interdisciplinary research has recently gained a central role in the advancement of science , leading to important achievements  .for instance , the 2014 nobel prize in chemistry was awarded to two physicists and a physical chemist , for a physical technique , developed with help from chemistry , that helps illuminate problems in biology " .  even though several definitions and metrics for interdisciplinarity have been proposed ,citation impact metrics accounting for this aspect of scientific research have not been defined yet .  on the other hand ,funding agencies have created specific calls for interdisciplinary projects , like the interdisciplinary programs funded by the national science foundation .the european research council explicitly encourages applications from scientists having published in multidisciplinary journals , and the evaluation criteria for the marie curie fellowships also include the interdisciplinary aspects of the research .consequently , there is significant need to evaluate projects and scholars by considering interdisciplinarity too .the difficulties in evaluating interdisciplinary research constitute a pressing controversy that leads many young scholars to remain on more traditional tracks , because the risks associated to undertaking an interdisciplinary career path seem too high  .this work addresses the issue of quantifying interdisciplinarity by proposing a method to rank scientific publications ( such as papers and patents ) and their producers ( scholars , inventors , institutions , companies and countries ) according to their scientific impact and its breadth over different scientific disciplines .the method is based on the detection of the most central elements of a complex bipartite interconnected multilayer network representing scientific producers and scientific citations within and across different fields .the citation network is composed of multiple layers , each representing a scientific discipline .accounting for this diversity instead of neglecting the information it provides by building an aggregated representation of the network allows to unveil the cross - disciplinary versatility of scientific publications and of their producers , and therefore to obtain a quantitative measure of their interdisciplinary scientific impact .since the seminal work of de solla price  and garfield  , scientists have put a great effort into trying to understand the patterns of citation distributions  and the non - trivial dynamics of scientific recognition  .this fundamental body of work has the ultimate goal of setting the basis for the definition of more accurate and fairer scientific impact metrics used for evaluation purposes .  in the last decades, several indices have been presented .they are based on the idea that we can quantify the impact of a scientific publication by counting the number of citations it has received over the years .a widely adopted indicator to evaluate scholars scientific impact is the h - index  ( a scholar has an index if of her / his publications have received at least citations each ) , and its numerous variants  .more recently , a different approach has been proposed .building networks that reconstruct the chains of scientific citations allows for a global understanding of the intrigued patterns of citations between publications or between producers .this representation allows to unveil the difference between , for instance , a publication that has received 10 citations coming from highly cited publications , and a publication that has received 10 citations too , but from low - cited papers .the two have the same number of citations but the former has clearly had a higher impact . to rank publications , journals or scholars according to their importance in the respective citation network, researchers have proposed diffusion algorithms that simulate the spreading of scientific credits on the network  . in practice , this is the same idea at the basis of the pagerank , i.e the algorithm that google uses to rank the pages of the world wide web  .and .if we discard the information about the scientific fields and consider the aggregated network shown in panel a , then the two authors centrality would be the same , because they authored the same number of papers , having an identical structure of incoming citations .however , the multilayer framework takes into account that one of s papers pertains to both physics and biology , and , moreover , had an impact in both fields ( one citation comes from a physics paper and the other from a biology one ) . therefore has a higher versatility than .,scaledwidth=50.0% ]  in this work , we propose to rank scientific publications and their producers employing the pagerank defined on a bipartite interconnected multilayer structure that accounts for citations within and across different disciplines .this is equivalent to ranking nodes according to their _ versatility _ on an interconnected multilayer network  .  to account for interdisciplinarity, we define a bipartite interconnected multilayer network representing citations between publications ( papers or patents ) and relations between publications and their manufacturers ( scholars , inventors , research institutions , companies or countries ) .given nodes and layers , the rank multilayer adjacency tensor is defined in the following way .let be the rank adjacency tensor encoding information about the relationship between layer and , where indicates the intensity of the relationship between node in layer and node in layer , and indicates the rank tensor that represents the canonical basis in the space ( note that when , represents the intra - layer adjacency tensor ) , then where indicates the rank tensor that represents the canonical basis in the space .this is the general formulation of an adjacency tensor representing a multilayer network .  to build our network we consider nodes ( where is the number of publications , and the number of manufacturers of the chosen type that produced the paperstherefore , given the ordered set of nodes , the first elements represent publication , and the other elements represent manufacturers .moreover , we consider layers , where is the number of scientific disciplines that the publications belong to .the 4 components of the rank adjacency tensor are defined as follows . and , with  ] , encode information about the relation between publications and their manufacturers , _i.e. _ if the chosen type of manufacturer is scholars , then if author is one of the authors of publication .if we consider research institutions , we connect each publication to the institutions to which its authors are affiliated ; if we consider countries , the connections are to the countries in which these institutions are based . finally , we define and to be zero tensors . tensors are null because we do not want the relations between publications and manufacturers to be symmetric , to avoid unrealistic paths to take place when computing the nodes centrality . is null because all the information is already encoded in the other tensors : we do not need to explicitly add citation edges between authors .  in this framework , for the citation layers ,each node is active on a given layer if and only if the publication it represents belongs to the corresponding field .for example , a monodisciplinary publication is active only on one layer , whereas an interdisciplinary publication , pertaining to both physics and biology , is active on two layers . as a consequence , a publicationwhose impact is restricted to only one discipline has intra - layer incoming edges only , whereas a publication that has influenced the work of researchers in more than one field has inter - layer incoming edges too , which represent the bridges between the different fields involved .therefore this framework allows for a natural representation of the interdisciplinarity degree of a publication .being our goal to rank publication producers too , we introduce in the network a second type of nodes , which , according to the specific need , represent scholars , inventors , research institutions , or countries .these nodes are active on a dedicated layer , and each publication has directed outgoing inter - layer edges pointing to each of its producers .previous works on ranking producers are based on one - mode projections of the bipartite network of publications and producers , whereas in this work we prefer to take advantage of the complete bipartite structure in order to avoid any information loss , as further detailed in the supplementary material .  on the proposed network, the ranking is obtained through a process of diffusion of scientific credits from paper to paper through citation edges within and across disciplines .producers are the sinks of this diffusion process , being represented by nodes with no outgoing edges , and incoming edges originated from the papers they have produced .a schematic representation of the proposed network is shown in figure [ explicative - figure ] .having defined the multilayer citation network , we propose to rank its nodes according to their pagerank versatility , which is given by the steady - state solution of the equation where is the time - dependent tensor that gives the probability to find a random walker at a particular node in a particular layer , and \mbox{,}\ ] ] being the number of nodes in the network , the number of layers , and the teleportation rate . denotes the rank tensor of transition probabilities for jumping between pairs of nodes and switching between pairs of layers , and is the rank tensor with all components equal to 1 .let be the eigentensor of the transition tensor , denoting the steady - state probability to find a random walker in node and layer .the tensor provides the pagerank of each node ( ) in each layer ( ) : it is crucial to remark here that this is not equivalent to calculate the pagerank in each layer separately , because our formulation accounts for the whole interconnected structure to solve the eigenvalue problem . to obtain the multilayer pagerank of each node ,regardless of the layer , we project the values obtained from its replicas in different layers , obtaining the multilayer pagerank vector where is the vector with all components equal to 1 .it has been shown  that this operation provides the same results that would be obtained by calculating pagerank by means of simulated random walkers that explore the multilayer structure according to transition rules encoded in .to illustrate the proposed ranking method , we test it on two case studies : the american physical society ( aps ) and the us patents datasets .  the first is a collection of papers published in the journals of the american physical society ( physical review letters , physical review and review of modern physics ) between 1985 and 2009 .we restricted the analysis only to papers with at most ten authors , to avoid biases due to the papers of experimental high - energy physics in which all the project collaborators are listed as co - authors . to disambiguate author s name, we used a simple technique introduced in previous studies  .meta - data in the dataset provide information about the topic of the papers through the specification of the assigned `` physics and astronomy classification scheme '' ( pacs ) code , developed by the american institute of physics ( aip ) and used in physical review since 1975 to identify fields and sub - fields of physics .we exploited this information to build a heterogeneous interconnected 10-layer network in which each layer represents a sub - field of physics , as defined by the pacs systems : _ general _ ; _ the physics of elementary particles and fields _ ; _ nuclear physics _ , _ atomic and molecular physics _ ; _ electromagnetism , optics , acoustics , heat transfer , classical mechanics , and fluid dynamics _ ; _ physics of gases , plasmas , and electric discharges _ ; _ condensed matter : structural , mechanical and thermal properties _ ; _ condensed matter : electronic structure , electrical , magnetic , and optical properties _ ; _ interdisciplinary physics and related areas of science and technology _ ; _ geophysics , astronomy , and astrophysics_. from the paper meta - data we also extracted the authors affiliation information , which allowed us to associate to each paper a list of ( one or more ) institutions and countries .the final dataset consists of 319816 papers , 204809 authors , 626 institutions and 54 countries .arguably , the aps dataset covers only physics , but note that physics is a vast field that spans from biological physics to astrophysics and although it may fall short of a full interdisciplinary analysis it is clear that this is a powerful indicator of multi - topic analysis that serves to proof the usefulness of the method .the second dataset contains the u.s .patents granted between january 1963 and december 1999 , and all citations made to these patents between 1975 and 1999 . to define the layers, we used the 6 categories proposed in previous studies  : _ chemical ( excluding drugs ) _ ; _ computers and communications _ ; _ drugs and medical _ ; _ electrical and electronics _ ; _ mechanical _ ; _ others_. each patent is assigned to one main class defined by the united states patent and trademark office ( uspto ) , and to any number of subsidiary classes .each class belongs to one of the listed categories , therefore each patent is associated with one or more layer according to its classes .however , the dataset only contains the information about the main class , therefore we complemented it by extracting the information about the other classes from the uspto patent grant full text .the final dataset contains 1574882 patents , 1142499 inventors , 138833 assignees ( i.e. corporations for the most part ) , and 127 countries .figure 2 shows the evolution of the interdisciplinary ranking of the world top physics departments , and of the world top companies , over time . this visualization allows to observe , for instance , the raise of the university of texas at austin during the 1990s , after the establishment , in 1985 , of the center for nonlinear dynamics , funded and directed by the boltzmann medal laureate harry swinney .compared to previously proposed algorithms of diffusion of scientific credit , the proposed method rewards