Jose C Jimenez-Lopez

B Karam

Singh

Alfonso Clemente

Jaroslaw Czubinski

Sergio Ochatt

Eric Von Wettberg

Petr Smy ´kal

Editorial: Legumes for global food security -volume II

Keywords: legumes, global food security, crop improvement, nutrient management, sustainable agriculture, food and feed security, climate resilience

Legumes for global food security -volume II

As the world faces an array of global challenges, including population growth, climate change, and the need for clean energy, the role of legumes in addressing these issues becomes increasingly significant. The Research Topic, "Legumes for Global Food Security, Volume II" aims to explore the diverse contributions of legumes in promoting sustainable agriculture and enhancing global food security.

Legumes play a crucial role in delivering vital services to societies worldwide. One of their primary contributions lies in their capacity to provide a wide range of food crops that serve as essential sources of plant-based proteins, addressing the challenge of food security for a rapidly growing population. Furthermore, grain legumes possess remarkable nutritional properties and act as cost-effective food choices, playing a pivotal role in achieving global food and feed security amid the growing world population. The significance of legumes extends beyond their role as a food source. Through rhizobial symbiosis, legumes have the unique ability to fix atmospheric nitrogen, enriching agroecosystems and boosting subsequent crop productivity by enhancing water and nutrient capture and recycling. Moreover, they play a key role in mitigating climate change, offering an alternative to synthetic nitrogenous fertilizers, which is energy-intensive to produce and release greenhouse gases upon breakdown. Furthermore, legumes contribute to the reduction of fossil fuel usage by providing biofuel feedstocks and industrial resources. Given the challenges posed by increasing climatic stresses, legumes' genetic diversity equips them to thrive in various environments, rendering them resilient and ideal for sustainable intensification on small-scale and resource-constrained farms. Moreover, they play a vital role as biocontrol agents, effectively combating pests and diseases that could otherwise cause significant agricultural losses.

This Research Topic aims to explore the multifaceted contributions of legumes in the development of robust and efficient agro-ecosystems, thereby enhancing global food security.

Here we summarize some of the highlights derived from the 22 articles published in this Research Topic, dividing them in four main topics, in order to better understand how research on legumes and related crops is contributing to crop improvement, adaptation, and nutrient management, ultimately aiming to address food security challenges and support sustainable agriculture.

Regarding crop adaptation to abiotic stress and improvement of abiotic stress tolerance, Gupta et al. assessed heat tolerance in urdbean genotypes to identify heat-tolerant cultivars suitable for cultivation during the summer season. In this paper, 97 diverse genotypes of urdbean (Vigna mungo L. Hepper) were assessed for yield under heat stress and non-stress conditions to identify heattolerant genotypes. Eight genotypes were highly heat tolerant, while 35 were highly heat sensitive. Physiological and biochemical characterization of selected genotypes under heat stress revealed variations in leaf nitrogen balance index, chlorophyll content, flavonols, and anthocyanin contents. Heat-tolerant genotypes exhibited higher membrane stability index and superior photosynthetic ability. Molecular characterization distinguished genetic differences between heat-tolerant (UPU 85-86) and heatsensitive (PKGU 1) genotypes.

Then, Diaz et al. investigated the genetic relationship between iron and zinc concentration and yield in common beans to develop biofortified varieties with higher micronutrient levels. In their study, they focused on common bean (Phaseolus vulgaris L.), an important legume that provides cost-effective proteins and micronutrients, especially iron and zinc. To combat malnutrition in developing countries, biofortification aims to develop varieties with higher Fe/Zn content. However, breeders faced challenges due to the negative correlation between Fe/Zn concentration and yield. Using QTL mapping and GWAS analysis on biofortified parent populations, they identified 79 QTLs and 23 hotspot regions that showed QTLs with opposing effects on yield components and Fe/Zn accumulation. This allowed them to select specific QTLs to enhance Fe/Zn levels without compromising yield in biofortified cultivars.

Alkaloids, a diverse group of cyclic nitrogen-containing secondary metabolites, are present in over 20% of plant species, including Lupinus albus (white lupin), which naturally contains quinolizidine alkaloids (QAs) in seeds. While QAs provide natural protection, lupin-breeding programs have unintentionally selected against them due to limited understanding of the QA biosynthetic pathway. The review by Osorio and Till discusses the current state of research, focusing on natural mutations like the pauper locus and the use of molecular markers and sequencing technology to identify candidate genes. Precision breeding of low-alkaloid, high-nutrition white lupin can be achieved by understanding QA biosynthesis, essential for sustainable agricultural productivity and high-quality protein for food and feed.

Then, research performed by Woo Choi et al. aimed to improve soybean food quality by breeding a new soybean line with penta null recessive alleles for five antinutritional and allergenic components: lipoxygenase, KTI, lectin, 7S a′ subunit, and stachyose. The breeding resulted in a soybean strain with the penta null (lox1lox2lox3/lox1lox2lox3-ti/ti-le/le-cgy1/cgy1-rs2/rs2) genotype, devoid of the mentioned proteins and with low stachyose content. The new strain exhibited desirable agronomic traits, including purple flowers, tawny pubescence, determinate growth habit, and yellow pods at maturity. The 100-seed weight was 31.1 g, and the yield was 2.80 t/ha, making it a promising soybean line with improved nutritional attributes.

Finally, the paper by Li et al. focused on the important area of plant responses phosphate deficiencies. Phosphorus is a key micronutrient for crops and large areas of arable land (30-40%) are limited by phosphorus availability. This study identified GmWRKY46 as crucial for enhanced phosphate starvation tolerance in soybean. It was induced in response to phosphate starvation, particularly in roots. Overexpression of GmWRKY46 in transgenic Arabidopsis and soybean composite plants showed its involvement in root development, leading to increased phosphate uptake and improved growth under phosphate starvation. RNAseq and ChIP-qPCR analysis identified differentially expressed genes, including AtED1, which improved tolerance to phosphate starvation when overexpressed in transgenic Arabidopsis. GmWRKY46 directly binds to a W box in the AtED1 promoter. Overall, GmWRKY46 represents a promising target for conventional breeding and transgenic approaches to enhance soybean's phosphate starvation tolerance.

The second main theme of this Research Topic is genetic variability and yield improvement. In this regard, a study performed by Susmitha et al. assessed the genetic variability of grain nutrients in 600 pigeonpea germplasms from the RS Paroda Genebank, ICRISAT, India, aiming to identify potential sources for biofortification. Field trials in 2019 and 2020 revealed significant differences in agronomic traits and grain nutrients. Germplasms showed wide variation in days to 50% flowering, days to maturity, 100-seed weight, and grain yield per plant. Grain nutrients like protein, minerals (P, K, Ca, Mg, Fe, Zn, Mn, Cu) exhibited substantial variation. Germplasms from Asia had diverse grain nutrient levels, while those from Africa showed high nutrient density. Some germplasms exhibited favorable nutrient profiles and can serve as promising sources for developing biofortified lines with improved agronomic traits. The study's phenotypic data can aid genetic improvement efforts through GWAS and SNP/haplotype-based approaches.

Then, Habtegebriel performed a study focused on genotype by environment interaction (GEI) in soybean, which affects breeding progress by hindering the selection of superior cultivars. The objective was to identify adapted and stable genotypes and explore potential mega-environments for future testing. Experiments were conducted over two years, evaluating yield components and other traits. Stability analysis using GGE biplot and AMMI model identified five genotypes as top performers. Genotypes JM-CLK/CRFD-15-SD (G8) and 5002T (G1) displayed the highest seed yield, making them suitable as parents for hybridization and commercial production. These genotypes are recommended for release as new soybean varieties for cultivation across various environments.

Rodas et al. investigated the SUPERMAN (SUP) gene, initially discovered in Arabidopsis thaliana, that maintains boundaries between reproductive organs, regulating stamen and carpel number in flowers. This study focuses on the SUP ortholog, MtSUP, in the legume Medicago truncatula. M. truncatula serves as a model to study unique developmental traits in legumes, like compound inflorescence and complex floral development. MtSUP plays a role in the genetic network controlling these processes, sharing conserved functions with SUP but also exhibiting contextspecific novel functions in legumes. MtSUP controls flower number and determinacy of ephemeral meristems, providing insights into compound inflorescence and flower development in legumes. Understanding these genetic controls can benefit legume breeding for improved crop production and food security.

The interest in plant proteins for human and animal nutrition continues to grow with legume crops being at the forefront of efforts to breed crops with higher protein levels and better quality. The paper by Zhou et al. assessed two important protein quality traits in pea (Pisum sativum L.), a major grain legume grown widely worldwide. They used field trials spread over 3 years and in multiple locations to assess the amino acid profiles and protein digestibility in a pea recombinant inbred line (RIL) population consisting of 110 lines. While pea has reasonable levels of protein, it is low in some amino acids such as tryptophan, methionine and cysteine. The authors used near-infrared spectroscopy to measure the amino acid profiles in the RILs and showed that this method had advantages for such studies over other methods used previously, being a high-throughput, low cost and non-destructive method. The use of the method, coupled with the use of multiple sites and a three-year period, helped to identify QTLS associated with key amino acids as well as QTLS associated with protein digestibility. These QTLs and the improved method for measuring seed proteins will help with efforts to breed pea cultivars with improved nutritional qualities.

Given that their production in the tropics is located on marginal, rainfed lands and hence yield is low, Smith et al. assessed the adaptation of 12 bush bean (Phaseolus vulgaris L.) genotypes to individual and combined stress induced by drought and low P availability through their physiological and chemical responses. Thus, seed weight and aerial biomass were very significantly decreased by both stress factors, whether applied individually or in combination, and was coupled with an also decreased photosynthesis and stomatal conductance. Differences between genotypes were apparent though, and the common bean genotypes SEF60 and NCB226 were more resilient and better adapted to stress than the commercial control DOR390 providing a higher yield, but there were no statistically significant differences in the concentration of mineral nutrients and amino acids within the seed at harvest. Under water deficit, carbon assimilation and water use in leaves were reduced throughout development, while combined low P and water deficit resulted in significant changes in the concentration of key nutrients and amino acids in the soluble leaf fraction but did not impact the seed. Those results suggest that common bean genotypes have a certain degree of resilience in terms of yield as expressed through pod harvest index and conservation of seed nutritional content.

