clarinetist Nicole Canham
email: n.canham@uq.edu.au

Carlos Lopez Charles
email: carloslopezcharles@gmail.com

Carlos Lopez

AN EVOLVING COLLABORATION : PERFORMER AND COMPOSER APPROACHES TO CREATING VISUAL MUSIC

This article aims to identify the key factors that contributed to an evolving collaborative practice between a performer (Canham) and a composer (Lopez Charles). Three new works for clarinet, electronics and video, presented as case studies, form the basis of this study. Journals, artifacts including sound recordings of experiments, rehearsals, performance documentation and joint reflection will allow the researchers to describe and reflect upon the evolution of their collaborative practice as it unfolded.

INTRODUCTION

Dannenberg suggests that the use of computers in music composition "leads to new ways of thinking about music composition and performance" [START_REF] Dannenberg | Interactive Visual Music: A Personal Perspective[END_REF]. In this project, two participant/researchers set out to document and observe their own collaborative process in the context of the development of a new work for clarinet, electronics and video which was based around interactive components facilitated by the use of computer technology. The aim was to track and identify key factorsincluding new ways of thinkingthat emerged during the process of creating three visual music works.

THE STUDY OF COLLABORATION

Collaboration has been a subject of study in numerous fields including business [START_REF] John-Steiner | The Challenge of Studying Collaboration[END_REF], the arts and sciences, [START_REF] John-Steiner | Creative collaboration[END_REF] in improvisational contexts in music and theatre [START_REF] Sawyer | Group Genius[END_REF] and in the field of performer/composer collaboration [see for example 6, 7, 11 and 12]. Performer/composer studies have addressed issues of the nature of collaborationwhich are often framed in terms of 'process'and the outcomes of the collaborative work environmentwhich Hayden refers to as 'output' [START_REF] Hayden | Collaboration and the Composer: Case Studies from the end of the 20 th century[END_REF]. Analysis and observations about collaboration, subsequently, can be made from a variety of angles which include detailed descriptions of processes involved [START_REF] John-Steiner | Creative collaboration[END_REF], or a focus on the employment of a collaborative approach with a certain outcome or output in mind.

John-Steiner suggests that "construction of knowledge is embedded in the cultural and historical milieu in which it arises [START_REF] John-Steiner | Creative collaboration[END_REF]". Her view provides an interesting challenge in the study of performer/composer collaborations in so far as these roles have often been perceived to be quite separate. In the western art music tradition, it has been argued that this segregation of roles is a product of the view of the score-as-object, an idea attributed to German music critic E.T.A Hoffman [START_REF] Goehr | Being True to the Work[END_REF]. Hoffman's notion of 'Werktreue', first proposed in the early 1800s, introduced a concept of musical works in which all other aspects of the presentation should be subservient to the score: "musical activities, be they of composition, performance, reception, evaluation, or analysis, should no longer be guided by extra-musical considerations of a religious, social, or scientific sort. They should now be guided by the works themselves [START_REF] Goehr | Being True to the Work[END_REF]".

The nineteenth century view of composition and performance has endured in two key ways. Both the direction and evolution of practice have been influenced. One of ways in which this influence can be perceived is in the limits the werktreue concept has placed on the musicological discourse [START_REF] Goehr | Being True to the Work[END_REF]. A preoccupation with the score has shaped the scope and focus of scholarship, with many other aspects of the compositional and music making process being overlooked. This has included a narrow view of the role the performer [START_REF] Small | Musicking : the meanings of performing and listening[END_REF] and, until recently, limited contribution from performers in academic research [START_REF] Crispin | Artistic Research and the Musician's Vulnerability[END_REF]. A second consequence of a scorebased approach has been the evolution of practice to reflect this emphasis, one example of which is the lengthy period in which improvisation for classically trained performers was not in vogue [START_REF] Small | Musicking : the meanings of performing and listening[END_REF].

In recent years, however, many of the limitations of a nineteenth century view of composition and performance have been addressed or challenged [START_REF] Blacking | How musical is man?[END_REF][START_REF] Goehr | Being True to the Work[END_REF][START_REF] Goehr | The imaginary museum of musical works an essay in the philosophy of music[END_REF][START_REF] Small | Musicking : the meanings of performing and listening[END_REF][START_REF] Taruskin | Sacred Entertainments[END_REF]. One advantage of the new directions in thinking and practice proposed has been the examination of a range of creative and artistic roles and relationships. Increasing research based upon a reconsideration of the work of the composer, the view of the performer and the opportunities offered by the addition of computer technology in the composition and performance workspace provide examples of a shifting dialogue.

CHALLENGING TRADITIONAL ROLES

A common theme in recent research into the activities of composers and performers has been the unmasking of processes that were not previously viewed as central to a discussion about creating new works. Studies of collaboration have been one area of scholarship in which more nuanced pictures of performer and composer have been proposed.

These studies also often reflect consideration of other elements influencing collaborative environments, including the use of digital technology.

With regard to composers, Hayden and Windsor [START_REF] Hayden | Collaboration and the Composer: Case Studies from the end of the 20 th century[END_REF] highlight the inaccuracy of the commonly held view of the composer as a lone genius when examined in the context of the creation of new works. Rather, they suggest that composers have always had some degree of interaction with performers, conductors or publishers, but that this has often been neglected in the discourse.

Performers, according to Crispin, [START_REF] Crispin | Artistic Research and the Musician's Vulnerability[END_REF] negotiate a reality in which many aspects of their practice are deliberately kept half-concealed: the years of training and discipline required in order to develop the connection between performer and instrument is often hidden in the desire to make performances appear effortless. Crispin suggests this reveals a preoccupation with the ideal at the expense of developing a "deep understanding of live performance in all its guises [START_REF] Crispin | Artistic Research and the Musician's Vulnerability[END_REF]".

Digital technology has also been recognized as a key compositional and collaborative tool in the twenty first century. Dannenberg, commenting on the rapid growth in capabilities of digital technology and the opportunity these present in creative contexts, suggests that "the biggest challenges ahead are artistic rather than technological. One of the attractions of this pursuit is that there are relatively few precedents and no established theory [START_REF] Dannenberg | Interactive Visual Music: A Personal Perspective[END_REF]".

In the absence of a single established theory, a range of ideas have been proposed in understanding the role of digital technology in artistic collaborations. Within the domain of music making, some studies have focused on the way technology has been used as an instrument for artistic practice -Partti's study of the 'digital musician' is one example [START_REF] Partti | Cosmopolitan musicianship under construction: Digital musicians illuminating emerging values in music education[END_REF]. Digital technology has also been harnessed to facilitate the sharing of information among musicians through web-based communities of practice. [START_REF] Partti | Reconceptualising musical learning: new media, identity and community in music education[END_REF] Literature concerning the design of computersupported co-operative work systems (CSCW) addresses the influence of social factors on systems design. [START_REF] Cockburn | Four principles of groupware design[END_REF] One aspect of CSCW which is pertinent to this study is the notion of tailorability. [START_REF] Steimerling | Distributed Component-Based Tailorability for CSCW Applications[END_REF] Although Dannenberg [START_REF] Dannenberg | Interactive Visual Music: A Personal Perspective[END_REF] emphasizes that creating visual music using technology is essentially an artistic challenge, questions of tailorability -how system design decisions are to be made, and who is to make them -remain an issue. These three elements of performer, composer and digital technology are focal points of this investigation of aspects of collaboration.

RESEARCH AIMS

Research questions

The study was built around a central question: What were the key factors that shaped this collaboration?

These factors included observation and consideration of the facilitators and constraints inherent in the study, which included:

-Practical and creative matters, including issues of systems design;

-The nature of the working process, identification of patterns of decision making, evaluation and discussion; and, -Discussion of the similarities and differences between performer and composer objectives as the collaboration progressed.

In acknowledging that the project united artists with different backgrounds and approaches, there were a series of sub-questions for both researchers which related directly to their own questions about their role in a collaborative work environment.

Composer-related questions:

How can different compositional approaches open up space for contrasting ways of collaborating with a performer? How can these approaches influence the relationships between sound and image in the creation of visual music?

Performer-related questions:

What makes a performer of an acoustic instrument necessary in the electroacoustic and visual framework?

To what degree can a performer make connections between visual and musical elements if they have no visual score or cannot watch the video during the performance?

To what degree does or should the performer have a role in the creative (compositional) process?

APPROACH

In this study, the authors and co-participants brought different areas of expertise to the research. Nicole Canham has commissioned numerous new works in her professional career as a performer, and has collaborated with a wide range of artists in diverse settings. Canham's doctoral research draws upon qualitative research methods, in particular that of case study. Carlos Lopez Charles is a composer whose work has focused mainly on electro-acoustic composition and computer programming. His current Phd study in the area of visual music is concerned with the compositional techniques in this field. The study presents three different examples of composer-performer collaboration which took place over a two-year period. During this time three new works for clarinet, video and electroacoustic music were created. The creation of each work, presented as a case study, reflects approaches to collaboration shaped by a variety of factors including the participants' varied expertise. The aim of presenting two cases of earlier collaborative works prior to the current project is two-fold: as a form of triangulation, and to provide a framework for evaluating to what degree the collaborative process evolved between our first and our most recent attempt.

Willis [START_REF] Thiétart | Introduction. Doing Management Research[END_REF] suggests that what we believe to be the "nature of truth (ontology) and what it means to know (epistemology)" (p.10) forms the basis of the ways in which research can be conducted and understood. Given the research aims, which are focused on gaining a greater understanding of multiple perspectives of the same situation, the ontological position of this study is that there are multiple versions of reality rather than a single truth, locating this study within the constructivist paradigm [START_REF] Allen | The Constructivist Paradigm[END_REF][START_REF] Hatch | Doing qualitative research in education settings[END_REF][START_REF] Willis | Chapter 1. World Views, Paradigms, and the Practice of Social Science Research[END_REF]. Reality, then, is constructed through the accounts and reflections of the researchers who were co-participants in the project [START_REF] Hatch | Doing qualitative research in education settings[END_REF][START_REF] Thiétart | Introduction. Doing Management Research[END_REF][START_REF] Willis | Chapter 1. World Views, Paradigms, and the Practice of Social Science Research[END_REF]. The research design reflects a close connection between the authors' researcher roles and their roles as participants as they worked together to co-construct an account of their evolving collaborative process.

