Franck Lihoreau

Relevant Alternatives Contextualism and Ordinary Contingent Knowledge

According to David Lewis' contextualist analysis of knowledge, there can be contexts in which a subject counts as knowing a proposition just because every possibility that this proposition might be false is irrelevant in these contexts. In this paper I argue that, in some cases at least, Lewis' analysis results in granting people non-evidentially based knowledge of ordinary contingent truths which, intuitively, cannot be known but on the basis of appropriate evidence.

In his 1996 paper 'Elusive Knowledge', David Lewis, one of the leading proponents of epistemological contextualism, offers an intriguing analysis of knowledge according to which there can be contexts in which a subject counts as knowing a proposition just because in these contexts every possibility that this proposition might be false is properly being ignored. This idea of an 'ignorance-based knowledge' is supposed to help us deal with some aspects of the problem of skepticism. In this paper I hold that it can be of no such help for it has an untenable consequence: in some cases it results in granting people non-evidentially based knowledge of ordinary contingent truths about which it is nonetheless intuitively absurd to think that they could be known in the absence of appropriate evidence.

Ignorance-based knowledge

According to so-called 'relevant alternatives analyses of knowledge', knowing requires excluding all relevant possibilities of error, where relevance is defined by some criterion to be made more precise. According to what I will call 'relevant alternatives contextualism', this criterion is a criterion of contextual relevance. In David Lewis' own version of relevant alternatives contextualism, given a context of knowledge attribution, a relevant alternative is characterized as a possibility of error that is not properly being ignored in that context:

[Subject] S knows that P iff S's evidence eliminates every possibility in which not-P -Psst! -except for those possibilities that we [knowledge attributors possibly distinct from S] are properly ignoring (Lewis 1996: 425), where S's 'evidence' mainly consists in the totality of her perceptual experience and memory, and where a possibility is 'eliminated' when the evidence at S's disposal in this possibility differs from the evidence she actually has.

In order to determine the domain of possibilities that may not properly be ignored in a context, Lewis puts forward the following set of 'relevance rules' 1 :

Actuality

The possibility that actually obtains for the subject may never properly be ignored, 1 The rules about to be mentioned are 'prohibitive' rules, telling us which possibilities may not properly be ignored. But Lewis also puts forward a set of 'permissive rules', telling us this time which possibilities may properly be ignored. By the Rules of Conservatism, Reliability, and Method, we may properly ignore (even though defeasibly), respectively: (1) all the possibilities normally and commonly known to be ignored by people around us, (2) the possibility of a deficiency in our normally reliable informational processes (perceptual experience, memories, other people's testimony, etc.), and (3) the possibility (a) that samples on which we base our inductive generalizations might not be representative or (b) that the best explanation of our evidence might not be the right one. But only prohibitive rules shall matter for the purpose at hand.

Belief

A possibility that the subject believes or ought to believe to obtain may not properly be ignored, Attention A possibility that is not being ignored simpliciter may not properly be ignored,

Resemblance

Given two possibilities resembling each other in a salient way, if one of them is not properly being ignored (in virtue of the previous three rules), neither is the other. Now, Lewis' definition of knowledge says that knowing requires having evidence against some error possibilities, namely those which may not properly be ignored in virtue of the four rules just mentioned. For instance, in ordinary contexts, in order for me to count as knowing that my car has not been stolen while I was in the bookstore, seeing it parked exactly where I left it a few hours ago will be enough. Of course, seeing it parked there does not eliminate the possibility that my car was stolen and put back in place by an excessively scrupulous thief while I was in the bookstore. But this possibility is not one that needs to be ruled out in most ordinary contexts. For in most ordinary contexts, provided that it is not actually the case that the car was stolen and put back in place, the possibility in question is neither one which is attended to nor one which I believe or ought to believe to obtain. So, it does not have to be eliminated by my evidence for me to count as knowing that my car was not stolen.

But elimination is not the only way in which one can count as knowing something. Lewis himself notices that if knowing requires excluding all contextually relevant alternatives to what is known, this requirement is also satisfied when no alternative is contextually relevant.

In other words, Lewis' account of knowledge allows that:

A subject S knows that P, if every possibility in which not-P is properly being ignored2 .

So, a possibility of error needs only properly be ignored, in accordance with the three rules of Actuality, Belief, and Attention, for us to know it not to obtain, and this is so in the absence of all appropriate evidence and thus in a purely a priori way -in a very broad sense of 'a priori' as 'evidence-independent'.

We can thus distinguish between two ways of coming to (count as) know(-ing) according to

Ignorance-based anti-skeptical knowledge

Lewis suggests that anti-skeptical knowledge be treated as an instance of ignorance-based knowledge. He says that some possibilities of error can never be eliminated but can only properly be ignored. According to him, in ordinary non-philosophical contexts, this is true of skeptical possibilities of massive error:

Knowledge by presupposing and ignoring is knowledge; but it is an especially elusive sort of knowledge, and consequently it is an unclaimable sort of knowledge.

[…] Simply mentioning any particular case of this knowledge, aloud or in silent thought, is a way to attend to the hitherto ignored possibility, and thereby render it no longer ignored, and thereby create a context in which it is no longer true to ascribe the knowledge in question to yourself or others (Lewis 1996: 438).

In general, when S knows that P, some of the possibilities in which not-P are eliminated by S's evidence and others of them are properly ignored. There are some that can be eliminated, but cannot properly be ignored.

[…] And there are other possibilities that never can be eliminated, but can properly be ignored. For instance, the possibility that Possum [the cat] is on the desk but has been made invisible by a deceiving demon falls normally into this class (though not when I attend to it in the special context of epistemology) (Lewis 1996: 438).

Of course, the possibility that actually obtains and a skeptical possibility of massive error (such as Descartes' Evil Demon possibility, or Putnam's Brain-in-a-vat possibility) resemble each other in respect of the fact that they cannot be eliminated by the subject's evidence. But as Lewis points out, the Rules of Actuality and Resemblance cannot be used to make a possibility W turn out relevant just because it resembles one's actuality in respect of one's evidence.

Here is what he says about such a misuse of these rules:

We must apply the Rule of Resemblance with care. Actuality is a possibility uneliminated by the subject's evidence. Any other possibility W that is likewise uneliminated by the subject's evidence resembles actuality in one salient respect: namely, in respect of the subject's evidence. That will be so even if W is in other respects very dissimilar to actuality.

[…] Plainly, we dare not apply the Rules of Actuality and Resemblance to conclude that any such W is a relevant alternative (Lewis 1996: 430).

In particular, a skeptical possibility of massive error cannot turn out relevant just because it resembles our actuality in respect of our evidence. This exception to the Rule of Resemblance is an exception that Lewis willingly admits to be ad hoc; but it is an exception without which there would never be any context in which one could truly be said to know anything (Lewis 1996: 430).

As a result, in most ordinary contexts in which we ignore a skeptical possibility W, in which we do not believe and have no reason to believe that W obtains, we do count as knowing that not-W. And we count as knowing that not-W in the absence of all empirical evidence against the possibility that W.

A powerful reason for rejecting Lewis' account of anti-skeptical knowledge is that the idea of an ignorance-based knowledge, on which the account relies, has an extremely undesirable consequence. It results in granting people non-evidentially based knowledge of contingent truths about which it is obviously absurd to think that they can ever be known but on the basis of appropriate evidence.

