# Capturing Humans’ Mental Models of AI: An Item Response Theory Approach

Markelle Kelly<EMAIL_ADDRESS>Department of Computer Science
University of California, IrvineIrvineCaliforniaUSA ,  Aakriti Kumar
<EMAIL_ADDRESS>Department of Cognitive Sciences
University of California, IrvineIrvineCaliforniaUSA ,  Padhraic Smyth
<EMAIL_ADDRESS>Department of Computer Science
University of California, IrvineIrvineCaliforniaUSA and  Mark Steyvers
<EMAIL_ADDRESS>Department of Cognitive Sciences
University of California, IrvineIrvineCaliforniaUSA

(2023)

###### Abstract.

Improving our understanding of how humans perceive AI teammates is an
important foundation for our general understanding of human-AI teams.
Extending relevant work from cognitive science, we propose a framework based
on item response theory for modeling these perceptions. We apply this
framework to real-world experiments, in which each participant works alongside
another person or an AI agent in a question-answering setting, repeatedly
assessing their teammate’s performance. Using this experimental data, we
demonstrate the use of our framework for testing research questions about
people’s perceptions of both AI agents and other people. We contrast mental
models of AI teammates with those of human teammates as we characterize the
dimensionality of these mental models, their development over time, and the
influence of the participants’ own self-perception. Our results indicate that
people expect AI agents’ performance to be significantly better on average
than the performance of other humans, with less variation across different
types of problems. We conclude with a discussion of the implications of these
findings for human-AI interaction.

theory of mind, mental models, human-AI interaction

††journalyear: 2023††copyright: rightsretained††conference: 2023 ACM
Conference on Fairness, Accountability, and Transparency; June 12–15, 2023;
Chicago, IL, USA††booktitle: 2023 ACM Conference on Fairness, Accountability,
and Transparency (FAccT ’23), June 12–15, 2023, Chicago, IL, USA††price:
15.00††doi: 10.1145/3593013.3594111††isbn: 979-8-4007-0192-4/23/06††ccs:
Human-centered computing HCI theory, concepts and models

##  1\. Introduction

With the recent rapid growth in interest in applying AI approaches111We will
use the term “AI” in this paper to refer to the broad spectrum of techniques
that are currently (in 2023) referred to as “AI,” including models built using
machine learning in particular. to a wide variety of decision and prediction
problems, there is an increasing realization that hybrid human-AI teams will
be an important component of how AI will be deployed in practice
(kamar2016directions; lai2021science). A decision making process that includes
humans and AI can, ideally, benefit from the strengths of each
(holstein2021k12; wang2016deep; kamar2012combining). Humans can act as a
safeguard for unpredictable or undesirable behavior in AI algorithms, and can
incorporate the type of contextual information and common sense reasoning that
AI often lacks (de2020case; cheng2022welfare; chandler2022applicability).
Conversely, the use of AI in prediction and decision making enables the
processing of more complex patterns and greater volumes of data than humans
alone can accommodate, for example in tedious and time-consuming work such as
fact-checking (la2022hybrid) and in high-stakes decision making such as
diagnostic medical imaging (Tschandl2020yv).

One important goal for human-AI systems is complementarity—achieving better
performance than either the human(s) or the AI agent(s) acting independently
(bansal2021does; hemmer2021human; donahue2022unfairness). A significant body
of literature has developed that aims to understand and achieve
complementarity across a variety of hybrid human-AI settings
(wilder2020complement; pmlr-v151-bordt22a; rastogi2022unifying;
steyvers2022bayesian). One of the lessons learned from this work is that
achieving complementarity is complex in practice; for instance, AI agents that
exhibit high performance (e.g., in terms of prediction accuracy) on their own
can actually harm overall team performance if their behavior is unpredictable
for humans (bansal2019updates; bansal2021teamwork). Thus, understanding
humans’ expectations of AI is essential for optimizing team performance, and
recent work in this area has called for a better understanding of how humans
perceive AI (lai2021science; steyvers_kumar_2022).

In particular, an important (and somewhat under-studied) topic in this context
is humans’ mental models of their AI teammates. If a person is deciding
whether or not to take the advice of an AI agent, they are likely to make
better decisions if they can accurately perceive its strengths and weaknesses,
that is, if they have developed an accurate mental model of the agent
(bansal2019beyond). For example, an AI agent could have particular
“blindspots” in terms of its expertise, even when the agent’s overall
performance on a particular task is comparable to or exceeds that of a human
(attenberg2015beat; d2022spotlight). In general, a better understanding of
human mental models of AI agents can help predict phenomena such as humans’
development of appropriate trust in AI agents, human behavior in deferring to
an AI agent, and how a human-AI team will function overall.

In this paper, we aim to improve our understanding of human mental models of
AI agents by building on prior theoretical and empirical work in cognitive
science that has analyzed how people form mental models of other people. To
this end, we present a general framework for modeling human mental models of
AI agents based on item response theory (IRT). In the traditional IRT
approach, given an agent’s performance on problem sets involving a particular
task, the IRT framework is used to estimate both problem set difficulties and
agent abilities. We build on this traditional IRT methodology to propose a new
framework that models human mental models both of themselves and of others, in
terms of perceived abilities and perceived problem difficulties. This enables
us to make comparative predictions about humans’ perceptions of AI agents, in
the context of their perceptions of themselves and of other people.

To highlight the use of this framework, we conduct experiments in the context
of question-answering where participants work alongside either another person
or an AI agent (in the form of a large language model). Participants estimate
the performance of their counterpart throughout the experiment, allowing us to
make inferences about participants’ perceptions (mental models) of the
abilities of their counterparts. In our analysis, we focus on two sets of
research questions that have not yet been explored in the literature:

  1. (1)

Multidimensionality of Mental Models: What is the dimensionality of humans’
mental models in the context of assessing task performance of other agents? In
particular, do humans’ mental models of others capture multiple different
abilities or areas of expertise, or do they estimate a single notion of
ability, a “general intelligence”? We investigate the specific structure of
these perceptions of ability, in particular, the correlations between
different abilities, and how our findings differ between mental models of
other humans and those of AI agents.

  2. (2)

Role of Self-Perception: What role does a person’s self-perception play as
they develop a mental model of another agent? For example, do people estimate
how another agent’s abilties differ from their own? We consider multiple
potential relationships between (a) humans’ perceptions of their own abilities
and experienced problem difficulties and (b) their perceptions of other
agents’ abilities and problem difficulties. Again, we explore how these
relationships differ between mental models of other humans and AI agents.

To address these questions we make two main contributions in this paper.
First, we present an extensive experimental dataset involving humans and AI in
a question-answering context. This data directly captures human perceptions of
self- and other-agent performance, providing insight into questions about
humans’ mental models of other agents. Second, we introduce a theoretical
model-based framework for directly modeling and analyzing humans’ mental
models of AI agents, and we use this modeling framework to gain insight into
our experimental data.222All our code and data, including the original trivia
questions, are available at https://github.com/markellekelly/AI_mental_models.

In Section 2, we review existing work on mental models and their role in
human-AI teams. In Section 3, we describe our experimental setup, and we
include empirical findings and data analysis in Section 4. Section 5
introduces our IRT-based framework, and in Sections 6 and 7 we present the
methodology and results for our two Research Foci—Section 6 on the
dimensionality of mental models and Section 7 on the influence of self-
perception. Finally, we discuss key takeaways in Section 8 and conclude in
Section 9.

##  2\. Related Work and Background

###  2.1. Mental Models and Collaboration

In general, mental models are simplified representations of the world that
people use to process new information and make predictions (Barnes1944-cg;
Smyth1994-qo). Our work focuses in particular on mental models of one’s self,
of other people, and of AI agents, and is informed by foundational work in
cognitive science in the areas of metacognition (livingston2003metacognition;
dunlosky2008metacognition), theory of mind (astington1995theory;
frith2005theory), and theory of machine (logg2022psychology), respectively.

The importance of mental models of other agents has received considerable
emphasis in prior work on collaboration, particularly for collaboration among
teams of humans. Specifically, the goal of shared mental models (SMMs)
(scheutz2017framework; merry2021mental; schelble2022shared) necessitates that
team members are aligned in terms of their perceptions of their team,
strategy, and the task at hand. Information about the skills and knowledge of
a teammate, which we refer to as an “other mental model” or OMM, is an
important component of these SMMs, promoting effective collaboration
(matheieu2000shared; sidera2018theory; miller2019explanation;
buehler2020theory; druce2021brittle). In the context of the focus of this
paper, namely hybrid human-AI teams, prior work has found that more accurate
perceptions of AI agents tend to result in better team performance
(gero2020mental; bansal2021teamwork) and more satisfying interactions for
humans (kulesza2012tell).

###  2.2. Understanding Mental Models of AI Agents

Given the importance of human perceptions of AI agents in human-AI
collaboration, an emerging body of work aims to understand these perceptions
(westermanit2020; wangtowards2021; chen2023understanding). Based on
experiments in cooperative game settings, (gero2020mental) delineated three
different components of mental models of AI: the agent’s knowledge
distribution, local behavior, and global behavior. There is evidence that
people develop a mental model of an AI agent’s knowledge distribution based on
their own knowledge (lee2005robots), and that this perceived intelligence can
be affected by provided explanations (paleja2021utility). It has also been
shown that humans develop perceptions of AI agents’ global behavior, but that
these perceptions weaken as error boundaries become more complex and
stochastic (bansal2019beyond) and can be biased by first impressions of the
agent (nourani2021anchoring). Finally, prior research has shown that people
can predict an AI agent’s local behavior, basing initial estimates on their
own abilities (bos2019mental), and that counterfactual examples can improve
these predictions (alipour2021improving).

In this paper, we characterize mental models of AI agents in terms of the
perceived ability of the agent and the perceived problem difficulty for the
agent. While we use perceptions of behavior to estimate these quantities, all
three of (gero2020mental)’s components of mental models are relevant. A
human’s perception of the abilities and problem difficulties for an AI agent
could be influenced by how that human perceives the agent’s knowledge
distribution. Perceived ability and problem difficulty could also be used to
predict both local and global behavior.

###  2.3. Contributions in the Context of Related Work

The focus of this paper differs from prior work on mental models of AI agents
in two important ways. First, we introduce a framework that directly models
OMMs, and thus can describe them in terms of relevant latent variables and
test hypotheses about their structure. In contrast, prior work generally has
not investigated OMMs directly, instead analyzing a proxy such as team
performance or humans’ predictions of AI behavior (e.g., (bansal2019updates;
alipour2021improving)). Second, in our experiments, we collect data on
participants’ mental models of other people. Directly comparing mental models
of AI agents to those of other people provides important context and helps
determine how human-AI collaboration relates to general cognitive science
research on teaming.