Methodology and Methods

Collaboration has been studied using a range of qualitative research methods, which have included interview [START_REF] John-Steiner | Creative collaboration[END_REF], observation [START_REF] Hooper | The start of performance, or, does collaboration matter?[END_REF], video interaction analysis [START_REF] Sawyer | Group Genius[END_REF] and reflective journal [START_REF] Hayden | Collaboration and the Composer: Case Studies from the end of the 20 th century[END_REF]. Case study is a research methodology that is often used in research based upon a constructivist paradigm, and has also been applied in studies of collaboration [START_REF] Hatch | Doing qualitative research in education settings[END_REF][START_REF] Hayden | Collaboration and the Composer: Case Studies from the end of the 20 th century[END_REF]. One of the advantages of case study is that it enables the researcher to employ a variety of different methods in the gathering of data [START_REF] Willis | Chapter 7. Methods of Qualitative Research[END_REF], which includes those methods stated above and other artifacts. In the case of this study, artifacts included meeting notes, task-based activities, scores with annotations, email correspondence, and recordings of rehearsals and performances.

Case Study

There are several different types of case study that researchers utilize, which Stake describes as intrinsic, instrumental and collective [START_REF] Stake | Qualitative Case Studies. The SAGE handbook of qualitative research[END_REF]. Willis [START_REF] Willis | Chapter 7. Methods of Qualitative Research[END_REF] offers two further sub-categories which are perhaps more reflective of the ways in which the data is expected to be used: "descriptive case study [START_REF] Willis | Chapter 7. Methods of Qualitative Research[END_REF]" is concerned with detailed descriptions of cases. "Interpretive case study" takes the descriptive case study work further, aiming to use the data generated to make statements about contexts beyond those being documented and described. "The focus is on understanding the intricacies of a particular situation, setting, organizations, culture, or individual, but that local understanding may be related to prevailing theories or models [START_REF] Willis | Chapter 1. World Views, Paradigms, and the Practice of Social Science Research[END_REF]".

Instrumental case study, like Interpretive case study, enables the researcher to develop detailed accounts of individual cases, but the purpose of the study is to use the case to provide insight about an issue.

This instrumental case study describes the creation of three new works in order to understand the nature of each collaborative context on an individual basis. These cases will then be considered together with a view to developing insights about the outcomes of an evolving collaborative process. Three examples of the creation of different works are used to illustrate ways in which creative and technical approaches to visual music from composer and performer perspectives evolved.

CASE STUDY 1 -NOT ALONE, FOR CLARINET, LIVE ELECTRONICS AND VIDEO

Nicole: Reading back over our emails at the time Not Alone was being written, I notice that the conversation had a lot to do with practical concerns. I liked the piece a great deal on first hearing and nothing in the score was technically problematic, so my feedback focused on page turns, breathing and some articulation issues. Not Alone was my first foray into working with live electronics in a surround sound environment. Perhaps as a result of this, the score Carlos provided me was highly detailed to ensure that the clarinet input would make the delay effect respond as envisaged. The video that accompanied Not Alone was created by a colleague after the music had been composed, almost as a creative response to the music.

In a way, it seems as though we were trying to superimpose different ways of working over the top of each other. I commissioned Not Alone as part of a multimedia installation/performance piece which involved a larger team of Australian and Mexican artists. Not Alone was a collaboration-within-a-collaboration in that sense, as I was also developing a cohesive 60minute long program. I wrote to Carlos about this on 4 May, 2011, about three weeks before the premiere: "I included a quote about time for each piece that inspires me on the musical interpretation side of things. In terms of the order, I've suggested it considering three different perspectives: 1. Ease of change of instruments, because I'm changing as we go; 2. Key [tonality] relationships between pieces so there isn't a jarring sensation going from one piece into the next; 3. Narrative possibilities of music in that order." Conceptually, bringing in the time quotes as my own interpretive aid had to do with the 'time' theme of the larger program. It was, however, quite different from the musical frame of reference that Carlos had first discussed with me. He was interested in delay effects used by electric guitarists in the 1980s, and was keen to find a way to transfer this idea into a piece for clarinet. Carlos was also left to consider all the technical matters that I didn't understand, or perhaps hadn't thought through coming from a background of mostly acoustic performance. Carlos' system design had a reactive function to performer and composer input: my sound was processed with a delay feedback effect and Carlos varied the volume and the spatialisation of the delayed clarinet sounds. In performance, I felt as though my playing and his live diffusion of the sound were similar to making chamber music, although I never experienced the full perspective of the spatialised performance.

CASE STUDY 2 -LAS FLORES Y LAS NUBES, FOR CLARINET, ELECTRONICS AND VIDEO

Carlos: My goal for this piece was to have a higher degree of integration between video and music than in Not Alone. Early discussions with Nicole focused on the technical aspects of how I wanted to approach this challenge. I also made a point of working with Nicole using improvisation as a basis for testing and developing ideas. We had two improvisation/recording sessions before a period of time apart. Although I intended to take a less conventional approach with this piece, I still made most of the compositional decisions on my own. I composed the video first and then the electroacoustic tape part, trying to make an audiovisual piece that could stand on its own and over which I could then add a clarinet part. To make the video, I used a custom-made program that allowed me to vary the size, movement and color of up to 12,000 points. Using this idea as a model for the composition of the tape, I created a patch to control different variables of a flow of short (15 ms. to 200 ms.) electronic sounds produced by a very sophisticated FM-based synthesizer.

At this stage, I was consciously avoiding the incorporation of clarinet sounds into the tape because I felt that approach (which is quite common) to be too standard. The Max/MSP patch that I made for the clarinet was programmed so that it changed its variables automatically at specific moments of the piece which meant that synchronization of parts was critical. For this reason, I included a click track that would enable the performer to synchronize with everything else very precisely. We didn't see each other after our improvisation sessions until 3 weeks before the premiere of the piece, so the clarinet part was completed by me and then discussed and revised in our final rehearsal period. I sent a very detailed score, along with a patch, a video and a click track to Nicole six weeks prior to our planned rehearsals. As it turned out, none of the work from our improvisation sessions was included in the finished score.

In our first rehearsal, I realized that the clarinet's sound did not blend as well as I wanted with the electronic sounds that I had chosen. I decided to integrate some transformed clarinet sounds into the tape part that would allow for a tighter connection between tape and clarinet. Adding these sounds, using a reverb effect and making minor changes to the score helped to achieve a higher degree of integration between the clarinet and the electronic part. We eventually used improvisation for the ending of the piece.

Nicole: Las Flores, like Not Alone, was premiered within a larger multimedia performance. The context of the premiere also reflected some of the broader technical and programming challenges that we were negotiating. This included practicalities of how the video was to be projected, the projection surface (black), the style of projection (via Matrox) resulting in a wrap-around style image, and the low level of light necessary in order to have the videos look most vibrant.

The technical and programming concerns surrounding the performance as a whole also impacted staging options. Although I performed almost the whole program from memory, I had to perform Las Flores with music in a fixed space so that I could access the click track to ensure that my part and the electronic part were together in performance. In order to accommodate the video requirements of the piece, I was obscured from view by some large pieces of black foil blocking the light coming from the sconce on my music stand. Transitions from one piece to the next were also critical in this performance. Carlos had left the ending of Las Flores quite open which gave us the opportunity to weave together a number of elements. This included some improvisation in performance on my part for the ending of the piece, which also incorporated mechanical instruments made by an artist-colleague. Whilst I felt sonically embedded in this piece, I didn't visually feel a part of it and it was difficult for me to be seen by the audience.

CASE STUDY 3 -POINTS, LINES, PLANE, FOR CLARINET, LIVE ELECTRONICS AND LIVE VIDEO

Nicole & Carlos: We had originally planned to rework Las Flores, and so we set aside some time for discussion and experimentation.

This included individual and joint reflection on what we felt worked well, and what might be improved based upon our experiences of both Not Alone and Las Flores. Feedback from Carlos concerned finding a better blend with clarinet and tape. Feedback from Nicole was concerned with having a better visual connection between performer and video, and exploring more meaningful ways of connecting with the visual element as a performer. Feeling able to make connections between the performative aspect of the visual component and other visual elements, namely the performer as visual component, was impeded by the necessity of performance with score and click track. We were also keen to set some new challenges and parameters for this collaboration. We wanted all the elements to be able to interact, as a basis for creating a collaborative environment where we would challenge our traditional roles as performer and composer. The goal was to develop a working process which might enable us to foster a collaborative environment where roles felt equal and to some degree interchangeable.

Nicole: One of our conversations stands out to me -I had been discussing with Carlos some of the problems I find with using a click track and he suggested, "what if we get rid of the click track?" We then moved on to considering how we might structure the score and this led to the thought… "what if we get rid of the score?" These two questions for me, and the possibilities opened up by leaving both click track and score behind, were the beginning of a completely different approach, and a new piece.

Carlos: Reflecting on my discussions with Nicole led me to realize that the issues that we were trying to address (in particular those of music-video and performer-video) could be solved by establishing interactions between electronic sounds, instrumental sounds and moving images through the use of the computer. Inspired by Josh Nimoy's "Bouncing Balls", I created an interactive system based on the idea of balls that make electronic sounds each time they bounce against lines drawn by the user. I programmed a patch in Processing to control the moving images and used Serge Le Mouton's samplor~ object in Max/MSP to play the audio samples each time a ball bounced. When bouncing against a line, the horizontal coordinate of a ball is mapped to a sound's position in a two-channel (left right) stereo field, while its vertical coordinate is mapped to its pitch (high to low). The communication between Max and Processing was established through the use of OSC messages.

Figure 3. Mapping of the balls' coordinates when colliding with a line

In a way, this system was already offering an answer to the sound-image integration in the piece, but the relationship between the performer and the visuals still had to be addressed. In order to do this, I used Miller Puckette's bonk~ Max/MSP object to detect the clarinet's percussive attacks and Tristan Jehan's analyzer~ object to track its loudness.