Ignorance-based knowledge of lottery propositions

Hawthorne 2002 does seem to put forth such an objection to relevant alternatives contextualism when he discusses Lewis' solution to the 'lottery paradox'. This paradox can be introduced in the following way, quoting from Lewis:

Pity poor Bill! He squanders all his spare cash on the pokies, the races, and the lottery. He will be a wage slave all his days. We know he will never be rich. But […] if we know that he will never be rich, we know that he will lose [the lottery]. But […] we cannot know that he will lose. All the possibilities in which Bill loses and someone else wins saliently resemble the possibility in which Bill wins and the others lose; one of those possibilities is actual; so by the rules of Actuality and Resemblance, we may not properly ignore the possibility that Bill wins (Lewis 1996: 443).

So, this instance of the lottery paradox consists of the following apparently inconsistent triad:

(1) We know that Bill will never be rich

(2) If we know that Bill will never be rich, then we (can) know that he will lose the lottery

(3) We do (/can) not know that Bill will lose the lottery. Now, here is Lewis' intended solution to this instance of the lottery paradox:

Salience, as well as ignoring, may vary between contexts.

[…] When

[…] explaining how the Rule of Resemblance applied to lotteries, […] the resemblance between the many possibilities associated with the many tickets was sufficiently salient. But […] when we were busy pitying poor Bill for his habits and not for his luck, the resemblance of the many possibilities was not so salient. At that point, the possibility of Bill's winning was properly ignored; so then it was true to say that we knew he would never be rich. Afterward I switched the context. I mentioned the possibility that Bill might win, wherefore that possibility was no longer properly ignored. […] I was no longer true that we knew he would lose. At that point, it was also no longer true that we knew he would never be rich (Lewis 1996: 443-4) 4 .

Let us note that Lewis' solution relies on the idea that there are contexts relative to which no evidence is needed in order to count as knowing that Bill will not win the lottery: these are contexts in which every possibility that Bill might win the lottery is properly being ignored and thus known not to obtain. According to Hawthorne (2002: 246-7), this aspect of Lewis' solution is what makes it problematic, as he tries to show by reflecting on the following case:

Suppose that 5001 lottery tickets are sold, that 5000 of the buyers are Ben's friends, and that the 5001 st buyer, John, is not one of Ben's friends. As it happens, John is the one who will win the lottery tonight.

Ben knows that each of his 5000 friends who bought lottery tickets has the bad habit of spending her money as soon as she manages to get some. Ben knows that each of them bought a lottery ticket, and he also knows that 5001 lottery tickets were sold. Ben even knows that John is the one who bought the 5001 st lottery ticket. Moreover, suppose that Arthur, another character, is as well informed as Ben is about the bad habit of each of Ben's friends, and that he may thus properly ignore every possibility that one of Ben's 5000 friends might win the lottery tonight.

Now, what if Arthur were to say that Ben knows that Bill, one of Ben's 5000 friends, will never be rich? According to Lewis' analysis and given its commitment to the possibility of an ignorance-based knowledge, Arthur's knowledge attribution to Ben would be true. Relative to the context in which Arthur would make the attribution, Ben would count as knowing that Bill will never be rich, for Arthur would then be ignoring every possibility that Bill might become rich, including those possibilities in which Bill wins the lottery. Similarly, if Arthur were to say that Ben knows that Harry, another one of Ben's 5000 friends, will never be rich, Arthur's knowledge attribution to Ben would also be true. And so on for each of Ben's other 4998 friends.

Thus, on Lewis' account, it is true in Arthur's context that Ben knows that none of his 5000 friends who bought tickets will ever be rich. It would therefore also be true in Arthur's context that Ben knows that none of his 5000 friends who bought tickets will win the lottery tonight. But then, it should also be true in Arthur's context that Ben knows that John, the 5001 st ticket buyer, is the one who will win the lottery. So, if Lewis' relevant alternatives contextualism is correct, then there can be cases in which the relevant context is such that one can count as knowing who the winner of a lottery will be. On Hawthorne's opinion, this result is clearly problematic for Lewis' analysis of knowledge. But is it really?

Everyone may not share Hawthorne's intuition on the Arthur and Ben case. After all, if Ben is informed that 5000 out of 5001 lottery tickets were sold to his friends and that John is the one who bought the 5001 st ticket, then if we take for granted from the start, just as Hawthorne does, that Ben is in a good enough position to be described in Arthur's context as knowing that none of his 5000 friends will win the lottery, why not think that Ben is also in a good enough position to be described in the same context as knowing that John will win the lottery?

For the context in which Arthur assesses Ben's epistemic position with respect to the proposition that none of his friends will win the lottery is supposed to be the same as the context in which Arthur assesses Ben's epistemic position with respect to the proposition that John will win the lottery. Since it is the same context, Arthur assesses Ben's epistemic position with respect to both propositions relative to the same epistemic standards, that is, against the same set of contextually relevant possibilities. Thus, one may reply to Hawthorne by saying this: Ben can truly be said to know that John will win the lottery relative to those standards which are lax enough for Ben to truly be said to know that none of his friends will win the lottery. After all, as Lewis himself says elsewhere (Lewis 1979: 245), if it is true enough in a context that Italy is boot-shaped, then it is also true enough in the same context that France is hexagonal. In a similar way, if it is true enough in a context that Ben knows that none of his friends will win the lottery, then it is also true enough in the same context that Ben knows that John will win the lottery. This is a reply which is easily available to the contextualist.

For this reason, I do not think that Hawthorne's objection to Lewis is conclusive. Even though it is intended to show that Lewis' analysis results in counting people as knowing things the knowledge of which we would not be inclined to grant them, Hawthorne's complicated example does not clearly indicate that relevant alternatives contextualism has the intended troublesome consequence. I nonetheless happen to think that such a consequence can clearly be drawn from Lewis' analysis of knowledge, but through much simpler examples, like the one put forward in the next section.

Ignorance-based knowledge of ordinary contingent truths

Besides lottery propositions and denials of skeptical hypotheses, there are 'ordinary contingent truths', e.g. that we have hands, that Grandma has moustaches, etc., which we would not claim knowledge of unless we did it in the light of appropriate evidence 5 . The problem is that Lewis' analysis of knowledge predicts that, in some cases, even ordinary contingent truths of the aforementioned kind can be known through mere ignorance, not elimination, of error possibilities.

5 Thinking differently would be absurd, since some people do not have hands or do not have a moustached

Grand mother, and therefore cannot know at all that they have hands or that they have a moustached grandmother. And this is true of many other propositions as well, especially those pertaining to the external world.

For instance, we could count as knowing what a person, whose existence we are not even aware of, had for lunch today. Let us consider the following case:

Mr. So-and-so6 had oysters for lunch today. But I do not know who Mr. So-and-so is; I do not even know him by name. I am not even aware of his existence, and I do not know anyone who knows him. Mr.

So-and-so and I live on opposite sides of the Globe. So, I do not have the slightest evidence about what he may have had for lunch today.

And honestly, I do not care.

Intuitively, the fact that Mr. So-and-so had oysters for lunch is an ordinary contingent truth which I cannot know in the absence of appropriate empirical evidence. And since I have not got any evidence regarding Mr. So-and-so, whose existence I do not even suspect, in all likelihood I cannot know what he had for lunch today. I take it that everyone will agree about this.

