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GALAXY COLLISIONS
Curtis STRUCK Department of Physics and Astronomy, Iowa State University, Ames, IA 50010, USA
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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Galaxy collisions Curtis Struck Department of Physics and Astronomy, Iowa State University, Ames, IA 50010, USA Received February 1999; editor: M. Kamionkowski Contents 1. Introduction to colliding galaxies 1.1. Overview 1.2. Orders of magnitude 1.3. Background and early history 1.4. The importance of collisions 1.5. Nature's galaxy experiments 2. Some phenomenology: what's out there? 2.1. Morphological classi"cation of collisional forms 2.2. Physical classi"cation? 2.3. The naming of things 3. Transient events I: some wave morphologies and their causes (Yxx) 3.1. Ring galaxies (YDe0) 3.2. Symmetric caustic waves 3.3. Ring relatives: bananas, swallows and others 3.4. From rings to spirals 3.5. Tidal spirals and oculars (YDx#) 3.6. Fan galaxies and one arms (YDx!) 3.7. Gas vs. stars in waves 4. Transient events II: death and trans"guration 4.1. Transient mass transfer and bridges 4.2. Complete collisional disruption 4.3. Transient summary 5. Coming back (Ixx) 5.1. Dynamical friction } bringing it back 5.2. Simulational examples of dynamical friction
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5.3. Halo braking 5.4. Tidal stretching: tails and antennae (IXd#) 5.5. Shells and ripples 5.6. Induced bars 5.7. Intermediate summary 6. Mergers: all the way back (Oxx) 6.1. Overview and historical highlights 6.2. Major merger dynamics 6.3. Minor mergers: disk heating and aging 6.4. New disks 6.5. Multiple mergers 7. Induced star formation 7.1. Color, H and other indicators of global ? enhancements 7.2. Spectral line diagnostics 7.3. SF region morphologies 7.4. Mechanisms and modes 8. Active galactic nuclei in collisional galaxies 8.1. Phenomenology 8.2. Fueling mechanisms 9. Environments and redshift dependences 9.1. Groups and compact groups 9.2. Dense clusters 9.3. High redshift collisions 10. Conclusions Acknowledgements References
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Abstract Theories of how galaxies, the fundamental constituents of large-scale structure, form and evolve have undergone a dramatic paradigm shift in the last few decades. Earlier views were of rapid, early collapse and formation of basic structures, followed by slow evolution of the stellar populations and steady buildup of the chemical elements. Current theories emphasize hierarchical buildup via recurrent collisions and mergers, separated by long periods of relaxation and secular restructuring. Thus, collisions between galaxies are now seen as a primary process in their evolution. This article begins with a brief history of how this once peripheral subject found its way to center stage. We then tour parts of the vast array of collisional forms that have been discovered to date. Many examples are provided to illustrate how detailed numerical models and multiwaveband observations have allowed the general chronological sequence of collisional morphologies to be deciphered, and how these forms are produced by the processes of tidal kinematics, hypersonic gas dynamics, collective dynamical friction and violent relaxation. Galaxy collisions may trigger the formation of a large fraction of all the stars ever formed, and play a key role in fueling active galactic nuclei. Current understanding of the processes involved is reviewed. The last decade has seen exciting new discoveries about how collisions are orchestrated by their environment, how collisional processes depend on environment, and how these environments depend on redshift or cosmological time. These discoveries and prospects for the future are summarized in the "nal sections. 1999 Elsevier Science B.V. All rights reserved. PACS: 98.54.!h; 98.62.!g; 98.58.Nk; 98.65.At; 98.65.Fz
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1. Introduction to colliding galaxies 1.1. Overview Galaxies were once seen as isolated, mostly unevolving island universes of a few characteristic types, and so rather like classical crystal structures. During the last few decades our view of galaxies has changed drastically. We now believe evolutionary processes are important even in isolated, undisturbed galaxies, and furthermore, that most galaxies are strongly, even violently a!ected by their environment. Speci"cally, as will be explained below, it is now thought that most galaxies experience several collisions or tidal interactions over the course of their lifetime which are strong enough to profoundly alter their structure and accelerate evolutionary processes (see Fig. 1). Thus, collisions and interactions are now generally believed to be one of the primary drivers of galaxy evolution. The processes of galaxy formation and evolution are intimately connected to star formation, and thus, to a variety of other problems of general interest, including: the buildup of heavy elements in the universe, the formation of planetary systems, and the production and distribution of galactic cosmic rays. Hence, subjects as diverse as the solar abundance of carbon, and the rate of biological mutations in `island universesa like our own depend to some degree on the large-scale environment and collisional history of the galaxy. For the moment, however, let us retreat to the realm of extragalactic astronomy. There is another aspect of galaxy collisions, that has not been considered much until recently. This is the use of galaxy collisions as a probe of galactic structure and stability. In brief, because of the universality of galaxy mass and kinematic distributions, it appears that the major structural components of galaxies are individually and collectively in quasi-equilibrium states. Secular evolutionary processes prevent the achievement of complete equilibrium. Moreover, the star formation rate and the distribution of the interstellar gas may be the result of dynamic self-regulation, rather than thermodynamic equilibrium. In any case, it is generally di$cult to learn about the nature of an equilibrium state simply because it is a single state. The usual way to probe such variables in a dynamical system is by studying the response of the system to a perturbation. As we will see in detail below, collisions are the natural disturbances to quasi-equilibrium galaxies and galaxy disks in particular, and nature provides an abundant variety of them. This author believes that the study of collisionally disturbed systems will become the primary means of learning about the physics of star formation which is orchestrated on large-scales, and of the phase balance and other thermohydrodynamical characteristics of the interstellar gas. This article is intended to provide an overview for students new to the subject and nonspecialists from other areas of physical science who would like an introduction to it. I emphasize simple physical descriptions of the phenomena wherever possible, and the physical relationships among di!erent phenomena. In many complex situations where this is not possible, published numerical simulations give us a view of the dynamics, and frequently also provide new insights. In some cases, the theory is not yet su$ciently well developed to allow a good model to be constructed, and it is not yet possible to describe a complete dynamical theory of galaxy collisions. Nonetheless, it appears that there is consensus on many general characteristics, as well as on many of the speci"cs, of such a theory. Therefore, the bulk of this article is structured along a path parallel to the generic dynamical histories of collisional galaxies. The last sections consider collisions in broader environments, but the development of these environments is another closely related temporal sequence.
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Fig. 1. The collisional galaxy system AM1724-622, nick-named the `Sacred Mushroom.a The strong ring wave of the primary galaxy was almost certainly induced by an interpenetrating collision. The structure of the companion galaxy was also strongly disturbed. The connecting `bridgea between the two is made up of stars torn o! one or both galaxies. (Digital Sky Survey image courtesy of AURA/STScI.)
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I will give relatively little attention to details of current numerical modeling and data analysis techniques, since these are emphasized in a number of excellent sources listed later. On the other hand, in a few sections there is a good deal of specialist detail. Newcomers who are not especially interested in such detail will generally "nd overview and summary material at the beginning and end of each major section. This "rst chapter consists mostly of a brief historical introduction, which addresses the questions } why does anyone study a topic so remote from everyday existence, and how did these studies get started? The literature on colliding and interacting galaxies has become too vast to fairly summarize in a single article. Whenever possible this will be a review of reviews. That is, if an up-to-date review exists already I will generally refer the reader there, and limit the discussion here. Inevitably, this means that important papers will be absent from the reference list, a circumstance that is both regrettable and necessary. There are several excellent books or review articles that each cover many aspects of this subject, including the conference proceedings edited by Sulentic et al. (1990), Weilen (1990), and Combes and Athanassoula (1993), and the articles of Schweizer (1986), Noguchi (1990), and Barnes and Hernquist (1992a,b). Two additional publications, containing numerous technical reviews were about to become available as this article was being completed (Friedli et al., 1998; Sanders, 1998). The author has had access to preprint versions of some of these articles, but not to the books as a whole. There are also popular books with some coverage of the subject, including Ferris (1980), Parker (1990), and Malin (1993). Parker's book, in particular, o!ers an interesting historical summary. 1.2. Orders of magnitude Before proceeding, we should de"ne the term `galaxy collisiona, which has so far been used quite loosely. In fact, it will be used in a very general sense in this article to indicate any close encounter that has a signi"cant e!ect on one of the galaxies involved. The term `tidal interactiona is more commonly used in the "eld, because the tidal gravity forces are responsible for the most signi"cant e!ects. These forces are able to generate spectacular e!ects without involving the actual intersection of the visible parts of the two galaxies at closest approach. A near miss is as good as a hit in this "eld. Still, it might seem that this general de"nition of the term collision is misleading. However, "ne distinctions are not necessary for present purposes, and in fact, may be misleading themselves. As we will see below almost all signi"cant `interactionsa involve the intersection of either the dark halos of the individual galaxies, or the mediation of a common group halo. It should also be noted that the adjective `signi"canta is highly context dependent, as will become clear below. Galaxy collisions involve a tremendous amount of energy. Two objects with masses of the order of 10 solar masses or 2;10 kg meet with typical relative velocities of about 300 km/s, so the collision energy is of order 10 J. This energy is equivalent to about 10\ supernovae, e.g., a number of supernovae that ultimately can be produced in the merger of the two galaxies. Despite the large energy, the modest encounter velocity (about 0.1% of the speed of light) means that this is not a high energy phenomenon in the usual collisional physics sense. Nonetheless, because the energies are comparable to the binding energies of the galaxies, collisions can have very important evolutionary e!ects.
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Galaxy collisions are extremely slow by terrestrial standards, with typical timescales of order 3;10 yr, or 10 s. There is little hope of observing any of the dynamics directly. Thus, when we look at the images of the hundreds of suspected collisional galaxies that are su$ciently nearby to observe in some detail what we are seeing is a snapshot gallery of systems characterized by a wide variety of structural and collisional parameters. Moreover, these systems are caught at random times in the interaction. This fact is one of the main reasons why it is so di$cult to interpret the observations, and arrange the systems in a physical classi"cation scheme. This fact reappears in many di!erent guises below. We shall see below that much of the collision energy is redistributed or dissipated over the interaction timescale. The dissipation rate is of order ¸"E/q+10 W. This is about the peak luminosity of one (bright) supernova. It is somewhat less than the output of a typical starburst resulting from the collision, and less than the luminosity of most quasars and other active galactic nuclei. One of the most fascinating aspects of galaxy collisions is the fact that most of the matter involved does not collide with anything. In the "rst place, most of the mass in a typical galaxy consists of collisionless dark matter. Thus, dark matter from the companion galaxy passes through that of the target with no e!ects except for those due to their collective gravitational forces. Similarly, there is only a very small probability for direct star-star collisions. The cross section of a star like the Sun is about 10 m, while the surface density of stars near the Sun is of order 10 per light year squared (10\ m\). This implies that the collision probability is of order 10\ for a typical star. The stellar density is much greater in the centers of galaxies, but the basic point is not changed. On the other hand, the warm neutral components of the interstellar gas in the disk of our galaxy have a large "lling factor (e.g. Dickey and Lockman, 1990; Combes, 1991). A similar conclusion is implied by the fact that the surface area of low density holes in the neutral hydrogen gas is small in other late-type galaxy disks (Brinks, 1990). Moreover, the "lling factor of the hot coronal or halo gas surrounding the thin cold disk in these galaxies is probably essentially unity (see e.g. McKee, 1993). Thus, there must be direct collisions between the various gas components when two gas-rich disk galaxies collide. The nature of collisions between gas elements in the two galaxies depends a great deal on their thermal state. Collisions between cold clouds will be highly supersonic, e.g. characterized by Mach numbers of order 300 for clouds with a mean temperature of 100 K colliding with a relative velocity of 300 km/s. At the other extreme, the sound speed in the coronal gas is of order 100 km/s, so collisions between gas haloes at such velocities will be transonic. The tidal forces in galaxy interactions which do not include direct collisions may drive waves at supersonic velocities relative to the cold gas within the disk, but the e!ects are less extreme. 1.3. Background and early history The Milky Way and the Magellanic Clouds may have experienced a tidal encounter within the last 10 years (e.g. Wayte (1991) for a brief review). Thus, there are naked eye colliding galaxies, though the e!ects on the Milky Way cannot be observed by simply stepping out into the backyard. One of the "rst `spiral nebulaea discovered with the telescope, the relatively nearby M51 system (albeit at a distance of about 9 megaparsec or 2.8;10 light years) has also been shown to be a collisional system (see Byrd and Salo (1995) for a review of current thoughts on the nature of
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Fig. 2. The Hubble `tuning forka galaxy classi"cation scheme (from Hubble, 1958).
the collision). The serious scienti"c study of galaxy collisions began in the wake of the early e!orts to discover a morphological classi"cation system for galaxies, and the great accumulation of imagery that resulted from these e!orts. I will not attempt a thorough account of the discovery of galaxy collisions here, though I think it would make a very fascinating subject for a trained science historian. However, I cannot resist some anecdotal sampling of the history, especially since it sets the scene for later developments (also see Parker, 1990). The work of Hubble and his colleagues in the "rst half of this century had two primary motivations. The "rst and best known was the desire to see whether the form of the relation describing the expansion of the universe, now called the Hubble Law, varied at greater distances and lookback times. The second motivation, which Hubble was interested in from an equally early date, focussed on "nding prototypes of important classes of galaxies, with the ultimate hope that, as in stellar astronomy and many other sciences, evolutionary connections between classes would become apparent. A major result of this work is well-known Hubble tuning fork scheme, which is essentially the periodic table of extragalactic astronomy. This system is reviewed in every elementary textbook, and described in detail in The Hubble Atlas of Galaxies (Sandage, 1961; Sandage and Bedke, 1994). The tuning fork diagram from Hubble's book (1958, originally 1936) is reproduced in Fig. 2. The handle of the fork consists of increasingly #attened elliptical galaxies, which are dominated by old stars and have little gas, dust, and young stars. The tongs of the fork consist of two parallel sequences of spiral or disk galaxies, one with a stellar bar component, one without. In each disk sequence the prominance of the stellar bulge component relative to the star-plus-gas disk decreases to the right (e.g., from Sa to Sc). Generally, the gas fraction and young star population increases from Sa to Sc. Galaxies in the transition class S0 and the Sa class are called early-type galaxies, while those in the Sc class are called late-type. Spiral arms tend to be more tightly wound about the center in the early types and more open, but also more irregular or #occulent in the late types. This capsule description does not represent either the original classi"cation criteria, or the modern understanding of these galaxies very well, but it is su$cient for present purposes. (See the review of Roberts and Haynes (1994) for a modern understanding.)
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In the classi"cation studies galaxies were discovered that did not "t into any of the major categories (e.g., any of the Hubble types), and whose morphologies were unusual, weird, or `peculiara. In later years it was demonstrated that while the `peculiara bin contained a wide variety of objects, it included a number of galaxies distorted by collisions. It is interesting to look at how some examples, most of which are now very well-known and well-studied collisional systems, were described at the time. It seems that Hubble put a number of these galaxies in his `Irregulara category, though he describes them as `highly peculiar objectsa (1958, p. 47). As examples he cites NGC 5363, NGC 1275, and M82. He further suggests that `Almost all of them require individual consideration but, in view of their very limited numbers, they can be neglected in preliminary surveys of nebular formsa (1958, p. 48). Another early general description of the classi"cation program is provided by Shapley's book Galaxies (1943, also revised edition 1961), and there we "nd a little section entitled `Remark on Freaksa. Shapley states, There are also plate spirals2 and frankly `pathologicala types, (as Baade calls such freaks) like NGC 51282 and the ring-tail system, NGC 4038-9, shown in Fig. 97. The theories that su$ciently explain the relatively simple looking Sc spiral, like Messier 33, and the most common galaxies in Virgo, must have su$cient #exibility to take care of these aberrant types. The interpreter may need to resort to the assuming of collisions to "nd satisfactory causes. He will "nd some justi"cation, because the individual galaxies are not so far separated but that encounters may have been fairly numerous, if the time scale has been long enough2 We are only at the threshold to the house of galactic knowledge, and within there are doubtless many dark and di$cult rooms to explore and set in order. Shapley's point about the frequency of galaxy companions was echoed by Baade: Hubble and I had a long-standing bet of $20 for the one who could "rst convince the other that a system which he found was single. We could never decide the bet; neither of us could pull out some distant fellow } in some cases there really was a companion and in other cases there could be. So single galaxies may be rare. (Baade, 1963) Shapley's comment about collisions probably was not a random speculation. It is likely that he was aware of Holmberg's (1941) article. This is evidently the "rst paper to present models of galaxy interactions. Holmberg's technique was to use essentially an analog computer consisting of light bulbs and photocells. The 1/r fallo! of light intensity was to represent gravitational forces. In modern terms this was equivalent to an N-body simulation with N"37 per galaxy (74 total), and crude time di!erencing. Nonetheless, the expected tidal deformations were con"rmed. This achievement (together with Holmberg's earlier paper (Holmberg, 1940)) can be taken as the beginning of the theory of galaxy collisions. This seems a fair assessment even though the work described in these seminal papers already had deep roots in the Scandanavian school. For example, Toomre (1977) provides a quote from Lindblad's (1926) conjectures on (gas dynamical) galaxy collisions. Zwicky (1959) also indicates that Holmberg's work carried on Lundmark's studies of multiple galaxies, which dates to around 1920. Toomre (1977) also emphasizes Chandrasekhar's early work on dynamical friction: `tucked away in several 1943-vintage appendices to Chandrasekhar's (1942) booka. However, it was some time before this work was applied to galaxy collisions (see Toomre, 1977).