Nicole: Our first session working with Carlos' new system gave me a lot of ideas in terms of how the clarinet and the video might relate in an interactive setting, particularly with the use of my instrument as controller. Visually, I was very taken with being able to trigger pitched, bouncing balls with my clarinet sounds. A secondary task, and one that came out of these sessions, was then to consider how I might sonically relate to the sound being produced by the balls once I had triggered them. This included my own 'homework', which was to develop techniques that in some ways mimic or approach the sound that the balls produce.

Carlos: A challenge of the instrument-as-controller approach is that it seemed to me to have some compositional and technical limitations. Incorporating a physical interface with different multi-slider controllers that would also let me manipulate the behavior of the balls in a more flexible way seemed to be a good solution. Using the multi-slider allowed me to improvise with Nicole and create a wider range of musical ideas than if I just let the system react to her input. It also gave me the possibility of an interpretive role in the piece, not only a compositional one.

ANALYSIS

Our framework for analysis of these descriptive accounts draws upon the work of John-Steiner, Weber and Minnis in two ways [START_REF] John-Steiner | The Challenge of Studying Collaboration[END_REF]. Firstly, we draw upon their definition of collaboration:

The principals in a true collaboration represent complementary domains of expertise. As collaborators, they not only plan, decide, and act jointly, they also think together, combining independent conceptual schemes to create original frameworks. Also, in a true collaboration, there is a commitment to shared resources, power, and talent: no individual's point of view dominates, authority for decisions and actions resides in the group, and work products reflect a blending of all participants' contributions [START_REF] John-Steiner | The Challenge of Studying Collaboration[END_REF].

We also adopt John-Steiner, Weber and Minnis' approach of "looking for commonalities and differences across settings, tasks, working methods, goals, and values" [START_REF] John-Steiner | The Challenge of Studying Collaboration[END_REF] as a tool in understanding and interpreting our collaborative process.

DISCUSSION

In reflecting upon the ways in which changes to our collaborative approach facilitated or constrained the development of an equal, interactive, creative working process we identified some key influential factors. These resonate throughout the Case Study descriptions offered here. We were able to observe similarities and differences from case to case in three main areas:

1. Challenging traditional roles 2. Changes to practice 3. Reflection: individual reflection and joint discussion 10.1.

Challenging traditional roles

Changes of approach to our roles within the collaborative environment highlight advantages and problems with the notion of true collaboration as outlined by John-Steiner, Weber and Minnis. Case Study 1 provided an example of a fairly standard commissioning arrangement. Within that framework, roles of composer and performer were essentially quite separate and the collaborative aspect of the work is best described as a 'layering' of these elements to produce the finished piece. Advantages of this approach within the context of Not Alone included differing concerns and levels of expertise that needed to be taken into account for both practical and artistic reasons. The performer, for example, was more than happy to defer to the views of the composer in relation to the presentation of the score and the live electronics environment because that was a new area in which she had limited experience. This was also the first time the artists had worked together. Limitations of this approach impacted the artistic outcome in the sense that the relationship with performer, electronics and video did not always appear to be connected in the performance of the piece.

In Case Study 2, our roles had begun to shift. One advantage of this shift was that discussion and early work sessions reflected a desire to let performer and composer into the domain of each other's practice. Interestingly, and perhaps unsurprisingly, the direction and timing of these conversations followed the development of the piece in relation to composer and performer rolesthat is, early conversations had a lot to do with Carlos' compositional ideas and approach, and later discussions just prior to, and after the premiere were more concerned with the technical challenges faced by Nicole and Carlos in performing the work. A limitation faced in Case Study 2 was the incompatibility of the timelines of composer and performer as they are traditionally understood: often a work can be almost complete before a performer becomes involved in a practical sense. Additionally, planning, development and rehearsal environments present different challenges to a live performance context which usually takes place later in the collaborative process, rather than at the beginning, for example.

Case Study 3 reflects a different approach again in that both performer and composer perspectives were included in the planning and discussion from the beginning, with systems design, compositional ideas and approaches to interpretation being given more consistent consideration. Advantages of this level of integration compared with the other two case studies are that both composer and performer felt more able to interact across the three key areas of the collaborative partnership. No one is excluded from a particular artistic relationship combination in this format. A limitation of this approach is that the amount of video and audio processing that can take place in real-time is determined by the system's computing capabilities [START_REF] Lopez Charles | La transduction dans la création de la musique visuelle en temps différé[END_REF]. This creates a situation in which the video and audio materials have to be designed within the system's constraints for the sake of a higher degree of interactivity. From the composer's perspective, this is not necessarily a drawback, given that composing interactive visual music is not only about combining sounds and images, but also about composing the relationships that will be established between them and the performer(s). However, reflecting on how to minimize the compromise between what a system can do in real-time and what the artist would want it to do is an important factor to consider in contexts like this.

Changes to practice

There were four notable changes to practice across the three case studies. Face to face interaction, joint decision making and regular and repeated collaborative work facilitated a number of changes to individual performer, composer and collaborative working approaches. Choices in the way in which computer systems were designed and employed also played a significant role in the evolution of a sense of joint practice.

Face to face interaction

Face to face interaction became an increasingly important factor in facilitating the development of new working processes between performer and composer. This has not only been the case for joint experiments, but also for determining the nature and structure of our independent work. Increased awareness of each other's individual skills and expertise brought about through our joint creative work has played a major part in shifting priorities reflected in the decision making and planning processes, allowing for the development of both more nuanced roles for composer and performer, and a reflective approach to systems design.

Joint decision making

Composer and performer jointly agreed what they would independently work on in Case Study 3. Resulting changes to practice have included more deliberate division of tasks and responsibilities, defining together pre-compositional material and determining parameters for performance. A fluid approach to systems design has greatly facilitated the conceptual shifts evident through these three cases of collaboration: the system has become an interface between composer and performer in which new roles, approaches and artistic outcomes have been defined and facilitated. In contrast, the approach taken in Case Study 1 was much less concerned with details of the role and tasks of the other with relation to the use of digital technology: it was designed by one artist for a specific purpose, and presented to the other without a great deal of prior discussion or consultation.

Regular and repeated collaborative work

The opportunity to undertake three different collaborative projects in a relatively short space of time (less than two years) is viewed by both participants as a significant factor in the development of their joint collaborative practice. In that sense, the case studies form a single example of an ongoing, evolving collaborative practice.

The possibility of the collaborative 'next step' in the form of ongoing performance opportunities has been a critical factorfrom both research and artistic perspectivesin providing an outlet and incentive in which to explore and implement changes to practice.

Changes to systems design/tailorability

Another observable feature of the evolving collaborative process is demonstrated in the systems design choices and/or use of digital technology in each case. In addition to discussions about the artistic possibilities and uses of different programs, an important consideration in the development of the system for Case Study 3 was the style of interaction between composer and performer that this system could afford. Notably different in this approach, as compared with case studies 1 and 2 was the objective to create a system that would facilitate jointly agreed artistic and collaborative objectives. This is in contrast to the idea of composer and performer supplying complementary, but separate components of the collaboration. In this sense, digital technology has a dual function, and in a way that mirrors changes to perfomer/composer interactions, a more nuanced role. It plays both a critical artistic part, and also functions as a tool for achieving new collaborative objectives.

10.3.

Reflection : individual reflection and joint discussion

Reflection upon a range of factors over the course of the collaboration was highly illuminating for both participants. Artistically, discussing and evaluating the strengths and weaknesses of each piece was an important factor in understanding quite different traditional roles and creative perspectives. Conducting a review of theories of systems design, creativity and collaboration for the purposes of this study provided us with ideas for the theoretical framework in which we could place our actions and decisions, or interrogate other understandings of collaboration.

In more general terms, reflection and discussion seemed to facilitate a blurring of boundaries between performer and composer roles as our conversations generated new levels of interest and ownership in a wider range of the creative tasks. As the collaboration unfolded, the consequences of these discussions and reflection is evident in the revised approaches to creative decision making, including the different approaches taken to systems design. Case Study 2 provides a good example of the composer increasing his frame of activity to music and visuals, whilst we see the performer increasing the scope of her concerns to include her relationship to the visual elements of the piece alongside her music making/interpretive role.

CONCLUSIONS

In this study we have aimed to document and describe key factors contributing to an evolving collaborative relationship between a performer and a composer. Three areas which most facilitated change in the collaborative contexts outlined were 1) challenging traditional roles, 2) changes to practice and 3) reflection (both separately and together).

As a contribution to understandings of collaboration beyond the scope of these three cases, both from the perspective as researchers drawing upon established theories of collaboration and our own professional experiences as artists, we suggest that our study is useful for:

Highlighting ways in which the challenging or blurring of traditional roles can facilitate new insights and approaches to practice; Appreciating the capabilities of systems design as both an essential artistic tool and facilitator of new levels of interaction between performer and composer in the composition and performance of visual music;

Recognising the value of collaborative work environments for artists as a place of ongoing development and learning.

To that end, discussing, reflecting upon and employing in practice a range of theoretical perspectives on collaboration may be highly useful for composers and performers. Within the visual music making framework, working with tailor-made computer technology in a collaborative environment offers great possibility as both creative tool and interface in new performer/composer relationships and interactions.

Figure 1 .

 1 Figure 1. Score excerpt, Not Alone

Figure 2 .

 2 Figure 2. Still image from Las flores y las nubes

Figure 4 .

 4 Figure 4. System design for the performance of Points, lines, motion


Carlos López Charles

NON-REAL-TIME TRANSDUCTION AND ITS USE AS A COMPOSITIONAL TECHNIQUE FOR THE CREATION OF VISUAL MUSIC

The computer allows us to simulate all sorts of behaviours that go beyond the physical, mechanical and optical possibilities imposed by traditional visual music instruments (Levin, 2000, p.33). Nevertheless, the amount of data and the resolution of the sound and image files an artist can use in real-time to create visual music are limited by the processing capabilities of the computer (or computers) that he works with.

