Kevin Fournier

Philippe Glorennec

Nathalie Bonvallot
email: nathalie.bonvallot@ehesp.fr

An exposure-based framework for grouping pollutants for a cumulative risk assessment approach. Case study of indoor semi-volatile organic compounds

Keywords: Cumulative exposures, Risk Assessment, Toxicity Reference Values, Toxic Equivalent Factors, Hazard index

An exposure-based framework for grouping pollutants for a cumulative risk assessment approach: case study of indoor semi-volatile organic compounds.

Introduction

Human beings use many chemicals in various circumstances, such as: manufacturing and use of products and articles, industrial processing, agriculture, etc. These chemicals and their byproducts are likely to be found in the environment and are the source of diffuse and multiple contaminations of the different compartments: pesticides in soils, water, diet or air, persistent organic pollutants in the food chain, urban air pollution, volatileand semi-volatile organic compounds in indoor and outdoor environments, etc. As a consequence, human beings areexposed throughoral, respiratory or dermal routesto a large variety of chemicalspresent in food, air, water and dust, generally at low doses.The presence of such pollutantsand their metabolites in human biological fluids has confirmed this hypothesis. For example, the National Health and Nutrition Examination Surveydesigned to assess the health and nutritional status of adults and children in the United Statesincluded biomonitoring surveysthat showed the presence of detectable levels of a large variety of chemicals and their metabolites in blood or urine of human people, such as metals, phthalates or pesticides' metabolites (NHANES, 2013).

Regulations related to the chemicals entering in European market, both for domestic and professional uses, are based on risk assessment and management (European Union, 2009, 2006). However, the laws on the use of chemicals (cosmetics, biocides, pesticides, medicines, food additives and more) do not allow a comprehensive and global risk analysis. Standard toxicological tools, in particular studies on laboratory mammals, are not suitable forstudying the effects induced by complex and low-dose chemical mixtures. Little knowledge is routinely generated on the hazards of these chemical mixtures. The real impact of dietary and environmentalcontamination is not known and therefore not fully taken into account in public health decisions. The implementation of cumulative risk assessment strategies for complex mixtures is therefore of high concern as statedby Sarigiannis and Hansen [START_REF] Sarigiannis | Considering the cumulative risk of mixtures of chemicalsa challenge for policy makers[END_REF].

Avariety ofcumulative risk assessment (CRA) methodologies have already been suggested to improve risk management decision-making. For example, toxic equivalent factors have been definedfor dioxins, furans, dioxin-like polychlorinated biphenyls (PCB-dl)andpolycyclic aromatic hydrocarbons (PAHs), on the basis of their ability to bind the Aryl hydrocarbon Receptor (Safe, 1990;[START_REF] Van Den Berg | The 2005 World Health Organization reevaluation of human and Mammalian toxic equivalency factors for dioxins and dioxin-like compounds[END_REF][START_REF] Van Den Berg | Toxic equivalency factors (TEFs) for PCBs, PCDDs, PCDFs for humans and wildlife[END_REF]. Similarly, the U.S. Environmental Protection Agencysuggestedusing an approach based onrelative potency factors(RPF) for organophosphate and carbamatepesticides on the basis oftheir ability to inhibit the acetylcholinesterase enzyme (US EPA, 2006[START_REF]Guidance on Cumulative Risk Assessment of Pesticide Chemicals That Have a Common Mechanism of Toxicity[END_REF]. More recently, theseapproacheswere extended to other molecules, such as pyrethroid pesticides [START_REF] Wolansky | Relative potencies for acute effects of pyrethroids on motor function in rats[END_REF]and phthalates [START_REF] Hannas | Dose-response assessment of fetal testosterone production and gene expression levels in rat testes following in utero exposure to diethylhexyl phthalate, diisobutyl phthalate, diisoheptyl phthalate, and diisononyl phthalate[END_REF]. However, these approaches generally take into account only one chemical family. Taking into account pollutantsfrom different chemical familieswould undoubtedly better reflect actualenvironmental human exposures.This issue was first addressed by Kortenkamp and co-workerswho proposed a CRAbased on the anti-androgenic abilities of 15 substances including pesticides (fungicides, insecticides), as well asphthalates, brominated compounds and parabens [START_REF] Kortenkamp | Combined exposures to anti-androgenic chemicals: steps towards cumulative risk assessment[END_REF]. The hazard index method was then chosen to afford more flexibility in dealing with various data quality issues by using varying uncertainty factors. However, the use of different critical doses(no observed adverse effect level versus lowest observed adverse effect level for example)led to difficulties in the interpretation of results. In addition, this pioneeringstudy was based on the mechanism of toxicity of substances that people are not typicallyexposed to.It seems therefore necessary to pursue the improvement of the CRA methodology.

The objective of this work is to propose an additional step in theCRA approach, based on grouping pollutants people are actually exposed to. Only the threshold method will be developed (for non-carcinogenic or non-genotoxic compounds). We illustrate this exposurebased CRA framework with a case study based on exposure to Semi Volatile Organic Compounds (SVOCs) in French indoor environments.

A brief review of cumulative risk assessment methods

The risk assessment process was initially defined by the NRC in 1983 (National Research Council U.S. Committee on the Institutional Means for Assessment of Risks to Public 1983).

In the traditional approach, four steps are used and described as follows: 1) Hazard identification, to determine whether a chemical is linked to one or morespecific health effects, 2) Dose-response assessment, to identify the relation between the magnitude of exposure and the likelihoodor occurrence of health effects in a population. In its simplest form, this step is the choice of a toxicity reference value (a reference dose for example)according to an exposure route and duration, 3) Exposure assessment, to determine the extent and the magnitude of human exposure, and 4) Risk characterization, to describe the nature and magnitude of human health risk, including uncertainties. This last step is the combination of the previous ones. It generally compares the estimated human exposure with the selected toxicity reference value. It results in a hazard quotient (Exposure / toxicity reference value) if the threshold assumption is used, or an excess risk (Exposure × toxicity reference value) in the case of a non-threshold assumption.

