Jonathan Burton

Alita Nandi

Lucinda Platt

Measuring ethnicity: challenges and opportunities for survey research

Keywords: Ethnic Identity, Ethnicity, Ethnic Groups, survey data, Census, Categories ethnic identity, ethnicity, ethnic groups, survey data, census, categories

published or not. The documents may come

Introduction

At least since Weber, extensive analytic and conceptual effort has been dedicated to defining ethnicity and its relationship to other components of identity, behaviour and expression. Much discussion highlights its fluid and contingent nature, the ways in which ethnic boundaries are formed or the conditions under which they come into being, and the situational nature of ethnicity (Wimmer, 2 By contrast, standard measures in surveys and censuses aim for stability and present the respondent with relatively crude fixed categories. Some analysts, working with qualitative insights into the complexity, temporal and geographical specificity and multiplicity of ethnic identifications, and frustrated by the lack of conceptual coherence in standard measures, have rejected census and survey measures as being inadequate to the task of informing us about ethnicity, or as used to buttress stratifying processes [START_REF] Zuberi | White Logic, White Methods: Racism and Methodology[END_REF]). Yet there remains substantial demand from research and policy perspectives for engaging with issues of ethnicity in survey research, and for carrying discrete measures that can be implemented in a survey context.

Rather than urging the separation of analysis of ethnicity from ethnic group survey measures, we suggest that there might be ways to bring them together more effectively [START_REF] Nobles | Racial categorization and censuses[END_REF]. By engaging both with qualitative insights into multiple affiliations and the multi-dimensionality of ethnicity and with the practical insights of survey methodology into the logic of question setting and the ways responses are shaped, we suggest that it may be possible to find a way of measuring ethnicity in surveys that avoids some of the pitfalls of current categorical questions, has greater conceptual rigour and is amenable to analysis across a wide range of disciplines and research interests.

To this end, we critically review why measuring ethnic identity has been problematic, and reflect on whether there is a way forward focusing on the UK.

While the issues around ethnic group measurement have been extensively 3 debated, and specialist surveys have set out to provide greater sensitivity and theoretical underpinning to the deployment of questions and the constructs measured [START_REF] Modood | Race", racism and ethnicity': a response to Ken Smith[END_REF], there have been few advances in how to address the measurement of ethnic identity in generalist surveys, nor the methodological issues that are necessarily implicated in such an attempt. This is the contribution of this paper. The UK is rich in such general survey data and most surveys carry some measure of ethnic group, typically the recommended census categories, which are used for a substantial amount of research. This is a reality it is important to engage with [START_REF] Aspinall | The challenges of measuring the ethno-cultural diversity of Britain in the new millennium[END_REF]; but there is a need for enhancing or complementing such measures to increase the potential of quantitative ethnicity research.

We identify three main issues in current survey and census questions.

First, the central issue is the problem of measuring a dynamic and multidimensional concept, ethnicity, using a single categorical question. Second, and related, is that researchers themselves bring different understandings of ethnicity and ethnic identity to their investigations and thus have different expectations of what any given measure can -or should -deliver. A researcher interested in individual acquisition of (subjective) identity will want to ask about ethnic identity in a different way to, for example, a population-level researcher looking at discrimination across 'objective' groups (those defined, for example, by first language, country of birth or migrations status) in the labour market.

The third key issue is the effectiveness of a given survey measure or set of measures. There are some practical measurement issues that need addressing if research interest in ethnicity. We conclude that asking series of questions across multiple domains, rather than offering simply a single category, and paying attention to question wording and context, it may be possible to go some way towards addressing both the multi-dimensionality of ethnicity and providing flexibility in meeting the varied demands of researchers within a multi-purpose survey.

The problem with categories

Issues of identity, ethnic diversity, and differences between the UK's ethnic groups are substantial research and policy interests. The UK also offers a wealth of multi-purpose surveys that can be used to analyse the experience of different ethnic groups and to highlight inequalities between them: surveys with minority boost samples, both cross sectional and longitudinal, and repeat cross-sectional surveys that can be pooled to increase sample sizes (see: http://www.dataarchive.ac.uk/). However, in general, they use a standard question developed for and utilised in the decennial population Census to define ethnic group. While the categories in this question broadly differentiate between histories of particular post-war immigrant experiences [START_REF] Coleman | a 'The ethnic group question in the 1991 Census: a new landmark in British social statistics[END_REF]Salt, 1996b, Peach, 1996), they have been subject to extensive criticism relating to both their conceptual basis and their operational use (Ballard, 1997[START_REF] Ballard | Negotiating race and ethnicity: exploring the implications of the 1991 Census[END_REF][START_REF] Ratcliffe | Introduction: social geography and ethnicity: a theoretical, conceptual and substantive overview[END_REF], Aspinall, 2001a).

Critics highlighted in particular the fact that different categories used different criteria for defining or conceptualising ethnicity, leading to conceptual Ballard (1997) argued that the question reinforced a racialised perception of the world that had no analytical acceptablity; and Ratcliffe was led to state that 'what is clear about the census ethnic group question is that the one thing it does not measure is ethnicity' (Ratcliffe, 1996 p.5).

The multi-dimensional nature of ethnicity

At the heart of these criticisms is the conundrum of attempting to divide the population into a set of mutually exclusive categories using a single question.

That is, treating a multi-dimensional, fluid, and contextually and relationally specific concept as if it were uni-dimensional, fixed and stable (Aspinall, 2001b[START_REF] Yancey | Emergent ethnicity: a review and reformulation[END_REF][START_REF] Mcdonnell | You're Brazilian, right? What kind of Brazilian are you? The racialization of Brazilian immigrant women[END_REF]. Instead we need to recognise that ethnicity and ethnic group identification are contingently located in one or more characteristic or expression of shared belonging. These 'dimensions' may include 'race' (or colour or visibility); national identity [START_REF] Jackson | Conceptualizing social identity: A new framework and evidence for the impact of different dimensions[END_REF]; parentage or ancestry [START_REF] Berthoud | Defining ethnic groups: origin or identity?[END_REF]; nationality; citizenship (Kymlicka, 1996, Hussain and[START_REF] Hussain | Citizenship, ethnicity and identity: British Pakistanis after the 2001 'riots[END_REF]; religion; language (Phinney et al., 2001, Vedder and[START_REF] Vedder | Language, ethnic identity, and the adaptation of Turkish immigrant youth in the Netherlands and Sweden[END_REF], and country of birth (or being an immigrant), as well as the problematic domain of 'culture' (Geertz, 1993, Ahmad and[START_REF] Ahmad | Locating ethnicity and health: exploring concepts and contexts[END_REF], which we consider here as behavioural and attitudinal and discuss separately from issues of identification [START_REF] Modood | Culture and identity[END_REF]. The extent to which any such 'dimensions' or combinations of dimensions are relevant to the construction of and identification with any group will vary according to group, to context and over time [START_REF] Nazroo | Patterns of identity among ethnic minority people: diversity and commonality[END_REF][START_REF] Modood | Changing Ethnic Identities[END_REF][START_REF] Warikoo | Gender and ethnic identity among second-generation Indo-Caribbeans[END_REF]. Researchers' and their conceptual constructions of ethnicity also differ in ways that imply particular demands on measures. Finding a simple way of meaningfully summarising ethnic group therefore invites problems as there are no 'essential' components and there is no continuous concept of single groups. It is virtually impossible to create single, mutually exclusive categories for ethnicity measures which are both conceptually coherent and which invite recognition and identification from respondents.