Transferring (or, to be more precise, transducing) data between digital representations of sounds, gestures and images is one of the main compositional techniques in today's interactive visual music. This article presents different techniques and software tools that can be useful in establishing transductions between digital sounds, images and control gestures in non-real-time. Three examples of the implementation of these ideas in different contexts will be presented in order to analyze the benefits of this approach. The first example, "Elastic Study", is an audiovisual piece in which different psychoacoustic characteristics of an acousmatic tape composition are analyzed and used to control a visual 3D particle system in non-real time. A second example, "The Flowers and the Clouds" is a visual music piece in which user's gestures are recorded and used to control a 3D particle system and an ensemble of synthesis sound processes in non-real time. The third and final example is a video in which a non-real time analysis of an HD video is used to control the parameters of a sound synthesis process.

INTRODUCTION

The term "visual music" has been used extensively to describe different forms of expression in which artists create analogies between sound and image through a constellation of ideas: such as the ideas of abstraction, synaesthesia and the aesthetic experiences of time, geometry and gesture. Several works have been created in this spirit by artists such as Kandinsky, Survage, Klee, Kupka, Delaunnay, Ruttmann, Eggeling, Fischinger, McLaren, Lye and the Whitney Brothers.

More recently, visual music artworks have been made with the use of digital systems in which heterogeneous elements are brought together and stabilised into functional units through the implementation of transduction [START_REF] Collins | kipp av: Live Algorithmic Splicing and Audiovisual Event Capture[END_REF] processes. Regardless of being used in real-time or non-real-time, transduction processes offer models for the creation of structural relationships between different representational spaces [START_REF] Emmerson | Living Electronic Music[END_REF] , where collections of discretely-defined symbols allow us to have a certain control over the forms and behaviour of sounds, images and different objects (patches, spectrograms, control algorithms, etc.) that can form part of a compositional environment. Consequently, within a digital environment, we can consider these transfers of information between different media to be transductions of symbols between different representational modes. The use of transduction as a tool for mediation becomes most effective in this context because it enhances the connection of the most diverse representational spaces. From the moment in which symbols in a digital environment acquire the role of generating new morphologies, they adopt -as Vaggione describes it -a role that is similar to music notation within a score (Vaggione, 2006, p. 14), becoming the elementary units of a language in which different characteristics of a represented object can be described, manipulated and articulated within a network of compositional operations. The use of transduction is an artistically pertinent approach because it sets specific criteria that can be used to endow objects in an audiovisual composition with a multiplicity of forms and behaviours. However, the use of this technique in non-real-time contexts has not been explored in depth.

THE USE OF TRANSDUCTION IN THE CONTEXT OF VISUAL MUSIC

In physics, transduction is defined as the conversion of one form of energy into another form of energy (University of Oxford, 2005, p. 672). An electroacoustic transducer, for example, is a device that allows us to convert electric energy into acoustic energy and viceversa. Condenser microphones and loudspeakers are examples of transducers used for the recording and playback of sounds.

Concerning audiovisual expression, transducers allow us to convert any kind of energy (radiant, mechanic, thermal, magnetic or chemical) into sounds and images [START_REF] Levin | Painterly Interfaces for Audiovisual Performance[END_REF] . In this way an input energy, like the process on a structure that operates from place to place: every defined structural region serves the next one as a constitutive foundation, even though a modification extends progressively at the same time as this operation takes place" mechanical energy of a human gesture [START_REF] Collins | kipp av: Live Algorithmic Splicing and Audiovisual Event Capture[END_REF] , can be transformed into electricity and, conversely, into magnetic and radiant energies that could control the variations of, for example, the pressure in a loudspeaker and the luminosity variations in a set of lights, respectively. In such cases, the trace of the input energy becomes a controller (or regulator) of the output energies that will be ultimately perceived as sounds or images.

Compared to analogical devices, digital technologies offer a lot of flexibility in the control of audiovisual materials. On the one hand, the use of digital-to-analog and analog-to-digital converters allows us to represent electric energy variations as a discrete (digital) collection of symbols. Conversely, the computer allows us to generate, store and manipulate any collection of symbols and create electric signals from them. This opens up the possibility of operating on different kinds of media not only from at the perceptual level, but also at the level of their code.

In summary, in computer-made visual music may be achieved by the use of input transducers, by generating a collection of symbols within the computer that will be eventually transduced into sounds or images (which has the advantage of allowing for more complex control procedures 2 ), and by a combination of these techniques.

NON REAL-TIME TRANSDUCTION

Not many tools have been developed to facilitate the circulation of data between different representational spaces (digital representations of sound, image, gesture, etc.) in the creation of visual music in fix video and cinema. As a consequence of this, artists have had to use tools that were originally conceived for real-time interaction. The drawbacks of this situation are considerable: the complexity of possible the transduction processes are limited to very simple operations (such as controlling the colour of the image with the amplitude of a sound, for example). If we wanted to stage a transduction process that involved heavy processing for the average computer (like, for example doing and FFT analysis and using the frequency of each bin as a control of a 100,000 element particle system in which each element took the colour of a high resolution video), there would be considerable latency issues or even unexpected program crashes and, with this, a decrease of the transformational capacity [START_REF] Levin | Painterly Interfaces for Audiovisual Performance[END_REF] (the ability of animating the What kind of software solutions could help in the creation of non-real-time visual music works that incorporate the possibility of establishing transduction relationships between different modes of representation?

STUDY CASE NO.1: SOUND AS THE CONTROLLER

The software for the piece "Elastic Study" was developed in the Max/MSP/Jitter programming environment. The music for this piece was created after testing how Tristan Jehan's object analyzer~4 (which makes real-time estimations of features like the brightness, noisiness, loudness, etc. of a given sound) could be used to control different variables of a particle system on a 3D scene created with the jit.p.shiva and jit.p.vishnu objects of the Jitter library. The sonic elements of this piece were composed in a way in which they could provide a very flexible control of the elements of this particle system. In order to provide more flexibility to the sound-image relationship of this piece, I integrated a system that would allow to vary the mappings automatically in particular sections of the piece (such as the amount of particles, their variations of colour, duration, inclination, the background colour, transparency and the camera's position). Even though the software presented a stable performance in real time, recording the output in high definition quality became impossible due to the amount of CPU consumption that the process demanded. To solve this problem, I created an abstraction called "dataSequencer" (figure 2), which let me 1) record the output values of the sound analysis, 2) sample them at a speed of 30 samples per second and 3) output these values in non-real-time speed so that the individual frames of a 30 frames-per-second movie could be generated individually, without surpassing the processor's capacity to render them.

Having generated the individual frames of the video in non-real-time, different Final Cut sessions were created to bounce the frames of each section into small videos that were finally mixed together into a final piece. This strategy made it possible to edit the frames produced by Max/MSP/Jitter and to correct any errors that could have been eventually produced through the use of the Jitter abstractions, which proved to be somehow limited for this kind of approach.

.

STUDY CASE NO.2: GESTURE AS THE CONTROLLER

"The Flowers and the Clouds" is a piece made using Processing 1 and Max/MSP in which a visual particle system adopts the colours of a background video in order to create abstract imagery that lies somewhere between the figurative and the non-figurative. The particle system was controlled with Max/MSP through the use of different sliders that allowed to modify different characteristics of their elements, including the number of particles, their speed, the randomness of their trajectories, their transparency, their size, etc. The communication between Max/MSP and Processing was made through the OSC protocol.

The control of the particle system proved to be very efficient within a certain threshold of complexity, but when the number of particles and their transparency increased beyond a certain number (for example, 12,000 particles), a slow down in the response of the system became evident and the level of performativity 2 was affected. To solve this situation, I used a similar solution to the one in "Elastic Study", but instead of using the analysis's data of a soundtrack as the control data, I used gestural data that that was 1) recorded from improvisations with a set of virtual sliders, 2) edited in non-real-time so that the undesired accidents in the 1 www.processing.org 2 LEVIN, Golan Painterly Interfaces for Audiovisual Performance, Massachussets Institute of Technology, 2000, p.33. defines performativity as the degree in which a system allows the users to establish a feedback loop in which they can explore their potential as actors in an open work.

improvisations could be corrected and 3) used to generate the individual frames of a movie.

The individual frames for this video were assembled into a HD Quicktime movie by using a custom patch made in Processing. In this case, no synchronisation problems occurred, as it happened with the use of the Jitter objects in the first example. My observation with regard to this is that open-code software like Processing allows for much more precise programming at a low level than that offered by software like Jitter, which translates into more creative flexibility in cases like this one. The soundtrack for this piece was made with a commercial FM synthesizer (FM8), whose different variables were also controlled in non-real-time through the use of recorded and edited gestural data from improvisations made in sync with the video, which allowed for a very precise writing of the sound materials and facilitated the establishment of synaesthetic relationships between the images and the music during the compositional process.

STUDY CASE NO.3:

"Miniature 3" is a short piece in which the movement of a particle system in a pre-recorded video controls the playback speed of two soundfiles (one for each audio channel) to create a synaesthetic effect through the use of synchronisation. Even though this synchronic approach could lead us to a situation in which we create sound-image relationships that seem too direct, it also provides a solid ground to achieve synaesthetic sensations that would be impossible to create otherwise [START_REF] Levin | Painterly Interfaces for Audiovisual Performance[END_REF] . The computational cost of creating the sound for this video in real-time was too high because the image analysis was very demanding. However, the nonreal-time solution proved once more to be very efficient within this context.

The audio for this video was created with Processing and Max/MSP through a process in which 1)

Processing divides a high-definition (1280 by 720 frames) video into two regions (left and right), 2) the amount of movement in each region is estimated by Processing (through the use of a frame-difference algorithm) and sent to Max/MSP as a sequence of numbers that Max saves into a text file and 3) the data in the text file is converted in non-real time into an audio signal which is finally used to control the playback speed of the sound files in the left and right audio channels.

CONCLUSIONS

Composing visual music through the use of computers enables the use of digital representations of sounds, images, movement or any kind of data -as well as the analyses that can be made from it-as models for the generation of forms and behaviours in different representational modes. Sounds, images and physical movement may present considerable perceptual differences between them, but represented as symbols, they can attain a certain degree of stability and the open up a space to establish translations (or, more precisely, transductions) between them. By using a transductive approach to visual music composition, the artist uses a representation of something as a model to create something new.