Keller et al. examined genetic variation for flowering time, yield, seed iron content, and growth habit in climbing common beans. In general, there has been paid less attention to the genetic basis of agronomic traits in climbing types than bush types of common bean, and the authors set out to correct this with an examination of 17 field trials and reanalysis of 16 previously published datasets from both the Meso-American and Andean genepools of common bean. In addition to finding a number of marker-trait associations, the authors show that genomic prediction is improved when bush bean information is included in models predicting climbing bean genotype-phenotype associations.

Finally, Zaki and Radwan studied the genotypic and phenotypic variance of five parental genotypes and six generated crosses in F1 and F2 generations. The results indicated a substantial degree of genotypic variability across the investigated variables, which was considerably greater than the phenotypic variance. The authors succeeded in obtaining crosses that possess promising potential for high yield.

The third topic includes five papers about improvement of nutritional quality of legumes. Salaria et al. reviewed the potential pathways for increase seed protein content, the role of antinutritional compounds, and the extent of genetic variation in cultivated lentil and its cross-compatible wild relatives. The authors discussed the need for careful phenotyping, and explored a range of breeding approaches considered as avenues for improvement including speed breeding, genomic selection, and genetic engineering.

To investigate the kinetics of iron (Fe) uptake and partitioning in chickpea, Jahan et al. used a hydroponic growth system and RNA-sequencing of six genotypes that vary in seed Fe content. A number of key transporters were found to be expressed in roots and leaves, with the genes FRO2 and IRT1 being important in roots in the presence of Fe, and GCN2 in low Fe conditions. Conversely, in leaves the genes NRAMP3, V1T1, YSL1 along with storage gene FER3 showed higher expression. The improved understanding of Fe dynamics provided by this work provides targets for efforts to increase chickpea seed Fe content under both high and low soil Fe conditions.

Carrillo-Pedromo et al. investigated cold tolerance in fava bean. They used two fava bean QTL mapping populations to investigate the genetic basis of cold tolerance. This investigation identified five genomic regions associated with improved overwintering tolerance. Investigation of synteny of these regions with the Pisum and Medicago genomes showed that these regions are also associated with cold tolerance across other closely related legumes, suggesting shared mechanisms of tolerance that could prove to be effective breeding targets.

Bautista-Expośito et al. analyzed the impact of germination on protein and phenolic compound profiles in lentil and fava bean seeds. In this paper, the authors decided to analyze how the profile of phenolic compounds in different seeds affects the duration of germination and digestibility of proteins. Hydrolysis of the main protein fractions (7S and 11S globulins) that occurred during germination resulted in a simultaneous increase in the content of peptides and free amino acids. In addition, the products of protein hydrolysis were tested by the authors for potential healthpromoting properties, i.e., antihypertensive and antioxidant activities. The results of this study clearly showed that regardless of the type of seed, the germination process contributes to increase of protease activity and reduction of the level of phytic acid, trypsin inhibitors, and tannins which are considered antinutrients. Furthermore, the key role of seed permeability on the speed of the germination process, which subsequently influences the degree of antinutrients and the change in seed microstructure and endogenous proteolysis, was clearly demonstrated.

Finally, the work performed by Kalve et al. evaluated an interspecific population of chickpeas in terms of genomeassociated stress tolerance indices. The approach used by the authors assumed the application of wild relatives as a source of novel alleles for adaptation to suboptimal environments. In particular, in their work, an interspecific population derived from Cicer reticulatum accessions was used as a donor of introgression of heat and drought tolerance. From the initial 600 interspecific lines that were generated and tested in terms of resistance to ascochyta blight, a 195-line subset was selected and studied. Subsequent analyses helped establish a set of individual lines that perform better under suboptimal conditions. The additional outcome of this research can be associated with identifying specific SNP markers that can be used as marker-assisted selection that can help identify genes underlying tolerance to abiotic stress.

The last five papers of this Research Topic were focused on the use of bio-inoculants and soil fertility improvement. Norris Savala et al. assessed the efficacy of bio-inoculants in soybean production in Mozambique. In this context, to improve soybean yield, farmers use bio-inoculants from various sources and agroecological adaptability. However, these bio-inoculants are often unavailable during planting time and vary in yield based on their source and handling. Mozambique relies on imported bio-inoculants from neighboring countries and even South America. In this study, seven Bradyrhizobium diazoefficiens strain-based bio-inoculants with different carrier materials were evaluated for their performance, adaptability, and soybean productivity. Inoculation significantly improved plant growth, nodulation, and yield, suggesting the potential of bio-inoculants to enhance soybean production in Mozambique.

Jacques et al. conducted two studies related to pea plant responses. The first study focused on the impact of mineral deprivation on nutrient content and remobilization. They imposed transient deprivation of 13 mineral nutrients during vegetative growth and observed preferential allocation of dry weight and elements to shoots, particularly tendrils. Different remobilization strategies were identified, and the study suggests strategies to enhance seed quality through precise fertilization during periods of mineral nutrient deficiency. The second study by Jacques et al. investigated pea plant responses to various water stress types and their effects on nutrient uptake and remobilization. Pea plants, being nutritionally important, face vulnerability to water deficits induced by climate change. Common responses to all water stress types were observed in shoots, with manganese (Mn) playing a significant role. Under continuous stress, boron (B) impacted root architecture. An "ecophysiological imprint" in the root system was also observed, leading to increased nodule numbers during the recovery period. These findings provide insights into plant strategies to cope with water stress and contribute to global food security and nutrient deficiencies while reducing reliance on animal products.

Agricultural soils are affected worldwide by acidic pH and high levels of aluminum (Al) contamination due to several factors including agriculture practices and climate change. Quinones et al. have reported the ability of lupin to tolerate and accumulate Al in the rhizosphere and inside the root cells, suggesting its use in the restoration of marginal acid Al-rich soils in temperate zones where other legumes are unable to grow. The authors describe several physiological and molecular mechanisms of tolerance, uptake and accumulation of Al by lupin; among them, cluster roots which are able to exudate organic acids anions and polyphenols as well as Al-tolerant rhizobia strains with ability to produce abundant exopolysaccharides. These adaptive mechanisms make lupin a suitable crop for acidic soils affected by Al toxicity.

Finally, Yu et al. identified the GmTic110a gene's role in chloroplast development and its impact on soybean growth. More concretely, in this study, a Glycine max pale green leaf 3-1 (Gmpgl3-1) mutant with reduced chlorophyll content, chloroplast defects, decreased yields, and fewer pods per plant was isolated from soybean. Bulked segregant analysis and map-based cloning identified a mutation in the chloroplast development-related GmTic110a gene. Knockout plants exhibited similar phenotypes to the mutant. GmTic110a was highly expressed in leaves and localized to the inner chloroplast membrane. Interaction experiments revealed GmTic110a's interaction with other proteins involved in chloroplast development (GmTic20, GmTic40a, and GmTic40b). These findings suggest the crucial role of GmTic110a in chloroplast development, impacting photosynthesis and soybean growth.

In conclusion, the Research Topic "Legumes for Global Food Security, Volume II" delves into the immense potential of legumes, paving the way for innovative and transformative solutions to address pressing 21st-century challenges. These 22 papers highlight the crucial role of legumes in providing essential plantbased proteins, supporting food and feed security in a cost-effective manner. Moreover, the diverse topics covered offer valuable insights to enhance crop improvement, adaptation, and nutrient management, ultimately contributing to global food security and sustainable agriculture. As a collective, these articles make a substantial contribution to understanding and breeding resilient legumes capable of withstanding climate challenges. Our hope is that these findings will significantly enhance global food security in the foreseeable future.

  original draft, Writingreview & editing. JC: Writingoriginal draft, Writingreview & editing. SO: Writingoriginal draft, Writingreview & editing. EW: Writingoriginal draft, Writingreview & editing. PS: Writingoriginal draft, Writingreview & editing.

Frontiers in Plant Science frontiersin.org

Funding JJ-L thanks the support of the European Research Program MARIE CURIE (FP7-PEOPLE-2011-IOF),grant ref. PIOF-GA-2011-301550; the Spanish Ministry of Economy, Industry and Competitiveness (Ramon y Cajal Research Program), grant ref. RYC-2014-16536; the CSIC intramural research program, grant ref. 202240I002; and the Spanish Ministry of Science and Innovation, grant ref. number CPP2021-008989. AC was supported by PY20_00242 grant, financed by the Consejeria de Trasnformacioń econoḿica, Industria, Conocimiento y Universidades de la Junta de Andalucıá (Spain) and FEDER funds A way to make Europe.

Author contributions

JJ-L: Writingoriginal draft, Writingreview & editing. KS: Writingoriginal draft, Writingreview & editing. AC: Writing -

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The author(s) declared that they were an editorial board member of Frontiers, at the time of submission. This had no impact on the peer review process and the final decision.

Publisher's note

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ORCID Evagelos D Lioutas
email: evagelos@agro.auth.gr

ORCID Anastasios Michailidis

Chrysanthi Charatsari

ORCID Dimitrios Aidonis
email: daidonis@ihu.gr

ORCID Paolo Prosperi
email: prosperi@iamm.fr

Reem El Khechen
email: elkhechen@iamm.fr

ORCID Maria Partalidou

Charisios Achillas
email: c.achillas@ihu.edu.gr

ORCID Stefanos Nastis
email: snastis@gaaps.auth.gr

ORCID Luca Camanzi
email: luca.camanzi@unibo.it

Reem El Khechen

646x

Green public procurement of agrifood products: a business model view

Keywords: green public procurement, business models, environmental value, social value, economic value

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The realization of the significant role that the public sector plays as a buyer has led policymakers to sharpen their focus on the ways the products and services bought by public organizations are produced and distributed. The World Bank (2020) estimates that public procurement accounts for about 9.5 trillion US dollars annually, whereas in the European Union the percentage of GDP spent on public procurement is 14% (European Commission, 2020). The question of how to purchase products and services (including work) in such a manner that ensures that public money is prudently spent was always pivotal for the public sector. In the scholarly literature, some first concerns about the value of public expenditures in sectors like education [START_REF] Briggs | Monosyllables: Clarifying the meaning of education[END_REF] or health [START_REF] Abel-Smith | The Cost of the National Health Service in England and Wales[END_REF] date back to the mid-20 th century. Nevertheless, today, beyond economic feasibility, researchers put equal emphasis on the environmental and social impacts of public procurement.