On the contrary, Lewis' analysis predicts that I should know this and that I should know it in a purely a priori manner. Indeed, in accordance with the above mentioned relevance rules, in the depicted circumstances I would properly be ignoring every possibility in which Mr. Soand-so did not have oysters for lunch. Firstly:

(1) The possibility that actually obtains is a possibility in which Mr.

So-and-so did have oysters for lunch, since we take for granted that he actually did have oysters for lunch; so here we have the Actuality Rule.

Secondly:

(2) I am ignoring simpliciter every possibility in which Mr. So-and-so did not have oysters for lunch, since after all, I am not even aware that Mr. So-and-so exists; so here we have the Attention Rule.

Thirdly:

(3) I do not believe and I have no reason to believe that Mr. So-and-so did not have oysters for lunch, since on the one hand I am not even aware of Mr. So-and-so's existence, and since on the other hand I have no evidence at all justifying me in believing that Mr. So-and-so did not have oysters for lunch; so here we have the Belief Rule.

Of course, since I have no evidence at all pertaining to Mr. So-and-so, a fortiori all the evidence I have speaks no more against any possibility in which Mr. So-and-so did not have oysters than it does against the possibility that actually obtains and in which he did have oysters; so the two possibilities resemble each other in respect of my evidence. But given the ad hoc exception to the Rule of Resemblance, which Lewis himself emphasizes, the Rules of Actuality and Resemblance cannot be used to render a possibility in which Mr. So-and-so did not have oysters relevant, in spite of the fact that the possibility in question and the possibility that actually obtains resemble each other in respect of my evidence, or more exactly, in respect of my lack of evidence regarding Mr. So-and-so.

Yet, one might reason in the following way: a possibility in which Mr. So-and-so did not have oysters for lunch may resemble the possibility that actually obtains in all respects, except for the fact that he did not have oysters but, say, mussels; thus, in virtue of the Actuality and Resemblance rules, the possibility that he did not have oysters may not properly be ignored and is relevant.

However, recall that for it to be relevant in virtue of these rules, its resemblance with the possibility that actually obtains must be salient. Now, remember that Lewis holds that salience, like ignoring, is a context-variable matter. We saw this earlier when we touched on Lewis' solution to the lottery paradox. Then, we saw that according to Lewis, the resemblance between the many possibilities of winning the lottery associated with the many tickets, and the possibility that actually obtains in which the subject will lose, could be salient when explaining the oddity of attributing knowledge that she will lose the lottery, and yet not salient when reflecting on the subject's bad habit of spending her money before it is earned.

In a similar way, the resemblance between the possibility that Mr. So-and-so did not have oysters and the possibility that actually obtains in which he did have oysters, though salient in the context in which you philosophically reflect on the oddity of attributing knowledge that he did have oysters to me, is not salient at all in my context, the ordinary one depicted in the example, in which I do not give a single thought to Mr. So-and-so. So, since the possibility that Mr. So-and-so did not have oysters for lunch does not resemble my actuality in a contextually salient way, it is not made contextually relevant by the Actuality and Resemblance rules, when the context of reference is my context as depicted in the example, the one of which I am the attributor (as well as the subject), that is, the only one that matters for the purpose at hand.

Therefore, in accordance with Lewis' relevance rules I may properly ignore every possibility in which Mr. So-and-so did not have oysters for lunch. Therefore, in accordance with Lewis' analysis of knowledge I should count as knowing that Mr. So-and-so had oysters for lunch, and I should know this just by properly ignoring and in the absence of all appropriate evidence. This, of course, goes against the intuitively expected result.

In order to defend Lewis' contextualist analysis of knowledge, one may try the following line of reply:

The Rule of Actuality stipulates that the subject's actuality is never properly ignored. And in my actuality -the one depicted in the little story -, it is the case that Mr. So-and-so had oysters for lunch. But in accordance with the little story, this is a possibility which I am ignoring simpliciter, for I am not attending to Mr. So-and-so at all. So, according to Lewis' analysis, I do not know anyway that Mr. So-andso had oysters.

Of course, if this reply is conclusive, the previous objection is short-circuited. I do not think it is a reply that can be drawn from Lewis' analysis, though. But anyway, let us suppose that it is a conclusive reply that can be drawn from Lewis' analysis. Well, all we have to do to come across the undesirable consequence of attributing non-evidentially based knowledge of ordinary contingent truths again is modify our previous story a little bit:

This time, the following information is given: Mr. So-and-so is the brother of Mr. Nobody's son-in-law, where Mr. Nobody7 is the person who invented a revolutionary device for controlling the fourth wheel of supermarket trolleys, you know, the "mad wheel" that systematically goes the wrong direction as if from its own will. Given how useful Mr. Nobody's invention is, everyone, including myself, has heard about Mr. Nobody. On the other hand, I do not know whether Mr. Nobody has a son-in-law, and I know even less that Mr.

So-and-so is the brother of Mr. Nobody's son-in-law. Like before, I do not know Mr. So-and-so, not even by name, and I do not know anyone who knows him either: I do not even suspect Mr. So-and-so's existence. He lives on one side of the Globe, I live on the other, and I

have no evidence at all about Mr. So-and-so.

But that day, I suddenly happen to consider out of the blue, as a mere possibility and without giving it any credence at all, the possibility that Mr. Nobody has a son-in-law and that his son-in-law has a brother, and I thus begin to consider the possible existence of Mr. Nobody's son-in-law's brother. I also happen to consider that the man might have ordered oysters for lunch.

Now, the question is this: do I know that Mr. So-and-so -whose possible existence I ignored until then but which I am now attending to through the description 'the brother of Mr.

Nobody's son-in-law' -had oysters for lunch? Intuitively no, since I do not have the slightest piece of evidence about Mr. So-and-so/the brother of Mr. Nobody's son-in-law and about what he might have had for lunch.

But once again, Lewis' analysis predicts a counterintuitive result. In the context of my own thought, I may properly ignore every possibility in which Mr. So-and-so did not have oysters for lunch. This is so because I ignore, I do not believe and I have no reason to believe that he did not have oysters for lunch. Moreover, this time, I am not ignoring -since I am attending to it -the only possibility that may not properly be ignored, that is, the possibility that actually obtains in which Mr. So-and-so, whom I am attending to through the description 'the brother of Mr. Nobody's son-in-law', did have oysters for lunch. And since Mr. So-and-so did have oysters in the only (and thus in every) possibility that I may not properly ignore, the conclusion to be drawn according to Lewis' analysis is the same as before: in the context in question, I count as knowing that Mr. So-and-so had oysters for lunch. Here again, we get the counterintuitive result of granting non-evidentially based knowledge of an ordinary contingent truth, but this time, the objection leaves no room for the above mentioned reply.

It is not difficult to think of similar cases involving other ordinary contingent propositions.

For instance, we can build contexts for ignorance-based knowledge of contingent propositions about the behavior of a new and still undetected virus, or about the life of the poor lonesome ant on the top of the Eiffel Tower, or about the state of preservation of the still unfound wreck of Commander La Pérouse's ship, etc. In short, the idea of an ignorance-based knowledge seems to have the consequence of granting us, in some contexts, a non-evidentially and ignorance-based knowledge of ordinary contingent truths about which it is nonetheless counterintuitive to think that they can be non-evidentially known, to think that they can be known without consulting all or part of our experience.

