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# Question Answering on Linked Data: Challenges and Future Directions

Saeedeh Shekarpour

Knoesis Research Center, USA<EMAIL_ADDRESS>Kemele M. Endris

University of Bonn, Germany<EMAIL_ADDRESS>Ashwini Jaya Kumar

Fraunhofer IAIS, Germany<EMAIL_ADDRESS>Denis Lukovnikov

Kuldeep Singh

Harsh Thakkar

University of Bonn, Germany<EMAIL_ADDRESS>Fraunhofer IAIS, Germany
<EMAIL_ADDRESS>University of Bonn, Germany<EMAIL_ADDRESS>Christoph Lange

University of Bonn / Fraunhofer IAIS, Germany<EMAIL_ADDRESS>
###### Abstract

Question Answering (QA) systems are becoming the inspiring model for the
future of search engines. While, recently, datasets underlying QA systems have
been promoted from unstructured datasets to structured datasets with
semantically highly enriched metadata, question answering systems are still
facing serious challenges and are therefore not meeting users’ expectations.
This paper provides an exhaustive insight of challenges known so far for
building QA systems, with a special focus on employing structured data (i.e.
knowledge graphs). It thus helps researchers to easily spot gaps to fill with
their future research agendas.

###### keywords:

Question Answering System, Research Challenge, Speech Interface, Query
Understanding, Data Quality, Distributed and Heterogeneous Datasets,
Interoperability of Components.

††conference: WWW’16 Companion, April 11–15, 2016, Montréal, Québec, Canada.

##  1 Introduction

The Web of Data is growing enormously (currently more than 84 billion
triples111observed on 14 October 2015 at http://stats.lod2.eu/). This figure
comprises both structured and unstructured data. Still, taking advantage of
this rapidly growing amount of data is challenging. Traditional information
retrieval approaches based on keyword search are user-friendly but fail to
exploit the internal structures of data due to their bag-of-words semantics.
For searching information on the Data Web we need similar user friendly
approaches, i.e. keyword-based interfaces, which rely on the internal
structure of the data.

Question Answering (QA) is a specialized form of information retrieval. A
Question Answering system retrieves exact answers to questions posed in
natural language by the user. While, recently, datasets underlying QA systems
have been promoted from unstructured datasets to structured datasets with
semantically highly enriched metadata, question answering systems are still
facing serious challenges and are therefore not meeting users’ expectations.

Question Answering systems consists of components that can be studied and
evolved independently. These components include (1) an input interface for
obtaining a query, (2) components for understanding, interpreting,
disambiguating and parsing the query, (3) components accessing and processing
the datasets employed (facing issues such as heterogeneity, quality and
indexing); thus, there are also issues of (4) interoperability among different
interacting components. In the following, we elaborately discuss challenges
related to each aspect and consider future research directions. We close with
a conclusion and a roadmap for future work.

##  2 Challenges

In this section we present question answering challenges from four different
aspects namely, (i) Speech-based interface challenge, (ii) query
understanding, interpreting, disambiguating and parsing challenges, (iii)
data-oriented challenges (iv) interoperability of QA components challenge.

###  2.1 Speech Interface

Interfacing speech to QA systems has become a focus of research for a long
time. But, the main focus of research effort so far has been spent on
interfacing speech to IR-based QA systems [41, 43, huspeechqoogle], and much
less on interfacing speech input to QA systems based on KGs (knowledge
graphs). Typical state-of-the-art IR approaches integrate a speech recognition
(SR) unit directly with the QA system. An effort beyond merely interfacing the
two units is required to enable natural conversation in question answering
system for both IR and KG methods.

An SR system mainly consists of an acoustic model and a language model, where
the main objective is to decode what is uttered by the user. In contrast, a
general IR based QA system comprises question processing (to extract the query
from the input and to determine the answer type), passage retrieval, document
retrieval, passage extraction, and finally answer selection depending on the
relatedness of the named entities found to the question keyword. The accuracy
of recognizing spoken words has a vital influence on the success of the whole
QA process. Ex: if ‘Jamshedpur’ (a city in India) is recognised as ‘game shed
poor’ (articulation style and duration of utterance is the key difference),
then the whole QA process is altered. The city name which constitute the
important answer type is not recognised by the QA system. This can be avoided
if there is a rich dataset to train a recogniser but it is not possible to
have acoustic training data for an open-domain. Hence speech recognisers are
usually built for a specified domain. The same applies for QA systems,
developing an open-domain QA is a challenge.

With the evolution of neural network based methods for speech recognition, the
whole conventional approach to speech recognition has changed. Generally,
acoustic model and language model were built as two independent units. The
development of single neural network architecture to transcribe an audio input
is a breakthrough in the speech recognition research
[DBLP:journals/corr/HannunCCCDEPSSCN14, graves2014towards, amodei2015deep].
The recognition accuracy has been tested for a character level transcription
and it is indicated that a word/sentence level transcription can be made with
the same architecture. In this type of single neural network based speech
recognition, language model is applied at the output of speech recogniser.
Following the same methodology, it is possible to build an end-to-end speech
interfaced QA system with deep neural networks. Current research direction is
towards exploring the interface of speech to knowledge graph using deep neural
networks.

###  2.2 Understanding Questions

####  2.2.1 Discussion

In the case of full-fledged QA over structured data, for example over a
knowledge base (KB) such as Freebase [8], the question must be translated into
a logical representation that conveys its meaning in terms of entities,
relations, types as well as logical operators. Simpler forms of QA can also be
achieved in other ways, however, approaches without formal translation can not
express certain constraints (e.g. comparison). The task of translating from NL
to a logical form (semantic parsing (SP)) is characterized by the mismatch
between natural language (NL) and knowledge base (KB). The semantic parsing
problem can be divided into two parts: (1) determining KB constituents
mentioned in the NL expression and (2) determining how these constituents
should be arranged in a logical structure. The mismatch between NL and KB
brings several problems. One problem is Entity Linking (EL), recognizing parts
of NL input that refer to an entity (NER) and determining which named entities
are meant by that part (disambiguation). A central challenge in EL is how to
take into account the context of an entity mention in order to find the
correct meaning (disambiguation). Another challenge is finding an optimal set
of suitable candidates for a mention, where the _lexicon_ (mapping between
words/phrases and entities) plays an important role. A problem bordering both
disambiguation and candidate generation is the large number of entities a word
can refer to (e.g. the thousands of possible “John”’s when confronted with
“John starred in 1984”).

Another problem is relation detection and classification. Given an NL phrase,
we want to determine which KB relation is implied by the phrase. Sometimes,
the relation is explicitly denoted by a NL constituent, for example verb-
mediated statements (e.g. “$X$ married $Y$”), in which case a lexicon can help
a lot to solve the problem. However, in general, a lexicon-based approach is
not sufficient. Sometimes there are no relation-specific words in the
sentence. Sometimes prepositions are used, for example “works by Repin” or
“cars from Germany” and sometimes the semantics of the relations and the
entities/types they connect are lexicalized as one, for example, “Russian
chemists” or “Tolstoy plays”. Such cases require context-based inference,
taking into account the semantics of the entities that would be connected by
the to-be-determined relation (which in turn is related to parsing).

Merely linking entities and recognizing the relations is not sufficient to
produce a logical representation that can be used to query a data source. The
remaining problem is to determine the overall logical structure of the NL
input. This problem becomes difficult for longer, more complex sentences,
where different linguistic phenomena, such as coordination and co-reference,
must be handled. Formal grammars, such as CCG [49], can help to parse NL
input. CCG in particular is well-suited for semantic parsing because of its
transparent interface between syntactic structure and underlying semantic
form. One problem with grammar-based semantic parsers is their rigidity, which
is not well-suited for incomplete input as often found in real-world QA
scenarios. Some works have explored learning relaxed grammars [60] to handle
such input.

The straightforward way of training semantic parsers requires training data
consisting of NL sentences annotated with the corresponding logical
representation, which are very cumbersome to obtain. Recent works have
explored different ways to reduce the annotation effort in order to bypass
this challenge. One proposed way is to train on question-answer pairs instead
[7]. Another way is to automatically generate training data from the KB and/or
from entity-linked corpora [39] (e.g. ClueWeb). Training with paraphrasing
corpora [7] is another technique explored in several works to improve the
range of expressions the system will be able to cover.

####  2.2.2 Future directions

Recently, impressive advances in different tasks in Artificial Intelligence
have been achieved using deep learning techniques. Embedding-based language
models, such as Word2Vec [34, 35] and GloVe [37], have helped to improve
performance in many NLP tasks. One of the most interesting and the most
promising future directions for semantic parsing and question answering is
further exploration of deep learning techniques in their context.