More specifically, our work differs from that of (bansal2019beyond), who also
experimentally investigated the capacity of humans to understand AI agents.
Their analysis focused on the relationship between the complexity of the AI
agent’s classification error boundary and the participant’s performance on the
task. This earlier work differs from the work in this paper in that it did not
directly capture or model participants’ OMMs, nor did the approach relate OMMs
to self-perceptions or mental models of other people. Our work also differs
from (nourani2021anchoring), who investigated experimentally how participants’
mental models of an AI agent were affected by their first impressions of the
agent. Their results demonstrated that people developed more accurate mental
models (i.e., had lower error in predicting model performance) when they had a
positive first impression. However, the mental models themselves were not
analyzed in terms of their structure, and they were not compared to self-
perceptions or OMMs of people.

Figure 1. Sample questions from each trivia category.

Finally, there has been recent prior work that has proposed analytic
frameworks for human mental models of other humans. In particular,
(kumar2023differentiating) developed an IRT framework for understanding these
OMMs, which we build upon in this paper. Further, (Westby2022CollectiveII)
introduced an AI agent that learns human OMMs of their human teammates for the
purpose of improving collective intelligence in a human-human teaming context.
This is relevant to the approach we propose in this paper in that it
demonstrates how a framework such as ours could be deployed in a team
scenario. However, these prior frameworks on modeling OMMs differ from our
approach in that they do not investigate mental models of AI agents.

##  3\. Experiments

In our experiments, participants complete a multi-category trivia question-
answering task, estimating their own performance and the performance of either
another human or an AI agent. Experiments were conducted using Amazon
Mechanical Turk.

###  3.1. Task

Each participant answered 16 sets of 12 trivia questions. We used a trivia
setting because it does not require specialized knowledge (and thus is doable
by Mechanical Turk workers), is discriminative (it is very unlikely humans or
AI will answer all trivia questions correctly) (graber2019trivia), and can be
broken up into distinct categories, allowing us to directly investigate
multiple dimensions of ability.

Using a dataset of trivia questions from The Question Company, we selected
four question topics: History of Art, Video Games, Cities, and Math. Sample
questions for each topic are shown in Figure 1. Using preliminary data, we
selected these topics to achieve (1) variation between participants for each
topic, (2) variation among topics for each participant, and (3) varying
correlations between pairs of topics across participants. In addition, we
selected topics that we expected people would perceive as being different,
e.g., requiring different types of knowledge.

For each category, we created four problem sets of 12 questions each, for a
total of 16 problem sets. Participants were presented with questions in four
rounds. Each round included one problem set from each trivia category. For
each participant, we randomized the order of questions within each problem
set, the order of categories within rounds, and the order of problem sets
across rounds. Examples of possible problem set orders are shown in Figure 2.

Figure 2. Example experiment configurations. Each circle represents a problem
set, identified by its ID (number) and topic category (color). Each round has
one problem set from each category and the order of these categories is
consistent across rounds. Both this order and the order of individual problem
sets is randomized for each participant.

###  3.2. Performance Assessment

After each 12-question problem set, participants were asked to estimate their
own performance, to capture self-assessment, and the performance of another
agent, to capture other-assessment. Participants were randomly assigned to
assess either an “AI system” or another person (see Appendix C for details).

Two-thirds of the participants were randomly selected to receive feedback
regarding their performance estimation: after they provided their estimates
for a given problem set, they were shown their own actual performance, and the
actual performance of the other agent, on that problem set. The remaining one-
third of the participants did not receive this feedback. This no feedback
group was included to capture prior expectations and to understand the
strategies people use to estimate the performance of another agent in the
absence of feedback.

###  3.3. “Other” Selection

To obtain the performance data for the other humans, we first performed a
pilot study ($n=34$) using the same 16 12-question problem sets. Based on
overall accuracy, we selected the top five (“high accuracy”) and bottom five
(“low accuracy”) participants in the pilot study, where each of the top five
and bottom five were then used as “other humans” in the main experiment.
Specifically, participants in the main experiment that were assigned to the
other human condition were shown performance data from one of these 10
participants. The name of this other human was randomly chosen from a set of
ten names drawn from a random name generator (e.g., “Anna” or “Felix”).

To reduce the possibility of performance as a confounding variable, we then
matched AI performance with the other human performance on a topic-wise basis.
This was done by running several variants of UnifiedQA (khashabi2020unifiedqa)
and Zero-shot-CoT (kojima2022large), which are large language models designed
to generalize to a range of tasks. We then chose two models for each topic,
one with similar performance to that of the high accuracy humans, and another
to match the performance of the low accuracy humans. (Details on the exact
model settings used, and the final topic-wise accuracies, can be found in
Appendix C.)

We include agents with both high and low accuracy to improve the
generalization of results; mental models could differ depending on whether the
other agent has higher or lower performance than the participant.

###  3.4. Setup

203 Amazon Mechanical Turk workers, all located in the U.S., participated in
the study, which was conducted in January 2023. Participants could only
complete the experiment once (and were disqualified if they had participated
in the earlier pilot study). To ensure high-quality participation, workers
were required to be AMT Masters and have a 95% approval rating; they were paid
$7 plus a bonus of up to $2. These incentive bonuses were based on the
participants’ other-assessment performance. The experimental protocol was
approved by the University of California, Irvine Institutional Review Board.

Participants were assigned to an agent type (other human or AI agent) and
agent category (high accuracy, low accuracy, or no feedback). Participants
were evenly divided between these six experimental condition combinations.

##  4\. Overview and Empirical Analysis of Experimental Data

We begin our analysis with an investigation of initial findings from our
experimental data, exploring the participants’ perceptions of their own (self)
performance, compared to their perceptions of the performance of others (both
human and AI).

Figure 3. Histograms of differences between other agent (AI or Human)
assessments and self assessments in the no feedback condition.

###  4.1. No Feedback Condition

We consider first the no feedback experimental condition, examining
participants’ OMMs when no performance information is available about the
other agent. We analyze aggregate results for all 1072 problem set assessments
in the no feedback condition. Each participant provided a score between 0 and
12 estimating their own performance (self-assessment) and a score estimating
the performance of another (human or AI) agent (other-assessment), in terms of
how many questions were answered correctly, for each of 16 problem sets.
Figure 3 shows a histogram plot of the differences between self-assessment and
other-assessment for both AI agents and other humans. Positive differences
indicate that a participant provided a higher score for the other agent than
for themselves.

The results in Figure 3 illustrate a striking difference between participants’
assessments of AI agents and their assessments of other humans. The mean
difference of the assessed performances of AI agents (relative to self) was
$+3.0$ points, compared to $+0.8$ points for other humans (the two means are
significantly different, and both are greater than 0, at a p-value threshold
of $\alpha=0.01$ under two-sample and one-sample one-tailed t-tests,
respectively). These results indicate that on average, people believe other
agents will perform better than themselves in trivia question answering, and
that AI agents will have much higher performance than other humans. Our
findings are consistent with previous research indicating that people expect
AI to be better at objective tasks, e.g., tasks that involve retrieving
factual information, when compared to humans (castelo2019task;
logg2017theory).

###  4.2. Feedback Condition

In order to investigate how participants’ OMMs adapt over time, given feedback
about the performance of the other agent, we analyze the experimental data for
all 2176 problem set assessments under the feedback condition. Of interest in
this context is how feedback about the other agent (provided after each self-
assessment, for 16 problem sets per participant) affects people’s perceptions
of the performance of the other human or of the other AI agent.

(a) High Accuracy Agents

(b) Low Accuracy Agents

Figure 4. Mean perceived performance of other agents and self at each problem
set. The results are separated by other agents with (a) high accuracy and (b)
low accuracy. Dashed lines show corresponding values of actual performance for
reference (for self, AI agents, and other humans, averaged across all
participants and all problem sets in the given experimental condition).
Results are smoothed across problem-sets to facilitate visual comparison.

Figure 4 illustrates how participants update their assessments over rounds of
feedback, starting from problem set 1 (when no feedback has been provided yet)
up to problem set 16 (when feedback has been provided about all previous 15
problem sets). Participants adapt their assessments of other agents (both AI
and human, both high and low accuracy) quickly within the first 4 to 6 problem
sets and then change relatively slowly after that. Even after feedback from 16
problem sets, for the high accuracy agents, participants still systematically
assess AI agents as being roughly 0.6 points more accurate than the human
agents, even though the two agents were selected to have approximately the
same accuracy (dotted lines). For the low accuracy agents there is also a
consistent bias in favor of the AI agent, of roughly 0.2 points.

We also note that feedback appears to have relatively little influence in
correcting self-assessed performance (in green). For example, for the low
accuracy agents, after feedback on 16 problem sets, even though the human and
AI agents are much worse (-3 points) in terms of actual performance than the
average participant (dotted lines), participants predict similar performance
for themselves and the low accuracy agents.

In summary, the experimental results show that OMMs are quite different for AI
agents and other humans for this question-answering task. People generally
expect, a priori, that AI agents outperform humans, and it takes considerable
evidence to adjust these expectations.

##  5\. IRT Framework

In this section we briefly outline our theoretical modeling framework for
mental models, which we then use in Sections 6 and 7 to investigate Research
Foci 1 and 2 posed in the Introduction. Our framework is based on Item
Response Theory (IRT) (fox2010bayesian; van2013handbook) which is widely used
in education (bichi2018item) and psychology (thomas2019advances) for modeling
observed performance on a task in terms of latent psychological factors. In
particular, we extend the hierarchical model of knowledge assessment proposed
in (kumar2023differentiating), which focused on people’s mental models of
their own knowledge and the knowledge of other humans, but did not investigate
mental models of AI agents.

In a standard IRT setup, we have $V$ problems per problem set $j$, where each
problem set $j$ has a latent (unobserved) difficulty $d_{j}$. For each
individual $i$ and each problem set $j$, we model the number of items answered
correctly $x_{i,j}$, ranging between 0 and $V$, as:

$x_{i,j}=f(\theta_{i,j})=f(a_{i}-d_{j})$

where $a_{i}$ is the latent ability of individual $i$ and $f$ is a function
that noisily converts the latent $\theta_{i,j}$ for individual $i$ and task
$j$ into an integer-valued $x_{i,j}$. The equation above represents how the
IRT model can simulate or generate data in a forward manner. Given observed
data $x_{i,j}$, for multiple participants $i$ and problem sets $j$, we can
then make inferences in the reverse direction about the latent abilities
$a_{i}$ and problem difficulties $d_{j}$ (e.g., using standard Bayesian
sampling techniques such as Markov Chain Monte Carlo sampling).

As in (kumar2023differentiating), our IRT framework analyzes perceived
performance in addition to actual performance. In our experiments, we ask
participants to estimate their own performance on each problem set. We denote
this self-perceived data $x_{i,j}^{s}$, using an $s$ superscript to signify
self-assessment, with
$x_{i,j}^{s}=f(\theta_{i,j}^{s})=f(a_{i}^{s}-d_{j}^{s})$. Participants are
also asked to assess the performance of another human or an AI agent,
capturing their OMM. We refer to this data with an $o$ superscript, i.e.,
$x_{i,j}^{o}=f(a_{i}^{o}-d_{j}^{o})$. Thus, using the other-assessment data
$x_{i,j}^{o}$, we can then estimate $a_{i}^{o}$, the perceived ability of the
other agent, as well as $d_{j}^{o}$, the perceived difficulty of problem $j$
for the other agent, all from the perspective of participant $i$. Further
details of our modeling framework are provided in Appendix B.