A dose-additivity assumption has been used for forty years [START_REF] Sprague | Measurement of pollutant toxicity to fish. II. Utilizing and applying bioassay results[END_REF]in the cumulative risk assessment process, where a threshold hypothesis is used. The simplest approach is to sum the hazard quotientobtained for each chemical (US EPA, 2000EPA, , 1986))to obtain a hazard index. The main limitation is the lack of comparability of the ratio obtained for each pollutant when, for example, the reference doses are not defined based on the same critical effect. This method could be used to identify the chemical of most concern in the mixture but cannotpredict the effect of the mixture. This method takes into account the most sensitive critical effect induced by each pollutant. The risk could be considered as negligiblewhen the hazard index is less thanone. If not, a more in-depth approach should be used. An improvement of this approach was proposedusingthe point of departure index which directly comparesthe exposure to a critical dose, or "point of departure", in which the same effect is selected for all substances,even if the point of departures are different:no observed adverse effect level, lowest observed adverse effect levelor benchmark dose). The point of departure indexis then compared to human exposure, leading to a margin of exposure that is interpretedkeeping in mind theuncertainties ofthe toxicological data used to derive the point of departure.Another approach is directly derived and extrapolatedfrom the toxic equivalent factor for dioxins, furans,PCB-dl and PAHs, based on their affinity to the arylhydrocarbon receptor. The RPF approach first described by the US EPA(US EPA, 2002)is based on the assumptions of a similar mechanism / mode of action and parallel dose-response curves. The toxic potency of each chemical is normalized relative to the best-known compound. These RPFs can be constructed by comparingthe toxicity indicators (benchmark doses based on the same mechanism for example, or an effective dose corresponding to a percentage of response or affinity). They are then used to weight the exposure of the different compounds,then added together to determine ahazard quotientbased on the toxicity reference valueof the index compound. This approach requires more accurate mechanistic information and can therefore be applied only to the most widely studied pollutants.Several reviews with a more in depth description of these approaches are available [START_REF] Kortenkamp | Combined exposures to anti-androgenic chemicals: steps towards cumulative risk assessment[END_REF][START_REF] Sarigiannis | Considering the cumulative risk of mixtures of chemicalsa challenge for policy makers[END_REF].These CRA methodologies have been used in various contexts. For example, Wolansky and co-workers derived relative potency factors for 11 pyrethroids after an acute exposure in rats, based on the prolongation of the open state of sodium channels [START_REF] Wolansky | Relative potencies for acute effects of pyrethroids on motor function in rats[END_REF]. Similarly, in 2002 and 2006, the US EPA proposed a CRA approach for 30 organophosphate pesticides based on brain cholinesterase inhibition in rats after exposure of21 days or longerviaoral, dermal or respiratoryroutes. RPFs were determined according to their respectivebenchmark dose (US EPA, 2006[START_REF]Guidance on Cumulative Risk Assessment of Pesticide Chemicals That Have a Common Mechanism of Toxicity[END_REF].In 2011 the same approach was suggested to assess the risk related to exposure to a mixture of phthalates during pregnancy, based on a decrease in testosterone production and alteration of gene expression by fetal testes [START_REF] Hannas | Dose-response assessment of fetal testosterone production and gene expression levels in rat testes following in utero exposure to diethylhexyl phthalate, diisobutyl phthalate, diisoheptyl phthalate, and diisononyl phthalate[END_REF].In 2013, Jensen and co-workerspublished a cumulative risk assessment for 4 anti-androgenic fungicides belonging to different chemical classes (triazoles, imidazoles, dicarboximides). They used the RPF approach based on benchmark dose calculation to assess two effects (increased gestation period in pregnant females and increased nipple retention in male offspring) [START_REF] Jensen | Probabilistic assessment of the cumulative dietary exposure of the population of Denmark to endocrine disrupting pesticides[END_REF].

In the case of situations where toxicological data are complete, with a deeper understanding of mechanisms and modes of action, it is possible to refine CRA by using toxicokinetic interactions and internal doses at the target site of action [START_REF] Haddad | PBPK modeling-based approach to account for interactions in the health risk assessment of chemical mixtures[END_REF]. Taking into account the toxicokinetic interactions was also developed by Sarigiannis and co-workers. they used a physiologically-based pharmacokinetic model to assess the internal dose of benzene following a co-exposure to four volatile organic compounds (benzene, toluene, ethylbenzene and xylene). These authors showed that benzene cancer risk estimate calculated from the combined exposure was lower than risk from benzene exposure alone. This was due to a metabolic competition [START_REF] Sarigiannis | Biology-based dose-response models for health risk assessment of chemical mixtures[END_REF].

If these methods should be developed in the future, as it was suggested by the authors [START_REF] Sarigiannis | Reflections on new directions for risk assessment of environmental chemical mixtures[END_REF], these situations where mechanistic data are understood are unfortunately still too rare and it is not possible to develop this type of approach in all cases.

An exposure-based conceptual framework

3.1.Initial assumptions

The concept of dose addition

The concept of dose addition is a long-established concept suggested in the risk assessment process to take into account chemicals with a common mode of action, in which it is assumed that each component in the mixture acts as if it were a simple dilution of another one. The fundamental assumption is that each chemical in a mixture exhibits a common toxic mode of action. Typically, it would be necessary to know all toxicokinetic data and biochemical and molecular pathways for each compound in order to verify this concept. But biochemical interactions (absorption, compounds-cell interactions) cannot be known for all possible mixtures and all given situation, which is why a pragmatic approach was finally recommended for risk assessment. This concept is recommended for CRA processes by several organizations as a default approach for decision making when no data prove the contrary (European Commission, 2012;US EPA, 2010[START_REF]Guidance on Cumulative Risk Assessment of Pesticide Chemicals That Have a Common Mechanism of Toxicity[END_REF][START_REF]Guidelines for the Health Risk Assessment of Chemical Mixtures[END_REF][START_REF]Guidelines for the Health Risk Assessment of Chemical Mixtures[END_REF]WHO/IPCS, 2009), based on the fact that interactions between chemicals are unlikely to occur at environmentally relevant concentration (SCHER, 2007). The dose addition concept has also been observed for some mixtures. In 2008, Howdeshelland co-workersvalidated this concept for a mixture of five phthalate esters able to inhibit fetal testicular testosterone production in rats [START_REF] Howdeshell | A mixture of five phthalate esters inhibits fetal testicular testosterone production in the sprague-dawley rat in a cumulative, dose-additive manner[END_REF]. In 2009, other authors validated the same hypothesis, finding that the combined effects of a mixture of four anti-androgens from different chemical families were dose additive (for changes in ano-genital distance, retained nipples and sex organ weights). Nevertheless, they also observed synergistic effects on malformations of external sex organs [START_REF] Christiansen | Synergistic disruption of external male sex organ development by a mixture of four antiandrogens[END_REF]. In 2012, Christen and co-workers showed that this dose-additivity concept is appropriate to assess themixture effect of antiandrogenic phthalates and bisphenol A [START_REF] Christen | Antiandrogenic activity of phthalate mixtures: validity of concentration addition[END_REF]. They also showed a tendency to synergism at high concentrations and antagonism at low concentrationsaccording to the model and the compound mixtures selected The authors explain thesedeviations from the dose-additivityby the mechanism of interaction at the androgen receptor (binding site and strength) rather than a toxicokinetic interaction. According to Evans and co-workers [START_REF] Evans | Additive mixture effects of estrogenic chemicals in human cell-based assays can be influenced by inclusion of chemicals with differing effect profiles[END_REF], these deviations can be explained by the type of chemical tested, the number of compounds in the mixture, and the design of the study. The authors designed experiments using mixtures of pollutants(plasticizers, phthalates, metals, PCBs, phytoestrogens, PAHs, heterocyclic amines, antioxidants, UV filters, musks, PBDEs and parabens) that address these factors. The first one was a "balanced" mixture with components present in proportion to a common effect concentration. The second one was a "non-balanced" mixture with components in proportion to potential human tissue concentrations. The second part of their study consistedof testing the individual and simultaneous ability of 16 potential modulator chemicals to influence the assay outcome produced by a reference mixture of chemicals.They concluded that the doseadditivity concept commonly provides a good prediction of mixture effects.