Using deep learning to better understand questions can be done by using
(possibly custom-trained) word (, word sense and entity) embeddings, which
capture their syntactic and semantic properties, as features to improve
existing workflows. However, a “deeper” approach would be to also devise new
models that provide the machine with more freedom to figure out how to
accomplish the task. An excellent and very recent example in NLP is the
Dynamic Memory Network (DMN [29]), that does not use any manually engineered
features or problem-tailored models, and yet achieves state-of-the-art
performance on all tested tasks, which are disjoint enough to leave one
impressed (POS tagging, co-reference resolution, sentiment analysis and
Question Answering on the bAbI dataset). The DMN is one of the works focusing
on attention and memory in deep learning that enables the neural network to
reason more freely. We share the belief that the investigation and application
of more advanced deep learning models (such as DMN and NTM [21]) could yield
impressive results for different tasks in AI, including question answering.

Recursive, convolutional (CNN) and recurrent (RNN) neural networks are widely
used in recent neural network-based approaches. Convolutional Neural Networks
(CNN), a special case of recursive NNs are well-explored for computer vision.
Recursive NNs have also been applied for parsing and sentiment analysis. RNNs
produce state-of-the-art results in speech processing as well as in NLP
because of their natural vigor for processing variable-length sequences. They
have been applied for machine translation (SMT) [13], language generation
(NLG) [50], language modeling and more and are also fundamental for the
success of the DMN and the NTM.

Even though the DMN has not yet been applied to our task of structured QA,
some recent works, such as the relatively simple embedding-based work of
Bordes et al. [bordes2014question] (which outperformed ParaSempre [7] on
WebQuestions) and the SMT-like SP approach of [dong2016language] seem to
acknowledge the promise of neural approaches with embeddings.

An additional interesting direction is the investigation of joint models for
the sub-problems involved in question interpretation (EL, co-reference
resolution, parsing, …). Many tasks in NLP depend on each other to some
degree, motivating the investigation of efficient approaches to make the
decisions for those tasks jointly. For example, co-reference resolution and EL
can benefit from each other as entity information from a KB can serve as quite
powerful features for co-reference resolution and co-reference resolution in
turn can improve EL as it transfers KB features to phrases where anaphora
refer to entities. Factor Graphs (and Markov Networks) are by nature very
well-suited for explicit joint models (e.g. [47]). However, a more internal
kind of joint inference could also be achieved within a neural architecture
(e.g. the DMN).

However, it is worth noting that training advanced neural models and explicit
joint models can be a difficult task because of the large number of training
parameters and co-dependence of these parameters. Deep learning typically
relies on the availability of large datasets. However, the whole task to be
solved can be divided in two parts, one focusing on representation learning,
which can accomplished in an unsupervised setting (with large amounts of data)
and the second part relying on and possibly fine-tuning the representations
obtained in the first part in a supervised training setting (requiring
annotated task-specific data). For explicit joint models, data capturing the
dependence between different task-specific parts of the models (e.g. annotated
for both EL and co-reference) are required and the efficient training of such
models is a very relevant current topic of investigation.

The concluding thought is that the further investigation of language [34, 35,
37, 24] and knowledge modeling [20, 56, 40, 48, 12] and powerful deep neural
architectures with self-regulating abilities (attention, memory) as well as
implicit or explicit joint models will continue to push the state of the art
in QA. Well-designed deep neural architectures, given proper supervision and
powerful input models, have the potential to learn to solve many different NLU
problems robustly with minimal customizations, eliminating the need for
carefully engineered features, strict formalisms to extract complex structures
or pipelines arranging problem-tailored algorithms. We believe that these
lines of research in QA could be the next yellow brick [5] in the road to true
AI, which has fascinated humanity since the ancient tales of Talos and Yan
Shi’s mechanical men.

###  2.3 Data challenges

####  2.3.1 Indexing Heterogeneous Datasets

A typical QA system is empirically only as good as the performance of its
indexing module [17]. The performance of indexing serves as an upper bound to
the overall output of the QA system, since it can process only as much data as
is being presented/served to it from the indices. The precision and recall of
the system may be good, but if all or most of the top relevant documents are
not indexed in the system, the system performance suffers and hence does the
end user.

Many researchers have compared effectiveness across a variety of indexing
techniques. Their studies show improvement if multiple techniques were
combined compared to any single individual indexing technique [38]. In the
present scenario, information retrieval systems are carefully tailored and
optimized to deliver highly accurate results for specific tasks. Over the
years, efforts of developing such task specific systems have been diversified
based on a variety of factors discussed in the following.

Based on the type of the data and the application setting, a wide range of
indexing techniques are deployed. They can broadly be categorized into three
categories based on the format and type of data indexed, namely: structured
(e.g. RDF, SQL, etc.), semi-structured (e.g. HTML, XML, JSON, CSV, etc.)
and/or unstructured data (e.g. text dumps). They are further distinguished by
the type of technique they use for indexing and/or also by the type of queries
that a particular technique can address. The different techniques inherently
make use of a wide spectrum of underlying fundamental data structures in order
to achieve the desirable result.

Most of the systems dealing with unstructured or semi-structured data make use
of inverted indices and lists for indexing. For structured datasets, a variety
of data structures such as AVL trees, B-Trees, sparse indices, IR trees, etc.,
have been developed in the past decades. Many systems combine two or more data
structures to maintain different indices for different data attributes. We
present a short survey of indexing platforms and data structures used in a
wide range of QA systems in table 1.

System |  Data structure used |  Platform used for indexing
---|---|---
SWSE/YARS2 [23] |  Sparse, Inverted Indices for RDF quads |  Lucene
Sindice [36] |  Inverted Index and On-disk persistent storage |  Solr
Sina [45] |  Bitmap index on RDF quads (in total 5 indices are maintained: 2 full RDF quad indices, 3 partial RDF quad indices) |  OpenLink Virtuoso
HAWK [54] |  *N/A |  *N/A
TBSL [53] |  Inverted Index |  Solr
Ephyra [44] |  Inverted index |  Lemur-Indri
PowerAqua [32] |  Inverted index |  Lucene (two indices are prepared taxonomically)
AquaLog [31] |  *N/A |  GATE MIMIR possibly with Lucene
Sig.ma [52] |  Inverted Index and On-disk persistent storage |  Solr
QUADS [57] |  Inverted index |  Lucene
MAYA [27] |  (key, value) pairs |  Traditional index with RDBMS
ESTER [4] |  Extended inverted index – inverted index with scores for each word ; combines prefix search and join operations |  Proprietary module
QAST [25] |  Inverted index with term weighting ((Minimal Span Weighting)) |  Lucene
FREyA [15] |  *N/A |  Sesame/OWLIM (aka GraphDB))
QAKIS [11] |  *N/A |  *N/A
MEANS [6] |  Inverted index |  Terrier
Watson/DeepQA [26, 18] |  Persistent disk caching |  Watson Explorer Engine XML (VXML)

Table 1: Comparison of the indexing platforms and data structures used by
different QA systems. *N/A = no data available

Table 1 is an excerpt from a table in our exhaustive survey of open QA
systems222The full data collection can be found at https://goo.gl/FM1LM9. Our
current work in progress is focusing on benchmarking different datasets such
as Wikidata [55], DBpedia [2], and FreeBase [9] against a wide spectrum of
indexing library platforms, such as
Indri333http://www.lemurproject.org/indri.php,
Solr444http://lucene.apache.org/solr/,
ElasticSearch555https://www.elastic.co/products/elasticsearch,
Sphinx666http://sphinxsearch.com/, Neo4j777http://neo4j.com/,
Titan888http://thinkaurelius.github.io/titan/, Xapian999http://xapian.org/,
and Terrier101010http://www.terrier.org/.

####  2.3.2 Data Quality Challenge

Recent advancements in the fields of Web of Data and Data Science have led to
an outburst of standards related to structured data111111The amount not only
of structured, but also of semi-structured and unstructured data available
online is also steadily increasing; however, for the purpose of our work we
assume that such data has first been translated to the RDF data model using
standard tools, e.g. from the Linked Data Stack [3]. such as RDF(a), Linked
Data, schema.org, etc., to an increasing amount of such data, and to a wide
range of tools to produce, manage and consume such data. To be available for
ready consumption, especially in open question answering systems, any such
data sources should meet a certain level of quality, e.g., defined by
benchmarks. Quality can generally be defined as “fitness for use”, but there
are a lot of concrete factors that influence a dataset’s fitness for use in
question answering121212In this section, we do not abbreviate “question
answering” as “QA” to avoid confusion with “quality assessment”. settings and
in specific application domains. Recently, a number of research activities
have been concerned with automating the assessment of linked data quality.
Debattista, who has developed one such tool (Luzzu [16]), provides an overview
of other state-of-the-art tools [16], including one by Flemming [19], as well
as Sieve [33], RDFUnit [28], TripleCheckMate [58], LinkQA [22], and LiQuate
[42]. In this section, we summarize the concrete criteria by which the quality
of linked data can be assessed, with a special focus on those criteria that
are relevant to question answering.