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A decade later Spitzer and Baade (1951) extended Holmberg's work by considering the removal of interstellar gas (a process now called `strippinga) in high velocity collisions, which they argued should be common in dense clusters of galaxies. The stage was set for what is argueably the seminal observational paper in this "eld, Baade and Minkowski's (1954) work announcing the discovery that `the radio source Cygnus A is an extragalactic object, two galaxies in actual collisiona. In the next few years a number of other bright radio sources were recognized as collisional systems (see review of Zwicky, 1959, Sec. V). Yet the full rami"cations of these discoveries (including galaxy mergers and super-starbursts) would not be appreciated until the late 1970s and the 1980s. There are many reasons for this, including the immaturity of infrared detector technologies, and the startling discovery of quasars, which received a great amount of attention in the 1960s and 1970s. Ironically, the morphology of Cygnus A was subsequently shown to be largely a result of dust obscuration of an active nucleus rather than an ongoing galaxy collision. Yet, recent observations suggest a past merger (Stockton et al., 1994). In fact, until recently the study of colliding galaxies has been a little traveled country lane even within the world of extragalactic astronomy. Many of the great names in the "eld in the "rst half of the century had contributed, but generally only as a spino! from other e!orts. (The exceptions are Holmberg and Zwicky, who devoted much e!ort to `multiplea galaxies.) The primary reason for this is the rarity of these morphologically peculiar galaxies. These `freaksa were not only too few to seem important, but they provided too few snapshots to enable a coherent picture of the dynamical processes to be synthesized. To a large degree this is still true, but now computer simulations can "ll in the missing frames. Many well-known collisional galaxies were discovered serendipitously, rather like dinosaur fossils, when selected areas or individual nebulae of unknown type were imaged with large telescopes. Many of the `nebulaea came from Dreyer's New General Catalog (NGC) and his later Index Catalog additions to it (see modern version of Sulentic and Ti!t, 1973). A systematic observational imaging program could have discovered many of the `freaksa at a much earlier date. However, no such search was performed before the Shapley}Ames photographic survey of all galaxies above a certain limiting brightness. (A "rst survey went to 13th magnitude, and a later partial survey to magnitude 17.6, see Shapley (1943, 1961).) Work on this catalogue has been continued by de Voucouleurs and de Vaucouleurs (1964), and later editions.) Shapley was clearly impressed with some of the forms discovered, like the `ring-tailsa NGC 4027 and NGC 4038/9 (now known as the Antennae and featured on the Nov. 3, 1997 cover of Newsweek magazine). Not all of the discoveries were NGC objects, one of the relatively early discoveries was the beautiful `Cartwheela ring galaxy discovered by Zwicky (1941). Zwicky was very interested in `interconnecteda galaxies, undertook his own surveys, and made many other discoveries (see Zwicky, 1959, 1961, 1971). Fig. 3 provides a summary of the morphologies he studied, and a preview of systems described throughout this article. However, with the completion of the Palomar all-sky Schmidt camera survey, which went deeper than the Shapley (deep) survey, it became possible to carry out new searches capable of discovering many `freaksa. H. Arp undertook the search for peculiar galaxies in the Palomar survey, and published his now famous atlas (Arp, 1966). A similar cataloging project was carried out by Vorontsov-Velyaminov and collaborators (1959), and also by Vorontsov-Velyaminov and Krasnogorskaya (1961), Vorontsov-Velyaminov (1977), though he was skeptical of the idea that most of these disturbed systems were the result of tidal interactions. A great many of the objects in Arp's
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Fig. 3. Montage of collisional forms, and speci"cally `bridges and "lamentsa from Zwicky's (1959) review article on `multiple galaxiesa.
beautiful photographic atlas are colliders, and it has provided a starting point for many subsequent studies. Once the still more sensitive southern sky survey was completed, Arp and Madore (1987) produced a southern hemisphere atlas, with many more objects. These works, especially the original Arp atlas, `launched a thousanda observational and theoretical studies, and remain invaluable resources in this "eld. On the theoretical side, Holmberg's exploratory work was followed up with the "rst computer models. Early works included the papers of P#eiderer and Siedentopf (1961), P#eiderer (1963), Tashpulatov (1969, 1970), and Yabushita (1971), which are reviewed in the Introduction of Toomre and Toomre (1972, 1974). It was the Toomre's work, which used the restricted three-body approximation to compute the e!ects on the orbits of disk stars in tidal interactions, that had the greatest impact. Although the Toomres noted that a number of their results were presaged in the earlier works, their work assembled all the available pieces to make a compelling case for the
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hypothesis that many peculiar galaxy morphologies were the result of tidal interactions. Other papers, including Wright (1972), Clutton-Brock (1972), and Eneev et al. (1973), also presented similar, and con"rming, numerical results. Most of these projects seem to have begun independently, though Wright acknowledges communications from the Toomres in his paper. Another important strand of this fabric was the analytic work on the Impulse Approximation of Alladin and collaborators in the 1960s and 1970s, which will be discussed below (see references in the review of Alladin and Narasimhan (1982)). From this point up to the present the "eld has grown very rapidly, and expanded in many directions, making it impossible to capture all the important developments in a brief historical summary. In fact, it is impossible to present all these developments in this modest review, so what follows will be a sampling. 1.4. The importance of collisions Before delving into the details of the more recent research, however, we should explicitly state some of the motivations for this activity. As described above, it was becoming clear by the mid-1970s that many of the morphologically `peculiara galaxies, i.e. those that didn't "t into the standard classi"cation schemes, could be accounted for as the result of tidal interactions. But from the beginning it was clear that these galaxies are rare, and so we might wonder, what is their importance? To roughly estimate their `rarenessa consider two catalogs based on the Palomar northern sky survey. The Zwicky (1961) catalog of `Galaxies and Clusters of Galaxiesa has some 30 000 objects, while the Arp atlas has 338 interacting pairs or groups. This implies that colliding galaxies are of order 1% of all galaxies. However, this estimate is too `rougha, as we will see later in this section. Toomre and Toomre (1972) o!ered some, at the time speculative, suggestions on these matters that generated much interest, and, in fact, ultimately became the dominant ideas in the "eld. These ideas were based on, but extended well beyond, the results of their collisionless (star-like) test particle simulations. To begin with, they noted that the observed tails and plumes were successfully reproduced in models involving close collisional encounters, and that such events were unlikely to be the result of galaxies approaching on random hyperbolic orbits. They argued that such collisions were more likely to occur between galaxies on eccentric, bound orbits. They then took the argument a step further and suggested that such large-scale tidal distortions must be formed at the expense of orbital energy, so that the two galaxies must inevitably merge (see discussion and early history in Toomre (1974)). A third deduction follows: 2Would not the violent mechanical agitation of a close tidal encounter } let alone an actual merger } already tend to bring deep into a galaxy a fairly sudden supply of fresh fuel in the form of interstellar material, either from its own outlying disk or by accretion from its partner? And in a previously gas-poor system or nucleus, would not the relatively mundane process of proli"c star formation thereupon mimic much of the `activitya that is observed? (Toomre and Toomre, 1972). E.g., interactions and mergers may funnel interstellar gas into the central regions of galaxies, and trigger enhanced star formation. Speci"c mechanisms are not described in detail, though a couple are implied. First of all, there are the direct tidal e!ects, the `mechanical agitationsa of the quote.
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Secondly, in the paragraphs following the quote they propose that the encounters might drive strong spiral waves as in M51. These waves could also enhance angular momentum transport and channel gas inward. As we will see below, there is now strong observational and theoretical support for these ideas about merging, fueling galactic centers, inducing proli"c star formation, and the more general notion that these processes can have a profound impact on the evolution of the individual galaxies. An important study by Larson and Tinsley (1978) provided early observational con"rmation of star formation enhancements in interacting galaxies. Larson and Tinsley studied the broad band optical (UBV) colors of the Arp atlas galaxies, and compared them to those of the Hubble atlas, assuming that the `normala galaxies of the latter could serve as a control sample. They further produced a grid of color evolution models for aging stellar populations with a variety of star formation histories. These ranged from cases with a constant star formation rate over 10 billion years (10 Gyr), to models of populations with all their stars formed in a relatively short burst (e.g. of duration 0.02 Gyr). The colors of the burst models, and of combination models with a signi"cant burst component, evolve signi"cantly in the "rst Gyr after the burst. Thus, large color variations were predicted in galaxies with signi"cant recent bursts of star formation, and indeed, they found that, Normal galaxies have colors that are consistent with a monotonically decreasing SFR2 In contrast, the peculiar galaxies have a large scatter in colors that is consistent with bursts as short as 2;10 yr involving up to (about) 5% of the total mass. Nearly all of this scatter is associated with galaxies showing evidence of tidal interaction2 These results provide evidence for a `bursta mode of star formation associated with violent dynamical phenomena. (Larson and Tinsley, 1978) In the succeeding years a great deal of evidence was obtained in a wide range of wavebands to support the conclusion that collisions and interactions frequently drive a much enhanced star formation rate, though there are exceptions to the rule. We will take up this topic again in a number of sections below; also see the reviews of Keel (1991), Barnes and Hernquist (1992a,b), Elmegreen (1992), Mirabel (1992) and Kennicutt Jr. (1998a). In the quote above, Toomre and Toomre raise another important issue } the connection between galaxy collisions and nuclear activity in galaxies. Like the question of collisionally induced star formation, this topic has received much attention in the last couple of decades, and we will summarize this story in Section 8. For the moment we merely note that such a connection has remained much more elusive than in the case of induced star formation. The question posed by Toomre and Toomre, whether extreme star formation in the central regions might mimic nuclear activity, has also been revived in recent years (e.g. Terlevich et al. (1992a, 1992b), Terlevich (1994), and references therein). For example, a vigorous debate developed around the question of what powered the ultraluminous, infrared galaxies discovered through the analysis of IRAS (Infrared Astronomical Satellite) data, enormous starbursts, active nuclei or both (see the review of Heckman (1990) and other papers in those proceedings). It is believed that these galaxies are primarily merger remnants, so in any case collisions were implicated. In sum, there is now nearly overwhelming evidence from observations, and numerical models that collisions can strongly disturb the morphology and evolution of the galaxies involved, both by direct gravitational `agitationa, but also indirectly by driving strong star formation. As we will see below, the latter process leads to the conversion of large quantities of gas to stars, the creation of
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whole new stellar populations, and massive changes in the distribution and thermal phase balance of the remaining gas. We will also see that there is now strong evidence to support the Toomres' conjecture that most collisional encounters are mere preludes to the eventual merger of the galaxies. However, for the moment we will leave the merger issue, to reexamine the second fundamental question } how uncommon are galaxy collisions? While they are rare on the sky, we now know that they profoundly a!ect the galaxies involved, so the relevant question is } how likely is it that a galaxy will experience a signi"cant collision in its lifetime. Toomre (1977) noted that in many of the systems with conspicuous tidal tails the centers of the two galaxies were very close. This fact, and the other available evidence, suggested that these galaxies were nearly merged. He provided 11 outstanding examples, and he estimated that of order 10% of the galaxies had participated in major merger sometime in their lives. Noting that this was close to the fraction of elliptical galaxies, and that induced star formation (and other processes) would tend to change a merger remnant to an earlier Hubble type than that of its predecessors, he speculated that most ellipticals might be formed from spirals in mergers. We will not be able to explore all of the huge literature that has grown up around this hypothesis in subsequent years, the reader is referred to the review of Hernquist (1993, and chapter 6). What is important for present purposes is the fact that, while on one hand this hypothesis generated a strong and long-lived debate (e.g., Ostriker, 1980; Parker, 1990, pp. 198}202, and the articles in the "nal section of Weilen, 1990), on the other hand, it fractured the concensus that strongly interacting galaxies were very rare, and thus, unimportant. The earlier concensus rested on the assumption that random galaxy collisions would be unlikely, since the mean distance between galaxies is large compared to their sizes. What the Toomres and others at the time discovered was that the evidence suggested that most collisions occurred between galaxies in groups that were at least loosely bound. I.e., collisions were built into the initial conditions, galaxies are born in groups. So why did they appear to be rare? Because the collision timescale is less than an order of magnitude of the age of the universe (Toomre (1977), results of more recent numerical work are reviewed in Barnes and Hernquist (1992a,b)). Thus, even if collisions happened at random times we would never see more than a fraction of them. Actually, as Toomre pointed out, there are good reasons to believe collisions were much more common in the distant past. We are in an age of increasing studies of high-redshift galaxies by the Hubble Space Telescope and a new generation of ground-based telescopes, and preliminary results indicate that this is indeed the case (see several relevant articles in Benvenuti et al. (1996) and Section 9.3). Finally, we note that while Toomre focussed on the extreme case of mergers between two (equally) massive progenitors, where the disruption and dynamical heating is great enough to form an elliptical galaxy, collisions and mergers between unequal partners are probably more common. Such collisions, and eventual mergers, can still have a dramatic e!ect on the evolution of the larger galaxy, as we will see below. It has also been realized in recent years that even the assumption that almost all collisions involve only two galaxies may be incorrect. Again this is not the result of chance, but of the overall collapse of (loosely) bound groups of galaxies (see chapter 9 and the recent papers of Governato et al. (1996), Weil and Hernquist (1996), and references therein). Indeed, Weil (1994) (Weil and Hernquist, 1996) has recently discovered that numerical simulations of multiple mergers can produce remnants that more closely match the structural and kinematic details of some ellipticals than binary merger remnants.
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These theoretical ideas have some profound observational consequences, most of which will be explored later in this article. First, the galaxies with extreme morphological disturbances may be only the tip of the iceberg as far as collisions are concerned, because although spectacular, this is a short lived phase. Then the question becomes, what evidences of collisions can be found later? The elliptical galaxies with faint shells or ripples around them, discovered by Malin and Carter (1980, 1983), and interpreted as tidal debris by Quinn (1982, 1984), provide one of the best examples (see Section 5.5). Seitzer and Schweizer (1990) found that 32% of the S0 galaxies and 56% of the elliptical galaxies in their sample have `ripplesa. As a second example Scoville (1994) estimates that in the last 10 yr 2% of the spiral galaxies became luminous infrared galaxies as a result of a merger or strong interaction. The arguments Toomre applied to elliptical galaxies would suggest that the overall fraction of spirals experiencing such an event in their lifetime might be at least 10 times larger. (On the other hand, if most of these spirals turned into ellipticals there would be uncomfortably many, so simple-minded extrapolation may be dangerous here!) A third and "nal example, Odewahn (1994) "nds that all but 4 in his sample of 75 Magellanic (very late type) spirals have close neighbors, so tidal interactions are likely to be very important in this class. In sum, collisions can profoundly in#uence the evolution of the individual galaxies involved in a cosmologically short time. While violently disturbed galaxies appear rare on the sky, a collision may occur between one and several times in the life of a typical galaxy. There are increasingly strong theoretical motivations and observational indications that collisions are one of the most important processes in galaxy evolution. 1.5. Nature's galaxy experiments There are many aspects of galaxies about which we know relatively little, including the dynamics and thermal processes in the interstellar gas, which are almost certainly coupled over a large range of scales by turbulence (e.g. Scalo, 1990). Similarly, there is a great deal to learn about the mechanisms of star formation in galaxy disks, especially large-scale, wave-driven star formation. Colliding galaxies can be viewed as nature's own experiments, ideally suited to probe structure and dynamics, in much the same way that accelerator experiments probe the micro-world. As Arp (1966) put it, The peculiarities of the galaxies2 represent perturbations, deformations, and interactions which should enable us to analyze the nature of the real galaxies which we observe and which are too remote to experiment on directly. 2 From this range of experiments which nature furnishes us, then, it is our task to select and study (those) which will give the most insight into the composition and structure and the forces which govern a galaxy. Collisional perturbations come in a range of strengths, depending on the mass and compactness of the galaxies, and the distance of closest approach (see Section 2). There is a perturbative limit, where a low-mass companion interacts with a massive primary. This case is especially interesting for studying interstellar gas dynamics and induced star formation, because the primary disk is disturbed, but not disrupted, in a single encounter. At the other end of the scale, there are mergers between nearly equal progenitors, which test the nonlinear stability of all components of the galaxies. The outcome of all types of collision depends on the structure of the dark matter halos, so
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at least statistically, the comparison of models and observation can provide information on these halos. Examples of all of these applications of the `experimental viewpointa will be given below. This viewpoint also comes naturally with numerical modeling, where we can do experiments on `galaxiesa. It should already be clear from the brief history above that the close interaction between computer modeling and observation has been the key to progress in this "eld, and it will be a recurring theme in the rest of this article.
2. Some phenomenology: what's out there? 2.1. Morphological classixcation of collisional forms The short answer to the question above is } a tremendous variety of forms. This is perhaps the place for the reader to put down this article and peruse the Arp or Arp}Madore atlas for an hour or so, if he/she has not done so before! These galaxies would not have been tossed into the `peculiarsa bin if they could have been "tted into simple scheme like the Hubble classi"cation. On the other hand, the instinct to classify and order is strong, and all of the major collectors of images of these systems } Zwicky, Verontsov-Velyaminov, Arp, Madore } developed some phenomenological system. For example, Arp (1966) arranged his 338 atlas galaxies into 37 descriptive catagories. Arp and Madore (1987) used 24 similar, but simpli"ed categories. Vorontsov-Velyaminov (1977) has a longer list of descripters. Many of the terms used in these atlases, along with a number of other anecdotal descriptions, are now used commonly, but in no uniform way. Schweizer (1990) has proposed a simple, elegant alternative. If the Hubble sequence from early-type ellipticals to late-type spirals is like a continuum line, then Schweizer's classi"cation of collisional galaxies consists of the plane of Cartesian pairs of progenitor galaxies. Basically, he uses a simpli"ed Hubble sequence of three types: ellipticals E, disk galaxies D, and gas-rich irregulars G. Given the di$culty in determining the detailed nature of the progenitors in many cases, this simpli"cation seems imminently justi"ed. Schweizer also adopts a clever device } upper case letters to indicate the large or primary galaxies in the system (e.g. DD to indicate a collision between two comparably sized disk galaxies), and lower case letter to indicate small companions (e.g. Ed for a collision between a small disk and a large elliptical). Empiricists could object that this scheme is not a direct classi"cation based only on features that are seen in the (optical) image of the galaxy. It requires an inference as to the type of the progenitors. Moreover, it assumes that the progenitors had a de"nite Hubble type, and that the distortions are mostly the result of the collision. However, the weight of the evidence overwhelmingly favors this assumption in most cases. In addition, in many cases the classi"cation doesn't depend as much on interpretation as on the use of other types of data, such as 21 cm observations of gas content, or kinematical data to detect a disk component. The advantage of the earlier descriptive systems is that they highlight the transient morphologies of the ongoing collision, whereas there are many morphologies within each category of Schweizer's scheme. Some of these morphologies are very short-lived, and others long-lived (see below), so (relative) lifetimes are also a natural parameter for classi"cation. In a study of color trends among tidal features in interacting systems Schombert et al. (1990) used a very simple scheme (bridges, tails, mergers) based on this idea. Schweizer and Seitzer (1990, 1992) have developed
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a semiquantitative system to measure tidal features in isolated (but presumably merged) ellipticals and S0 galaxies. Other classi"cation systems have been developed for speci"c purposes, but these examples illustrate both the di$culty of developing an all encompassing system, and the utility of specialized systems. 2.2. Physical classixcation? Nonetheless, should it not be possible to give a relatively complete physical characterization (including the nature of the progenitors and the collision parameters) of any collisional system by comparison to computer simulations? In principle this may be true, though there are many practical di$culties. Uniqueness theorems apply to the relevant stellar and gas dynamical equations, if the initial conditions are given with in"nite accuracy. In practice, uniqueness at practical particle, spatial and velocity resolutions remains unproven, except in a few speci"c applications. Moreover, the theorem does not apply to two-dimensional projections (onto the sky) of threedimensional results. In many cases this point is a mere academic quibble } there is no di$culty in distinguishing an Arp}Madore `Sacred Mushrooma type collisional ring galaxy from a welldeveloped merger remnant in either observations or simulations. However, less extreme cases are not so clear (see below). The question of uniqueness is also evolving as resolution increases in both observational and modeling realms. However, even assuming that collisional forms are unique functions of progenitor orbital and structural parameters, their great multiplicity is a problem. Consider this multiplicity from a `"rst principlesa viewpoint. The orbital parameters include a couple of angles of attack specifying the approach direction, and an impact parameter specifying the closest approach of the two centers. The amplitude of the collisional e!ects depends on the mass ratio of the two galaxies. Absolute timescales for the evolution of these e!ects depend on actual masses, but the actual values are not important to identify morphologies, so we only need to include the mass ratio among our parameters for the present. Most collisions are relatively quick and impulsive, but the magnitude of the e!ects does depend on the relative velocity at closest approach (see e.g. Binney and Tremaine (1987, Section 7.2), henceforth BT). Finally, the orientation of the spin axis of each galaxy disk, relative to the axis of the relative orbit, also e!ects the outcome. This introduces a bare minimum of one more parameter, assuming only one galaxy has a disk. (Two orientation angles would obviously be better, but the primary e!ects depend on whether the orbit is prograde or retrograde relative to the disk spin.) Thus, we have a total of 6 or more orbital parameters. Next we must consider structural parameters. The most important of these are the mass ratios of di!erent components, e.g. bulge/disk/stellar bar/halo. The gas fraction in the disk is also important. The scale length of the various components, or relative compactness, is another important parameter. There are many other structural parameters, e.g. the velocity dispersions (or temperatures) of the various components, or their density pro"les. Velocity dispersion is probably not independent of the other parameters, in fact, masses and scale lengths may not be independent. Moreover, the density pro"les may be su$ciently universal that they can be omitted as a parameter. Nonetheless, we have a minimum of about 7 structural parameters. We see that, if we consider a minimal grid of models, with say, 10 values for each parameter, we would need of order 10 computational runs. Then, of course, time is a crucial parameter. Simulations generally show great temporal variations over the course of e.g. a Gyr run, so we