Legislative frameworks, such as the European Procurement Directives, along with the growing sustainability awareness of citizens, led governments worldwide to integrate green practices in their procurement procedures [START_REF] Brammer | Sustainable procurement in the public sector: An international comparative study[END_REF]. Under the overarching label "green public procurement" (hereafter GPP) fall a series of environmental criteria that define an environmentally responsible procurement [START_REF] Li | Environmentally responsible public procurement (ERPP) and its implications for integrated product policy (IPP)[END_REF] and should be used in the process through which public authorities purchase products or services [START_REF] Cheng | Green Public Procurement, missing concepts and future trends-A critical review[END_REF].

These criteria involve a broad range of technical specifications, eco-labels, standards, and performance clauses (e.g., related to energy waste and/or greenhouse gas pollution), which can frame the process of candidate suppliers' selection or exclusion [START_REF] Halonen | Is public procurement fit for reaching sustainability goals? A law and economics approach to green public procurement[END_REF][START_REF] Rainville | Standards in green public procurement-A framework to enhance innovation[END_REF][START_REF] Diófási | Step by step towards mandatory green public procurement[END_REF]. Although not centered on the social dimension of sustainabilitylike its kin term "sustainable public procurement" [START_REF] Sönnichsen | Review of green and sustainable public procurement: Towards circular public procurement[END_REF] -GPP aims to produce positive outcomes for society by promoting sustainable production and consumption [START_REF] Pouikli | Towards mandatory Green Public Procurement (GPP) requirements under the EU Green Deal: Reconsidering the role of public procurement as an environmental policy tool[END_REF] and fostering sustainability-related attitudes [START_REF] Wang | Green public procurement as a promoter for green consumption: From the perspective of individual's knowledge[END_REF].

Although the relevant research shows enormous progress, the green procurement of agrifood products has not yet enjoyed considerable attention. Nevertheless, the public sector buys high quantities of such products to cover the needs of public schools and kindergartens, universities, elderly houses, hospitals, prisons, military bases, etc. Embracing the greening of food procurement can, thus, substantially contribute to the achievement of sustainability targets, also ensuring that public money is spent with an emphasis on the common good. Although there is a lack of financial data, [START_REF] Caldeira | Public Procurement of Food for Health[END_REF] confirm that in some European countries, the expenditure for covering the public sector's needs in food products is considerably high. For instance, in Finland and Ireland, the annual expenditures for meeting public food procurement amount to €395 and €195 million, respectively.

However, what is the value of GPP schemes, and how is it created? To answer this question, in the present study, we theoretically developed and empirically evaluated a business model for GPP schemes, focusing on three different facets of value: economic, environmental, and social. Using data from a sample of French experts in GPP schemes, we evaluated the dimensions of this model. France was selected because it represents a country with many good practice examples in GPP, both at the policy/institutional [START_REF] Vidal | Public contracts and sustainable development in France[END_REF] and the practice level (EU-FPC, 2021).

Developing a business model for green public procurement schemes

A brief conceptualization of business models

A business model is a mechanism through which a unit of interest (organization, company, or supply chain system) produces value. It is, simultaneously, a set of strategies [START_REF] Markides | Pioneering and first mover advantages: The importance of business models[END_REF], a "story" portraying the (potential) customers, their needs, and the most efficient ways to address them [START_REF] Magretta | Why business models matter[END_REF], and a prototype showing how business is done [START_REF] Baden-Fuller | Business models as models[END_REF]. In the economic sense, a business model illustrates the value proposition offered to the consumers and the resources needed to create it [START_REF] Morris | The entrepreneur's business model: Toward a unified perspective[END_REF]. [START_REF] Osterwalder | Business Model Generation: A Handbook for Visionaries, Game Changers, and Challengers[END_REF], were the first to develop a business model canvas presenting the essential components of economic value creation. The central element of their business model is the value proposition (Figure 1), i.e., the benefits the unit of interest conveys to its customers. To transform their value propositions into real value, during the production phase, organizations exploit their resources, enact value-creating activities, and develop partnerships. However, it is equally important to use effective distribution strategies by targeting the proper consumers, developing functional relationships with their customers, and choosing efficient distribution channels. At the bottom of their model, they added the sum of costs and revenues that the application of the model can generate.

Joyce and Paquin (2016) took a step forward, incorporating the environmental and social value to their "Triple Layered Business Model Canvas" (TLBMC). In their view, organizations and companies produce (or not) through their action value in the form of environmental and social benefits. Hence, they developed two extra layers following the operationalization of [START_REF] Osterwalder | Business Model Generation: A Handbook for Visionaries, Game Changers, and Challengers[END_REF]. The first layer refers to environmental value (or, in their words, "functional value"), which concerns the sum of functional units consumed in a given timeframe.

Functional value depends on both core production activities (presented on the left side of the canvas), including the process of producing a product or service and the environmental footprint of the materials and supplies used during it, and the post-production management of the products (illustrated at the right side), referring to the distribution process, consumer use of the product, and the positive or negative environmental outcomes of the end-of-life phase. Finally, environmental impacts and benefits are aggregated at the bottom of the layer. In the same vein, the social layer focuses on social value, which defines the bundle of benefits that the unit under consideration creates for the community of stakeholders and the wider society. On the left and right sides of social value, the developers of the model entered two parts. The first one includes dimensions referring to stakeholders (like the employees and the local communities), their engagement with the unit, and the governance structures that define stakeholders and their level of involvement in value co-creation processes. The second refers to the social outcomes extending beyond the unit's cycle of activities, involving the "scale of outreach" elementwhich [START_REF] Joyce | The triple layered business model canvas: A tool to design more sustainable business models[END_REF], p. 1180-1181) describe as "the depth and breadth of the relationships an organization builds with its stakeholders through its actions over time," the impacts of the society through influences on its societal culture, and the value absorbed by end-users of the products. The sum of social impacts and benefits is presented at the foot of the canvas.

Developing a triple layered business model canvas for green public procurement

In GPP systems, contracts signed between actors refer to the food production process and the compliance with environmentally sound behaviors, the products used during production, and may also involve the need for specific certifications (e.g., organic certification) [START_REF] Lindström | How Green Public Procurement can drive conversion of farmland: An empirical analysis of an organic food policy[END_REF][START_REF] Cerutti | Carbon footprint in green public procurement: Policy evaluation a case study in the food sector[END_REF]. In that sense, the buyer (the public sector) pays an amount of money to buy greenor greener than those produced and distributed through conventional practicesproducts, which are then consumed by the end users (e.g., students of public schools and universities).

Hence, GPP schemes emerged having as their central value proposition the supply of public organizations with high-quality products which are produced in an environmentally sound manner and meet specific environmental criteria. The main segments to which these supply systems are targeted are public authorities, municipalities, universities, and kindergartens/nurseries [START_REF] Fuentes-Bargues | Analysis of Green public procurement of works by Spanish public universities[END_REF][START_REF] Neto | The use of green criteria in the public procurement of food products and catering services: A review of EU schemes[END_REF][START_REF] Testa | Drawbacks and opportunities of green public procurement: An effective tool for sustainable production[END_REF].

To build relationships with these segments, the suppliers can offer complaint forms and traceability systems [START_REF] Bucea-Manea-Țoniș | Green and sustainable public procurement-An instrument for nudging consumer behavior. A case study on Romanian green public agriculture across different sectors of activity[END_REF]. As Figure 2 highlights, the channel used is direct selling to public agencies. That is to say, GPP schemes connect individual producers or farmers' groups (e.g., cooperatives) to public organizations.

To meet their objectives, GPP systems use on-farm resources (land, labor, capital) combined with certification schemes (organic, ethical, or fair-trade certification) [START_REF] Cerutti | Modelling, assessing, and ranking public procurement options for a climate-friendly catering service[END_REF]. Hence, for GPP, sustainable branding is another intangible resource contributing to their economic performance. The activities performed include the production of food products, logistics and transportation required to produce these products, and the quality costing since, in public procurement, the cost is always a critical factor for choosing among potential suppliers [START_REF] Schotanus | Developing a typology of organisational forms of cooperative purchasing[END_REF]. In such systems, farmers develop partnerships with suppliers of seeds, pesticides, and other agro-supplies. However, the creation of partnerships between farmers and brokers is also possible. Farmers' revenues come from selling products to public sector organizations, while subsidieswhen applicablecan also increase the economic performance of GPP schemes. The costs involve the budget for the production at the farm level, the transportation and storage expenses, the fees for certification, and the costs for the development and implementation of traceability systems.

The environmental layer of the TLBMC (Figure 3) has at its core the functional value of GPP, that is, the total amount of products purchased through these systems and consumed by public authorities over one year. Aside from the energy needed at the farm level, the production activities include logistics operations, which also have an environmental cost. The materials used include those integrated within agricultural production (agrochemicals, propagation material, farm equipment, irrigation water) and transportation vehicles. Beyond the core of the system, the energy used for transportation, the production of transportation vehicles and agricultural machinery, and the electricity needed also produce environmental externalities. [START_REF] Sonnino | Food waste, catering practices and public procurement: A case study of hospital food systems in Wales[END_REF] produce high quantities of food waste. Since products are transported, food waste during transportation can considerably affect the environmental performance of GPP supply chains. Besides, the distribution might involve middle or even long-distance transportation, also having environmental costs [START_REF] Cerutti | Carbon footprint in green public procurement: Policy evaluation a case study in the food sector[END_REF], whereas packaging is another source of environmental burden. However, since packaging in GPP is expected to comply with green practices [START_REF] Lundberg | Green public procurement and multiple environmental objectives[END_REF], its externalities are limited.