In a comprehensive review of literature and systems, Savors et al. [59] have
identified the dimensions of linked data quality and categorized them as
follows:

  * •

Accessibility dimensions: This category covers aspects related to retrieving
and accessing data, which includes full or partial access and different
technical means of access (e.g. the possibility to download a data dump vs.
the availability of a SPARQL endpoint, i.e. a standardized query interface).

    * –

Availability is generally defined as the ease of access with which particular
information is obtainable or rapidly retrievable for readily consumption. In a
linked data context, availability can be referred to as the accessibility of a
SPARQL endpoint or RDF dumps or dereferenceable URIs.

    * –

Interlinking is relevant as it refers to the data integration and
interoperability. The output of interlinking is a _linkset_ , i.e. a set of
RDF triples linking subjects and recognized related objects.

    * –

Security denotes the degree to which a particular dataset is resistant to
misuse or alteration without appropriate user access rights.

    * –

Verifiability, usually by an unbiased third party, addresses the authenticity
and correctness of the dataset. Verifiability is typically enabled by
provenance metadata.

  * •

Intrinsic dimensions: This category covers aspects that are independent of the
user’s context, or the out of the application context – such as accuracy and
consistency.

    * –

Accuracy refers to the degree of a dataset correctly representing the captured
real world facts and figures in the form of information with high precision.

    * –

Consistency refers to the independence from logical, formal or
representational contradictions of a dataset with respect to others.

    * –

Completeness is referred to as the degree to which information in the dataset
is complete or not missing. The dataset should have all the required objects
or values for a given task in order to be considered as complete. Thus,
arguing intuitively, completeness is one of the concrete metrics for linked
data quality assessment.

  * •

Contextual dimensions: This category is concerned with the context of the task
being pursued.

    * –

Timeliness is concerned with the freshness of data over time or timeliness,
i.e. the regularity of updates or merges and so on.

    * –

Understandability can be achieved by providing appropriate human readable
annotations to a dataset and its entities, and by consistently following a
certain regular expression as a pattern for forming entity URIs.

    * –

Trustworthiness is concerned with the reliability or trustworthiness of the
data and its source.

  * •

Representational dimensions: This category is concerned with the design and
representation of the data and its schema. For instance, understandability and
interpretability.

    * –

Interpretability refers to adhering to the standard practice of representing
information using appropriate notations, symbols, units and languages.

Data quality dimensions in all of these categories can be relevant in question
answering scenarios. In a preliminary study [51], we evaluated few selected
metrics mentioned above on two popular datasets of linked data namely,
Wikidata and DBpedia131313Freebase used to be another popular cross-domain
dataset but support for it has expired, which is why we did not consider it;
cf. https://www.freebase.com/.. We evaluated the subsections extracted from
these datasets on categories such as “politicians”, “Movies”, “Restaurants”
and “Soccer players”. The slices have been made available to the common public
and can be found: https://goo.gl/Kn6Fom (DBpedia slices),
https://goo.gl/5aTkLp (Wikidata slices). The detailed information regarding
the data slice statistics can be found from the work [51] which is selected
from a detailed report made public at https://goo.gl/ignzzI. For this
evaluation we have obtained four slices of both DBpedia and Wikidata, namely
Restaurants, Politicians, Films and Soccer players. From this preliminary
study, we could so far observe that Wikidata dominates DBpedia for the
considered setting. The detailed scores and discussion can be found from the
mentioned work and the spreadsheet.

Our next step is (i) implement and evaluated the pending metric from the above
work and (ii) to identify more systematically what other dimensions and
metrics of data quality are specifically relevant in the typical application
domains of question answering, or sufficient for determining a dataset’s
“fitness” for question answering. Having identified such dimensions, we have
two goals: (a) identifying datasets that are suitable for question answering
at all, and (b) evaluate these metrics on a major part of the LOD
Cloud141414LOD Cloud: http://lod-cloud.net/ datasets, identifying more
specifically what quality problems they still suffer from.

Regarding implementation, in our recent study [51], we evaluated the results
on DBpedia and Wikidata slices using the metrics that the Luzzu linked data
quality assessment framework already provides. We look forward to extending
Luzzu to use in a question answering setting, that further existing
implementations of metrics in Luzzu can be specifically adapted to make them
suitable for quality assessment related to question answering, and that,
finally, Luzzu’s flexible extensibility even enables us to implement new
metrics that may be required. In summary, our near-future work will be
concerned with defining a generally and flexibly applicable framework for
automating the process of rigorously assessing the quality of linked datasets
for question answering by identifying, formalizing and implementing the
required metrics.

####  2.3.3 Distributed Heterogeneous Datasets

The decentralized architecture of the Web has produced a wealth of knowledge
distributed across different data sources and different data types. Question
answering systems consume different types of data: structured, semi-structured
or unstructured data. Most question answering systems uses either of these
types of data to answer user queries. Only few systems exploit the wealth of
data on the Web by combining these types of data. Hybrid question answering
systems are able to answer queries by combining both structured and
unstructured types of data. HAWK [54], for instance, provides entity search
for hybrid question answering using Linked Data and textual data. HAWK is able
to achieve an F-measure of up to 0.68 on the QALD-4 benchmark.

Most question answering systems today uses a single source to answer users
question. It should rather be possible to answer questions imposed by a user
by combining different interconnected sources. The challenges imposed by the
distributed nature of the Web are, on the one hand, finding the right sources
that can answer user query and, on the other hand, integrating partial answers
found from different sources. Source selection is one of the challenges in
federated question answering approaches. In [46], the authors presented an
approach to construct a federated query from user supplied (natural language)
questions using disambiguated resources.

Answers may come from different sources which have different data quality and
trust levels, ranking and fusion of data should be applied to select the best
sources.

The amount of data to be used to answer users’ queries should also be balanced
with the response time.

###  2.4 Interoperability Challenge

The field of QA is so vast that the list of different QA systems can go long.
Many Question Answering systems Based on specific domains have been developed.
Domain-specific QA systems, for example [1] are limited to a specific
knowledge, for example medicine. They are known as _closed domain_ QA systems.
However, when scope is limited to an explicit domain or ontology, there are
less chances of ambiguity and high accuracy of answers. It is also difficult
and costly to extend closed domain systems to a new domain or reusing it in
implementing a new system.To overcome the limitations of closed domain QA
systems, researchers have shifted their focus to _open domain_ QA systems.
FREyA [14], QAKiS [10], and PowerAqua [30] are few examples of open domain QA
systems which use publicly available semantic knowledge for example DBpedia
[2].

While many of these system achieved significant performance for special use
cases, a shortage was observed in all of them. We figured out that the
existing QA systems suffer from the following drawbacks: (1) potential of
reusing its components is very weak, (2) extension of the components is
problematic, and (3) interoperability between the employed components are not
systematically defined. There is little, but a work towards interoperable
architecture, e.g. QA archiecture developed by OKBQA151515Open Knowledge base
and Question Answering (http://okbqa.org). Interoperability of different QA
tools and components is required to enhance QA process which is still missing
at the conceptual level and currently more focused on implementation details.
Therefore, there is a need for a descriptive approach that define a conceptual
view of QA systems. This approach must cover all needs of current QA systems
and be abstracted from implementation details. Moreover it must be open such
that it can be used in future QA systems. The generalized approach for
architecture or ontology of a QA system and semantic search must focus to
bring all state-of-the advancement of QA under a single umbrella
[singhtowards]. We envisioned that a generalized vocabulary for QA will be an
abstraction level on top of all the existing QA approaches and will provide
interoperability and exchangeability between them. This generalized vocabulary
can be further used to integrate different components and web services within
a QA system [bothqanary].

##  3 Conclusion and Future Roadmap

In this paper, we presented an exhaustive overview of all the open challenges
being still controversial for developing a question answering system. The
intuition is that Linked Data which provides advantages such as semantic
metadata and interlinked dataset can influence all of the four major elements
(i.e. interface, parsing, data and component interoperability) which play a
key role in Question Answering systems. As our future research agenda, we are
steering our research on all of the discussed issues with the focus of
employing Linked Data technology to promote question answering capabilities.

Acknowledgments Parts of this work received funding from the European Union’s
Horizon 2020 research and innovation program under the Marie Sklodowska-Curie
grant agreement No. 642795 (WDAqua project).