##  6\. Research Focus 1: Multidimensionality of Mental Models

We next investigate whether people develop multidimensional OMMs, and more
specifically, if people develop estimates of multiple different abilities of
another agent. In the context of our experiments, we explore whether
participants assess the strengths and weaknesses of the other agents across
different topics. We are also interested in the correlational structure of
these perceived abilities: do people expect other agents to have expertise in
specific topics, or do they expect other agents to have more generalized
intelligence? Finally, we investigate how these mental models develop over
time and how these mental models differ between other humans and AI agents.

###  6.1. Methods

We address the question of multidimensional models of ability by comparing
one-dimensional and multidimensional IRT models for other-assessment on our
experimental data. Multidimensional item response theory (MIRT)
(reckase1997future; Ackerman1994-lk) models ability as a vector with multiple
dimensions. For example, a student’s performance on a standardized test might
be related to their reading, writing, and mathematical abilities. In our
framework, we use a between-items MIRT setup, which assumes that the
probability of success for a specific item is affected by only one of the
ability dimensions (Sheng2007-mi; HARTIG200957).

In particular, we associate each of the four trivia topics (history of art,
video games, cities, and math) (see Section 3.1) with an ability dimension.
Thus, each component of $\mathbf{a}_{i}^{o}$ corresponds to the perceived
ability of the other agent in a specific trivia category. To understand the
dimensionality of perceived abilities, we can then compare the one-dimensional
and multidimensional models:

$\begin{aligned} &\text{One-dimensional}\\\
&\theta_{i,j}^{o}=a_{i}^{o}-d_{j}^{o}\\\
\end{aligned}\qquad\qquad\qquad\qquad\begin{aligned}
&\text{Multidimensional}\\\
&\theta_{i,j}^{o}={a}_{ij}^{o}-d_{j}^{o}\end{aligned}$

where, in the multidimensional model, ${a}_{ij}^{o}$ is the component of the
$k$-dimensional ability vector $\mathbf{a}_{i}^{o}$ that corresponds to
problem $j$.

Further, the MIRT model estimates a $4\times 4$ matrix $\Sigma^{o}$ of latent
linear correlations between ability dimensions (see details in Appendix B).
These estimated correlations capture the structure of perceived abilities,
allowing us to quantify how participants expect these topic-wise abilities to
be related.

###  6.2. Results

(a) AI agents

(b) Other humans

(c) True correlations

Figure 5. Latent ability correlations for the four trivia categories in the
assessment of AI agents (a) and other humans (b), and for the true performance
of both humans and AI agents (c). The results are based on the feedback
condition.

####  6.2.1. Dimensionality of mental models

To determine the dimensionality of participants’ OMMs, we compare the simpler
one-dimensional model with the multidimensional model using three widely-used
statistical model selection techniques: (i) held-out log-likelihood, (ii) WAIC
score, and (iii) LOO score (vehtari2017practical; luo2017performances). 333In
the main paper, for all model comparisons, we present only the held-out log-
likelihood scores for brevity. See Appendix A for additional details; all
three scores agree in terms of which models were selected in all model
comparisons. For each participant, we hold out the final four problem sets
they completed, and compute the log-likelihood over those four sets. Average
held-out log-likelihoods across problem sets and participants are shown in
Table 1. For reference, throughout the paper we contrast the performance of
the IRT models with a discrete uniform baseline on $[0,12]$.

Table 1. Held-out log-likelihood (higher is better) of the baseline, one-dimensional, and multidimensional models for other humans and AI agents, in the feedback condition. | Humans | AI
---|---|---
Baseline | -2.56 | -2.56
One-dimensional | -1.81 | -1.82
Multidimensional | -1.74 | -1.72

The held-out log-likelihoods (and WAIC and LOO scores) indicate that the
multidimensional model is a better model for both other-human and AI agent
assessment. These results suggest that people can, and do, develop ideas of
another agent’s strengths and weaknesses; mental models are not limited to a
single ability.

####  6.2.2. Latent correlations of ability dimensions

Given that there is evidence that perceptions of others’ abilities are
multidimensional, we are interested in their specific correlational structure.
For instance, people might expect agents to have specific pockets of expertise
or to exhibit a more general intelligence (spearman1904general;
HUMPHREYS1979105).

To this end, we compare the latent correlations between trivia topics for
other assessment, shown in Figure 5. Here we include only participants in the
feedback condition, capturing differences in structure even when participants
have observed the agent’s actual performance across these topics. We also
include the true correlations across both humans and AI agents. (Note that
these true correlations are similar between other humans and AI agents; the
correlations split by agent type can be found in Appendix A.)

In summary, we find that the perceived correlations between abilities are
significantly higher when assessing AI agents than when assessing other
people. This means that participants expect the abilities of AI agents across
different trivia categories to be highly correlated with each other, much more
so than the correlations across abilities of another person. Note that this
does not contradict the earlier findings of multidimensionality in OMMs; the
perceived abilities per topic are correlated, but they are still distinct from
each other.

####  6.2.3. Mental models over time

In Section 4 we saw that other-assessment (on average) tends to stabilize,
without converging to the other agent’s true performance, over the first 4 to
6 problem sets, with some continued drift for low accuracy agents. In this
section we explore how other-assessment develops at the per-topic level, that
is, how multidimensionality develops over time. In Figure 6, other-assessment
data for low accuracy agents is plotted across rounds (where a round consists
of 4 problem sets for a participant, one problem set from each topic).
(Similar plots for high accuracy agents can be found in Appendix A.)

Figure 6 reflects that participants’ estimates of agent performance do not
converge to the true agent performance at the per-topic level. Although in
Round 4 there are differences, on average, between performance estimates for
different topics (i.e. perceptions of ability are multidimensional), the
estimates do not appear to have stabilized, and are not spread out enough to
match true per-topic performance, particularly for AI agents. Instead, they
appear to be anchored by the overall average performance (shown in black).

(a) AI agents

(b) Other humans

Figure 6. Average other-assessment for low accuracy agents in the feedback
condition. In both figures, the solid lines plot the average other-assessed
performance across the four rounds. The average true performances of the other
agent are shown as dotted horizontal lines, with the overall average in black
and per-topic averages in their respective colors.

In line with the findings of Section 6.2.2, Figure 6 also reflects that people
expect the abilities of AI agents to be highly correlated. Initially,
participants expect similar performance across history of art, video games,
and cities questions for AI agents (see the cluster of points in Round 1 in
Figure 6(a)). For other humans, the estimates are more spread out across
topics. After this first round, participants’ estimates of AI agent
performance exhibit a similar decrease across all four categories—even though
initial estimates of math and video games performance were, on average,
underestimates. This reflects high perceived correlations between abilities:
people expect AI agent scores to be closely related across topics, more so
than for other humans (Figure 6(b)).

##  7\. Research Focus 2: Influence of Self-Perception

In this section, we explore the role a person’s self-perception plays when
forming a mental model of another agent. In particular, we investigate whether
the role of self-perception in developing an OMM differs between AI agents and
other people, as well as how this changes as more information about the other
agent becomes available. Developing an understanding of the differences
between one’s own capabilities and those of an AI agent is essential for
improving cooperation (steyvers_kumar_2022), and explicitly comparing one’s
own performance with the performance of an AI agent can promote appropriate
selective reliance on the algorithm (liang2022adapting).

###  7.1. Methods

We test the fit of three different hierarchical IRT structures connecting the
true (“underlying”) performance, self-assessment, and other-assessment data.
We assume that self-assessment is (noisily) related to true performance. More
specifically, we assume that self-assessed ability is a function of a person’s
underlying ability, and self-perceived difficulty is a function of the
underlying problem difficulty. We then test the relationship between self-
assessment and other-assessment latent parameters, following a three-tier
hierarchical structure.

We refer to the three setups as undifferentiated, differentiated by ability,
and fully differentiated. Figure 7 depicts the graphical models for each of
these structures. In the undifferentiated structure, the participant uses the
same mental model to understand their own performance and the other agent’s
performance; the model does not allow for differentiation between the
participant’s own abilities and difficulties and those of the other agent. In
the differentiated by ability setup, the participant learns a difference
$\delta$ between their own ability and the ability of the other agent, but
problem difficulties remain undifferentiated. Finally, in the fully
differentiated structure, the person does not use their own ability or
difficulties to estimate those of the other agent; the self and other mental
models are independent.

(a) Undifferentiated

(b) Differentiated by ability

(c) Fully differentiated

Figure 7. Assumed structures for the three different hierarchical models
tested.

We test the fit of each of these three hierarchical models to determine which
matches most closely with the true relationships between parameters. Because
we have evidence that these mental models are multidimensional (see Section
6), we use a multidimensional structure for the underlying, self-assessed, and
other-assessed abilities; $\delta$ is a $k$-dimensional vector capturing
topic-wise ability differences.

###  7.2. Results

####  7.2.1. Role of self-perception

For both AI agents and other people, we train three different MIRT models, one
for each hypothesized hierarchical structure. To capture mental model
development over time, we evaluate models based on next-round predictions,
that is, we compute the log-likelihood for round $t$ using a model trained on
data from rounds 1 to $t-1$. These next-round log-likelihoods, under the
feedback condition, are shown in Table 2.

Table 2. Held-out next-round log-likelihoods (higher is better) for self differentiation models in the feedback condition. | Humans | AI
---|---|---
Baseline | -2.56 | -2.56
Undifferentiated | -2.69 | -3.36
Differentiated by Ability | -2.00 | -2.18
Fully Differentiated | -2.13 | -2.13

When feedback on performance is provided, the undifferentiated model has the
worst fit overall, especially for AI agents—in fact, the undifferentiated
model performs worse than random guessing (the baseline model). Thus, there is
strong evidence that people differentiate between themselves and other agents.
For perceptions of other humans, the differentiated by ability model fits the
data best, suggesting that participants perceive other humans’ abilities in
relation to their own abilities. In contrast, the fully differentiated model
best explains the perceived scores of AI agents; this provides evidence that
participants’ mental models of themselves were less relevant in developing
mental models of AI agents (in comparison to those of other humans).

####  7.2.2. Ability differential

The differentiated by ability model learns a parameter $\delta$, estimating
the differential between a person’s self-assessed ability and their perception
of the other agent’s ability. These estimates, for high-accuracy other agents,
are shown in Table 3. Higher values of $\delta$ correspond to higher perceived
abilities of the other agent (relative to self-perceived ability). For
reference, for the cities topic, $\delta=0.79$ corresponds to a 16-percentage-
point increase in the latent perceived probability of a correct answer (i.e.,
the participant perceives the other agent’s latent probability of success to
be 16 percentage points higher than their own), whereas $\delta=1.72$
corresponds to a 27-percentage-point increase in that probability.444These
interpretations are computed using the median values of self-perceived ability
and difficulty (0.54 and -0.14, respectively).