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would probably want to look at 30 or more snapshots to represent the whole evolution. More if we want to see more than one orthogonal view at each timestep. This is beginning to make for a very large family photo album! We have also been assuming that collisions only involve two galaxies at a time, but encounters involving three or more galaxies are probably not uncommon in groups and clusters (see chapters 1, 9). Consideration of three body collisions would clearly enormously increase the number of runs needed for a complete catalog. It seems that direct physical classi"cation by reference to a complete numerical atlas with adequate resolutions is not the way to go. There are too many possible computer realizations to assemble a complete atlas of all collisional systems. This result has a number of nasty corollaries, including } the perfect computer match to a given system may be lost in an unexplored part of phase space. This can be true even if you've already found a `perfecta match, because uniqueness isn't guaranteed at any given level of resolution. Another interesting corollary } a hypothetical computer atlas like that described above would contain far more snapshots of collisional galaxies than will be observed on the sky within the foreseeable future. In fact, since there are of order 0.01 average galaxies per cubic megaparsec, and the volume of the observable universe is of order 10 Mpc, it would contain orders of magnitude more images than there are galaxies in the universe. All the more so since not all galaxies are interacting, and we only get one viewing angle of each system. These realities certainly in#uence the choice of problems studied, and the progress made on these problems. For example, the multiplicity and uniqueness appear to signi"cantly complicate studies of collisional shell galaxies, where observation-simulation comparisons are an intrinsic part of the research (see Section 4.3). They did not have much of a role in early simulations of mergers (see Section 4.2), where the basic questions of how long does merging take, and what is the approximate structure of the remnant, did not depend on many details. Now that this research has advanced to a much higher level of detail (see Hernquist, 1993; Barnes, 1998), they may be becoming important. Nonetheless, the use of distinctive landmark morphologies and common sense rules make it possible to divide up the parameter space into more tractible regions. This judgement seems to be con"rmed in the work of Howard et al. (1993), who actually assembled a quite extensive atlas of N-body simulations with 86 runs and 1700 snapshots. (Even so, they had to incorporate several substantial simpli"cations, including a rigid, inert gravitational potential for the companion galaxy and the halo of the primary galaxy, a two dimensional disk, i.e., not including warps and distortions, and an extremely approximate treatment of gas dynamics in the disk.) Howard et al. discussed a number of generalizations and rules of thumb derived from these simulations, especially with regard to prograde versus retrograde and direct (perpendicular) collisions. Other extensive simulation projects have been carried out and will be discussed below. Some very useful insights into the role of the collision parameters (and into the uniqueness problem) are provided by Gerber and Lamb's (1994) work. This paper was primarily a comparison of semi-analytic kinematic models to fully self-consistent simulations, in the restricted setting of collisions between a small companion approaching on an orbit nearly perpendicular to the primary disk, though with a range of impact parameters. Gerber and Lamb pointed out that the perturbation in the kinematic models (which were found to match the simulations well at early times), depended on four dimensionless parameters. The "rst of these is just the time scaling. Another two are the dimensionless impact parameter, and a dimensionless measure of the compactness of the companion. Finally, there is a strength parameter equal to 2GM /(b
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the companion, b the impact parameter, < the relative velocity at impact, and v the circular velocity in the primary disk (assuming a #at rotation curve, i.e. v "constant). This parameter is the velocity disturbance derived from the Impulse Approximation (see Binney and Tremaine, 1987, Section 7.2 and references therein), divided by the disk orbital velocity, which is proportional to the escape velocity. Because it was derived for a restricted application, this set of dimensionless parameters is incomplete, and also some assumptions about the structure of the two galaxies are embedded in the parameters. However, it is a very good example of how to attack a limited part of the problem. 2.3. The naming of things I will conclude this section with an attempt at a classi"cation scheme that tries to pull together some of the insights from the older morphological systems as well as those derived from analytic and simulation studies of the e!ects of varying collision parameters. The speci"c motivation for attempting this here is to provide simple categories, which highlight the relationships between and the natural ordering of the many individual cases discussed in the rest of this article. The system is essentially a modi"cation of Schweizer's system, with the addition of a few variables that are directly related to both the physical quantities characterizing the collision and to observable characteristics. Because there are a number of collisional forms that are easily identi"able as either short-lived or old, the "rst variable is time. The actual development timescale of collisional features depends on the masses of the galaxies, and galaxy masses can range over at least 6 orders of magnitude. Thus, the presence or absence of specixc features can only yield relative timescales. Age determinations of collision-induced stellar populations might be more accurate in some cases, especially when a single short starburst dominates. (However, in many cases the enhanced star formation may be of long duration, or there may be multiple bursts.) But whatever dating technique is used, for present purposes we limit this variable to three values: young, intermediate, and old (Y, I, O). In collisions the dynamically cold disks respond the most strongly and the most quickly to the disturbance, so the relative mass of the disk component is signi"cant as a "rst gauge of the magnitude of the collisional e!ects. If there is a substantial disk, the gas fraction in the disk is also important. The dynamically cool/hot component ratio and the gas fraction are related, and related to other quantities (as in the Hubble sequence), so it su$ces to adopt Schweizer's Hubble pairs scheme. However, I would prefer to modify his de"nition of category D to include only early-type disks (S0-Sb), and of category G to cover all the later Hubble types. (There is increasing evidence that most gas-rich galaxies have a rotating disk, and are not irregular, see e.g., the recent review of Skillman (1996) on dwarf irregulars.) I propose only two variables to describe the collision itself, one for the magnitude and one for the directionality. For the former we can use Gerber and Lamb's dimensionless strength parameter. As de"ned, this parameter is not an observable, but it can be brought closer to observables with some additional assumptions, which are valid for the most common cases. Most collisions probably involve members of bound galaxy groups. If the mass of the collisional system is dominated by a dark matter halo encompassing the whole group, or by the halo of the primary galaxy (e.g. a small companion), it is reasonable to assume that <+v (relative impact velocity approximately equals circular velocity in primary disk). Then using the equation of centrifugal balance in the disk, we can
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Table 1 Classi"cation variables Timescale
Y (young) 410 yr
I (intermediate) 10}10
O (old) '10 yr
Hubble Type for each galaxy (disk component, gas fraction)
E elliptical, spheroidal
D signi"cant disk
G gas disk or irregular
Companion mass, perturbation amplitude
Small (10% denote by subscript
Intermediate +10}50% lower case
Large '50% upper case
Spin-orbit coupling (for each galaxy)
# Prograde
0 Orthogonal
! Retrograde
substitute G/v+R/M , where M ("M (R)) and R are the mass and disk radius of the primary. Then the strength parameter becomes,
M 2GM +2 S" M b
R . b
(1)
The ratio b/R is still not an observable, but in most collisions with signi"cant e!ects, its value ranges from a bit less than unity up to of order a few. (In heat-on collisions b is replaced with an e!ective `softeninga length.) Thus, S is roughly equal to a factor of order unity times the mass ratio M /M , so at a coarse level of description we can use the two interchangeably. Note: these approximations are not valid for high-velocity collisions, such as to occur in large galaxy clusters. In this article a qualitative estimate of S, or just an indication of whether it is small, medium or large will su$ce. In the latter class are collisions between nearly equal mass galaxies. We can de"ne the `smalla class as consisting of collisions with companions of such low mass that the companion is likely to be disrupted or stripped of substantial mass, e.g. S(0.1. (Of course, this consequence is determined by more than just the mass ratio, a complication we accept as part of the grey area between qualitative classes.) We again adopt a modi"ed form of Schweizer's notation, using upper case letters in the Hubble variable for the primary galaxy and comparable companions, lower case for medium strength companion interactions, and subscripts for low strength encounters. Thus, E represents a disruptive collision between an elliptical and a small disk. The last classi"cation variable is for directionality or angular momentum coupling. For each disk component involved we assign a value of #, 0, or !, depending on whether the encounter is prograde, perpendicular (or head-on), or retrograde, relative to the spin of that disk. Thus, a recent planar encounter between two comparable disk galaxies, at an early stage might have a classi"cation of YDD#! (or YDD## or YDD!!). A variable of unknown value will either be omitted, or highlighted with an x (e.g., YDDxx). Another example, an elliptical shell galaxy would be OE; a model of this galaxy might be OE >. In the remainder of this article, where appropriate, the collision categories described in each section will be given in parenthesis in the section title. The classi"cation variables and the range of values considered are summarized in Table 1.
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3. Transient events I: some wave morphologies and their causes (Yxx) This and the following chapter are devoted to the consideration of short term responses in galaxy collisions, i.e. those that occur shortly after closest passage. The examples below will be presented more or less in order of increasing disruptiveness. We begin with waves induced in disks, then take up large-scale mass transfer events, and "nally the disruption of small companions. As discussed above, the often bizarre morphologies produced shortly after closest approach had a substantial impact on the early morphological groupings in catalogs; most of the nicknames below derive from those systems. Less obvious is the role of these transient features in the larger issues of collisionally enhanced (or suppressed) star formation, the merger process, and fueling active galactic nuclei (AGNs). In fact, their role is probably signi"cant in all these cases, but they also provide unique signposts to help identify the nature of the interaction and the structure of the precursors. For example, the waves discussed below can provide a kind of seismological probe of the collisional galaxy. In this chapter we consider in turn the types of waves induced in galaxy disks as a function of the spin-angular momentum coupling (whether #, ! or 0). There is a great variety of disk waveforms, but the general structure of these waves depends more strongly on the angular momentum coupling than any other variable. We consider these waves `transientsa because, generally, wave propagation times in disks are shorter than the companion return or merger timescales. Moreover, as we will discuss in later chapters, the processes of disk heating and phase mixing generally guarantee that waves will damp or disperse within a few propagation times. There are several reasons for limiting our discussion to waves in disks. First of all, the constituents of bulge and halos have much larger random velocities than those found in disks, so unless the disturbance is large, waves rapidly di!use. Here `largea disturbance means one that generates velocity perturbations comparable to the thermal velocities in these dynamically hot components. Such large disturbances are probably at least partially disruptive, and so, belong with the cases considered in the following sections. Secondly, halo oscillations have relatively long characteristic timescales. The discussion in this chapter will also concentrate on two-dimensional waves in thin #at disks. Disk warping is undoubtedly an important e!ect of most collisions. We will also focus on stellar waves in this section, or on wave behavior that is common to both gas clouds and stars, but important di!erences will be noted in context. 3.1. Ring galaxies (YDe0) P.N. Appleton and the author have recently completed an extensive review of this subject (Appleton and Struck-Marcell (1996), henceforth AS96), so I will limit this to a brief summary. (References are minimal in this section, but the reader can "nd many sources of more detailed information in that review.) Collisional ring galaxies are rare. They are the product of a nearly head-on collision between a D-type primary and a substantial companion, i.e. one with mass in the range 10}100% of the primary. A companion with much less mass would not have much of an e!ect on the primary, while one more massive than the `primarya is possible, but evidently unusual. The basic theory was worked out by Lynds and Toomre (1976), also see Theys and Spiegel (1976, 1977), and Toomre (1978). As the companion approaches and passes through the primary disk,
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stars and gas clouds assumed to be in circular orbits before the collision, are drawn inward by the extra gravity. As the companion moves away, the unbalanced centripetal force drives an outward rebound. The response is faster in the inner disk and slower in the outer disk, so stars still moving inward meet rebounders moving out, producing a compression wave which propagates outward. If the impact parameter is small this wave is a circular ring. The Cartwheel galaxy, mentioned above, was probably the "rst ring galaxy discovered, and still is regarded as a prototype. This despite the fact that its progenitor was an usually late-type galaxy. The outer disk shows no evidence of old stars, though there is plenty of gas. It is also unusual in having two prominent rings, and the so-called spokes } spiral segments between the two rings (see Fig. 4). However, there are a couple dozen collisional ring galaxies that have been studied in some detail, with many more candidates awaiting further study. Their progenitors span the whole range of Hubble disk types, e.g., from the `Sacred Mushrooma system, AM 1724-622, studied by Wallin and Struck-Marcell (1994, see Fig. 1) with a very early-type progenitor (e.g. an S0 galaxy) to the Cartwheel. Three fundamental facts make collisional ring galaxies a very important class, despite their rarity. The "rst is the symmetry of the collision that produces them. Because this symmetry is needed to produce a circular or nearly circular ring wave, once a collisional (e.g. expanding) ring is identi"ed we immediately know a great deal about the collision. More precisely, there is a growing literature of comparisons between the collisional model and observation, and the general conclusion is that the collision theory is doing very well in accounting for observational features. (Though we note that there are other mechanisms for producing rings in galaxy disks, and we must have su$cient data to distinguish rings produced by these mechanisms before making detailed comparisons to collisional models.) On the other hand, nature has not missed a chance for an ironical twist. In a number of ring systems, the companion galaxy has not been identi"ed. This is often because there are several possible suspects, which is not surprising since galaxies are commonly found in small groups. The second fact is that the ring compression wave drives strongly enhanced star formation. Theys and Spiegel (1976) discovered that the rings in their modest sample frequently had blue colors indicative of mass young stars. Jeske (1986) and Appleton and Struck-Marcell (1987a) found that ring galaxy systems were relatively strong far-infrared emitters on the basis of IRAS observations. Now, observations of a number of systems in a variety of wavebands con"rm the enhanced star formation in almost all cases, except those where the precursor was evidently an early-type, gas-poor disk (see Appleton and Marston, 1997; Appleton, 1998, for an update of AS96). Insofar as density wave-driven star formation is understood (see e.g., the reviews of Elmegreen, 1992, 1994b) this is not a surprise, strong compressions are supposed to trigger star formation. However, the details of this process are not well understood, and ring waves provide a relatively clean way to study them. In this case the Cartwheel is a prime example. The evidence suggests (e.g. Higdon, 1993, 1995) that the ring wave is driving some of the "rst star formation to occur in the outer disk of the Cartwheel. Moreover, in the Cartwheel, Arp 10 and other systems, the intensity of the current star formation varies around the ring, which models suggest is the result of a variation in wave-strength following a slightly o!-center collision. Thus, the ring waves provide a nearly direct con"rmation that waves can induce vigorous star formation (i.e. a nonlinear response), and even within a single wave there are indications of a variation of response as a function of wave amplitude. Of course, this is also true in the much more common spiral density waves, but the generally more
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complicated spatial-temporal variations of those waves, and the e!ects of various resonances, make it useful to have a very di!erent case like the ring waves to compare to. The range of companion masses implies a corresponding range in the strength of ring waves in di!erent systems, potentially providing a great deal of information about wave-driven start formation from comparisons between systems. The third fundamental fact is that if the collision is impulsive, and the companion relatively small, then the structure of the ring waves is primarily a function of the distribution of matter in precursor. Thus, ring seismology is possible. The amplitude, width, spacing between successive rings, and azimuthal variations in the case of o!-center collisions, can be used to deduce the distribution of dark matter in the precursor. For example, widely spaced rings are a good indication of the presence of a massive halo (Struck-Marcell and Lotan, 1990). In most ring galaxies we only see one ring, so the rings are de"nitely widely spaced, and most of the precursors probably had substantial halos. Models (Struck-Marcell and Higdon, 1993) suggest that the Cartwheel is dominated by a large halo. If the collision is not impulsive (e.g., the relative velocity is low), ring seismology should still be possible, but the time-dependent perturbation will have to be modeled. Let us return to the second point, star formation in rings, for a moment. At high resolution, such as that obtained by the Hubble Space Telescope observations shown in Fig. 4, we have detailed information about where in the ring star clusters are formed (Appleton, private communication). Even at lower resolutions, information can be obtained on the relative positions of young stars, old stars, and the gas clouds (Marston and Appleton, 1995; Appleton and Marston, 1997). For example, these authors "nd evidence that in large rings the ionized gas is concentrated on the outer edge of the old star wave. This data can provide powerful constraints on theories of the star formation process, and the gas/star wave dynamics. One important complication, however, is that young star activity, i.e., winds, radiation and supernova explosions, may provide nonlinear feedbacks to the gas dynamics. For example, pushing some gas to the front of the wave, or out of the disk. Fig. 4 provides some direct visual evidence for such e!ects, i.e. the interstellar gas represented by emission and re#ection nebulae seems very frothy (to borrow the term of Hunter and Gallagher, 1990). The "laments, arcs and shells are all likely consequences of the activity of the young star clusters (e.g., Heiles et al. (1996), and references therein), and in aggregate give a clear impression of turbulence. There are a myriad questions waiting to be addressed: How is this turbulence di!erent from that in the interstellar gas of undisturbed galaxies (model examples of which are given in Passot et al., 1995)? How welldeveloped is it, and over what range of scales does it extend before it is damped or frozen out in the rarefaction region behind the compression wave? How does the turbulence e!ect the star formation process which generated it? How does it e!ect the thermal phase balance in the gas? These are very di$cult questions, relevant to many types of collisional galaxy, which have hardly begun to be explored. The relative simplicity of ring waves makes them an attractive locale for addressing them. 3.2. Symmetric caustic waves At this point, we will retreat from the complexities of turbulent gas dynamics, and review the simple theory of symmetric stellar waves excited in a planar disk by a collision. Special attention was given to this topic in the review of AS96, so I will omit many details. However, since the key
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Fig. 4. Hubble Space Telescope image of the `Cartwheela, a prototypical collisional ring galaxy (courtesy P.N. Appleton and NASA).
concepts can be generalized and carried over to many other cases, a self-contained overview is needed here. There are three key elements to this theory: (1) an impulsive disturbance (e.g. Alladin and Narasimhan, 1982), (2) followed by epicyclic kinematic motions (Lynds and Toomre, 1976), and (3) the development of nonlinear, caustic waveforms (Struck-Marcell and Lotan, 1990). The "rst item
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actually has two parts: that the disturbance occurs very rapidly compared to other relevant timescales, and that the disturbance can be decoupled from the subsequent evolution. Simple (e.g. analytic) models are based on the idealization of an instantaneous disturbance, but they remain interesting even if this is only approximately true. If the disturbance is persistent, it cannot be treated as part of the initial conditions of the dynamical equations, and in general, no conceptually simple model can be constructed. (However, this case can be treated with the perturbation theory described in Chapter 5.) The "rst condition above is necessary, but not su$cient, for the second condition. The validity of a kinematic approximation to the motions of stars, gas clouds, and dark matter particles, depends not only on the prompt disappearance of the disturber, but on the constancy of the gravitational potential they move in. In principle, this potential is also perturbed by the collision. However, if it is dominated by a dynamically hot component, like the dark matter halo, and the perturber is not too massive, the halo disturbance may be small compared to that experienced by dynamically cold disk particles. Henceforth, I will refer to assumptions (1) and (2) together as the KIA (kinematic impulse approximation), and (1) as the IA. If these approximations are valid, then we only need a description of the (kinematic) particle orbits to complete the theory. Depending on the form of the potential, the orbit equations will generally involve elliptical integrals (e.g., Grossman, 1996). However, the ancient greeks developed a planetary orbital model that provides a very convenient conceptual and analytic tool here too. This is the famous epicyclic model, in which, the particle is assumed to orbit on a (small) circle, whose center orbits the potential center on a larger orbit. The epicyclic model was "rst extensively applied to galactic dynamics by Lindblad (1959 and references therein). If we assume circular orbits in the target disk before the collision, and that the impulsive disturbance in the symmetric collision is small (perturbative limit), then the e!ect of the collision on the orbit will be a sinusoidal oscillation about the initial, `guiding centera radius. That is, an epicycle. This is only an approximation when the disturbance is of "nite amplitude, but the comparison of analytic and numerical models suggests that it can be a good one for transient waves. The epicyclic orbit equations for a star are, r(q, t)"q!A(q)q sin(i(q)t) ,
(2)
Rr v " "!Aq i cos(it) , P Rt
(3)
where r(q, t) is the instantaneous particle radius, q is the precollision orbital radius, and A(q) is the amplitude of the epicyclic oscillations. In the IA it is assumed that the collision is so rapid that the particles do not move during it, but the force and acceleration induce a velocity change. Thus, the initial radial velocity amplitude is the velocity impulse, and the amplitude A is found by setting t"0 in Eq. (3), A"!*v /qi. The epicyclic frequency i depends on the structure of the gravitational potential (Binney and Tremaine, 1987, Section 3.2), and generally goes in the sense of longer periods at larger radii. This is the origin of the ring compression, which, as noted above, results when outwardly rebounding particles meet infalling particles from larger radii (as a result of the longer epicyclic periods of the latter). The radial motions of a collection of such particles, in a representative gravitational potential, are shown in Fig. 5. In the particular case shown in Fig. 5 it is assumed that the companion to
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Fig. 5. Radius versus time for representative stars in a kinematic model for a collisional ring galaxy as described in the text. Fig. 6. Phase diagram of radial velocity versus radius (r!v ) from the kinematic calculation of Fig. 5, at dimensionless time t"20. The loops are the result of orbit crossing in the inner ring, while the positive velocity wave between radii of r"3.0}4.0 shows the orbit-crowding outer ring.