Atrans-family multi-pollutant approach

The selection of compounds is a critical step of in CRA. It would make sense to groupchemicals on the basis of chemical similarities because of dose additivity, but in real life situations, people are exposed toa variety of pollutants from different chemical families.

This issue needs to be addressed,as pointed out by [START_REF] Sexton | Assessing cumulative health risks from exposure to environmental mixtures -three fundamental questions[END_REF][START_REF] Sexton | Assessing cumulative health risks from exposure to environmental mixtures -three fundamental questions[END_REF][START_REF] Sexton | Assessing cumulative health risks from exposure to environmental mixtures -three fundamental questions[END_REF]. In the framework we propose, the chemicals to be included in the CRA should not be limited to onechemical family (e.g. phthalates) or onechemical use (e.g. fungicides) but should be selected based on observed exposures or environmental contaminations.

Data availability

The risk assessment approach is currently used to address health risk policy issues related to environmental contaminants when epidemiological data are insufficient to establish causal relationships and calculate attributable risks to humans. The process has been defined by the US General Accounting Office as "a systematic, scientific description of potential adverse effects of exposures to hazardous substances or situations" and "a complex but valuable set of tools helping agencies to identify issues of potential concern, select regulatory options and estimate the range of a forthcoming regulation's benefit"(US GAO, 2001). Thus the main objective of a risk assessment approach is to assist policymakersin the decision-making process.Because the risk assessment process is implemented even if scientific data are lacking, a number of default assumptions are usually applied.A default option has been defined by the NRC as "the option chosen on the basis of risk assessment policy that appears to be the best choice in the absence of data to the contrary" (National Research Council, 1983). The main assumptions involve the value of using animal data to assess the risk in humans (same types of effects expected in both species), the use of a threshold approach for non-carcinogenic or non-genotoxic carcinogenic chemicals and the idea that humans are more vulnerable than laboratory animals. In the case of a CRA with a threshold approach, dose-additivity is also a default assumption. If new data becomes available, the default assumptions are not taken into account and the risk assessment process shouldbe more accurate leading to less conservative conclusions.

3.2.Structuring the hazard identification of the CRA

For a given situation, usingactualexposures for a CRArequiresa comprehensive view of pollutants, i.e.selecting pollutantsfrom a large variety. This isthe firstsub-step of hazard identificationin exposure-based CRA. In the second sub-step, the effects,modes and mechanisms of action of all the detected compounds are identified. In the last sub-step,the pollutants are grouped according to their similarity of action on the organism. At the end of these three sub-steps, the dose response assessment consists inderiving multi-pollutant toxicity reference values for each exposure situation, leading to the last two steps of the conventional approach, i.e. exposure assessment and risk characterization. The framework is presented in Table 1. Each of the 3 sub-steps is described below.

Identification of pollutants people are exposed to

This identification of pollutants people are exposed to can rely on human biological monitoring and/or on environmental monitoring. These data can originate from biomonitoring programs (NHANES, 2013;[START_REF] Schulz | Twenty years of the German Environmental Survey (GerES): human biomonitoring-temporal and spatial (West Germany/East Germany) differences in population exposure[END_REF], from total diet studies [START_REF] Arnich | Dietary exposure to trace elements and health risk assessment in the 2nd French Total Diet Study[END_REF]orfrom environmental studies in general, indoor or particular environment [START_REF] Lucas | Lead contamination in French children's homes and environment[END_REF][START_REF] Mompelat | Contamination levels of human pharmaceutical compounds in French surface and drinking water[END_REF][START_REF] Rudel | Endocrine disrupting chemicals in indoor and outdoor air[END_REF]. Main limitations at this step are related to false negative (a pollutant is not identified although it should be).The presence of false negatives may be due to high limits of detection or by the transformation of the parent compoundinto unknown metabolitesin the environment or in the body.

Identification of effects, modes and mechanisms of action

From a biological point of view, an effect due to exposure to one or more pollutants is caused by a series of mechanisms of action or "key-events", defined by Boobisand co-workersas measurable events that are critical to the induction of the effect [START_REF] Boobis | IPCS framework for analyzing the relevance of a noncancer mode of action for humans[END_REF][START_REF] Boobis | IPCS framework for analyzing the relevance of a cancer mode of action for humans[END_REF]as shown in figure 1.On the basis ofthisstatement, several levels of effects can be defined:  Clinical effect, which corresponds to an alteration of a body function observable by a clinical diagnosis. For example, alteration of cognitive or behavioral functions, such as hyperactivity or attention-deficit, could be diagnosed based on standardized testslike the Behavior Assessment Systems for Children;

 Impairment of the target organ. The clinical effect previously described is usually induced by an effect on an organ. For example, in the case of behavioral impairment, the effect on a target organ could be an alteration of the thyroid or the brain;  Effect at a cellular level: at this scale, cell damage can cause alteration of the target organ. These effects can be observed directly in exposed laboratory animals (after autopsy, microscopicobservations) or by in vitro tests (cell cultures). Examples include swelling, necrosis or cell death, including changes in the dendritic morphology of hippocampal neurons or abnormal plasticity of the synaptic junctions.

 Effect at a molecular level, which corresponds to the direct interaction between the macromolecules responsible for cell function (proteins, aminoacids, lipids, nucleic acids) and the chemicals (or metabolites).A variety of intracellular mechanisms of action can be disrupted. These effects are observable by in vitro tests or with highthroughput techniques such as transcriptomics or proteomics.Examples include phosphorylation of the N-methyl-D-aspartate receptor or a direct interaction with the estrogenreceptors. These types of effects arerarelyelucidated. Nevertheless, the molecular consequences of these interactions can be identified, such as changes in Nmethyl-D-aspartate receptor expression, a decrease in neurotransmitters or enzymes levels (dopamine, etc.) or over-expression of a specific gene.In our framework, we consider the molecular consequence of direct interactions as an effect at a molecular level.