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# Finite temperature effects in magnetic dipole transitions

Amandeep Kaur<EMAIL_ADDRESS>Department of Physics, Faculty of Science,
University of Zagreb, Bijenička c. 32, 10000 Zagreb, Croatia  Esra Yüksel
<EMAIL_ADDRESS>Department of Physics, University of Surrey, Guildford,
Surrey GU2 7XH, United Kingdom  Nils Paar<EMAIL_ADDRESS>Department of Physics,
Faculty of Science, University of Zagreb, Bijenička c. 32, 10000 Zagreb,
Croatia

(August 31, 2024)

###### Abstract

Finite temperature effects in electromagnetic transitions in nuclei contribute
to many aspects of nuclear structure and astrophysically relevant nuclear
reactions. While electric dipole transitions have already been extensively
studied, the temperature sensitivity of magnetic transitions remains largely
unknown. This work comprises the study of isovector magnetic dipole
excitations (M1) occurring between spin-orbit (SO) partner states using the
recently developed self-consistent finite temperature relativistic
quasiparticle random phase approximation (FT-RQRPA) in the temperature range
from $T=$ 0 to 2 MeV. The M1 strength distributions of 40-60Ca and 100-140Sn
isotopic chains exhibit a strong temperature dependence. The M1 strength peaks
shift significantly towards the lower energies due to the decrease in SO
splitting energies and weakening of the residual interaction, especially above
the critical temperatures ($T_{\textrm{c}}$) where the pairing correlations
vanish. By exploring the relevant two-quasiparticle ($2qp$) configurations
contributing to the M1 strength of closed- and open-shell nuclei, new proton
and neutron excitation channels between SO partners are observed in low- and
high-energy regions due to the thermal unblocking effects around the Fermi
level. At higher temperatures, we have noticed an interesting result in
40,60Ca nuclei, the appearance of M1 excitations, which are forbidden at zero
temperature due to fully occupied (or fully vacant) spin-orbit partner states.

††preprint: APS/123-QED

##  I Introduction

A comprehensive understanding of magnetic dipole (M1) nuclear response is
essential to various aspects of nuclear structure phenomena, such as isospin-
mixing, isospin-splitting, and ground-state correlations [1, 2, 3]. It also
aids in the study of radiative neutron capture, which has a key role in the
production of neutron-rich elements in hot stellar environments [4, 5, 6, 7].
Several experimental and theoretical studies have revealed intriguing behavior
in $\gamma$-ray strength functions ($\gamma$SF), and a notable enhancement is
observed in the strength function toward lower transition energies [8, 9, 10,
11, 12, 13, 14, 15]. The fine structure of strength functions in the
$E_{\gamma}=$ 5-8 MeV region is generally determined using the low-energy part
of the electric dipole (E1) response; however, some studies also suggest an
anomalous increase below 4 MeV in $\gamma$SF, which is attributed to the M1
strength [16, 17, 18, 19]. Thus, more research is required to explore the role
of M1 excitations in the $\gamma$SF. The M1 spin-flip excitations are obtained
at higher excitation energies around 8 MeV, which has so far been difficult to
measure experimentally [1]. Further, a so-called scissors mode of M1
excitations in deformed nuclei is observed at energies around 3 MeV due to the
scissors-like motion of neutrons and protons relative to each other [20, 21,
22, 17].

M1 excitations exhibit a wide range of characteristics related to their
energies, transition strengths, and decay properties, which are expected to be
highly sensitive to the extreme conditions of temperature ($T$), isospin
(N/Z), and deformation ($\beta$) of nuclei. Recently, the relativistic
quasiparticle random phase approximation (RQRPA) has been utilized to
investigate the role of residual interaction, spin-orbit (SO) splitting,
pairing correlations, and neutron excess on M1 response [23, 24, 25, 26, 27].
However, the knowledge about thermal effects on M1 excitations is rather
limited. Temperature can significantly impact the electromagnetic nuclear
response, which in turn modifies astrophysically relevant quantities such as
neutron capture cross-section, nuclear reaction rates, and element abundances.

In previous studies, temperature effects in electric multipole excitations
have been investigated using several extensions of the random phase
approximation (RPA) [28, 29, 30, 31, 32]. The finite temperature quasiparticle
RPA based on the Skyrme functional was used in studies of the electric
multipole responses of hot nuclei in Refs. [28, 29, 30]. A self-consistent
finite temperature relativistic RPA (FT-RRPA), based on meson-exchange
interaction, was successfully employed to study the evolution of isoscalar and
isovector electric multipole modes with temperature; however, the pairing
correlations were not implemented in this approach [31]. Furthermore, finite-
temperature relativistic time-blocking approximation (FT-RTBA) approach, based
on Matsubara Green’s function formalism, has been developed to investigate the
electric nuclear response in excited nuclei [33, 34]. Lately, electromagnetic
strength distributions in both closed-shell and open-shell nuclei have been
obtained from first principles at zero and finite temperatures, taking into
account pairing and deformation effects [15]. Additionally, a fully self-
consistent finite-temperature relativistic quasiparticle RPA (FT-RQRPA) has
been introduced for non-charge exchange excitations at finite temperatures,
which also includes pairing correlations to address open-shell nuclei [35]. It
has been employed to study the thermal effects on isovector E1 excitations,
demonstrating how the isovector giant dipole resonance and low-energy dipole
excitations evolve with increasing temperature. It has been shown that new
low-energy states appear due to the thermal unblocking effects. Consequently,
the FT-RQRPA provides new perspectives for the microscopic calculation of
$\gamma$SF at finite temperatures associated with nuclear phenomena in stellar
environments. Therefore, it is of particular interest to analyze the thermal
effects on M1 excitations as well as their contributions to $\gamma$SF
relevant for nucleosynthesis. In this work, we focus on M1 excitations,
characterized by unnatural parity transitions, for which additional
modifications of the residual FT-RQRPA interaction are required. More details
about the RQRPA formalism for magnetic transitions and their properties in the
zero-temperature limit can be found in Refs. [25, 23, 24, 26, 27, 36].

The novelty of the present study lies in its exploration of the temperature
dependence of isovector (IV) M1 ($J^{\pi}=1^{+}$) excitations using the newly
developed FT-RQRPA framework for nuclei in the 40-60Ca and 100-140Sn isotopic
chains. To study the unnatural parity excitations of M1 type, the FT-RQRPA
residual interaction is further extended by introducing the relativistic
isovector-pseudovector (IV-PV) contact interaction [25]. The primary goals of
this work are (i) to investigate the evolution of M1 strength distributions
with increasing temperature and isospin of nuclei; (ii) to study the impact of
temperature on the SO splitting energies and their relation with the M1
nuclear response; (iii) to examine the isovector M1 non-energy-weighted and
energy-weighted summations as a function of temperature and mass number; and
(iv) to identify the new M1 proton and neutron excitations in the low-energy
as well as the high-energy region.

The paper is organised as follows: a brief description of the FT-RQRPA
framework used in this study is given in Sec. II. The extension of the
residual interaction for unnatural parity M1 excitations is explained. Sec.
III presents the FT-RQRPA results and discussions of M1 excitations in Ca and
Sn isotopic chains in the temperature range $T=0-2$ MeV. Finally, a summary of
the results and conclusions are outlined in Sec. IV.

##  II Methodology

In this work, we extend the FT-RQRPA from Ref. [35] for the study of unnatural
parity excitation of M1 type. The nuclear properties are described within the
finite temperature Hartree-Bardeen-Cooper-Schrieffer (FT-HBCS) framework [37,
38]. In both the FT-HBCS and FT-RQRPA, the relativistic energy density
functional (REDF) with point coupling DD-PCX interaction is implemented [39].
The point-coupling REDF is determined from the Lagrangian density,

$\mathcal{L}=\mathcal{L}_{\textrm{PC}}+\mathcal{L}_{\textrm{IV-PV}},$ (1)

where $\mathcal{L}_{\textrm{PC}}$ includes fermion contact interaction terms
as isoscalar-scalar, isoscalar-vector and isovector-vector channels, for the
detailed information see Refs. [39, 40]. The Lagrangian density (1) also
includes the relativistic isovector-pseudovector (IV-PV) contact interaction,
which is necessary for the FT-RQRPA residual interaction for the unnatural
parity excitations of M1 type [25],

$\displaystyle\mathcal{L}_{\textrm{IV-PV}}=-\frac{1}{2}\alpha_{\textrm{IV-
PV}}[\bar{\Psi}_{N}\gamma^{5}\gamma^{\mu}\vec{\tau}\Psi_{N}]\cdot[\bar{\Psi}_{N}\gamma^{5}\gamma_{\mu}\vec{\tau}\Psi_{N}].$
(2)

The coupling strength parameter $\alpha_{\textrm{IV-PV}}=0.63$ MeVfm3 for DD-
PCX [39] parameterization is obtained by minimizing the relative error
$\Delta$ $\lesssim$ 1 MeV between experimentally determined M1 peak position
and theoretically calculated centroid energies for magic nuclei 48Ca and 208Pb
[25, 36]. Note that the pseudovector type of interaction has been modeled as a
scalar product of two pseudovectors, which leads to the parity-violating mean
field at the Hartree level. Thus, it does not make a contribution to the
solution of natural-parity states, including the 0+ nuclear ground state.
However, $\mathcal{L}_{\textrm{IV-PV}}$ has a finite contribution in the FT-
RQRPA residual interactions for M1 excitations.