researchers that have carried out interdisciplinary works or have had an impact in different scientific areas . to show this , we first compare it with the science author rank algorithm ( sara )  .we find that the rankings of aps authors obtained using sara and using the proposed method have a spearman s rank correlation of 0.77 ( confidence level ) .the high value of correlation is to be expected since both methods rank researchers simulating a diffusion process on a citation network .however , the proposed method gives higher ranking to versatile researchers such as the self - organised criticality pioneer per bak ( + 21 positions gained ) , econophysics co - founder eugene stanley ( + 56 ) , complex networks pioneer shlomo havlin ( + 104 ) , and complex systems professor leonard m. sander ( + 93 ) .we show that the proposed method is in fact able to capture two fundamental aspects of interdisciplinary research : intrinsic multidisciplinarity ( i.e. publishing papers or patents pertaining to different areas ) on the one hand , and effective interdisciplinarity , i.e. being credited by different scientific areas , on the other .we define the _ topical interdisciplinarity _ of an author ( who could be a scholar or an inventor ) as the average number of different scientific areas her / his publications pertain to , i.e. : where is the number of publications authored by , and is the number of fields that publication belongs to .moreover , for each publication we define an entropy metrics based on the distribution of its incoming citations across the different fields represented by the different layers : where the sum is over the different fields ( layers ) and is the proportion of edges incident in that come from layer . therefore if a publication is only cited by other publications belonging to its own field ,whereas a publication that has received citations from different fields has , and the higher the number of fields it has had impact on , the higher its entropy . for each author we then compute her / his _ citation interdisciplinarity _ as the average entropy of her / his publications :   we find a strong positive correlation between the gain in rank that scholars and inventors obtain when evaluated using the proposed method instead of a method based on a flat representation of the citation network , and their topical interdisciplinarity ( figure 3 , panels ( a ) and ( b ) ) .moreover , we find that the rank gain is also positively correlated with the disciplinary diversity of scholars and inventors incoming citations ( figure 3 , panels ( c ) and ( d ) ) . to control for the effects of productivity , i.e. the fact that a researcher that produces more papers has more chances to publish in more areas or to be cited by papers in many different areas , we perform the same analysis on two subsets of the data by considering in each case only authors with a fixed number of publications ( and ) .the correlation coefficients found in these subsets are consistent with those found using the whole dataset , demonstrating that the proposed method is not biased by productivity .the results are reported in the supplementary material .coefficient , and setting the statistical significance at .solid lines represent density gradient contours , and dashed lines represent linear regression models estimated via maximum - likelihood.,scaledwidth=50.0% ]in this paper , we propose a methodology to assess the citation impact of scientific publications and their producers that intrinsically accounts for their interdisciplinarity .this aspect was not included in previous citation impact indicators .  even though numerous metrics have been proposed to assess citation impact ,several issues have been raised .these include the accounting of self - citations  , the choice of the appropriate citation time window  , field normalisation  , and author credit allocation  . despite the vast literature on the subject, consensus is still lacking on how to solve these issues . here , we propose a method whose objective is to account for interdisciplinarity , and we do not enter those debates .however , the bipartite interconnected multilayer networks of citations and disciplines that we introduce can be adapted to take into account specific needs .for example , edges connecting papers to their authors could be weighted differently to take into account non - homogeneous allocation of credit , or a specific time window could be chosen a priori to select the papers constituting the network .going beyond the presented assessment of the benefits produced by interdisciplinarity , the method proposed in this work could constitute a tool for funding agencies and academic hiring decision makers to quantify the impact of interdisciplinary research and its producers , for a faster advancement of excellent science .a.a . , e.o . and m.d.d .acknowledge financial support by the spanish government through grant fis2015 - 38266 .m.d.d . also acknowledges financial support from the spanish program juan de la cierva ( ijci-2014 - 20225 ) .a.a . also acknowledges partial financial support from the european commission fet - proactive project multiplex ( grant no .317532 ) , icrea academia and james s. mcdonnell foundation .the idea of ranking scholars simulating a diffusion process was already introduced in  and previously in to rank scientific publications but the approach proposed in this work considers a different kind of network a bipartite interconnected multilayer network . in this sectionwe motivate the choice of taking into account the complete bipartite structure instead of its one - mode projection .  for the sake of simplicity, we will consider a bilayer version of the network .our focus here is not in fact the multilayer aspect of the network capturing interdisciplinarity , but rather its bipartition .therefore , let us consider nodes and 2 layers .the 4 components of the rank adjacency tensor are now defined as follows . encodes information about citing relations between papers , _ i.e. _ if paper cites paper . encodes information about paper authorship , _ i.e. _ if author is one of the authors of paper . finally , we define and to be zero tensors , consistently with the representation introduced in s.1 . for the sake of simplicity , since in the rest of the section we will be dealing only with rank-2 tensors , we will make use of the simpler classical matrix notation instead of the tensorial one . therefore we will denote as and as . in the author citation network proposed in ,each node represents an author , and if there exist at least one publication , of which is an author , that cites a publication of which is an author .each such publication gives a contribution to ( where is the number of authors of publication , and is the number of authors of ) so that the total contribution of each citation is equal to 1 .let us consider an adjacency matrix of size , encoding the citation links between papers . can be built from by means of multiplication with a rectangular matrix of size such that for , and all the other elements are equal to 0 .then using we can also build from the projection matrix , of size , where if is one of the authors of paper : let us define as the normalised version of , _i.e. _ ( where is the number of authors of paper ) , then the adjacency matrix representing the network of citations between authors can be obtained performing two successive matrix multiplications : _ proof : _ means that is a sum over the papers authored by that cite paper , where each paper gives a contribution of 1 over the number of authors .then : each element is therefore a sum over all the pairs of papers such that is an author of and of , and each element of the sum gives a contribution equal to , as indeed defined in .  we have demonstrated that the adjacency matrix of the author citation network is obtained performing two operations of matrix multiplication involving and .matrix multiplications consists in multiplications and summations of the matrix elements , which inevitably lead to information loss .the supra - adjacency matrix of the network proposed in this paper is instead the sum of the two expansions of and , _ i.e. _ and , respectively .this guarantees that no information is loss , and this why we chose to consider the whole bipartite structure .figure [ toy - model-1 ] shows an example in which the information loss characteristic of the author citation network leads to a less fair ranking of authors compared to the ranking based on the network introduced in this paper . using our approach ( top figure ), the most central author is , who is the author of both the most central paper and of one of the second most central papers .however , in the author citation network framework , the most central author is ( to understand why , we recall that the pagerank centrality is based not only on the number of incoming edges , but on the importance of the nodes from which these edges originate ) .this is due to the fact that in the author citation network all the information about an author s incoming citations from different papers is aggregated , and therefore benefits from the importance of without any distinction between the importance coming from papers that actually cite , and that coming from s other papers . in this case most central ( and cited ) paper is not the one citing s paper , but this information is lost in the author citation network . as a consequence , the resulting ranking does not always reflect the real importance of the different authors .an alternative approach is to use the pagerank method to get the centrality of papers in , and then compute the author centrality as a properly normalised sum of the centralities of the papers she / he has authored . in matrix terms , the author pagerank centrality vector can be obtained by simply applying a linear transformation to the paper pagerank centrality vector : however , this solution involves another kind of aggregation which can lead to misleading results too .an example is shown in figure [ toy - model-2 ] . using the sum approach ,author becomes more central than author , because she / he authored two papers , even though they are the two most marginal papers in the network ( note that the only citation to paper 4 is a self - citation ) . on the contrary , is the author of a very central paper , and in fact our approach correctly classifies her / him as more central than .the issue with this alternative approach is that the pagerank is a diffusion process , which is not a linear dynamics .therefore summing over the centralities of different nodes is also an aggregation process through which some information on the system is lost .in figure 3 of the main text , we show that we find a strong positive correlation between the gain in rank that scholars and inventors obtain when evaluated using the proposed method instead of a method based on a flat representation of the citation network , and their topical interdisciplinarity ( panels ( a ) and ( b ) ) .moreover , we show that the rank gain is also positively correlated with the disciplinary diversity of scholars and inventors incoming citations ( panels ( c ) and ( d ) ) . to control for the effects of productivity , i.e. the fact that a researcher that has produces more papers has more chances to publishes in more areas or to be cited by papers in many different areas , we perform the same analysis on two subsets of the data by considering in each case only authors with a fixed number of publications . figure [ control_20papers ] shows the result for the subset of authors and inventors with publications , and figure [ control_50papers ] for authors and inventors with publications. the results are consistent with those obtained using the whole dataset .publications obtain when evaluated using the proposed method instead of a method based on a flat representation of the citation network , and two measures of their interdisciplinarity level .the x - axis represents , in panel ( a ) and ( b ) , scholars and inventors topical interdisciplinarity , defined as the average number of different scientific areas their publications pertain to , and , in panel ( c ) and ( d ) , their diversity in terms of disciplines of the scholars and inventors incoming citations ( citation interdisciplinarity ) .correlations are calculated using pearson s coefficient , and setting the statistical significance at .solid lines represent density gradient contours , and dashed lines represent linear regression models estimated via maximum - likelihood.,scaledwidth=50.0% ]   publications obtain when evaluated using the proposed method instead of a method based on a flat representation of the citation network , and two measures of their interdisciplinarity level .the x - axis represents , in panel ( a ) and ( b ) , scholars and inventors topical interdisciplinarity , defined as the average number of different scientific areas their publications pertain to , and , in panel ( c ) and ( d ) , their diversity in terms of disciplines of the scholars and inventors incoming citations ( citation interdisciplinarity ) .correlations are calculated using pearson s coefficient , and setting the statistical significance at .solid lines represent density gradient contours , and dashed lines represent linear regression models estimated via maximum - likelihood.,scaledwidth=50.0% ]    bornmann , lutz , mutz , rdiger , & daniel , hans - dieter .are there better indices for evaluation purposes than the h index ?a comparison of nine different variants of the h index using data from biomedicine ., * 59*(5 ) , 830837 .de domenico , manlio , sol - ribalta , albert , cozzo , emanuele , kivel , mikko , moreno , yamir , porter , mason a , gmez , sergio , & arenas , alex .. mathematical formulation of multilayer networks ., * 3*(4 ) , 041022 .glnzel , wolfgang , debackere , koenraad , thijs , bart , & schubert , andrs .( 2006 ) . a concise review on the role of author self - citations in information science , bibliometrics and science policy . ,* 67*(2 ) , 263277 .wagner , caroline s , roessner , j david , bobb , kamau , klein , julie thompson , boyack , kevin w , keyton , joann , rafols , ismael , & brner , katy .approaches to understanding and measuring interdisciplinary scientific research ( idr ) : a review of the literature ., * 5*(1 ) , 1426 .