In this step, it is important to address the biological plausibility of the effects.The mode of action observed in animals mustbe plausible to humans and higher plausibility is associated with a higher confidence level [START_REF] Meek | A framework for human relevance analysis of information on carcinogenic modes of action[END_REF].Thus, it is important to take into account the species tested (in vivo, in vitro), the window, route and duration of exposure, etc.

Moreover, as pointed out by Borgert et al, a detailed characterization of the pharmacodynamics, pharmacokinetics and slope of dose response curves may be important criteria in evaluating whether the toxicity of a mixture can be predicted by the mode of action of its component chemicals [START_REF] Borgert | Can mode of action predict mixture toxicity for risk assessment?[END_REF].

Grouping pollutantsbased on similaraction

The available information on effects, modes and mechanisms of action for each pollutant canbe gathered into a unique dendrogram recapitulating the 4 levels of effects (clinical, organic, cellular, and molecular). Figure 2 describes this framework that groups information according to theselevels. Pollutants inducing the same effects on organisms or having the same mechanism of action at a cellular and molecular level can be identified and grouped. This grouping is also a way to classify pollutants according to the availability of data. Itdrives the choice of a CRA method:hazard index, point of departure index, or RPF.According to the available information, the hazard indexor the point of departure indexapproachescan be used when we have information only on a clinical effect (e.g.reduced fertility, behavioral changes) or a target organ or cell, depending on the application of uncertainty factors (for each pollutant or for the mixture). When the mechanism of action is known (e.g. binding to a receptor) or the consequence of this mechanism of action is known (e.g. decreased protein synthesis),an RPF approachcan be used. This exposure-based CRA framework was tested on a case study presented below.

Case study: semi-volatile organic compounds in French indoor environments

4.1.Context

Semi-volatile organic compounds (SVOCs) such as phthalates, PBDEs and PCBs are ubiquitous in indoor environments because of the increasing use of new materials and substances (pesticides, plasticizers, flame retardants, preservatives) [START_REF] Mercier | Cumulative indoor exposures to Semi-Volatile Organic Compounds in France: measurement methods in settled dust[END_REF]and most of them are considered as endocrine disruptors [START_REF] Rudel | Endocrine disrupting chemicals in indoor and outdoor air[END_REF].Due to their physicochemical properties, SVOCs partition betweenair, airborne particles, surface and settled dust, making them easily bioaccessible, whether viarespiratory, oral or dermal routes.

Children are particularly vulnerable to exposure (vulnerable biological systems and vulnerable behavior with hand-to-mouth activity).

Step 1a: Pollutant identification based on real exposures

A list of SVOCs was prioritized on the basis of their contamination and toxicity levels (Bonvallot et al, 2010). Nationwide sampling campaigns were carried out in French dwellingsto characterize the contamination of indoor air and settled dust by more than 55 SVOCs [START_REF] Blanchard | Cumulative indoor exposures to Semi-Volatile Organic Compounds (SVOCs) in France: measurements of endocrine disruptors in 30 French dwellings[END_REF][START_REF] Glorennec | Cumulative indoor exposures to Semi-Volatile Organic Compounds (SVOCs)in France: the ECOS project[END_REF]measured together in a specific multiresidue analysis [START_REF] Mercier | Cumulative indoor exposures to Semi-Volatile Organic Compounds in France: measurement methods in settled dust[END_REF]2012). For CRA, we selected the SVOCswith concentrations in air or settled dust above the limit of quantificationin more than 25% of dwellings, on the basis of preliminary results (n=235). We identified21 SVOCsout of 55 measured including 8 phthalates (di(2-ethylhexyl)phthalate (DEHP), dibutyl phthalate (DBP), di-isononyl-phthalate (DiNP), di-isobutyl-phthalate (DiBP), bentyl benzyl phthalate (BBP), di-ethyl phthalate (DEP), bis(2-methoxyethyl)phthalate (DMEP)anddi-methyl phthalate (DMP)), 4 PBDEs (BDE-47, -99, -100 and-209), 3 pesticides (lindane, permethrin and cypermethrin),3 PCBs (PCB-101, -138 and -153), 2 musks (galaxolide andtolanide) and bisphenol A (BPA). (thyroid hormones), which are strongly involved in the nervous system and brain development (see Supplemental Material, figure 4).

4.4.Step 1c: Grouping pollutants

Toxicologicaldatawere tabulated indendrogramsto visualize groups of pollutants according to their effects or mechanisms of action for each system (reproductive and nervous) (figures 3 and 4). This also showstheaccuracy of the available toxicologicalinformation (from target organ to molecular level).Five groups of pollutants were identified (Table 2)witha common molecular consequence of their mechanisms of action (decrease of testosterone synthesis, INSL3 or connexin 43 inhibition, decrease of T4 or dopamine levels). The data areaccurate enough to supporta hypothesis on the consequence of the molecular interaction between the SVOC and the organism. For these groups, aRPFapproach could be suggested. Most of these groups contain different chemical familiessuch as phthalates, pesticides, PCBs and PBDEs.

Two other groups of pollutants described in Table 3 have common cellular or clinical effects (neuronal cell impairment and embryotoxic effects). The data are notaccurate enoughto elucidate the mechanism of action. For these groups, a Point of Departure Index could be suggested.

In all, considering indoor SVOCs and the reproductive and nervous systems, 7 groups of pollutants can be considered for a cumulative risk assessment: 5 groups for a RPF approach, and 2 for a point of departure index approach. They are composed of 5 to 12 pollutants from different chemical families in most cases.

Discussion -Conclusion

This framework adapted the traditional CRA approach to a trans-family multi-pollutant approach by taking into account data availability and actual exposures. Our proposed conceptual framework consists of adding a step to thetraditional cumulative risk assessment approach. Based on actual exposures, it groups pollutants according to their effect, on the basis of their mode or mechanism of action.Applying the concept to a real case allowed us to identify issues and to assess the value of the approach.

This approach involves a number of important methodological issues.First, selection of pollutantsmust be based on criteria of interest previously defined to achieve the objective (frequency of quantification in our case study). During this step, we eliminated a number of chemicals. Second, the available information varies according to the body system studied, since there is more mechanistic information onpollutants affecting reproduction than for other systems. The RPF approach wouldtherefore be easier to implement for reproductive effects.