At finite temperature, the occupation probabilities of single-particle states
are given as

$n_{i}=v_{i}^{2}(1-f_{i})+u_{i}^{2}f_{i},$ (3)

where $u_{i}$ and $v_{i}$ are the BCS amplitudes. The temperature-dependent
Fermi-Dirac distribution function is given by

$f_{i}=[1+exp(E_{i}/k_{\textrm{B}}T)]^{-1},$ (4)

$k_{\textrm{B}}$ is the Boltzmann constant and $T$ is the temperature. The
quasiparticle energy of a state is given by

$E_{i}=\sqrt{(\varepsilon_{i}-\lambda_{q})^{2}+\Delta_{i}^{2}},$ (5)

where $\varepsilon_{i}$ denotes the single-particle energies and $\lambda_{q}$
represents the chemical potentials for either proton or neutron states.
$\Delta_{i}$ refers to the pairing gap of the given state. A separable form of
pairing interaction is introduced both in the FT-HBCS and FT-RQRPA, also
including the same relativistic point coupling interaction, DD-PCX [41]. The
values of critical temperatures ($T_{\textrm{c}}$), where the pairing
correlations vanish, are calculated using the FT-HBCS with DD-PCX interaction.
For instance, the $T_{\textrm{c}}$ values of open-shell 44Ca, 52Ca, 56Ca,
64Ca, 108Sn, 116Sn, 124Sn, and 140Sn nuclei are obtained as 0.862, 0.528,
0.743, 0.700, 0.872, 0.834, 0.764, and 0.644 MeV, respectively.

The non-charge exchange FT-RQRPA matrix is given by

$\left({\begin{array}[]{cccc}\widetilde{C}&\widetilde{a}&\widetilde{b}&\widetilde{D}\\\
\widetilde{a}^{+}&\widetilde{A}&\widetilde{B}&\widetilde{b}^{T}\\\
-\widetilde{b}^{+}&-\widetilde{B}^{\ast}&-\widetilde{A}^{\ast}&-\widetilde{a}^{T}\\\
-\widetilde{D}^{\ast}&-\widetilde{b}^{\ast}&-\widetilde{a}^{\ast}&-\widetilde{C}^{\ast}\end{array}}\right)\left({\begin{array}[]{cc}\widetilde{P}\\\
\widetilde{X}\\\ \widetilde{Y}\\\
\widetilde{Q}\end{array}}\right)=E_{w}\left({\begin{array}[]{cc}\widetilde{P}\\\
\widetilde{X}\\\ \widetilde{Y}\\\ \widetilde{Q}\end{array}}\right),$ (6)

where $E_{w}$ denotes the excitation energies, and
$\widetilde{P},\widetilde{X},\widetilde{Y},\widetilde{Q}$ depict the
eigenvectors. The detailed description of the finite temperature QRPA matrices
is given in [42, 29, 30]. The FT-RQRPA matrices are diagonalized in a self-
consistent way, allowing for a detailed analysis of each excitation on a
state-by-state basis. We note that the $A$ and $B$ matrices contribute at $T=$
0 and at $T\neq$ 0 as well; however, other matrix elements begin to contribute
only at finite temperature due to the changes in occupation factors as well as
the temperature factors. The reduced transition probability is given by

$\begin{split}B(\textrm{M}J)=\bigl{|}\langle
w||\hat{F}_{J}||\widetilde{0}\rangle\bigr{|}^{2}&=\biggl{|}\sum_{c\geq
d}\Big{\\{}(\widetilde{X}_{cd}^{w}+(-1)^{j_{c}-j_{d}+J}\widetilde{Y}_{cd}^{w})(u_{c}v_{d}+(-1)^{J}v_{c}u_{d})\sqrt{1-f_{c}-f_{d}}\\\
&+(\widetilde{P}_{cd}^{w}+(-1)^{j_{c}-j_{d}+J}\widetilde{Q}_{cd}^{w})(u_{c}u_{d}-(-1)^{J}v_{c}v_{d})\sqrt{f_{d}-f_{c}}\Big{\\}}\langle
c||\hat{F}_{J}||d\rangle\biggr{|}^{2},\end{split}$ (7)

where $|w\rangle$ is the excited state and $|\widetilde{0}\rangle$ is the
correlated FT-RQRPA vacuum state, and $\hat{F}_{J}$ is the transition operator
of the relevant excitation. To evaluate the magnetic dipole strength, we use
the IV-M1 operator for $k$th nucleon as

$\hat{\mu}_{w}^{M1,IV}=\mu_{N}\sqrt{\frac{3}{4\pi}}\sum_{k\in
A}[g_{s}^{IV}\hat{s}_{w}(k)+g_{l}^{IV}\hat{l}_{w}(k)]\hat{\tau}_{0}(k),$ (8)

including the spin $\hat{s}_{w}$ and orbital angular momentum $\hat{l}_{w}$.
The isospin convention is used as $\hat{\tau}_{0}(k)$ = 1 (-1) for protons
(neutrons). $\mu_{N}=e\hbar/(2m_{N})$ denotes the nuclear magneton, and the
nuclear spin and orbital $g$ factors for the IV-M1 mode are $g_{s}$ = 4.706
and $g_{l}$ = 0.5 (see Refs. [25, 43] for more details). The total M1
transition strength can also be written in a compact form to identify the role
of particular proton or neutron configurations,

$B(\textrm{M1},E_{w})=\biggl{|}\sum_{c\geq
d}\big{[}b^{\pi}_{cd}(E_{w})+b^{\nu}_{cd}(E_{w})\big{]}\biggr{|}^{2}.$ (9)

Here, $E_{w}$ is excitation energy obtained from the FT-RQRPA discrete M1
spectra. The $b^{\pi}_{cd}$($E_{w}$) and $b^{\nu}_{cd}(E_{w})$ represent the
proton ($\pi$) and neutron ($\nu$) partial contributions for a specific
configuration. The two-quasiparticle cut-off energy for the configuration
space $E_{cut}$ is selected as 100 MeV, to provide a sufficient convergence in
the M1 excitation strength. For the presentation of the results, the discrete
FT-RQRPA spectrum of M1 response is smoothed with a Lorentzian function of
$\Gamma=1.0$ MeV width [43].

##  III Results

###  III.1 Calcium isotopes

The M1 excitation at the leading one-body operator level would take place
between the spin-orbit (SO) partner orbits, provided the independent single-
particle picture is a good approximation [24]. Thus, the M1 response provides
important information on the underlying SO splittings. However, in the REDF
framework, the M1 properties depend not only on the SO splittings but also on
the effects of the residual interaction in the RQRPA [24]. The SO splitting
energies are calculated as

$\Delta\varepsilon_{LS}=\varepsilon_{nlj_{<}}-\varepsilon_{nlj_{>}},$ (10)

where ($nlj$) are the quantum numbers of major SO-partner single-particle
states, and $j_{<}=l-1/2$ and $j_{>}=l+1/2$. In Figs. 1(a) and 1(b), the SO
splitting energies are displayed both for neutrons and protons as a function
of temperature for 52Ca. We note that the single-particle energies can easily
be affected by the inclusion of temperature in the calculations, which, in
turn, modifies the SO splitting. It is clearly seen from Fig. 1 that the SO
splitting energy is temperature-dependent. While the SO splitting energy
remains almost constant up to the critical temperature $T_{\textrm{c}}$ =
0.528 MeV, it gradually starts decreasing at higher temperatures. This
reduction of the SO splitting energies above $T_{\textrm{c}}$ will modify the
M1 response in both the low-energy and high-energy regions, as we will discuss
below.

Figure 1: The spin-orbit splitting energies $\Delta\varepsilon_{LS}$ of
neutron (a) and proton (b) configurations as a function of temperature for
52Ca nucleus.