As Simon Emmerson points out (Emmerson, 2007, p. 42), the compositional choices that an artist has to make in using models in his creative process are at least four: [I would number these] the identification of the model and its behaviour, the choice of the variables that will correspond to the chosen model and the definition of the scale of their relationships. However, by choosing to compose digital visual music with a transductive approach, another decision has to be made: the nature of the compromise that will take place between the performativity of the system and the quantity and quality of the materials that will be used in the compositional process. In that regard, a non-real time approach seems to be a good option for the creation of visual music.

Figure 1 .

 1 Figure 1. Still image from "Elastic Study

Figure 2 .

 2 Figure 2. Basic use of the DataSequencer object

Figure 3 .

 3 Figure 3. Still image from "The Flowers and the Clouds"

Figure 4 .

 4 Figure 4. Movement detection on "Miniature 3".Movement on each side of the screen (white pixels) is used to control the playback speed of sound files in non-real time.

  VOLPI, G. et al. Gesture Analysis dans Enaction and Enactive Interfaces : a Handbook of Terms edité par Annie Luciani et Claude Cadoz, Enactive System Books, ACROE, Grenoble, France, 2007. define gesture as a human action 2 Portillo O. et Polviani I., "Control, Digital" in Enaction and Enactive Interfaces : a Handbook of Terms, édité par Annie Luciani et Claude Cadoz, Enactive System Books, ACROE, Grenoble, France, 2007, p.58. 3 VAGGIONE Horacio, "Vers une approche transformationnelle en CAO Actes des Journées d'Informatique Musicale (JIM) 1996. Les cahiers du CNRS. CNRS-Université de Caen. Electronic version available at http://jim.afimasso.org/jim96/actes/vaggione/VaggioneTEXTE.html#Tahil objects in our compositional network with different forms and behaviours) of the system.

[START_REF] Collins | kipp av: Live Algorithmic Splicing and Audiovisual Event Capture[END_REF]

SIMONDON, Gilbert L'individuation à la lumière des notions de forme et d'information, Éd. J. Millon, coll. Krisis, Grenoble,

2005, p. 32. explains transduction in the following way: "We understand transduction as a physical, biological, mental, social operation by which an activity is spread within a domain, founding this spreading

VAGGIONE, H. Symboles, signaux, opérations musicales, CICM-Université Paris VIII, unpulbished article, 2006, p.14

[START_REF] Levin | Painterly Interfaces for Audiovisual Performance[END_REF] EMMERSON, Simon Living Electronic Music, Ed. Ashgate, Anglaterre, 2007, p.117-142 offers a very complete overview of the different transducers used in electroacoustic music.

JEHAN, Tristan. Disponible sur le site électronique personnel http://web.media.mit.edu/~tristan/

COLLINS N. et OLOFSSON F., kipp av : Live Algorithmic Splicing and Audiovisual Event Capture, Computer Music Journal 30:2,, Massachusetts Institute of Technology, 2006, p. 8-18.


A Hassairi
email: abdelhamid.hassairi@fss.rnu.tn

Gaussian representation of a class of Riesz probability distributions

Keywords: Chi-square distribution, Gaussian distribution, Wishart probability distribution, Riesz probability distribution, Laplace transform AMS Classification : 60B11, 60B15, 60B20

whether they are published or not. The documents may come

Gaussian representation of a class of Riesz probability distributions

Introduction

Let V be the space of real symmetric r × r matrices, Ω be the cone of positive definite and Ω the cone of positive semi-definite elements of V . We denote the identity matrix by e, the trace of an element x of V by tr(x) and its determinant by ∆(x). We equip V by the scalar product θ, x = tr(θx).

Denoting

L µ (θ) = V e θ,x µ(dx)
the Laplace transform of a measure µ, it is well known (see [START_REF] Faraut | Analysis on Symmetric Cones[END_REF]) that there exists a positive measure µ p on Ω such that the Laplace transform of µ p exists for -θ ∈ Ω and is equal to Ω exp(tr(θx))µ p (dx) = ∆ -p (-θ), (1.1) if and only if p is in the so called Gindikin set

Λ = 1 2 , 1, . . . , r -1 2 ∪ r -1 2 , +∞ . (1.2)
When p > r-1 2 , the measure µ p is absolutely continuous with respect to the Lebesgue measure and is given by

µ p (dx) = 1 Γ Ω (p) ∆ p-r+1 2 (x)1 Ω (x)dx,
where

Γ Ω (p) = (2π) n-r 2 r k=1 Γ(p - k -1 2 ),
where n is the dimension of V . And when p = j 2 with j integer, 1 ≤ j ≤ r -1, the measure µ p is singular concentrated on the set of elements of rank j of Ω. For p ∈ Λ and σ ∈ Ω, (1.1) implies that the measure W p,σ on Ω defined by

W p,σ (dx) = ∆ -p (σ) exp(-tr(xσ -1 ))µ p (dx) (1.3)
is a probability distribution. It is called the Wishart distribution with shape parameter p and scale parameter σ. Its Laplace transform exists for θ ∈ σ -1 -Ω and is equal to Ω exp(tr(θx))W p,σ (dx) = ∆ -p (eσθ).

(1.4)

When p > r-1 2 , the Wishart distribution W p,σ is given by

W p,σ (dx) = ∆ -p (σ) Γ Ω (p) exp(-tr(xσ -1 ))∆ p-r+1 2 (x)1 Ω (x)dx (1.5)
Further details on the Wishart distribution on Ω may be obtained from [START_REF] Casalis | The Lukacs-Olkin-Rubin characterization of the Whishart distributions on the symmetric cones[END_REF] or from [START_REF] Letac | Quadratic and inverse regressions for Wishart distributions[END_REF]. It is however important to mention that the Wishart probability distribution has initially been defined as a multivariate extension of the chi-square distribution. In fact, taking U 1 , ..., U p random vectors in IR r with distribution N (0, σ), the distribution of the matrix X = p i=1 U t i U i is a Wishart matrix. This kind of Wishart probability distributions may be defined using the Gaussian random matrices. Let u = t (u 1 , ..., u r ) be a random vector with distribution N r (0, σ). Suppose that we have a random sample of u i.e., a sequence of independent and identically distributed random variables with distribution equal to the distribution of u, and consider the r × s Gaussian matrix

U =        u 1,1 . . . u 1,s . . . . . . . . . . . . . . . u r,1 . . . u r,s        . The distribution of U is N r,s (0, σ) equal to 1 (2π) rs 2 ∆ s 2 (σ) e -1 2 u,σ -1 u ,
where u, σ -1 u = tr(σ -1 u t u).

We have

Proposition 1.1 If U ∼ N r,s (0, σ), then the matrix X = U t U has the Wishart probability distribution W r ( s 2 , σ -1 2 ). Proof The Laplace transform of X estimated in θ ∈ ( σ -1 2 -Ω) is given by E(e θ,X ) = 1 (2π) rs 2 ∆ s 2 (σ) IR rs e θ,u t u -1 2 u,σ -1 u du = 1 (2π) rs 2 ∆ s 2 (σ) IR rs e -1 2 u,(σ -1 -2θ)u du = ∆ s 2 (σ -1 -2θ) -1 ∆ s 2 (σ) = ∆ s 2 ( σ -1 2 -θ) -1 ∆ s 2 ( σ -1 2 ) -1
.

It is then the Laplace transform of a W r ( s 2 , σ -1 2 ) distribution. ✷
This distribution is absolutely continuous with respect to the Lebesgue measure if and only if s 2 > r-1 2 , that is if and only if the size s of the sample is greater or equal to the dimension r of the space of observations. If we consider p independent random matrices U 1 , ..., U p with distribution N r,s (0, σ), then the distribution of the matrix

X = p i=1 U i t U i ∼ W r ( ps 2 , σ -1
2 ) Of course the class of Wishart distributions defined by mean of the Gaussian vectors or the Gaussian matrices doesn't cover all the Wishart distributions, however the fact that this random matrix is expressed in terms of Gaussian random vectors allows to establish many important properties of this subclass of the class of Wishart probability distribution. Using a theorem due to Gindikin which relayes on the notion of generalized power, Hassairi and Lajmi [START_REF] Hassairi | Riesz exponential families on symmetric cones[END_REF] have introduced an important generalization of the Wishart distribution that they have called Riesz distribution. This distribution in its general form is defined by its Laplace transform. We will show that some among the Riesz distributions may be represented by the Gaussian distribution using samples of Gaussian vectors with missing data.

Riesz distributions

For 1 ≤ i, j ≤ r, we define the matrix µ ij = (γ kℓ ) 1≤k,ℓ≤r such that γ ij = 1 √ 2 and the other entries are equal to 0. We also set

c i = √ 2µ ii for 1 ≤ i ≤ r, and e ij = (µ ij + µ ji ), for 1 ≤ i < j ≤ r.
With these notations, an element x of V may be written

x = r i=1 x i c i + i<j x ij e ij .
In particular, for 1 ≤ k ≤ r, we set

e k = c 1 + • • • + c k . Now consider the map P k from V into V defined by x = r i=1 x i c i + i<j x ij e ij -→ P k (x) = k i=1 x i c i + i<j≤k x ij e ij .
Then the determinant ∆

(k) (P k (x)) of the k × k bloc P k (x) is denoted by ∆ k (x)
, it is the principal minor of x of order k. The generalized power of an element x of Ω is then defined

for s = (s 1 , • • •, s r ) ∈ IR r , by ∆ s (x) = ∆ 1 (x) s 1 -s 2 ∆ 2 (x) s 2 -s 3 • • • ∆ r (x) sr . Note that ∆ s (x) = ∆ p (x) if s = (p, • • •, p) with p ∈ IR. It is also easy to see that ∆ s+s ′ (x) = ∆ s (x)∆ s ′ (x). In particular, if m ∈ IR and s + m = (s 1 + m, • • •, s r + m), we have ∆ s+m (x) = ∆ s (x)∆ m (x)
. Now, we introduce the set Ξ of elements s of IR r defined as follows:

For a real u ≥ 0, we put ε(u) = 0 if u = 0 and ε(u

) = 1 if u > 0. For u 1 , u 2 , • • • , u r ≥ 0, we define s 1 = u 1 and s k = u k + 1 2 (ε(u 1 ) + • • • + ε(u k-1 )), for 2 ≤ k ≤ r.
We have the following result due to Gindikin [START_REF] Gindikin | Analysis on homogenous domains[END_REF], it is also proved in the monograph by Faraut and Korányi [START_REF] Faraut | Analysis on Symmetric Cones[END_REF].