Table 3. Latent per-topic differences between self-assessed and other-assessed ability for high-accuracy agents. | Humans | AI
---|---|---
History of Art | 0.83 | 1.63
Video Games | 0.38 | 1.19
Cities | 0.79 | 1.72
Math | 0.75 | 1.69

The variation of $\delta$s between agent types suggests substantial
differences between perceptions of other humans and AI agents. Specifically,
the values of $\delta$ are larger for AI agents than for other people, despite
the true actual abilities of the AI agents and other people being very similar
(by design) on each topic. This aligns with the findings from Section 4, in
particular, that participants generally expect AI agents to perform at a
significantly higher level than (other) humans.

####  7.2.3. Relationship with self-perception

In this section we investigate the overall relationship between self- and
other-assessment in the absence of feedback. Note that these results, without
feedback, are solely focused on each person’s perception of the other agent,
as a function of their self-perception; the true performance of the other
agent does not play any role since no feedback is provided. Figure 8 compares
self- and other-assessment scores in the no feedback condition.

Figure 8. Relationship between self-assessed score and other-assessed score in
the no feedback condition. The dotted black diagonal line represents equality
between self- and other-assessment.

Figure 8 reflects a positive correlation between self-assessment and other-
assessment when evaluating other humans. In comparison, the relationship
between self-assessed performance and the perceived performance of AI agents
is less pronounced. When participants perceive that they have done poorly on a
particular problem set, they accordingly reduce their expectations of other
humans. In contrast, participants predict similar scores for AI agents across
self-assessed scores. Overall, this aligns with our findings from Sections
7.2.1 and 7.2.2, namely, that participants expect the performance of AI agents
to be more different from their own performance, compared to the performance
of other humans.

We also note that for lower self-assessed scores, participants expect other
agents to score higher than them, while for higher self-assessed scores,
other-assessment falls slightly below self-assessment (i.e., below the
diagonal). This observation aligns with existing work in cognitive science,
which has found that people believe they are better than others at easier
tasks, but worse than others on more difficult tasks (moore2007overconfidence;
dunning2011dunning). Our experiments suggest that this finding, in particular
that people believe they are worse than others on more difficult tasks, may be
especially pronounced when the other is an AI agent.

##  8\. Discussion

First, we outline three key takeaways from our experiments and discuss how
they might affect human-AI collaboration.

##### 1\. Over-differentiation of AI from self

We observe that mental models of an AI agent’s ability are highly
differentiated from self-perceived ability (e.g., Figures 3 and 4 and Table
3). On average, participants expect AI agents to perform very differently from
themselves, especially in the absence of feedback. This bias could lead to
under- or over-reliance on an AI agent in a team setting; additional work is
needed to better understand and counteract it.

##### 2\. “General intelligence” bias

While we observe that people can pick up on the strengths and weaknesses of AI
agents (see also (bansal2019beyond; nourani2021anchoring)), in our experiments
participants expect a more unified, single intelligence from an AI agent than
they do another person, even after observing evidence to the contrary. In
particular, the other agents in our experiments have near-identical inter-
topic performance variations, but participants perceive much higher
correlations between topics for AI agents (see Section 6.2.2). This can look
like a failure to recognize how well the AI agent performs in its strongest
areas, and how poorly it performs in its weakest areas (see Section 6.2.3),
potentially resulting in over- or under-use of an AI decision making aid.
Again, further research is needed to determine the extent of this bias and how
it might be counteracted in human-AI teams.

##### 3\. Incomplete development of mental models

In our experiments, participants’ other-assessments did not fully converge to
AI agents’ true performances, even given feedback (see Figure 4). This
extended to the per-topic level; participants did not accurately estimate AI
agents’ strengths and weaknesses after 16 rounds of feedback (see Figure 6).
This phenomenon could serve as motivation to give the teammates of an AI agent
extra information to aid in OMM development, e.g. a “primer” or onboarding
process (cai2019uncovering) or prediction explanations (paleja2021utility;
alipour2021improving).

Looking ahead, we believe our modeling framework could be useful for capturing
people’s OMMs in the context of hybrid human-AI teams. In this paper, we
investigate how a person perceives an AI agent, in terms of their different
abilities, or strengths and weaknesses, and the difficulties of specific
problems for the agent. Our findings and framework could help predict when a
person is likely to defer to an AI agent (and thus help predict overall team
performance) and identify biases that could lead to over- or under-use of the
agent.

##### Limitations

Our experiments and results are limited to a single task and setting, and
involve only Amazon Mechanical Turk workers, who are not necessarily
knowledgeable on the task or on AI in general. In future work, it will be
important to investigate these research questions across other settings, e.g.,
for image classification or other tasks, and with other users, e.g., human
experts who already interact with an AI agent on a regular basis.

##  9\. Conclusion

In this paper, we present an experimental dataset capturing participants’
mental models of themselves, other humans, and AI agents and introduce a
framework for analyzing these mental models. Our findings indicate that (1)
people tend to over-estimate the performance of AI agents relative to their
own performance; (2) people expect the different abilities of AI agents to be
highly correlated, even after observing evidence otherwise; and (3) these OMMs
fail to develop completely, particularly in capturing agents’ different
strengths and weaknesses. We anticipate that our modeling framework, and these
findings, will be useful in both understanding and improving interaction in
hybrid human-AI teams.

###### Acknowledgements.

This research was supported by NSF under awards 1900644 and 1927245, by the
Irvine Initiative in AI, Law, and Society, and by the Hasso Plattner Institute
(HPI) Research Center in Machine Learning and Data Science at the University
of California, Irvine.

##  Appendix A Additional Results and Figures

###  A.1. From Section 6

Here we include other metrics (WAIC and LOO scores) for the one-dimensional
and multidimensional IRT models. Scores are computed on the log-score scale,
i.e., a higher score is better. Note there are no standard errors for the
baseline model, as no parameters are learned.

Table 4. WAIC scores and standard errors of one-dimensional and multidimensional models for other humans and AI agents, across both feedback conditions. | Humans | AI
---|---|---
Baseline | -4227.04 | -4144.96
One-dimensional | -3404.6 $\pm$ 28.0 | -3348.5 $\pm$ 28.7
Multidimensional | -3024.9 $\pm$ 33.7 | -2855.4 $\pm$ 40.3
Table 5. LOO scores and standard errors of one-dimensional and multidimensional models for other humans and AI agents, across both feedback conditions. | Humans | AI
---|---|---
Baseline | -4227.04 | -4144.96
One-dimensional | -3405.1 $\pm$ 28.0 | -3349.1 $\pm$ 28.7
Multidimensional | -3031.5 $\pm$ 33.9 | -2859.8 $\pm$ 40.4

(a) AI agents

(b) Other humans

Figure 9. True ability correlations for per-topic performances, split by agent
type. These are the latent correlations between abilities, computed from the
true other agent performance data.

(a) AI agents

(b) Other humans

Figure 10. Average other-assessment for high accuracy agents (AI agents in
(a), other humans in (b)) in the feedback condition. In both figures, the
solid lines plot the average other-assessed performance across all four
rounds. The average true performances of the other agent are shown as dotted
horizontal lines, with the overall average in black and per-topic averages in
their respective colors.

###  A.2. From Section 7

Here we include additional scores, averaged over each participant, for each of
the three hierarchical models.

Table 6. Held-out next-round log-likelihoods (higher is better) for self differentiation models in the no feedback condition | Humans | AI
---|---|---
Baseline | -2.56 | -2.56
Undifferentiated | -2.68 | -5.80
Differentiated by Ability | -1.89 | -1.72
Fully Differentiated | -2.07 | -1.77
Table 7. Average LOO scores for self differentiation models, across both feedback conditions | Humans | AI
---|---|---
Baseline | -41.04 | -41.04
Undifferentiated | -42.58 | -67.89
Differentiated by Ability | -29.49 | -30.29
Fully Differentiated | -30.58 | -28.82
Table 8. Average WAIC scores for self differentiation models, across both feedback conditions | Humans | AI
---|---|---
Baseline | -41.04 | -41.04
Undifferentiated | -42.58 | -67.89
Differentiated by Ability | -29.38 | -30.17
Fully Differentiated | -29.94 | -28.25

##  Appendix B IRT Model Details and Priors

All models were fit using Stan. The underlying model (B.1) was used to model
other-assessment data $x_{i,j}^{o}$ in Section 6 as well as true performance
$x_{i,j}$ in the top level of the hierarchy in Section 7. The second level of
the hierarchy was modeled by the self-assessment model (B.2). The three other-
assessment models are detailed in B.3.

The underlying models were run with 800 warm-up iterations, 1500 samples, and
three chains. Each of the self-assessment and other-assessment models (one for
each participant) were run with 600 warm-up iterations, 1000 samples, and 2
chains. These hyperparameters were chosen based on chain convergence plots.

To convert latent scores $\theta_{i,j}$ to discrete scores, we used:

$\displaystyle p_{i,j}=\frac{1}{1+\text{exp}(-\theta_{i,j})}$ $\displaystyle
x_{i,j}\sim\text{OrderedProbit}(p_{i,j},v,\sigma)$

where $v$ is an array of cutoff points for conversion to discrete scores. We
used 13 equally-spaced bins between 0 and 1 (converting into a score between 0
and 12, the number of questions in each problem set).