In Fig. 2, we display the M1-transition strength distributions of 40-60Ca
nuclei with increasing temperature. First, we consider the results at $T=0$
MeV for 40Ca $(Z,N=20)$, showing the absence of any M1-response, as
($1p_{3/2}$, $1p_{1/2}$) and ($1d_{5/2}$, $1d_{3/2}$) states are fully
occupied for protons and neutrons; therefore, no SO partners are available for
M1 transition. Similarly, for 60Ca with $N=40$ neutrons, states up to
$1f_{5/2}$ and $2p_{1/2}$ are fully occupied, and hence the M1 transition is
forbidden. Thus, no M1-response is obtained for these two nuclei at zero
temperature. For 44-56Ca, one can see a strong peak in each isotope that
attributes to the M1 excitation of valence neutron transitions
$\nu$($1f_{7/2}$ $\rightarrow$ $1f_{5/2}$), whereas M1 transitions are not
present for protons due to the shell closure at $Z=20$. A low-energy M1 peak
is also observed in 52,56Ca nuclei due to the $\nu$($2p_{3/2}$ $\rightarrow$
$2p_{1/2}$) transition. More details about the evolution of M1 strength along
the 40-60Ca isotopic chain at zero temperature are given in Ref. [24].

Figure 2: The isovector M1 transition strength distributions of 40-60Ca isotopes, calculated using the FT-RQRPA at temperatures $T=0$, $0.5$, $1$, and $2$ MeV. Table 1: The isovector partial contributions $b_{2qp}^{\pi(\nu)}$ [$\mu_{N}$] of protons ($\pi$) and neutrons ($\nu$) to the dominant high-energy M1 excitations in 52Ca nucleus at $T=0$, 1 and 2 MeV are listed at corresponding excitation energies ($E$). The total $B(M1,E)$ values are obtained using Eq. (9), by summing over all proton and neutron configurations, also including those not listed in the table. The major contributions are highlighted in bold. Configuration |  $T=0$ MeV |  $T=1$ MeV |  |  $T=2$ MeV
---|---|---|---|---
|  $E=$ 9.09 MeV |  $E=$ 8.84 MeV |  $E=$ 6.85 MeV |  |  $E=$ 8.71 MeV |  $E=$ 8.24 MeV |  $E=$ 6.68 MeV
$\nu$(1f7/2$\rightarrow$1f5/2) | -3.363 | -3.207 | 0.216 |  | -2.397 | -1.694 | 0.382
$\nu$(2p3/2$\rightarrow$2p1/2) | -0.029 | -0.012 | -0.003 |  | -0.005 |  | -0.003
$\nu$(1f5/2$\rightarrow$2f7/2) |  | -0.006 |  |  | 0.012 | 0.021 | -0.003
$\nu$(1g9/2$\rightarrow$2g7/2) |  | -0.002 |  |  | -0.082 | 0.039 | 0.004
$\pi$(1f7/2$\rightarrow$1f5/2) |  | -0.117 | 0.005 |  | -0.615 | 1.009 | 0.060
$\pi$(1d5/2$\rightarrow$1d3/2) |  | -0.027 | -0.557 |  | -0.115 | -0.028 | -1.189
total $B(M1,E)$[$\mu_{N}^{2}$] | 11.262 | 11.032 | 0.115 |  | 10.045 | 0.416 | 0.565
Table 2: Same as Table 1, but for the low-energy M1 excitations in the 52Ca nucleus for $E<5$ MeV. Configuration |  $T=0$ MeV |  $T=1$ MeV |  $T=2$ MeV
---|---|---|---
|  $E=$ 2.98 MeV |  $E=$ 2.14 MeV |  $E=$ 4.36 MeV |  $E$=1.86 MeV
$\nu$(2p3/2$\rightarrow$2p1/2) | -1.726 | 1.520 | -0.002 | 1.057
$\nu$(1f7/2$\rightarrow$1f5/2) | 0.192 | -0.123 | 0.028 | -0.067
$\nu$(1g9/2$\rightarrow$1g7/2) |  |  | -0.364 | -0.002
$\pi$(1d5/2$\rightarrow$1d3/2) |  | -0.006 | 0.005 | -0.018
$\pi$(1f7/2$\rightarrow$1f5/2) |  | -0.004 | 0.004 | -0.013
$\pi$(2p3/2$\rightarrow$2p1/2) |  |  |  | -0.012
total $B(M1,E)$ [$\mu_{N}^{2}$] | 2.336 | 1.923 | 0.108 | 0.892
Figure 3: Major high-energy ($E_{\text{peak}}^{\text{High}}$) and low-energy
($E_{\text{peak}}^{\text{Low}}$) M1 peak excitation energies, along with the
SO splitting energies ($\Delta\varepsilon_{LS}$) of the respective primary
contributing neutron configurations in 52Ca. The results are shown as a
function of temperature in the range from $T=0$ to $2$ MeV. Figure 4: (a) Non-
energy-weighted $m_{0}$ and (b) energy-weighted $m_{1}$ moments of M1 response
in 40-60Ca between $T=0-2$ MeV.

Further, it is clearly evident from Fig. 2 that M1 transition strength
distributions are remarkably sensitive to changes in temperature. At $T$ = 0.5
MeV, the results do not change much in Ca isotopes. By increasing temperature
further, at $T$ = 1 and 2 MeV, an interesting outcome is observed for the case
of 40Ca and 60Ca. The M1 transition strength suddenly appears for these
isotopes due to the emergence of new transitions in the $\nu,\pi$ ($1d_{5/2}$,
$1d_{3/2}$) and $\nu,\pi$ ($1f_{7/2}$, $1f_{5/2}$) configurations. This occurs
because particles are promoted to higher-energy orbits as a result of
temperature effects, leading to the thermal unblocking of forbidden M1
transitions. It is also noticed in Fig. 2 that M1 response shifts up to
$\approx$ 2 MeV to the lower energies for 44,48,52,56Ca with increasing
temperature. In addition to the decrease in the pairing correlations for open-
shell nuclei and softening in the repulsive residual interaction, the changes
in the M1 response are also linked to the SO splitting energies, which reduce
with increasing temperature, as shown in Fig. 1. Due to the thermal unblocking
of states, new proton and neutron transitions also become possible in both
high-energy ($E>$ 5 MeV) and low-energy ($E<$ 5 MeV) regions of open-shell
nuclei 44,48,52,56Ca. In the high-energy region of these nuclei, proton and
neutron ($1f_{7/2}$$\rightarrow$$1f_{5/2}$) transitions are the major
contributors to the M1 strength at $T=1$ and 2 MeV. A finite contribution from
$\pi$($1d_{5/2}$$\rightarrow$$1d_{3/2}$) transition is also observed at $T=2$
MeV in 40-60Ca nuclei. Also, new smaller peaks arise in the low-energy region
($E<$ 5 MeV) of neutron rich 48-60Ca isotopes as a result of neutron
transitions, e.g., $\nu$($2p_{3/2}$$\rightarrow$$2p_{1/2}$) and
$\nu$($1f_{7/2}$$\rightarrow$$1f_{5/2}$).

Tables 1 and 2 display the proton and neutron isovector partial contributions
($b_{2qp}^{\pi(\nu)}$) to the M1 transition strength, as defined in Eq. (9),
for the 52Ca nucleus. Table 1 pertains to the high-energy region, whereas
Table 2 is provided for the low-energy region. It is evident from the tables
that the number of excitation energies and related contributing transitions in
the M1 response increases as one moves from $T=0$ to 2 MeV, signifying the
thermal unblocking of transitions. The reduced transition probability for a
specific excited state, denoted as $B(M1,E)$, is determined by summing up the
contributions of each configuration while considering their relative signs.
First, we observe that temperature leads to the fragmentation of the excited
states, impacts the contribution of the main configurations, and causes a
decrease in the total $B(M1,E)$ value for a particular state with increasing
temperature. Second, we found that new states appear; however, the
contributions of these new configurations are either too small, or they
interfere destructively, which, in turn, results in low strength. While
temperature leads to the scattering of nucleons into the continuum, this
effect is less pronounced for protons due to the Coulomb barrier. Hence,
neutron transitions dominate in the low-energy region of M1 strength for
neutron-rich Ca nuclei.

For further illustration, Figure 3 displays the temperature dependence of the
major high-energy $E_{\text{peak}}^{\text{High}}$ and low-energy
$E_{\text{peak}}^{\text{Low}}$ peaks in the M1 strength distribution for 52Ca.
In this case, $\nu$($1f_{7/2}$, $1f_{5/2}$) and $\nu$($2p_{3/2}$, $2p_{1/2}$)
transitions are the primary contributors to the high-energy and low-energy
peaks, respectively. Therefore, in Fig. 3, the SO splitting energies
$\Delta\varepsilon_{LS}$ of these two respective configurations are shown for
comparison relative to the M1 excitation energies. For the
$E_{\text{peak}}^{\text{High}}$, we observe a decrease around the critical
temperature $T_{\textrm{c}}$ = 0.528 MeV, which then continues to decrease
gradually with increasing temperature. The SO splitting energy also decreases
slightly. Although the impact of pairing is subtle in this region, its effect
can be seen in the comparably rapid decrease in energy around the critical
temperature. At higher temperatures, the slight decrease in the
$E_{\text{peak}}^{\text{High}}$ is mainly related to the decrease in the SO
splitting energy and weakening of the repulsive residual interaction. On the
other hand, we observe a rapid decrease in the $E_{\text{peak}}^{\text{Low}}$
near the critical temperature, which is similar to the pairing phase
transition, demonstrating the subtle interplay between the temperature effects
and pairing correlations. Although the SO splitting energies are obtained as
almost constant up to $T_{\textrm{c}}$, the decrease in the pairing
correlations with increasing temperature leads to a sharp reduction of the
energy. In other words, the pairing plays a significant role in the low-energy
region of the M1 response of open-shell nuclei below $T_{\textrm{c}}$. At
higher temperatures, $E_{\text{peak}}^{\text{Low}}$ continues to gradually
decrease, but at a faster rate compared to $E_{\text{peak}}^{\text{High}}$
since the SO splitting energy of the relevant transition decreases more
rapidly.