primary galaxy mass ratio is 0.25 and that both galaxies have massive dark halos. Speci"cally, the potential assumed for the primary galaxy gives a rotation curve of the form v"v(c) (r/c) L, where c is a constant scale-length. A large value of n (n"20) is used to make v nearly constant. It is further assumed that the amplitude of the collisional disturbance is constant with radius. Dimensionless units are used, where the scale-length of the gravitational potential c, and the product GM(c) have been set to unity. As stellar orbits at di!erent radii get more out of phase, the orbit crowding phenomenon becomes orbit crossing, and thus, the second and third rings are broader than the "rst. (Ultimately, the rings overlap and become e!ectively smoothed out by this `phase mixinga.) This is shown in Fig. 6, which is a r!v phase diagram (after "gures in Struck-Marcell 1990a,b) with a phase mixed P center, an orbit crowding outer ring, and an isolated orbit-crossing ring between. The orbit crossing rings are bounded by sharp edges. They are in fact caustics, formal singularities in the stellar density. The conservation of mass in a thin cylindrical annulus implies that the density is given by o (q) o" , r Rr q Rq
(4)
where o (q) is the initial, unperturbed density pro"le. Eq. (4) applies to regions with a single star stream. In oribit crossing zones (e.g., the inner rings in Figs. 5 and 6), the right hand side must be
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replaced with a sum over terms for each star stream. Singularities occur wherever Rr/Rq"0, i.e. where the particles some initial radial range *q are squeezed into zero volume, *r"0. Then, formally, oPR, though in"nite densities will not occur in real galaxies with "nite numbers of stars. Substituting the orbit equation (2) into the caustic condition, allows one to derive an equation for the location of the caustic edges. This, together with Eqs. (2)}(4), provides the basis for a complete analytic model of stellar waves in ring galaxies (see AS96 for details). We will see in subsequent sections how the three elements of this theory can be generalized to less symmetric cases. Generally, we do not expect the same wave structure in the collisional (see Section 1.2) gas component. The stellar orbit crossing zone and its caustic edges should be replaced by a dissipative shock wave. The gas will be heated in this shock, but cooling times are short, and so as a "rst approximation the shock can be assumed to be isothermal. In the "rst ring this shock may be relatively weak, because the epicyclic motions in adjacent radial zones are still nearly in phase. The shock in the second ring waves is likely to be much stronger (Appleton and Struck-Marcell, 1987b; Struck-Marcell and Appleton, 1987). Even so, the thermal physics and the observables are likely to be dominated by the compression induced star formation behind the shock. Sometimes, however, the gas may behave more like the collisionless stars, as discovered by Gerber et al. (1992) in their simulational study of Arp 147. This is the result of a very interesting e!ect, when the disk is warped by the collision, so gas clouds at di!erent radii are able to execute their radial epicyclic oscillations in di!erent vertical planes, thus becoming collisionless. The gas behavior in any particular situation depends on whether the ratio of the radius of curvature of the warp to the local epicyclic excursion is greater or less than unity. 3.3. Ring relatives: bananas, swallows and others When collisions become less than perfectly cylindrically symmetric, that is, as the impact parameter increases, the diversity of waveforms increases rapidly. In this Section I will illustrate this with a few examples, and note how the theory described in the previous section is generalized. The consequences of a small increase in the impact parameter (relative to the scale length of the gravitational potential) are not terribly dramatic. The result is an asymmetric or partial ring, which looks like a crescent or banana (see Appleton and Struck-Marcell, 1987b; Chatterjee, 1986). Theoretically, these crescent waves are nearly as simple as the symmetric rings, at least for points at radii greater than the impact radius. There, the impulse is still primarily radial, but generally with an amplitude that depends on distance from the impact point. However, the radius of the compression wave depends primarily on the epicyclic frequencies (in the perturbation limit), so it is little di!erent from the symmetric case. Thus, in this approximation, the wave is still nearly circular, but with an amplitude that varies with azimuth around the ring. (See AS96 for a more complete description.) For the stellar component this means that the caustic wave may not extend to all azimuths, i.e. there may be orbit crossing on the `stronga side, but only orbit crowding on the weak side. As a result the two circular caustic edges of the symmetric wave are replaced by the crescent. Similarly we expect variable compression and shock strengths around the ring. As the ring propagates outward, the ratio of the impact radius to the ring radius decreases, so the perturbation is more symmetric, and the crescent ends join to form a (weaker) ring. This is shown in Fig. 7 (from Appleton and Struck-Marcell, 1987b). Other numerical models of asymmetric rings can be found in Appleton and James (1990), Gerber (1993), and Struck-Marcell and Higdon (1993).
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Fig. 7. Contour maps of the gas density for a hydrodynamical simulation of an o!-center galaxy collision (Appleton and Struck-Marcell, 1987b). Solid contours indicate densities above the initial unperturbed value and dotted contours show lower densities.
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The overwhelming majority of real ring galaxies are asymmetric in appearance. This includes both the Cartwheel and the very similar VII Zw 466 ring. Appearances can be a bit deceiving here, since optical/infrared observations usually re#ect the number of massive young stars and clusters. The local star formation rate (SFR), and perhaps the stellar mass function, are most likely nonlinear ampli"ers of the wave compression. Examples of galaxies with apparently strong crescents include the `Sacred Mushrooma AM1724-622 (see Fig. 1), and most of the objects on Page 6.1 of the Arp-Madore photographic atlas. The crescent is an evolving or `metamorphosinga caustic structure, and thus, we can learn more about it from singularity or catastrophe theory (see e.g. Poston and Stewart, 1978; Arnold, 1986). While singularity theory is not widely used in astronomy, it has found a couple of niches. One of these is the Zeldovich `pancakea approximation for galaxy and large-scale structure formation (Arnold et al., 1982), where as a result of gravitational collapse, collisionless dark matter particles form a full range of three-dimensional caustics or singularities. An early, less well known application, is found in the work of Hunter (1974) on spiral density waves. Hunter found that wave characteristics converged to singularities. Caustics occur in (models of) a number of di!erent types of collisional galaxies. AS96 reviewed the application of singularity theory to collisional galaxies, and suggested that the theory is more generally relevant because, In two and three spatial dimensions it o!ers a complete classi"cation of the generic, nonlinear waveforms and their possible evolutions2 . These include cusps, swallowtails, and pockets or purses (see Arnold, 1986), and overlapping combinations. 2 (Moreover,) it signi"cantly extends our conceptual model. It takes us from models for individual stellar orbits to the structure of the nonlinear density waves, 2 It makes us aware of `elementarya waveforms that are more complicated than rings or spirals, and yet not intractably complex2 For example, beginning with a model for the orbits, like the kinematic impulse approximation, we can derive analytic expressions for the location of the caustics, which provide a skeletal outline of the waves for any particular set of structural and collision parameters. (AS96) The procedure for "nding ring edge caustics described in the previous section can be generalized with the goal of mapping the edges or `skeletal outlinea of the more general waveform. However, even limiting consideration to waves in a thin, unwraped disk, three e!ects complicate this formalism: (1) the azimuthal dependence of the perturbation amplitude A, (2) the fact there is now an azimuthal component to the velocity impulse (or a torque), and (3) even in the limit that the gravitational potential is dominated by a rigid halo or bulge, there is an impulse on the potential center. Items (2) and (3) may have a small magnitude, but still have important e!ects. Because of e!ect (3) halo particles at radii smaller than the impact radius will receive a net impulse toward the intruder, while particles in a spherical shell at large radii will not. Thus, the halo is apparently broken into two kinematic components, though even this dichotomy is too simpli"ed to realistically represent the time-dependent potential. It seems that perturbation theory is rather shakey unless the typical velocity impulse is smaller than the mean halo particle velocities, i.e. the halo is too hot to be signi"cantly perturbed.
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We will adopt this hot halo assumption for the moment (but see the analysis of Gerber and Lamb (1994) which does not). The following AS96 we can write the approximate kinematic equations as, r"q !Aq sin(i t#t ),
v "!Aq i cos(i t#t ) , P (5) h"h #u (q )t, v "v #*v . F F F These epicyclic orbit equations are very similar to Eqs. (2) and (3). The chief di!erence is that, as a result of the azimuthal velocity impulse, the epicycle is centered on a new guiding center of radius q . The initial particle radius q is generally di!erent from q , so this introduces an initial phase t. The epicyclic frequency is i "i(q ), and the mean angular velocity is u (q ). The amplitude A is a function of radius and azimuth. Using the initial conditions (in the IA), and the force balance equation for the guiding center, the perturbed (primed) quantities can be eliminated in favor of pre-collision values and the velocity impulses (see AS96). Then, in this two-dimensional case, the in"nite density caustic condition is given by setting the Jacobean determinant equal to zero,
Rr Rr Rq Rh "0 . (6) Rh Rh Rq Rh AS96 describe a limiting case of this equation that has `crescenta solutions, and probably contains other waveforms as well. Struck-Marcell (1990a), Donner et al. (1991), and Gerber and Lamb (1994) all used numerical models to study the creation of waves in disks of particles following kinematic orbits like those of Eq. (5). Donner et al. also numerically solved the caustic determinant equation (6) in speci"c cases to compare to numerical models. Donner et al. (in planar collisions), and Gerber and Lamb (in collisions nearly perpendicular to the target disk), both found good agreement between the kinematic models and self-consistent N-body simulations at early times. Thus, providing evidence that the e!ects of the time varying gravitational potential take some time to accumulate. Struck-Marcell (1990a) and Donner et al. both found evidence for the development of higher order caustics like the cusp, swallowtail and pocket/purse (see Arnold 1986) in their calculations. Perusal of the photographic atlases shows many systems that appear to match the nonlinear waveforms, but this is very circumstantial evidence. M. Kaufman and collaborators (including the author) are investigating two galaxies that have the appearance of swallowtails (NGC 3145 and NGC 5676) with radio (21 cm) and optical observations, but the results are not yet complete. We are speci"cally searching for the high velocity dispersions or gaseous shock waves that would characterize the stellar orbit crossing region. Wallin and Struck-Marcell (1994) compared caustic models to the broad crescent wave in the early-type ring galaxy AM1724-622 (the `Sacred Mushrooma). Interestingly, the crescent was best matched by the models with a declining rotation curve in the precursor. Though somewhat unusual, this might be the result of the gravitational potential in the disk being dominated by the strong bulge component. It was hoped that it would be possible to determine if the broad ring edges were in fact sharp caustics, or if not, obtain some measure of the e!ect of di!usive smoothing processes. Ultimately, such processes, together with phase mixing (see previous section) and the
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overlap of multiple caustics, will erase all traces of caustics waves in the disk. However, the published simulations indicate that these processes do not strongly e!ect the "rst couple of waves. Because of the limited resolution of the data, and the fact that dozens of foreground stars cover this galaxy, which lies near the galactic plane, this study was not de"nitive. Hubble Space Telescope observations might be able to surmount the di$culties. The caustics theory for nonlinear waves in collisional disks is still relatively young and untested. Many questions have not yet been fully addressed, including how far can the analytic models be pushed before they become too inaccurate or too algebraically complex to be useful? 3.4. From rings to spirals Toomre (1978) discovered that, as the impact point in a vertical collision is moved out from the disk center to its edge, the ring wave metamorphoses into a spiral. Toomre used a sequence of restricted three-body models (reproduced in Fig. 8) to demonstrate this. Several remarkable and instructive points can be taken from this "gure. The "rst, noted by Toomre, is the disks are not destroyed in any of these cases, though they do su!er signi"cant time-dependent warping or #apping. This result, along with most of the other qualitative features of Fig. 8 are con"rmed in an analogous series of fully self-consistent star-plus-gas simulations run by Gerber (1993). There are di!erences, but they are modest given the di!erent simulation techniques and the di!erent structural characteristics in the initial galaxies. (I.e., a rigid point-mass potential in Toomre's versus an initial #at rotation curve structure in Gerber's.) Secondly, these simulations clearly show caustic edges, and in some cases they appear quite complex. We would hope that these waveforms could be explained by the KIA-caustics theory, but to date no detailed analysis has been done. However, some qualitative aspects can be explained by this theory with little e!ort. To begin with the line connecting the target disk center and the impact point is a key division. Ahead of this line one component of the velocity impulse will be directed against the particles' rotation velocity, so these particles will be slowed and fall inward. Behind this line particles will have their tangential velocity increased, so they will #y outwards. The infalling particles will compress, forming a region of enhanced density. In fact, since the backward impulse increases to some maximum as azimuth increases from the division line, we can expect an orbitcrossing zone in this compressed region if the impulse is great enough. Thus, in general, a `lipsa (crescent) caustic will form and shear into the leading edge of a spiral wave. This feature can be seen in the more o!-center of Gerber's model collisions, and in his model Ring4, the wave is strong and caustic-like edges are especially apparent in the dynamically cooler gas particles (see Fig. 9). We can sketch a quick derivation of this wave from the KIA caustics theory (neglecting the perturbation of the potential center of the disturbed galaxy). The azimuthal impulse can be written, r *v "*v H sin(h) , F R
(7)
where h is the azimuth of a disk particle relative to the center-to-impact point line, and r is the H impact radius, and R the distance between disk particle and the impact point (see SM90, AS96). R is related to r , and the particle's unperturbed radius q by the law of cosines, H R"(r #q!2qr cos(h)) . (8) H H
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Fig. 8. Toomre's ring-to-spiral transition is illustrated by a sequence numerical model evolutions with progressively decreasing companion impact radii. Each row shows a di!erent model. See text and Toomre (1978) for details.
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Fig. 9. A self-consistent, N-body plus Smoothed Particle Hydrodynamics simulation from R.A. Gerber's thesis, showing an incipient spiral in a collision like those that produce ring galaxies. In this case the trajectory of the companion galaxy was nearly perpendicular to the primary disk, and the point of closest approach was at the edge of that disk (see Gerber (1993) for details).
At each radius, and on each side of the center-to-impact line, there is some azimuth where "*v " is F a maximum. Physically, particles located at these extremal azimuths feel the strongest pull backward (or forward) in their orbits, and so, would seem to be likely participants in the formation of an orbit crossing zone. More formally, the term Rh/Rh in the caustic condition (6) is propor tional to R*v /Rh . Therefore, if the cross terms in Eq. (6) are small, the zeros of these derivatives F yield caustics.
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For simplicity, assume that the potential of the perturber is that of a simple point mass. Then the equation R*v /Rh "0 reduces to a simple quadratic for cos(h), F r r r H cos(h)# 1# H cos(h)!2 H"0 , (9) q q q
which depends only on the parameter r /q. *v itself depends on the mass of the perturber and H F other variables. In the small radius limit, qP0, the solution is h"$903. When r /q is about H unity, cos(h)+(3!1, or h+453. The full solution curve to equation (9) spirals out from small radii to a nearly linear form at larger radii. Rotational shear will turn this linear feature into a spiral. Qualitatively, this result seems much like the behavior of the numerical models. For impacts within the disk, this spiral kinematics is superimposed on the radial kinematics discussed above. Fig. 8 shows that as the impact point moves out of the disk, the radial ringing diminishes in importance, and tidal stretching becomes more important. Note that this is a very nonstandard description of a spiral wave in a galactic disk. The caustic spiral is a nonlinear transient, so there is no obvious connection to the classical quasi-stationary (linear) density wave theory (see e.g. Binney and Tremaine, 1987, ch. 6; Palmer, 1994, ch. 12). Swing ampli"cation is another process that plays a very important role in interaction driven waves (see Toomre (1981) and below), but in this case there has not been time for signi"cant ampli"cation. This wave is simply the result of correlated initial conditions, and a special aspect of the direct collision is that it is truly impulsive. This example also provides an interesting illustration of the transition from the study of ring waves to more standard tidally induced waves. 3.5. Tidal spirals and oculars (YDx#) The class YDx#includes encounters in which the companion galaxy #ies by in the plane of the primary disk, and in the same orbital sense as the disk rotation, and with a point of closest approach generally located outside the disk (i.e., prograde collisions). It seems odd at "rst that these collisions can result in more damage than nearly head-on vertical collisions, but the tidal perturbation is sustained for a longer time in this case. The result is the formation of the great bridges, tails and strong spirals hinted at in Holmberg's work, emphasized by Zwicky, and shown convincingly to be tidal remnants in the work of the Toomres and others in the early 1970s (see Chapter 1). In this section we will consider the strong spirals, and leave bridges and tails to later sections. The great `Whirlpoola galaxy M51 is the prototype of these collisional spirals (see Fig. 10 and Section 9 of the Arp}Madore atlas). It is also a bridge/tail galaxy, with a connected companion that has attracted much attention in the interacting galaxy literature. At the same time, it has also been a prototype in the spiral density wave literature. The possible connection between the inner and outer phenomena was discussed by the Toomres (1972, 1974), but the idea encountered di$culties (Toomre, 1978). While there was some evidence for tidal in#uence on the inner spirals in the 21 cm observations, the restricted three-body simulations did not produce any such waves. However, by 1980 Toomre had discovered the missing piece of the puzzle in `swing ampli"cationa. Swing ampli"cation, as described by Toomre (1981), depends on the near commensurability of the shearing timescale and the epicyclic (compression) timescale, which is common in galaxy disks. This commensurability works to keep stars in the overdense region for relatively long times, which
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Fig. 10. Optical image of the `Whirlpool galaxya M51, whose beautiful spiral arms are likely a result of the ongoing collision, see text. (Digital Sky Survey image courtesy of AURA/STScI.)
gives self-gravity the time needed to greatly amplify the density contrast of a spiral wave. Toomre used self-consistent N-body simulations to demonstrate the operation of the process, and simple, `shearing sheeta, kinematic models to illustrate the role of the three main component processes: epicyclic `shakinga, wave shear and self-gravity. The original paper gives a very clear presentation and other pedagogical summaries can be found in Athanassoula (1984) and Binney and Tremaine [(1987), ch. 6.3]. The discussion here will be very brief, and more details on all the topics covered can be found in these sources. For a recent technical review see Tagger et al. (1993). The amount of ampli"cation is surprising } factors of between a few and a hundred resulting from external perturbations of a few percent or less. Toomre points out that the basic process was understood in work on spiral waves carried out more than a decade earlier, but its ampli"cation ability was not. However, the nonlinear wave is generally short-lived, building up to maximum amplitude within about one disk rotation time, and then winding up and decaying on a comparable
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Fig. 11. The development of a `swing-ampli"eda trailing spiral wave from an initially leading wave from Toomre (1981). Contours represent "xed fractional excess surface density, and the time between snapshots is one half of the rotation period at the corotation point.
timescale. These points are illustrated in Fig. 11 from Toomre's paper. If the structure of the galaxy disk is such that it possesses an inner Lindblad resonance, then the wave is `absorbeda at the radius of the resonance. Lindblad resonances occur where the epicyclic frequency is commensurate with the wave pattern frequency in the local rotating frame. If there is no inner Lindblad resonance, the trailing wave can propagate to the disk center, and reemerge as a leading wave. This leading wave is sheared around into a trailing wave, and in the process, another round of swing ampli"cation occurs. The amount of ampli"cation depends on two parameters. The "rst is the Toomre Q parameter, which is a measure of how close to gravitational
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instability the unperturbed disk is. The second parameter Toomre called X"j/j , where j is the (unwrapped) spiral pattern wavelength, and j is critical wavelength for gravitational instability. Note that Q, X and j are all de"ned as local quantities, though they typically do not vary drastically across a galaxy disk. X and j depend on the number of spiral arms in the global pattern, and since the ampli"cation is very sensitive to the value of X, signi"cant ampli"cation can only occur when there are no more than a few arms. If the wave can traverse the ampli"cation feedback loop, numerical models show that the end result is a global bar instability. This is an extremely important consequence of a collisional disturbance. However, it is not a transient event, so we defer further discussion to later chapters. Another feature of the models in Toomre's paper was that the spiral wave started from the center and moved out to include a large part of the disk. This phenomenon o!ers hope that there would be time for large-scale tidal features to develop, as in M51, before the prominent spirals disappear. In sum, these discoveries bode well for the idea that strong inner spiral waves can be driven by tidal encounters. Unfortunately, the M51 story has not yet ended happily ever after. Extensive 21 cm observations (Appleton et al., 1986; Rots et al., 1990) revealed unexpected HI features, including a long southern tail coming o! the (outer) western arm, and gas clumps north of the companion. These discoveries coincided with the implementation of a new generation of N-body and gas simulation codes (see e.g. Sellwood, 1987; Barnes and Hernquist, 1992a), so it is not surprising that a number of new modeling e!orts were initiated. These include Hernquist (1990), Howard and Byrd (1990) and Sundelius (1990), and more recently, Toomre (1994) and Byrd and Salo (1995). The models of Barnes, Hernquist and Toomre are based on a passage of a companion on a high eccentricity orbit, while Byrd and collaborators favor two disk passages to account some of the morphological details of the system. Barnes and Hernquist (1992a) conclude: `At present, however, none of the calculations o!er a really convincing reconstruction of M51's spiral structure2a (see also the discussion of Barnes, 1998). To compound the modeler's di$culties, new high resolution, multiwaveband observations are being acquired at a steady rate, and they reveal not only more detail, but new phenomena. These include the distribution of ionized, atomic and molecular gas, and star formation across the disk, and especially in the spiral waves (see reviews of Rand and Tilanus, 1990; Casoli, 1991; as well as the other observational reports in Combes and Casoli, 1991). Casoli (1991) discusses how the sequence of dense cloud buildup, star formation, and subsequent cloud disruption is displayed as expected across one spiral arm, but does not follow this sequence elsewhere. Another example is the recent discovery from infrared imaging that the spirals go deep into the central regions, and wind through three full revolutions (Zaritsky et al., 1993a). The infrared observations also revealed a small bar in the inner regions. These phenomena, together with the large-scale structures, will undoubtedly continue to challenge modelers for a long time yet. We should not, however, let the details of the M51 system distract us from the general result } that even moderate collisional disturbances can stimulate the formation of strong spiral waves via nonlinear ampli"cation processes. With perfect hindsight, we can see that this is just what was required for M51 types, not only to explain the waves, but also their presence in a disk that does not appear highly disturbed (except in the outer regions). Stronger disturbances bring more wholesale distortions. This point is well illustrated by the ocular galaxies, "rst studied by Elmegreen et al. (1991). Elmegreen et al. de"ne the ocular as `a bright oval approximately one-half the size of the galaxy centered on the nucleus with a right angled vertex at each end of the major axis, and spiral arms
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Fig. 12. The collisional system NGC 2207/IC 2163 illustrates the ocular waveform. Speci"cally, the disk of the smaller galaxy (IC 2163) has the characteristic eyelid shape and the double-branched spiral arm. The other spiral arm has been hidden or disrupted by the larger galaxy. (Digital Sky Survey image courtesy of AURA/STScI.)