Theorem 2.1 There exists a positive measure R s on V such that the Laplace transform L Rs is defined on -Ω and is equal to ∆ s (-θ -1 ) if and only if s is in Ξ. The measures R s , defined in the previous theorem in terms of their Laplace transforms are divided into two classes according to the position of s in Ξ. In the first class, the measures are absolutely continuous with respect to the Lebesgue measure on Ω. More precisely, we have (see [START_REF] Hassairi | Riesz exponential families on symmetric cones[END_REF]) that when s

= (s 1 , • • • , s r ) in Ξ is such that for all i, s i > i-1 2 , R s (dx) = 1 Γ Ω (s) ∆ s-r+1 2 (x)1 Ω (x)dx where Γ Ω (s) = (2π) n-r 2 r j=1 Γ(s j - j - 1 2
).

The second class corresponds to s in Ξ \ r i=1 ] i-1 2 , +∞[. In this case the Riesz measure R s is concentrated on the boundary ∂Ω of Ω, it has a complicated form which is explicitly described in [START_REF] Hassairi | Singular Riesz measures on symmetric cones[END_REF]. For s = (s 1 , • • • , s r ) in Ξ and σ in Ω, we define the Riesz distribution R r (s, σ) by

R r (s, σ)(dx) = e -σ,x ∆ s (σ -1 ) R s (dx).
The Laplace transform of the Riesz distribution is defined for θ in σ -Ω by

L Rr(s,σ) (θ) = ∆ s ((σ -θ) -1 ) ∆ s (σ -1 ) (2.6) If s = (p, • • • , p) such that p ∈ Λ = {0, 1 2 , • • • , r-1 2 }∪] r-1 2 , +∞[, then R r (s, σ
) is nothing but the Wishart distribution with shape parameter p and scale parameter σ. The definition of the absolutely continuous Riesz distribution on the cone of positive definite symmetric matrices Ω relies on the notion of generalized power. It is defined for

σ in Ω and s = (s 1 , • • •, s r ) ∈ IR r such that s i > i-1 2 by R(s, σ)(dx) = 1 Γ Ω (s)∆ s (σ -1 ) e -σ,x ∆ s-r+1 2 (x)1 Ω (x)dx.

Main result

We now show that some of the Riesz distributions have a representation in terms of the Gaussian distribution generalizing what accours in the case of the Wishart distribution.

For simplicity, we take σ equal to the identity matrix. Suppose that we have s r independent observations of the vector (u 1 , ..., u r ) with Gaussian distribution N r (0, I r ), and that some data in the observations are missing. The sample is then partitioned according to the number of available components in the observation. More precisely, we suppose that there exist integers 0 < s 1 ≤ s 2 ≤ ... ≤ s r such that we have s 1 observations on (u 1 , ..., u r ) where all the components of the observation are available, (s 2s 1 ) observations on (u 2 , ..., u r ), the first component of the observation is missing, (s i+1s i ) observations on (u i+1 , ..., u r ), the i first components of the observation are missing, 2 ≤ i ≤ r -1. The number s i is in particular equal to the number of times the component u i appears in the data. Define

p 1 = sup{i ≥ 1; s i = s 1 }, p 2 = sup{i > p 1 ; s i = s p 1 +1 }, p l+1 = sup{i > p l ; s i = s p l +1 }, p k = inf{i; s i = s r } -1, so that we have s 1 = ... = s p 1 , s p 1 = s p 1 +1 , s p 1 +1 = ... = s p 2 , s p 2 = s p 2 +1 . . . s p l +1 = ... = s p l+1 , . . . s p k = s r s p k +1 = ... = s r .
We have that p k+1 = r . Replacing the missing components by the theoretical mean that is by zero, we obtain the matrix

U =                               u 1,1 . . u 1,
                              (3.7)
This matrix U consists of blocks defined recursively in the following way :

U (1) = (u i,j ) 1≤i≤p 1 1≤j≤sp 1 U (l+1) = U (l+1) 1 0 U (l+1) 21 U (l+1) 0 , 1 ≤ l ≤ k (3.8) with U (l+1) 1 = U (l) U (l+1) 21 = (u i,j ) p l +1≤i≤p l+1 , 1≤j≤sp l U (l+1) 0 = (u i,j ) p l +1≤i≤p l+1 , sp l +1≤j≤sp l+1 .
We have that U (k+1) = U .

Theorem 3.1 The distribution of X = U t U is R(s, Ir 2
) with s = ( s 1 2 , ..., sr 2 ).

Proof We use the following Peirse decomposition of an r × r matrix ω.

For 1 ≤ l ≤ k, we set ω (l) = (ω i,j ) 1≤i≤p l 1≤j≤p l ,
and we write ω (l+1) as

ω (l+1) = ω (l) t ω (l+1) 21 ω (l+1) 21 ω (l+1) 0 . (3.9)
From Proposition 1.1, we have that U (1) t U (1) has the W p 1 (

sp 1 2 , Ip 1
2 ) distribution. On the other hand, since s 1 = ... = s p 1 , we have that

W p 1 s p 1 2 , I p 1 2 = R p 1 s 1 2 , ..., s p 1 2 , I p 1 2 . It suffises to show that if U (l) t U (l) is R p l s 1 2 , ..., sp l 2 , Ip l 2 , then U (l+1) t U (l+1) is R p l+1 s 1 2 , ..., sp l+1 2 , Ip l+1 2 .
Again we use the Laplace transform. For θ ∈ ( Ir 2 -Ω) and with the decomposition (3.9), we have

L U (l+1) t U (l+1) (θ (l+1) ) = E exp θ (l+1) , U (l+1) t U (l+1) = E exp tr θ (l+1) 0 U (l+1) 0 t U (l+1) 0 E exp tr θ (l) U (l) t U (l) + t θ (l+1) 21 U (l+1) 21 t U (l) + U (l) t U (l+1) 21 θ (l+1) 21 + θ (l+1) 0 U (l+1) 21 t U (l+1) 21 . We have that U (l+1) 0 is a (p (l+1) -p (l) ) × (s p (l+1) -s p (l) ) Gaussian matrix with distribution N (0, I (p (l+1) -p (l) ).
According to Proposition 1.1, the matrix U

(l+1) 0 t U (l+1) 0 has the Wishart distribution W r ( sp (l+1) -sp (l) 2
,

I (p (l+1) -p (l) ) 2
). Hence

E exp tr θ (l+1) 0 U (l+1) 0 t U (l+1) 0 = ∆ sp (l+1) -sp (l) 2 ( I (p (l+1) -p (l) ) 2 -θ (l+1) 0 ) -1 ∆ sp (l+1) -sp (l) 2 2(I (p (l+1) -p (l) ) )
.

Concerning the second expectation, we have that

tr t θ (l+1) 21
Finally, we obtain

L U (l+1) t U (l+1) (θ (l+1) ) = ∆ sp (l+1) -sp (l) 2 I (p (l+1) -p (l) 2 -θ (l+1) 0 -1 ∆ sp (l+1) -sp (l) 2 2(I (p (l+1) -p (l) ) ) ∆ sp (l) 2 I (p (l+1) -p (l) 2 -θ (l+1) 0 -1 ∆ sp (l) 2 2(I (p (l+1) -p (l) ) ) ∆ s 1 2 ,..., sp l 2   Ip l 2 -θ (l) -t θ (l+1) 21
I (p (l+1) -p (l) 2 -θ (l+1) 0 -1 θ (l+1) 21 -1   ∆ s 1 2 ,..., sp l 2 (2I p l ) ,
that is ) .

L U (l+1) t U (l+1) (θ (l+1) ) = ∆ sp (
Therefore U (l+1) t U (l+1) is a Riesz random matrix with distribution

R

l+1) 2 I 2 (sp (l+1) 2 2 2 , 2 (2I p l )∆ s p l +1 2 , 1 2

 2222221 (p (l+1) -p (l) 2I p l ) .But s p l +1 = ... = s p l+1 , then∆ (I (p (l+1) -p (l) ) ) = ∆ s p l +1 (p (l+1) -p (l) ) ) , (p (l+1) -p (l) ) ) = ∆ s (p (l+1) ) ) , it follows that L U (l+1) t U (l+1) (θ (l+1) (p (l+1) )

  +1,1 . . u p 1 +1,sp 1 . . u p 1 +1,sp 2 0 . . . ,1 . . u p 2 -1,sp 1 . . u p 2 ,sp 2 0 . . .

			sp 1	0 .	.	. . . .	0
	.	. .	.	. .	.	. . . .	.
	.	. .	.	. .	.	. . . .	.
	0 . u p 1 0 . . . . . 0 u p 1 ,1 . . u p 1 ,sp 1
	.	. .	.	. .	.	. . . .	.
	.	. .	.	. .	.	. . . .	.
	u p 2 0
	.	. .	.	. .	.	. . . .	.
	.	. .	.	. .	.	. . . .	.
	.	. .	.	. .	.	. . . .	.
	u p k +1,1 . .	.	. .	.	. . . . u p k +1,sr
	.	. .	.	. .	.	. . . .	.
	.	. .	.	. .	.	. . . .	.
	u r,1	. .	.	. .	.	. . . .	u r,sr


Philippe Gabriel

Development Of Employability Strategy And Improvement Of The Offer Of Study Programmes In Connection With The Needs Of Local And International Economies

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


R Penionzhkevich

D-S Anne

C Baiborodin

Z Borcea

D Dlouhy

A-S Guillemaud-Mueller

R Fomichev

M Kalpakchieva

O B Tarasov

Yu E Penionzhkevich\

R Anne

D S Baiborodin

C Borcea

Z Dlouhy

D Guillemaud-Mueller

A S Fomichev

R Kalpakchieva

M Lewitowicz

S M Lukyanov'