From a mechanistic point of view, the system for which there is the least information is the immune system. For this system, a point of departure indexapproach is likely to be promoted, emphasizing effects rather than mechanisms. The case study also illustrated that there is a trade-off between the number of pollutants we wish to consider and the accuracy level for dose-additivity. Indeed, because of the accuracy of available information, the more pollutants we want to include, the less likely it is that the RPF approach can be used. Third, uncertainty plays an important role in risk assessment. In the case of pollutants grouping for CRA approaches, the main sources of uncertainty include the selection of compounds (no pollutant must be neglected) and the knowledge of mechanisms or modes of action, which is fragmented. Nevertheless, the use of this approach provides a wider perspective on human health risks and results in a lower uncertainty than the traditional risk assessment.

Finally, this case study demonstrated the value of an exposure-based CRA, as we determined that pollutants with similar effects or mechanisms of action donot all belong tothe same chemical family.The approach has also been shown to befeasiblealthough the greatest limitation is its dependence on the availability of toxicological data.

In conclusion, the proposed methodology for creating groups of substances with the same targets and modes or mechanisms of action can formthe basis for exposure-based cumulative risk assessment for many situations with complex exposures. The main challenge is still to generate toxicologicalor epidemiologicaldata on the largest possible number of pollutants that are relevant to human exposure. Traditionally, the hazard index approach is used to assess the health risk of a substance. The point of departure index approach is used when more accurate information (physiological effect, target organ or target cells) is available. The Relative Potency Factor (RPF) approach is used when even more accurate information (mode or mechanism of action) is available.

4. 3 .

 3 Step 1b: Hazard identification, effects and mechanisms of action Hazard identification of the 21 SVOCs was performed by means of a literature review focusing on two well-documented systems: the reproductive and nervous systems. Selection criteria included the description of mechanism of action of these SVOCs, and the description of the target organs/cells. When there were a great many articles, priority was given to the most recent ones.4.3.1. Reprotoxic effectsAccording to the time of exposure (pre or post-natal), some SVOCs may induce malformations of the genital tract orreducedfertility (alteration of sperm production) in mammals due to their impact on Leydig and Sertoli cells. Different mechanismsof levels (StAR, P450scc),leading to decreased testosterone synthesis;inhibition ofFollicle-Stimulating Hormone (FSH) action, leading to inhibition of Androgen Binding Protein (ABP) and/or inhibinα (INHα) synthesis andstopping Sertoli cell proliferation;induction of ROS production, leading to the death of Sertoli cells; inhibition ofconnexin 43 (protein acting as a link between Sertoli and germ cells) with a release of Fas-L death signal. Reduced female fertility due to lower aromatase activity, and higher testosterone, androstenedione and progesterone secretionwas also observed (see Supplemental Material, figure3).4.3.2. Neurotoxic effectsTwelve of the 21 selected SVOCs act on the nervous system and are able to interfere with neuronal development (DEHP, BDE -47, -99, -209, BPA, PCB-101, -138, -153, lindane, permethrin and cypermethrin). Different modes of action were suggested: an increase in intracellular calcium which induces an increase of the frequency of the postsynaptic current and promotes the release of neurotransmitters;oxidative stress inducing neuronal apoptosis and a decrease in dopamine levels, inducingneurodegeneration.The selected SVOCs are also able to act on the thyroid in females and newborns,by inhibiting synthesis of T3 and T4

Figure 1 :

 1 Figure 1: Effect of pollutants at different body scalesWe are exposed, via different routes, to a number of pollutants which penetrate the body and reach the cells of target tissues and organs. Key molecular events are responsible for organic dysfunction affecting the exposed population.

Figure 2 :

 2 Figure 2: Suggested way of presenting toxicologicalinformation in a cumulative risk assessment

Figure 3 :

 3 Figure 3: Mechanisms of action and toxic effects of selected indoor semi volatile organic compounds (SVOCs)on the reproductive system 15 pollutants act at different levels of the reproductive system. This dendrogramshowsthe accuracy of the information for each one.It allows us to identifythose acting together at different scales of this system, which can be grouped to determine toxicityreference values.

Table 1 :

 1 Comparison of the proposed exposure-based cumulative risk assessment framework with the conventional cumulative risk assessment.

	Conventional CRA approach	Exposure-based CRA approach
		1a. Identification of pollutants people are
		exposed to
	1. Hazard identification	1b. Hazard identification: effects and mechanisms of action of each pollutant
		1c. Grouping of pollutants with similar
		effects and/or mechanisms of action
	2. Selection of toxicity reference values	2. Derivation of "multi-pollutant"toxicity reference values
	3. Exposure assessment
	4. Risk characterization

Table 2 :

 2 Groups of semi volatile organic compounds (SVOCs)for which the relative potency approach is suggested

Table 3 :

 3 Groups of semi volatile organic compounds (SVOCs)for which the point of departure indexapproach is suggested

	Group of SVOCs	Description	Clinical effects	Cellular effects	Mechanisms
		BDE 47, BDE 99, BDE
		100, BDE 209, BPA, PCB	Nervous system:	Neuronal
	Group F	101, PCB 138, PCB 153,		cell	?
		lindane, permethrin,	Neurodegeneration	impairment
		cypermethrin
			Reproduction:
	Group G	DEHP, DBP, DMEP, BPA,lindane	and/or teratogenic Embryotoxic, fetotoxic	?	?
			effects

Acknowledgments

We would like to thank the FrenchSchool of Public Health (EHESP) and its LERES Laboratory for having provided SVOC contamination data.

We should like to thank the School of Public Health for funding.

Competing financial interest declaration

The authors declare they have no competing financial interests.