extend smoothly from each of the #atter sides of the oval2a At "rst glance the ocular seems to be a relatively pure result of tidal forces, with the oval resulting from tidal compression, while the spiral arms result from tidal stretching plus shear (see Fig. 12 and the images in Elmegreen et al., 1995). The fact that the ocular persists for only about one rotation reinforces this impression. However, the ocular is not simply the result of a homologous compression. Unlike the initial unperturbed disk the ocular has sharp, caustic edges (in the models). The ocular is formed in prograde collisions where the perturbation is relatively strong, so the azimuthal impulse is substantial. Moreover, it is clear that the sharp edge on the companion side forms "rst, and it appears in the quadrant ahead of the line connecting the companion to the primary center. Thus, it appears that stars are pulled ahead in the near side quadrant behind the line of centers, and they swing out to apoapse in the leading quadrant, where they are also pulled backwards by the companion. A mirror image process occurs in the other half of the disk, but as a result of the interaction between the stars and the swinging potential center of the primary. It would be interesting to see the trajectories of stars that make up the ocular, but these have not yet been presented in the literature. It would also be interesting to see if the ocular form could be captured in a KIA model. This would probably require an instantaneous approximation to the swing imparted by the primary center as well as the companion. Elmegreen et al. also found that the formation of an ocular depends on a tidal strength parameter,
S"
M M
R *¹ , R ¹
(10)
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where *¹ is the time it takes the companion to travel through one radian relative to the primary, and ¹"(R /GM ) . (Compare to Eq. (1) derived from the IA.) Their two-dimensional simu lations showed that oculars only formed when the value of this parameter is greater than 0.019. For lower values, substantial spiral waves are still produced, but there is not enough transverse compression to produce the eye shape. They also found that when S'0.038, the ocular galaxy evolves into a barred galaxy, and beyond the YD#stage considered here. 3.6. Fan galaxies and one arms (YDx!) Retrograde collisions have never inspired the same interest as prograde collisions. They do not form beautiful two-armed spirals. Toomre and Toomre (1972) included a retrograde encounter among their four numerical examples, but concluded that the e!ects were `remarkably milda. Eneev et al. (1973) presented a simulation that gives a very di!erent impression, probably because the perturbation was stronger. However, they did not discuss the morphology of this model. In summarizing the retrograde encounters in their atlas of N-body simulations of galaxy interactions, Howard et al. (1993) state that they `produce only broad fanlike global patterns, but rich small-scale internal structure.a This is not to say that retrogrades were entirely overlooked in the colliding galaxy renaissance of the 1970s. Kalnajs (1975) and Athanassoula (1978) studied the idea that a companion in a retrograde orbit could stimulate the formation of a leading spiral wave (i.e. one whose outer end points in the direction of disk rotation). Kalnajs (1975) presented evidence that M31, the Andromeda galaxy, possessed such a one-arm. This work was followed up by the study of Thomasson et al. (1989), which included analytic work, numerical studies, and a comparison to observation. These works suggested that the one-armed spiral wave is the result of the 1 : 1 orbital resonance, at which orbits close after one radial oscillation. This resonance plays a role similar to that of the 2 : 1 Lindblad resonance (two radial oscillations to closure) in the case of the two-armed trailing pattern generated in prograde collisions. On the face of it the preceeding sentences seem strange. If the radial epicyclic oscillation period depends primarily on the intrinsic mass distribution in the disk galaxy, then how can the number of radial bounces depend on the orientation of the perturber's orbit. This would be no problem if the two types of wave appeared in di!erent parts of the disk, but the simulations show that they both can involve a considerable fraction of the disk. The answer to the paradox is that the orbit closure statements refer to a reference frame rotating with the wave pattern. In general, the stellar orbits are precessing ellipses that do not close in an inertial frame, but as Fig. 3 of Thomasson et al. shows closure is nearly achieved in the wave frame (also Fig. 16 of Athanassoula, 1984). It is worth a little further digression on this point, which is a matter of fundamental kinematics akin to others considered in this chapter. Consider, for example, a #at rotation curve galaxy, in which the rotation velocity v"constant, the rotation frequency X"v/r, and the epicyclic frequency i"(2X. This means that in an inertial frame a star goes around a bit less than 3/4 of a circle (2553) in one radial period. Let us suppose, that the wave is de"ned by stars at a given epicyclic phase, e.g., at minimum radius (point of greatest radial compression). In the case of the leading one-arm this means that the wave pattern merely has to travel (counter to the rotation) through an angle of about 1053 to meet the star again at minimum radius, and thus, close the orbit
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in the wave frame. The wave must be leading because stars at larger radii take longer to traverse their 2553 of azimuth, by which time the inner wave has traveled more than 1053 (i.e., farther `backwardsa). In the two-armed case, the star begins at minimum radius in one arm, and meets the second arm at the next minimum if that arm has also advanced (in the same direction) by about 753. Because this wave is moving in the rotation direction, it must be trailing to maintain coherence at all radii. We have discussed above how the tidal perturbation in a prograde collision induces the two-armed wave. The one-arm mode is probably excited in retrograde collisions simply because it is the lowest order leading mode. Thomasson et al. (1989) "nd that the one-arm persists for several disk rotation times, i.e. about as long as the typical collision-induced two-arm pattern. The question then arises } why are there so few examples to be found in the observations. The authors consider a variety of possible answers. They favor a somewhat indirect explanation. A large halo-to-disk mass ratio makes a galaxy stable against Swing Ampli"cation of the m"2 mode. On the contrary, they suggest that this mass ratio may commonly be low, enough to give the m"2 mode a competitive advantage. It appears that the halo they are referring to is that contained within the radius of the stellar disk. This `haloa should also include a bulge if present. Thomasson et al. also "nd that the retrograde disturbance has a steeper dependence on separation (1/r) than the usual tidal force, and that it takes a substantial disturbance to produce the leading arm. Retrograde collisions with small disturbances produce m"2 trailing arms or combined m"1 leading and m"2 trailing patterns. `Rich internal structurea indeed. Another possible example of this richness is the galaxy NGC 4622, modeled by Byrd et al. (1993). This galaxy possesses an inner leading arm, a ring, and the two outer trailing arms. Byrd et al. found that such features could be produced following a small impact parameter collision with a low mass companion, orbiting in either direct or retrograde senses. However, the retrograde collisions produced the better match to the observed morphology. A "nal note on leading one-armed waves } Lotan-Luban (1990) carried out a series of restricted three-body simulations which showed that head-on, low impact parameter collisions (like those that produce ring galaxies) can produce a long-lived one-arm spiral. Generally, this spiral becomes prominant after several ring waves have propagated through the disk, and the ringing has pretty well phase-mixed away. By varying the potential structure she con"rmed Thomasson et al.'s result that a substantial halo component is needed to produce the one-arm. She also varied the companion to target mass ratio and found that intermediate mass companions produced the strongest wave. This is not too surprising } high-mass companions caused much disruption in the test particle disk, and low-mass companions did not produce a su$ciently strong perturbation. Thirdly, she carried out a series of simulations with varying impact angle from head-on (small impact parameter) to in-plane retrograde (with impact parameters greater than the disk radius). Of these, the head-on small impact parameter collisions produce the strongest one-arm waves. This helps account for the fact that some of her simulations seem to make stronger and longer lived waves than those of Thomasson et al. The retrograde planar waves are similar in both works. Unfortunately, Lotan-Luban's work has not been redone with self-consistent N-body simulations. This is especially important for testing the longevity of these waves.
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3.7. Gas vs. stars in waves Most of the discussion above, and in the literature, on collisional wave morphologies concerns stellar waves. Low amplitude, transient waves mainly depend on kinematics, so to "rst order there is no signi"cant di!erence in the behavior of the interstellar gas and stars. However, in nonlinear waves we expect caustic waveforms to develop in the stars and dissipative shocks to develop in the gas. Dissipation could lead to some separation of the two components, unless the gas disk is highly warped or distorted. However, observations have not yet provided any compelling examples of separation in waves, perhaps because strong perturbations lead to highly distorted disks. Extreme separations may occur on large scales in mergers, and large quantities of gas can be funneled to the center (e.g., Negroponte and White, 1983; Noguchi, 1987, 1990; Barnes and Hernquist, 1991, 1992a), at the same time that gas and stars can be thrown out to great distances to form separate shells and ripples (e.g. Hibbard, 1995; Hibbard and van Gorkom, 1996, and references therein). These topics are discussed below. A large-scale, but less violent, example is provided by the galaxy NGC4747 (Arp 159) studied by Wevers et al. (1984), whose HI disk seems to have been twisted relative to its stellar disk by 113 in projection! Exactly how this occurred remains a mystery, but given the distortions of the outer HI disk of its companion NGC4725 it may well be that a direct collision with a modest impact parameter was involved. In that case, direct cloud collisions might have contributed to the separation, as well as dissipative accretion of gas from the partner, described in the next section. There we will also meet some milder examples of gas-star separation in waves and tidal structures. In nonmerging collisional galaxies important di!erences between stellar and gas dynamics result from heating and cooling e!ects. Young stars winds, UV photoheating, and supernova blasts can heat and push the gas, and disrupt cold clouds. At the least this can boost gas to greater heights above the disk, i.e. making a thick gas disk. This is evident in recent simulations of ring galaxies that include heating and cooling (Struck, 1997), though it is a transient e!ect in that application. The disk gas cools and settles on a timescale comparable to the wave passage time. A more spectacular heating phenomenon is that of superwinds generated by nuclear starbursts (Heckman et al., 1993; Lehnert and Heckman, 1996). Collisional galaxies may frequently experience a nuclear starburst phase, driven by gas in#ow resulting from dissipation in waves and induced bars (see below).
4. Transient events II: death and trans5guration 4.1. Transient mass transfer and bridges Most collisions or near misses involve some exchange of material between galaxies. Under speci"c circumstances described below the amount of material exchanged can be considerable. In this subsection we will consider cases where the collision does not end in a prompt merger, and does not result in the large-scale disruption of the partners. Even in these cases the mass transfer can signi"cantly in#uence the evolution of both galaxies. There are two general reasons for this. (1) A large fraction of the material transferred is usually ISG (interstellar gas), the seminal material of galaxy evolution. (2) The transfer, and the forces that drive it, upset the thermo-hydrodynamical (quasi-)equilibrium of the gas disks, initiating a long-term relaxation process. These points apply
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not only to mass exchanges between galaxies, but also to the case of mass forced out of galaxy disks into large tidal structures, such as bridges and tails. In both cases material falls onto the galaxy disks on an extended timescale. Transient mass transfer can also be viewed as a large scale experiment in ISG dynamics, and thus, can be used to further our understanding of these processes. There are basically two modes of mass transfer in these situations: hydrodynamic and tidal. By the former, I mean the case of a direct collision between the gas disks of the two galaxies. (In this category we also include the collision of the hot gas halo components, but this is generally of minor importance for the two galaxies, see Section 1.) In the second mode material is pulled out of at least one galaxy by gravitational forces. Speci"cally, this mode usually works by increasing angular momentum via tidal torques, rather than by direct gravitational attraction. Thus, it is the dynamically cold, spinning disk material that is most vulnerable to the exchange process. The resulting tidal features often have a large gas fraction because the neutral hydrogen disks of most galaxies are larger than the stellar disk, and the outer parts are more loosely bound. We will consider each of these modes separately. 4.1.1. Splashes: bridges and infall in direct collisions (YDd0) As discussed in Section 1, when two gas disks collide, the di!use gas clouds within them will collide at highly supersonic speeds. Thus, we can expect an extensive gas splash, and the observable e!ects of shock heating and radiative cooling. We do not have to look far for examples of supersonic collisions involving external clouds. Our Milky Way galaxy contains a population of so-called high velocity clouds (HVCs), which unlike other interstellar clouds, are not con"ned to the disk, and which have very di!erent kinematics. In recent years several studies have provided good evidence for recent or ongoing collisions between some HVCs and disk clouds (e.g., Cabrera-Cano et al., 1995; Tamanaha, 1995), possibly including the nearby Gould's Belt region in Orion (Comeron and Torra, 1994). In addition, Mirabel and Morras (1990 and references therein) have argued that a stream of clouds is impacting the disk in the anticenter direction. These impacts are usually discovered in observations of the #ux and distribution of the radio continuum emission from hot gas, and the distribution and kinematics of the (infalling) neutral hydrogen gas. The galactic disk HVC collisions may also be an example of mass transfer induced by a galaxy collision. Mirabel and Morras' stream may be connected or related to the Magellanic stream, a partial ring of HI gas around the galaxy, and connected to the Magellanic Clouds (and the bridge between them). Recent models support the old conjecture that the Magellanic stream may be a bridge-tail structure resulting from the interaction between the Magellanic Clouds and the Galaxy (e.g., Lin et al., 1995; and references therein). Stunning new maps of the HI gas in these structures have recently been presented (Stavely-Smith et al., 1998, and in progress), and constraints from these data should allow the construction of much more detailed models in the near future. Hydrodynamic mass transfer in many colliding galaxies involves cloud collisions on a much larger scale than the HVC-disk interactions in the Milky Way. (Despite the fact that the latter involve regions of size greater than 1.0 kpc.) Bridges or plumes in these systems have gas masses comparable to or greater than the Magellanic stream. Let us consider a few speci"c examples from the recent research literature. HI observations have also provided kinematic evidence for a cloud-disk collision in the nearby galaxy M101 (van der Hulst and Sancisi, 1988). According to Kamphuis (1993, and see
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van der Hulst, 1996) the kinematically disturbed region `is over 20 kpc in size and contains a few 10 solar masses of HI and reaches velocities up to 160 km/s above the local rotation.a This is certainly the most impressive example of the phenomenon among the nearby galaxies. Another very appropriate example is the system UGC 12914/5, called the `Ta!ya galaxies by Condon et al. (1993). (The term was also used by Zwicky in 1956 for such `stickya systems, according to TT.) These authors note that `almost half of the (radio continuum) #ux2 arises from the gap between UGC 12914 and 12915, across which radio contours are drawn like "laments of Ta!y.a Besides radio continuum emission, and the magnetic "elds and cosmic rays assumed to produce it, there is also abundant HI gas in the bridge connecting the two galaxies. From the spatial distribution of these emissions, and the continuous kinematics along the bridge, the authors conclude that this is, in fact, the splash following the `interpenetrationa of two galaxy disks. Interestingly, there is little recent or ongoing star formation in the bridge. It is also interesting that one of the two galaxies is a ring galaxy. Thus, the Ta!y system is probably a close relative of galaxies in the `Sacred Mushrooma class of Arp and Madore (1987). (Even if the culinary association does not sound promising!) A Sacred Mushroom is de"ned by a stem consisting of an edge-on or disrupted companion connected to a ring galaxy cap. The fact that the two galaxies are still in contact (at least in projection) implies that the collision is not quite over. The companion orientation is di!erent in the Ta!y system, but the collision may not be any older. The prototype Sacred Mushroom is the AM 1724-622 system (see Fig. 1). Unfortunately, the primary in this system was apparently gas poor, and thus, it is not a good splash example (see Wallin and Struck-Marcell, 1994). Arp 284 (NGC 7714/5) provides a better example, according to the HI observations and modeling of Smith and Wallin (1992), even though it has a rather small cap-to-stem ratio, see Fig. 13. This system is especially interesting because the Smith and Wallin model suggests that the bridge may be the result of the combined action of gravitational and hydrodynamic procesess. The recent high-resolution HI observations (see Smith et al., 1997) show that the bridge gas and HII regions are both spatially o!set from the stellar bridge by a large amount, which, along with new hydrodynamic models, provides support the notion that they are the result of di!erent physical processes. The large amount of HI gas in this bridge also shows how e!ective these processes can be. Recent HI observations have revealed long HI bridges or plumes in two classic ring galaxies indicating that they are at least closely related to the `Mushrooma class. The Cartwheel ring appears to have a fully connecting bridge (see Fig. 14 from Higdon, 1996), while the plume of the ring galaxy in the VII Zw 466 system is not so complete (Appleton et al., 1996). In both cases the long bridges have thus far only been seen in neutral hydrogen emission. Like the Ta!y system, all of these other systems show little or no new star formation, with the exception of Arp 284, the transition system involving both tidal and hydrodynamic mass transfer. It is not surprising that splashes inhibit star formation since they disrupt disk clouds, and lead to an overall rarefaction of disk gas. As we will discuss below, our understanding of the physics of large-scale star formation is very incomplete. It seems clear that compression is a key, though it may only be the initiator of a multi-step process. On the other hand, direct triggering of star formation in high velocity cloud-cloud collisions is another popular idea that is very relevant here. This is emphasized by the fact that the Milky Way HVC galactic disk collisions seems to have triggered star formation (e.g., Cabrera-Cano et al., 1995; Comeron and Torra, 1994). Nonetheless,
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Fig. 13. The bridge connecting the two galaxies of the Arp 284 system. A VLA multi-array intensity map of the HI gas (grayscale) is superposed on a narrowband red continuum image (contours) smoothed to 12 arcsec resolution (see Smith et al. (1997) for details). The o!set in the bridge between the gas (dark ridge) and the old red stars is about 10 arcsec.
this induced star formation seems to be very modest compared to that triggered in large-scale compression waves. 4.1.2. Models and splash physics Probably the "rst authors to describe the gas dynamics of collisions between two galaxy disks were Spitzer and Baade (1951, see Section 1.3). However, this paper was not followed up by many observational studies until the recent examples above, in part because the detector technology wasn't adequate (but see Fabbiano and Trinchieri, 1983). Spitzer and Baade were decades ahead of their time. Because of the nonlinear, dissipative shock hydrodynamics, and the wide range of scales involved, it is also a di$cult process to simulate numerically, so theoretical progress was impeded too. There is a fairly extensive literature on supersonic collisions between interstellar clouds, and between clouds and large-scale shocks, dating from the early 1970s. This work provided a basis for the simulations of Tenorio-Tagle (1981), and Tenorio-Tagle et al. (1986, 1987) on collisions between HVCs and the galactic disk. In this series of papers it was demonstrated that such collisions had enough energy and momentum to generate the largest loops or supershells of HI seen in our galaxy. These models also demonstrated, that depending on the density and velocity of the impacting clouds, they could either dump most of their infall energy in the galactic disk, or punch a cylindrical
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Fig. 14. The HI gas bridge in the Cartwheel system from Higdon (1996).
hole in the disk and retain part of their energy. It is possible that such holes and other manifestations of cloud infall out of a bridge have been observed in recent HST observations of the Cartwheel ring galaxy (Struck et al., 1996). However, the high-resolution observations were only made in two broad bands (B and I), so shock emission could not be distinguished from other sources. Harwit et al. (1987) presented a paper that can be regarded as an update of Spitzer and Baade, detailing the infrared emission expected in disk-disk collisions. This paper was inspired by then new IRAS discoveries of ultraluminous galaxies, and suggested that some might be the results of such collisions. The subsequent realization that many of the ultraluminous galaxies were probably merger remnants, led many researchers to persue studies of the spectacular merger dynamics, where gas splashes play a secondary role (see below). Recent merger simulations do include the gas component in the disks of both galaxies (see Section 6), but the discussion focusses on the "nal gas distribution in the merger remnant, rather than the transient splash dynamics. However, the recent paper of Thakar and Ryden (1996) does discuss gaseous infall onto disks, and the formation of counter-rotating disks (discussed below). This paper highlights the role of infall or collisions with dwarf companions in dynamically heating galaxy disks, thus demonstrating one long-term evolutionary e!ect.