Conclusion

Lewis' relevant alternatives contextualism allows knowledge without elimination, knowledge by proper ignoring. This idea is meant to be put to use to account for the possibility of antiskeptical knowledge. But it is problematic, for it leads us to grant a priori ignorance-based knowledge of ordinary contingent truths, of which intuitively we can only have a posteriori, elimination-based knowledge. For this reason, I do not simply think that Lewis' ignorancebased knowledge cannot be exploited in order to account for anti-skeptical knowledge. I think that a piece of 'cheap' knowledge for which the standards would be as liberal as those for Lewis' ignorance-based knowledge would be no 'knowledge' at all8 .

Here is what Lewis says about this:

See Black 2003 for an attempt to immunize Lewis' relevant alternatives contextualism against Schaffer's missed clues objection, by claiming that the latter can be dealt with by appealing to the Rule of Belief.

For other contextualist solutions to the lottery paradox, see[START_REF] Cohen | Contextualist Solutions to Epistemological Problems: Skepticism, Gettier, and the Lottery[END_REF][START_REF] Cohen | Reply to Baumann[END_REF][START_REF] Derose | Knowledge, Assertion, and Lotteries[END_REF]. For a discussion of such solutions, see chapter 2 of[START_REF] Hawthorne | Knowledge and Lotteries[END_REF]

Here, 'Mr. So-and-so' is supposed to be a proper name.

Here, 'Mr. Nobody' is supposed to be a proper name too.

I am extremely grateful to Berit Brogaard, Pascal Engel, Jonathan Schaffer, and the audience at the University of Geneva, May 12, 2006, for helpful comments on earlier versions of this paper.


Weslley Florentino
email: weslley.florentino@ifnmg.edu.brgiancarloqpellegrino@cefetmg.br

D E Oliveira

Giancarlo Queiroz Pellegrino

Chaos-based Potentials in the One-dimensional Tight-binding Model Probed by the Inverse Participation Ratio

Keywords: aperiodic potentials, tight-binding model, inverse participation ratio, Lyapunov exponent

Chaos-based potentials are defined and implemented in the onedimensional tight-binding model as a way of simulating disorder-controlled crystalline lattices. In this setting, disorder is handled with the aid of the chaoticity parameter. The inverse participation ratio (IP R) probes the response of the system to three different such potentials and shows consistent agreement with results given by the Lyapunov exponent Ly: the greater Ly(r) for the chaotic sequence as a function of the chaoticity parameter r, the greater the asymptotic value IP R(r) for the large-system ground state.

Introduction

Growing crystalline structures and new materials which would show previously desired properties -or avoid undesired ones -is a long-standing task. As a particular characteristic, disorder is an inherent and mostly uncontrolled feature of natural materials. It is acceptedly one of the main responsible facts for resistivity, be it electrical, thermal, or signal resistivity; and its very notion is based on the randomness of its occurrence. Contrary to common sense, controlled disorder can be made an interesting property, although perhaps difficult to achieve. The advent of quasicrystals made a notable step towards disorder control, due to the possibility of using binary substitution sequences to guide deposition-based crystalline growth. Although disorder was not fully controlled, it could be classified as an ascending feature for different quasicrystals, i.e., different substitution sequences. Remarkably, the early and well-known chaotic maps can give rise to binary sequences where the disorder is controlled by their chaoticity parameter and even measured by their associated Lyapunov exponent. Here, such chaos-based sequences are constructed from three known maps -logistic, tent, and Gaussian maps -and used as potentials in the one-dimensional Schrödinger equation. It is seen that a system simulated in this way not only responds unambiguously to more or less disorder in the potential but also allows for a disorder control in an experimental realization.

The remainder of this article is organized as follows. Section 2 states the main ideas, scope, and objectives of this work. In section 3, there appear the necessary definitions and model. Section 4 brings results and comments. In section 5, we discuss the consistency of our proposals in comparison with results known for disordered potentials. A short section 6 concludes this article.

Theory

Aperiodic sequences of different types have recently been used to simulate disordered one-dimensional crystalline structures, as potentials entering the Schrödinger equation. One of the main purposes of this effort is to simulate the physics of quasicrystals, which can be viewed as intermediate systems between the well-understood cases of periodic and disordered lattices. The latter ones are fairly well described by Bloch theorem and Anderson localization phenomenon respectively. In the case of periodic potentials, the eigensolutions of the Schrödinger operator can be chosen to exhibit periodic absolute values [START_REF] Ashcroft | Solid State Physics[END_REF], thus providing good conducting properties. At the other extreme, one-dimensional disordered potentials afford localized probability densities and, in this way, insulating properties [START_REF] Anderson | Absence of diffusion in certain random lattices[END_REF].

In the context of one-dimensional systems, quasicrystal issues have been addressed by using almost-periodic substitution sequences to mimic the ordering of the atoms in a crystalline lattice [START_REF]QUASICRYSTALS: The State of the Art[END_REF][START_REF]Beyond Quasicrystals[END_REF]. Substitution sequences and the properties of the Scrödinger operator have been extensively studied both from the theoretical [START_REF] Queffélec | Substitution Dynamical Systems -Spectral Analysis[END_REF][START_REF] Luck | Cantor spectra and scaling of gap widths in deterministic aperiodic systems[END_REF][START_REF] Dulea | Trace-map invariant and zeroenergy states of the tight-binding Rudin-Shapiro model[END_REF][START_REF] Ryu | Extended and critical wave functions in a Thue-Morse chain[END_REF][START_REF] Dulea | Unusual scaling of the spectrum in a deterministic aperiodic tight-bindind model[END_REF][START_REF] Oh | Band-structural and Fourier-spectral properties of one-dimensional generalized Fibonacci lattices[END_REF][START_REF] Piéchon | Analytical results for scaling properties of the spectrum of the Fibonacci chain[END_REF][START_REF] Oliveira | Quantum return probability for substitution potentials[END_REF] and the experimental [START_REF] Merlin | Quasiperiodic GaAs-AlAs Heterostructures[END_REF][START_REF] Todd | Synchrotron X-Ray Study of a Fibonacci Superlattice[END_REF][START_REF] Axel | High-Resolution X-Ray-Diffraction Spectra of Thue-Morse GaAs-AlAs Heterostructures: Towards a Novel Description of Disorder[END_REF][START_REF] Mizoguchi | Observation of coherent acoustic phonons in Fibonacci superlattices[END_REF] points of view. Some examples of these sequences are the early Fibonacci substitution sequence, as well as Thue-Morse, perioddoubling, paper-folding, Rudin-Shapiro sequences, and others [START_REF]Beyond Quasicrystals[END_REF]. The main interests in these studies are the physical transport properties [START_REF] Dulea | Localization of electrons and electromagnetic waves in a deterministic aperiodic system[END_REF][START_REF] Iochum | Resistance of one-dimensional quasicrystals[END_REF][START_REF] Ryu | Electronic properties of a tight-binding and a Kronig-Penney model of the Thue-Morse chain[END_REF][START_REF] Roy | Landauer resistance of Thue-Morse and Fibonacci lattices and some related issues[END_REF][START_REF] Katsanos | Quantum electron dynamics in periodic and aperiodic sequences[END_REF][START_REF] Roy | A study of Landauer resistance and related issues of the generalized Thue-Morse lattice[END_REF][START_REF] Piéchon | Anomalous diffusion properties of wave packets on quasiperiodic chains[END_REF][START_REF] Roche | Conductivity of quasiperiodic systems: A numerical study[END_REF][START_REF] Salejda | The Landauer conductance of generalised Fibonacci superlattices[END_REF][START_REF] Pellegrino | Persistent current and Drude weight in onedimensional rings with substitution potentials[END_REF] and the mathematical spectral types [START_REF] Queffélec | Substitution Dynamical Systems -Spectral Analysis[END_REF][START_REF] Bovier | Remarks on the spectral properties of tightbindind and Kronig-Penney models with substitution sequences[END_REF][START_REF] Allouche | Schrödinger operators with Rudin-Shapiro potentials are not palindromic[END_REF] of the corresponding Schrödinger operators.