Next, we study the non-energy-weighted $m_{0}$ and energy-weighted $m_{1}$
moments of the M1 transition strength, calculated using the FT-RQRPA method.
In Figure 4, we can see the behavior of $m_{0}$ (a) and $m_{1}$ (b) as a
function of the mass number $A$ for Ca isotopes at temperatures ranging from
$T=0$ to $2$ MeV. The M1 response does not appear when the SO-partner states
are either fully occupied or completely empty, a characteristic stemming from
the use of the one-body operator in Equation (8). Consequently, the M1
response is absent at $T$=0 MeV in 40,60Ca and appears predominantly in other
Ca isotopes through neutron transitions, e.g., $\nu$($1f_{7/2}\rightarrow
1f_{5/2}$). It can be seen from Fig. 4(a) and 4(b) that the moments $m_{0}$
and $m_{1}$ have the highest magnitude for 48Ca and 52Ca, since the
$\nu(1f_{7/2})$ (and $\nu(2p_{3/2})$ for 52Ca) state is nearly filled, and the
corresponding M1 excitation strength is higher. For heavier isotopes of
56,60Ca, as the $\nu$($1f_{5/2}$) state begins to fill, the transition
strength decreases due to the blocking of the $\nu$($1f_{7/2}\rightarrow
1f_{5/2}$) transition, as can be seen in Figs. 2 and 4 at zero temperature.
Similar results are also obtained in Ref. [24].

With the increase in temperature, the $m_{0}$ and $m_{1}$ moments of 40,60Ca
nuclei show a rapid increase above $T=$ 0.8 MeV due to the opening of new
proton and neutron excitation channels. Although it is less pronounced
compared to 40Ca and 60Ca, we also obtain an increase in the moments of 44Ca
and 56Ca at higher temperatures. These results are also consistent with the
results obtained in observation of Fig. 2 and illustrate substantial
contributions of new transitions which become allowed at higher temperatures.
On the other hand, we obtain a moderate decrease in the moments for 48Ca and
52Ca at high temperatures. The main reason for the decrease in the $m_{0}$ and
$m_{1}$ moments in 48Ca and 52Ca is the impact of the temperature on the
$\nu$($1f_{7/2}$) and $\nu$($2p_{3/2}$) states. As can be seen from Tables 1
and 2, the contribution of the $\nu$($1f_{7/2}$, $1f_{5/2}$) and
$\nu$($2p_{3/2}$, $2p_{1/2}$) transitions to the M1 strength decreases
substantially with increasing temperature, and the contribution of the new
states is found to be subtle since their partial magnitudes are small and
interfere destructively, resulting in a decrease in the M1 strength.

Figure 5: The same as in Fig. 2, but for 100-140Sn isotopes. Table 3: Same as table 1 but for the high-energy M1 excitations in 140Sn nucleus. The excitation energies ($E$) are given in units of MeV. Configuration |  $T=0$ MeV |  |  $T=1$ MeV |  |  $T=2$ MeV
---|---|---|---|---|---
|  $E$=7.12 |  $E$=9.16 |  |  $E$=6.97 |  $E$=8.41 |  $E$=8.46 |  |  $E$=6.79 |  $E$= 8.04 |  $E$=8.20 |  $E$=8.27 |  $E$=8.98
$\nu$(1h11/2$\rightarrow$1h9/2) | -1.049 | 3.664 |  | 1.136 | 2.382 | -2.743 |  | 1.041 | -1.857 | 2.301 | -1.963 | 0.437
$\nu$(2f7/2$\rightarrow$2f5/2) | 0.062 | 0.046 |  | -0.019 | 0.013 | -0.013 |  | -0.008 | -0.003 | 0.006 | -0.005 | 0.003
$\nu$(2f7/2$\rightarrow$3f5/2) | -0.020 | 0.037 |  | 0.031 | 0.016 | -0.016 |  | 0.014 | -0.007 | 0.012 | -0.009 | 0.003
$\nu$(1h9/2$\rightarrow$2h11/2) | 0.001 |  |  | -0.003 | -0.028 |  |  |  | -0.013 | 0.111 | 0.145 | -0.003
$\nu$(1i13/2$\rightarrow$i11/2) |  |  |  | -0.008 | 0.004 | -0.003 |  | -0.049 | -0.008 | 0.015 | -0.013 | 0.005
$\nu$(1i13/2$\rightarrow$2i11/2) | -0.001 |  |  | 0.005 | -0.029 | 0.031 |  | 0.022 | 0.038 | -0.087 | 0.089 | 0.890
$\pi$(1g9/2$\rightarrow$1g7/2) | 3.913 | 0.828 |  | -3.227 | 0.748 | -0.806 |  | -3.183 | -0.281 | 0.505 | -0.477 | 0.158
$\pi$(1h11/2$\rightarrow$1h9/2) |  |  |  |  | 0.006 |  |  | 0.029 | 0.609 | 0.298 | -0.179 | 0.019
total $B(M1,E)$ [$\mu_{N}^{2}$] | 8.448 | 20.585 |  | 4.420 | 9.627 | 12.403 |  | 4.647 | 2.313 | 9.979 | 5.811 | 2.281

###  III.2 Tin isotopes

The temperature dependence of isovector M1 strength distributions is also
systematically studied in even-even 100-140Sn isotopes, as shown in Fig. 5.
First, we consider the $T=0$ MeV case. The M1 excitations are expected due to
the proton $\pi(1g_{9/2}\rightarrow 1g_{7/2})$ and the neutron
$\nu(1g_{9/2}\rightarrow 1g_{7/2})$, $\nu(2d_{5/2}\rightarrow 2d_{3/2})$,
and/or $\nu(1h_{11/2}\rightarrow 1h_{9/2})$ transitions until the higher state
in the SO configuration becomes fully occupied to block the M1 transitions. In
closed shell nuclei 100Sn and 132Sn, the pairing correlations do not
contribute, hence, the M1-excitation energy is determined mainly by the SO
splitting energy as well as the residual interaction. For 100Sn at zero
temperature, the M1 strength is obtained as a single peak at 9.47 MeV, which
is mainly formed with the proton and neutron ($1g_{9/2}$ and $1g_{7/2}$)
transitions. As the number of neutrons increases along the Sn isotope chain,
the M1 response exhibits two peaks in the high-energy region. The first peak
at lower energy is primarily due to proton transitions, while the second, at
higher energy, is dominated by neutron transitions. Similar results are
obtained for the $T=$ 0.5 MeV case, where M1 strength distribution shifts
slightly to the lower energies, as shown in Fig. 5. A thorough study of M1
transitions in 100-136Sn isotopes has been carried out at $T$ = 0 MeV using
the RQRPA framework, and a detailed description is given in Ref. [26].