Subjective identification and preconceived groups

As [START_REF] Weber | Economy and Society[END_REF] pointed out, what is central to the definition of ethnic groups is 'a subjective belief in their common descent', even though 'it does not matter whether or not an objective blood relationship exists' (Weber 1978: 389).

Despite some debate on 'objective' measures versus self selection [START_REF] Smith | Some critical observations on the use of the concept of "ethnicity[END_REF][START_REF] Modood | Race", racism and ethnicity': a response to Ken Smith[END_REF], this emphasis on self-conscious identification is widely accepted including by census bureaux and statistical offices (Martin andGerber, 2006, Office for National Statistics, 2003), but may serve to introduce confusion where the actual object of a question has little to do with expressive identity.

For the UK's first ethnic group question in a census, in 1991, the question was framed as a measure of subjective ethnic group, and about identification.

However, both commentary on the UK 1991 question and the approach used to collect the ethnic group information emphasised that what was of interest was in fact the 'non-white' population [START_REF] Coleman | a 'The ethnic group question in the 1991 Census: a new landmark in British social statistics[END_REF]Salt, 1996a). Thus the logic seemed close to understandings of race questions in US questions despite the explicit rejection of the language of race in favour of a concentration of the Not only governments but also many analysts are interested in stable measures which help them understand population trends and experiences across population subgroups including ethnic (or in US terms, racial) groups [START_REF] Kertzer | Censuses, identity formation, and the struggle for political power[END_REF]. There are important policy drivers for such stable measures, such as having sufficient information to meet the requirement to monitor policy and service delivery for different ethnic groups (Bulmer 1986). A reporting requirement is also behind the race and ethnicity questions in the US census (Anderson and Fienberg, 1999). In both countries, the construction of groups is based on politically salient categories, though of course what is politically salient is highly locally specific, and renders international consideration of groups on the basis of ostensibly self-ascribed categories essentially meaningless.

To equate such monitoring systems with individuals' identification is highly problematic [START_REF] Bonnett | White Identities: Historical and International Perspectives[END_REF]. Official categorisations represent systems of power and authority which prejudge who are minorities or 'others' and the implicit or explicit bases on which they are to be distinguished -even if such presumptions are later contested [START_REF] Howard | Constructing the Irish of Britain: ethnic recognition and the 2001 UK Censuses[END_REF]. For example, the ways in which 'write-in' answers in the UK censuses are reallocated to the main categories on offer clearly illustrates the prioritisation of official views on where people 'fit' (Office for National Statistics, 2003). Legal and political considerations and mobilisation frame the circumstances under which categories adjust over time. The need of ethnic minority groups to be heard in a democracy [START_REF] Urla | Cultural politics in an age of statistics: numbers, nations, and the making of Basque identity[END_REF]. Certain groups may lobby for inclusion and the recognition that offers [START_REF] Howard | Constructing the Irish of Britain: ethnic recognition and the 2001 UK Censuses[END_REF]. Conversely, there may be political arguments against the introduction of new categories: the multiple response race question in the U.S. census was met with opposition from certain black interest groups who feared that this would lead to a reduction in the proportion of respondents reporting as black [START_REF] Skerry | Counting on the Census? Race, Group Identity, and the Evasion of Politics[END_REF][START_REF] Farley | Racial trends and differences in the United States[END_REF], as 6.8 million people took up the opportunity to chose two or more races (LEA AND BEAN).

Even if respondents provide answers to the categories offered, they may not be providing even approximate information on their self-identification, since responses can be 'learnt', without necessarily increasing the relevance of categories to analysts [START_REF] Kertzer | Censuses, identity formation, and the struggle for political power[END_REF]. [START_REF] Martin | Methodological Influences on Comparability of Race Measurements: Several Cautionary Examples[END_REF] point out that pre-designated categories imply that these are the 'correct' categories and may constrain people to report within categories that they do not identify with. When asking an open-text unprompted ethnicity question and the existing Census ethnicity question, [START_REF] Pringle | Practicality of recording patient ethnicity in general practice: descriptive intervention study and attitude survey[END_REF] found that there was an exact or near exact match in only 28 per cent of the valid responses; while [START_REF] Lopez | Collecting and tabulating race/ethnicity data with diverse and mixed heritage populations: A case-study with US high school students[END_REF] showed that two-fifths of respondents would give different or additional responses if faced with a multiple response question rather than a single-response set of categories.

It is clear then that even if current census categories have a role, they cannot be expected to provide an adequate understanding of the diversity and development of subjective ethnic identities.

Improving on categories

For some, the perceived problems of using one single-response categorical question have resulted in a rejection of survey research of ethnicity, or to its construction as malign or dangerous. See, for example, the discussions in [START_REF] Blum | Resistance to identity categorization in France[END_REF], [START_REF] Bhavnani | Tackling the Roots of Racism: Lessons for Success[END_REF] [START_REF] Bonnett | Fitting into the categories or falling between them? Rethinking ethnic classification[END_REF]. Proxies for ethnic group membership based on family origins have also been subject to critiques [START_REF] Smith | Some critical observations on the use of the concept of "ethnicity[END_REF][START_REF] Cole | Ethnicity, 'status groups' and racialization: a contribution to a debate on national identity in Britain[END_REF]. While we recognise the criticisms, we argue that the alternative is to focus on developing better measures.

What is important is that respondents are offered the opportunity to define themselves -and express strength of identification -in relation to those aspects that are meaningful and matter to themselves. Allowing multiple responses across differing bases of potential affiliation, has been shown to be popular with respondents and may help to achieve this. Such measures are also relatively transparent and do not imply that they are being asked to fit into a single 'box'. They also allow researchers to ascertain what it is that divides and joins 'groups' along these dimensions and facilitate the construction of conceptually coherent 'groups' for research purposes. Identifying the content of each of the dimensions (the respondent's country of birth, her religion her parents' country of birth, and so on) will not only enable an empirical understanding of groupness, based on strength of identification; also better meet the needs for 'objective' measures that are relatively stable than questions that ask for self-identification. For some analysts particular dimensions will be of interest in their own right, and to be able to focus only on those dimensions -and Such an approach would, then, allow flexibility for researchers both in their constructions of ethnicity and in the research questions they are able to address. There will, of course remain questions that are not susceptible to empirical survey analysis, and social surveys can never address the full range of questions that are of concern to ethnicity researchers. But that is not to say that they cannot do a better job and cover far more than they do currently. Increasing the flexibility of responses so that researchers can focus on the dimensions that are of interest may also serve to enable meaningful cross-national comparison, given that ethnicity is differently defined and categorised according to national context. For example, the UK and the US operate with very different conceptions of ethnicity, which, in the US, is distinguished from race. This makes harmonisation of categorical questions well-nigh impossible. The proposed approach based on collecting information about multiple general dimensions could be deployed in contexts where concepts (and language) of race and ethnicity and very different.