Abstract: nowadays , scientific challenges usually require approaches that cross traditional boundaries between academic disciplines , driving many researchers towards interdisciplinarity . despite its obvious importance , there is a lack of studies on how to quantify the influence of interdisciplinarity on the research impact , posing uncertainty in a proper evaluation for hiring and funding purposes . here we propose a method based on the analysis of bipartite interconnected multilayer networks of citations and disciplines , to assess scholars , institutions and countries interdisciplinary importance . using data about physics publications and us patents , we show that our method allows to reward , using a quantitative approach , scholars and institutions that have carried out interdisciplinary work and have had an impact in different scientific areas . the proposed method could be used by funding agencies , universities and scientific policy decision makers for hiring and funding purposes , and to complement existing methods to rank universities and countries .


in -body simulations of stellar systems the gravitational interaction is modified for curing the newtonian divergence at short distances .basically , such modifications are introduced through a soft cut - off : the softening length .however , the precise form in which they are implemented can vary . since gravity plays a fundamental role in these systems and the gravitational interactionis modified precisely where it becomes singular , the dynamical effects of softening should be well understood when designing experiments and interpreting their results .this dynamical problem has recently stimulated considerable interest ( e.g. , hernquist & barnes 1990 ; hernquist & ostriker 1992 ; kandrup et al .1992 ; pfenniger 1993 ; pfenniger & friedli 1993 ; gurzadyan & pfenniger 1994 ; romeo 1994 , hereafter paper i ; byrd 1995 ; gerber 1996 ; merritt 1996 ; weinberg 1996 ; sommer - larsen et al . 1997; theis 1997 ; see also goodman et al .1993 ; farouki & salpeter 1994 ) . for extensive overviewssee the above - mentioned pfenniger & friedli ( 1993 ) , gurzadyan & pfenniger ( 1994 ) and paper i.  in paper i we have investigated the stability problem in the case of 2-d models with plummer softening , which are commonly employed in simulations of disc galaxies .the basic message is that the effect of softening becomes strongly artificial for , being the typical radial wavelength , which means half an order of magnitude below the expected value .the major results are summarized in the form of a criterion of approximate physical consistency for and a stability criterion for the toomre parameter .( other important aspects of the stability problem have been considered by byrd 1995 . )  in the present paper we carry out five extensions , as is discussed below .  1 .we generalize the stability analysis of paper i to an _ arbitrary _ isotropic form of softening .this is a natural extension since types of softened gravity different from the standard plummer softening are becoming more and more commonly employed ( e.g. , combes et al .1990 ; palou et al .1993 ; shlosman & noguchi 1993 ) . in particular , the alternatives proposed by hernquist & katz ( 1989 ) and pfenniger & friedli ( 1993 ) reflect an interesting idea , viz .that softening should be as localized as possible since there is no clear reason for modifying the gravitational interaction at long distances , and it is tempting to explore its dynamical consequences . from a more general point of view , this and the following extensions provide the tools for comparing experiments that employ different types of softened gravity .we investigate the implications of our stability analysis for the classical relaxation problem ( rybicki 1972 ; white 1988 ) .relaxation and stability are intimately related in self - gravitating systems , and even simple treatments reveal their strong coupling through random motions . on the other hand ,the contribution of velocity dispersion to the relaxation time has been understood only in part and , because of that , the classical argument favouring the choice of large values of is wrong .we revise this argument and conclude that neither small nor large values of are convenient .surprisingly , there exists an intermediate choice of that _ optimizes _ the ` dynamical resolution ' of the model , i.e. its faithfulness in simulating the dynamics of 3-d discs with newtonian gravity , especially in situations near to the stability threshold .we identify the optimal characteristics , and show how to evaluate them for a given type of softened gravity .in addition to investigating this aspect of the relaxation problem , we explain how effectively softening reduces noise on various scales .3 .  we complete the examination of 2-d models with isotropic softening by investigating the equilibrium problem for an axisymmetric state with epicyclic motions .in particular , we explain how significantly the circular speed and related quantities deviate from their newtonian behaviours at various distances from the centre .we consider 3-d models with isotropic softening and examine two limiting cases : discs and the simple , yet instructive , jeans problem .an extension to 3d has been encouraged by friedli ( 1994 ) and junqueira & combes ( 1996 , see the interesting remarks in sect .real stellar systems have several gravitationally interacting components . both -body simulations and theoretical worksare forced to use simplified models , which do not necessarily provide faithful representations of the complexity of such systems .our motivation is to understand the basic differences between the dynamical effects of softening in 3d and 2d , in the presence of a single stellar component ( 3-d vs. 2-d discs and jeans problem vs. discs ) . in particular , we point up the _ strong _ modifications introduced by a homogeneous geometry and the absence of rotation . 5 .we complete the examination of 3-d models by discussing the basic dynamical effects of softening anisotropy .the idea underlying this extension is that softening should be anisotropic in simulations of stellar systems where significantly higher spatial resolution is required along a certain direction , such as in disc galaxies .the alternative family of softening recently proposed by pfenniger & friedli ( 1993 ) reflects such an important idea , and it is tempting to explore the dynamical relations between its members .( an analogous idea has been discussed in the context of smoothed particle hydrodynamics by shapiro et al .1994 , 1996 and fulbright et al . 1995 ; hernquist , private communication , has remarked that in models with anisotropic smoothing there may be a significant tendency for angular momentum not to be conserved . )these extensions all together form a method for exploring the dynamical effects of softening in -body simulations of stellar systems .our method is described in sect . 2 , and is structured as in the previous discussion .the two applications mentioned in the same context are shown in sect . 3( see also appendix a ) .the conclusions of this paper are drawn in sect .4 , where we present our contribution in a more general perspective and motivate future applications .in 2-d discs with isotropic softened gravity , a given surface - density perturbation induces a potential perturbation where is the point - mass potential , which depends on the softening length [ is also known as the softening kernel ] . in analysing the poisson equation ( 1 ), we adopt the lowest - order wkbj approach , as in paper i , but with a different albeit asymptotically equivalent spectral representation . in simple terms , the new feature consists in considering perturbations with a radial dependence rather than , where is the radial wavenumber and denotes the bessel function of the first kind and order ( see , e.g. , abramowitz & stegun 1972 ) .the bessel - hankel representation is more convenient than the fourier representation because it allows factorizing the convolution in the poisson equation directly , without any assumption on the form of : (k)\,\,\tilde\sigma_1\ , , \ ] ] where denotes the hankel transform of order ( see , e.g. , sneddon 1972 ; bracewell 1986 ) : (k)=\int_0^\infty\!g(r)\,{\rm j}_\nu(kr)\,r\,{\rm d}r\ , .\ ] ]  [ [ the - reduction - factor . ] ] the reduction factor .+ + + + + + + + + + + + + + + + + + + + +  equation ( 2 ) admits of a simple interpretation : the potential perturbation induced by a given surface - density perturbation is weakened by a factor (k ) }           { { \cal h}_0[\varphi_{\rm n}(r)](k)}\ , , \ ] ] being the newtonian point - mass potential .correspondingly , the contribution of self - gravity to the dispersion relation is weakened through a reduction of the active unperturbed surface density by the same factor .thus , provides complete information about the effect of isotropic softening on the stability of 2-d discs , and its evaluation is the starting - point of our method .there are two complementary routes to .one is to calculate it analytically from : (k)\ ] ] ( use , e.g. , the comprehensive table of integrals by gradshteyn & ryzhik 1994 ) .this equation follows directly from eq .