The average neutron multiplicity before scission, $\langle n\rangle$, is shown
on the right Y-axis for the same set of isotopes. The increasing value of this
quantity with neutron excess shows that mc-$\beta$df occupies a substantial
amount of real estate after the $N=184$ shell closure. Additional neutrons may
emitted during scission which we have not considered here that will provide
more late-time neutrons for capture in the astrophysical $r$ process
Mumpower+12 .

We now focus our discussion on the global results shown in Fig. 5. The
$r$-process highway, nuclei with $S_{\textrm{n}}\sim 1$ to $2$ MeV, is denoted
by black dots to guide the eye. This neutron-rich pathway may extend into the
region of large neutron emission as shown in panel (a). Towards the neutron
dripline roughly five to six neutrons are emitted on average during the decay,
similar to other model predictions in the literature Marketin+16 . The average
neutron energy after $\beta$-decay is plotted in panel (b). Neutron energy for
these heavy nuclei is smaller for nuclei that exhibit mc-$\beta$df as compared
to those surrounding nuclei resulting in a faster thermalization timescale in
the context of nucleosynthesis.

Several additional features of importance for the $r$ process are shown in
panel (c) which highlights the cumulative probability for $\beta$df. Beyond
the $N=184$ shell closure, many nuclei have $P_{j\textrm{\small f}}=100$%.
This means that the $\beta$-decay chain always ends in fission rather than the
population of the ground state of any of the daughter generations. When this
occurs, the nuclear flow of the $r$ process can no longer increase in proton
number preventing the production of superheavy elements. Nuclear flow may
proceed at the dripline, subject to mc-$\beta$df, but network simulations show
a termination point around $A\sim 300$ Mumpower+18a . We find this region to
extend further than previous calculations Thielemann+83  suggesting that a
neutron-rich pathway to populate superheavies via $\alpha$-decay is unlikely
Meldner+72 ; Petermann+12 . Multi-chance $\beta$df, outlined by the solid
black boundary in Fig. 5, that occurs towards the neutron dripline is another
feature of interest for $r$-process nucleosynthesis. These nuclei have large
$Q_{\beta}$ and large neutron multiplicity after $\beta$-decay due to small
separation energies. Unique to this decay process is the observation that the
production of light fission products come from a superposition of the yield
distributions of the heavy daughter nuclei. Details of the impact of first-
chance and multi-chance $\beta$df on $r$-process abundances can be found in
Ref. Mumpower+18a .

Figure 5:  (Color Online) (a) The average neutron multiplicity, $\langle
n\rangle$, after $\beta$-decay for heavy neutron-rich nuclei computed with the
QRPA+HF framework. (b) Average $\beta$-delayed neutron emission energy in
units of MeV. (c) The cumulative probability for $\beta$df to occur in the
region. Nuclei with large cumulative probability and neutron multiplicity are
classified as multi-chance $\beta$df, denoted by solid black border. The black
dots denote the $r$-process highway, roughly $S_{\textrm{n}}\sim 1$ to $2$
MeV, using FRDM2012 masses.

##  IV Nuclei of interest close to stability

The bulk of the discussion has been centered around nuclei with extreme
neutron excess. We now turn to the possibility of future experimental
measurements probing neutron-rich $\beta$df branching ratios for actinides and
superheavies. We remind the reader that closer to stability model calculations
are more susceptible to deviations, for example from the $\beta$-strength
function, than at high neutron excess as has been pointed out in previous work
Moller+03 .

To isolate potentially interesting nuclei, and reduce dependence on any single
model, we account for several variations in our inputs. We probe the
differences that arise in the $\beta$-strength from three models reported in
our previous work Moller+97 ; Moller+03 ; Moller+19 . We additionally account
for variation in the nuclear mass surface by using various models: the 2012
FRDM Moller+16 , Duflo-Zuker Duflo+95 , Hartree-Fock-Bogoliubov (HFB-27)
Goriely+13 , UNEDF1 Kortelainen+12 , Wood-Saxon Wang+14  and KTUY05 Koura+05 .
Finally, we also consider variation in the estimated fission barrier heights
of the models FRLDM Moller+15 , ETFSI Mamdouh+98 , HFB-14 Goriely+09  and KTUY
Koura+14 . This totals 72 model variations in which we can probe the
differences in predicted $\beta$-delayed neutron and $\beta$df probabilities.
We note that many more additional model variations were performed using other
mass models and those results are consistent with and do not alter the
conclusions presented below.

Figure 6 shows the resultant model variations for nuclei near stability with
the extent of NuBase (2020) in the region shown for reference NUBASE2020 . The
colorbar of this figure indicates the percentage of models that predict
greater than one percent $\beta$df probability for the given species. Near
measured nuclei there is little predicted $\beta$df branching. However, with
increasing neutron excess there is substantial agreement among the models that
neutron-rich nuclei undergo at least some $\beta$df.