C Vecchini

A Bombardi

L C Chapon

G Beutier

P G Radaelli

S Park

S.-W Cheong

Magnetically induced femtoscale

Keywords: numbers: 75.50.Ee, 75.30.Gw, 75.85.+t, 78.70.Ck

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INTRODUCTION

In all magnetic systems, the onset of magnetic order leads to magnetostriction effects, which can also induce the onset of charge modulation in materials [1]. Density waves in the charge distribution could be due either to a periodically distorted lattice, with each ion retaining its equilibrium charge, often referred as strain wave, or by a periodic excess and deficit of charge on the sites of an undistorted lattice (charge density wave). Strain waves have been studied since the '70s for example in chromium [2][3][4] and can provide a very convenient tool for the study of magnetic domain populations [5] as well as magnetic propagation vectors [6] or multi-k structures [7]. In magnetoelectric multiferroic materials [8], where magnetism and ferroelectricity coexist and couple, the study of strain waves and their relation with the magnetic structure can be of particular interest for the fundamental understanding of ordering phenomena. The system presented in this study belongs to the RMn 2 O 5 family (R=rare-earth or Y, Bi), whose members have a complex orthorhombic crystal structure (space group Pbam). At low temperature, they exhibit a rich magnetic phase diagram due to a complex pattern of magnetic interactions. Upon cooling, at T N 1 ∼ 40K, HoMn 2 O 5 becomes antiferromagnetic with an incommensurate (ICM) magnetic order which locks into a commensurate (CM) structure at T N ∼36K, characterized by anomalies in the dielectric constant and the presence of spontaneous electric polarization. A further phase transition to a reentrant incommensurate phase appears at T LT -ICM ∼18K, which is accompanied with an abrupt decrease of spontaneous polarization. Symmetric magnetic exchange has been recognized to be the primary origin of the spontaneous electric polarization (P) observed in the commensurate magnetic phases of RMn 2 O 5 systems [9,10], although a smaller antisymmetric exchange contribution has also been observed (R=Y,Ho,Tb) [11]. In this family of compounds, polar displacements due to the structural relaxation induced by competing Heisenberg terms (J ij S i • S j ) and by Dzyaloshinskii-Moriya terms (D ij • S i × S j ) in the magnetic Hamiltonian lead to small P values compared to proper ferroelectrics. The ionic displacements due to magneto-striction are predicted to be of the order 10 -4 Å in magnitude [12] and as such are not measurable by laboratory crystallography instruments. A recent experiment to determine the induced lattice distortions in the perovskite multiferroic TbMnO 3 has been successfully performed by Walker et al. [13] by using a combination of electric and magnetic fields (>2T) to produce a single magnetic domain state and to switch on the linear magnetoelastic coupling [14]. In these conditions, an interference between charge and magnetic x-ray scattering arose which encoded the amplitude and phase of the displacements. When combined with a theoretical analysis, the data measured at the first harmonic k m (k m representing the magnetic propagation vector), which correspond to the biquadratic coupling term in the free energy expansion (k m • k c ) 2 , where k c represent the lattice modulation, allowed Walker's team to resolve the ionic displacements at the femtoscale. Nowadays it is possible to determine femtoscale distortions occurring at 2k m originating from the third order linear-quadratic term k 2 m • k c , using synchrotron radiation and large area photon counting detectors. This approach can be applied to all magnetic systems showing magnetostriction effects or to materials where secondary order parameters lead to subtle lattice distortions. The magnetostrictive interactions, and therefore the corresponding displacements, are invariant in a general reversal of the moments, and for a magnetic structure with propagation vector k m , the period of the charge modulation is half the period of the magnetic modulation [START_REF] Lovesey | X-Ray Scattering and Absorption by Magnetic Materials[END_REF][START_REF] Lawiński | [END_REF]. In the RMn 2 O 5 case, it has been shown that the k c (k c = 2k m ) modulation is induced by the main (magnetic) order parameter. In a previous experiment [17] the temperature dependence of the (4 4 0.5) satellite reflection was measured and the quadratic law dependence with respect to a magnetic (2.5 3 0.25) reflection agrees with the time-reversal symmetry invariance requiring the lattice distortion L ∝ M 2 at the leading term. Therefore the structural displacements induced by the magnetic structure can be investigated by the modeling of such satellite charge reflections. Here a systematic measurement of those superlattice reflections was performed. A displacement model to determine the modulation pattern arising from the frustration of the underling magnetic structure was found. The magnetic transition breaks the inversion symmetry and in the CM phase the system goes into a polar group. Several experiments support a lowering of symmetry at least to the space group Pb2 1 m, however as the distortions are tiny, no direct evidence of a symmetry breaking transition could be obtained from x-ray diffraction till now [18,19].

EXPERIMENTAL

A high quality single crystal (1 x 1 x 3mm 3 ) of HoMn 2 O 5 grown using flux method [11] was used for the x-ray scattering experiment, performed at the I16 beamline (Diamond Light Source Ltd., UK) [20,[START_REF] Collins | AIP Conference Proceedings[END_REF]. The sample was mounted on the beamline Kappa-diffractometer in a closed circle refrigerator (CCR) with base temperature of ∼ 5K. The crystal was oriented with the diffraction face perpendicular to the (110) direction. The diffractometer was operated in the vertical plane scattering mode with linear horizontal polarization of the incident beam and the photon energy was tuned to 14keV. This energy was chosen in order to maximize the scattering volume and the access to reciprocal space while still maintaining a good detector efficiency of about 60% (320µm thick Si sensors). Two Rh coated mirrors were used to focus the beam to 30 × 180 µm 2 and high harmonic contaminations suppression. A Dectris R Pilatus 100k single-photon-counting area detector has been used throughout the whole experiment. This detector combines high resolution with extremely small point-spread and a large dynamic range (20 bits), essential requirements to be able to collect weak superlattice reflections as well as the strong Bragg charge reflections. At this energy, harmonic contaminations can be omitted in the calculations: the mirrors transmission for the second harmonic (E=28 keV) is < 1% whereas the detector efficiency decreases to ∼ 10%. Moreover the flux provided by the ondulator will be much decreased due to its optimization for 14 keV and of the peak intensity of the flux provided by the synchrotron itself, which leads to a further decrease of intensities of more than one order of magnitude. Finally the ratio of the monocromator structure factor for the Si333 and the Si111 is about half, while for the Si222 is vanishingly small. We measured the integrated intensities of 455 reflections at position (h, k, l+1/2) in the commensurate (CM) magnetic phase at T=22K. A dataset in the paramagnetic phase at T=45K was also collected using the same statistics for background subtraction. Due to the CCR vibrations together with the sphere of confusion of the K-diffractometer (∼ 80 µm 3 ) and the possible non-homogeneity of the sample surface, it is possible to switch between different grains or even probe several grains at the same time, making the data analysis extremely challenging. Therefore most care was given to probe the sample surface by collecting 85 reflections in a different position on the sample surface, again both in the CM and in the paramagnetic phases, to be used as a consistency check. Moreover we measured several charge reflections with the purpose to determine the scaling factor for quantitative analysis. The data were then corrected for background and possible harmonic contaminations by pixel-by-pixel subtraction of the high temperature data and then corrected for geometrical (self-)absorption and Lorentz factor, multiplying them by the

factor A(Q, ψ) • L(Q) = 1 + sin α(Q,ψ) sin β(Q,ψ)
• sin 2θ(Q) where α(Q, ψ) and β(Q, ψ) are the incident and exit angles with respect to the crystal surface, with no need for polarization correction in this geometry. The intensities were all treated as not equivalent, regardless of symmetry operations. The corrected integrated intensities were then used to analyze the distortion pattern with the FullProf program suite [START_REF] Rodriguez-Carvajal | [END_REF], by using the propagation vector formalism, in which the displacements for all sites are expanded in Fourier series. For the superstructure with propagation vector k c , the displacement u n of an atom in the n th cell is calculated as:

u n = D n (-1) n (1)
The components of the vectors D are defined in term of cartesian components D x , D y , D z for each n, which are sufficient to fully determine the superstructure. Constraints exists between D n for symmetry related sites, these constraints are determined by representation analysis and will be described below.