Constraints on identification

However, it is not simply in potentially multiple dimensions of identity and identification that ethnicity resides. Returning to the issue of subjective identification, as well as defining ethnicity by "a sense of belonging", others have defined groups in relation to how they construct difference from others and (Wimmer, 2008, Min and[START_REF] Min | Ethnic and sub-ethnic attachments among Chinese, Korean, and Indian immigrants in New York City[END_REF]. This is a perspective that is associated with Barth's work [START_REF] Barth | Ethnic Groups and Boundaries: The Social Organisation of Culture Difference[END_REF]; but has also been stressed by the acculturation framework, which focuses on the juxtaposition of two cultures in rendering the concept of ethnic identity meaningful. Individual ethnic identification is limited by external forces that shape the options, feasibility and attractiveness of various ethnicities [START_REF] Nagel | Constructing ethnicity: creating and recreating ethnic identity and culture[END_REF][START_REF] Salway | Labour market experiences of young UK Bangladeshi men: Identity, inclusion and exclusion in inner-city London[END_REF][START_REF] Ryan | Who do you think you are? Irish nurses encountering ethnicity and constructing identity in Britain[END_REF]. These external factors include attitudes and responses of others [START_REF] Golash-Boza | Latino racial choices: the effects of skin colour and discrimination on Latinos' and Latinas' racial selfidentifications[END_REF], but also the use of official categories in censuses and monitoring forms themselves [START_REF] Jenkins | Rethinking ethnicity: identity, categorization and power[END_REF]. It is therefore important that the external constraints and impulses to particular forms of identification are also captured, both through recognising -and repeatingexisting forms of labelling, and through attempting to measure own and other group attitudes as well. Thus questions on apparently 'innocent' dimensions of identity also need to be complemented by measures of prejudice and of ownership of ethnic identity.

Lived ethnicity

Finally, in addition to relational aspects of ethnicity, there are also behavioural aspects: how people use their time, who they associate with and so on. To assume that identification with a particular category implies particular sets or patterns of behaviour of association is potentially to overload those categories yet further. Instead, behaviour or 'involvement' could be conceived of as an additional 'dimension' that may be present in some research formulations of what an 'ethnic group' means -or may be investigated in relation to particular constructions of ethnic group (Nazroo andKarlsen, 2003, Min and[START_REF] Min | Ethnic and sub-ethnic attachments among Chinese, Korean, and Indian immigrants in New York City[END_REF]. When considering the survey measurement of ethnicity, then, it is important to consider not only multi-dimensional measures of identity and identification, but also to collect complementary information on intergroup relations and on patterns of behaviour. [START_REF] Phinney | Ethnic Identity in Adolescents and Adults: Review of Research[END_REF] identifies five components of ethnicity, which are often merged or overlap in usage of the terms 'ethnicity' or 'ethnic group': ethnicity by which she means a person's heritage, parents' ethnicity, country of origin; self-identification as a member of an ethnic group; ethnic belonging, which refers to a sense of belonging to the selfidentified ethnic group/s; ethnic involvement or participation and practice of being a member of an ethnic group; and ethnic attitude, which refers to one's opinion towards one's self-identified ethnic group. Though we might dispute the precise components or the terminology, the attempt to break down ethnicity into elements which can be separately measured and which can be jointly or separately analysed appears to be a positive way forward for the operationalisation of ethnicity within social surveys [START_REF] Mckenzie | Describing race, ethnicity and culture in medical research[END_REF][START_REF] Senior | Ethnicity as a variable in epidemiological research[END_REF][START_REF] Singh | Ethnicity in psychiatric epidemiology: need for precision[END_REF], even to those somewhat sceptical about the feasibility of unpacking ethnicity [START_REF] Mckenzie | Ethnicity in psychiatric epidemiology[END_REF].

The fluidity of ethnicity

To capture issues of context, contingency and change, it is also important (2009) on the benefits of longitudinal surveys for measurement.) Much of the psychological literature on ethnic identity focuses on young people who are assumed to be at the most flexible stage of their development [START_REF] Phinney | Ethnic Identity in Adolescents and Adults: Review of Research[END_REF][START_REF] Phoenix | The identity of mixed parentage adolescents[END_REF], Hitlin et al., 2006[START_REF] Hsiao | Acculturation among three racial/ethnic groups of host and immigrant adolescents[END_REF]; but it would be valuable to explore the temporal processes operating among adults, such as migration or the arrival and departure of family members, about which we know much less.

Asking ethnic group questions: issues and practice

What, then, are the practical implications in collecting measures of identification across multiple dimensions and of experience and perception across multiple domains? Many of the insights into the multidimensionality of individuals' experience and expression of ethnicity have come from qualitative research. To translate these insights into effective survey measures requires recognition of the constraints of survey questions, the need to ensure validity of responses relative to the concept being measured, as well as ensuring acceptability and maximising response. To this end we can take stock of the lessons that have been learnt in Moreover it is important to recognise that not even well-designed questions can facilitate research in all areas of interest to understanding ethnicity.

Issues relating to personal narratives, to the role of wider family kin and to peer groups in forming personal identities, as well as the investigation of very specific groups, such as transnationally adopted children, will continue to remain best addressed by qualitative research. But even with such research, a more general understanding of identification may provide important context and starting points.

Measurement issues

We noted above that census bureaux stress the subjectivity of responses even while attempting to measure an assumed objective reality. Part of the emphasis on subjectivity is a genuine recognition of the conceptual discussions relating to ethnicity and that it is not fixed. Part is a recognition that it would be impossible to consider any other way of ascertaining people's ethnic group other than by asking them: co-operation from respondents is a prerequisite for collection and thus acceptability becomes a crucial consideration for the specification of categories [START_REF] Sillitoe | Ethnic group and the British Census: the search for a question[END_REF]White, 1992, Maylor, 2009). This raises the general point that if categories or domains used in questions do not make sense to the respondents then there will be a lot of item non-response or 'other' reporting [START_REF] Bonnett | Fitting into the categories or falling between them? Rethinking ethnic classification[END_REF]. [START_REF] Bates | Questionnaire effects on measurement of race and Spanish origin[END_REF] also reveal the substantial frustration that can arise when ethnic categories are felt to deny preferred identifications. And when interviewers themselves find questions ambiguous they try to resolve these issues differently, resulting in interviewer effects on the responses [START_REF] Martin | Methodological Influences on Comparability of Race Measurements: Several Cautionary Examples[END_REF]. Questions therefore need to reflect concepts that are straightforward and meaningful to respondents, whether or not they identify with them. Dimensions such as country of birth and religious affiliation are relatively clear, others concerning main/first language, colour or nationality may require some care in matching the intended concept to the word or phrase used in a question.

Another potential source of measurement error that we discussed is 'learning' the expected response. When people 'learn' a response through exposure to the Census question (or alternatively learn to resist it), this potentially increases the stability, but not necessarily the meaningfulness of the data. The acceptability of, and even preference for, multiple response categories has been regularly demonstrated [START_REF] Lopez | Collecting and tabulating race/ethnicity data with diverse and mixed heritage populations: A case-study with US high school students[END_REF][START_REF] Callister | Measuring ethnicity in New Zealand: developing tools for health[END_REF]; and illustrates that such questions offer a preferable means to improve reliability than through attempting to shoe-horn respondents into preconceived matches.