( 4 ) , being and (k)=1/|k| ] . the relative quadratic deviations of and from their newtonian behaviours depend on the model of mass distribution and on , being the disc scale length . in order to estimate the magnitude of these deviations, we can set and find that , which is strictly valid for [ the approximate formula for the circular speed is : .more important are the deviations that result near the centre .they imply a change in the number and/or location of the inner lindblad resonances , for a given pattern speed , together with larger natural scales for , , , and .the contribution of a massive bulge to the rotation curve makes the system more robust against such modifications , but in certain respects this modelling is unclear and we are not yet in a position to draw quantitative conclusions ( cf .bertin 1996 ; bertin & lin 1996 ; junqueira & combes 1996 , sect .llllllllllllll & &  & &  & + & abbr . & & & & & & & & & & & & + plummer & p & & 0.03 & 0.12 & 0.25 & 0.20 & 0.37 & & 1.70 & 0.05 & 0.39 & 1.00 & 0.37 + cubic spline & cs & & 0.06 & 0.23 & 0.44 & 0.45 & 0.61 & & 0.99 & 0.11 & 0.30 & 0.44 & 0.46 + homogeneous sphere & hs & & 0.07 & 0.27 & 0.52 & 0.53 & 0.71 & & 0.83 & 0.12 & 0.30 & 0.38 & 0.47 + homogeneous oblate spheroid & hos & & 0.18 & 0.55 & 0.94 & & 1.21 & & 0.44 & 0.32 & & 0.04 & 0.61 + homogeneous prolate spheroid & hps & & 0.03 & 0.11 & 0.24 & 0.18 & 0.37 & & 1.73 & 0.05 & 0.39 & 1.13 & 0.33 +  values corresponding to one half the optimal relaxation level .  optimal choice .safety threshold for approximate physical consistency.  critical value .correction factor for the rybicki - white relaxation time .optimal relaxation level .temperature anisotropy corresponding to the stability threshold for the optimal value of the effective thickness parameter .conversion factor , being the effective thickness parameter ( below the safety threshold for approximate physical consistency ) .critical radial wavelength .hos has and hps has , where and are the planar and vertical softening semi - axes , respectively , and is regarded as the softening length of reference .the dynamics of 3-d discs with isotropic softened gravity is difficult to investigate because the effects of softening combine with those of vertical random motion in a complicated form .nevertheless , we do in part understand how to benefit from the additional degree of freedom introduced into such models .a simple scale argument suggests that we should choose values of sufficiently smaller than the newtonian characteristic scale height , otherwise softening would significantly affect the vertical structure at equilibrium , i.e. _ both _ the mass distribution _ and _ the thickness scale .likewise , the 2-d stability analysis suggests that in 3d we should choose , otherwise softening would significantly interfere with the effect of thickness , as far as density waves are concerned .note that choices of have the same consequences as in 2d , since the critical stability characteristics do not depend on the temperature anisotropy .we mention that the vertical structure at equilibrium and the stability of 3-d discs with newtonian gravity have been investigated in previous papers ( romeo 1990 , 1992 ; see also paper i ) .the 3-d jeans problem with isotropic softened gravity can be solved by using the technique of fourier transforms ( see , e.g. , sneddon 1972 ; bracewell 1986 ) . in cartesiancoordinates , this is the natural approach for factorizing the convolution in the poisson equation and investigating the dynamical effects of softening , as pfenniger & friedli ( 1993 ) have previously emphasized . in the following discussion we concentrate on the reduction factor and compare the jeans problem with the dynamics of discs .the reduction factor for the volume density can be calculated analytically from : (|\vec{k}|)\ , , \ ] ] where is the wavevector of the perturbation and denotes the fourier sine transform : (|\vec{k}| ) = \int_0^\infty\!g(r)\,\sin(|\vec{k}|r)\,{\rm d}r\ , .\ ] ] equation ( 21 ) follows from the definition of and the spherical symmetry of , which allows expressing the 3-d fourier exponential transform in terms of .two alternative equations that are more convenient for numerical computation can be obtained by first singling out the newtonian behaviour of at long distances , and then integrating by parts ( the integrand can be split in two ways ) . as regards the magnitude of , at small and at large .the stability properties are weakly affected by softening . in particular , significant deviations of the marginally stable wavelength from the jeans wavelength would occur for values of comparable to . for example, in the case of plummer softening this can be shown by using the simple numerical approximation for [ the analytical formula for is reported in appendix a ] .weinberg ( 1993 ) has shown that in analogous models with newtonian gravity the dominant source of noise is represented by fluctuations on scales of stability interest .this fact and our stability analysis suggest that the collective relaxation properties are also weakly affected by softening .thus , there is a _sharp _ contrast between the jeans problem and the dynamics of discs : a flattened geometry strengthens the effects of softening , and rotation makes them critical for values of that are an order of magnitude below .the dynamical properties of 3-d discs are roughly decoupled parallel and perpendicular to the plane , and a satisfactory modelling with isotropic softening may impose significantly different requirements on .pfenniger & friedli ( 1993 ) have discussed an ingenious way of coping with this difficulty , viz . to introduce a further degree of freedom into such models : softening anisotropy .we assume that the point - mass potential is of the form , and depends on the planar and vertical softening lengths and , respectively .the particular form in which softening anisotropy is implemented defines and , and specifies their meaning in the context of the dynamical requirements .the 3-d jeans problem shows the basic dynamical effects of softening anisotropy .the reduction factor may be evaluated from ( cf .2-d discs with isotropic softened gravity ) : (k_\perp)\ , , \ ] ] where is an abbreviation for (k_\parallel)$ ] and denotes the fourier cosine transform .this equation follows from the axial and planar symmetries of , which allow expressing the 3-d fourier exponential transform in terms of and .supposing that , then we expect that for , which means that the stability and collective relaxation properties are more affected parallel than perpendicular to the plane of reference .as a first application of our method , we compare the dynamical effects of three isotropic types of softened gravity : plummer softening ( aarseth 1963 ; miller 1970 ) , cubic - spline softening ( hernquist & katz 1989 ) and homogeneous - sphere softening ( pfenniger & friedli 1993 ) ; hereafter abbreviated to p , cs and hs , respectively .each name means that the softened point - mass potential can be viewed as the newtonian potential generated by a spherical mass distribution of that type ( but the force is evaluated regarding the test particle as a point mass ) . in the following discussionwe do not refer to this finite - sized particle interpretation of softening , unless otherwise specified .the behaviours of these types of softened gravity are shown in fig . 2 ( left ) . in pthe gravitational interaction is softened mainly at short distances and the newtonian behaviour is recovered asymptotically , whereas in cs and hs softening is perfectly localized , i.e.beyond a certain cut - off distance the gravitational interaction is identically newtonian . the form in which localization is implemented differs in the two types .in particular , cs has a rather soft cut - off at , being the nominal softening length , whereas hs has a sharper cut - off at .the results of the dynamical comparison are presented in fig. 3 ( left ) and table 1 ( top ) , which are discussed below , and in figs . 4 ( left ) and 5 .useful analytical formulae are reported in appendix a.  figure 3 ( left ) shows that the effective scale height varies significantly with softening type , but apart from that the stability properties are analogous on scales larger than .this can be shown by rescaling in terms of and noting that is similar in the three types for . for , changes sign and starts to oscillate , in contrast to .an analogous behaviour occurs in , but is less noticeable .negative values of mean that a given surface - density perturbation is in phase with the induced potential perturbation and , correspondingly , that the contribution of self - gravity to the dispersion relation is stabilizing .however , this feature does not affect the stability properties on scales comparable to the inverse of the typical radial wavenumber , since changes sign well beyond the critical point [ cf .table 1 ( top ) ] .the related oscillations of mean that in certain ranges of noise is suppressed more effectively on larger than smaller scales , and correspond to oscillations of the short - wave branch of the dispersion relation .on the other hand , this feature has not significant consequences for the relaxation properties , because the corrected optimal relaxation level is approximately the same as in p [ cf .table 1 ( top ) ] .table 1 ( top ) shows that the stability and relaxation characteristics vary significantly with softening type .the largest variations occur between hs and p , concern and , and are up to a factor of 3 .in contrast , the corrected optimal relaxation level , and remain approximately constant .the variations between hs and cs are within 20% .   as a second application of our method, we consider an anisotropic generalization of hs , the family of homogeneous - ellipsoid softening ( pfenniger & friedli 1993 ) , and compare the dynamical effects of two representative spheroidal members : one oblate with , and being the planar and vertical softening semi - axes , and the other prolate with ; hereafter abbreviated to hos and hps , respectively . regarding as the softening length of reference, gives a measure of softening anisotropy .the behaviours of these types of softened gravity are shown in fig . 