RESULTS AND DISCUSSION

Superlattice reflections are typically weaker than the fundamental ones by a a factor larger than 10 6 . The rocking curve of a fundamental charge (8 8 2) and a very strong superlattice (SL) (7 9 0.5) reflections are shown in figure 1. The FWHM for a rocking curve on a typical superlattice reflection on HoMn 2 O 5 is σ ∼0.03 • (in θ), value that is comparable to the FWHM observed for the fundamental ones, hence the ionic displacements show long range order. The temperature dependence of two superlattice reflections (7, 9, 0.5) and (6, 7, -0.5) have been collected in the temperature range 5K<T<45K and is shown in figure 2. The appearance of the k c reflections coincides with the onset of the long range magnetic order (T N 1 ∼40K), which develops in a CM magnetic structure at ∼36K where an abrupt change of slope is observed. Further cooling in this phase leads to the observation of maximum intensities at about 18K for both measured reflections, followed by a decrease due to the low temperature CM to ICM magnetic phase transition. These features are in very good agreement with previously reported neutron and magnetization measurements [23]. The change in the signal observed in the low temperature phase (T<T LT -ICM ) is also in agreement with the expectation of a modified ionic displacements pattern due to the change in the magnetic structure. At 14 keV, the separation between e.g. (4.96 8 0.56) and (4.96 8 0.44) is ≤ σ. This doesn't allow a complete deconvolution of the four split ICM peak positions (although the separation between (0.96 0 0.56) and (1.04 0 0.44) is large, >5σ and can be separated). The change in intensity in this phase is smoother that the previously reported one for the TbMn 2 O 5 case [24] but in good agreement with the previously reported data on HoMn 2 O 5 [17]. An initial attempt to use an unconstrained model to fit the data has been performed but due to the large number of free parameters, a stable fit could not be achieved: the completely unconstrained model involves up to 97 free parameters: 32 atoms per unit cell with x,y,z displacements components and a global scale factor. The latter has been determined with the use of few charge reflections in the refinements, which have been corrected for self-absorption. In order to reduce the number of free parameters during data modelling, we used the symmetry mode analysis method. The symmetry properties of a distortion mode are characterized by an Irreducible representation (Irrep) of the high-symmetry space group, defining the mode transformation properties under the operations of this group. Therefore we first determined a basis of symmetry modes of the parent phase (P bam space group) compatible with the low symmetry phase, and then we decomposed the structural distortion as a sum of the contributions of all of them. We used the computer program BasIreps, part of the FullProf suite, to perform the symmetry analysis [START_REF] Rodriguez-Carvajal | [END_REF] and the programs ISOTROPY [25] and Amplimodes [26], the latter from the Bilbao Crystallographic Server [27][28][29].

As shown in table I, there are 8-one-dimensional Irreps allowed for the Pbam space group for the wave vector k c =(0 0 0.5). For each Irrep, the number of free parameters varies between 12 and 14. An initial leastsquare refinement for each of the 8 Irreps lead to very poor agreement factors. The best fits (with a single Irrep) were obtained with Z + 1 and Z - 4 with R(F)= 51.6% and 39.6% respectively. However a mixing of the two (Z + 1 and Z - 4 ), with a total of 16 free parameters, lead to excellent agreement factors: R(F) factor and χ 2 are 10% and 10.3 respectively. Figure 3 shows the results of the final refinement, where the experimental structure factors are plotted against the calculated ones. Moreover, as a consistency check, we used the same model with no free parameters to fit the reflections collected in a different position on the sample surface, with excellent agreement. It is therefore important to notice a single irrep distortion component is not sufficient to explain the full symmetry break of the transformation. A combination of two distortions belonging to different Irreps is indeed necessary. Z - 4 is the main distortion mode associated with the principal magnetic order parameter and has only four active symmetry operations: two fold axis along b, which is compatible with the observation of spontaneous electric polarization in the b direction in this phase and two mirror planes (m xy and m yz ), while it doesn't contain inversion symmetry. Z + 1 is the totally symmetric and is a secondary order parameter; in this representation all symmetry operations of the high symmetry space group are allowed. Although Z + 1 is not related to the polar distortions which are expected to occur at the Γ point (not reported in this manuscript due to the presence of charge peaks and multiple scattering that make the measurement very hard to perform), Z - 4 and Z + 1 together break inversion and the remaining symmetry is consistent with the point group symmetry m2m, in agreement with the previously suggested point group symmetry for the low temperature CM/ferroelectric (FE) phase, magnetic point symmetry m2 ′ m [23,[START_REF] Radaelli | [END_REF].

Figure 4 shows the in plane (ab) ionic displacements pattern (table II), where the arrows (magnified by 1000 to be visible) represent the ionic displacements. The Ho ions displace in the ab plane. This configuration is very stable and even a refinement with no restraint on these sites leads to the same configuration reported here (error bars ∼ 10 -4 ). The Ho displacement belongs to the II) lie in the same mirror plane as Ho ions and share the same in plane displacement as Ho (Wyckoff label 4g), however because of the large difference in the x-ray atomic form factors between these two species, only the a component shows error bars smaller than the displacement itself. The same is true for the Mn 3+ ions and O 2-(labeled respectively Mn2 and O3 in table II), lying in the mirror plane at z=0.5 (Wyckoff site 4h). In this case the error bars are of the same order of magnitude of the displacements and therefore we conclude no displacement on these sites can be measured from the available data. A fit leaving the oxygens sites, labeled O4 (general Wyckoff site 8i ), to move freely lead to good agreement factors, but constraining them with regard to symmetry and in the basal plane, lead to a considerably better fit and smaller error bars. The Mn 4+ ions (Mn1) sitting in the octahedral sites, displace mainly in the z direction, with a smaller in-plane component, whereas the O1 sites show very large in-plane components, roughly orthogonal to the Mn 3+ -O-Mn 3+ chains. The ionic displacements observed are of the order of 10 -3 Å with error bars in the Ho sites of 10 -4 Å.

The out of plane component observed on the Mn 4+ sites is shown in figure 5, where the effect of the k c propagation vector is shown by the alternating + + --periodicity along c. Despite the large number of reflections measured, due to the small sensitivity of the x-ray to light elements, ionic displacements of the oxygens sites show large error bars ( 30%). However, these displacements are essential to the model stability.