At the use phase, the preparation of meals (when the food is not ready-to-eat) and the storage of products contribute to the environmental footprint of the system. For instance, research on schools indicates that electricity and/or gas needed for preparing and serving meals is an issue that should be taken into account [START_REF] Batlle-Bayer | An explorative assessment of environmental and nutritional benefits of introducing low-carbon meals to Barcelona schools[END_REF][START_REF] García-Herrero | Food waste at school. The environmental and cost impact of a canteen meal[END_REF].

Among sources of environmental footprint, the transportation and logistics activities and the end-of-life component seem to have notable negative impacts. Nevertheless, GPP systems are based on operating mechanisms that conform to environmentally sound practices. The European Union has already developed a relevant legal framework [START_REF] Mélon | More than a nudge? Arguments and tools for mandating green public procurement in the EU[END_REF][START_REF] Kunzlik | Green public procurement-European law, environmental standards and 'what to buy'decisions[END_REF] defining what criteria should be met by suppliers. Such a structure of guidelines and regulations helps to reduce the environmental impacts of GPP schemes.

Moving to the social layer of TLBMC (Figure 4), the system seems to have the potential to generate "green" communities when suppliers and purchasing organizations belong to the same community. Another essential impact of the system is the development of institutions between communities and public authorities. The governance mechanisms of GPP schemes are a priory defined by legislative regulations. The rules are clear, and the decision-making processes (e.g., while choosing a supplier) are pre-described, explicit, and accepted by all the involved actors.

That characteristic represents a pivotal difference from other types of agrifood supply chains, compatible with the fundamental social ambition of GPP: the creation of social value through the promotion of both green production and responsible consumption [START_REF] Wang | Green public procurement as a promoter for green consumption: From the perspective of individual's knowledge[END_REF][START_REF] Pacheco-Blanco | Green public procurement as an initiative for sustainable consumption. An exploratory study of Spanish public universities[END_REF]. In pursuing these targets, the system under consideration infuses a societal culture of responsibility in production and consumption. Some scholars argue that applying green practices during public procurement can promote the transition towards sustainable production paradigms [START_REF] Borsato | Use of multiple indicators to compare sustainability performance of organic vs conventional vineyard management[END_REF][START_REF] Lindström | How Green Public Procurement can drive conversion of farmland: An empirical analysis of an organic food policy[END_REF]. Nevertheless, they can also promote an elitist culture since farmers who are not certified with organic, ethical, or fair-trade certifications might be excluded by GPP chains.

The social outcomes of GPP for end-users refer to the access to healthy food products that meet sustainability standards; the development of awareness on issues like environmental sustainability, ethical food production, and fair trade; and the information the public authorities can have on the food production standards. Finally, the exclusion of non-certified farmers from green public procurement chains and the potential transformation of green, sustainable production to a marketing mechanism that overemphasizes the market benefits and undervalues the real meaning of sustainable agrifood production represent significant negative social impacts of GPP schemes. On the other hand, the promotion of sustainable and responsible food production and consumption is a valuable social benefit. Hence, it can be argued that the depth of outreach is high, but the breadthbeing limited to a regional or national levelis medium.

Methods

Measures

To evaluate the developed business model, we constructed an instrument consisting of 72 items, referring to the sub-dimensions of each layer's components. To examine the content and face validity of the questionnaire, we conducted a preliminary analysis by inviting two experts with experience in the topics under study to assess the instrument's general quality, the complexity and wording of the items, and their matching with the relevant literature. After making some modifications suggested by the experts, the instrument took its final form. To measure items, we used a Likert scale, ranging from 1 (strongly disagree) to 5 (strongly agree).

The questionnaire also included a section aiming at collecting socio-demographic data.

To confirm that the sets of developed items belonged to the theoretically expected dimensions of the TLBMC, we performed a series of principal axis factor analyses after recoding negatively worded items. In all cases, the process revealed unidimensional structures for each component. Nevertheless, four items with loadings lower than 0.4 were eliminated. Hence, the final analysis included 68 items.

The variance explained by the new factors was quite high for all nine components. Eigenvalues were above 1 for all factors. Cronbach's alphas were sufficient in all cases, especially considering the small sample size. Alpha values ranged between 0.57 and 0.76 for the economic (Table 1), 0.50 to 0.91 for the environmental (Table 2), and 0.61 to 0.85 for the social layer (Table 3).

Participants

For this study, we drew on data from a sample of 20 experts in GPP from France.

Participants were farmers (20%), farmer cooperative members (65%), wholesalers involved in GPP (5%), and consumers buying products sold through these schemes (10%). Among respondents, 13 were men, whereas 75% of the sample were aged between 41-60 years, 20% belonged to the age group of 21-40 years, and 5% were above 60 years. Half of the participants (50%) had secondary education, 30% had completed post-secondary education, 15% graduated from a tertiary institute, and only one (5%) had only primary school education.

Data analysis procedure

To present data, we used mean scores and standard deviations. Differences between components of the TLBMC were examined using paired samples t-tests. We also used Spearman's correlation coefficient to uncover significant associations between the components of TLBMC. In all cases, significance level was defined at p<0.05.

Results

According to the mean scores (Table 4), GPP schemes have an average to high value proposition (M=3.97) while they lead to moderately high revenues (M=3.10). The dimension "costs" received a relatively low mean score (M=2.57). Given the positive wording of the items belonging to the component, we can argue that the cost represents an issue for GPP schemes.

Aside from this, there are many spaces for improvement since none of the remaining dimensions yielded a mean score higher than 3.38. As Table 4 highlights, the margins for improving the effective use of resources (M=3.08) and targeting customer segments (M=3.18) are broad. The same is true for the component referring to the weaving of functional customer relationships (M=3.22).

Testing for significant differences between the dimensions referring to the production process, we discovered that t values ranged from 0 to │1.78│ (p>0.05 in all cases). Following the same procedure for the dimensions associated with the distribution of products, we found no significant differences (0.31≤t≤1.09, p>0.05).

For the environmental layer, the mean scores revealed that, despite the fact that the environmental benefits of GPP received a relatively high mean score (M=3.65), there are aspects related to the environmental performance of those schemes that can be seriously improved. The production dimension yielded a mean score of 3.05, indicating that it represents a factor limiting the environmental performance of these schemes. The mean score for the dimension concerning suppliers and outsourcing (M=3.32) was also somewhat low. Nevertheless, the functional value of green public procurement is just above the reference line of 3.00 (M=3.27), suggesting that some aspects of GPP are not as "green" as hoped.

The production component had a significantly lower mean score compared with all the post-production dimensions. Paired samples t-tests revealed that these differences were significant at the 0.01 level for distribution (t=-2.86, p=0.010) and at the 0.05 level for end-oflife (t=-2.83, p=0.011) and use phase (t=-2.22, p=0.039). Moreover, we observed that environmental benefits correlate with dimensions belonging to core productionnamely materials (ρ=0.47, p=0.036) and suppliers/outsourcing (ρ=0.45, p=0.049)as well as with the post-production components of distribution (ρ=0.59, p=0.006) and use phase (ρ=0.45, p=0.045).

On the contrary, the production process did not correlate with environmental benefits (ρ=0.07, p=0.775) and impacts (ρ=0.28, p=0.224). Finally, the mean scores for the social layer confirmed that GPP schemes do have a relatively high ability to generate social value (M=3.82) and produce high social benefits (M=3.87). The mean score for social externalities (M=2.90) also showed that GPP perform relatively well in the social layer. Among the other aspects of the model, the highest means were observed for the dimensions referring to end users (M=3.65), the scale of outreach (M=3.60), and societal culture (M=3.52), which concern the broad social impacts of GPP operation.

Interestingly, governance mechanisms had a somewhat low mean score (M=3.17), potentially suggesting that adaptations to the current governance structures can lead to higher levels of social value.

Another notable finding was that the components pertaining to the social impacts that extend the boundaries of GPP had higher mean scores than those reflecting stakeholders' involvement in the social value creation process. However, paired samples t-tests uncovered that the only significant difference was that between the end user and governance components (t=2.92, p=0.009), while in all other cases, the differences were not significant. Notably, Spearman's correlations demonstrated that social value modestly but significantly correlated with all the dimensions that regard broad social impacts, i.e., societal culture (ρ=0.55, p=0.013), the scale of outreach (ρ=0.49, p=0.029), and end user (ρ=0.48, p=0.033). On the other hand, in the general category that includes stakeholders, the only significant correlation was noticed for local communities (ρ=0.58, p=0.007). Governance (ρ=0.22, p=0.348) and employees (ρ=0.27, p=0.257) did not correlate with social value.

The comparison of the three layers led to some interesting notes. The functional value of GPP schemes was found to be significantly lower than both economic value proposition (t=-3.07, p=0.006) and social value (t=-3.24, p=0.004) while no significant differences were detected between the latest two constructs (t=1.20, p=0.249). On the other hand, the revenues had a significantly lower mean score than both environmental and social benefits (t=-2.57, p=0.019 and t=-2.79, p=0.012, respectively), between which no significant difference was observed (t=-1.06, p=0.304). A final remarkable finding was that, although functional value positively correlated with social value (ρ=0.62, p=0.003), it did not show a significant correlation with the (economic) value proposition (ρ=0.19, p=0.432), pointing out that the capacity of a GPP scheme to produce environmental value is not associated with its ability to deliver value propositions.

Discussion and conclusions

Adding to the growing body of literature on GPP schemes, the current study developed a business model canvas illustrating the essential components that contribute to the production of economic, environmental, and social value through the operation of GPP supply chains in France. Our analysis uncovered an average to high value proposition for the schemes under consideration, showing a clear need to improve both the production and distribution processes, and increase revenues. On the other hand, when looking at the social layer of the TLBMC, some silver linings emerge. GPP schemes seem capable of producing social value that extends beyond the boundaries of the buyer-seller dyad, thus confirming the contention of Bucea-Manea-Țoniș et al. ( 2021) that green procurement of agrifood products by the public sector serves social purposes.