2 ( right ) .anisotropy is implemented in a form consistent with the finite - sized particle interpretation of softening , i.e. through a spheroidal deformation of the field particle .this mainly corresponds to a spheroidal transformation of the surface on which the force peaks , but it also strongly influences the behaviour in the plane .in particular , the degree of softening localization , the sharpness and magnitude of the force peak differ from those in hs [ cf.fig . 2 ( left ) ] , and are higher in hos than hps .so a 2-d analysis is required even though the isotropic case has already been investigated .the results of the dynamical comparison are presented in fig . 3 ( right ) and table 1 ( bottom ) , which are discussed below , and in fig . 4 ( right ) .the major points are then generalized in the final discussion .useful analytical formulae are reported in appendix a.  figure 3 ( right ) shows that , for , is concave , in contrast to .the transition behaviour occurs in [ cf .fig . 3 ( left ) ] .concavity of mainly means that softening has a stronger tendency to cause artificial stabilization for a given effective scale height , for we know that the reduction factor of 3-d discs with newtonian gravity is convex ( cf.paper i , fig . 1 ) .in fact , hos does not mimic the effect of thickness for realistically large values of , since the safety threshold for approximate physical consistency and the optimal value are ill - defined [ cf .table 1 ( bottom ) ] .concavity of also means that softening causes a moderate degree of ` redshift ' . for , and analogous to , and analogous conclusions can be drawn concerning the stability and relaxation properties .table 1 ( bottom ) shows that the stability and relaxation characteristics differ considerably in the two types .the major qualitative differences concern and , and have been pointed out above [ cf .discussion of fig . 3 ( right ) ] .excluding , which has a restricted range of applicability in hos , the largest variation concerns and is beyond half an order of magnitude .in contrast , note the low sensitivity of the corrected optimal relaxation level , which remains roughly constant .finally , instead of comparing those results with the isotropic case , let us clarify the dynamical relations between _ all _ the spheroidal members of the family of homogeneous - ellipsoid softening and the standard type of softened gravity [ cf .table 1 ( top ) ] .  1 .a general feature of the spheroidal members is that is determined by alone : .this is not intuitive since a more natural softening length in the plane is expected to be .2 .  all the oblate members with have an ill - defined , i.e.  , and thus differ fundamentally from p ( cf .3 .  as increases ,such a difference becomes progressively less important ( cf .hs and hps ) .indeed , the prolate member with , i.e.  , turns out to be hardly distinguishable from p. ( values of are not employed in simulations . )modelling gravity is a fundamental problem that must be tackled in -body simulations of stellar systems , and satisfactory solutions require a deep understanding of the dynamical effects of softening .this problem has deserved special attention both in the past ( e.g. , miller 1970 , 1976 ) and in more recent times ( e.g. , efstathiou et al . 1985 ; hernquist & katz 1989 ; pfenniger & friedli 1993 ) .viewed in this general perspective , our contribution has a threefold practical importance in addition to the points emphasized in sect . 1 .  1 .  the two present applications of our method reveal the dynamical differences between the most representative types of softened gravity : the isotropic p , cs and hs ( abbreviated as in sect .3.1 ) , and the anisotropic hos and hps ( abbreviated as in sect .the major conclusions concerning their dynamical resolution , i.e. their faithfulness in simulating the newtonian dynamics , are summarized below . 1 .as regards the isotropic types , the dynamical resolution is comparable .this results from the fact that , even though the spatial resolution and the effectiveness in reducing noise differ significantly in p , cs and hs for the same nominal softening length , those differences can largely be removed by considering a more appropriate softening length of reference .as regards the anisotropic types , the dynamical resolution is significantly coupled parallel and perpendicular to the plane . in the plane , it decreases in quality from hps to hos , and the transition occurs in the oblate members for a softening axial ratio ( hps is dynamically similar to p ) .these disadvantages result from the finite - sized particle implementation of softening anisotropy . on the other hand, they have less importance than the advantage of introducing such a degree of freedom into 3-d simulations of disc galaxies , which has been emphasized by pfenniger & friedli ( 1993 ) and in our method .3 .  last but not least , when employing these types of softened gravity in simulations of disc galaxies , we should recall that the dynamical resolution depends _ critically _ on two quantities : , or for a given , and the safronov - toomre parameter ( cf .paper i ) .the choice of or should be checked vs. the profiles of the characteristic values , and , which are tabulated in the applications . the choice of should be checked vs. the profiles of the stability threshold and level , which can be evaluated as is explained in the method .our method can be applied for testing new ideas about softening .there are two features that encourage such future applicationsone is the _ unified _ approach adopted for investigating stability , relaxation and equilibrium . as a result , full information about the dynamical effects of softening is contained in a single quantity : the reduction factor .2 .  the other is the _ modular _ structure of the method .we describe step by step how to extract detailed information concerning the dynamical properties , starting from and pointing out the quantities of major interest .3 .  but our method can be applied in another , more fruitful , way : for developing new ideas about softening . indeed , it opens a _ direct _ route to the discovery of optimal types of softened gravity for given dynamical requirements , and thus to the accomplishment of a physically consistent modelling even in the presence of a cold interstellar gaseous component. such a future application will be the objective of a ` twin ' paper .this paper is dedicated to my parents francesco and grazia .it is a great pleasure to thank the referee , daniel pfenniger , whose papers about softening have represented an invaluable source of inspiration for both developing and maturing my ideas . in addition , i am very grateful to francoise combes and lars hernquist for strong encouragement and valuable suggestions on a previous draft of this paper .i am also very grateful to john black , daniel friedli , cathy horellou , henry kandrup , richard miller , juan muzzio , masafumi noguchi and jan palou for strong encouragement and useful discussions . the financial support of the swedish natural science research council is greatly appreciated .           aarseth s.j . , 1963 ,mnras 126 ,223 abramowitz m. , stegun i.a . , 1972 , handbook of mathematical functions with formulas , graphs , and mathematical tables .dover , new york bender c.m ., orszag s.a . , 1978 , advanced mathematical methods for scientists and engineers .mcgraw - 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( eds . ) , 1994 , ergodic concepts in stellar dynamics .springer - verlag , berlin hernquist l. , barnes j.e ., 1990 , apj 349 , 562 hernquist l. , katz n. , 1989 , apjs 70 , 419 hernquist l. , ostriker j.p ., 1992 , apj 386 , 375 hockney r.w ., eastwood j.w . , 1988 ,computer simulation using particles .adam hilger , bristol junqueira s. , combes f. , 1996 , a&a 312 , 703 kandrup h.e . , smith h. jr . , willmes d.e . , 1992 ,apj 399 , 627 masset f. , tagger m. , 1996 , a&a 307 , 21 merritt d. , 1996 , aj 111 , 2462 miller r.h . , 1970 , j. comput .phys . 6 , 449 miller r.h . , 1976 , j. comput .21 , 400 palou j. , jungwiert b. , kopeck j. , 1993 , a&a 274 , 189 pfenniger d. , 1993 , order and chaos in -body systems . in : combes f. ,athanassoula e. 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Abstract: two questions that naturally arise in -body simulations of stellar systems are :  1 .  how can we compare experiments that employ different types of softened gravity ? 2 .  given a particular type of softened gravity , which choices of the softening length optimize the faithfulness of the experiments to the newtonian dynamics ?   we devise a method for exploring the dynamical effects of softening , which provides detailed answers in the case of 2-d simulations of disc galaxies and also solves important aspects of the 3-d problem . in the present paper we focus on two applications that reveal the dynamical differences between the most representative types of softened gravity , including certain anisotropic alternatives . our method is potentially important not only for testing but also for developing new ideas about softening . indeed , it opens a _ direct _ route to the discovery of optimal types of softened gravity for given dynamical requirements , and thus to the accomplishment of a physically consistent modelling .