The almost-periodic sequences cited above have no definite period and appear as intermediate cases between periodic and disordered structures. Although these substitution sequences could be classified in a hierarchy of ascending disorder [START_REF] Pellegrino | Persistent current and Drude weight in onedimensional rings with substitution potentials[END_REF][START_REF] Steuer | Entropy and optimal partition for data analysis[END_REF][START_REF] Gong | A measurement of disorder in binary sequences[END_REF], this is a static classification, since each sequence is generated by a fixed iteration rule. It would be greatly desirable both for the physical and mathematical possibilities to have a variable disorder, eventually controlled by a continuous variation of a parameter. This is the main purpose of this article. We start exploring here the properties of the Schrödinger operator triggered by a new type of possibly disorder-controlled potential, which will be based on iteration maps appearing in Chaos Theory [START_REF] Hilborn | Chaos and Nonlinear Dynamics -An Introduction for Scientists and Engineers[END_REF] (hereafter called chaotic potentials) 1 . As it is well known those maps are not periodic in the chaotic region, and chaoticity can be controlled by a continuous parameter r and measured by a Lyapunov exponent [START_REF] Hilborn | Chaos and Nonlinear Dynamics -An Introduction for Scientists and Engineers[END_REF].

To that end, our main probing tool will be the Inverse Participation Ratio (IP R) of a given eigenvector of the Schrödinger operator. This ratio is a real number defined and used in Solid State Theory to characterize localizationdelocalization transitions and their consequences to the transport properties [START_REF] Evers | Fluctuations of the Inverse Participation Ratio at the Anderson Transition[END_REF][START_REF] Ludlam | Localisation of the Vibrations of Amorphous Materials[END_REF][START_REF] Monthus | Anderson localization of phonons in dimension d = 1, 2, 3: Finite properties of the inverse participation ratios of eigenstates[END_REF][START_REF] Murphy | Generalized inverse participation ratio as a possible measure of localization for interacting systems[END_REF]. It will be seen that IP R for the ground-state eigenvector, as a function of the chaoticity parameter, shows very good agreement with the structure of chaotic regions and periodicity windows characteristic of these chaotic maps. In this way, one could seek to control the effects of the disorder in the potential sequence appearing in the Schrödinger operator just by varying the chaoticity parameter.

3 Definitions and model

The tight-binding model

We consider here the time-independent Schrödinger operator in the tightbinding approximation

(Hψ) n = ψ n+1 + ψ n-1 + V n ψ n , (1)
where one assumes the potential V , created by a lattice of atoms placed at positions n, to be concentrated in the regions close to the atoms, being negligible in the regions between them [START_REF] Ashcroft | Solid State Physics[END_REF][START_REF] Oliveira | Quantum return probability for substitution potentials[END_REF]. In this approximation, the eigenfunctions are given by the values ψ n at sites n, when subjected to a potential with values V n . In this model, one easily switches different potentials just by choosing appropriate sequences of real numbers {V n } ∞ n=1 .

Chaotic sequences and potentials

The sequences to be used as potentials in eq. ( 1) will be obtained from three iteration maps w n+1 = f r (w n ) commonly encountered in Chaos Theory [START_REF] Hilborn | Chaos and Nonlinear Dynamics -An Introduction for Scientists and Engineers[END_REF], namely

• the logistic map

w n+1 = rw n (1 -w n ) r ∈ [3.4, 4.0] (2)
• the tent map

w n+1 = r 1 -2 w n - 1 2 r ∈ [0.4, 1.0] (3)
• the Gaussian map

w n+1 = e -bwn 2 + r r ∈ [-1, 0.1] (4)
In all these maps, r is a continuous parameter which controls in a sense the chaoticity of the map. In the expressions above, the intervals are chosen to cover the regions of chaos and periodicity of the maps. For the sake of the point we want to make, plots for these maps will appear in the next section along with our results. It suffices to state here that, for r in the chaotic regions, these sequences neither have definite period nor are completely random, once they are constructed with fixed rules. Chaoticity, which will give us the desired disorder, is incremented by going deeper into the chaotic region and is measured by the corresponding Lyapunov exponent [START_REF] Hilborn | Chaos and Nonlinear Dynamics -An Introduction for Scientists and Engineers[END_REF]

Ly (r) = 1 N N n=1 ln f r (w n ) . (5)
In order to allow an easier comparison with results found in the literature for almost-periodic substitution potentials, and also to set an experimental perspective based on crystal growing [START_REF] Axel | High-Resolution X-Ray-Diffraction Spectra of Thue-Morse GaAs-AlAs Heterostructures: Towards a Novel Description of Disorder[END_REF], we take here binary chaotic potentials as given by

• V n = 1 if w n ≤ 0.5 and V n = 2 if w n > 0.
5 in the cases of logistic and tent maps;

• V n = 1 if w n ≤ 0 and V n = 2 if w n > 0 in the case of the Gaussian map.
The Lyapunov exponent, as given by eq. ( 5), is not directly applicable to these binary sequences [START_REF] Steuer | Entropy and optimal partition for data analysis[END_REF][START_REF] Gong | A measurement of disorder in binary sequences[END_REF]. It is, however, remarkable the fact that the Inverse Participation Ratio, to be defined in the following, captures indeed the features of that exponent even after the transposition to the corresponding binary sequence. Details of this accordance will be seen in the next section.

Inverse Participation Ratio

Perhaps the most conspicuous effect of a one-dimensional random potential is the localization of the wavefunctions, solutions of the Schrödinger equation, within finite regions in space [START_REF] Anderson | Absence of diffusion in certain random lattices[END_REF][START_REF] Kramer | Localization: theory and experiment[END_REF]. At the other extreme, periodic potentials give rise to wavefunctions which are extended over the lattice [START_REF] Ashcroft | Solid State Physics[END_REF]. The Inverse Participation Ratio of a given wavefunction, defined as a function of the system size N by

IP R (N ) =        N n=1 |ψ n | 4 N n=1 |ψ n | 2 2        -1 , (6)
reflects the regions where the wavefunction has appreciable amplitudes and, therefore, its distribution over the lattice [START_REF] Evers | Fluctuations of the Inverse Participation Ratio at the Anderson Transition[END_REF][START_REF] Ludlam | Localisation of the Vibrations of Amorphous Materials[END_REF][START_REF] Monthus | Anderson localization of phonons in dimension d = 1, 2, 3: Finite properties of the inverse participation ratios of eigenstates[END_REF][START_REF] Murphy | Generalized inverse participation ratio as a possible measure of localization for interacting systems[END_REF]. In this way, one can verify that IP R(N ) tends to zero as 1/N for large enough system sizes N in the case of periodic potentials. For the random case, IP R(N ) attends a constant value k > 0 as N goes to infinity [START_REF] Ludlam | Localisation of the Vibrations of Amorphous Materials[END_REF][START_REF] Monthus | Anderson localization of phonons in dimension d = 1, 2, 3: Finite properties of the inverse participation ratios of eigenstates[END_REF]. Aperiodic potentials, such as the almost-periodic substitution potentials, are expected to fall between these extreme cases [START_REF] Oliveira | Characterization of spectrum and eigenvectors of the Schrödinger operator with chaotic potentials[END_REF]. The large-N values of IP R(N ) could, therefore, probe the effect of a greater or smaller disorder in the potentials; the higher the asymptotic value of IP R(N ), the greater the disorder of the potential. Here we take the eigenvector associated with the lowest energy eigenvalue for the Schrödinger operator and calculate its large-N value for IP R. This asymptotic value, being a function of the parameter r, will be denoted here simply IP R(r). This will be our probing tool in order to see whether chaotic potentials, as defined here, could provide a way in which disorder is implemented in a more controlled procedure, as compared to other known aperiodic potentials.