From these model variations we form a subset (indicated by the space between
NuBase (2020) and the black solid line in Fig. 6) that may be accessible in
future experimental undertakings. Our criterion for future accessibility is:
(1) neutron-rich nuclei with $90\leq Z\leq 120$ (2) $P_{\mathrm{f}}\geq 1\%$
and (3) no more than 5 neutrons away from the last neutron-rich value in
NuBase (2020). This criterion produces a set of 35 nuclei close to stability
which are predicted to be most likely to have a measurable $\beta$-delayed
fission branching. We summarize this information in Table 1. The percent
models (column 5) in the table represents the number of variations out of 72
total with $P_{\mathrm{f}}\geq 1\%$. The minimum fission probability,
$P^{\textrm{min}}_{\textrm{f}}$ (column 6), represents the smallest
$P_{\mathrm{f}}$ value among the subset of models with $P_{\mathrm{f}}\geq
1\%$ and likewise for the maximum fission probability,
$P^{\textrm{max}}_{\textrm{f}}$ (column 7).

Several nuclei in this list are only a few neutrons from the last neutron-rich
isotope found in the recent evaluated data of NuBase (2020). A straightforward
way to assess the top experimental candidates is to multiply the percentage of
models in agreement of $\beta$df with the minimum $\beta$df probability,
$P^{\textrm{min}}_{\textrm{f}}$. This information can be found in columns 5
and 6 of Table 1. From this estimate one finds primarily superheavies with
282Bh ($Z=107$) topping the list, but such nuclei may be rather hard to
measure owing to low production. In the actinide region, the odd-$Z$ isotopic
chains are of interest with Am ($Z=95$), Bk ($Z=97$), Es ($Z=99$), Md
($Z=101$) and Lr ($Z=103$) hosting several candidates respectively. These
nuclei may be reachable in the future with fusion evaporation, transfer
reactions, or similar techniques.

Table 1:  Nuclei with $\beta$df branching ratios greater than 1% ($P_{\mathrm{f}}=P_{0\mathrm{f}}\geq 1\%$) among 72 model variations (see text for details). These nuclei represent prime experimental candidates and they reside no more than 5 neutrons away the most neutron-rich isotope in NuBase (2020). We limit the element number by $80\leq Z\leq 120$. Symbol | $Z$ | $N$ | $A$ |  $\%$ Models | $P^{\textrm{min}}_{\textrm{f}}$ | $P^{\textrm{max}}_{\textrm{f}}$
---|---|---|---|---|---|---
Pa | 91 | 153 | 244 | 14 | 1 | 4
Pa | 91 | 155 | 246 | 25 | 11 | 26
Np | 93 | 155 | 248 | 14 | 2 | 3
Np | 93 | 157 | 250 | 35 | 1 | 36
Am | 95 | 157 | 252 | 8 | 1 | 3
Am | 95 | 159 | 254 | 75 | 4 | 25
Bk | 97 | 159 | 256 | 58 | 2 | 32
Bk | 97 | 161 | 258 | 75 | 7 | 72
Bk | 97 | 162 | 259 | 18 | 1 | 5
Cf | 98 | 163 | 261 | 17 | 1 | 2
Es | 99 | 163 | 262 | 56 | 1 | 17
Es | 99 | 164 | 263 | 4 | 1 | 1
Md | 101 | 163 | 264 | 8 | 1 | 2
Md | 101 | 165 | 266 | 50 | 41 | 84
Md | 101 | 166 | 267 | 47 | 1 | 28
No | 102 | 167 | 269 | 39 | 1 | 12
Lr | 103 | 165 | 268 | 50 | 4 | 85
Lr | 103 | 166 | 269 | 21 | 2 | 6
Lr | 103 | 167 | 270 | 50 | 90 | 100
Lr | 103 | 168 | 271 | 50 | 2 | 99
Rf | 104 | 167 | 271 | 17 | 2 | 63
Rf | 104 | 169 | 273 | 17 | 59 | 100
Db | 105 | 167 | 272 | 17 | 91 | 94
Db | 105 | 168 | 273 | 14 | 95 | 100
Db | 105 | 169 | 274 | 17 | 94 | 95
Db | 105 | 170 | 275 | 17 | 99 | 100
Bh | 107 | 173 | 280 | 25 | 86 | 94
Bh | 107 | 174 | 281 | 14 | 48 | 100
Bh | 107 | 175 | 282 | 75 | 87 | 100
Bh | 107 | 176 | 283 | 62 | 1 | 100
Hs | 108 | 177 | 285 | 12 | 38 | 100
Mt | 109 | 175 | 284 | 25 | 74 | 93
Mt | 109 | 176 | 285 | 4 | 96 | 99
Mt | 109 | 177 | 286 | 17 | 86 | 94
Mt | 109 | 178 | 287 | 25 | 100 | 100