###  B.1. Underlying Model

$\begin{aligned} &\textbf{Multidimensional}\\\
&x_{i,j}=f(\lambda_{j}\cdot\mathbf{a}_{i},d_{j},\sigma)\\\
&\sigma\sim\text{Cauchy}(0,2)\\\ &d_{j}\sim\text{N}(\mu_{d},\sigma_{d})\\\
&\qquad\mu_{d}\sim\text{N}(0,2)\\\ &\qquad\sigma_{d}\sim\text{Cauchy}(0,5)\\\
&\mathbf{a}_{i}\sim\text{MVN}(\mathbf{0},\Sigma_{L})\\\
&\qquad\Sigma_{L}=L_{\text{std}}\cdot L_{\Omega}\\\ &\qquad
L_{\Omega}\sim\text{lkj\\_corr\\_cholesky}(1)\\\ &\qquad
L_{\text{std}}\sim\text{N}(0,2.5)\\\
\end{aligned}\qquad\qquad\qquad\qquad\begin{aligned} &\textbf{One-
dimensional}\\\ &Y_{i,j}=f(a_{i},d_{j},\sigma)\\\
&\sigma\sim\text{Cauchy}(0,2)\\\ &d_{j}\sim\text{N}(\mu_{d},\sigma_{d})\\\
&\qquad\mu_{d}\sim\text{N}(0,2)\\\ &\qquad\sigma_{d}\sim\text{Cauchy}(0,5)\\\
&a_{i}\sim\text{N}(0,1)\\\ &\qquad\\\ &\qquad\\\ &\qquad\\\ \end{aligned}$

###  B.2. Self-Assessment Model

$\begin{aligned} &\textbf{Multidimensional}\\\
&x^{s}_{i,j}=f(\lambda_{j}\cdot\mathbf{a}^{s}_{i},d^{s}_{j},\sigma^{s})\\\
&\sigma^{s}\sim\text{Cauchy}(0,2)\\\ &d^{s}_{j}\sim\text{N}(\gamma\cdot
d_{j}+\Lambda,\sigma_{d,i})\\\ &\qquad\gamma\sim\text{N}(0,1)\\\
&\qquad\Lambda\sim\text{N}(0,1)\\\
&\qquad\sigma_{d,i}\sim\text{Cauchy}(0,2)\\\
&a^{s}_{i,k}\sim\text{N}(a_{i,k},\sigma_{a,i})\\\
&\qquad\sigma_{a,i}\sim\text{Cauchy}(0,2)\\\
\end{aligned}\qquad\qquad\qquad\qquad\begin{aligned} &\textbf{One-
dimensional}\\\ &x^{s}_{i,j}=f(a^{s}_{i},d^{s}_{j},\sigma^{s})\\\
&\sigma^{s}\sim\text{Cauchy}(0,2)\\\ &d^{s}_{j}\sim\text{N}(\gamma\cdot
d_{j}+\Lambda,\sigma_{d,i})\\\ &\qquad\gamma\sim\text{N}(0,1)\\\
&\qquad\Lambda\sim\text{N}(0,1)\\\
&\qquad\sigma_{d,i}\sim\text{Cauchy}(0,2)\\\
&a^{s}_{i}\sim\text{N}(a_{i},\sigma_{a,i})\\\
&\qquad\sigma_{a,i}\sim\text{Cauchy}(0,2)\\\ \end{aligned}$

###  B.3. Other-Assessment Models

####  B.3.1. Undifferentiated

$\begin{aligned} &\textbf{Multidimensional}\\\
&x^{o}_{i,j}=f(\lambda_{j}\cdot\mathbf{a}^{s}_{i},d^{s}_{j},\sigma^{s})\\\
&\text{Input data: }\mathbf{a}^{s}_{i},d^{s}_{j},\sigma^{s}\\\
\end{aligned}\qquad\qquad\qquad\qquad\begin{aligned} &\textbf{One-
dimensional}\\\ &x^{o}_{i,j}=f(a^{s}_{i},d^{s}_{j},\sigma^{s})\\\ &\text{Input
data: }a^{s}_{i},d^{s}_{j},\sigma^{s}\\\ \end{aligned}$

####  B.3.2. Differentiated by Ability

$\begin{aligned} &\textbf{Multidimensional}\\\
&x^{o}_{i,j}=f(\lambda_{j}\cdot\mathbf{a}^{o}_{i},d^{s}_{j},\sigma^{s})\\\
&a^{o}_{i,k}=a^{s}_{i,k}+\delta_{i,k}\\\
&\qquad\delta_{i,k}\sim\text{N}(0,1)\\\ &\text{Input data:
}\mathbf{a}^{s}_{i},d^{s}_{j},\sigma^{s}\\\
\end{aligned}\qquad\qquad\qquad\qquad\begin{aligned} &\textbf{One-
dimensional}\\\ &x^{o}_{i,j}=f(a^{o}_{i},d^{s}_{j},\sigma^{s})\\\
&a^{o}_{i}=a^{s}_{i}+\delta_{i}\\\
&\qquad\delta_{i}\sim\text{N}(\mu_{\delta_{i}},\sigma_{\delta})\\\
&\qquad\mu_{\delta_{i}}\sim\text{N}(0,1)\\\
&\qquad\sigma_{\delta}\sim\text{Cauchy}(0,2)\\\ &\text{Input data:
}a^{s}_{i},d^{s}_{j},\sigma^{s}\\\ \end{aligned}$

####  B.3.3. Fully Differentiated

$\begin{aligned} &\textbf{Multidimensional}\\\
&x^{o}_{i,j}=f(\lambda_{j}\cdot\mathbf{a}^{o}_{i},d^{o}_{j},\sigma^{s})\\\
&d^{o}_{j}\sim\text{N}(\mu^{o}_{d},\sigma^{o}_{d})\\\
&\qquad\mu^{o}_{d}\sim\text{N}(0,2)\\\
&\qquad\sigma^{o}_{d}\sim\text{Cauchy}(0,5)\\\
&a^{o}_{i,k}\sim\text{N}(0,1)\\\ &\text{Input data: }\sigma^{s}\\\
\end{aligned}\qquad\qquad\qquad\qquad\begin{aligned} &\textbf{One-
dimensional}\\\ &x^{o}_{i,j}=f(a^{o}_{i},d^{o}_{j},\sigma^{s})\\\
&d^{o}_{j}\sim\text{N}(\mu^{o}_{d},\sigma^{o}_{d})\\\
&\qquad\mu^{o}_{d}\sim\text{N}(0,2)\\\
&\qquad\sigma^{o}_{d}\sim\text{Cauchy}(0,5)\\\ &a^{o}_{i}\sim\text{N}(0,1)\\\
&\text{Input data: }\sigma^{s}\\\ \end{aligned}$

##  Appendix C Experiment Setup

The instructions provided to participants at the beginning of the experiment
are shown in Figures 11 through 14. The second page of instructions depends on
whether the participant was assigned another human or an AI agent to assess.

Figure 11. First page of instructions (for all participants). Figure 12.
Second page of instructions shown to participants in the “other human”
condition. Figure 13. Second page of instructions shown to participants in the
“AI agent” condition. Figure 14. Third page of instructions (for all
participants). Figure 15. Example performance estimation questions. In this
example, the participant estimates their own performance and the performance
of another person, Sonia (left). They receive feedback about their own, and
Sonia’s, actual performance (right).

To match AI agent performance to that of human performance, we first chose the
five highest-accuracy and five lowest-accuracy participants in our pilot
study. We then ran six models: three versions of UnifiedQA (base, large, and
3B, see documentation) and three versions of Zero-shot-CoT (one with GPT3-XL
version 1 and method “zero_shot,” two with method “zero_shot_cot” and GPT3-XL
versions 1 and 3, respectively; see documentation). Testing this wide variety
of models enabled us to match accuracy relatively closely to the humans. The
models used for each topic are shown in Table 9.

Table 9. Models used for AI Agent | Topic | Human | Model | UQA (base) | UQA (large) | UQA (3B) | ZS (001) | ZSC (001) | ZSC (003)
---|---|---|---|---|---|---|---|---|---
High accuracy | Art | 75.4 | 77.1 |  |  |  | ✓ |  |
| Video Games | 73.8 | 72.9 |  |  |  |  | ✓ |
| Cities | 71.3 | 70.8 |  |  | ✓ |  |  |
| Math | 90.0 | 89.6 |  |  |  |  |  | ✓
Low accuracy | Art | 30.0 | 29.2 | ✓ |  |  |  |  |
| Video Games | 51.3 | 52.1 |  | ✓ |  |  |  |
| Cities | 33.8 | 31.3 | ✓ |  |  |  |  |
| Math | 60.0 | 62.5 |  |  |  | ✓ |  |