Results

In Figures 123we show in plots (a) the original chaotic maps for different values of r, prior to the conversion into binary sequences. In plots (b) there appear the Lyapunov exponents for the original sequences and the average asymptotic large-N values IP R(r) for the corresponding binary sequences. In all these plots, we use sequences of size N = 1000 and in the calculation of IP R(r) we take averages over the results for 500 realizations of the sequences with different initial seeds w 0 .

In plots (a) we see the characteristic features of these chaotic maps, with regions of chaos and, eventually, windows of periodicity. We have chosen these three maps due to their different routes to chaos. The logistic map undergoes a sequence of period-doublings for its fixed points until entering a region of chaos at r ≈ 3.6. This chaotic behavior is observed up to r = 4.0, with periodicity windows appearing at precise values of r. For the tent map, the behavior is quite different, the sequence entering directly the chaotic region without passing a period-doubling transition. Finally, for the Gaussian map, we see the period-doubling transition to chaos, as in the case of the logistic map, but also a period-undoubling transition back to periodicity. The corresponding plots in (b) show that IP R(r) accompanies this structure, with positive values for r in the chaotic regions and vanishing values for r in the periodicity windows.

This agreement is in fact remarkably emphasized by the comparison between IP R(r) and the Lyapunov exponent Ly(r), both shown in plots (b). Expression (5) yields corresponding forms for the Lyapunov exponents • for the logistic map

Ly (r) = 1 N N n=1 ln |r (1 -2w n )| (7)
• for the tent map

Ly (r) = ln |2r| (8)
• for the Gaussian map In the regions where Ly(r) ≤ 0, IP R(r) vanishes as well, pointing out a periodicity window. On the other hand, Ly(r) is positive in the regions of chaotic behavior, a feature captured by IP R(r). Moreover, one can see that IP R(r) follows on average the increasing-decreasing behavior of Ly(r). A closer look reveals that IP R(r) increases smoothly as the maps undergo the bifurcation process, and drops down abruptly when the maps enter a periodicity window. This behavior can be seen, for example, at r ≈ 3.83 for the logistic map, or at r ≈ -0.4 for the Gaussian map. On the other hand, one observes some abrupt jumps in the IP R curve for the tent map, which are apparently not associated with a particular behavior of the corresponding map. The wavefunction for these values of the chaoticity parameter, however, shows accordingly a more localized distribution. This is suggestive of a consistent estimative, via IP R(r), of the response of the system to the disorder of the potential. In references [START_REF] Pellegrino | Persistent current and Drude weight in onedimensional rings with substitution potentials[END_REF][START_REF] Gong | A measurement of disorder in binary sequences[END_REF] it is pointed out that different approaches in order to produce a hierarchy of disorder for aperiodic sequences frequently end at inconsistent results.

Ly (r) = 1 N N n=1 ln 2bw n e -bw 2 n (9)
We stress the fact that this consistent agreement between Ly(r) and IP R(r) is seen here despite the conversion of the original sequences into binary ones, Ly(r) being directly calculated for the former and IP R(r) being indirectly calculated for the ground state generated by the latter ones. This agreement is found also for the excited states, with the same overall behavior of IP R. In the next section we give more details on this point.

Discussion

Much work on disordered potentials in one-dimensional tight-binding models has been done in the last decades. Since the potentials proposed here are less studied, we compare our results with those ones known for disordered potentials, in order to check for the consistency of IP R as indicative of the effects of randomness.

The correlation between the ratio IP R and the Lyapunov exponent Ly of the sequences which generate these chaotic potentials is clearly seen in Figure 4, where we plot the varying of IP R(r) with Ly(r). In fact, wherever Ly > 0 one has IP R > 0. Also, IP R increases on average with the increase of Ly. For logistic and Gaussian potentials, the many-valued character of IP R is due to the periodicity windows in the original maps, where Ly goes back to negative values and reincreases with the next period-doubling process, followed by IP R which goes back to zero and reincreases jointly. In this respect, the tent map shows a neat single-valued aspect, since there are no periodicity windows above the chaos threshold. In a situation where one would seek controlled disordered, the corresponding tent potential could be a good choice.

For potentials with increasing randomness, the energy-band structure seen for periodic potential is gradually destroyed. It is known that, in this process, the eigenstates associated with band-edge energies are more localized than those associated with band-center energies. This delocalization feature is also captured by the inverse participation ratio, eigenstates with greater IP R being more localized. We verified that this agreement is found for the excited states all along the spectrum, and we illustrate it with the logistic potential in the chaotic region (r = 4). In Fig. 5 we show the inverse participation ratio IP R for all the eigenstates. The central region of the spectrum have on average eigenstates with lower values for IP R. From these results we choose three eigenstates (k = 101, 510, 511), and plot in Fig. 6 the corresponding probability densities |ψ k | 2 over the lattice. It is seen that even small variations of IP R indicate corresponding variations in the localization length of states.

In this work, chaotic maps were intentionally chosen due to their intricate dynamics, which provided the desired disorder, and to the positive Lyapunov exponents in chaotic regions, which provided the desired control. Nonetheless, there is a whole class of systems, so-called strange nonchaotic attractors (SNA) [START_REF] Grebogi | Strange attractors that are not chaotic[END_REF], which present a fractal nature, with complicated dynamics, but for which all the Lyapunov exponents are either zero or negative. It would be interesting to test the behavior of the participation ratio IP R for tight-binding solutions generated by binary sequences built from such attractors. In what follows, as an example, we take the quasiperiodically forced logistic map [START_REF] Heagy | The birth of strange nonchaotic attractors[END_REF][START_REF] Prasad | Intermittency route to strange nonchaotic attractors[END_REF][START_REF] Prasad | Strange nonchaotic attractors[END_REF] given by

w n+1 = r [1 + cos (2πθ n )] w n (1 -w n ) ( 10
)
θ n+1 = θ n + φ (mod1) , (11)
where φ is an irrational number (we take the golden mean φ = √ 5 -1 /2), and the usual restriction 0 ≤ r ≤ 4 imposes 0 ≤ ≤ (4/r -1). It was shown that for = r / (4 -r) = 0.95 this system has a SNA behavior in the approximate interval 2.75 ≤ r ≤ 3. For r far below this interval, one has (quasi)periodic dynamics, which becomes more and more complicated as r approaches the SNA region. For r > 3 one has two regions of chaotic behavior separated by a narrow (quasi)periodic region around r ≈ 3.4. In Fig.

Figure 5: Inverse participation ratio of all the eigenstates ψ k for the logistic potential with r = 4.0. In the boxes there appear the choices used in Fig. 6.