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Fig. 15. Three orthogonal views at two times, immediately before and after the collision of two gas-rich, disk galaxies in a numerical hydrodynamical simulation. As in observed systems the collisional splash produces a substantial gas bridge, while leaving the primary disk largely intact. See Struck (1997) for details.
A small grid of Smoothed Particle Hydrodynamics simulations carried out by the author are complementary to those of Thakur and Ryden, in that direct collisions with an intermediate mass companion (about 1 : 3 mass ratio) were studied (Appleton et al., 1996; Struck, 1996, 1997). One of the main motivations for these simulations was to model the HI bridges in the ring galaxies VII Zw 466 and the Cartwheel, which are very gas rich, so the model disks were purely gaseous (and were embedded in a rigid halo potential). Cooling and heating processes were also included. The orbital trajectories were such that following the collision the companion continued outward to a distance of several primary disk diameters, before falling back for a second collision. Not surprisingly, the initial impact did indeed produce a substantial splash, which was then stretched out into a long bridge between the two galaxies (Fig. 15). Because of the size di!erence much of the primary disk remained intact following the collision, but in all cases the companion was highly disrupted. In some cases it was nearly completely disrupted. After an initial delay, in which most of the bridge gas stretches away from both galaxies, infall onto both galaxies accelerates. This (re)accretion generally occurs out of distinct streams within the bridge gas. The accretion rate peaks and then declines before the two galaxies fall back together. In all cases the companion reaccretes less gas than it had initially, though the recaptured gas generally
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forms a smaller, more compact disk, a likely site of later star formation. The gas mass in the bridge, and its galaxy of origin, are found to depend sensitively on the orientation of the companion disk at the time of impact. The bridge can consist mostly of gas from the primary disk (face-on impact), gas from the companion (companion disk in its orbital plane), or a nearly equal mixture of both. The rami"cations for the short-term evolution of the companion are extreme. E.g., partial to complete removal of its gas disk, and reformation with a very di!erent structure and composition. These, and analogous trans"gurations of stellar bulge and dark halo components (not modeled in these simulations), will have important e!ect on subsequent collisions and merger. This is also true of changes in the primary, though they are less extreme. The accretion onto the primary is only of order 10% of its gas mass, but the infalling material has relatively low angular momentum, so it tends to concentrate in the central regions. Infall heats gas in the central regions. Thus, it initially inhibits star formation there, according to the algorithm used in these simulations, which requires dense, cool-to-warm gas for star formation (henceforth SF). Moreover, the accreted gas initially settles into a second inner disk, with a signi"cantly di!erent orientation than the primary disk, like the `counter-rotatinga disks of Thakar and Ryden. However, as this gas settles in it provides a reservoir to fuel (delayed) starbursts and nuclear activity. The timing of this activity relative to the return time of the companion will determine how much gas is available during the "nal merger, which in turn in#uences the "nal structure of the remnant. A small, but important, coda to this story } most published galaxy collision simulations, which include a gas component, employ an isothermal equation of state. This approximation is based on the fact that free cooling times in the interstellar gas are very short compared to other relevant timescales in galactic dynamics. Yet, this neglects the fact that in star-forming regions, stellar winds, high UV photon #uxes, and supernova explosions provide continuing sources of heat, and so the local thermal balance is probably dynamical. In the Struck (1996,1997) simulations comparisons were run between isothermal and cooling/heating models. Most of the large-scale tidal morphologies and wave structures are very similar. There are small di!erences that could be attributed in some cases to the di!erent jump conditions in the impact shock (i.e., isothermal vs. locally adiabatic). However, there are also larger di!erences, like the fact that star formation cannot occur in some regions with high gas density, because the region was heated by infall or previous star formation. The treatment of thermal e!ects and even the resolution of cloud structures is highly approximate in present-day simulations, but there are clearly many interesting e!ects to study as the state of the art advances. 4.1.3. Slings: tidally torqued bridges (YDx#) Bridge formation and mass transfer in the tidal process a!ects stars and gas more equally, and has been studied longer and in more detail than the splash process. In fact, the tidal stretching and spin-up that generates a mass transfer bridge is the same process that makes the tidal spirals discussed above. Thus, the "rst models of tidal bridges were in fact the earliest interaction models. The process is clearly illustrated in the papers of the Toomres (1972) and Eneev et al. (1973). The former contains the "rst systematic investigation of the process, and presents the basic laws governing it. The "rst and most important is that the most mass transfer from the primary to the companion occurs when the encounter is in the plane of the primary disk, and prograde relative to it. More generally, TT concluded that collisions with inclinations of less than 303 produced `reala (mass transfer) bridges, and those with inclinations of more than 603 did not. Wallin and Stuart (1992)
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(henceforth WS) carried out a huge grid of restricted three-body simulations to model tidal mass transfer as a function of interaction parameter. Their Fig. 5 con"rms the TT result, and shows quantitatively the rapid decrease to transferred mass in the intermediate inclination range. Retrograde encounters do produce tidal tails, but they either form too late, or on the wrong side of the galaxy to connect with the companion near closest approach. Similarly, high inclination companions have generally moved too far from the primary disk plane by the time tails, which are potential bridges, form. Thus, qualitatively, the criterion for mass transfer is that the companion must orbit roughly along with particles in the primary for a signi"cant fraction of their orbit. This not only provides time for bridge formation, but allows enhanced gravitational torques from the companion to propel the particles outward. Another law of bridge formation from TT is that the bridges (and tails) are not three dimensional objects. They tend to form on two dimensional surfaces, which look like a continuously deformed or tilted version of their original orbital plane. From the considerations of the previous paragraph it is reasonable to suppose that the accelerations in the direction perpendicular to this plane will be relatively small, unless the perturbed particle is very close to the companion (or is swung above or below it). For encounters with a "xed inclination, the amount of mass torn from the primary disk also depends sensitively on the mass ratio of the two galaxies, and the closest approach distance. These dependences were studied by WS in the case of parabolic encounters on zero inclination orbits. If the companion mass is less than the primary, and its closest approach distance is more than twice the outer radius of the primary disk, then WS found that the primary will lose less than 5% of its disk mass. If the closest approach distance equals the primary disk radius, then it was found that up to about 40% of the primary disk mass is removed. In such extreme cases the restricted three-body approximation is almost certainly inaccurate, and so too the derived mass loss rates. Both disk self-gravity and distortions of bulge and halo components, among other e!ects, are probably important. However, as long as the encounter is rapid and impulsive, these factors may a!ect the amount of mass removed less than they a!ect where it ends up. These variables will also be determined by the structure and extent of the primary halo. With a scale length of 1}2 disk radii, the halos used by WS were of moderate extent. They found that typically about half of the (stellar) particles removed from the primary became bound to the companion, with the remainder becoming unbound to either galaxy. With dissipative gas one would expect a smaller fraction to escape. The "gures in WS seem to suggest that some relatively simple scaling relationships might exist for the mass loss as a function of the various parameters. Having such relations would facilitate the comparison of their results to more detailed, self-consistent simulations, and perhaps, to observation. (Though determining interaction parameters, let alone the amount of mass transfer, from observation is an extremely di$cult undertaking.) The simplest scalings can be derived from the KIA formalism described above, with the additional assumption that the impulsive acceleration is imparted while the companion travels over a small angular sector *h of its orbit near closest approach. For the parabolic companion orbits usually considered, this sector can be approximated by a circular arc centered on the primary. Moreover, the relative velocity of the companion will not vary much in the time interval *t during which the segment is traversed. This velocity will
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approximately equal the escape velocity from the primary at the companion radius,
2GM . (11) v + r If r is in the range 1.0}2.0 disk radii, then this is also about equal to the circular velocity of disk particles located just inside the outer radius. The conclusion is that the disk particles in the sector *h will travel roughly parallel to the companion during the interval *t, and maintain a rougly constant distance, *r. Then the velocity impluse is about
GM v *h GM *t+ , (12) *r r *r where we apply this to primary disk particles following the companion. Particles which lead the companion around the primary disk will have their angular momentum reduced by the azimuthal impulse. They will then fall inward unless the radial impulse is larger (see Section 3.4). To estimate the mass loss let us assume that particles with impulses greater than *v"v do escape. The escape velocity is constant as a function of radius for a #at rotation curve galaxy. Then, that part of the primary disk contained within the circle centered on the companion, with a radius of, *v+
GM *t , (13) v will be lost. Without calculating the overlap area of this circle and the disk, we can assume that it, and the mass loss scale as *r, so *r"
v *h GM . (14) MQ J r v Despite the many simpli"cations, this scaling seems to capture several of the WS results. First of all, with other parameters "xed, both this formula and the simulations agree that mass loss scales nearly linearly with companion mass. Secondly, the anlge *h should not change with inclination angle i as long as the shape of the parabolic orbit doesn't. If this is true, then its projection onto the primary disk plane, which determines the impulse, scales as cos(i). Interestingly, the WS curve of mass loss versus inclination is quite well approximated by the function 0.5 cos(i), except near i"03, and i"903. The factor 0.5 is the fraction of the disk that is positively torqued, and thus, roughly the maximum mass loss. However, the third dependence, on the closest approach distance, is not clari"ed by the formula above. If the WS results are approximated by a power law, i.e., mass loss scales as 1/rL, it appears that n+ 3}4. Even given that v Jr\ , a steep dependence of *h on r is needed for the above formula to match this. Such a steep dependence seems unlikely, and more likely the assumption that the particle removal zone is a simple sector is not correct in the range of closest approach distances where the mass loss changes rapidly. In summary, signi"cant tidal mass transfer only occurs when the companion orbit inclination is less than about 453, and the closest approach distance is less than about 2.0 primary disk radii. Hydrodynamic splashes can occur in collisions with any inclination, but the distance of closest
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approach must be less than the sum of the disk radii. The dependences of mass loss on the companion mass may not be too di!erent in the two cases. Thus, since the "rst two factors are o!setting, the relative importance of the two processes for net mass transfer between galaxies may be about equal. This is somewhat surprising because systems with companions that appear to be connected by tidal bridges are very common in the peculiar galaxies atlases (e.g., `companions on armsa in Arp (1966), M51 types in Arp and Madore (1987)). Obvious splash systems are not so common. However, there are a couple of straightforward explanations. The "rst is that the cross sections and impact parameters for generating tidal arms or tails are much larger than those for mass transfer, and the probability for apparent superpositions of arms and companions is not small. The second is that ongoing star formation is frequently lights up tidal tails, while splashes are usually invisible optically. 4.1.4. Observations of tidal bridges and star formation M51, the `Whirlpoola galaxy, is the most conspicuous example of a galaxy with a companion on the end of an arm (see Fig. 10). Unfortunately, TT argued that the connection is in fact only apparent, and not a physical connection. They found that their models could not reproduce the detailed arm morphology and other tidal features in the brief time the two galaxies were connected. Also reddening observations suggest that the arm lies in front of the companion. Later, fully self-consistent simulations agree that it is di$cult to produce the observed morphology during a brief parabolic passage (e.g., Hernquist 1990). Howard and Byrd (1990) suggested that the companion is on a bound orbit, and that two disk crossings account for the observations best. A review of the recent models, and a more detailed version of the two-crossing model is presented in Byrd and Salo (1995). In any case, it appears that under close scrutiny M51 has become too complicated to be a true prototype for the simplest forms of transient tidal mass transfer. Laurikainen et al. (1993) and Salo and Laurikainen (1993) have recently presented detailed studies of another M51-type system, Arp 86, which may be simpler. In the Arp atlas it appears very simple compared to its atlas neighbour Arp 85"M51. Salo and Laurikainen carried out a set of detailed N-body, star-gas simulations of this system, and found a best-"t model that "ts both the observed morphology and kinematics remarkably well. In this model the companion mass is about 10% of the primary, and it has a low inclination orbit, i"203. However, it is a bound, low eccentricity (quite circular) orbit. Salo and Laurikainen "nd that the disturbed morphology is well accounted for as the result of the tidal disturbances on the last half-orbit, but this involves two close passages, with the earlier one playing the larger role in generating the present features. Deja vu M51. Salo and Laurikainen also found an interesting result by running their simulation far into the future of this system, i.e., through three more companion orbits. The model shows that the companion is connected to the primary for a considerable fraction of this time, though at late times the bridges have a very messy appearance. The authors make the reasonable conjecture that the original orbit was parabolic, but that dynamical friction (see below) led to capture and circularization of the orbit. Given the relative durations of the initial encounter versus the sum of later encounters it seems likely that this is true of many M51-type systems, and that the catalog objects are probably distributed along a generic sequence of transient encounters, rather than being primarily "rst encounter objects.
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Some record of this sequence of events might be found in the stellar populations of these systems, if the perturbations produced at closest approach generate transient episodes of enhanced star formation. A great many numerical models of prograde, low inclination interactions have been produced over the last few decades, and they all agree that there is enhanced compression in the tidally induced arms, tails and bridges, which probably drives enhanced SF. In fact, there is a good deal of observational evidence for ongoing, and enhanced, SF in tidal bridges (and tails). Schombert et al. (1990) presented photometric imaging of tidal features in 25 Arp systems. They found relatively blue mean colors and large color dispersions indicative of ongoing SF in many of these features, and especially in the bridges. The Ha image of M51 of Thronson et al. (1991) shows that its (apparent?) bridge is also forming new stars. So too is Arp 86 according to the color data of Laurikainen et al., with an extremely blue region at its end. Salo and Laurikainen's models for this system provide a detailed history of the wave compression. There is qualitative agreement with the broad band color data, assuming SF occurs in compressed regions, but this type of data is not su$cient to allow a detailed reconstruction of the SF history and spatial distribution. Nonetheless, this result is in sharp contrast to the splash bridges, which evidently expand and disperse gas, and suppress SF. Taken together, the results on these two kinds of bridge support the idea of a strong gas density dependence of SF in tidal features analogous to the surface density dependence in isolated disks (e.g., Kennicutt, 1989, 1990). It seems very likely that more precise conclusions about the SF process can be derived from careful study and comparison of di!erent types of bridge. Let us return to the Arp 86 system, for a moment. In addition to wave compression, the mass transfer onto the companion (NGC 7752) is likely to result in gas accumulation at its center, eventually triggering enhanced SF. The companion is in fact experiencing a strong burst of SF, and it has su$cient gas to continue forming stars for 1}2;10 yr (Laurikainen et al.). Moreover, this gas is only a small fraction of the total in the primary, so mass transfer in future encounters could provide more fuel. Thus, this system provides a fairly extreme example of how multiple encounters can lead to considerable galaxy evolution long before the "nal merger. On the other hand, the M51 companion shows little current SF activity (Thronson et al., 1991), though it shows Balmer absorption lines thought to indicative of a `post-starbursta stellar population (Ho et al., 1995). Joseph et al. (1984) carried out a near-infrared study of interacting systems that included six M51-type systems, and found most had quite high infrared luminosities, though none were found to be as active as Arp 86. If there is suppression and delay in the accretion process, the timescale is evidently smaller than the duration of enhanced SF. Mihos (1994) has recently reminded us of another factor a!ecting the net SF in M51-type systems. Simulations have shown that prograde interactions frequently induce the formation of a bar component in the primary, which, in turn, funnels gas to the center (also Noguchi 1990, and see Sections 3.5 and 5.4). Mihos' simulations suggest that bar-driven SF is much greater than the wave-driven SF in these systems. However, the timescale for the bar-driven SF is generally much longer than that of the waves and bridges. Nonetheless, like the `transientsa considered in these chapters, this process could have a substantial e!ect before the colliding galaxies merge. One "nal topic to touch on here is mass transfer onto ellipticals or very early-type disk galaxies, which are very gas poor. de Mello et al. (1995, 1996 and references therein) have studied a number of these so-called E#S mixed pairs, with the goal of determining what fraction are truly physically associated, and what e!ects they have had on each other. Although the statistics are still small,
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the answer seems to be that, among the physical pairs, tidal morphologies are common. So too is evidence for a young stellar component in the ellipticals, probably fueled by tidal mass transfer. Part of the motivation for such studies was the discovery that early-type galaxies are not devoid of gas. Space does not allow a recital of the history of this discovery, or an explanation of why it was surprising to many. However, it is now well-established that many ellipticals have at least a small gas fraction, and S0 galaxies often have an (extensive) annular neutral hydrogen disk located at a much larger mean radius than the stellar disk (e.g., Lees et al. (1991, 1992), Lees (1992) and references therein). Tidal mass transfer was one of a number of possibilities considered as the source of this gas, though it currently seems unlikely that this process is responsible in most cases. On the other hand, the de Mello et al. studies, and an S0 example of Appleton (1983), suggest that it is important in some systems. 4.2. Complete collisional disruption Complete collisional disruption is the ultimate transient event. As elsewhere in nature, big ones eat little ones, i.e., small companions generally su!er the most from tidal forces, and the transfer of orbital energy to internal energy. This is certainly the case in the interaction between the Magellanic Clouds and the Milky Way. However, the classical Roche disruption criterion for a satellite orbiting a planet teaches us that other factors, like the relative compactness (density), also play a role. However, the IA theory is again more appropriate than the classical theory of a secular process. Its application dates back to Spitzer (1958), who studied the tidal destruction of star clusters due to perturbations from passing interstellar clouds. Spitzer used the IA to estimate the internal energy change of the cluster, and compared this to its total binding energy. Disruption is assumed to result when the magnitude of this ratio is of order unity or greater. McNamara et al. (1994) have given a scaled version of this ratio appropriate to the case of disruption of dwarf galaxies by large primaries,
M *E 54 10M E
R 5 kpc
P \ < \ < \ , 15 kpc 1200 km s\ 60 km s\
(15)
for disruption. In this expression, M is the mass of the primary, R is the radius of the dwarf, P is the closest approach distance, < is the relative velocity at closest approach, and < is the `stellar velocity dispersion of the dwarf prior to the collision.a Detailed analytic formulae for the energy transfer have been derived in a variety of special cases, much more discussion of this topic can be found in the review of Alladin and Narasimhan (1982), and references therein. Note that the IA estimate for the tidal capture, due to transfer of orbital energy to internal energy, is very similar, but uses the ratio of the orbital energy and its change. IA analytic estimates of the transfer of angular momentum have also been derived in special cases, see e.g., Sunder et al. (1990), and references therein. In light of modern N-body simulations, which we will discuss in a moment, the impulsive energy transfer criterion for disruption seems unrealistically simple. It clearly neglects many processes, including angular momentum coupling and dynamical friction (see below). If the perturbation is nonlinear, the assumption that it is instantaneous is also unrealistic: e!ects like tidal distension and dynamical friction will develop during the encounter, and a!ect its outcome. Finally, the simple
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energy criterion "nesses the fact that galaxies consist of multiple components. Massive dark halos bind galaxies much more tightly than was thought in the early days of this subject, so at the least the primary mass factor in formula should include the halo contribution. Nonetheless, Spitzer's formula remains useful if only as a quantitative measure of how nonlinear is a given perturbation. Yet, we still face a number of questions, and a possible paradox, concerning collisional disruption. On one hand, there is observational evidence for its occurrence. On the other hand, the existence of massive halos makes it seem an unlikely event, and recent N-body simulations con"rm the di$culty of destroying a companion in a single collision. On the observational side, the case is easiest to make for very low mass companions like the Magellanic Clouds and other dwarf companions of the Milky Way. Dwarf companions are di$cult to detect, so it is hard to "nd systems actually undergoing disruption. But some possible examples have emerged. One such is the Virgo cluster pair consisting of the Virgo elliptical NGC 4472, and its extremely damaged dwarf companion UGC 7636 studied by McNamara et al. (1994). In the optical, the latter galaxy shows signs of extreme tidal stress, including a large tail of debris. An HI cloud lies between the two galaxies. McNamara et al. conjecture that the HI has been removed from the dwarf by ram pressure stripping by the hot gas in the halo of the giant elliptical. Radio observations indicate that the cloud is not self-gravitating, and will ultimately sink into the elliptical. They also conclude that the dwarf will be disrupted and dispersed. McNamara et al. note the apparent contradiction between their "ndings on this system and Gallagher and Hunter's (1989) "nding that, in a large sample of Virgo cluster dwarf irregulars, there was little evidence for major alterations or destruction. However, they note that `only interactions which result in2 an observable "nal statea could be detected. Events like that the NGC 4472/UGC 7636 are relatively short, and probably leave little trace afterwards. Gallagher and Hunter's result is consistent with this system if dwarfs are generally able to avoid their fate for a long time, but the result is quick when it comes. Another possible example, is suggested by the long HI plume extending out of the ring galaxy in the Arp 143 system. (Appleton et al. 1987a, b) suggested that this may in fact be the remains of a recently despatched gas-rich dwarf, destroyed in the collision which formed the ring. Unfortunately, the presence of another large (early type) galaxy in the group complicates the interpretation. There are virtually no clear examples of companion disruption among equal mass pairs, where we expect disruption to be di$cult, but merging to be rapid. The most intriguing cases should be those involving an intermediate mass companion, but again few examples can be found in the literature. Appleton et al. (1990) have provided an intricate example of a possible near miss for an intermediate mass galaxy in the double elliptical system NGC 5903/5898. Their 21 cm mapping showed a large mass (4.5;10 solar masses) of gas in a large region around NGC 5903. This gas is in a state of extreme disequilibrium, and probably recently accreted. The other elliptical, NGC 5898, contains little HI, but has an ionized gas disk that rotates in the opposite sense of its stars, and so was also probably accreted at an earlier time. Appleton et al. consider the possibility of a double accretion event. It seems unlikely that a single dwarf would su!er such a spectacular two part death. However, it is also possible that a disturbed spiral to the northeast was massive enough to survive two collisions. The HI "laments near NGC 5903 do point in its direction. In sum, observations provide evidence that prompt disruption may occur, but there are ambiguities in all particular cases. N-body simulations inspire a less sanguine view of the possibility of complete disruption in a single collision. Unfortunately, the simulations are the most useful
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(and e$cient) for modeling collisions between galaxies of similar mass, rather than those with mass ratios 3}4 orders of magnitude smaller, as in dwarf-giant collisions. Yet we can expect simulations to provide very useful information in the case of intermediate mass ratios, which is perhaps the most interesting case. The new simulations of head-on collisions by SeH quin and Dupraz (1996) provide especially good examples because they study cases with quite small mass ratios, i.e., 1/10 and 1/20. A weakness of these simulations is that each galaxy only contains one component, but if this is assumed to be the dark halo, it seems unlikely that conclusions about disruption will be greatly a!ected by the inclusion of other components. Their description of two particular runs is of interest here. These runs use a 1/10 mass companion, and are identical except that the companion in run F3 (di!use) has twice the scale length of that in run F1 (compact). If the di!use run there is a great deal of destruction in the "rst collision, and the companion is nearly destroyed. The core of the compact companion in the other run survives several collisions, and merger and disruption appear to occur simultaneously. The authors analyse the roles of various processes (e.g., tidal shock, distension, and dynamical friction) in detail, but they "nd that the amount of mass loss in each encounter depends sensitively on the density ratio of the two galaxies at closest approach. This conclusion is very reminiscent of both the classical Roche criterion, and various factors in the Spitzer formula. The paper of Walker et al. (1996) also studies the merger of a 10% companion, but from an orbit with an inclination of only 303. Multi-component galaxies (bulge-disk-halo) were used in this work. The companion was quite compact, and its core did survive the merger, though the authors "nd that the classical tidal radius (see BT Section 7.3) did not predict the core size very well. This `sinking satellitea situation, with little tidal shock, is much milder than the head-on collision case (see Section 5.2). While these N-body papers con"rm that reality is much more complex than a simple criterion for tidal disruption, that criterion stands up as a qualitative predictor. The N-body simulations also imply that the probable resolution to the `disruption paradoxa is that it may take several encounters to completely destroy a di!use companion even in favorable cases, where there is strong tidal shocking. Nonetheless, it is possible to do so before the merger occurs. Yet research up to the present still leaves open the question of how important this process, versus merger, is for dwarf companions. 4.3. Transient summary We have seen that the e!ects of a single impulsive collision, i.e., the transients, are complex, and have great morphological and dynamical variety. They can also have a major e!ect on the structure of the galaxies involved, especially the small galaxy in an unequal collision. However, summed over the universe, their net impact on galaxy evolution is less than that of mergers. Yet they are the inevitable prelude to merger. The "rst collisions leading up to the merger can greatly modify the structure and gas content of the merging partners. These modi"cations will be re#ected in the detailed structure of the merger remnant, as well as that of the tidal debris. Thus, we expect merger remnants to be morphologically diverse, even if their progenitors were not. Studies of transients also provide clues to and checks on theories of how merging happens. If the merging process had ended long ago, our understanding of it would be much more incomplete and speculative. It is a great advantage to be able to observe the diverse set of processes involved at many di!erent stages.