Further reinforcement that future experimental campaigns may reach measurable
$\beta$df branchings comes from the observation that the number of nuclei with
$P_{\textrm{f}}>1$% increases quadratically with increasing neutron number
beyond what has been measured. This behavior arises from the above model
variations and is shown in Fig. 7. The functional form can be written as

$C_{P_{\textrm{f}}>1\%}\approx 0.8866\times n^{2}+3.355\times n-3.618\ ,$ (15)

where the coefficients arise from a fit to the observed trend and $n$ measures
the distance from the maximum neutron number in NuBase (2020). The neutrons
counted beyond NuBase (2020) should be $n\geq 1$ when using this equation,
owing to the negative value of the last coefficient. While $n$ implicitly
depends on proton number, the ensemble averaged quantity, represented by
$C_{P_{\textrm{f}}>1\%}$, is insensitive to it. This finding comes from the
increase of $Q_{\beta}$ values, the decrease of $S_{1\mathrm{n}}$, and rather
smooth behavior of predicted fission barriers along isotopic chains. With
sufficient enough neutron excess, this observed behavior will break down as
the predicted $N=184$ shell closure is breached. Nevertheless, Equation 15
provides a simple picture for how close experimental efforts are to probing
significant delayed fission probabilities.

Figure 6:  (Color Online) The percentage of models that predict $\beta$df with
$P_{\textrm{f}}>1$% just outside the range of current measurements (see text
for details). The grey region indcates the extent of the NuBase (2020)
evaluation and the black line is 5 neutrons from it. Shaded nuclei in region
between these two bounds represent the first potential candidates for future
exploration; these nuclei are summarized in Table 1.

Figure 7:  (Color Online) Using model variations, the number of nuclei with at
least 1 % chance for $\beta$df increases quadratically (dotted gray) with
neutrons beyond NuBase (2020) (green). See text for details.

##  V Summary

We have provided the theoretical underpinnings of the Los Alamos Quasi-
particle Random Phase Approximation plus Hauser-Feshbach (QRPA+HF) approach to
$\beta$-delayed fission ($\beta$df). This approach seeks to combine
microscopic nuclear structure information with statistical Hauser-Feshbach
theory allowing for a description of the competition between neutrons,
$\gamma$-rays and fission during nuclear $\beta$-decay.

We have calculated $\beta$-delayed neutron emission and $\beta$-delayed
fission probabilities for all neutron-rich nuclei from near stability to
extreme neutron excess using this framework. We find a large region of nuclei
which exhibit a propensity for $\beta$df. A subset of these extremely neutron-
rich nuclei have the potential for multi-chance $\beta$df where a cascade of
neutron emission and subsequent fission may occur in each daughter generation
after the parent nucleus $\beta$-decays.

We use variations in our model inputs ($\beta$-strength distribution, mass
model and fission barriers) to study the likelihood that future experimental
campaigns may observe a measurable $\beta$df branching. We show that the
number of nuclei with $\beta$df branching increases quadratically with neutron
excess relative to current experimental reach. From this observed behavior we
are hopeful that future endeavors will reveal exciting insights into the
nature of delayed neutron and fission branchings among heavy and superheavy
elements.

###### Acknowledgements.

The authors thank Ani Aprahamian for encouraging the initial development of
this work. The authors would also like to thank Arnie Sierk, Jørgen Randrup,
Sean Liddick and Nick Esker for useful discussions. The authors finally thank
P. Möller for his useful comments and the QRPA $\beta$-strength used in this
work. This work was carried out under the auspices of the National Nuclear
Security Administration of the U.S. Department of Energy at Los Alamos
National Laboratory. LANL is operated by Triad National Security, LLC, for the
National Nuclear Security Administration of U.S. Department of Energy
(Contract No. 89233218CNA000001). M.M, T.K. and T.S. were supported in part by
the U.S. Department of Energy under Contract No. DE-AC52-07NA27344 for the
topical collaboration Fission In R-process Elements (FIRE).

##  Appendix A Supplemental Data

We provide the calculated $\beta$-delayed neutron emission and $\beta$-delayed
fission probabilities in ASCII format which can be used in suitable
applications. We additionally provide a copy of Table 1 in ASCII format for
ease of parsing. All of our calculated results can be found online in
association with this submission and is released by Los Alamos with number:
LA-UR-21-31670. Additional model variations can be obtained from the authors
via email.

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