MODEL FOR THE PROPAGATION ALONG THE C -AXIS

In contrast with the complexity of the ionic displacement pattern in the ab-plane, the out of plane Mn 4+ displacement follows a simple ordering. This periodicity could be used, together with the magnetic ordering information, to determine the stabilization conditions for the magnetic propagation vector k z =1/4 and to find the relative phases of the magnetic and crystal structures, effectively linking the magnetic and the crystallographic structure by removing the inherent phase ambiguity arisen from structural refinements alone. Here we use an approach based on the linear-chain model, in which only the c and a components, respectively of displacements (δ) and magnetic nearest-neighbor (NN, J 1 , J 2 , where J 1 is the Mn 4+ -Mn 4+ intracell exchange constant, J 2 is the intercell Mn 4+ -Mn 4+ exchange) and nextnearest-neighbor (NNN, J 3 ) interactions are considered (fig. 6) [33]. The Mn 4+ in-plane spins are defined as:

s 1 = S • cos(2πq z R L + φ) s 2 = S • cos(2πq z R L + φ) (2)
where R L is the unit cell index, q z is the magnetic propagation vector component (along z ) and φ is the phase between the structure and the magnetic sublattice. φ being expressed in radians, so it relates to the tabulated values as φ = 2πΦ (Φ are site dependent relative phases [11]). We can construct an idealized onedimensional model, coupled through Nearest-Neighbour interactions (J 1 and J 2 ) and Next-Nearest-Neighbour interactions (J 3 ).

J 1 = J 1 J 2a = J 2 (1 -δ) J 2b = J 2 (1 + δ) J 3a = J 3 (1 -δ) J 3b = J 3 (1 + δ) (3)
where the J 2 and J 3 exchange constants are split as function of the ionic displacement δ. The total energy per unit cell is:

E = 1 N R L J 1 s 1 • s 2 + 1 2 J 2a s 2 • s ′ 1 + J 2b s 1 • s ′ 2 + + 1 2 J 3a s 1 • s ′ 1 + J 3b s 2 • s ′ 2 (4)
where the primed spins belong to the neighbouring unit cells. By replacing the expressions of the spins and the exchange integrals in equation 4 and minimizing the energy with respect to φ, we obtain the relative phases between the magnetic sublattice and the crystal structure: the coupling of the ionic displacements with J 2 and J 3 leads to the stabilization of the magnetic structure at a phase φ = π/4 with respect to the crystal structure (Φ = 1/8). This phase allows for even moment magnitudes throughout the system, in contrast for example to a phase of Φ = 1/4, which leads to an alternating stack along c of [+0 -0] magnetic moments.

CONCLUSIONS

In this work we develop a method to measure and refine tiny collective ionic displacements in a complex magnetic system. Although the crystal studied is a multiferroic compound, this method is not restricted to this class of materials. In fact no external perturbation (electric/magnetic field) was applied therefore no magnetoelectric domain have been selected. We have demonstrated that very high precisions in the determination of the magneto-striction induced ionic displacement pattern (∼ 100 fm, light atoms) are achievable. Moreover this work shows the first direct evidence of a subtle symmetry breaking transition in the CM magnetic phase of HoMn 2 O 5 . Finally, we have also shown that by knowing both the magnetic structure and the magnetically induced strain waves, it is possible to determine the global phase between them removing any phase ambiguity in the magnetic structure determination. These high accuracy measurements could be used to both convalidate and challenge ab initio theoretical predictions of displacement modes due to competing (super)-exchange interactions in (f electrons) frustrated magnetic materials.

Work at Rutgers was supported by the NSF Grant No.

DMR-1104484

Symmetry Operations Irreps {1|000} {200z|000} {20y0|

1 2 1 2 0} {2x00| 1 2 1 2 0} {1|000} {mxy0|000} {mx0z| 1 2 1 2 0} {m0yz| 1 2 1 2 0} Z + 1 1 1 1 1 1 1 1 1 Z - 1 1 1 1 1 -1 -1 -1 -1 Z + 2 1 1 -1 -1 1 1 -1 -1 Z - 2 1 1 -1 -1 -1 -1 1 1 Z + 4 1 -1 1 -1 1 -1 1 -1 Z - 4 1 -1 1 -1 -1 1 -1 1 Z + 3 1 -1 -1 1 1 -1 -1 1 Z - 3 1 -1 -1 1 -1 1 1 -1 TABLE I.
Action of the symmetry operations of the paramagnetic group on the distortion modes associated to different irreducible representations. Symmetry operations are shown in the Seitz notation [30]. The Irreps symmetries are given in Miller and Love notation [START_REF] Miller | Tables of Irreducible Rep-resentations of Space Groups and Co-Representations of Magnetic Space Groups[END_REF]

FIG. 1 .

 1 FIG. 1. Peak profiles of the (7 9 0.5) and (8 8 2). The line is a fit to the data with a Lorentian profile function.

FIG. 2 .

 2 FIG. 2. Temperature dependence of integrated intensities of SL peaks: (6 7 -0.5) (upper panel) and (7 9 0.5) (bottom panel).

FIG. 3 .

 3 FIG. 3. Results of the refinement with kc=(0 0 0.5) for the ionic displacement for HoMn2O5 compound. The experimental structure factors are plotted against the calculated ones. The agreement factors are R(F) factors and χ 2 are 10% and 10.3 respectively.

FIG. 4 .

 4 FIG. 4. (Color online) Representation of the observed ionic displacements (kc=(0 0 0.5)) in HoMn2O5 with arrows (magnified by 1000). The structure is shown in one unit-cell, marked by thin black lines, along the a-and b-axis. The projection is separately shown for ionic displacements sites belonging to adjacent crystallographic cells along the cdirection. The purple (small and light grey), grey and blue (black) arrows represent ionic displacements on M n, O 2+ and Ho 3+ sites respectively. Corresponding Mn-O polyhedra are also shown.

FIG. 5 .

 5 FIG. 5. (Color online) Representation of the observed ionic displacements (kc=(0 0 0.5)) in HoMn2O5 with arrows (magnified by 1000). The structure is shown in four unit-cells, marked by thin black lines (only one cell), along the a-and c-axis. The projection is separately shown for ionic sites belonging to the first zig-zag chain (top) and the second chain (bottom). The purple (small and light grey), grey and blue (black) arrows represent ionic displacements on M n, O 2+ and Ho 3+ sites respectively. The green (light grey) horizontal arrows inside the M n 4+ polyhedra represent the spin ordering of M n 4+ from reference [11]. Corresponding Mn-O polyhedra are also shown.

FIG. 6 .

 6 FIG.6. (Color online) Labeling scheme and relevant interactions for the linear-chain model. Purple (dark grey) arrows represent ionic displacements while green (light grey) arrows represent the spin ordering from reference[11]

The Fourier coefficients for each site in the primitive unit-cell are given along crystallographic direction (x,y,z) and labeled D. Error bars are given for all parameters within parenthesis.