However, quite unexpectedly, we observed that functional value was rather low and significantly lower than economic and social value. A plausible explanation for the limited capacity of GPP schemes to achieve high levels of functional value, therefore attesting to their "green" character, is the environmental inefficiency of the production component. Indeed, the results seem to support this conjecture. The production dimension was found to have a relatively low mean score, indicating the need to improve the environmental performance at the initial nodes of GPP supply chains. Of course, to meet that purpose, GPP policy frameworks require farmers to apply green practices [START_REF] Lindström | Green public procurement: An empirical analysis of the uptake of organic food policy[END_REF][START_REF] Mélon | More than a nudge? Arguments and tools for mandating green public procurement in the EU[END_REF]; nevertheless, even sustainabilityoriented farm production systems have questionable environmental outcomes [START_REF] Mcgee | Does certified organic farming reduce greenhouse gas emissions from agricultural production?[END_REF][START_REF] Tuomisto | Does organic farming reduce environmental impacts?-A meta-analysis of European research[END_REF].

Although our study offers some interesting insights, we are well aware that the small sample size reduces the power of the statistical analyses. Another potential limitation involved the fact that we did not focus on a specific GPP system (e.g., schemes supplying hospitals or public organizations' canteens). Diverse sectorsand even organizations within the same sector apply different criteria when purchasing "green" food products [START_REF] Neto | The use of green criteria in the public procurement of food products and catering services: A review of EU schemes[END_REF], thus impacting the production process and, hence, the value-generating capacity of GPP supply chains.

In light of these limitations, our work should be viewed as a preliminaryand, to the best of our knowledge, the firstattempt to estimate GPP schemes' ability to create economic, environmental, and social value. Future researchers can lean upon our theoretical framework, using the TLBMC developed in the present study to conceptually depict and empirically evaluate how GPP schemes produce and deliver value and what impedes them from reaching their full value-creating potential. In addition, our findings raised an intriguing question: are GPP schemes really green? Our data did not provide a positive answer, revealing a relatively low functional value for French GPP of agrifood products. Is that the case in other countries and/or sectors?

This question awaits further research.

Figure 1 .

 1 Figure 1. A triple layered business model canvasbased on Joyce and Paquin's (2016) TLBMC

Figure 2 .

 2 Figure 2. The economic layer of TLBMC

Figure 3 .

 3 Figure 3. The environmental layer of TLBMC

Figure 4 .

 4 Figure 4. The social layer of TLBMC

Table 1 .

 1 Components of the economic layer, number of items, percentages of variance explained and

	Cronbach's alphas
	Component	No of items	Example item	Explained (%) variance	α
	Value	2	are able to offer high quality agrifood products,	76.71	0.69
	proposition		produced with environmentally sound practices
	Activities	3	are characterized by high production effectiveness	56.76	0.60
	Partners	3	are based on functional collaborations between	65.83	0.72
			farmers and suppliers of seeds, pesticides, fertilizers
	Resources	3	are based on the effective use of land, labor, and	68.30	0.76
			capital
	Customer	2	operate based on effective traceability systems	71.47	0.57
	relationships
	Channels	2	distribute effectively products through direct sales to	72.07	0.61
			local authorities
	Customer	5	target effectively municipalities and municipal services	76.71	0.69
	segments
	Costs	2	have low production cost at the farm level	56.76	0.60
	Revenues	2	offer farmers an extra revenue due to the associated	65.83	0.72
			subsidies
	Note: Items endorse the statement "green public procurement schemes in my region…"

Table 2 .

 2 Components of the environmental layer, number of items, percentages of variance explained and

	Cronbach's alphas
	Component	No of items	Example item	Explained (%) variance	α
	Functional	2	have a low environmental footprint per unit of product	86.30	0.84
	value		sold in public bodies
	Production	2	are based on efficient energy use at the farm level	64.28	0.61
	Materials	2	use agricultural supplies (seeds, fertilizers and	66.53	0.50
			pesticides) that do not harm the environment
	Suppliers/	3	use farm machinery that requires high amounts of	65.97	0.70
	outsourcing		energy to be produced*
	End-of-life	2	produce waste of products during transportation*	67.96	0.53
	Distribution	3	are based on middle-distance transportation that does	68.26	0.53
			not harm the environment
	Use phase	2	consume high amounts of energy during the storage of	91.91	0.91
			products (e.g., in public authorities' refrigerators)*
	Impacts	2	are based on energy-consuming procedures across the	65.09	0.46
			chain*
	Benefits	2	mitigate the environmental impacts due to the	84.41	0.82
			compliance with environmental practices and
			processes

Notes: Items endorse the statement "green public procurement schemes in my region…" Negatively worded items are marked with an asterisk.

Table 3 .

 3 Components of the social layer, number of items, percentages of variance explained and

	Cronbach's alphas
	Component	No of items	Example item	Explained (%) variance	α
	Social value	2	promote the idea of green agrifood production	77.21	0.70
	Local	2	develop institutions between local communities and	83.10	0.80
	communities		public authorities
	Governance	2	decision-making processes are prespecified and widely	86.67	0.85
			accepted
	Employees	3	offer employees opportunities for training in	69.57	0.73
			sustainable production
	Societal	2	cultivate a culture of responsibility in production and	72.04	0.61
	culture		consumption
	Scale of	3	offer benefits that extend beyond local communities	71.57	0.80
	outreach
	End user	3	offer consumers (employees in public authorities,	58.77	0.63
			public universities' students) access to healthy food
			products
	Social	2	transform sustainable food production into a marketing	73.98	0.65
	impacts		mechanism*
	Social	2	promote sustainable and responsible consumption	81.14	0.75
	benefits
	Notes: Items endorse the statement "green public procurement schemes in my region…" Negatively
	worded items are marked with an asterisk.

Table 4 .

 4 Mean scores and standard deviations for the components of the TLBMC

	Economic layer	Environmental layer	Social layer
	Component	Mean score (S.D.)	Component	Mean score (S.D.)	Component	Mean score (S.D.)
	Value	3.97 (0.62)	Functional	3.27 (0.92)	Social value	3.82 (0.61)
	proposition		value
	Activities	3.38 (0.59)	Production	3.05 (0.65)	Local	3.37 (0.87)
					communities
	Partners	3.38 (1.02)	Materials	3.65 (0.63)	Governance	3.17 (0.75)
	Resources	3.08 (0.95)	Suppliers/	3.32 (0.66)	Employees	3.40 (0.65)
			outsourcing
	Customer	3.22 (0.68)	End-of-life	3.70 (0.75)	Societal culture	3.52 (0.70)
	relationships
	Channels	3.37 (0.84)	Distribution	3.72 (0.87)	Scale of	3.60 (0.70)
					outreach
	Customer	3.18 (0.81)	Use phase	3.57 (0.81)	End user	3.65 (0.72)
	segments
	Costs	2.57 (1.07)	Impacts	2.55 (0.60)	Impacts	2.90 (0.87)
	Revenues	3.10 (1.01)	Benefits	3.65 (0.74)	Benefits	3.87 (0.82)

Acknowledgment

This study has been realised in the framework of the project "Data-enabled Business Models and Market Linkages Enhancing Value Creation and Distribution in Mediterranean Fruit and Vegetable Supply Chains -MED-LINKS" (ID 1591). Financial support to the project has been provided by PRIMA, a program supported by the European Union, and co-funding has been provided by the Italian Ministry for University and Research (MUR), the Egyptian Academy of Scientific Research and Technology (ASRT), the French National Research Agency (ANR), the Greek General Secretariat for Research and Technology (GSRT) and the Moroccan Ministry of Higher Education, Scientific Research and Professional Training (MESRSFC).


R Chakma
email: rikel.chakma@ganil.fr

A Lopez-Martens

K Hauschild

A V Yeremin

O N Malyshev

A G Popeko

Yu A Popov

A I Svirikhin

V I Chepigin

E A Sokol

A V Isaev

A A Kuznetsova

M.S M L Chelnokov

I N Tezekbayeva

Izosimov

O Dorvaux

B Gall

S Z Asfaril

Investigation of isomeric states in 255 Rf

Keywords:

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I. INTRODUCTION

Superheavy nuclei (SHN) with Z≥ 104 materialize because of the stabilizing shell correction energy. This shell correction energy creates a fission barrier that is nonexistent for these nuclei in the liquid-drop model because of the massive Coulomb repulsion between the protons. The notion of the stabilizing effects of shells points to the possible existence of a hypothetical island of stability inhabited by long-lived spherical superheavy nuclei. However, the predicted localization of this island based on different theoretical approaches diverge. For instance, Macroscopic-Microscopic models (using Nilsson, Woods-Saxon, Folded-Yukawa potentials) predict the center to be around Z = 114, N = 184 [1,2] while the self-consistent models using Skyrme, Gogny, or Relativistic Mean Field effective nuclear interactions predict stronger shell effects at Z = 120, 124, 126 and N = 172, 184 [3][4][5][6].

Therefore, a large body of experimental data of single and collective states gathered through prompt and decay spectroscopy is desirable to benchmark the theories that aim to describe the underlying nuclear structure in the superheavy mass region. The SHN are typically produced in fusion evaporation reactions with very low production cross-sections (≈ 10 nb in A = 250 region). The cross-section diminishes even more as a function of atomic number [7]. As a result, the available spectroscopic data in the transfermium region is rather scarce [8]. Performing detailed spectroscopy in the lighter transfermium region is thus more practical and one step closer to understanding the properties of the SHN. In this region, K isomers are of specific interest as they give information about the single-particle states near the Fermi level and their internal decay can populate states that are not normally accessible through alpha decay. In axially deformed nuclei, the K quantum number is the projection of the total angular momentum onto the symmetry axis. High-K isomerism occurs due to hindrance in electromagnetic transitions involving a large change in the K quantum number (from high-K to low-K for example). K isomers have been identified and/or suggested in several isotopes in this region (see Ref. [9][10][11][12][13][14][15][16][17][18]) and are also expected in 255 Rf. The ground-state configuration of 255 Rf is thought to be based on the 9/2 -[734] Nillsson orbital as in the lighter isotones (such a configuration has been unambiguously determined in 253 No through laser spectroscopy [19]).

In a recent study from GSI, two high-K isomers have been observed with half-lives T 1/2 = 38 +12 -7 µs and T 1/2 = 15 +6 -4 µs and were suggested to have spin ≥ 17/2ℏ [20]. The excitation energies of these isomers were estimated to be 1.15-1.45 MeV and 0.9-1.2 MeV. In another study at GSI, an isomeric state with a half-life of T 1/2 = 50 ± 17µs and an excitation energy of around 135 keV [21] was observed in alpha decay of 259 Sg to 255 Rf. The 5/2+ [622] configuration was assigned to this isomer from the systematics in the lighter N = 151 isotones. Besides the 5/2+[622] spin isomer, two other excited states 1/2 + [620] around 510 keV and 11/2 -[725] around 600 keV were also tentatively suggested in their work.