# Shape Coexistence at Zero Spin in 64Ni Driven by the Monopole Tensor
Interaction

N. Mărginean111Corresponding author<EMAIL_ADDRESS>Horia Hulubei
National Institute of Physics and Nuclear Engineering - IFIN HH, Bucharest,
077125, Romania  D. Little  Dept. of Physics and Astronomy, University of
North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3255, USA
Triangle Universities Nuclear Laboratory, Duke University, Durham, North
Carolina 27708-2308, USA  Y. Tsunoda  Center for Nuclear Study, The University
of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  S. Leoni222Corresponding
author<EMAIL_ADDRESS>Dipartimento di Fisica, Universit$\grave{a}$
degli Studi di Milano, I-20133 Milano, Italy  INFN sezione di Milano via
Celoria 16, 20133, Milano, Italy  R. V. F. Janssens333Corresponding author:
<EMAIL_ADDRESS>Dept. of Physics and Astronomy, University of North
Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3255, USA  Triangle
Universities Nuclear Laboratory, Duke University, Durham, North Carolina
27708-2308, USA  B. Fornal444Corresponding author<EMAIL_ADDRESS>Institute of Nuclear Physics, PAN, 31-342 Kraków, Poland  T.
Otsuka555Corresponding author<EMAIL_ADDRESS>Department of
Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
RIKEN Nishina Center, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan  KU Leuven,
Instituut voor Kern- en Stralingsfysica, 3000 Leuven, Belgium  C. Michelagnoli
Institut Laue-Langevin (ILL), 71 Avenue des Martyrs, 38042 Grenoble, France
L. Stan  Horia Hulubei National Institute of Physics and Nuclear Engineering -
IFIN HH, Bucharest, 077125, Romania  F. C. L. Crespi  Dipartimento di Fisica,
Universit$\grave{a}$ degli Studi di Milano, I-20133 Milano, Italy  INFN
sezione di Milano via Celoria 16, 20133, Milano, Italy  C. Costache  Horia
Hulubei National Institute of Physics and Nuclear Engineering - IFIN HH,
Bucharest, 077125, Romania  R. Lica  Horia Hulubei National Institute of
Physics and Nuclear Engineering - IFIN HH, Bucharest, 077125, Romania  M.
Sferrazza  Département de Physique, Université libre de Bruxelles, B-1050
Bruxelles, Belgium  A. Turturica  Horia Hulubei National Institute of Physics
and Nuclear Engineering - IFIN HH, Bucharest, 077125, Romania  A. D.
Ayangeakaa  Department of Physics, United States Naval Academy, Annapolis,
Maryland 21402, USA  K. Auranen666Present address: Department of Physics,
University of Jyvaskyla, P.O. Box 35, 40014, Jyvaskyla, Finland Physics
Division, Argonne National Laboratory, Argonne, Illinois 60439, USA  M. Barani
Dipartimento di Fisica, Universit$\grave{a}$ degli Studi di Milano, I-20133
Milano, Italy  INFN sezione di Milano via Celoria 16, 20133, Milano, Italy
Institut Laue-Langevin (ILL), 71 Avenue des Martyrs, 38042 Grenoble, France
P. C. Bender  Department of Physics and Applied Physics, University of
Massachusetts Lowell, Lowell, Massachusetts 01854, USA  S. Bottoni
Dipartimento di Fisica, Universit$\grave{a}$ degli Studi di Milano, I-20133
Milano, Italy  INFN sezione di Milano via Celoria 16, 20133, Milano, Italy  M.
Boromiza  Horia Hulubei National Institute of Physics and Nuclear Engineering
- IFIN HH, Bucharest, 077125, Romania  A. Bracco  Dipartimento di Fisica,
Universit$\grave{a}$ degli Studi di Milano, I-20133 Milano, Italy  INFN
sezione di Milano via Celoria 16, 20133, Milano, Italy  S. Călinescu  Horia
Hulubei National Institute of Physics and Nuclear Engineering - IFIN HH,
Bucharest, 077125, Romania  C. M. Campbell  Nuclear Science Division, Lawrence
Berkeley National Laboratory, Berkeley, California 94720, USA  M. P. Carpenter
Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
P. Chowdhury  Department of Physics and Applied Physics, University of
Massachusetts Lowell, Lowell, Massachusetts 01854, USA  M. Ciemała  Institute
of Nuclear Physics, PAN, 31-342 Kraków, Poland  N. Cieplicka-Oryǹczak
Institute of Nuclear Physics, PAN, 31-342 Kraków, Poland  D. Cline  Department
of Physics and Astronomy, University of Rochester, Rochester, New York 14627,
USA  C. Clisu  Horia Hulubei National Institute of Physics and Nuclear
Engineering - IFIN HH, Bucharest, 077125, Romania  H. L. Crawford  Nuclear
Science Division, Lawrence Berkeley National Laboratory, Berkeley, California
94720, USA  I. E. Dinescu  Horia Hulubei National Institute of Physics and
Nuclear Engineering - IFIN HH, Bucharest, 077125, Romania  D. Filipescu  Horia
Hulubei National Institute of Physics and Nuclear Engineering - IFIN HH,
Bucharest, 077125, Romania  N. Florea  Horia Hulubei National Institute of
Physics and Nuclear Engineering - IFIN HH, Bucharest, 077125, Romania  A. M.
Forney  Department of Chemistry and Biochemistry, University of Maryland,
College Park, Maryland 20742, USA  S. Fracassetti777Present address: KU
Leuven, Instituut voor Kern- en Stralingsfysica, 3000 Leuven, Belgium
Dipartimento di Fisica, Universit$\grave{a}$ degli Studi di Milano, I-20133
Milano, Italy  INFN sezione di Milano via Celoria 16, 20133, Milano, Italy  A.
Gade  Department of Physics and Astronomy, Michigan State University, East
Lansing, Michigan 48824, USA  National Superconducting Cyclotron Laboratory,
Michigan State University, East Lansing, Michigan 48824, USA  I. Gheorghe
Horia Hulubei National Institute of Physics and Nuclear Engineering - IFIN HH,
Bucharest, 077125, Romania  A. B. Hayes  National Nuclear Data Center,
Brookhaven National Laboratory, Upton, New York 11973-5000, USA  I. Harca
Horia Hulubei National Institute of Physics and Nuclear Engineering - IFIN HH,
Bucharest, 077125, Romania  J. Henderson  Lawrence Livermore National
Laboratory, Livermore, California 94550, USA  A. Ionescu  Horia Hulubei
National Institute of Physics and Nuclear Engineering - IFIN HH, Bucharest,
077125, Romania  Ł. W. Iskra  INFN sezione di Milano via Celoria 16, 20133,
Milano, Italy  M. Jentschel  Institut Laue-Langevin (ILL), 71 Avenue des
Martyrs, 38042 Grenoble, France  F. Kandzia  Institut Laue-Langevin (ILL), 71
Avenue des Martyrs, 38042 Grenoble, France  Y. H. Kim  Institut Laue-Langevin
(ILL), 71 Avenue des Martyrs, 38042 Grenoble, France  F. G. Kondev  Physics
Division, Argonne National Laboratory, Argonne, Illinois 60439, USA  G.
Korschinek  Technische Universitat Munchen, Munchen, Germany  U. Köster
Institut Laue-Langevin (ILL), 71 Avenue des Martyrs, 38042 Grenoble, France
Krishichayan  Triangle Universities Nuclear Laboratory, Duke University,
Durham, North Carolina 27708-2308, USA  M. Krzysiek  Institute of Nuclear
Physics, PAN, 31-342 Kraków, Poland  T. Lauritsen  Physics Division, Argonne
National Laboratory, Argonne, Illinois 60439, USA  J. Li888Present address:
Department of Physics and Astronomy and National Superconducting Cyclotron
Laboratory, Michigan State University, East Lansing, Michigan 48824, USA
Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
R. Mărginean  Horia Hulubei National Institute of Physics and Nuclear
Engineering - IFIN HH, Bucharest, 077125, Romania  E. A. Maugeri  Paul
Scherrer Institut, Villigen, Switzerland  C. Mihai  Horia Hulubei National
Institute of Physics and Nuclear Engineering - IFIN HH, Bucharest, 077125,
Romania  R. E. Mihai  Horia Hulubei National Institute of Physics and Nuclear
Engineering - IFIN HH, Bucharest, 077125, Romania  A. Mitu  Horia Hulubei
National Institute of Physics and Nuclear Engineering - IFIN HH, Bucharest,
077125, Romania  P. Mutti  Institut Laue-Langevin (ILL), 71 Avenue des
Martyrs, 38042 Grenoble, France  A. Negret  Horia Hulubei National Institute
of Physics and Nuclear Engineering - IFIN HH, Bucharest, 077125, Romania  C.
R. Niţă  Horia Hulubei National Institute of Physics and Nuclear Engineering -
IFIN HH, Bucharest, 077125, Romania  A. Olăcel  Horia Hulubei National
Institute of Physics and Nuclear Engineering - IFIN HH, Bucharest, 077125,
Romania  A. Oprea  Horia Hulubei National Institute of Physics and Nuclear
Engineering - IFIN HH, Bucharest, 077125, Romania  S. Pascu  Horia Hulubei
National Institute of Physics and Nuclear Engineering - IFIN HH, Bucharest,
077125, Romania  C. Petrone  Horia Hulubei National Institute of Physics and
Nuclear Engineering - IFIN HH, Bucharest, 077125, Romania  C. Porzio
Dipartimento di Fisica, Universit$\grave{a}$ degli Studi di Milano, I-20133
Milano, Italy  INFN sezione di Milano via Celoria 16, 20133, Milano, Italy  D.
Rhodes  Department of Physics and Astronomy, Michigan State University, East
Lansing, Michigan 48824, USA  National Superconducting Cyclotron Laboratory,
Michigan State University, East Lansing, Michigan 48824, USA  D. Seweryniak
Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
D. Schumann  Paul Scherrer Institut, Villigen, Switzerland  C. Sotty  Horia
Hulubei National Institute of Physics and Nuclear Engineering - IFIN HH,
Bucharest, 077125, Romania  S. M. Stolze  Physics Division, Argonne National
Laboratory, Argonne, Illinois 60439, USA  R. Şuvăilă  Horia Hulubei National
Institute of Physics and Nuclear Engineering - IFIN HH, Bucharest, 077125,
Romania  S. Toma  Horia Hulubei National Institute of Physics and Nuclear
Engineering - IFIN HH, Bucharest, 077125, Romania  S. Ujeniuc  Horia Hulubei
National Institute of Physics and Nuclear Engineering - IFIN HH, Bucharest,
077125, Romania  W. B. Walters  Department of Chemistry and Biochemistry,
University of Maryland, College Park, Maryland 20742, USA  C. Y. Wu  Lawrence
Livermore National Laboratory, Livermore, California 94550, USA  J. Wu
Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
S. Zhu  National Nuclear Data Center, Brookhaven National Laboratory, Upton,
New York 11973-5000, USA  S. Ziliani  Dipartimento di Fisica,
Universit$\grave{a}$ degli Studi di Milano, I-20133 Milano, Italy  INFN
sezione di Milano via Celoria 16, 20133, Milano, Italy

###### Abstract

The low-spin structure of the semi-magic 64Ni nucleus has been considerably
expanded: combining four experiments, several 0+ and 2+ excited states were
identified below 4.5 MeV, and their properties established. The Monte Carlo
shell model accounts for the results and unveils an unexpectedly-complex
landscape of coexisting shapes: a prolate 0+ excitation is located at a
surprisingly high energy (3463 keV), with a collective 2+ state 286 keV above
it, the first such observation in Ni isotopes. The evolution in excitation
energy of the prolate minimum across the neutron $N=40$ sub-shell gap
highlights the impact of the monopole interaction and its variation in
strength with $N$.

In mesoscopic systems with many degrees of freedom (e.g., molecules, atomic
nuclei, etc.), deformation is a common phenomenon resulting from symmetry
breaking associated with quantum-mechanical states (practically) degenerate in
energy. The concept was originally introduced by Jahn and Teller who
demonstrated that, in non-linear molecules, coupling between degenerate
electronic states and collective vibrations can destroy the system’s original
symmetry JT37 . In atomic nuclei, the appearance of ellipsoidal deformation is
a realization of this effect with specific superpositions of spherical single-
particle states (e.g., Nilsson model BM-2 ) induced by deformed mean
potentials (mean-field approaches) Rei84 ; Naz94 , or by quadrupole
correlations (shell-model descriptions) Uts12 ; Ots19 , highlighting the
interplay between single-particle states and collective modes.

Among the features associated with deformation figures shape coexistence: a
phenomenon ubiquitous throughout the nuclear chart Woo92 ; Hey11  where
different shapes are present at comparable excitation energies. A clear-cut
signature for its occurrence in even-even systems is the presence of low-lying
0+ excitations residing in local minima of the nuclear potential energy
surface (PES) in deformation space.

Over the past two decades, studies of neutron-rich nuclei have highlighted the
contribution of the monopole component of the tensor force to the evolution of
the structure of exotic nuclei Ots10 ; Ots19 , especially in the change in
single-particle (or shell) structure with neutron excess, with some magic
numbers vanishing and other, new ones appearing Ots20 . Besides such single-
particle properties, its role in driving the nuclear shape was subsequently
identified Tsu14 ; Ots16 ; Tog16 ; Mar18 ; Sel19 ; Ots19 , specifically in
connection with shape coexistence.

Neutron-rich even 28Ni isotopes are a noteworthy example of shape coexistence:
68Ni exhibits a spherical ground state, while the 1605-keV, 0${}^{+}_{2}$ and
2511-keV, 0${}^{+}_{3}$ levels are understood as oblate and prolate
excitations Tsu14 ; Cri16 ; Suc14 ; Rec13 ; Fla15 ; Mue00 ; Chi12 . In 70Ni, a
prolate 0${}^{+}_{2}$ state is found at 1567 keV above the spherical ground
state Pro15 . Finally, four 0+ levels are known below a 4 MeV excitation
energy in 66Ni, where the ground state and the 2664-keV 0${}^{+}_{3}$ level
are interpreted as spherical, while the 0${}^{+}_{2}$, 2445-keV and the
0${}^{+}_{4}$, 2945-keV states are of oblate and prolate character Leo17 .