7 we can see that the inverse participation ratio IP R(r) signals all these features. We take as an example a center-band delocalized eigenstate, ψ 300 . IP R(r) presents small values for r in the periodic region near r = 2 and attains increasing values as r approaches the SNA region and the dynamics becomes more complicated. Even in the SNA region, where the (largest) Lyapunov exponent Ly

(r) = 1 N N n=1 ln |r [1 + cos (2πθ n )] (1 -2w n )|
oscilates slightly around zero, IP R(r) correctly points the intricateness of the dynamics, with a corresponding localization process of the eigenstate (not shown here). It could be interesting to note that this results corroborates in a sense the choice of chaotic maps as a recipe for controled disorder since, then, the vanishing IP R in periodic regions clearly distinguishes it from the chaos-given disordered behavior with IP R(r) > 0. Finally, we just mention that, although it is known that the strength of the potential influences the localization properties of the eigenstates, the inverse participation ratio consistently keeps the agreement seen in the cases presented here. We have checked this consistency for different values of the potential amplitude V n in the interval [0, 1].

Conclusion

In this work we have proposed the use of new binary sequences, based on three different chaotic maps, as a way of introducing variable-disordered potentials in the Schrödinger operator. We have seen that the large-N value of the Inverse Participation Ratio (IP R(r)) for the ground-state of the Schrödinger operator, as a function of the chaoticity parameter r, correctly indicates the windows of periodicity and the regions of chaos pointed out by the Lyapunov exponents Ly(r). Moreover, we have seen in figures 1-3 (b) that IP R(r) follows the increasing (decreasing) behavior of Ly(r), providing a measure of chaoticity and disorder for the binary sequences, where the Lyapunov exponent is not directly applicable.

Finally, we mention that the techniques of growing superlattices by epi- taxy [START_REF] Axel | High-Resolution X-Ray-Diffraction Spectra of Thue-Morse GaAs-AlAs Heterostructures: Towards a Novel Description of Disorder[END_REF] could in principle be applicable to the chaotic potentials considered in this work. Experimental, and eventually technological, exploration of systems based on these chaotic potentials could then be considered.

Figure 1 :

 1 Figure 1: Logistic Map. (a) Bifurcation Diagram for r ∈ [3.4, 4.0]. (b) IP R(r) and Ly(r) multiplied by 0.1 for better viewing.

Figure 2 :

 2 Figure 2: Tent Map. (a) Bifurcation Diagram for r ∈ [0.4, 1.0]. (b) IP R(r) and Ly(r) multiplied by 0.1 for better viewing.

Figure 3 :

 3 Figure 3: Gaussian Map. (a) Bifurcation Diagram for r ∈ [-1.0, -0.1]. (b) IP R(r) and Ly(r) multiplied by 0.1 for better viewing.

Figure 4 :

 4 Figure 4: Correlation between Ly and IP R, both obtained as functions of the chaoticity parameter r.

Figure 6 :

 6 Figure 6: Logistic Map. Probability density distributions over the lattice sites n for the eigenstates ψ 101 , ψ 510 , and ψ 511 .

Figure 7 :

 7 Figure 7: Forced Logistic Map. IP R(r) and Ly(r) multiplied by 0.1 for better viewing. Dotted guiding lines mark zero for IP R and Ly, and the approximate SNA region.

To the best of the authors' knowledge, this was first proposed by C. R. de Oliveira (personal communication,

2002).

Acknowledgements

The authors thank sincerely Luciano Coutinho dos Santos and Luiz Argel Poveda Calvino for discussions on nonchaotic attractors. Also, the authors thank very much the anonymous Referees for their invaluable advice.


E Carvou

J L Le Garrec

E Yee

Kin Choi

J.-M Jouvard

M Kerkar

J B A Mitchell
email: mitchell@univ-rennes1.fr

Particle Size Determination in Electrical Arcs with Ag and AgSnO Electrodes using X-Ray Scattering

Keywords: arcs, nanoparticles, X-ray Scattering

This article discusses recent synchrotron radiation based measurements of nanoparticle formation between arcing contacts. The studied material is typical in contact applications, Ag and AgSnO 2 .

INTRODUCTION

During arcing between electrical contacts, there is a movement of material between the two electrodes that leads to degradation of the switching function, over time. This phenomenon is responsible for the erosion of electrodes. In the case of pure silver and pseudo-alloy AgSnO 2 rivets, this material transfer depends on the electrodes gap among several other parameters such as the current, the voltage, the contact geometry, etc. [START_REF] Yee Kin | Optimization of material erosion and welding performance by metal oxides and magnetic particles[END_REF]. Several regimes of mass change are highlighted in this phenomenon: anodic erosion, the compensation phase, cathodic erosion and bilateral losses.

In order to better understand the mechanism of arc formation and the influence of the arc on the electrode material, it is useful to examine the solid material that is transferred. In an effort to further understand this phenomenon, we have used Small Angle X-Ray Scattering (SAXS) to study the size of particles formed during the arcing process. Due to the arc temperature, T > 5000 K [START_REF] Slade | Electrical Contacts: Principles and applications[END_REF], the nanoparticles are probably formed on the periphery of the plasma where the temperature gradient allows condensation processes to occur. We have already reported our results for carbon, silver and silver-tinoxide (AgSnO 2 ) contacts [START_REF] Carvou | [END_REF][START_REF] Carvou | Particle size determination in electrical arcs using X-ray scattering[END_REF][START_REF] Carvou | [END_REF][START_REF] Carvou | th Int. Conf. Elec. Contacts (ICEC)[END_REF] where we found that the size and character of the nanoparticles were a clear function of the materials. While AgSnO 2 electrodes have been found to generate particles with smooth surfaces in the range of 30-40 nm in diameter, pure silver contacts produce aggregated particles. With carbon, the particles were much larger and had rough surfaces.

In a subsequent experimental study at the Soleil Synchrotron in Saclay, France [7], we have continued our examination of this phenomenon with a wider range of materials including silver (Ag) contacts produced by differing (proprietary) metallurgical methods, AgSnO 2 , AgZnO, AgC, Cu, Al, Ti and brass. The results of these measurements concerning silver based contacts are presented in this article. A fuller comparative discussion of the other materials will be presented elsewhere.

EXPERIMENTAL METHOD

A. Sample synthesis The silver samples were fabricated using 4 different metallurgical processes. The first was casting, which involved melting silver grains and casting them directly in order to produce a wire. The three last processes involve different size of silver grains in their fabrication. One process was atomization of grains, the latter being disintegrated by a highpressure jet and a shock wave. With this method, one obtains a powder with a grain size of 45 µm. The two remaining processes are chemical and electrolytic methods. They allow a much smaller grain size to be obtained compared with those obtained by atomization. The chemical method involves the reduction of a silver compound by a chemical agent forming silver with very fine grain size (5 µm) and a sub-product which is be eliminated. The last method was electrolysis of a silver salt, which produced a precipitate with a grain size of about 25 µm. This is summarized in Table 1.

B. Apparatus

The present experiment was performed on the SWING beamline of the Soleil synchrotron [START_REF] Carvou | [END_REF] dedicated to the SAXS technique.