In this paper, we confirm the existence of two high-K isomers and the spin isomer in 255 Rf. We report on their half lives and excitation energies and suggest their most likely spins and parities. We also present a decay sce-nario, whose simulation in Geant4 [22] reproduces all of the spectroscopic observables and provides a measurement of the isomeric ratios.

II. EXPERIMENT

In this study, rutherfordium nuclei were produced in the fusion evaporation reaction of 207 Pb( 50 Ti,2n) 255 Rf in two experimental campaigns at the FLNR facility of JINR in Dubna, Russia. The 50 Ti beam having intensities ≈ 337-370 pnA in the first campaign and ≈ 360-396 pnA in the second and an average energy of ≈ 253 MeV in both was delivered by the U400 cyclotron. The PbS targets (enriched with 207 Pb -96.2 %, 208 Pb -2.6%, 206 Pb -1.1% and 204 Pb -0.1%) were 0.4 µg/cm 2 thick, deposited on 1.5 µm Ti backings, and were mounted on a rotating target wheel. The evaporation residues (ERs) that emerged out of the target were separated using the velocity filter SHELS [23] and transported to the focalplane detection setup, GABRIELA [24,25].

The ERs were implanted into a large-area double-sided silicon strip detector (DSSD) with the corresponding electronics [26] set to measure conversion electrons (up to 2 MeV) and alpha particles (up to 25 MeV) simultaneously. The fission fragments (typically > 60 MeV) thus were detected as overflows. The DSSD provided position and time correlations between the implanted ions and their successive decays. In the DSSD, isomeric decays were detected through Jones' calorimetric method [27] i.e., by measuring the low-energy signals (called CE henceforth) produced from the summation of total or partial energy depositions in the DSSD by internal conversion electrons (ICEs), X rays, Auger and Coster-Kronig electrons. The calorimetric method is a powerful technique to evidence isomers whose decay proceeds by a cascade of highly converted transitions. In the case of short lifetimes and short cascades, however, the isomeric CE signals may be missed due to the dead-time and thresholds of the electronics. In this study, the average threshold of the DSSD was between 60-100 keV, and the dead-time was ≈ 32 µs. To detect the ionizing particles (mostly ICEs) that escape the DSSD, a tunnel composed of eight small DSSDs is placed in the upstream direction of the large DSSD. Four coaxial, large volume Germanium (Ge) detectors were mounted on each lateral side of the DSSD array and one clover detector [28] behind the DSSD to measure the X rays and gamma rays. The average threshold of the tunnel detector was also between 60-100 keV and the energy resolution was 14.4 (12) keV (Full Width at Half Maximum) at 320 keV. For the gammaray detectors these corresponding values were ≈ 15 keV and 2.26 (17) keV at 1332.5 keV. The relevant regions of the plot has been marked from I-VII and they correspond to random correlations (I), 255 Rf fission-decay events (II), 256 Rf fission-decay events (III), 254 Rf fission-decay events (IV), 255 Rf full-energy alpha-decay events (V), alpha-decay events of transfer products (VI) and CE signals from isomeric-decay events (VII).

III. RESULTS

A. Ground-state properties

In the two experimental campaigns, we produced a total of ≈ 7880(112) 255 Rf nuclei. Fig. 1 shows a recoildecay correlation plot, where the time between a decay event and an ER signal is plotted in log 2 scale as a function of the decay energy. The half-life of the ground state of 255 Rf was measured from the time distribution of its characteristic alpha decay events (in the 8660-8850 keV range) and fission events, and they are T α 1/2 = 1.67 ± 0.05 s and T f 1/2 = 1.69 ± 0.03 s respectively. The corresponding estimated branching ratios are B α = 49.1 ± 1.3% and B f = 50.9 ± 1.1%. These values are compatible with the reported values given in the literature [29][30][31][32]. Since no 255 Lr alpha decay was identified, an upper limit of the β + /EC decay branch of 255 Rf was deduced as B EC/β+ = 0.06 ± 0.02%.

B. Isomers

Detection of two isomeric transitions in cascade

After the detection of an ER and before the detection of the ground-state alpha or fission decay of 255 Rf, two CE signals in a cascade were detected in the DSSD. These decay chains are denoted as ER-CE-CE-255 Rf, and a total of 27 such decay chains were detected. The energy and the time distributions of the first and second CEs are compared in Fig. 2, where one can notice:

(a) the first isomer (iso1) is relatively longer-lived than the second (iso2) and (b) the detected signal of iso1 is of much smaller in energy than that of iso2. The lifetime of iso1 was measured with respect to the ER signals that precede them and that of iso2 with respect to the iso1. Incidentally, no coincident gamma rays or ICEs (in the tunnel detectors) were detected with the CE signals of iso1. However, six gamma rays and eight ICEs were detected in coincidence with the CEs of iso2. The recorded energies of these gamma rays are 166 keV, 281 keV, 470 keV, 543 keV, 551 keV, and 778 keV and the electrons have energies below 160 keV. In the decay of iso2, by summing the energies of CEs with the energies of the coincident gamma rays and electrons on an event-by-event basis, the total calorimetric energy (E calor ) was found to be ≈ 1 MeV. For ios1, E calor is ≈ 200 keV as no coincident gamma ray or ICEs were detected.

In some cases where only one CE signal in between an ER and the ground-state decay of 255 Rf was detected, it was still possible to assign two isomeric decays in a cascade using the germanium and tunnel detectors. Gamma rays and/or ICEs were detected in between the ER and CE signals, and, these decay chains are denoted as ER-[*]-CE-255 Rf, where [*] = γ for gamma rays and [*] = e -for ICEs. We observed nineteen ER-γ-CE-255 Rf and twenty-three ER-e --CE-255 Rf correlations. In the ERγ-CE-255 Rf correlations, there were L X rays around 26 keV and a distinct 102 keV transition. No gamma rays above 102 keV were observed, and the absence of K X rays in the gamma-ray spectrum suggests that no transition in the decay from iso1 to iso2 lies above the K-shell binding energy (156.3 keV) of rutherfordium. In the electron spectrum, a peak around 80 keV, possibly from the L conversion of 102 keV transition and/or another transition with similar energy, was observed. From all these correlations, the half-lives of iso1 and iso2 were measured to be 49 +13 -10 µs and 29 +7 -5 µs respectively. Our data confirms the results of Ref. [20] that there are at least two isomers: iso1 and iso2; iso1 at excitation energy around 1.2 MeV and iso2 around 1 MeV above the ground state, and iso1 feeds iso2.

Detection of single isomeric transitions

A total of 701 CEs were detected between the ER implantation signals and 255 Rf ground-state decays. Out of these, 38 were preceded by gamma rays or ICEs or both and were discussed in the previous section. The energy and time distribution (in log 2 scale) of the remaining 663 CEs of the ER-CE-255 Rf decay chains are shown in Fig. 3. The bulk of these CEs is concentrated around 145 keV, whose lifetime seems slightly higher than those of the higher energy counterparts. The average half-life of these CEs is T 1/2 = 52 ± 2 µs. In coincidence with the CEs in the decay chains of ER-CE-255 Rf, 103 gamma rays and 69 ICEs (in the tunnel detectors) were observed (see Fig. 4). In the gamma-ray spectrum, the most intense line is the 778 keV transition and L X rays around 24 keV. The 778 keV, 543 keV and 170 keV transitions that are distinguishable in the gamma-ray spectrum of ER-CE-255 Rf were also seen in coincidence with iso2 in the ER-CE-CE- 255 Rf cascades (see the previous section). In the electron spectrum, two noticeable structures were observed: one below 100 keV and the other around ≈ 145 keV. Very few electrons were observed with energies between 200 and 400 keV. The measured E calor involved in the de-excitation of the supposed isomer was found to be ≈ 1.1 MeV. This excitation energy corresponds to the one of iso2 in the two-CEcascade correlations, however, the half-life is longer but is in line with the half-life of the 5/2 + spin isomer measured at GSI [21]. The spectrum of single isomeric decays is most likely complex, involving contributions from iso1, when iso2 goes undetected and vice-versa, from iso2 alone and also from the 5/2 + spin isomer. When no CEs are recorded in the implantation detector, thanks to the Ge and tunnel detectors again, single isomeric decays can still be detected in these detectors by looking at the gamma rays and ICEs that follow an ER signal with a subsequent alpha or fission decay in the same pixel (ER-γ/e --255 Rf). The E calor obtained in this case from the summed energies of the coincident gamma rays and electrons is ≈ 1 MeV, in agreement with the previous values. The energy spectra of these gamma rays and electrons observed in this case are shown in Fig. 5. In the electron spectra, a peak around 110 keV is visible, but, it is impossible to resolve other peaks and associate them to the observed gamma-ray transitions.

In the gamma-ray spectrum, we notice two prominent gamma-ray lines: one at 778 keV and the other at 170 keV. Some weaker transitions of energy 708 keV, 666 keV, 543 keV, 185 keV, 150 keV and 102 keV are also observable. Here one could argue that the 150 keV line is due to radiative process while filling vacancies of the K shell of rutherfordium. However, only ≈ 2 counts of K β1 = 150 keV are expected from an estimation using the intensity of the K α2 (126.3 keV) in contrast to 12 counts in the gamma-ray spectrum suggesting that the 150 keV line is indeed a real gamma-ray transition.