The present paper focuses on 64Ni, the heaviest, stable nucleus in the
isotopic chain, and reveals a complex landscape in deformation that was not
anticipated by mean-field calculations Hil07 ; Ber91 ; Dec80 ; Mol04 ; Mol09 ,
which predicted a single, spherical minimum, the development of a secondary
prolate one occurring only in heavier isotopes. In contrast, recent Monte
Carlo Shell-model (MCSM) calculations Tsu14  indicate coexistence of spherical
and deformed oblate and prolate 0+ states already in 62,64Ni. This coexistence
originates from the action of the monopole tensor force which shifts effective
single-particle energies, already at the valley of stability, weakening
resistance against deformation Tsu14 ; Ots16 ; Ots19 . This Letter reports
extensive tests of these MCSM predictions. Besides the customary data on level
energies, spins and parities, comparisons also extend to state lifetimes,
transition probabilities, branching and multipole mixing ratios. Evidence is
given for three coexisting shapes, with the prolate 0+ state at $\sim$3.5 MeV,
an excitation energy reproduced only by MCSM calculations incorporating the
monopole tensor interaction. To achieve the required experimental sensitivity,
four experiments; i.e., transfer reactions, neutron capture, Coulomb
excitation and nuclear resonance fluorescence had to be carried out at the
IFIN-HH Tandem Laboratory (Bucharest, Romania), the Institut Laue-Langevin
(ILL, Grenoble, France), the Argonne National Laboratory (ANL, Argonne, USA)
and the Triangle Universities Nuclear Laboratory (TUNL, Duke Univ., USA),
respectively. Results from the first three techniques are reported below (for
the last one, see Ref. Fri21 ).

Prior to this work, two excited 0+ states had been identified in 64Ni, at 2867
and 3026 keV, following $\beta$-decay and (t,p)-reaction studies Paw12 ; Dar71
. These levels were subsequently confirmed in deep-inelastic reaction
measurements Bro12 , and their $\gamma$ decay to the 1346-keV 2${}^{+}_{1}$
state was observed. Candidates for other, higher-lying 0+ levels have also
been reported Har92 .

At IFIN-HH, 64Ni was populated by 62Ni(18O,16O) two-neutron (2n) transfer on a
5 mg/cm2-thick target, with a 39-MeV beam energy; i.e., just below the Coulomb
barrier in order to reduce competition from fusion-evaporation. Transitions of
interest were measured with ROSPHERE, an array of 25 Compton-suppressed HPGe
detectors with $\sim$2$\%$ total efficiency at 1.3 MeV Buc16 . The same
reaction, but with a thinner, 0.92 mg/cm2 target and a 5 mg/cm2-Ta stopper,
placed at six distances from the target (10, 17, 25, 45, 100 and 150 $\mu$m),
was employed for lifetime measurements via the recoil-distance technique. The
sub-barrier one-proton (1p) transfer reaction 65Cu(11B,12C)64Ni at 26 MeV on a
7.22 mg/cm2-thick target was performed as well Stan20 .

The coincidence spectrum, from the thick target 2n-transfer reaction, gated on
the 1346-keV 2${}^{+}_{1}\rightarrow$0${}^{+}_{1}$ 64Ni transition, is
presented in Fig. 1. All visible lines correspond to 64Ni transitions which
deexcite states below 4.6 MeV, with spin up to 7- Bro12 . The 1521- and
1680-keV $\gamma$ rays, depopulating the known 0${}^{+}_{2}$ and 0${}^{+}_{3}$
states, are clearly visible. Their respective half-lives, as measured in the
plunger experiment, are $T_{1/2}$=1.4(6) and 3.6(1.2) ps (inset of Fig. 1)
Stan20 . A search for transitions from higher-lying candidate 0+ states proved
inconclusive in the IFIN-HH data sets.

Figure 1: 2n-transfer reaction: partial spectrum in coincidence with the 64Ni
$2^{+}_{1}\rightarrow$0${}^{+}_{1}$ transition (thick-target data). Inset:
decay curves from the plunger experiment for the 1521- and 1680-keV
transitions linking the 0${}^{+}_{2}$ and 0${}^{+}_{3}$ levels to the
2${}^{+}_{1}$ state.

A twenty-day experiment was then conducted at ILL Mar20 , where 64Ni was
populated via thermal-neutron capture on a 2 GBq 63Ni sample (extracted from a
larger CERN-nTOF sample Led14 ), where 12.1 mg of NiO grains were glued
between two 6 $\mu$m-thick Al foils and contained $\approx 8$% 63Ni (T1/2 =
101.2(15) y), 69% 62Ni, $<$3% 63Cu and other Ni isotopes, and 20% O. The
$\gamma$ rays were detected with FIPPS Mic18 , a $\sim$3.7$\%$ efficient array
of 16 clover detectors (8 on loan from IFIN-HH) arranged in a 4$\pi$ geometry,
particularly suitable for angular correlation studies in view of the large
number (71) of relative angles between the germanium crystals.

The 1/2- spin-parity of the 63Ni ground state results in a 0- or 1- 64Ni
capture state at 9657.47 keV, and strong population of 0+, 1+ and 2+ levels
through $E1$ primary $\gamma$ rays is expected. Direct feeding of the known
0${}^{+}_{2}$ and 0${}^{+}_{3}$ states, at 2866.9 and 3025.5 keV, is clearly
visible in coincidence spectra gated on respective 6791- and 6632-keV primary
transitions. Figure 2(a) displays the 6632-keV gated spectrum where both the
0${}^{+}_{3}\rightarrow$ 2${}^{+}_{1}$, 1680-keV transition and a new, weaker
(3.6(2)$\%$ of the former) 749-keV, 0${}^{+}_{3}\rightarrow$ 2${}^{+}_{2}$
decay branch are observed. The angular correlation for the
0${}^{+}_{3}\rightarrow$2${}^{+}_{1}\rightarrow$ 0${}^{+}_{1}$ cascade (inset,
Fig. 2(a)) agrees with the 0+ spin-parity assignment to the 3026-keV level. A
search for additional 0+ excited states was undertaken by considering every
primary transition in coincidence with the 1346-keV ground-state transition,
and also feeding levels in the 3-6 MeV excitation energy range. Five such
high-energy transitions, at 6194, 5801, 5389, 4954 and 3889 keV, populating
states at 3463.1, 3856.0, 4268.1, 4703.9 and 5768.6 keV were found to exhibit
decay patterns only consistent with 0+ spin-parity assignments 999For the
3856.0-keV level, the 0+ assignment was established via the 702-3154 keV
correlation cascade, involving the intermediate 1+ state at 3154 keV, first
identified in this work Mar20 ).. Further, the angular correlation analysis
yielded firm 0+ assignments for the states at 3463.1, 4268.1, 4703.9 and
5768.6 keV by considering in each case pairs of $\gamma$ rays composed, on the
one hand, of the decay branch to the 2${}^{+}_{1}$ state and, on the other, of
the 2${}^{+}_{1}\rightarrow$0${}^{+}_{1}$, 1346-keV transition. The relevant
analysis for the 0${}^{+}_{4}$, 3463.1-keV level is illustrated in Fig. 2(b).

The partial level scheme is given in Fig. 3. The 0${}^{+}_{4}$ state at 3463
keV is of particular interest; it was observed earlier with tentative (2+/-,
3-) Bro12  and (0+, 1, 2, 3-) spin-parity assignments Har92 , but is firmly
assigned here. It should be emphasized that this level is not populated in
64Co $\beta$-decay Paw12 , in contrast to all other 0+ states, up to
0${}^{+}_{6}$, which are fed in this process. This observation already points
to a marked difference in structure for this excitation, and is reminiscent of
that occurring in 66Ni Leo17 , where the prolate-deformed 0${}^{+}_{4}$ state
at 2974 keV was also the only 0+ excitation not fed in the $\beta$-decay of
the spherical 66Co ground state Str18 . Further inspection of the ILL data
revealed three 2+ states (firmly established in this work) at 3647.9, 3749.1
and 3798.7 keV, which complement four such excitations, at 1345.8, 2276.6,
2972.1 and 3276.0 keV, reported in Refs. NUDAT  (see Fig. 3). The angular
correlations for the transitions deexciting 2${}^{+}_{4}$, 2${}^{+}_{5}$ and
2${}^{+}_{7}$ levels toward the 2${}^{+}_{1}$ state all indicate a dominant
$M1$ character, with only a small $E2$ admixture. This is illustrated through
the representative data for the 2${}^{+}_{5}\rightarrow$
2${}^{+}_{1}\rightarrow$ 0${}^{+}_{1}$ cascade of Fig. 2(c). A notable
exception to this trend is the 2${}^{+}_{6}\rightarrow$
2${}^{+}_{1}\rightarrow$0${}^{+}_{1}$ sequence where the 2403-keV $\gamma$ ray
exhibits a pronounced $E2$ character, as strikingly illustrated by comparing
Figs. 2(c) and (d). The corresponding mixing ratio was determined to be
$\delta$ = +1.23(10). Furthermore, from the line shape observed for this
2403-keV $\gamma$ ray in spectra following 1p transfer, a lower limit of 0.5
ps was obtained for the 2${}^{+}_{6}$ state half-life, which results in upper
limits to the respective $B$($E2$) strengths of 0.02, 0.4, and 0.02 W.u. for
the 2${}^{+}_{6}\rightarrow$ 0${}^{+}_{1}$,
2${}^{+}_{6}\rightarrow$2${}^{+}_{1}$ and 2${}^{+}_{6}\rightarrow$
2${}^{+}_{2}$ transitions. These all indicate that deexcitation from the
2${}^{+}_{6}$ state is significantly hindered.

Figure 2: Neutron-capture data: $\gamma$-ray spectra in coincidence with
primary transitions (energy indicated in each panel). Relevant transitions are
also labeled. Insets: measured angular correlations for the cascades
indicated.

Figure 3: Data (left) and MCSM calculations (right) for 64Ni level properties
derived in this work. Transition strengths are from CE, branchings and mixing
ratios from neutron capture. In red, decays from prolate structures.