Learning also requires time and familiarity. So, while it may deliver the 'right' responses in some cases, it is less likely to do so for new or recent immigrants -groups potentially of great interest for governments as well as researchers. Those who have recently migrated initially retain the concept of ethnicity of their country of origin. For example, [START_REF] Martin | Methodological Influences on Comparability of Race Measurements: Several Cautionary Examples[END_REF] find that persons from Central and South America and the Caribbean do not think race and ethnicity are separate concepts and would like to choose the country 16 they have come from as their race/ethnicity (see also [START_REF] Massey | Latinos, poverty and the underclass: a new agenda for research[END_REF] McDonell and de Lourenco ( 2009)). Over time, classification systems can shape not only responses but also self-identification [START_REF] Itzigsohn | Immigrant incorporation and racial identity: racial self-identification among Dominican immigrants[END_REF], given the relevance of how others see you (or your perception of it) to your identity. But to allow investigation of new populations and the process of identity formation and development, it is important that measures and the categories they employ are comprehensible to new migrants as well as established populations. Avoiding questions which directly invoke concepts of race or ethnicity, which are known to vary widely across contexts is one potentially important strategy [START_REF] Bhopal | Revisiting race/ethnicity as a variable in health research[END_REF]. Such a strategy is eased in context of multiple domains rather than mutually exclusive categories.

But achieving reliable responses and the acceptability of questions is not simply a matter of the categories asked. Positioning of questions is explicitly used to increase the numbers completing the 'right' box [START_REF] Bates | Questionnaire effects on measurement of race and Spanish origin[END_REF][START_REF] Martin | Context effects for census measures of race and hispanic origin[END_REF]. In the UK 2011 census, the proposed placing of a national identity question prior to the ethnic group question is designed to allow all those who wish to assert their Britishness to have an initial opportunity to do so, with the intended result that fewer minorities will 'incorrectly' subsequently select the British sub-option of 'white' or use the write-in sections to define themselves as British. Sometimes composite categories are used for the same purpose. In the 2001 Census "Black British" and "Asian British" options were included as a way of allowing people to state their Britishness in addition to their ethnicity. This returns us to issues of acceptability and maximising response, by offering options that explicitly match on to people's preferred ways of seeing themselves. Without overloading respondents it is clearly important to provide sufficient domains to meet their requirements for expressing their sense of who they are.

Rather than the continual increase in single categories that has categorised the UK Census questions, allowing responses over multiple shorter measures, should reduce list effects and the effort required from respondents.

There are of course other general issues of question design that we need to be aware of. Responses may differ depending on the method of the interview (face-to-face, telephone or self-completion); and mode effects may be specifically related to dimensions of ethnicity. For example, those who do not understand English are less likely to complete a mail questionnaire and are thus interviewed face to face, which affects response; but those with limited grasp of English also tend to report ethnicity differently from those who do have a strong grasp of English [START_REF] Martin | Methodological Influences on Comparability of Race Measurements: Several Cautionary Examples[END_REF]. Understanding the impact of such variation may be important in designing new questions for multiple or single modes.

As well as 'recency' or 'primacy' effects -a danger with categorical ethnic group questions that may have up to seventeen categories -there may also be different responses according to whether there are other people present during the interview who could influence their responses. Studies have found differences in responses depending on the interview location such as at home vs. at school [START_REF] Harris | Who is Multi-racial? Assessing the Complexity of Lived Race[END_REF]. Persons of mixed parentage may choose different ethnic group identities depending on which parent is present at the time of the interview [START_REF] Martin | Methodological Influences on Comparability of Race Measurements: Several Cautionary Examples[END_REF][START_REF] Xie | The racial identification of biracial children with one Asian parent: evidence from the 1990 Census[END_REF], Phinney, 1992 [START_REF] Holloway | Place, scale and the racial claims made for multiracial children in the 1990 US Census[END_REF]. Others may wish to claim more than one of the offered categories and may resist being forced to choose [START_REF] Aspinall | The challenges of measuring the ethno-cultural diversity of Britain in the new millennium[END_REF] or allocate themselves arbitrarily between the options proffered. It is important both to know the context and to be able to standardise it across respondents as far as possible (for example, by giving respondents a questionnaire to complete privately), and to ensure that multiple affiliations are possible.

Nevertheless, techniques to improve the question design may not resolve consistency issues (i.e., stable responses over time) that arise because these questions are asking people to pick their ethnic identity which is inherently subjective and therefore subject to change with the context of the question and the questioning. For example, a person's choice of ethnic group label may be correlated with how she views her ethnic group vis-à-vis the mainstream dominant group [START_REF] Hecht | Sociological roots of ethnic identity: a look at black America[END_REF]. Changes in identification may occur through the development of ethnic identity (along with other aspects of identity) over a person's life [START_REF] Phinney | Ethnic Identity in College Students from Four Ethnic Groups[END_REF].

If ethnic groups are considered as composites of characteristics and affiliations, it may be possible to create consistency in questions rather than categories over time, while the combination and salience of the responses differ over individual life courses and as a result of demographic change. In addition, while what is typically at stake for census categorical question measurement is consistency, 'inconsistencies' may be central to a concept of ethnicity as fluid and contingent, and therefore be of interest in their own right. Thus some attempts to reduce inconsistency may be unnecessary or even unhelpful. These sources of 'inconsistency' may be of interest -as long as it is possible to measure the relevant correlates (out-group attitudes, location, age, sex, income, interview context and so on). For example, Hecht and Ribeau. [START_REF] Hecht | Sociological roots of ethnic identity: a look at black America[END_REF] find differences in the choice of ethnic label by gender, age and family income. [START_REF] Travassos | The concept and measurement of race and their relationship to public health: a review focused on Brazil and the United States[END_REF]) also find differences in ethnic group identification by socio-demographic characteristics. In Brazil wealthier people classify themselves and their children -and are classified by others -in lighter-skinned categories [START_REF] Schwartzman | Does money whiten? Intergenerational changes in racial classification in Brazil[END_REF] see also [START_REF] Golash-Boza | Latino racial choices: the effects of skin colour and discrimination on Latinos' and Latinas' racial selfidentifications[END_REF]. [START_REF] Telles | Does it matter who answers the race question?: Racial classification and income inequality in Brazil[END_REF] report similar findings for interviewer reported race/ethnicity with respect to education and income. Under our proposed approach these become not problems for categories but part of the understanding of groups and group identification in their own right.

The implication is that to understand ethnic identity and to construct meaningful groups or categories, requires additionally a range of contextual information: information that is typically gleaned in social surveys. By these means, it may be possible to complement focused psychological studies of ethnic identity with broader social contextual information as well as to bridge the gap between current qualitative approaches to ethnicity research and the statistical analysis of ethnic difference. But to achieve this, care needs to be given to how questions are asked and how they relate to other question content.

A way forward

Historically the interest in measuring ethnicity has been from census bureaux and other official sources for the purpose of assigning people into ethnic groups to groups. With this objective, the focus has been identifying single measures of ethnicity which yield consistent and reliable results, i.e., stable ethnic groups.

Our main conclusion is that it is not possible to measure a complex, multidimensional, fluid concept using a single question at one point in time.