V Z Maidikov

A C Mueller

Yu Ts

Oganessian

M G Saint-Laurent

N K Skobelev

O Sorlin

V D Toneev

W Trinder

Yu E Penionzhkevich

R Kaipakchieva

M Lewitowiczi

Smlukyanov\ V Z Maidikov\

M G Oganessian\

Saint-Laurent

PRODUCTION OF NUCLEI IN ^^S-INDUCED REACTIONS IN THE ENERGY RANGE 6^75 MEV/A

Introduction

Heavy-ion induced reactions are shown to be one of the most efficient tools for producing nuclei far from the stability line. The first experiments carried out in Dubna using ^Ne and 40 Ar beams of 7 MeV/A resulted in the observation of about 20 new isotopes lying significantly away from stability [1,2]. In these experiments, a new heavy-ion-reaction mechanism was discovered which later on was called deep inelastic reactions [2]. This process, corresponding to a transitional regime between complete fusion and direct reactions, has been well studied at energies up to 10 MeV/A. It was shown that in deep inelastic processes the production yields of different isotopes could be well described using statistical models [START_REF] Norenberg | [END_REF], and could also be explained by the reaction ß-vahie taking into account pairing corrections (Qggsystematics) [2]. However, the advent to the region of nuclear instability by using deep inelastic reactions was limited by the low production cross sections of the sought-after exotic nuclear species. Later, reactions with high-energy heavy ions were used, where as a result of the projectile fragmentation on a thick target many new neutron-rich light-element isotopes were produced with relatively high cross sections [4]. The advent into the region of more neutron-rich isotopes and of heavier elements required beams of heavy projectiles with higher intensities. In the late 1970s, the availability of new heavy ion accelerators of intermediate energies provided further possibilities of using reactions induced by heavy ion beams of E = 50-Î-100 MeV/A for the production of new exotic nuclei. Very successful was proven to be the use of 36 S and ^Ca beams in the fragmentation of which many extremely neutron-rich nuclei were observed. A series of experiments using a ^Ca beam, performed within the ties of over 20 new isotopes lying close to the limit of particle stability [5].

There is evidence that, although for a certain domain of nuclei the fragmentation process is dominant already at 27MeV/A [6], the multinucleon transfer reaction plays still a noticeable role. For example, in 58 Ni(Z=28)-induced reactions at 55MeV/A new neutron-deficient Cu isotopes were observed [7]. A similar effect, viz. production of charge pickup products (Z>50), has been observed also in 112 Sn-induced reactions at 63MeV/A [8], The continuous investigation of the mechanism of nuclear reactions is closely connected with the projects for radioactive nuclear beam facilities, which will open new possibilities for the study of exotic nuclei and where, for the generation of radioactive secondary beams, primary beams of very different energies will be used [9], e.g., lOOMeV/A (GANIL), 20MeV/A (Dubna) and up to 1 GeV/A (GSI and RJKEN). Questions arise concerning the extent of coexistence of different reaction mechanisms (e.g., multinucleon transfer reactions and fragmentation) at various energies, the dependence of the production rates on the projectile and target isospin, etc.

The present work was undertaken in order to obtain experimental information on the trends in the formation of different isotopes with 6<Z<14. Use was made of reactions induced by 32 -34 ' 36 S beams in the energy range 6 <E< 75 MeV/A.

Experimental Procedure

The experiments with 32>34 S beams of energies 6<£<20MeV/A were carried out at the U400 cyclotron of the Flerov Laboratory of Nuclear Reactions (JINR). The yields of the various isotopes were measured at a laboratory angle 0ub =4° using the MSP-144 magnetic spectrometer [10] with a position-sensitive ionization chamber as the focal plane detector [11]. The solid angle was 0.85 msr; and the momentum acceptance, ±4.1%. In the measurements, targets of ""C (400 u.g/cm 2 thick) and 197 Au (200 ug/cm 2 thick) were used. The beam monitoring was performed by means of a Faraday cup. The Z and A identification of the reaction products was done using the measured energy-loss (dE), total energy (£) and position (x) in the focal plane. The relations used are:

i-, (2)

A where A, Z, and q are the mass, the atomic number and the ionic charge of the nucleus, respectively, B -the magnetic field of the spectrometer and k -a constant. For a better presentation of the identified isotopes a two-dimensional plot of production yields as a function of J4-2Z and Z was used. One example is shown in Fig. 1. As can be seen from the figure, in the given experimental set-up, a good separation of the isotopes ranging from C to Cl was achieved, which in turn allowed unambiguous identification of the reaction products.

Several settings of the magnetic spectrometer were used in order to obtain the energy spectra of the produced isotopes. Fig. 2 represents the energy distributions of Mg isotopes obtained in the reaction 32 S (14.5 MeV/A) + C.

The 36 S(75 MeV/A) beam was provided by the accelerator complex GANIL (France) and the isotope yields were measured [12] with the help of the fragment separator LISE [13]. (A-2Z, Z) of the products of the 32 S(14.5 MeV/A)+C reaction obtained at a magnetic field B = 0.7975 T. The left solid line passes through the completely stripped nuclei with zero isotopic spin. The right curve discriminates between regions of nuclei with charge q = Z and q = 2-1.

-1 0

A-2Z

Experimental Results and Their Analysis

The resuks discussed in the present work have been obtained in reactions induced by j ons m a vef y |j roa( j energy range.

The distributions of carbon, oxygen, neon, magnesium and silicon isotopes produced in the 36 S (75 MeV/A)-projectile fragmentation on two targets ("^C and 181 Ta) are shown in Fig. 3. To describe the experimental data on the production cross sections of the isotopes with ,,,,;,,,» ,,,T,,.,,,,,,, 16 20 24 28 32 Fig. 3. Experimental isotopic distributions of carbon, oxygen, neon, magnesium and silicon nuclei obtained in the fragmentation of a 75 MeV/A ^S beam on two targets -l2 C (closed symbols) and 181 Ta (open symbols) [12]. The solid line is the result of calculations for the Ta-target based on a modification of the empirical parameterization [14].

atomic numbers 6<Z< 14 an empirical parameterization similar to the one in [14] was used.

The calculations for the Ta-target, denoted by the solid line in Fig. 3, show a satisfactory agreement with the experimental data. It can also be seen that at an energy of 75 MeV/A the production cross sections of neutron-rich nuclei are higher for the Ta than for the C-target.

Figs. 4aandb show the experimental isotopic distributions from the 32 S (9.1 and 16.1MeV/A) beam for two targets ("t and 197 Au), and Fig. 4c-the calculations for 32 S at 75 MeV/A, obtained as for Fig. 3 using the same set of parameters. From the comparison of the experimental data with the calculations it follows that the isotopic distributions can be described by a smooth curve close to a Gaussian distribution. The widths of these distributions at 9.1 and 16.1MeV/A differ strongly for the two targets (""'C, 197 Au), at least for the isotopes of elements far from the projectile (oxygen and neon). At the same time, the yield of the neutron-rich isotopes of these elements in the case of the Au-target is significantly higher than in the case of the C-target.

The comparison (Fig. 5) of the differential production cross sections as a function of the atomic number Z of the products for different targets, projectile energies and projectile neutron excess has made it possible to draw the following conclusions about the trends in the formation of various nuclei in S-induced reactions.

At energies 7+10 MeV/A for the light target, a rather large drop in cross section for smaller Z-values (or greater number of transferred protons) is observed: from 700 mb/sr, in the case of two stripped protons, to 1 mb/sr, for the stripping of 7 protons (see Fig. 5a). At higher bombarding energies this difference decreases and at intermediate energies is negligible (Fig. 5b). Comparison of reactions induced by projectiles of different mass shows that thecross sections obtained for 32 S and 34 S beams are very close to each other. Significant difference can be noticed only for products with Z higher than the projectile atomic number, which corresponds to the case of pick-up of protons by the projectile (Fig. 5c). A comparison of the centers of gravity of the isotopic differential-cross-sectiondistributions of Fig. 4, corresponding to neutron excess values (N-Z), is shown in Fig. 6 for the studied reactions as a fiinction of the atomic number Z. In the case of the heavy target, one can observe on the average a shift of the centers of gravity in the direction of the neutronrich region when decreasing the bombarding energy (Fig. 6a). On the contrary, the displacement is towards the proton-rich side for the light target (Fig. 6b). The observed trend is in a qualitative agreement with the microscopic theory of multinucleon exchange [15]. The direction of the flow of nucleons in the system is determined by the difference in the corresponding Fermi energies, with the nucleons "flowing" from the nucleus with a higher Fermi energy to the nucleus with the smaller one. The decreased shift in the centers of gravity with increasing the projectile energy in the case of the heavy target can be explained by the high excitation, which can lead to the emission of a larger number of neutrons. Moreover, at small incident energies oscillations are observed in the centers-of-gravity distributions, probably due to oddeven effects, which are smoothed with increasing the energy. Figs. 6c,d show the Z dependence of the most probable neutron excess of the produced isotopes for the two ion beams, 32 S and 34 S, at an energy of 14.5MeV/A (experimental data) and at 75MeV/A (calculated values). From Fig. 6c it follows that, from the point of view of the production of neutron-rich isotopes, there is only a small advantage in using a j4 S beam compared to a beam of 32 S at an energy of 14.5MeV/A. However, at high energies some advantage can be expected, but only for the production of nuclei close to the projectile with Z> 12.

As it was mentioned earlier, the yields of isotopes in deep inelastic processes are well described by the reaction energy (Qgg). The cross sections are determined by the relation [2]:

G~exp{[Qgg + AE c -5]/T}, ( 3
)
where Qgg is the energy necessary for the rearrangement of the nuclei in the input channel into the nuclei in the exit channel, AE C -the change in the Coulomb energy of the system due to the redistribution of protons between the nuclei and to the deformation of the system, 6 -the nucleon pairing corrections, and T-the temperature of the dinuclear system. Figure 7 presents the experimental differential cross sections for producing oxygen isotopes as a fiinction of Qgg for different target-projectile combinations at various projectile energies. It can be seen that there is an overall good agreement between the experimental points and the expected from relation [START_REF] Norenberg | [END_REF]. Making use of the Qgg systematics, it is possible to estimate the yields of nuclei lying far from the stability line.