The efficiency corrected intensities of the above transitions relative to the 778 keV line are given in table I. From the gamma-ray intensities, we can exclude the possibility of the 170 keV line being an M1 transition as we do not observe enough K X-rays. If it were an M1 transition, we expect to detect ≈ 106 counts at the energy of the Rf K α2 energies, whereas only 8 counts were detected. Looking at the electron spectrum, we can also rule out the possibility of the 170 keV line to be an E2 transition. If it were an E2 or an M1 transition, we expect to detect ≈ 80 LMN+ ICEs with energies ranging from 139-170 keV in the tunnel detector. But, in the electron spectrum, we have only a total of 110 counts in the 130 -180 keV range, which include the tail of the 110 keV main peak. If, on the other hand, the 170 keV line is an E1 transition, we expect to detect only 1-2 counts. Intensities were normalized to the intensity of the 778-keV transition. The multipolarity of the 170 keV transition has been deduced from experiment while the counterparts for the other transitions are from the proposed decay scheme shown in Fig. 7. Spin assignments and branching ratios (br γ ) to these transitions are also from the proposed scheme (see section IV for details). To get a hint of the states involved in the 778 keV, the 666 keV, the 543 keV, the 170 keV and the 98 keV (discussed later) transitions, their reduced hindrance factors f ν were calculated. We have calculated the f ν values for E1, M1 and E2 electromagnetic multipolarities undergoing ∆K = 3, 4 and 5 in each case. f ν is defined as:

f ν = F 1/ν W , F W = (T γ 1/2 /T W 1/2 ) (1)
where F W is the hindrance of a given gamma-ray transition, T γ 1/2 is the partial gamma-ray half-life determined experimentally, T W 1/2 is the half-life calculated from Weisskopf single-particle estimates and ν is the degree of Kforbiddenness expressed as ν = ∆K -L in which L is the multipolarity of the emitted radiation and ∆K is the change in K quantum number in transitioning from an initial state I i to a final state I f . Knowing the total internal conversion coefficient α T ot of a gamma-ray transition, its T γ 1/2 can be calculated from the experimentally measured half-life T exp 1/2 using the following relation:

T γ 1/2 = T exp 1/2 × (1 + α T ot ) br γ (2)
T γ 1/2 depends on the energy E γ and the multipole order L of the emitted photon, the type of the transition σ (electric or magnetic) and on the nuclear structure entailed in the reduced transition probability B(σL, I i → I f ) [START_REF] Ring | The Nuclear Many-Body Problem[END_REF]. Thus, F W is a measure of the deviation of the true nuclear transition rate from the approximate single particle model in which a single nucleon in an average potential de-excites the nucleus. Using the measured half-life of iso2, branching ratios given in table I and internal conversion coefficients from BRICC [START_REF] Kibédi | [END_REF], the f ν values for the 778 keV, 666 keV, 543 keV and 170 keV transitions were calculated. For the 98 keV transition, we have used half-life of iso1 and assumed 100 % branching ratio in the decay of iso1 to iso2 as shown in Fig. 7. These measured f ν values are plotted in Fig. 6a in which the band shows the phenomenological range (f ν = 20, 300) usually referred to for interpreting decay properties and assignment of spin/parity. In Fig. 6a, it is indicative that the transitions emanating from iso2 can most likely involve ∆K = 5. However, the types of these transitions can not be determined. Fig. 6a also suggests that they could be M1 or E2 transitions. But, on the other hand, it is also not uncommon to have large hindrances (f ν ≈ 1000) for E1 transitions and it should also be emphasized that this phenomenological range is arbitrary as the experimental data points on which it is based mainly in the works of Rusinov [35] and Löbner [36] are considerably scattered. Löbner pointed out that there is a dependence of hindrances on multipolarities. Kondev et al. with more experimental data later quantified these multipolarity-dependent offsets in their work [37] (see table C of Ref. [37]). They have also shown that f ν is much smaller (represented here by f ν = µ) compared to the commonly used Rusinov's empirical rule f ν ≈ 100, i.e., the hindrance increases by a factor of approximately 100 with an increase of ∆K by 1 unit. We have compared the measured F W with the expected values obtained after factoring in the multipolarity-dependent offsets on hindrances in Fig. 6b. The figure shows two lines obtained using the relation log 10 F W = log 10 F 0 +ν log 10 f 0 [37]: a) when f ν = µ, using the fit parameters of Ref. [37] and b) when f ν = 100 expected from Rusinov's empirical rule but with the same F 0 parameter of a) to take into account the multipolarity dependency. Several trends can be noticed in

IV. DISCUSSIONS

To interpret the experimental results, we have relied heavily on Geant4 simulations. The ground-state band of 255 Rf was built based on the ground-state band of the N = 151 neighboring isotone i.e., 253 No [38]. The rotational g-factor g R was taken as g R = 0.7Z/A = 0.285 where we have used a quenching factor of 0.7 to account for pairing effects [39]. The intrinsic quadrupole moment was taken as Q 0 = 12.42 eb using the predicted quadrupole deformation of β 2 = 0.252 [40]. We assumed these values to remain constant for all other excited states. The single-particle g-factors g K in this work were obtained from deformed shell model calculations with a Woods Saxon potential using the prescription that the nucleon spin g-factor is taken to be 0.7 × g s f ree . For the 9/2 -[734] configuration, g K was taken to be -0.25. To build a rotational band, we also assumed a constant moment of inertia in eq. 3

E K (I) = E K (I 0 ) + ℏ 2 2J (I(I + 1) -K(K + 1)) (3)
where E K (I 0 ) is the band head energy, I is the spin of the rotational state, and K is the total spin projection onto the symmetry axis. For the ground-state band, ℏ 2 2J was taken as 5.8 to have one in-band E2 transition around 185 keV, a peak observed experimentally at this energy. Similarly, for the 11/2 -[725] configuration, the g K was taken as -0.22 in accordance with the theoretical value (g K -g R )/Q 0 = 0.040 [18], and the rotational parameter was taken as ℏ 2 2J = 6.7 to reproduce the rotational band observed in 257 Rf. The excitation energy of the 11/2 - state was taken to be around 600 keV as per the study of alpha decay of 259 Sg to 255 Rf [21]. We have adjusted the energies of the rotational states by a few keV to accommodate the observed gamma-ray transitions. The M1 and E2 transition rates and the mixing ratios were computed using the standard formulas [START_REF] Ring | The Nuclear Many-Body Problem[END_REF]. The conversion coefficients were obtained from BRICC [START_REF] Kibédi | [END_REF], and for the intra-band M1/E2 transitions, the conversion coefficients have been corrected using the relation:

α mixed (M 1, E2) = α(M 1) + δ 2 α(E2) 1 + δ 2 (4)
where δ is the mixing ratio and is derived from the square root of the ratio of the transition rates T i→f :

δ(E2/M 1) = T i→f (E2) T i→f (M 1) . ( 5
)
For the two observed isomers at excitation energies around ≈ 1.1 -1.25 MeV, there are several possibilities: the odd neutron (9/2 -[734] ν ) can couple to the 5 -,8 -or 3 + two-quasi-proton states observed and/or suggested in 254 No and 254,256 Rf. The 5 - and 8 -two-proton configurations are expected to be the lowest (see Table VII in Ref. [18]) and lie close in energy. So, the possible high-K and low-lying 1neutron ⊗ 2-proton configurations for the isomers are therefore

{9/2 -[734] ν ⊗ 1/2 -[521] π ⊗ 9/2 + [624] π } and {9/2 -[734] ν ⊗ 7/2 -[514] π ⊗ 9/2 + [624] π } yielding states with K π =19/2 + and 25/2 + .
Considering the constraints set by observables such as E calor , T 1/2 and the intensities in the γ-ray, electron, and CE spectra, many decay scenarios were simulated and compared with the experimental results. The decay scenario shown in Fig. 7 is the final decay scheme whose simulation reproduced the experimental spectra very well.

In this scenario, the 19/2 + is placed at 1103 keV as per the E calor of iso2. We assumed that the isomer decays to the ground-state (∆K = 5) and to an intermediate structure 11/2 -[725] (∆K = 4) via E1 transitions. From intensity arguments (see table 1), the two most intense lines (778 keV and 170 keV transitions) must be parallel. We speculate that the 778 keV transition along with two other high energy lines (666 keV and 543 keV) populate the 17/2 -, 19/2 -and 21/2 -states of the ground-state band. The 170 keV transition populates the 17/2 -of the intermediate structure, which then decays to the ground-state band via two high energy M1 transitions. All E1 transitions have hindrances f ν < 900 (610 +42 -29 , 857 +63 -47 and 688 +52 -39 for the 778 keV, 666 keV and 543 keV transition respectively) except for the 170 keV transition with f ν = 1700 +164 -121 . These values are of the order of the values obtained for three E1 transitions de-exciting the 21/2 + isomer in 257 Rf [18] and ∆K = 5 52 keV E1 transition from the 8 -isomer in 254 No [13].

Furthermore, the 708 keV and the 610 keV M1 transitions from the 17/2 -state of the 11/2 -band to the 17/2 -and the 15/2 -states of the ground state band seem to obey the Alaga intensity rule [41]. The expected branching ratio of these two transitions is < (12) experimentally. Note that to fit the high-energy lines, we have placed the 11/2 - band at 632 keV, which is slightly higher than what was inferred from alpha spectroscopy.

Regarding the 25/2 + isomer, E calor ≤ 200 sets a constraint on how far the 25/2 + isomer can be in energy from the 19/2 + isomer. It was assumed that a low energy E2 transition deexcites the 25/2 + (1238 keV) and populates the 21/2 + state of the 19/2 + band with a low energy E2 transition of 98 keV. Then an M1 transition of 102 keV to the 19/2 + state (observed experimentally) follows. To reproduce the 102 keV energy spacing, the rotational parameter ℏ 2 2J was taken as 4.9, which is acceptable for a band built on a 3-quasiparticle configuration with reduced pairing. The parameters used to build the rotational bands are summarized in table II. The 25/2 + isomer cannot be lower than the 21/2 + , otherwise it would then have to decay to the 19/2 + band head via an E3 transition, thus, having a much longer lifetime. Assuming 100 % branching ratio for the 98 keV transition from K=25/2 to K=19/2, a K hindrance of f ν = 3183 +844 -650 was obtained. Geant4 simulations of the decay of iso1 and iso2 reproduce quite well the E calor (see Fig. 8) and the gamma-ray spectrum except for a deficit of L X rays. A deficit was also noticed in the simulated electron spec-

FIG. 8: E calor in the decay of (a) iso1 and (b) iso2.

trum around 110 keV and a deficit in the CE spectrum around 148 keV. These deficits were interpreted as the contribution of the 5/2 + [622] isomer (iso3) that decays to the ground state via an M2 transition, not included in the simulations thus far. The excitation energy which best fits the excess counts in the experimental spectra places the isomer at an excitation energy of 150 keV, just below the K-shell binding energy of rutherfordium. The significantly intense 150 keV line whose nature was not understood a priori and missing in the simulated spectrum was also recovered. This excitation energy explains the lack of K X rays observed at GSI [21]. After adding the contribution of the 5/2 + isomer, the shape and intensity of the experimental CE and electron spectra are well reproduced. Fig. 9, 10 and 11 show the contribution of each isomer in the gamma-ray, electron and CE spectra, respectively. Note that in Fig. 9d, since there is 0 100 200 300 400 500 600 700 800 (keV) no background in the simulations, a normalized experimental background has been added to the total simulated spectrum to compare with the experimental counterpart (see Appendix A). The iso1, given its low E calor does not seem to contribute significantly to the electron and CE spectra. Despite this uncertainty of the excitation energy of iso1, simulations of the decay scheme reproduce the experimental spectra. The properties (spin and parity, excitation energy and half-life) of these three isomers obtained in this work are compared with the values from literature in table III.