Nevertheless, we do think it is possible for survey research to provide a much more sensitive measure of ethnicity, where variation over time and between parents and children in elements of a multi-dimensional measure are seen as assets rather than as problems for consistency.

For most surveys, a categorical question in the form of the Census question would be important not only for range of research purposes concerned with analysis of patterns of association between ethnic groups and other characteristics, as well as detailed understanding of population characteristics and their relationship over time, but also be important for purposes of comparison and calibration with other sources, both census and other surveys. It also enables the relationship between categories and other measures of groupness to be investigated. However, it is important to address the fact that such a categorical question does not represent ethnic identification or belonging and would not meet the research needs of social psychologists or of those interested in understanding in more detail the meaning and coherence of groups. For such purposes it is surely important to develop questions which tackle who people think they are and how important they consider that to be. multiple dimensions would also allow researchers interested only in specific dimensions (religion, national belonging etc.) to use responses directly rather than using ethnic group categories as a proxy for other 'group' concepts. In addition, these dimensions of identity may need to be considered in relation to other aspects of the respondent's experience. For example, we might envisage a series of questions asking about country of birth, nationality, religion, perceived skin colour etc., which would be followed by a question asking which of country of birth, nationality, and all the other dimensions were part of their identity, and for each dimension acknowledged there could be a question attempting to gauge the strength or importance of that dimension, possibly on a scale or rated by terms such as 'very', 'not very' etc. Individuals may quite appropriately fit into a number of different populations according to the focus of the research and all of these might reflect valid aspects of their self-identity. For example, a respondent might see herself simultaneously as Scottish (upbringing, language/accent, politics, local affiliation), black (politically used to express solidarity with minority, non-white experience), Asian (to suggest antecedents (parents /grandparents or earlier generations from South Asia) and British (to express nationality, citizenship etc.). All these might be important to her, but some aspects might be more to be her 'Scottishness'. Additionally, the relative importance could easily vary with context: whether she is in Glasgow or London, whether she is experiencing harassment, whether she is voting, whether she is visiting her parents or being interviewed by a non-Asian interviewer etc. (Hall, 2000, Aspinall, 2002[START_REF] Hall | Introduction: who needs identity?[END_REF]. And it could vary over time, in relation to recent national or international events such as wards or personal events such as marriage or bereavement.

Clearly not all the subtleties of an individual's identifications could be captured by particular survey questions, but surveys could certainly move towards being able to address more of them and their dynamic relationship with experiences and social context.

It is important not only to measure relevant dimensions (relevant to respondents and researchers), but also to consider the practical impact of acceptability, phrasing, position, mode, and the wider context in which the questions are being asked. The nature of surveys is that there is no opportunity to interrogate responses, and so it remains important to avoid arbitrary responses or to lead respondents to particular answers to ensure that it is possible to measure meaningful concepts robustly. Lessons from methodological research into achieving 'correct' responses from categorical census questions, therefore remain pertinent to more flexible survey measures underpinned by different aims and research agendas.

In thinking about the implementation and long-term value of a new measure of ethnic identity, we would also highlight the relevance of surveys Surveys which can track individuals over time (and, again, the UK is well resourced in such surveys) are likely to be most fruitful for the exploration of contingency, context and change in ethnic identity. Finally, it is important to stress that to identify and investigate the correlates and attributes of ethnic groups does not rely solely on being able to capture self-perception of ethnicity, ethnic identity or dimensions of ethnic identity. Ethnicity is also expressed through patterns of behaviour, friendship and association that are best measured separately. For example, rather than asking if people tend to associate with those of the same ethnicity, we can ask about respondents leisure and friendship activities, their nature and their composition, and relations and contact with countries of origin. Moreover, an inescapable dimension of understanding groups, how they evolve and how they are formed, is the perceptions and attitudes of others. Any survey that aims to define ethnic groups will therefore need information not only on respondents' ethnicity, but also on their patterns of association and on the views and values of the wider population.

To get a valid measure of ethnicity which covers the issues discussed in this paper is a challenge. But the prize is worth the effort.

  improve the extent to which they meet existing and evolving

  in the UK, and the very different solutions to the construction of ethnic group questions.

  formation of new ethnic identities or redefinition of existing ones

  for whom they are salient -will promote conceptual clarity and the 'fit' of measures to particular research agendas.

  to consider the survey context, or the type of survey in which ethnic identity might most effectively be measured. Having responses from different family members, specifically parents (partners) and children and their siblings allows a detailed exploration of the how those sharing a common family background subjective identifications. Qualitative studies are beginning to demonstrate how within family, specifically between siblings, identification can vary; and surveys that, as many do, collect information on multiple, cohabiting family members can offer a valuable complement to such research. Longitudinal surveys can enable research into ethnic identity development and distribution of identities both over time and for different cohorts. (See Umana_Taylor et al.

  standard categorical questions and be alert to the potential pitfalls.

  change and life opportunities for certain ethnic

  to recognize both instabilities in affiliation and multiple affiliations. Such questions would therefore need to incorporate flexibility, through multiple response options, and allow respondents to give some weight to the different elements specified. Incorporating and weighting

  example, she may consider the most critical element of her identity

  on multiple family members, both parents and children and siblings and ideally, though this is more unusual, of non-co-resident family.

  If we are able to get the measure(s) right, then this will open up the rich resources of survey research to a wide range of analysts from different disciplines and allow theoretically robust research.

  ,

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	18


Vladimir A Mukhanov

Oleksandr O Kurakevych

Vladimir L Solozhenko

Hardness of materials at high temperature and high pressure

Keywords: hardness, inorganic materials user supplied): superhard materials, high temperature, high pressure

Intrinsic character of correlation between hardness and thermodynamic properties of solids has been established. The proposed thermodynamic model of hardness allows one to easily estimate hardness and bulk moduli of known or even hypothetical solids from the data on Gibbs energy of atomization of the elements or on the enthalpy at the melting point. The correctness of this approach has been illustrated by an example of the recently synthesized superhard diamond-like BC 5 and orthorhombic modification of boron, γ-B 28 . The pressure and/or temperature dependences of hardness have been calculated for a number of hard and superhard phases, i.e. diamond, cBN, B 6 O, B 4 C, SiC, Al 2 O 3 , β-B 2 O 3 and β-rh boron. The excellent agreement between experimental and calculated values has been observed for temperature dependences of Vickers and Knoop hardness. Besides, the model predicts that some materials can become harder than diamond already at pressures in the megabar range.

Keywords: superhard materials, theory of hardness, high pressure, high temperature.

Introduction.

Hardness describes the abrasive properties of materials and is understood as the ability of a material to resist an elastic and plastic deformation or brittle failure [1][START_REF] Poliakov | Diamonds and superhard materials[END_REF][START_REF] Povarennykh | Hardness of minerals[END_REF]. Interest in the study of hardness in a wide temperature range covers many fields [START_REF] Novikov | [END_REF]; i.e. modeling the mechanical behavior of materials in technological processes, assessing the performance of hard tools under extreme conditions, studying the elementary steps of the plastic deformation and fracture of hard materials, etc.