the influence of space charges in proportional counters has been studied theoretically and experimentally and is described in various articles . there , the main emphasis was put on the drop of the gas gain which is important for measuring charges .+ for drift chambers in which no charge measurement is foreseen , the gain drop is of secondary importance and the drift time is the relevant information .variations in the drift time due to a disturbed electric field lead to a loss in the spatial resolution .+ this work was done in the context of the development of the atlas muon spectrometer where high background rates coming from neutrons and photons are expected .the detector should still work at background rates of 500hz/ ( 5 times the expected background) with a spatial resolution for a single tube better than 100 m .+ the atlas muon spectrometer will be built from 3 cm diameter drift tubes with a 50 m wire in the middle .the gas pressure is raised to 3bars absolute in order to reach the desired spatial resolution at the nominal gas gain of .if not stated otherwise , all the measurements and calculations described here refer to these operating conditions .+ important criteria for the choice of the gases are given by the need of a non - flammable gas with a maximum drift time well below 1 .up to an accumulated charge deposition of 0.6c per cm wire , ageing effects should be excluded . +the outline of this article is as follows : in section2 a calculation of the space charge effects is shown , section3 gives a description of the experimental setup in the test beam .section 4 shows how the data readout and analysis was done .section5 gives results of the gas gain reduction , followed by the treatment of the changes in the drift time in section6 .the event - to - event fluctuations which are responsible for the irreducible loss of the spatial resolution are described in section7 .the electric field inside a drift tube with radius and wire radius held at potential is given by : this formula is only correct if one can neglect the positive ions that are produced in the avalanche processes .they drift towards the cathode and disturb the electric field .they screen the positive potential at the wire and lead to a reduction of the electric field near the wire .thus one expects a lower gas gain .since the total voltage between wire and tube is kept constant , the electric field at large radii is increased if the field near the wire is decreased .one expects a change in the drift time because the drift velocity for electrons depends on the electric field .+ the density of the space charges coming from positive ions can be calculated assuming a homogeneous irradiation within one tube , thus neglecting all effects of event - to - event fluctuations .these fluctuations will not change the mean value of the charge density but affect the resolution as will be shown in the last section . +the drift velocity of the positive ions is proportional to the reduced electric field where is the gas pressure , the mobility of the gas and the electric field is given by ( [ eq : ef ] ) .the maximum ion drift time ( which is the drift time for almost all ions because nearly all of them are produced at the wire ) is obtained by integrating ( [ eq : mue ] ) , note that for the calculation of the above formula the electric field ( [ eq : ef ] ) of the undisturbed tube is used . for high rates , the electric field will change and the maximum ion drift time has to be corrected , as will be shown below .+ from ] one can see that the density of the ions is independent of the radial distance from the tube centre in case of a cylindric tube geometry for a homogeneous irradiation .since the density of the ions is : with = the number of primary ion pairs ( the number of pairs before gas amplification ) , is the gas amplification factor and is the particle rate per wire length .the potential can now be calculated with poisson s equation and the boundary conditions and . \end{aligned}\ ] ] thus \nonumber  \end{aligned}\ ] ] this formula has to be taken instead of ( [ eq : ef ] ) .+  figure [ fig = efield ] ( top ) shows an example of the electric field inside a tube with and without the addition of space charges coming from a muon rate of 3000 hz/ . a 3 cm diameter tube with a 50 m diameter wireis operated at a gain of at a pressure of 3bar with the gas ar - ch-n 91 - 5 - 4% .one can see that the field for drifting electrons is changed .the difference plot ( bottom ) shows clearly that the field is reduced in the region near the wire , but increased for radial distance mm .as the drift velocity depends on the electric field , the time to drift a certain distance will be different : the r - t relation changes with rate .the magnitude of this effect depends on the variation of the drift velocity with e , which is strongly dependent on the gas . + for calculations of the gas gain , only the field near the wire has to be known . in this case, the above formula can be simplified to the electric field in the avalanche region behaves as if the effective voltage at the wire was .+ taking now a model for the gas amplification where the gas gain is a function of the applied voltage , one should be able to describe the dependence of the gas gain reduction as a function of the rate .we will use the diethorn formula  } \ \ .\label{eq : gain}\ ] ]  the diethorn model has two parameters and which are characteristic for every gas mixture . is the potential difference an electron passes between two successive ionisations and is the critical value of where ionisation starts . is the ratio of the actual pressure to the reference pressure where k was determined .+ for calculating the actual gas gain at rate , one now has to replace by in ( [ eq : gain ] ) where is given by ( [ eq : dv ] ) . depends on the gain and vice versa .the two equations have been solved in an iterative way to get a self consistent gain for every rate .+ one more correction has to be made for high rates : it was ignored in ( [ eq : t+ ] ) that the ion drift time depends on the rate . for a better approximation of in ( [ eq : dv ] ) , one can insert into ( [ eq : mue ] ) the electric field ( [ eq : newfield ] ) which leads to the formula }  \label{eq : newt+ } \  .\ ] ] in the limit of rate , this is identical to ( [ eq : t+ ] ) .the change of with rate is plotted in fig.[fig = iondrift ] for the gas ar - ch-n 91 - 5 - 4 % . +to measure effects of high rates , a specially designed chamber was built for operation in the m2 muon beam at cern .the task of this setup was to see only effects coming from space charges and to avoid measuring electronic effects such as saturation of preamplifiers , baseline shifts etc . a schematic drawing and a photoare shown in figs .[ fig = schematic ] and [ fig = photo ] , respectively .the chamber consists of 24 drift tubes : two bundles of eight tubes with 3 cm diameter , two tubes with 2 cm and two tubes with 4 cm diameter .the tubes are glued together precisely , and the two bundles are staggered to improve track fitting . + for measuring the second coordinate , four tubes are placed perpendicular to the others .all tubes are 30 cm long , but the active length is only 4 cm in the middle part of the tubes for the precision coordinate and 10 cm for the second coordinate ( fig . [fig = schematic ] bottom ) . in order to passivate the rest of the tube ,the wires have been coated galvanically with silver to increase the wire diameter from 50 m to 150 m in order to avoid gas amplification .the trigger is made of a coincidence of four scintillation counters which cover an area of 1 cm ( along the wires ) 6 cm (= two tubes in height ) and are placed in the middle of the active part of the tubes .+ the whole setup can be moved horizontally by a stepper motor within a range of 1 m from the centre of the muon beam . by moving the apparatus with respect to the beamone can adjust the trigger rate from 10hz/ up to more than 10khz/ .whereas in atlas the tubes will be read out with tdcs , here the full analog pulse information was read out with 250 mhz flash adcs .this is necessary for measuring pulse height reductions and to be independent of a fixed discriminator threshold .+  an important property of the data acquisition was the ability to record at least 8 successive tracks without dead time .this is necessary for the calculation of time distances between consecutive hits from which the particle rate is calculated .it is also used to observe effects coming from event - to - event fluctuations where the `` history '' of the tube has to be known .thus it can be studied how previous hits influence the properties of the detector for the following hits .+ in order to achieve this readout without dead time , a trigger logic has been set up that does not need any readout cycle between the first and eighth event trigger . the memory depth of the flash adc is 2048 bytes per channel corresponding to 8192 ns in time which is large enough to store the waveforms of 8 pulses .the digitization of the flash adcs is started for each new muon trigger and stopped after 1 .+  the time differences between consecutive triggers were measured with a setup of slow tdcs ( clock ) covering a time range of 20 ms . with this information it is possible to compute for every tube the time differences to the last hit .the distribution of these times is exponential ( fig .[ fig = zeitvert ] ) .the mean value is , in agreement with the number of triggers per second counted with a scaler . with this measurementswe know both , the time difference between two consecutive hits and ( from the second coordinate measurements ) the spatial distance of the muon tracks along the wire . +a typical size for the data that had to be read out after every cycle ( 8 events ) was 40 kb .the data were processed by a vme processor operated under os9 and stored on an exabyte tape . to maximize the readout speed , all time critical routineshave been written in assembler code .the maximum readout rate that could be achieved ( if the particle rate was high enough ) was events within one spill of the cern sps accelerator ( 2.4s beam followed by a pause of ) and was limited by the speed of the tape drive .the time needed to take a high statistics run ( 800.000 triggers ) at highest rates was 4 hours .+ a slow control system was installed to monitor the relevant parameters such as temperature , gas flow , gas mixture , pressure and the high voltages of the tubes .+ the measurement of drift times with this kind of flash adc needs a special setup .the time difference between incoming start signal and start of the digitization can vary randomly between 0 and 12ns . in order to get a precise time marker ,the trigger signal starting the readout was delayed by 800 ns and fed into the test input of the preamplifiers fbpanik-04 .thus , every drift chamber pulse was followed by a reference pulse coming always with the same delay with respect to the muon trigger .the time difference between the pulse and the reference pulse is the drift time plus a constant offset t .+ the time bins of the flash adcs have a width of 4 ns . to get a better time resolution of the pulse , the dos ( difference of samples ) method was used which was developed for the analysis of the flash adc data of the opal experiment .this method is numerically rather simple but very efficient in both finding pulses and calculating the drift time with an accuracy better than 1ns .first , the pulse shape is differentiated by subtracting the pulse heights of successive bins .then pulses can be found if the differentiated pulse reaches a certain threshold .finally the drift time is calculated as the weighted mean of the differentiated pulse around the position of the maximum of the differentiated pulse . + an event display of a muon passing the chamber is shown in fig .