The experimental apparatus (fig. 1) is a fully automated device, described in previous papers [START_REF] Carvou | [END_REF][START_REF] Carvou | Particle size determination in electrical arcs using X-ray scattering[END_REF][START_REF] Carvou | [END_REF]. The apparatus is 2 mainly composed of an electrical part (power supply, variable resistance) and a mechanical part using a stepping motor coupled with a sliding plate, which ensures a controlled separation between the fixed and the moving contact. A digital oscilloscope is used to sample and store the voltage and current characteristics of the arc during the break, i.e. V a (t) and I a (t). The whole system (power supply, motor and oscilloscope) is computer controlled. This apparatus performs contact breaking under initial direct current supply conditions of 400V and 8A. This voltage/current pair was chosen in order to maintain a stable arc with a large electrode gap of 2 mm during the X-ray exposure. In these experimental conditions, the material transfer mode corresponds to the bilateral one. The arc begins at t=0 when the electrodes begin to separate, with a voltage between the electrodes, initially of 12V (a typical value for metallic electrode arcs maintained by the resistance of the plasma), then rising to a constant level of 25V, as the electrodes reach a fixed separation of 2 mm in less than 0.5 second. The current decreases during this time. The mean arc duration during the various measurements was 5s. The arc extinguishes (indicated by the sudden rise in voltage, sharp drop in current) when the power supply is switched off, followed by a more gradual fall-off of the voltage to zero volts (fig. 2).

C. Small Angle X-Ray scattering

As soon as the 2 mm gap is reached, the intense beam of Xrays (12 keV, λ ≈ 1Å) from the Soleil synchrotron, passes through the arc established between the electrodes (and thus through the arc) before entering an evacuated detector chamber where they are collected and measured by a 2-D detector [7], so that their scattering pattern can be determined. A typical scattering pattern is shown in figure 3 for Ag contacts. The symmetric scattering patterns are integrated radially to produce graphs of Scattering Intensity (I) as a function of the momentum transfer q = (2π/λ) sin (θ/2) where θ is the scattering angle with respect to the primary beam direction and λ is the wavelength of the x-rays (see figure 4). The distance between the arc center and the detector in these experiments was held fixed at 4.075 m which allows measurements made over a q range from 0.003 -0.27 Å.

Figure 4 Schematic layout of a SAXS setup (from ref. [START_REF] Narayanan | Synchroton small-angle x-ray scattering[END_REF]). One can make an approximation for the size of the particles [START_REF] Guinier | X-ray Diffraction in crystals, Imperfect Crystals and Amorphous Bodies[END_REF] using the expression D ≈ π/q and so the particle size range that we could examine lay between 1.2 nm and 100 nm. A series of measurements are taken in series with exposure times of 250 ms separated by a deadtime (for data readout) of 250 ms. During this series, the arc is lit and then extinguishes when the voltage on the contacts is cut off. By subtracting the measurement without arcing at the end of the sequence from those with arc present, the influence of background air scattering and thermal effects can be eliminated.

RESULTS AND DISCUSSION

For isolated particles, such as nanoparticles in the gas phase, the scattering from a given structural level is described by a Guinier's law and a structurally limited power law, I(q) ∝ q -p , depicting a Porod region. The power law exponent, p, gives information on the morphology of the particulate system. For instance, p=4 signifies a sharp interface between the particle and its surrounding medium whereas p=2 is related to a Gaussian polymer chain [START_REF]Modern Aspects of Small Angle Scattering[END_REF]. The Porod region is observed at large q (i.e. large angles) where the scattered intensity is proportional to the surface area per unit volume. The Guinier law is described by [START_REF] Guinier | X-ray Diffraction in crystals, Imperfect Crystals and Amorphous Bodies[END_REF]:

𝐼 𝑞 = 𝐼 ! exp (- 𝑞 ! 𝑅 ! ! 3 )
where I 0 is the scattered intensity not related to the shape of the particle and R g is the radius of gyration. This law is valid only for small angles, i.e. qR g <<1 and provides information on both the radius of gyration of the scattering particles and the target number density. Therefore, on a log-log plot, the scattered intensity is reflected by a knee at low q (Guinier region) and a linear region at higher q (Porod region).

Figure 5 shows I vs q diagrams for Ag and AgSnO 2 contacts and it is seen that their scattering behavior is quite different in the Guinier region (low q) though very similar in the high q (Porod) region. At large q, the slope of the curve (i.e. Porod exponent p) in these experiments was found to be systematically q -4 which indicates so-called Porod scattering which means that the interface between the particle surface and the surrounding air is smooth. In essence this implies that the particles are spherical.

The data was collected, processed and analyzed using the inhouse software on the Swing beamline (Foxtrot -3.036-3-Swing).

Fig. 5: Intensity vs q plots for pure silver (Ag) contacts (orange) and AgSnO2 contacts (Red). It should be noted that the Guinier region for the Ag contacts occurs at much higher q (around 10 -2 A -1 ) than for the AgSnO2 contacts (around 0,006 10 -3 A -1 ) indicating the particles are smaller in the Ag case.

CONCLUSION

The results found here confirm our earlier studies which were mainly concentrated on AgSnO 2 , namely that the particles sizes were in the 30-40 nm range and that they exhibited smooth surfaces. Furthermore, in the case of pure silver contacts, the particle sizes are generally considerably smaller though there seems to be some variation depending upon the fabrication methods for the actual contacts, with pure silver yielding larger particles than those formed using powder and chemical based methods.

Fig. 1 :

 1 Fig. 1: Photo of the experimental apparatus mounted on the Swing beamline at the Soleil synchrotron in Saclay, France. X-rays exit the beamline through the tube on the right, pass through the inter-electrode gap (through the arc when it is lit) and enter the detector chamber through the cone on the left. The detector is located inside a 10 meter long evacuated chamber (left in the picture) and set at 4.075 m from the arc center.The contacts used in the present experiment were made from commercial (Metalor Corporation [8]) silver (Ag) and AgSnO 2 materials. The moving electrode had a spherical shape of 4 mm diameter while the fixed electrode was flat.

Fig 2 :

 2 Fig 2: Arc Voltage during time for analysed materials.

Fig. 3 :

 3 Fig. 3: Image of scattering signal as measured on the 2-D x-ray detector. The color indicates the intensity of the x ray scattered beam.

Table I .

 I Grains size as a function of silver samples production method

	Metallurgical method	Ag grain size
	Atomization	45 µm
	Chemical	5 µm
	Electrolytic	25 µm
	Concerning AgSnO

2 , the metallurgical process was atomization of Ag (45 µm) and SnO 2 (4 µm) grains. Two different percentages by weight of SnO 2 were used in this study: 3.3 % and 14 %

Table 2

 2 Radius of gyration of nanoparticles formed in electrical arcs according to the base material and preparation method.It can be seen directly from figure4that the particles formed with silver (Ag) contacts are smaller than those formed by AgSnO 2 contacts. In fact in these experiments, silver produced by different methods was used and two different AgSnO 2 samples had 3.3% and 14% respectively of SnO 2 . The results for these different materials are listed in table I.

	Base Material	Preparation Method	Rg (nm)
	Ag	Casted	24-26
	Ag	Atomized	18.5
	Ag	Chemical	17-19
	Ag	Electrolytic	17.5-19.5
	AgSnO 2	3.3% SnO 2	30-45
	AgSnO 2	14% SnO 2	35-38

ACKNOWLEDGMENT

The authors wish to thank Synchrotron SOLEIL for financial and technical support of these measurements, performed on the SWING beamline. Thanks are also due to Metalor Corporation for supplying the electrode materials.