The theories of hardness and design of novel superhard materials are great challenge to materials scientists till now. Many attempts to predict hardness have been made using the structural data and such characteristics as bulk (B) and shear (G) moduli, specific bond energy, band gap (E g ), density of valent electrons (i.e. the number of valent electrons per unit volume N e ), etc. [1][START_REF] Poliakov | Diamonds and superhard materials[END_REF][START_REF] Povarennykh | Hardness of minerals[END_REF][5][6][START_REF]Synthesis of superhard materials[END_REF][START_REF] Teter | [END_REF][9][10]. Up to date the best correspondence between the calculated and experimental values of hardness has been achieved in the recent papers [5,6]. In both cases, the final accuracy is about 10% for hard phases, i.e. at the level of experimental errors. However, the temperature and pressure dependencies of hardness can be hardly derived from any known model.

The universal model of hardness should also take into account the microstructure of materials (grain size, inter-grain boundaries, etc.) [START_REF] Povarennykh | Hardness of minerals[END_REF]11,12]. However, these factors are usually ignored in theoretical simulations, so that the calculated values correspond to socalled "chemical" hardness that is usually observed only for single crystals and well-sintered polycrystalline bulks. Here we will deal with the "chemical" hardness only.

The purpose of present work was to establish an intrinsic relationship between hardness and thermodynamic parameters of solids; that would allows one to calculate the hardness of materials under extreme pressure-temperature conditions. According to our concept, the hardness of a phase is proportional to the atomization energy, which may be considered as a characteristic of the bond rigidity (for clarity, we will use the standard values of Gibbs energy of atomization ∆G°a t ), and is in inverse proportion to the molar volume of a phase [START_REF]for carbon phases the linear dependence between the hardness and density has been established in Ref[END_REF] and to the maximal coordination number of the atoms. The value defined in such way has the dimensions of pressure. The plasticity of materials is taken into account by the empirical coefficient α. In general case the polarity of bonds leads to the hardness decrease, which may be clearly seen in the sequence of isoelectronic analogues of diamond, i.e. diamond (115 GPa) [14,15] -cubic boron nitride cBN (62 GPa) [16] -BeO (13 ГПа) [START_REF] Povarennykh | Hardness of minerals[END_REF]5,[START_REF]Synthesis of superhard materials[END_REF] -LiF (1.5 ГПа) [START_REF] Povarennykh | Hardness of minerals[END_REF]5,[START_REF]Synthesis of superhard materials[END_REF]. This factor has been evaluated by empirical coefficient β, which is the measure of the bond covalency.

Hardness as a function of Gibbs energy of atomization.

The formula that allows calculating the Vickers hardness (H V ) of crystals at 298 K is

αβε VN G H at V °∆ = 2 , (1)
where V -molar (atomic) volume (cm 3 mole -1 ); N -maximal coordination number; αcoefficient of relative (as compared to diamond) plasticity; β -coefficient corresponding to the bond polarity (see below); ε -ratio between the mean number of valent electrons per atom and the number of bonds with neighboring atoms (N) [START_REF]The use of this coefficient allows to establish the hardness of the A I B VII (ε = 1/N) and A II B VI (ε = 2/N) compounds, i.e. LiF, NaCl, BeO, ZnS, MgO, etc[END_REF]; ∆G°a t -standard Gibbs energy of

atomization (kJ mole -1 ) of compound X m Y n . n m n m Y X f Y at X at Y X at G G n G m G °∆ - °∆ + °∆ = °∆ ; (2)
where

n m Y X f G°∆ -standard Gibbs energy of formation of X m Y n , X at
G°∆ and Y at G°∆ standard Gibbs energy of atomization of elements X и Y.

Coefficient α has been estimated from the experimental values of H V for diamond, d-Si, d-Ge and d-Sn. For the elementary substances and compounds of second period elements α equals 1, while for other periods (≥3) α makes 0.7. This coefficient reflects the difference in the bond strength [6] for the elements of different periods.

Coefficient β (square of the covalency f) has been calculated by the equation

2 2         + = X Y Y χ χ χ β ; ( 3
)
where χ X , χ Y -electronegativities of the elements by Pauling, χ X > χ Y [START_REF] Emsley | The Elements[END_REF]. For elementary substances β = 1.

For the refractory crystalline compounds the values of hardness calculated by equation ( 1) are in a very good agreement (less than 4 GPa of discrepancy, i.e. < 7%) with the experimental values [START_REF] Novikov | [END_REF][5][6][START_REF]Synthesis of superhard materials[END_REF][14][15][16][START_REF] Emsley | The Elements[END_REF][START_REF] Domnich | [END_REF][START_REF] Kurdyumov | man, Polymorph modifications of carbon and boron nitride[END_REF][START_REF] Rizzo | [END_REF][START_REF] Rabinovich | Small handbook of chemistry[END_REF][START_REF] Bulakh | Thermodynamic methods in mineralogy[END_REF][START_REF]Thermodynamic properties of inorganic materials[END_REF][START_REF] Gurvich | Thermodynamic properties of individual substances[END_REF][START_REF]Chemical Encyclopedia, Bol'shaia Rossiyskaia Entsiklopediya[END_REF][START_REF] Mcmillan | [END_REF][28][29][30][31] (Fig. 1a [START_REF]The considered compounds/phases are diamond, Si, Ge, d-Sn, SiC, cBN[END_REF]). One more advantage of the proposed method is the possibility to estimate the hardness of various forms of boron and its compounds (B 4 C, B 6 O, B 13 N 2 ), that is rather complicated by using other methods because of extreme complexity of boron-related structures.

The experimental values of hardness for the α-B 12 (H V = 42 GPa) and β-B 106 (H V = 45 GPa) phases [33] are in good agreement with the values (39.2 and 43.8 GPa, respectively) calculated in the framework of the thermodynamic model of hardness. The hardness of recently synthesized superhard high-pressure boron phase, orthorhombic γ-B 28 [34], was found to be 50 GPa [33], which also well agrees with the calculated value of 48.8

GPa. Our model suggests that γ-B 28 has the highest hardness among the known crystalline modifications of boron because of its highest density (2.544 g/cm 3 ).

In our calculations for boron-rich compounds we have taken the mean value of electronegativities of all atoms connected to B 12 icosahedron as a χ value for anion. Thus, the calculated values of Vickers hardness for B 4 C and B 6 O are 44 and 38 GPa, respectively; that is in a very good agreement with the experimental data for single crystal B 4 C (H V = 45 GPa) [START_REF] Domnich | [END_REF] and polycrystalline B 6 O (H V = 38 GPa) [START_REF] Rizzo | [END_REF]. The lower value of hardness for B 6 O as compared to B 4 C may be explained by the higher ionicity of the B-O bonds than that of B-C bonds. The estimation of hardness for the recently synthesized rhombohedral boron subnitride B 13 N 2 [35,36] has given H V = 40.3 GPa [START_REF]The 2∆G°a t /NV value has been set to a mean (~51 GPa) of corresponding values for B 6 O and B 4 C[END_REF] that allows ascribing B 13 N 2 to superhard phases.