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However, the study of transients does inevitably emphasize diversity. Secular, dissipative, long-term dynamical processes stabilize galaxies and can push them towards uniformity. We will consider these processes in the following chapters. 5. Coming back (Ixx) This chapter is about the evolution in systems where the interaction does not end with a single impulsive collision. Speci"cally, we will consider morphologies that are either produced in multiple collisions, or formed on comparable timescales. We also begin the discussion of several long term dynamical processes like dynamical friction and collisionally induced bars. 5.1. Dynamical friction } bringing it back Dissipative collective e!ects are an important part of the reason why galaxy mergers are common, and a key process in galaxy evolution. Dynamical friction is the primary dissipative e!ect. In astrophysics this process was discovered by Chandrasekhar (1942, 1943), and completely developed for the case of a massive object moving through an in"nite, homogeneous sea of low mass objects with which it interacts gravitationally. The basic idea is that the massive object pulls the low mass objects towards itself as it moves through them, generating an overdense wake. The gravitational pull of the wake in turn decelerates the massive object, much like a frictional force. Chandrasekhar derived a simple analytic formula for this force; a brief version can be found in BT (Section 7.1). His "rst application was to stars in star clusters that randomly acquired speeds greater than the escape speed from the cluster. These stars might be retained, and the cluster's life prolonged, as a result of dynamical friction. Another application was to star clusters moving through a background of `"elda stars in a galaxy. Chandrasekhar's formula was not applied to colliding galaxies until the 1970s. First, it was recognized as the cause of sinking satellites. In terms of the basic theory this is the same problem as the star cluster in the galaxy, but with a small satellite galaxy instead (see the review of Ostriker, 1977). Then, Toomre (1977) argued for its operation in collisions between two galaxies in order to explain the results of early N-body simulations. While the sinking satellite studies showed it was a mechanism of satellite accretion, Toomre suggested it was the agent of what are now called major mergers. This stimulated renewed interest in the process, and in Chandrasekhar's formula in particular. There are several problems in the application of this formula, however (see e.g., Tremaine and Weinberg, 1984; Sequin and Dupraz, 1996). The "rst is that in its derivation, the force integration over all the in"nite sea of low mass `starsa diverges unless minimum and maximum impact parameters are selected. The ratio of these impact parameters appears in the so-called Coulomb logarithm term (from plasma physics) in the formula. This means the formula depends on how these terms are selected, and is essentially phenomenological. On the other hand, the quantitative dependence on the parameters in this term is weak. Moreover, there is a natural choice for the values of the minimum and maximum parameters as the satellite and primary galaxy sizes. The second di$culty is that the derivation suggests no way to modify or generalize the assumption of an in"nite, homogeneous background, e.g., to a spherical galaxy with a declining
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density pro"le. The third di$culty is that even though the friction is the result of a long range force, an obvious generalization is to identify the background density term with the local density. However, across a sharp density edge, this would give a discontinuous change in the friction. Finally, Tremaine and Weinberg noted a surprising fourth problem, the formula is simply wrong in some cases: Kalnajs (1972) has computed the dynamical friction force in an arti"cial but exactly soluble galaxy model consisting of a uniformly rotating sheet of stars. This is the only analytic model in which the collective e!ects due to the self-gravity of the disc stars can be included. Kalnajs "nds that collective e!ects modify the disc response so that there is no dynamical friction. (Tremaine and Weinberg, 1984, p. 730) As more N-body studies were published in the early 1980s it became increasingly clear that collective (feedback) responses were as important in determining the friction as the summed e!ects of individual star-satellite interactions (e.g., White, 1983). The paper of Tremaine and Weinberg (1984, henceforth TW) presented a new analytic approach to dynamical friction which was capable of capturing global responses, and overcame many of these di$culties. Chandrasekhar's formula derives from direct integrations over all two-body (star-satellite) interactions. TW's approach was a perturbation analysis of the Poisson and Hamilton equations for a star moving in the galaxy potential, with the additional perturbing potential due to the satellite. Only after the stellar accelerations were computed in the perturbation approximation were the integrations over all stars carried out. The two approaches were shown to yield the same result for the in"nite, homogeneous medium, but the perturbation analysis can be carried out in the spherically symmetric case as well. Speci"cally, TW applied the analysis to the case where the perturbing potential was a weak bar component, and derived the `LBK torque formulaa (after Lynden-Bell and Kalnajs, 1972) for the `frictionala torque exerted on the bar. Though this formula and the procedure from which it is derived are very complex, no logarithmic divergences result from this approach. At the conclusion of this calculation, TW note two important e!ects that were not included: strong or resonant interactions, and the continuous change in the motion of the bar or satellite. The resonances are generally transient, and TW were able to calculate their e!ects in a representative case. They found that the resonant interactions dominate the acceleration, but that in the case of fast resonances the LBK formula is correct. In the case of slow resonances, reversible torques operate, and the LBK and Chandrasekhar formulae are not applicable. Resonant interactions also provide a way for friction to operate even when the satellite is located outside the galaxy. In the end TW provide theoretical resolutions to all the problems of the Chandrasekhar formula, and equally importantly, clarify when it is applicable. Weinberg (1986, 1989) followed up this work with examples of explicit perturbation calculations of the wake structure and frictional drag on satellites in circular orbit about spherically symmetric primary galaxies. The second paper, in particular, calculated the self-gravitational response of the galaxy to the satellite wake in a speci"c case. Weinberg found that in this case the orbital decay time was increased by a factor of 2}3, primarily due to the barycentric motion. This motion is a global e!ect, which induces a counterwake in the primary galaxy. This e!ect provides a clear illustration of why the approximation that dynamical friction cannot depend only on the local density within a rigid galaxy.
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SeH quin and Dupraz (1993, 1994, 1996) have studied dynamical friction in the case of head-on galaxy collisions, using both N-body simulations and an appropriately taylored version of the Weinberg formalism. The 1993 reference presents a summary and comparison between the analytic and numerical results. Again global e!ects, like the tidal distortion of both galaxies are found to be important. Moreover, radial encounters `are both qualitatively and quantitatively di!erenta from near circular encounters. The transient response in the radial case is one reason. Secondly, in the spherical harmonic expansion of the density and potential of the primary galaxy, the monopole and dipole terms were found to be more important in the radial case (see the 1993 paper). In fact, these authors "nd that in the radial case, almost all of the dynamical friction is due to the "rst few harmonic terms. Very recently, another perturbative approach, based on `linear response theorya applied to the disturbing forces has been developed by Colpi and Pallavicini (1998) (Colpi, 1998), and Nelson and Tremaine (1997). When applied to the satellite accretion problems this method has comparable successes compared to the other recent approaches. Moreover, this method makes clear the connection between this problem and the #uctuation}dissipation theorem of statistical physics. In sum, a much more complete understanding of the phenomena of dynamical friction, a key ingredient in the merger process has been achieved in the last two decades. However, it has come with a high price in terms of theoretical and analytical complexity. The latter is obvious to the reader of the references above. By theoretical complexity I mean that many di!erent cases must be considered separately, and we cannot rely on a single, simple formula. However, there is a silver lining to this dark cloud } the new, more complex theory is intimately connected to, and provides insights into, the theories of tidal disruption and bar evolution. 5.2. Simulational examples of dynamical friction The theory described in the previous section, and in fact our understanding of all intermediate timescale processes, are hard to check observationally. The tidal morphologies are usually messy and complex after multiple encounters, and so, it is very di$cult to decipher their speci"c histories to the degree possible for those produced by single collisions. On the other hand, the systems have not yet settled down to quasi-equilibrium states like older merger remnants. Thus, the comparison of theory and numerical experimentation is very important for understanding dynamical processes operating on this timescale, while observational comparisons play a smaller role at present. Aside from merger simulations, which we defer to the next chapter, the simulational studies relevant to dynamical friction theory are concentrated in two areas } the sinking satellite problem, and its orthogonal counterpart, which we might call the `bobbinga satellite. The former are generally assumed to be in circular orbits initially, and in the latter case the satellites are on nearly radial orbits. Let us consider the `sinkersa "rst. 5.2.1. Sinking satellites (IDe#/!) Most of the sinking satellite studies have concentrated on one of two particular cases, either the satellite is assumed to be in a circular orbit around a spheroidal (or spherical) primary, or if the primary is a disk galaxy, the orbital plane of the satellite is assumed to have a small inclination to that of the primary. For brevity, I'll refer to the latter case as the planar case. A number of N-body studies were carried out in the 1980s, as were comparisons between them and the analytic works,
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notably the work of Hernquist and Weinberg (1989). This literature was succintly reviewed by Athanassoula (1990). In fact, this seems to have been an especially good moment for a review, since several issues had recently been resolved. In the spheroidal case, a controversy in the literature over the role of self-gravity was settled, in part by the realization of the importance the motion of the primary galaxy about the center of mass. More generally, the analytic work of Weinberg discussed above demonstrated the global response of the interacting system. Hernquist and Weinberg's comparisons of analytic models and simulations showed generally good agreement, particularly on the result that sinking times are longer (e.g., a few times 10 yr) than predicted by the Chandrasekhar formula. There was excellent agreement in limiting cases, and discrepancies could be understood as the result of nonlinear e!ects not adequately included in the analytic model. In the planar case, the realization of the role of resonance orbits (a la Tremaine and Weinberg), and how their interactions can work in the opposite sense of standard friction, was an important breakthrough. Among the important results cited by Athanassoula are the more rapid decay of heavy satellites (a clear result of global e!ects), and the slower sinking of satellites in retrograde and highly inclined orbits as compared to those in prograde planar orbits (e.g., about 10 yr in the recent prograde model of Walker et al., 1996). Since most of these e!ects were considered in the previous section, we will not discuss them further. However, before leaving the sinking satellites, we will brie#y note two related e!ects. The "rst is the Holmberg (1969) e!ect, which is the tendency he found, in a modest sample (58) of edge-on disk galaxies, for small companions to be found more frequently near the poles of the disk galaxy than near the disk plane. Quinn and Goodman (1986) drew the connection with our topic by hypothesizing that the e!ect was the result of more rapid sinking rates for satellites near the disk plane (especially the prograde population). Unfortunately, their simulations did not reveal a large enough di!erence between polar and planar orbits to account for Holmberg's result. This prompted a re-examination of the e!ect itself, by means of several catalog searches, but because of the small sample sizes a statistically signi"cant result could not be obtained. Zaritsky et al. (1993b) have looked at the subject again with a `homogeneousa sample of nearly 100 late-type spirals. They found evidence for a `weaka Holmberg e!ect for satellites within a (projected) radius of about 50 kpc, with an isotropic satellite distribution at larger radii. Although the satellite numbers are still too small for reliable conclusions, a weak result would seem more in accord with the theory and simulations. These authors also reported another, Holmberg-like e!ect } `satellites on prograde orbits tend to be brighter than those on retrograde orbitsa. The second e!ect of planar sinking satellites is disk thickening via dynamical heating as the satellite settles through the disk. The basic e!ect is straightforward } as long as the satellite is not completely disrupted it serves as a scattering center for individual disk stars. This e!ect was also studied by Quinn and Goodman. They pointed out that, while the thickening e!ect could potentially be very large based on the energy available, in fact, it was modest in their simulations. Instead they found that most of the dynamical heating occurred in the plane, i.e., the radial velocity dispersion of disk stars increased much more than the vertical dispersion. Yet the e!ect was signi"cant, and a number of studies in the last decade have con"rmed and elaborated the result. Quinn and Goodman used a restricted N-body method in their simulations, but the subsequent study of Quinn et al. (1993) was fully self-consistent with three components: a primary halo and disk, and a single component satellite. The later paper also considered higher inclination orbits, where the companion was found to sink into the plane faster than it sank in radius. The satellites on
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Fig. 16. Sequence of face-on and edge-on views of a numerical N-body simulation of a minor merger from Walker et al. (1996). Note the waves that develop in the face-on view, and the disk thickening in the edge-on view. (Time indicated in units of about 125 million years for typical galaxy scales.)
higher inclination orbits increased the vertical velocity dispersion as much as the other directions in the outer disk. The most ambitious simulational study to date is that of Walker et al. (1996), using large-scale N-body simulations of three components as in Quinn et al. With 500 000 particles they were able to use a relatively small companion (10% of the primary mass), but still have adequate particle resolution of all components. In their "ducial model a companion placed on a 303 inclination orbit reached the center after 1.0 Gyr, retaining 45% of its mass. The primary disk was thickened by 60%, and all velocity dispersions increased by comparable amounts. Fig. 16 shows the interesting structure that developed through the course of this `minor mergera. Another recent N-body study, by Huang and Carlberg (1997), focusses on the fate of low-density sinkers, and "nds such companions can be destroyed without contributing signi"cantly to disk thickening. On the observational side, Reshetnikov and Combes (1996, 1997) have recently compared a sample of 24 edge-on interacting disk galaxies to a control sample with 7 objects. They "nd that
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the interacting galaxies have a much lower ratio of disk radius to disk thickness, which they attribute to the combined e!ects of disk thickening and radial truncation. Further studies with larger samples would be very interesting. 5.2.2. Bobbing satellites (IXx0) Satellites sinking on radial orbits, the `bobbinga satellites, have been most studied as means to form shell galaxies, in which case the satellite is usually assumed to have a very low mass, and be readily disrupted. Radial orbits have also been used in early studies of mergers between nearly equal mass galaxies. Both of these case are discussed in the next chapter. In their recent papers on dynamical friction SeH quin and Dupraz (1993, 1994, 1996) present N-body simulations of the case of head-on collisions with intermediate mass companions (5}10% of the primary mass). The basic result is that satellites on parabolic orbits do not bob for long. They are generally merged within a few crossings, with the exception that `di!usea satellites are highly disrupted, as discussed above. There are some interesting di!erences relative to the circular case. The most important of these is the fact that the radial case is much more time-dependent; dynamical friction occurs primarily in discrete jumps during each interpenetration. This is because the transient response of both galaxies enhances the friction nonlinearly. E.g., according to the authors, `between radial and circular orbits the relative contribution of the monopolar term (contraction during the collision) to the total drag undergone by the satellite decreases from dominant to zeroa. Clearly, the Chandrasekhar formula misses the dominant e!ect in this case. As noted in Section 4.2, SeH quin and Dupraz's simulations include only single component galaxies. The question of what happens to the disk in multiple radial collisions has not yet been well studied, but the indications are that it is a very interesting area. First of all, the simulations of Taniguchi and Noguchi (1991) show that if the relative orbit lies in the plane of the primary disk the consequences can be incredibly violent, producing a highly distorted `winga galaxy. On the other hand, Lotan has studied the case where the relative orbit is near the symmetry axis of the primary disk, i.e., the ring galaxy case, but with multiple collisions (see Lotan-Luban (1990) and discussion in AS96). She used restricted three-body simulations with the inclusion of the Chandrasekhar formula for dynamical friction. Thus, the results cannot be regarded as accurate, but they reveal a fascinating e!ect whose existence should not depend on the details of the friction. If the companion returns while ring waves are still propagating through the disk, it imparts a new impulsive radial perturbation onto the pre-existing pattern of radial phase oscillations. The result is like wave interference, with enhanced wave amplitudes at some radii, and near cancellation at others. Qualitatively, it would seem that the strong impulsive friction found by SeH quin and Dupraz would speed the return and enhance this e!ect. In conclusion, simulation results and the current analytic theory of dynamical friction are in good accord. Because of the complexity of the latter, simulations can still produce surprises. The history of the Holmberg e!ect illustrates just how di$cult comparison to observation can be when subtle dynamical e!ects are involved. 5.3. Halo braking This topic is closely related to the previous one, since both deal with an intermediate time e!ect that draws galaxies together. Galaxy and galaxy group halos play a key role in accelerating the
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pace of successive collisions and mergers. Actually, there are several speci"c e!ects involved here. The "rst is just the fact that since galaxies have massive halos, they are much more likely to be gravitationally bound at a given separation and relative velocity than in the absence of such halos. There is evidence that galaxies in small groups are often contained in an extensive dark halo. For example, Ramella et al. (1989), in a study of groups in the Center for Astrophysics redshift survey, found median mass-to-light (M/¸) ratios of about 180 in solar units, which is much larger than typical values of individual galaxies. They also found a median halo scale size of about 500 kpc. On the other hand, the M/¸ ratios in Hickson compact galaxy groups are smaller, though still not small (about 50, according to Hickson et al. (1992)). The collisional ring galaxy VII Zw 466 is part of a small group, which was recently estimated to have M/¸+70 (Appleton et al., 1996). Granted this result, we would expect that the dynamical friction on a companion from this halo is a major e!ect, but this turns out to be an oversimpli"cation. The review of Barnes and Hernquist (1992a, henceforth BH92a) provides a good brief summary, as follows. Roughly speaking, a pair of spherical, interpenetrating dark halos interact as if they were single-component systems: the orbital angular momentum of the two halos is transferred to internal degrees of freedom, imparting spin and creating broad tails. 2embedded disks and/or bulges, are not much barked by the tidal forces retarding the dark matter; instead, these components lose orbital angular momentum mostly by interacting with their own surrounding halos, once the latter have been decelerated. These results are basically con"rmed by recent N-body simulations of mergers in galaxy groups. E.g., Governato et al. (1996) also "nd that friction operates "rst on halos. They also note the interesting result that galaxies falling into a group can be stripped of their halos, which can delay merger with the group galaxies. The multiple merger simulations of Weil and Hernquist (1996) also show the basic phenomenon of prompt halo merger. Their Fig. 3 clearly illustrates the dynamical heating of the halo in the merger process. In sum, these results inspire a much more optimistic view of the probability of collisions and mergers, compared to the general opinion when Toomre proposed it in the 1970s. Now, since most galaxies are born in groups, collisions and mergers appear to be almost inevitable on a group free-fall timescale. 5.4. Tidal stretching: tails and antennae (IXd#) Long tidal tails, jets, and plumes that stretch out to several galaxy diameters or more, are some of the most spectacular structures formed in galaxy collisions. The Arp atlas contains a number of famous examples, including well-known systems like the `Micea, the `Antennaea, and the `Atoms for Peacea. Table 1 of Schweizer (1983) contains a list of nearly two dozen. Melnick and Mirabel (1990) presented a number of additional southern examples, including the `Superantennaea with a 350 kpc long tail (Mirabel et al., 1991)! Fig. 17 shows Appleton et al.'s HI map of the Leo plume, drawn out from the galaxy NGC 3628. Because of its proximity, this structure stretches an incredible 403 across the sky. We have already covered the mechanisms responsible for producing these tails: it is simply tidal stretching and torques, and the slinging of outer disk components that result from the motion of the potential center. TT "rst demonstrated that these mechanisms can produce such long
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Fig. 17. The giant tidal plume extending out 403 from the galaxy NGC 3628 in Leo. The angular size of this plume is very large because it is relatively nearby. It is otherwise a representative tidal plume. (Unpublished image courtesy P.N. Appleton.)