[ fig = eventdisplay ] . +in order to measure the correct gain drop that is only due to particle rate one has to take care of several effects that influence the measured pulse height .first of all , only data from the same tube are compared to exclude systematic errors like dependence on preamplifiers or slightly different gas mixtures .+ one important quantity is the temperature of the gas .the pulse height increases with temperature if the pressure is kept constant ( fig.[fig = gainmeas ] ) .the order of magnitude of this effect is some percent per kelvin .since the data were taken in a non - air - conditioned hall with temperature variations of up to 5k within one measurement , temperature corrections of the pulse height had to be done .gas gain measurements at different temperatures have been done with the setup described in for the gases that were tested in beam measurements .thus it was possible to apply the correct temperature corrections and to get the diethorn parameters needed for the calculation .+ however , the most important point is to compare only pulses with equal drift times for the primary electrons .as can be seen in , the measured pulse height increases with drift time ( with distance from the wire ) .pulses with a short drift time have a lower maximum and a longer tail compared to those with long drift time although the primary ionisation is lower due to the shorter track length inside the tube .therefore , a cut on the drift time was applied to ensure that only pulses with approximately the same drift distance were compared .+  [ fig = height_tuberad ]  after these cuts , the maximum of each pulse was determined and histogramed .because the muons are minimum ionising particles , the distribution of the pulse heights follows a landau function which was fitted to the data .this is shown in fig .[ fig = landaufit ] . to describe the distribution of the pulse heights , the most probable pulse height ( which is one parameter of the fit ) was used and not the mean value of the data sample ( which is bigger ) . in the following pulse heightmeans the most probable pulse height of the landau fit .[ fig = height_rate ] shows the behaviour of the pulse height with increasing rate for three different settings of the high voltage , corresponding to gas gains of , and at zero rate .+ the solid lines show a calculation of the pulse height using the diethorn formula ( [ eq : gain ] ) to calculate the gain drop . for the calculation of the electric field ( [ eq : newfield ] ) and the actual gain , several iterations of the calculation of ( [ eq : dv ] ) , the gas gain ( [ eq : gain ] ) and the maximum ion drift time ( [ eq : newt+ ] ) are necessary to obtain an accurate value for the charge density ( [ eq : rho ] ) . to compare measured pulse height and calculated gain , one needs to know the ion mobility which was fitted to and found to be consistent with the value of 1.53 for ar in ar from .+ one can see that an initially high gas gain drops rather fast with increasing rate , going down to 30% of its initial value at 10khz/ . for nominal atlas conditions with a gas gain of 2 and photon rates / ( corresponding to a muon rate of 200hz/ ) the behaviour is rather uncritical , the reduction is some 10% .+ however , at very high rates , the pulse height is rather independent of the high voltage , the electric field is reduced to the same limiting value .the order of magnitude of the gain drop with rate is approximately the same for different gases , but the gain drop is larger for gases with low working point ( low high voltage for the same gas gain ) . in that case , the ion drift time is longer and so the space charge density is higher .+ the same study was done for the 2 cm and 4 cm tubes . because the primary ionisation of a muon is proportional to its path length inside the tube , the signals of a 4 cm tube are higher than those of 3 cm and 2 cm tubes if all are operated at the same gas gain . in order to compare the behaviour of the different tubes , measurements at the same charge per track ( not at the same gas gain ) at zero ratewere compared . for easier comparison , the charge for a 3 cm tube at a gas gain of set to 1 in fig.[fig = height_tuberad ] .for the 4 cm tube , no measurement was available for charge=1 , so a calculation is shown ( dashed line ) together with a measurement for lower charge .one can clearly see that the gain drop is increasing for bigger tubes at the same irradiation .it is almost negligible for the 2 cm tube , but even for the 4 cm one , the gain drop is in the order of 20% at a muon rate of 1khz/ and the nominal gas gain . + the gain reduction could be totally compensated by increasing the high voltage by ( [ eq : dv ] ) corresponding to the actual rate assuming that the irradiation is homogeneous along the tube .the electric field for drifting electrons is changed with rate according to ( [ eq : newfield ] ) . whereas the field changes are more or less the same for different gases at the same rate ( at least if they have to be operated at the same high voltage ) , the effects to the electron drift times are different . the drift time is given by the change in drift time depends on the slope of the drift velocity with the electric field .a gas becomes faster with rate if the slope is positive and slower if it is negative .+ one can see here that in order to keep the drift time changes small , one has to look for gases with constant drift velocity , so called linear gases ( linear because in this case , the r - t relation is a straight line ) .[ fig = velo ] shows the drift velocities and the corresponding r - t relations for the three gases that have been studied in this experiment : the linear gases ar - ch-n 91 - 5 - 4% , and ar - c-n-co 94 - 3 - 2 - 1% and the rather non - linear ar - c-n-co 92 - 3 - 2 - 3% . for comparison, the very non - linear gases ar - co 80 - 20% and ar - co are added .+ to study the effects of the modified electric field , we look at the change of the maximum electron drift times at different irradiation rates . + the maximum drift time can be calculated using ( [ eq : t_r ] ) with radius of the tube . to do this, one needs to know the drift velocity of the gas mixture .the magboltz program was used to calculate the drift velocity as a function of the reduced electric field .( [ eq : newfield ] ) was taken to calculate the correct electric field and then the new r - t relation was integrated using ( [ eq : t_r ] ) . + for using the equation ( [ eq : newfield ] ) the same numbers for the charge density ( and the ion mobility ) were used as for the calculation of the gain drop in the previous section .the calculated maximum drift time is shown as lines in fig .[ fig = drift_calc ] , together with the measured points .+ the maximum electron drift time of the measured data can be determined from the falling edge of the distribution of the drift times . because of diffusion and vanishing ionisation at the tube wall ,the tail is smeared . to get a precise and reproducible number , a fermi function has been fitted to the distribution ( ) .the four parameters of the fit are the upper level ( ) , the time at half height ( ) , the slope of the fall ( ) , and an offset ( ) .parameter is taken as the maximum drift time .the physical maximum drift time may be different from this value by a constant offset .however , we are primarily interested in changes of the maximum drift time .an offset between measurement and calculation was fitted and compensated .+  three gases are shown in the left plot of fig.[fig = drift_calc ] for a 3 cm tube : the linear gas ar - ch-n 91 - 5 - 4% , the non - linear gas with the same maximum drift time ar - c-n-co 92 - 3 - 2 - 3% and the linear gas ar - c-n-co 94 - 3 - 2 - 1% .for the first one , the maximum drift time is stable up to very high rates with a tendency to increase with rate .the non - linear one shows already differences of several nanoseconds at rather low rates ( ) . for comparing the 2 cm and the 4 cm tubes to our standard 3 cm one , fig.[fig = drift_calc](right ) shows the change of the maximum drift time with rate in each case .+ the data points for the 3 cm tube are the same as in the left plot for the gas ar - c-n-co 92 - 3 - 2 - 3% .the solid lines show the calculation done in the same way as for the left plot .the datasets used to compute the right plot are the same as those for the right plot of fig.[fig = height_tuberad ] .+ as for the gain reduction , one can see here that space charges become more and more important if the tube diameter is increased .the 2 cm tube might be operated with a non - linear gas if that was already considered impracticable for a 3 cm tube .for the 4 cm tube , the drift characteristics are dramatically changed at particle rates of 1khz/ or more .it is surely not advisable to use it for high precision measurements at high rates .+ surprisingly , both measurement and calculation show that the gas with the most linear r - t relation is not the one that is most insensitive to rate .the reason for this is the fact that the operation point of ar - c-n-co 94 - 3 - 2 - 1% is much lower compared to the others ( 2575v instead of 3300v for ar - ch-n 91 - 5 - 4% and a 3 cm tube .this lower voltage has two effects :  * the amount of space charge is proportional to the inverse of the voltage . because of ( [ eq : rho ] ) and ( [ eq : t+ ] ) . *the relative change of the electric field is proportional to the inverse square of the voltage + + + + because of ( [ eq : newfield ] ) .both points show that an optimum gas should have a high operational voltage .the reason for the rather low working point of this gas is the use of c as quencher . by replacing the ethane with methane , the voltage for the same gas gain will increase by 500v keeping the r - t relation almost unchanged .[ fig = terr ] shows the same calculation as fig .[ fig = drift_calc ] for low rates in a linear scale .additionally , calculations for the gases ar - co 80 - 20 % and 90 - 10 % are shown for comparison with non - linear gases .+ instead of comparing time shifts it is more adequate to compare the resulting shift in position .[ fig = terr ] bottom shows this position shift ( = the difference between real an assumed position of the particle ) when using the r - t relation suited for rate=0 at the actual rate .this value is plotted for the maximum drift radius ( 1.5 cm ) where for most gases this shift has a maximum .it is much smaller for smaller drift radii .the reason for this is the slope of the drift velocity with the electric field ( fig.[fig = velo ] ) .if it is zero , there is no shift because the drift velocity stays constant .if the slope is monotonous the shift increases with drift radius .this is the case for most gases . for the gas ar - ch-n 91 - 5 - 4%, the slope is very low and the sign changes twice in the relevant range of e / p so the integrated effect is almost zero for the maximum drift distance . + calculating the rms of the values for all radii at given rate leads to fig.[fig = rmserr ] .this is a good measure for comparing the rate behaviour of different gases .typical values are in the order of 50 - 100 m for a muon rate of 1khz/ , but for linear gases 30 m can be reached which is almost negligible compared to the intrinsic resolution of 70 m of the tube . +  the position shift does not necessarily decrease the resolution of the drift tube .if the rate is stable , one can calculate an adapted r - t relation where the position shift is compensated .such a r - t relation can be calculated directly from the data with the autocalibration method .[ fig = rtdiff ] shows the difference between two r - t relations calculated from data at rate=150hz/ and rate=1200hz/ .in this section we will show that the resolution becomes worse at high particle rates even if one uses the r - t relation with the position shift compensated .it will be shown that the reason for the resolution loss is the fluctuation of the times between two hits which was ignored up to now where only the mean value was taken into account .fig.