Table 4: Same as Table 1 but for the low-energy M1 excitations in 140Sn nucleus. Configuration |  $T=0$ MeV |  $T=1$ MeV |  $T=2$ MeV
---|---|---|---
|  $E$=3.57 MeV |  $E$=2.26 MeV |  $E$=2.17 MeV
$\nu$(2f7/2$\rightarrow$2f5/2) | 2.078 | -2.267 | -1.434
$\nu$(1h11/2$\rightarrow$1h9/2) | -0.138 | 0.102 | 0.079
$\nu$(2d5/2$\rightarrow$2d3/2) |  |  | -0.166
$\pi$(2d5/2$\rightarrow$2d3/2) |  | -0.016 | -0.131
$\pi$(1g9/2$\rightarrow$1g7/2) | -0.267 | 0.155 | 0.092
total $B(M1,E)$[$\mu_{N}^{2}$] | 2.815 | 4.088 | 2.403

Similar to the findings in Ca isotopes, the M1 strength starts to shift
downward, and new excited states appear both in the low-energy and high-energy
regions with increasing temperature. Besides the downward shift in the M1
strength, we also observe a change in the 108,124,140Sn nuclei from a two-peak
to a single-peak structure of M1 strength as the temperature increases. Due to
the vanishing of pairing correlations at critical temperatures, the
configuration energies between the proton and neutron components start to
approach one another, which results in the merging of two peaks into a single
peak at higher temperatures. To elucidate further, the structure of M1
strength distributions at finite temperatures is explored by analysing all
transitions that contribute to these states for each considered Sn nucleus at
$T=1$ and 2 MeV. This study has revealed that along with the major
$\nu,\pi$$(1g_{9/2}\rightarrow 1g_{7/2})$ transitions,
$\nu,\pi$$(1h_{11/2}$$\rightarrow$$1h_{9/2}$) transitions also start
contributing in the high-energy region of 100-140Sn nuclei at higher
temperatures. A finite contribution from
$\nu$($1i_{13/2}$$\rightarrow$$2i_{11/2}$) and
$\nu$($1h_{9/2}$$\rightarrow$$2h_{11/2}$) transitions are also obtained in the
neutron-rich Sn isotopes due to the thermal unblocking. Moreover, a new low-
energy M1 peak below $E=$ 5 MeV appears in all Sn isotopes at finite
temperatures. In the lower-mass Sn isotopes, the
$\nu$($2d_{5/2}$$\rightarrow$$2d_{3/2}$) transition is the major contributor
to the low-energy peak, and $\nu$($2f_{7/2}$$\rightarrow$$2f_{5/2}$)
transition also begins contributing in the neutron-rich nuclei. A thorough
analysis of the partial contributions of protons and neutrons in the high-
energy ($E>5$ MeV) and low-energy regions ($E<5$ MeV) is given in Tables 3 and
4, respectively, for the neutron-rich 140Sn nucleus at $T=0$, 1, and 2 MeV.

Figure 6: The isovector M1 non-energy-weighted sum $m_{0}$, energy-weighted
sum $m_{1}$ and the centroid energy ($m_{1}/m_{0}$) in 100-140Sn for the low-
energy region $E<$ 5 MeV [panels (a)-(c)] and high-energy region $E>$ 5 MeV
[panels (d)-(f)] as a function of temperature.

This investigation extends further to the M1 non-energy-weighted $m_{0}$ and
energy-weighted $m_{1}$ summations, along with the centroid energies
($E_{c}=m_{1}/m_{0}$), as functions of temperature for 100-140Sn isotopes, as
shown in Fig. 6. The upper panels (a-c) display moments for the low-energy
region ($E<5$ MeV), while the lower panels (d-f) represent these summations
for the high-energy region ($E>5$ MeV). This energy cut-off is selected
because it clearly separates the low- and high-energy regions in the M1
strength distributions for all Sn isotopes, as shown in Fig. 5. We then
elaborate on the variation of moments in the low-energy region with increasing
temperature, as presented in Figs. 6(a-c). For closed-shell 100,132Sn nuclei,
the moments and centroid energy remain nearly zero up to $T=0.8$ MeV because
the spin-orbit partner states are unavailable for M1 transitions in the low-
energy region. Then, $E_{c}$ values increase linearly with temperature due to
the thermal unblocking of these states. The rate of increase in the moments is
higher for 132Sn compared to 100Sn because it is neutron-rich, and with
increasing temperature particles are more easily scattered to the continuum
region. Additionally, we observed a slight increase in the centroid energies
with increasing temperature. However, the behavior of the $m_{0}$ and $m_{1}$
moments in open-shell nuclei with increasing temperature is not
straightforward and displays dependence on the particular shell structure. In
all open-shell nuclei, a sharp increase is observed in the moments above $T=$
0.4 MeV and up to the critical temperatures. This change is mainly related to
the decrease in the pairing correlations. At higher temperatures, pairing
correlations disappear and a decrease is observed in the $m_{0}$ and $m_{1}$
moments for 108,116,140Sn. Only for 124Sn, the moments continue to increase
with increasing temperature. For instance, in 140Sn, we found that the impact
of the major configuration $\nu$(2f7/2$\rightarrow$2f${}_{5/2})$ decreases
with increasing temperature due to the weakening of the residual interaction,
and the partial contributions from the new configurations interfere
destructively. As a result, we observe a decrease in the moments, as
illustrated in Table 4. Similar results are also obtained for 108,116,140Sn.
For 124Sn, there is a negligible M1 response below $E<5$ MeV at very low
temperatures due to the inaccessibility of single-particle (SP) partners for
M1 transitions in the low-energy region. However, due to thermal unblocking,
new excitation channels primarily emerge between the $\nu$($2d_{5/2}$,
$2d_{3/2}$) configurations, and the sum of moments $m_{0}$ and $m_{1}$ also
increases with temperature. It is also observed that the difference between
the $m_{0}$ and $m_{1}$ moments of different nuclei decrease as the shell
effects diminish at higher temperatures. Finally, the centroid energy first
decreases sharply up to the critical temperatures and then exhibits only
slight changes as the temperature increases further.

In the high-energy region ($E>$ 5 MeV), the isovector variables $m_{0}$,
$m_{1}$, and $E_{c}$ smoothly reduce for closed-shell nuclei 100,132Sn. This
means that at higher temperatures, a portion of the M1 strength shifts to the
low-energy region. However, a transition in the variation of moments and
centroid energies is observed around $T\approx$ 0.8 MeV (or critical
temperatures) for open-shell Sn isotopes in the high-energy region. At higher
temperatures, we observed either a slight increase or decrease in the $m_{0}$
and $m_{1}$ moments, depending on the particular nuclei. For all open-shell
nuclei, the centroid energy decreases with increasing temperature up to the
critical point, after which it changes only slightly with further temperature
increases. As discussed earlier in the explanation of Fig. 5, this transition
occurs due to the interplay of pairing and thermal effects. The behavior of
the M1 strength is governed by subtle effects related to single-particle
structure, pairing correlations, respective occupation probabilities, and the
FT-RQRPA residual interaction.

##  IV Summary

A novel approach, the fully self-consistent FT-RQRPA framework based on the
relativistic energy density functional, has been applied to study the
temperature dependence of isovector M1 excitations. The properties of 40-60Ca
and 100-140Sn isotopes were calculated at finite temperatures within the
relativistic framework using the FT-HBCS with the DD-PCX functional and
separable pairing interaction. The FT-RQRPA is extended for the study of
unnatural parity excitations of M1 type.

By employing the FT-HBCS, we studied the evolution of spin-orbit (SO)
splitting energies in the open-shell nucleus 52Ca at temperatures both below
and above the critical temperature $T_{\textrm{c}}$. We observed a reduction
in the spin-orbit splitting energies as the temperature increased. The FT-
RQRPA calculation of M1 response in both Ca and Sn isotopes showed a
systematic shift of the strength distributions to lower energies in both Ca
and Sn isotope chains as the temperature increased. The changes in the M1
response are influenced not only by the weakening and disappearance of the
pairing correlations and the softening of the repulsive residual interaction
but also by the decreasing SO splitting energies. Remarkably, we found that M1
strength suddenly emerges as the temperature increases in nuclei with
initially blocked M1 transitions. This phenomenon is attributed to the thermal
unblocking of new proton and neutron excitations between the spin-orbit
partner states. The thermal unblocking of states also results in the formation
of new M1 peaks in the low-energy region.

The results of our study are also quantitatively assessed by analyzing the
temperature dependence of the non-energy-weighted sum $(m_{0})$, the energy-
weighted sum of the transition strength $(m_{1})$, and the centroid energy in
the considered isotope chains. The changes in the moments $m_{0}$ and $m_{1}$
of Ca and Sn nuclei with increasing temperature indicate that the isotopic
evolution of the M1 response lacks a uniform pattern, at least at low
temperatures. This lack of uniformity can be attributed to the influence of
multiple contributing factors, such as the particular shell structure, pairing
correlations, and residual interaction. In general, we observe a sharp change
in the variation of moments and centroid energies with temperature for open-
shell Sn nuclei. This change results from a phase transition from a superfluid
state to the normal state that occurs near the critical temperatures, where
pairing effects decrease sharply. At high temperatures, the shell effects also
diminish, and the differences between the $m_{0}$ and $m_{1}$ moments decrease
for the considered isotope chains. In conclusion, we have found that
temperature effects can considerably modify the magnetic dipole response.
Further studies on the possible contributions of magnetic transitions at
finite temperature in the $\gamma$-strength functions for $(n,\gamma)$
reactions are required. These studies are left for future work.

##  V Acknowledgements

This work is supported by the QuantiXLie Centre of Excellence, a project co-
financed by the Croatian Government and European Union through the European
Regional Development Fund, the Competitiveness and Cohesion Operational
Programme (KK.01.1.1.01.0004). E.Y. acknowledges support from the Science and
Technology Facilities Council (UK) through grant ST/Y000013/1.

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