The obtained experimental results are of great interest from the point of view of the reaction mechanism involved in the formation of nuclei and because they give a possibility of estimating the usefulness of heavy ion induced reactions as a way to produce high-intensity beams in radioactive nuclear beam factories. For this purpose it is necessary to consider the total reaction cross sections, which means that the angular distributions of the products have to be taken into account.

The angular distributions at low energies 6<E<20 MeV/A were calculated in the framework of the dynamical model of deep inelastic collisions [16,17]. This model takes into account both the dissipation of the kinetic energy of the colliding nuclei and the fluctuations of the motion of the nuclei with respect to the classical trajectories, as well as the effect of deformation of the fragments in the exit channel. The results of the calculations for producing oxygen nuclei in the 32 S + 197 Au reaction are presented in Fig. 8. The solid and dashed curves correspond to deep inelastic fragments produced at different ion kinetic energy losses (AE>50 MeV and AE>E-V C ). In Fig. 8, two experimental points from the present experiment are also included. They are in agreement with the calculation corresponding to the small kinetic energy losses. This implies that the differential cross sections can be transformed to total cross sections by integrating over the angular distributions for the small kinetic energy losses.

In the intermediate energy domain, the maximum of the angular distribution of the reaction products lies at 9 = 0°. The transformation of the differential into total cross sections was made on the basis of the angular distributions calculated by the method used in ref. [18]. In the laboratory coordinate system, the angular distributions of products at intermediate energies can be described according to [18] by the expressions:

dQdE A
where A and E A are the mass number and the kinetic energy of the fragment, respectively, E is the most probable value of the energy and c p is the width of the fragment momentum distribution.

The total production cross sections for some isotopes of oxygen obtained by integrating over the angular distributions are shown in Fig. 9 as a function of the energy of the projectiles 32 > 34>36 s. As can be seen from Fig. 9a the cross sections rise for incident energies up to 15-^20 MeV/A. For higher energies it can be speculated that they either go to a plateau ( 15 O) or pass through a maximum after which they tend to decrease ( 19 O, 21 O). The dashed lines present the calculated cross sections of transfer reaction products as a function of the projectile energy. The calculation of the yields of the primary fragments was carried out using the microscopic transport model assuming a binary character of the reaction [15]. The isotopic distributions of the final (experimentally observed) nuclei were calculated within the framework of the statistical theory of decay of excited primary fragments [START_REF] Barashenkov | High Energy Interaction of Particles and Nuclei with Atomic Nuclei[END_REF]. The comparison of the experimental data and the calculations shows (see Fig. 9a) that the contribution from deep inelastic reactions to the cross section of producing both neutron-rich and neutron-deficient isotopes is dominant at low energies, while at intermediate energies the main contribution comes from fragmentation reactions. Nevertheless, as one can see from the figure, at intermediate energies the contribution of multinucleon transfer reactions is still noticeable. 21 O on a gold target by projectiles of different neutron excess, including an experimental point at 75 MeV/A for the S beam; (c) 'Ne on different targets by the 32 S beam. The cross sections for 32> S(75MeV/A) fragmentation were calculated using a modification of the empirical parameterization [14] and are denoted by arrows.

-(E A -2COSQ,JE A E ( 4
The solid curves in the figure are the results of smoothing. The dashed lines present the calculated cross sections of transfer reaction products (normalized to I9 O) as a function of the projectile energy.

On the basis of the experimental data shown in Fig. 9, the yields of different oxygen isotopes as a function of projectile energy were calculated, assuming total absorption in the target of the beam having an initial energy Eo, using the relation: (5) where Vc is the Coulomb barrier, a(£) is the cross section in units of [cm 2 ], Nbeam is the intensity of the beam [pps], M target is the mass of the target nucleus in [mg] and dE/dX corresponds to the stopping power of the projectiles in the target [MeV/ (mg/cm 2 ). The yields of oxygen isotopes, calculated using the expession (5), produced in S-induced reaction on a gold target at a beam intensity of 1 e^A are presented in Fig. 10 as a function of the projectile energy. In the upper panel (a) one can see the yields of 15 O, 19 O and 21 O produced by a 32 S beam; while in the lower panel (b) the yields of the isotope 21 O produced using beams of 32 S, 34 S, and 36 S. From Fig. 10a it follows that an increase in the projectile energy leads to an abrupt rise in the yield of neutron-deficient nuclei ( 15 O), while the increase in the yield of neutron-rich nuclei ( 21 O) is negligible. In Fig. 10b, it can be seen that at energies of up to 20MeV/A the difference between the yields of 21 O produced with different beams is small. At 75 MeV/A this difference increases and amounts to an order of magnitude for the 32 S and 36 S beams.

Taking into account that the intensity of the 32 S (20 MeV/A) beam in FLNR exceeds by an order of magnitude the intensity of the 36 S (75 MeV/A) beam at GANIL, one can expect comparative yields of neutron-rich isotopes within several mass units from the stability line for the beams stopped in the target. However, for producing extremely neutron-rich nuclei with 12 <Z< 15 the use of a 36 S beam is preferable (Fig. 6d).

In addition, as it was mentioned earlier, the isotope production cross sections at energies E<20MeV/A depend on the energy of the reaction (Qgg) and are highest for the smallest absolute values of Ogg (Fig. 7). The latter depend on the target-projectile combination and for the case presented in Fig. 10 ( 32 S +Au) Qgg is not the optimal one (Qgg< -50MeV) for deep inelastic reactions. The target can be chosen so as to have Qgg equal to about -30MeV (e.g. Nb) and then the expected isotope yields may increase several times.

Conclusions

On the basis of the obtained data the following conclusions can be drawn:

• For the energy range 7^-10 MeV/A a quite abrupt decrease in cross section is observed for the light target when the number of transferred protons is increased. At high energies this difference decreases and at intermediate energies it is negligible.

• As far as the dependence of the isotope yields on the target is concerned, there is evidence that at small projectile energies in the case of the heavy target with a large ratio N/Z=\A9, on the average a shift of the centers of gravity of the differential-cross-section distributions occurs in the direction of the neutron-rich region, and vice versa for the light target (N/2=\). At intermediate energies this peculiarity of the isotope yields practically vanishes.

• At small incident energies oscillations are observed in the centers-of-gravity distributions. These oscillatons are smoothed with increasing the energy.

• At high energies the isotopic content of the projectile plays a dominant role for the production of nuclei close to the projectile (Z > 12), while at energies E < 20 MeV/A the production cross sections for the 32 S and 34 S beams are comparable. A difference in cross sections is observed only in the region of nuclei heavier than the projectile, where pick-up reactions prevail.

• The isotope production cross sections are seen to rise for energies up to about 15-^20 MeV/A, after which they either flatten or pass through a maximum and drop for energy regions where fragmentation is expected to prevail.

It is worthwhile noting that for the further investigation of the reaction mechanism it is very important to obtain new information on the yields of reaction products and their angular distributions using 32 S and 34 S beams in the energy range 3OV75 MeV/A and a 36 S beam at 10-50MeV/A.

Fig. 1 .

 1 Fig.1. Identification matrix (A-2Z, Z) of the products of the 32 S(14.5 MeV/A)+C reaction obtained at a magnetic field B = 0.7975 T. The left solid line passes through the completely stripped nuclei with zero isotopic spin. The right curve discriminates between regions of nuclei with charge q = Z and q = 2-1.

Fig. 2 .

 2 Fig. 2. Energy spectra of magnesium isotopes from the reaction 32 S (14.5 MeV/A)+C. The horizontal error bars indicate the energy acceptance at the corresponding magnetic rigidity of the spectrometer. The vertical arrow indicates the primary beam energy. The curves are the result of smoothing

Fig. 4 . 18 Fig. 5 .

 4185 Fig. 4. Experimental isotopic distributions of oxygen, neon, magnesium and silicon isotopes obtained for two targets, 12 C (closed symbols) and 197 Au (open symbols), at a laboratory angle fyatr^0 and at two different energies of 32 S, 9.1 MeV/A (a), and 16.1 MeY/A (b). Calculated distributions for 75 MeV/A are presented for 81^=0° in (c). The curves on (a) and (b) are the result of smoothing.

6 - 14 Fig. 6 .Fig. 7 .

 61467 Fig.6. Centers of gravity, expressed as (N-Z), of the differential distributions of Fig.4, as a function of the atomic number Z: Energy dependence for the 32 S beam for the gold target (a), the carbon target (b); dependence on the projectile neutron excess at an energy of 14.5 MeV/A for a carbon target (c), and calculations for 32,34,36 S at ?

Fig. 8 .

 8 Fig.8. Angular distributions for oxygen isotopes obtained in the reaction 32 S + Au at 16 MeV/A (a) and 9.1 MeV/A (b), calculated in the framework of the dynamical model of deep inelastic collisions [17]. The solid and dashed curves correspond to deep inelastic fragments produced at projectile kinetic energy losses AE>50 MeV and AE>E-V C . The two points are from the present experiment

Fig. 9 .

 9 Fig.9. Total production cross sections for different isotopes as a function of energy. Cross sections for producing of: (a) oxygen isotopes in the irradiation of a gold target by the 32 S beam; (b)21 O on a gold target by projectiles of different neutron excess, including an experimental point at 75 MeV/A for the S beam; (c) 'Ne on different targets by the 32 S beam. The cross sections for 32> S(75MeV/A) fragmentation were calculated using a modification of the empirical parameterization[14] and are denoted by arrows.The solid curves in the figure are the results of smoothing. The dashed lines present the calculated cross sections of transfer reaction products (normalized to I9 O) as a function of the projectile energy.

Fig. 10 .

 10 Fig. 10. Yields of oxygen isotopes in S-induced reactions on a gold target at a beam current of 1 e^A vs projectile energy, (a) Yields of 15 0, 19 O and 2I O in the case of a 32 S beam, (b) Yields of 21 O in the case of S, M S, and ^S beams. Calculations are performed assuming full absorption of the beam in the target.

Acknowledgements

The authors would like to express their gratitude to A. V. Belozerov for the help in carrying out the experiments, and to A.V. Antonenko and T.N. Shneidman for fruitful discussions.

The present work was carried out with the support of the Russian Foundation for Fundamental Research (RFFI) under grant No.96-02-17381a and of the Bulgarian Foundation for Scientific Research under grant No. F503.