[fig = zeitvert ] shows that the times between two hits are distributed exponentially . the time constant of the exponential function is defined by the rate . that example for rate 80hz/ also shows that for roughly half of the events the last hit was more than 4ms ago , which is the time of the ion drift for a 3 cm tube . in that case , all space charges are gone and the tube behaves as if the rate was zero . on the other hand , we know that for a rate of 80hz/ , the mean position shift is not negligible for most gases . if the time distance between two events is much shorter than 4ms , one has to take into account all hits within this time window in order to get the correct r - t relation .this is impossible for most applications .+ with our setup , it is possible to see the influence of previous hits .fig.[fig = dt_tl ] shows how the maximum drift time depends on the time since the last event passed for the gas ar - c-n-co 92 - 3 - 2 - 3% .all tracks were used that were within one cm along the wire coordinate .a tighter cut did not change the behaviour significantly .+ the second example ( fig.[fig = ph_dz ] ) shows the pulse height versus the distance along the wire to the previous track .one can clearly see a decrease of the pulse height in the immediate vicinity ( 1 mm ) of the previous track , but the effect is very small ( % ) .+  for a better understanding of the space charge fluctuations , a simulation study was done and afterwards compared to the measured data . in a first step ,it was investigated , how an ion cloud produced at the wire evolves with time in three dimensions .a typical charge cloud consists of 10 ions ( primary ion pairs multiplied by gas gain ) .ions were generated near the wire with an initial radial coordinate distributed exponentially within 50 m from the wire .the position of the coordinate along the wire ( z ) was distributed like a gaussian with mm .the same was true for the azimuthal angular distribution with .the ions were represented by 1000 points , each one carrying 1/1000 of the total charge .the effective electric field ( coming from the applied voltage and the field of the point charges ) was calculated at each of these points .now , one can calculate the drift velocity of the ions and from this the new position after the time dt . to describe the status of the whole cloud at the time t after creation , the mean value and the rms of the coordinates r , and z were calculated .+ in the simulation , the extension of an ion cloud does not exceed mm ) over the whole drift path .but within a z range of 5 mm , field distortions cause changes in the drift time of more than 1ns for non - linear gases ( e.g. ar - c-n-co 92 - 3 - 2 - 3% ) . on the contrary , for the pulse height ( or the gain )only the electric field in the immediate vicinity of the wire is relevant .the field distortions coming from an ion cloud that is some mm away are negligible compared to the very high field near the wire .+  in a second step of the simulation , the time distances t between two hits were generated according to the given rate .ions were generated at positions according to the results of step one for all previous tracks within the time of the ion drift .fig [ fig = ioncloud ] gives an example of the distribution of space charges coming from different events .now , electrons were put at radii r=0.5 mm ...r=14.5 mm ( or the respective tube radius for the 2 and 4 cm tubes ) and the drift in the ( static ) field resulting of all relevant space charges and the applied voltage was calculated .the time needed to arrive at the wire was histogramed for each radius. +  fig .[ fig = shiftdemo ] ( left ) shows the distribution of these times for radius=14.5 mm .the right plot shows the position shift resulting from inserting the drift times in the r - t relation ( for rate=0 ) and subtracting the real position where the electron started .the big , narrow peak belongs to tracks where no charges were inside the tube because the last hit was longer than the ion drift time ago .the distributions become more and more gaussian for higher rates . + from the histogram of the position shifts ,two parameters are extracted : the mean value which tells how wrong the r - t relation is for the actual rate and the rms which gives the contribution to resolution decrease .this number has to be added quadratically to the resolution of the drift tube at zero rate to get the correct value for the resolution at the investigated rate .+  for comparison with the measurements and the analytical calculation done in the previous section , the mean position shifts for rate=0 ... 1khz/ have been calculated ( fig.[fig = shiftsim ] , top ) .they agree within 5% with the values shown in fig .[ fig = terr ] ( bottom ) for the gas ar - c-n-co 92 - 3 - 2 - 3% .[ fig = shiftsim ] bottom shows the corresponding rms values of the position shift histogram .this is the contribution to the resolution loss coming from fluctuations .the numbers are non - negligible in the case of high rates and for high precision drift tubes like those of the muon spectrometer for atlas . in that case , the resolution of the tubes is dominated by this effect for a non - linear gas .+ calculations of drift chamber properties can be done very precisely using the garfield program .it also gives reliable numbers for the expected resolution ( at zero rate ) .it was used to compute the resolution of the gases ar - c-n-co 92 - 3 - 2 - 3% and ar - ch-n 91 - 5 - 4% with parameters discriminator threshold , noise etc set as expected for atlas conditions .the result is shown in fig .[ fig = totres ] ( open circles ) . on the contrary to the results shown before ,the gas gain was doubled to in order to have the same amount of space charges at the same rate as expected for the photons of the atlas background . also the measured data that will be shown belowrefer to muon data with a gas gain of .+ the results of the resolution loss for a rate of 500hz/ are shown as squares . the total resolution ( filled circles )is then given as a quadratical sum of both contributions .for the first gas , space charges play an important part for radii 8 mm , becoming the dominant contribution .for the second gas , the resolution is not affected by space charges even at very high rates .the different behaviour of the two gases is in agreement with the expectations : fig.[fig = drift_calc ] already showed that the drift time is nearly uneffected due to space charges for ar - ch-n 91 - 5 - 4% .for this linear gas , the drift velocity stays constant with changing electric field - in contrast to the non - linear gas .+ an experimental proof of these simulations is shown in figs .[ fig = res129 ] , [ fig = res040 ] .tracks have been fitted through the chamber and the resolution of a single tube was calculated for different conditions . for the gas ar - c-n-co 92 - 3 - 2 - 3% , one can see in both , measurement and simulation a degradation of the resolution for big radii in the case of high rates .it is not visible if one takes only low rate data or as in shown in fig.[fig = res129](bottom ) selects from a high rate run only those events where the previous track was longer than the ion drift time ago and all space charges are gone .+ the degradation of the resolution is also not visible for a linear gas ( fig.[fig = res040 ] ) where data were taken at the same rate and no cut on the event selection was applied .+ the measured resolution is slightly worse ( also for low rate ) than the simulation .this is due to the higher noise and a higher threshold for identification of the pulses needed for our flash adcs .the simulation shown refers to tdc measurements as foreseen for atlas .increasing the relevant parameters brings the simulation in accordance with our measurements .+ finally , the expected resolution of 2 , 3 , and 4 cm tubes shall be compared using a non - linear gas ( for a linear one , the differences are mostly negligible in all cases ) .the 3 cm results are shown in fig.[fig = totres ] , the others in fig.[fig = totres24 ] .the 2 cm tube is nearly unaffected by the high irradiation , a resolution loss occurs only in the last 2 mm . on the contrary , for a 4 cm tubethe resolution is dominated by the space charge contribution and the mean spatial resolution becomes much worse than 100 m .gain reduction and changes of drift times have been measured as a function of the rate . for the gases that have been tested ,a good agreement between measurements and calculation was found .predictions of the high rate behaviour for new gases can be made without performing measurements . + the gas gain reduction due to space charges is similar for most gases and unavoidable . for a constant rateit could be compensated by increasing the high voltage .+ the electric field for drifting electrons is increased because of space charges for radii larger than 4 mm ( in a 3 cm tube ) . for linear gases ( like ar - ch-n 91 - 5 - 4% )this may have almost no effect on the r - t relation even at very high rates . for non - linear onesthere are already big effects at rates of 100hz/ .whereas the r - t relation can be adapted for a given irradiation rate , it is ( for practical applications ) impossible to correct for the field fluctuations resulting from the random distribution of the times between two hits .this gives a contribution to the total resolution of the drift tube which for non - linear gases becomes the dominant limitation of the spatial resolution at large distances from the wire .we would like to thank our colleagues from the atlas muon collaboration , especially the people from the test beam group for their support and many helpful discussions at any time of the day or night .special thanks to rob veenhof for some very useful ideas .we appreciate the hospitality extended by the smc collaboration during the taking of these data in their muon beam . this work has been supported by the german bundesministerium fr bildung , wissenschaft , forschung und technologie .hendricks , the review of scientific instruments * 40 * , 1216 ( 1969 ) e. mathieson , nucl . instr . and meth .a249 ( 1986 ) 413 the atlas collaboration _ muon spectrometer technical design report _cern / lhcc/97 - 22 ( 1997 )  w. diethorn , _ a methane proportional counter system for natural radiocarbon measurements , u. s. aec rep .nyo-6628 ( 1956 ) b. struck tangstedt / hamburg _ technical manual dl 515 flash adc vme module _ schematics and layout : f / b - muonchamber pre amp fbpanik-04 , nikhef 1996 d. schaile , o. schaile and j. schwarz , nucl .instr . and meth . a242 ( 1986 ) 247 s.m .tkaczyk et al . , nucl .instr . and meth .a270 ( 1988 ) 373 k. handrich _ aufbau einer apparatur und messung der gasverstrkung in den driftrhren des atlas - myondetektors , thesis , university freiburg 1998 f. sauli _ principles of operation of multiwire proportional and drift chambers _ , cern 77 - 09 ( 1977 ) e.w .mcdaniel and e.a .mason , _ the mobility and diffusion of ions in gases _ , wiley , new york ( 1973 ) landolt - boernstein , vol 4/3 _ eigenschaften des plasmas _ , 6 . ed . , springer ( 1957 ) a. biscossa et al . , atlas internal note muon - no 196 ( 1997 ) _ _  g. schultz , g. charpak and f. sauli , rev .appl.(france ) * 12 * , 67 ( 1977 ) s.f .biagi , nucl .instr . and meth . a283 ( 1989 )716 c. bacci et al .atlas internal note muon - no 135 ( 1997 ) rob veenhof , garfield users manual , cern program library w5050

Abstract: this article describes calculations and measurements of space charge effects due to high rate irradiation in high resolution drift tubes . two main items are studied : the reduction of the gas gain and changes of the drift time . whereas the gain reduction is similar for all gases and unavoidable , the drift time changes depend on the kind of gas that is used . the loss in resolution due to high particle rate can be minimized with a suitable gas . this behaviour is calculable , allowing predictions for new gas mixtures .