Using equation ( 1) it is possible to calculate the hardness of dense phases with threedimensional structures that have not been synthesized to present time, e.g. C 3 N 4 with Si 3 N 4 structure [START_REF] Teter | [END_REF], CO 2 with α-SiO 2 structure, hp-B 2 O 3 with Al 2 O 3 structure [START_REF] Mukhanov | [END_REF] and diamond-like phases of the B-C system [39,40] (see Table 1). The advantage of the proposed method is that only the maximal coordination number is used as a structural data. In all cases the molar volumes have been calculated from the covalent radii of the elements, while ∆G°f values (usually the negligible term as compared with ∆G°a t of the elements) of the phases have been fixed to the standard Gibbs energies of formation of known compounds in the corresponding binary systems, i.e. C 2 N 2 , CO 2 , B 4 C, β-B 2 O 3 [START_REF] Kurdyumov | man, Polymorph modifications of carbon and boron nitride[END_REF][START_REF] Rabinovich | Small handbook of chemistry[END_REF][START_REF] Bulakh | Thermodynamic methods in mineralogy[END_REF][START_REF]Thermodynamic properties of inorganic materials[END_REF][START_REF] Gurvich | Thermodynamic properties of individual substances[END_REF][START_REF]Chemical Encyclopedia, Bol'shaia Rossiyskaia Entsiklopediya[END_REF]41]. The applicability of this method for estimating the hardness of hypothetical compounds has been recently illustrated by the example of diamond-like BC 5 (c-BC 5 ), a novel superhard phase synthesized under high pressures and temperatures [42]. Vickers hardness of this phase has been calculated to be 70.6 GPa (Table 1), which is in excellent agreement with the experimental value H V = 71 GPa [42].

Hardness as a function of enthalpy at a melting point.

We have also established that instead of Gibbs energy of atomization, the heat content of a phase at a melting point may be used. The corresponding equation for calculation of Vickers hardness

H V is F o r P e e r R e v i e w O n l y V N dT T C H m T K p V δ ∫ = 298 ) ( 2 . 13 , (4)
where С р -molar heat capacity (kJ mole -1 K -1 ); N -maximal coordination number; V -molar volume (cm 3 mole -1 ); δ -empirical coefficient. The main advantage of equation ( 4) as compared to equation ( 1) is that in the vicinity of the melting point, the calculated hardness tends to zero; therefore, one can expect that the better correspondence between the experimental temperature dependence of hardness and equation ( 4). However, equation ( 4) can be hardly used for prediction of hardness for hypothetical phases.

The experimental hardness of covalent crystals [START_REF] Novikov | [END_REF][5][6][START_REF]Synthesis of superhard materials[END_REF][14][15][16][START_REF] Emsley | The Elements[END_REF][START_REF] Domnich | [END_REF][START_REF] Kurdyumov | man, Polymorph modifications of carbon and boron nitride[END_REF][START_REF] Rizzo | [END_REF][START_REF] Rabinovich | Small handbook of chemistry[END_REF][START_REF] Bulakh | Thermodynamic methods in mineralogy[END_REF][START_REF]Thermodynamic properties of inorganic materials[END_REF][START_REF] Gurvich | Thermodynamic properties of individual substances[END_REF][START_REF]Chemical Encyclopedia, Bol'shaia Rossiyskaia Entsiklopediya[END_REF][START_REF] Mcmillan | [END_REF][28][29][30][31] is in a good agreement with the calculated values (Fig. 2 [START_REF]The considered compounds/phases are diamond, Si, Ge, d-Sn, cBN, SiC, B 4 C, β-rh B, α-Al 2 O 3 , quartz, Ga 2 O 3 , Si 3 N 4 , GeO 2[END_REF]) (the heat capacity data have been taken from Refs. [START_REF]Thermodynamic properties of inorganic materials[END_REF][START_REF] Gurvich | Thermodynamic properties of individual substances[END_REF]. For covalent compounds δ ≈ 1, while for ionic compounds δ > 3. The metals show very good agreement between calculated and experimental values of hardness at δ ≈ 12.

Correlation between hardness and bulk modulus.

In the framework of our approach, the compressibility K of a phase at 298 K is proportional to the molar volume V and is in inverse proportion to Gibbs energy of atomization ∆G°a t [44], so

at G f V g K °∆ = 3 , (5)
where

Y X Y f χ χ χ β + = = 2
-covalency of chemical bonds. The empirical coefficient "3" in equation ( 5) has been evaluated using the experimental data on the compressibility of cBN, d-Si and d-Ge [START_REF] Baron | Small handbook of physico-chemical values[END_REF][START_REF] Babichev | Physical values. Handbook, Energoatomizdat[END_REF], while g is a correction coefficient usually fixed to 1 (see below). For the majority of the closely packed covalent compounds and metals there is a good agreement between the values of K exp и K theor , however, for the phases with anisotropic lattices, alkali and some alkali-earth metals the calculated values are lower than the experimental ones. For transition metals of periods V and VI, g = 0.625 in equation (5). Fig. 1b [START_REF] Bc | The considered compounds/phases are diamond, Si, Ge, d-Sn (α-Sn), β-Sn, SiC, сBN[END_REF] shows the comparison between experimental and theoretical values of bulk modulus for various compounds [START_REF] Solozhenko | [END_REF]49]. The remarkable deviation (g ~ 1.4) is observed only for three most hard phases containing carbon, i.e. diamond, cubic BC 2 N and diamond-like BC 5 .

By combining equations ( 1) and ( 5), obtain that illustrates the famous non-monotone correlation between bulk modulus B (the value inverse to compressibility K) and hardness H V [50][51][52].

Hardness at high temperature.

Equation ( 1) also allows to calculate the values of hardness at various temperatures by introducing the linear approximation of temperature dependence of ∆G at (T), i.e.

∆G at (T) = ∆G at (300)•[1-(T-300)/(T at -300)], (7)
where T at -temperature of atomization [START_REF]For diamond and cBN the corresponding temperatures of sublimation are 4300 K and 3300 K, respectively[END_REF]; as well as by introducing the temperature dependences of molar volumes V(T). Fig. 3a shows the temperature dependences of Vickers and/or Knoop hardness for diamond, cBN, ReB 2 and Al 2 O 3 in comparison with experimental data [START_REF] Novikov | [END_REF][START_REF] Kurdyumov | man, Polymorph modifications of carbon and boron nitride[END_REF][START_REF] Krell | [END_REF]55]. The theoretical lines have been calculated by equation

) ( ) 300 ( ) 300 ( ) ( ) 300 ( ) ( T V G V T G H T H at at ⋅ ∆ ⋅ ∆ ⋅ = . (8)
At relatively high temperatures (~ 0.3-0.5 T at ) this equation gives 10-15% higher values than the observed ones (bold lines on Fig. 3a), that should be attributed to the increase of materials' plasticity due to the intensification of the surface and bulk diffusion [START_REF] Trim | Design of Industrial Catalysts[END_REF]. The influence of the temperature on plasticity (coefficient α) can be taken into account by the following empirical equation:

        - ⋅ =         ∞ - ⋅ = - T T melt e k T k T 3 / 2 1 ) 300 ( ) ( ) ( 1 ) 300 ( ) ( α α α (9)
that supposes the Arhenius-type temperature dependence of the dislocation propagation constant k(T) (following Ref. 57, the activation energy was set to 2/3RT melt ). This term allows decrease the discrepancy between experimental and calculated data down to the level of experimental error (dashed lines on Fig. 3a). Equation ( 4) may be also generalized for calculation of the temperature dependence of where H T -the phase enthalpy at temperature T, while H melt -the enthalpy of solid phase at a melting point. Fig. 3b shows the calculated values of hardness for diamond, B 4 C and SiC (the data on the volume expansion with temperature have been taken from Refs. 58,59) in comparison with experimental data [START_REF] Novikov | [END_REF][START_REF] O'connor | A High Temperature Semiconductor[END_REF][START_REF] Marinescu | Handbook of Ceramic Grinding and Polishing[END_REF].

hardness H V (Т) [or H K (Т)], i.e. T T melt melt V V V H H H H V H T H - - = 298 298 ) 298 ( ) ( , ( 10
) T T V V V H H N H T H 298 2 . 13 ) 298 ( ) ( - - = δ , (11)
Both equations ( 8) and (10) [or (11)] suggest that the strong decrease of diamond hardness with temperature in comparison with other materials is due to the lower thermal expansion of diamond. We should also note that according to equation ( 6), temperature dependence of hardness should closely follow the temperature dependence of bulk modulus and plasticity (in vicinity of 300 K), i.e.

) ( ) ( ) ( T B T const T H V ⋅ ⋅ = α (12) or ) 300 ( ) 300 ( ) ( ) ( ) 300 ( ) ( B T B T H T H V V ⋅ ⋅ ⋅ = α α (13)
Hardness at high pressure.

Previously some suggestions have been made on the increase of hardness with pressure [START_REF] Nellis | [END_REF]. Because of the lack of reliable data on ∆ G at and ∆ H melt at very high pressures, the prediction cannot be easily made using equations (1) or (4) (the ab initio calculations of corresponding thermodynamic parameters could be useful in this case). However, according to equation (6), pressure dependence of hardness is the same (up to a constant) that pressure dependence of bulk modulus, i.e.

) ( ) ( p B const p H V ⋅ = (14)
or, applying the Murnaghan equation of state [START_REF]other equation of state having finite value of B 0 ' at high pressure[END_REF] 1

0 0 1 ) ( ) 0 ( 1 ) 0 ( ) ( B V V V V p V H p B B H p H -         ⋅ =         + ⋅ = (15)
Equation ( 15) allows one to suggest that the hard phases with relatively low bulk moduli should show remarkable hardness increase with pressure. From Fig. 4 it can be clearly seen that some compounds with relatively high hardness at ambient pressure and relatively low bulk modulus become harder more rapidly than diamond under pressure; that allows some of them to reach the diamond hardness (as well as diamond's compressibility) [START_REF] Nellis | [END_REF] at very high pressures. It is interesting to note that graphite, a very soft material at ambient conditions, may reach diamond hardness at lower pressure than many other materials. This fact is in excellent agreement with results reported by Mao et al. [START_REF] Mao | [END_REF] on formation of "superhard graphite" that can scratch the single-crystal diamond. One can suggest that other ordered [64][65][66] and disordered [67][68][69][70] graphite-like phases should show similar behavior under high pressure, even if the "compressed state" is not always quenchable down to ambient pressure [66,67,70].

Conclusions.

Thus, it has been shown that the hardness of solids is directly related to their thermodynamic and structural properties. The formulated equations may be used for a large number of compounds with various types of chemical bonding and structures. The proposed method allows estimating the hardness and compressibility of various hypothetical compounds using the data on the Gibbs energy of atomization of elements and covalent/ionic radii. The capacity of this approach to predict hardness has been illustrated by examples of the recently synthesized superhard diamond-like BC 5 [42] and orthorhombic modification of boron, γ-B 28 . In the framework of proposed method we have calculated the temperature dependencies of hardness for diamond, cBN, B 4 C, SiC, ReB 2 and α-Al 2 O 3 . Besides, it has been shown that in the megabar pressure range some phases can become harder than diamond at the same pressure. under assumption that α = const (solid line) and using equation (9) for α (dashed line).

(b) The lines correspond to the calculations using equations (10) (dashed line) and (11) (solid line).

  Hardness of materials at high temperature and high pressureV.A. MUKHANOV, O.O. KURAKEVYCH and V.L. SOLOZHENKO * LPMTM-CNRS, Université Paris Nord, Villetaneuse, France

Fig. 1

 1 Fig. 1 Hardness and bulk modulus as a function of Gibbs energy of atomization. (a) Comparison of experimental values of Vickers hardness of various phases with corresponding values calculated in the framework of the model proposed in the present paper [equation (1)]. (b) Comparison of experimental bulk moduli of various phases with values calculated by equation (6).

Fig. 2

 2 Fig. 2 Hardness as a function of enthalpy at a melting point. Comparison of experimental values of Vickers hardness of various phases with corresponding values calculated using equation (4).

Fig. 3

 3 Fig. 3 Temperature dependence of hardness of single-crystal diamond, polycrystalline cBN (mean particle size of 5 µm), single-crystal ReB 2 , and B 4 C-, SiC-and Al 2 O 3 -based ceramics. The symbols represent the experimental data obtained by static indentation [4,20,54,55,59,60]. (a) The lines show the results of calculation using equation (8)

Fig. 4

 4 Fig.4Prediction of pressure dependence of hardness using experimental data on bulk moduli and their pressure derivatives [equation(15)].

Table 1 .

 1 Theoretical values of Vickers hardness for some hypothetical superhard high-pressure phases

	Solids a	-∆G°f, kJ mole -1	-∆G°a t kJ mole -1	V, cm 3 mole -1 N	2	NV G at °∆ GPa	,	χ X	χ Y	β	H V theor , GPa
	C 3 N 4	60 b	3896.0	35.45 b	4		55.0		3.04 2.55 0.8393	41.7
	c-BC 5	96 b	3971.1	21.32	4		93.1		2.55 2.04 0.7903	70.6
	c-BC 3	62 b	2594.6	14.09 b	4		92.1		2.55 2.04 0.7903	73.2
	d-B c	0 b	518.8	4.242 c	4		61.2		2.04 2.04	1	61.2
	hp-B 2 O 3 1272.9 b 3005.7	22.29 d	4		67.4		3.44 2.04 0.5543	37.4
	hp-B 2 O 3 1272.9 b 3005.7	21.0 e	6	47.07		3.44 2.04 0.5543	26.4
	CO 2	294.0 b 1429.0	14.5 b	4		49.3		3.44 2.55 0.725	35.7
	(α-SiO 2 )
	"d-C 2 О" f 148.7 b 1722.9	10.64	4		81.0		3.44 2.55 0.725	58.7
	d-CO	37.0 b	940.0	5.90	4		79.7		3.44 2.55 0.725	57.8
	a the calculations have been performed with ε = 1; thermodynamic data from Refs.
	18,20,22-26;
	b the values have been estimated using the standard Gibbs energies of formation of known
	compounds in the corresponding binary systems;
	c the length of B-B bond taken as 1.66 Å;
	d molar volume of β-B 2 O 3 phase;
	e estimation for the lowest possible limit of the molar volume of B 2 O 3 according to the
	covalent radius data [18];
	f buckled layers of graphite are connected by oxygen atoms.

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Acknowledgements.

The authors are grateful to Agence Nationale de la Recherche for financial support (grant ANR-05-BLAN-0141).