To gain further insight into the nature of the 0+ and 2+ excitations in 64Ni,
a multi-step Coulomb excitation (CE) measurement was carried out at the ATLAS
facility at ANL. A 0.5 mg/cm2 isotopically-enriched 208Pb target was bombarded
by a 64Ni beam of 272 MeV, a value 14$\%$ below the Coulomb barrier.
Transitions were detected with the GRETINA tracking array Pas13  in
coincidence with the two reaction partners measured by the CHICO2 compact
heavy ion counter Wu16 , as described in Ref. Aya19 . The resulting yields
were analyzed with the semi-classical Coulomb excitation code, GOSIA Czo12 ,
which provided electromagnetic matrix elements for transitions from 13 states
Lit20 . For the present paper, the following reduced transition probabilities
are relevant: $B$($E2$; 2${}^{+}_{1}\rightarrow$0${}^{+}_{1}$) = 140(20)
$e^{2}$fm4, $B$($E2$;2${}^{+}_{2}\rightarrow$ 2${}^{+}_{1}$) = 73(8)
$e^{2}$fm4, $B$($E2$; 0${}^{+}_{2}\rightarrow$ 2${}^{+}_{1}$) = 48(3)
$e^{2}$fm4, $B$($E2$; 0${}^{+}_{3}\rightarrow$ 2${}^{+}_{1}$) = 10(1)
$e^{2}$fm4, $B$($E2$; 0${}^{+}_{4}\rightarrow$ 2${}^{+}_{1}$ ) $<$ 1.3
$e^{2}$fm4 and $B$($E2$; 2${}^{+}_{6}\rightarrow$ 2${}^{+}_{2}$) $<$ 3.2
$e^{2}$fm4. In terms of single-particle estimates, these values translate into
strengths of 9(1), 4.8(5), 3.2(2), 0.65(7), $<$0.08 and $<$0.2 W.u.,
respectively. Good agreement is noted with the lifetime data for the
0${}^{+}_{2}$ and 0${}^{+}_{3}$ decays to the 2${}^{+}_{1}$ state with
$B$($E2$) values of 3.3(14) and 0.8(3) W.u., respectively (Fig. 1). The limits
for the 2${}^{+}_{6}$ and 0${}^{+}_{4}$ states are also consistent, but
somewhat tighter in the CE case.

The left part of Fig. 3 provides a 64Ni level scheme encompassing states of
positive parity with spins 0, 1, 2, up to 4.26 MeV excitation energy. The
information gathered on mixing, branching ratios and transition strengths has
been added to enable extensive comparisons with calculations. Such additional
tests of theory are generally difficult, if not impossible, especially for
neutron-rich nuclei requiring experiments with radioactive beams.

The right side of Fig. 3 presents the level scheme from MCSM calculations,
performed with significantly extended MCSM basis vectors as compared to
earlier studies of 66-78Ni Leo17 ; Tsu14 . The model space includes protons
and neutrons in the full $fp$ shell with, in addition, the $g_{9/2}$ and
$d_{5/2}$ orbitals, and the Hamiltonian is based on the A3DA-m effective
interaction Tsu14 . The transition probabilities were obtained with standard
effective charges ($e_{p}$ = 1.5 $e$, $e_{n}$ = 0.5 $e$), a spin quenching
factor of 0.7 and an isovector orbital $g$-factor of 0.1 Hon09 . State
energies are reproduced satisfactorily $-$ the rms deviation is $\sim$300 keV,
commensurate with expectations for shell-model calculations. For the first
three 0+ excitations, the computed decay patterns mirror the data: the decay
to the 2${}^{+}_{1}$ level dominates the deexcitation from 0${}^{+}_{2}$ and
0${}^{+}_{3}$ states, and the branching ratios between the four transitions
from the 0${}^{+}_{4}$ state are qualitatively reproduced, with the
0${}^{+}_{4}\rightarrow$ 2${}^{+}_{2}$ one being strongest. The relative
$B$($E2$) strengths calculated for the 0${}^{+}_{2,3,4}\rightarrow$
2${}^{+}_{1}$ decays (i.e., 12, 2.4 and 5 10-4 W.u.) are consistent with the
data, even though the absolute strengths are larger. Finally, the MCSM
calculations also reproduce the lack of feeding of the 0${}^{+}_{4}$ state in
$\beta$ decay, when compared to that of the other 0+ levels.

A sequence of relatively close-lying 2+ levels is also predicted with
deexcitation patterns and transition probabilities agreeing with observations,
at least when the calculated 2${}^{+}_{7}$ state is associated with the
2${}^{+}_{6}$ experimental one $-$ the 2${}^{+}_{6}$ and 2${}^{+}_{7}$ levels
are computed to lie only 235 keV apart; i.e., within the expected accuracy of
the A3DA-m interaction. Theory also reproduces (i) relative variations in
$B$($E2$) values between the 2+ levels (including the retardation for the
transitions out of the 2${}^{+}_{6}$ state, which agrees with the observed
small $B$($E2$) upper limits), and (ii) the strong $E2$ component in the
$\Delta I$ = 1, 2${}^{+}_{6}\rightarrow$ 2${}^{+}_{1}$ transition, where the
measured mixing ratio $\delta$($E2/M1$) = +1.23(10) (vs. $\delta_{\rm
MCSM}$=2.6) contrasts those for similar transitions from the other 2+
excitations (Fig. 2 (c)).

According to the MCSM calculations, the first four 0+ states reside in
spherical, oblate, spherical and prolate minima, respectively, in the PES
obtained for the A3DA-m Hamiltonian by the constrained Hartree-Fock method
Tsu14 ; Ots16 . Thus, the 0${}^{+}_{4}\rightarrow$2${}^{+}_{1}$ decay is a
prolate-to-spherical shape-changing transition, resulting in significant
retardation, in line with the $B$($E2$) limit of $<$0.08 W.u. The same picture
applies to the computed 2${}^{+}_{7}$ level, which theory also locates in the
prolate minimum. The observed decay pattern, the limits on the decay strengths
and the dominant $E2$ character of the 2${}^{+}_{6}\rightarrow$ 2${}^{+}_{1}$
transition argue in favor of this interpretation for the observed
2${}^{+}_{6}$ state. Hence, based on the consistency between data and theory,
this 2${}^{+}_{6}$ level represents the first observation in the Ni isotopes
of a 2+ excited state in a well-isolated prolate potential minimum. The
“shape-isomer”-like properties of the 0${}^{+}_{4}$ excitation in 64Ni mirror
closely those found in 66Ni Leo17  with, in addition, the observation of the
first element of a rotational sequence. Using the Raman systematics, linking
2+ energies to transition strengths Ram01 , this results in a computed
$\beta_{2}$ deformation of $\sim$0.4, in agreement with the MCSM result (Fig.
4(a)). The low-energy 286-keV in-band 2${}^{+}_{6}\rightarrow$ 0${}^{+}_{4}$
transition, even with a calculated $B$($E2$) strength of $\sim$40 W.u., cannot
be observed: the flux proceeds through high-energy ($>$1 MeV) $E2$ $\gamma$
rays due to the $E_{\gamma}^{5}$ factor.

With these new, extensive data in 64Ni, the evolution in energy of the prolate
minimum with $N$ can now be traced in the Ni isotopes, revealing a sharp
contrast with that exhibited by the 2${}^{+}_{1}$ levels of spherical nature.
The latter are all in the 1250 - 1450 keV range, with the exception of the
2033-keV value for 68Ni due to the $N=40$ sub-shell closure. In contrast, the
prolate 0+ excitation rises from 1567 keV in 70Ni, to 2511 and 2905 keV in
68Ni and 66Ni, and 3463 keV in 64Ni. This behavior for $N<40$ differs markedly
from the lowering of deformed intruder states when moving away from a
(sub)-shell closure, observed in the Hg and Pb nuclei Woo92 ; Hey11 , for
example. Low-lying prolate intruder states in the aforementioned neutron-rich
Ni isotopes reflect the action of the monopole tensor force which is often
referred to as Type II shell evolution Ots16 ; Tog16 ; Leo17 , and involves
particle-hole excitations of neutrons to the $g_{9/2}$ unique-parity orbital
from the $fp$ shell. Extra binding for such intruder states is provided
largely by the monopole tensor part of the nucleon-nucleon force (the proton
$f_{5/2}$ \- $f_{7/2}$ spin-orbit splitting is reduced, favoring proton
excitations across the $Z=28$ shell gap), which stabilizes isolated, deformed
local minima in the PES (Fig. 4(a)). This additional binding is reduced for
lower $N$ values as there are progressively fewer neutrons which can be
excited to the $g_{9/2}$ orbital. The deformed minimum rises in excitation
energy as a result. As demonstrated in Fig. 4(b), by deactivating components
of the monopole interaction (i.e., monopole frozen Ots19 ), a nearly vanishing
prolate minimum would reside at even higher excitation, in line with mean
field predictions Hil07 ; Ber91 ; Dec80 ; Mol04 ; Mol09 .

Figure 4: 64Ni potential energy surfaces with (a) full, original interaction
used in MCSM calculations Tsu14 , and (b) monopole-frozen interaction (i.e.,
the monopole component is subtracted from the proton-neutron interaction, and
single-particle energies are adjusted to original effective values of the
spherical minimum Ots19 ).

The present work has unveiled an unexpectedly complex landscape of nuclear
deformation at zero spin in stable, semi-magic 64Ni. This includes the first
identification, in Ni isotopes, of a 2+ excitation in the prolate minimum. The
new results provide, for the first time, a complete picture of the mechanisms
underlying the appearance of deformation and shape coexistence in the Ni
isotopes. They highlight the impact of the monopole tensor interaction in
driving deformation at zero spin, even in 64Ni, a nucleus within the valley of
stability.

This work was partially supported by the European project for Nuclear Physics
ENSAR2 (project number: 654002), by the Italian Istituto Nazionale di Fisica
Nucleare, by the Polish National Science Centre under Contract No.
2014/14/M/ST2/00738 and 2013/08/M/ST2/00257 and by the Fonds de la Recherche
Scientifique – FNRS Grant Number 4.45.10.08. It was supported by the Romanian
Nucleu Project PN-19-06-01-02 and IFA-CERN-RO-06-ISOLDE. The MCSM calculations
were performed on the K computer at RIKEN AICS and Oakforest-PACS operated by
JCAHPC (hp160211, hp170230, hp180179, hp190160, hp200130). This work was
supported in part by MEXT as “Priority Issue on Post-K computer” (Elucidation
of the Fundamental Laws and Evolution of the Universe) and “Program for
Promoting Research on the Supercomputer Fugaku” (Simulation for basic science:
from fundamental laws of particles to creation of nuclei) and by JICFuS. This
work was funded by the U.S. Department of Energy, Office of Science, Office of
Nuclear Physics, under Contracts No. DE-AC02-06CH11357 (ANL), DE-
AC52-07NA27344 (LLNL), DE-AC02-05CH11231 (LBNL), and under Grants No. DE-
FG02-97ER41041 (UNC), No. DE-FG02-97ER41033 (TUNL), No. DE-FG02-08ER41556 and
DE-SC0020451 (MSU), No. DE-FG02-94ER40848 (UML), and No. DE-FG02-94ER4084
(Maryland) and by the National Science Foundation under Grants No. PHY-1565546
(MSU) and No. PHY-1502092 (USNA). GRETINA was funded by the U.S. DOE, Office
of Science, Office of Nuclear Physics under the ANL and LBNL contract numbers
above. A. D. A. acknowledges support by Office of Naval Research (ONR) through
the Naval Academy Research Council (NARC). This research used resources of
ANLs ATLAS facility, a DOE Office of Science User Facility. The open access
fee was covered by FILL2030, a European Union project within the European
Commission’s Horizon 2020 Research and Innovation program under grant
agreement No. 731096.

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