For the simulations, we took the average threshold of the implantation detector to be 100 keV. After taking into account the threshold, the dead-time associated with the implantation detector, and the lifetime of the isomers, we have estimated the number of the three isomers produced during the reaction (see table IV). Although these suffer from uncertainties in the dead-time and thresholds, they are consistent with known populations of low-lying 1-particle and 3-quasiparticle states. The difference in isomeric ratios of the 2 high-K isomers may reflect the average spin populated in the reaction [42]. The goodness of the reproduction of all the experimental spectra gives confidence in the main features of the decay scheme shown in Fig. 7 respectively. A tentative decay scheme has been proposed. Geant4 simulations of this decay scheme were found to reproduce all the observables fairly well. Despite some uncertainties in the energy of the E2 transition of the 25/2 + isomer deexciting to the 19/2 + band (corresponding to a very large hindrance), the conclusions of this study should remain valid as it is constrained by the measured E calor ≤ 200 keV. The extracted isomeric ratios (which include dead-time and threshold effects) correspond to the ratios expected for 1-particle and 3-quasiparticle states in a reaction.

VI. OUTLOOK

The proposed decay scheme should be confirmed by repeating the experiment with better adapted electronics, perhaps with digital electronics coupled to faster and low-noise preamplifiers to lower the thresholds and the dead-time. In particular, the prompt high energy transitions could be ascertained by performing prompt spectroscopy at the target for 255 Rf as in the case of 253 No. This could be attempted using JUROGAM at RITU in Jyväskylä. However, as the cross-section is low and the flux is fragmented such an experiment will probably require the next-generation gamma-ray spectrometer such as AGATA [44]. One could also repeat the experiment on the alpha decay of 259 Sg to 255 Rf to measure the decay properties of 11/2 -and 5/2 + states similar to what was done in the case of 257 Rf to 253 No [45].

Appendix A: Sources of background in the experimental spectra

The random correlations in the DSSD (Fig. 1), tunnel (Fig. 12a) and Ge (Fig. 12b) detectors mainly occur from the decay of beam/target-like transfer products, long-lived nuclei produced in the calibration reactions or during previous irradiation. In the gamma-ray spectrum, there are also contributions from terrestrial sources such as 40 K (1460 keV), 214 Bi (610 and 1120 keV), 228 Ac (912 and 966 keV), 208 Tl (584 keV), 212 Pb (238 keV), 232 Th, 238 U and electron and positron annihilation radiation (511 keV). To exclude the random correlations from the gamma-ray and electron spectra the search time was limited to log 2 ∆T = 8.6 = 388 µs, the lifetime limit seen in Fig. 2a. This gate removes the random correlations in the electron spectrum but not completely in the gammaray spectrum especially on the lower energy side. To compare the simulated gamma-ray spectrum with the experimental one in Fig. 9d, the events in the time window from log 2 ∆T = 9 = 512 µs to log 2 ∆T = 12 = 4096 µs in the energy-time matrix shown in Fig. 12b as the background. This projected energy spectrum was then normalized to the experimental one where no peaks are observed.

FIG. 1 :

 1 FIG.1: Recoil-decay correlations measured in the DSSD for 207 Pb( 50 Ti,xn). The relevant regions of the plot has been marked from I-VII and they correspond to random correlations (I),255 Rf fission-decay events (II),256 Rf fission-decay events (III),254 Rf fission-decay events (IV),255 Rf full-energy alpha-decay events (V), alpha-decay events of transfer products (VI) and CE signals from isomeric-decay events (VII).

FIG. 2 :

 2 FIG. 2: (a) Time and (b) energy distribution comparisons between the CEs observed in the 27 ER-CE-CE-255 Rf decay chains. The dotted lines in (a) are for guiding the eye.

FIG. 3 :

 3 FIG. 3: Energy and time distribution plot of the CE signals observed in the decay chains ER-CE-255 Rf. The inset shows the fit of the time distribution.

FIG. 4 :

 4 FIG. 4: Spectrum of (a) electrons in the tunnel detectors and (b) gamma rays observed in coincidence with the CEs in the ER-CE-255 Rf decay chains.

FIG. 5 :

 5 FIG. 5: Spectrum of (a) electrons in the tunnel detectors and (b) gamma rays observed in the ER-γ/e --255 Rf decay chains.

FIG. 6 :

 6 FIG. 6: (a)Measured f ν of the prominent gamma-ray transitions observed in coincidence with the decay of iso1 and iso2, for ∆K = 3, 4 and 5, having E1, M1 and E2 multipolarities in each case. The band gives the phenomenological range, f ν = 20 -300, usually used to make spin/parity assignments. (b) Their corresponding F W compared with the expected F W after taking into account the multipority-dependent offset shown by f 0 = µ and f 0 = 100 line. The lines are there for guiding the eye.

  Fig. 6b: compared to a), b) overestimates the hindrances in all cases, b) favours ∆K = 4 E1 and M1 transitions and a) favours ∆K = 5 E1 and M1 transitions and ∆K = 4 E1 transitions.

FIG. 9 : 2 FIG. 10 :FIG. 11 :

 921011 FIG. 9: Comparison of experimental gamma-ray spectrum with simulations showing the contributions of (a) iso1, (b) iso2, (c) iso3 and (d) the total including normalized experimental background.

FIG. 12 :

 12 FIG. 12: Energy and time distributions of (a) electrons (in the tunnel detectors) and (b) gamma rays detected after the detection of an ER but before the detection of ground state alpha/fission decay event of 255 Rf in the same DSSD pixel.

TABLE I :

 I Detector efficiency corrected intensities (I γ ) of the prominent gamma-ray transitions following the decay of the isomeric states observed in this work.

  Proposed partial level scheme for 255 Rf. The half-lives of the 19/2 + and 25/2 + are from this work and that of the 5/2 -level is taken from Ref.[21].

	T 1/2 = 49 +13 -10 μs	1205 1238 25/2 + 98 E2	21/2 + 9/2 -[734] ν ⊗ 7/2 -[514] π ⊗ 9/2 + [624] π
	T 1/2 = 29 +7 -5 μs	1103	19/2 +	E1 9/2 -[734] ν ⊗ 1/2 -[521] π ⊗ 9/2 + [624] π
						E1	170	934	17/2 -
						E1 666 E1 M1 543	819 719	13/2 -15/2 -
					437 560	19/2 -21/2 -	708 610 778 M1	632	11/2 -
					324	17/2 -
	T 1/2 = 50 ± 17 μs	150	5/2 +	139 226	13/2 -15/2 -
			150 M2	64	11/2 -
					0	9/2 -
					255 Rf
	FIG. 7:

TABLE II :

 II Parameters used to build the rotational bands.

	I π	gR	gK	ℏ 2 2J (keV)	Q0 (eb)
	9/2 -	0.285	-0.25	5.8		12.42
	11/2 -	0.285	-0.22	6.7		12.42
	19/2 +	0.285	0.47	4.9		12.42
	0	50	100	150	200	250 (keV)

TABLE III :

 III Comparison of the properties (spin and parity I π , excitation energy E * and half-life T 1/2 ) of the three isomers from this work with the values from literature.

		This work			Literature
	I 25/2 +	1.24	49 +13 -10		1.15-1.45 38 +12 -7	[20]
	19/2 +	1.1	29 +7 -5	> 17/2 0.9-1.2	15 +6 -4	[20]
	5/2 +	150		5/2 +	≈ 135	50 ± 17 [21]

π E * (keV) T 1/2 (µs) I π E * (keV) T 1/2 (µs) Ref.

TABLE IV :

 IV Isomer populations and their isomeric ratios. To calculate these values, we varied the DSSD threshold from 60 keV to 100 keV in steps of 5 keV (bin width of the electron and CE spectra).

	Isomer	I π	Population	Isomeric ratios (%)
	iso1	25/2 +	256 +109 -79	3.3 +1.4 -1.0
	iso2	19/2 +	700 +236 -197	8.9 +3.0 -2.5
	iso3	5/2 +	984±170	12.5±2.2

  and in the proposed configurations for the high-K isomers, which are in line with what is known in heavier rutherfordium isotopes. This decay scheme also suggests that the 11/2 -configuration may contain some vibrational component through mixing of the 11/2 -with the particle-phonon [7/2 + [624] ⊗ 2 -] . Such a vibrational component has been predicted in the quasiparticle-phonon model[43].From this study, we confirm the existence of three isomeric states in 255 Rf which are interpreted as the 5/2 + [622] ν spin isomer and two high-K π 19/2 + and 25/2 + states interpreted as the coupling of 9/2 -[734] ν with {1/2 -[521] π ⊗ 9/2 + [624] π } 5-and {7/2 -[514] π ⊗ 9/2 + [624] π } 8-

	V. CONCLUSION

ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research (projects nos. 18-52-15004), the French national Research Agency (projects nos. ANR-06-BLAN-0034-01 and ANR-12-BS05-0013) and the IN2P3-JINR collaboration agreement no. 04-63.