structures, speci"cally, in their models of the Antennae and the Mice systems. The result has been veri"ed in more recent years with N-body simulations, e.g., by Barnes and Hernquist (1992b), Elmegreen et al. (1993) and Mihos et al. (1993). Since tails are formed in essentially the same way as the `tidal transientsa, the reader might wonder why they are discussed here, rather than in chapter 3? The answer is that the kinematic timescales of such large structures are long. For example, consider the free-fall timescale, qJo\ J(r/M(r)) . Within a dark matter halo M(r)Jr, so qJr. Thus, at the end of a 100 Mpc long tail, this timescale is of the order of few Gyr., or longer if the halo does not extend that far. This helps understand the result of Barnes' (1988) N-body simulations that tails are still evident after the galaxies that produced them have merged. Thus, the presence of tails can reveal the collisional history of a galaxy when other traces have disappeared in the more settled central regions. The speci"c topic of the formation of star clusters or dwarf galaxies in tidal tails deserves further discussion. The idea that clusters or dwarf galaxies can be formed in bridges and tails, and subsequently, with the dissolution of the tail, be freed to orbit the galaxy dates to Zwicky (1956, also 1959). It seems likely that the idea was viewed as very novel at the time. Schweizer's (1978) discovery of the young cluster at the end of one of the `Antennaea made it much more plausible by providing a good example. Mirabel et al. (1992) studied this object further, and concluded that it was indeed a young, low-surface-brightness dwarf irregular galaxy. Since then a number of other possible examples have been discovered, including the Superantennae which has young star clusters distributed along its great length. Arp 105 may have formed two dwarfs in its collision(s), a blue compact one to the south, and a Magellanic irregular in the north at the end of a long plume (Duc and Mirabel, 1994). The interacting cluster galaxy NGC 5291 is another example (Malphrus et al., 1997; Duc and Mirabel, 1997, 1998). A number of other examples have been studied recently (Hibbard and van Gorkom, 1996; Duc et al., 1998; Deeg et al., 1998). Hunsberger et al. (1996) found 47 candidate dwarf galaxies associated with tidal tails in a survey of 42 Hickson compact groups of galaxies. They estimate that as many as one half of the dwarf galaxies in compact groups may be `the product of interactions among giant parent galaxiesa. However, most of these objects have not been studied in detail, so we lack evidence that they have
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the detailed characteristics of dwarf galaxies, and speci"cally, are su$ciently strongly selfgravitating to become independent entities (see discussion in Schweizer (1998), and the models of Barnes (1998)). The phenomenon of star formation in tails is also of general interest because SF is unusual so far from the center of the parent galaxies. At "rst sight this seems mysterious } should not gas stretched out of a disk be less likely to form stars? However, there are several interesting e!ects that make it possible. The "rst, discussed above in Section 4.1.3, is the tendency of bridges and tails produced by tidal torques to remain on a two-dimensional surface, rather than spreading into the third dimension. The second is that, as discussed explicitly by Wallin (1990), and as is apparent in early simulations, intersecting caustics and compression waves propagate along tidal spirals and tails. (This is true even when the tail itself is primarily a material wave.) These compressions may stimulate massive cloud formation and star formation. Furthermore, Elmegreen et al. (1993) have argued that because large velocity dispersions are found in these tidal structures, local gravitational instabilities assemble very large gas clouds. Both large clouds and large velocity dispersions were found in their N-body simulations, as was the tendency for an `extended gas poola to accumulate at the end of the tidal arms. Several of these phenomena can be understood as natural results of the basic fact that in these types of collision, the orbit shapes of the gas clouds in the outer disk are changed from nearly circular to primarily radial. The dispersion of radial phases generates the propagating caustic waves as described in Section 3. The fact that there is an outermost excursion radius for the radial particle orbits accounts for pileups at the ends of tails. At other locations within the tail, particles are moving both radially inward and outward. It is reasonable to expect that collisional interactions between gas clouds moving in opposite directions may convert some of that motion into enhanced velocity dispersions. Simulational studies of tail dynamics are advancing, but many of these conjectures await testing with higher resolution, gas dynamical models. Even without gas there are complications. For example, several recent studies have clari"ed the role of halo structure in determining the morphology of stellar tails, and how this dependence might be exploited to probe halos (Dubinski et al., 1996; Mihos et al., 1998, and references therein). In any case, this is a very interesting environment for studying details of the gravitational instability, both numerically and observationally, if a wide range of length scales could be resolved. 5.5. Shells and ripples The story of shell ellipticals is dramatic. Their surprising discovery initiated a decade of intense activity, after which the phenomenon is almost taken for granted as another merger diagnostic, see Fig. 18. Presently, work on a variety of outstanding questions continues more quietly. Malin's (1979) initial discovery of huge, sharp, edge or shell-like structures wrapped around the `normala elliptical M89 was all the more surprising because it was the result of a little known photographic technique, called unsharp masking (see Malin, 1993). Malin and Carter (1980, 1983) soon presented more examples, and then a catalog. Schweizer (1980, 1983) realized that shells could be produced in mergers, and produced a list of merger candidates with `ripplesa, his preferred term. For detailed descriptions of shell morphologies and systematics see the review of Prieur (1990), and references therein.
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Fig. 18. The prototypical shell galaxy NGC 3923. Several sharp shell edges are visible in this negative image, even without special processing. (Digital Sky Survey image courtesy of AURA/STScI.)
At the same time Quinn (1982, 1983) and Toomre (as quoted in Schweizer, 1983) produced the "rst numerical simulations. These works showed that it was possible to produce sharp-edged features from the disruption of a small, disk galaxy companion via multiple crossings of a large elliptical primary. Quinn (1984) presented several simulations and describing in detail how the `phase wrapping of the dynamically cold disk materiala produced the shells. The basic idea of this process is well illustrated by his Fig. 5, reproduced here as Fig. 19. (The "gure shows that again, as in Section 3, we are dealing with radial caustics). Quinn also found that nearly radial collisions did the best job of reproducing observed interleaved shell morphologies. High angular momentum collisions produced `spatially wrappeda shells whose appearance depended strongly on viewing angle, and which crossed and overlapped in ways that the shell ellipticals did not. This suggested that shells were only produced in a rather narrow class of collisions. Since observations suggested that shells were not uncommon around ellipticals, this appeared to be a weakness of the collisional theory. Alternate models had been proposed, and the interested reader can "nd a summary of them in the review of Prieur (1990). However, the situation didn't lead to
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Fig. 19. Evolution of the phase and con"guration space distribution of a set of test particles falling from rest into a rigid (isochrone) galaxy potential, representing the formation of a shell galaxy (from Quinn, 1984). The time in units of the radial period of the most tightly bound particles is shown in the phase plots. Positions are given in units of the potential scale length.
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Fig. 20. Prieur's shell galaxy type classi"cation is illustrated by schematics of two systems. NGC 3923 on the left is an aligned system (type 1), and 0422-476 on the right is all-round, type 2 system (from Prieur, 1990).
wider acceptance of these alternatives, but rather to a broader investigation of the collisional model, and a number of new simulational studies. In several papers, Hernquist and Quinn (1987, 1989) described, e.g., how shell structure depended on the density pro"les and shapes of dark matter halos. These topics were also treated by Dupraz and Combes (1986). There was agreement that shells could be produced in a variety of di!erent potentials. Thus, Hernquist and Quinn (1989) concluded that `it will not be possible to unambiguously determine the mass distribution of the primary galaxy from observations of the morphology of shell systems alonea, ending an earlier hope. More positively, both Dupraz and Combes (1986), and Hernquist and Quinn (1988) generalized the initial studies by "nding that shell formation was possible in nonradial encounters, and with early type (elliptical) companions. This removed the major problem of the merger theory, that it required "ne-tuning. Hernquist and Quinn (1989) also discovered that the curious X-structures observed in some galaxies, were produced in some of their simulations, and so could be the result of accretion in some cases (see BH92a, and for a recent view, Mihos et al., 1995). These simulations were of the restricted three-body, or similar test particle, types, and as Dupraz and Combes pointed out, they did not resolve one di$cult problem. This is the huge range of radii,
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up to a factor of 30 in the extreme case of NGC 3923 (see Figs. 20 and 18). Dupraz and Combes conjectured that dynamical friction would play an important role in producing such systems, and thus, self-consistent N-body simulations would be needed. At the turn of the decade, several papers showed that shells, in fact, could be produced in self-consistent N-body simulations (e.g., Barnes, 1989; Salmon et al., 1990). Heisler and White (1990) made a detailed study of the radial structure with simulations that included a self-consistent treatment of the companion (in a low order, spherical harmonic approximation), and a rigid potential for the primary. They found that `there is a substantial transfer of energy in the various parts of the satellitea in the disruption process. This is the key that allows some stars to be thrown out to great distances, while other stars are left tightly bound to the primary center. Moreover, `the most extensive shell systems are formed for satellites of relatively high mass with mean densities similar to that of the primarya. Heisler and White did not quite succeed in getting as large a range of shell radii as observed, but they (also) speculated that the inclusion of dynamical friction might complete the job. In light of the impulsive dynamical friction found by Sequin and Dupraz (1996, and see Sections 5.1, 5.2 above), in radial encounters with modest mass companions, this seems highly probable. The large-scale N-body simulation of Salmon et al. (1990) also showed a large range of shell radii, suggesting this is correct. Another possible solution is that the deep shells might, in fact, be internal ripples in a small inner disk in the elliptical (Thomson and Wright (1990), also see discussion in BH92a). However, the continuity of shell intensities and colors with radius in NGC 3923 (Pence, 1986; Fort et al., 1986), argues against separate origins for some shells (although these works could not treat the deepest shells). Yet internal oscillations may be responsible for some of the less regularly rippled systems. This was shown by the models of Wallin and Struck-Marcell (1988) of direct, o!-center collisions, similar to those that make ring galaxies, but with gas-poor primary disks. At long times, compared to the propagation time of an individual ring wave, the asymmetric ringing produces shell-like features. These ripples are not as aligned or well-ordered as in classical shell systems like NGC 3923, but more like the Type 2 systems of Prieur (1990, see Fig. 20). They are, however, phase and not material features. These models were inspired by observations of Arp 227, whose primary is a rippled S0, which has not merged with its companion (Schombert and Wallin, 1987). However, a ring-ripple interpretation of this system is based in part on the belief that the colors of the ripples and primary are similar, but McGaugh and Bothun (1990) suggest that with proper subtraction of background starlight the shells are somewhat bluer. This might suggest that they were accreted. Recent work on Arp 10, a ring galaxy with outer ripples indicates it may be a better ring-ripple example, whose companion is almost merged (Charmandaris et al., 1993; Charmandaris and Appleton, 1996). The shell galaxy story is largely based on optical observations of the stars, but what about the gas? We expect that multiple, near radial collisions, which disrupt the stellar component, and presumably the halo, of the companion to have a similarly catastrophic e!ect on the gas component. In the only substantial simulation study to date, Weil and Hernquist (1993) showed that this is indeed the case. Moreover, because of dissipation in the gas, it rapidly became separated from the stars. While the stars continued their radial, shell-making oscillations, the gas settled into a compact inner ring or annulus. The cleaness of this result may, in part, be due to the adopted approximation that the companion halo vanishes at closest approach in the "rst collision, i.e., it is assumed to be of such low mass that it is completely disrupted. Nonetheless, the result is both
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reasonable, and as the authors point out, supported by observations of compact disks in shell ellipticals. Yet the recent HI mapping studies of van Gorkom (e.g., Hibbard et al., 1995; Schiminovich et al., 1995) and collaborators have shown that gas shells, ripples or streamers can also be found in interacting or merged systems. This is not to suggest a contradiction with the models or the theory. The gas structures resemble the messier ripple systems (Prieur Type 2), with usually no more than a few features per system. There are no (Type 1) examples with numerous, interleaved gas shells. The implication is that they are spatially wrapped shells produced in high angular momentum collisions. Another possibility found by the author in unpublished simulations is that detached streamers can be formed in o!-center collisions involving two gas disks, when there is partial overlap at impact. Very recently, Kojima and Noguchi (1997) have carried out N-body simulations of the merger of a gas-rich spiral and an elliptical galaxy, using sticky particle hydrodynamics. The stars from the disk galaxy make shells, the gas is dispersed over a wider part of the volume containing the shells, and star formation is terminated by the gas dispersal. New simulations of Charmandaris and Combes (1998, private communication) emphasize that a large part of the morphological di!erences between the gas and star shells produced in a given system are the result of the much greater radius of the initial gas disk, and subsequent collisional kinematics. These tentative results suggest that this will be an exciting area for continuing work. 5.6. Induced bars The collisionally induced formation of stellar bars is one of the most important intermediate timescale process in terms of its long-term consequences. Hints of the importance of this process appeared early, but since it is a collective e!ect, and requires fully self-consistent (N-body) modeling, it took longer to realize its full rami"cations. For example, Lynden-Bell (1979, 1993; Earn and Lynden-Bell, 1996, for updated pedagogical discussions) demonstrated that an oval perturbation can readily induce bar formation in the rising rotation curve region of a galactic disk, because of the nearly solid-body kinematics of such regions. The work of Toomre and Zang discussed above (Section 3.5) showed the swing ampli"cation process could also lead to bar formation. However, the dramatic rapidity of bar formation in prograde collisions, and the e!ectiveness of such bars in funneling gas into the central regions was "rst, clearly demonstrated in Noguchi's simulations (Noguchi, 1987, 1988; also see his review (Noguchi, 1990), and the reviews of Athanassoula, 1994; Combes, 1994). 5.6.1. Collisional bar formation The tidal forces in all non-central collisions naturally produce oval distortions, but as kinematic features they would be expected to phase mix and disappear relatively quickly. The prograde planar interactions studied by Noguchi (1987) produce a relatively strong and synchronous perturbation, as discussed above in connection with tidal spirals. Noguchi's models had two components, a halo and a stellar disk. The mass was initially distributed such that there was a signi"cant rising rotation curve region, where Lynden-Bell's work would suggest induced bar formation was likely. The size of this region, and the disk-to-halo mass ratio were varied. The only models that didn't form bars were ones with small disk masses or very small rising rotation curve
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regions. In most cases the perturbation was strong enough, and the disk su$ciently susceptible, that self-gravity was able to organize a long-lived bar. Similar but weaker, results were found by Byrd et al. (1986), whose #at rotation curve disk was less susceptible to bar-making. Noguchi (1987) also found a number of interesting trends in his models. Firstly, bars formed more rapidly, and rotated faster, as the disk to halo mass was increased. Secondly, the same results were obtained when the halo was made more centrally concentrated. These results are not too surprising since if the central mass is increased, then the fundamental gravitational timescale is decreased. Thirdly, the bars formed more rapidly and were much longer when the companion galaxy mass was increased. While these trends are very interesting in their own right, but the general result is that bars should be even more common in interacting galaxies than in isolated disk galaxies. This provides a mechanism for understanding analogous observational results. For example, Elmegreen and Elmegreen (1982) found that fraction of barred galaxies was 81% in binary systems, as compared to 63% for isolated galaxies. In a sample of 48 paired galaxies, Schweizer (1987) found a higher percentage of systems containing at least one barred galaxy among those systems with two disk galaxies, as compared to systems with one disk and on spheroidal galaxy. This last result provides some indirect evidence for the mechanism of bar formation induced by prograde interactions of disk galaxies. However, Noguchi's (1987, 1988) results had two signi"cant limitations: the models were two dimensional and the perturbing companion masses were greater than or equal to that of the target galaxy. One might worry if collisionally induced bar formation would be weakened if stars could be scattered into the third dimension, or if a very massive perturber was required to induce bar formation in a galaxy that was stable against the bar instability before the interaction. (See BT Sections 6.3, 6.5 for a discussion of bar formation in isolated galaxies.) The work of Gerin, Combes and Athanassoula (1990), included three dimensional simulations with 50% mass companions, and thus, easied both limitations. This work both supported and extended Noguchi's results. These three dimensional simulations had very modest numerical resolution by current standards. However, the basic result has been reproduced in many subsequent simulational studies, and the idea of collisionally induced bar formation is now a commonplace. 5.6.2. Ewects of induced bars Before Noguchi's collisional models, studies of bars in isolated galaxies had shown the profound e!ects they can have on disk structures, including: dynamical heating of the stellar disk, driving of spiral waves (outside the region containing the bar), and inducing the radial #ow of gas (see the reviews of Athanassoula (1994, 1996b) and Combes (1994)). The latter is the most important e!ect in the case of collisionally induced bars because, as Noguchi (1988) demonstrated, it is a means of funneling large amounts of gas into the central regions. This gas can, in turn, fuel nuclear starbursts and active galactic nuclei, which are commonly found in collisional systems. The simulations of Noguchi and others have left little doubt about the e!ectiveness of this mass transfer process when a bar is present (e.g., Noguchi, 1988, 1990; Barnes and Hernquist, 1991; Mihos and Hernquist, 1996). The question is rather, what fraction of central SF and nuclear activity in collisional galaxies is mediated by bars. Noguchi suggested that this might be the primary mechanism responsible for the high frequency of such activity in interacting galaxies. He speci"cally argued that this mechanism was much more e$cient at moving gas to the center than tidally induced spiral waves. Recently, Mihos (1994) has presented simulation results indicating
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that `under a wider range of interaction scenariosa the bar mediated process is more e!ective in inducing nuclear activity than another competing process } mass transfer between galaxies. Of course, the latter process, discussed above, is generally much more transient. Thus, with the caveat that mergers, in which time-dependent torques and #uctuating gravitational "elds persist for some time, can generate the most intense nuclear activity, Noguchi's conclusion has held up well. It is worth emphasizing that these simulations all involve parabolic, and so, relatively impulsive collisions. Thus, the longer timescale of the central activity has little to do with the collision, whose main role is to trigger bar formation. The nature of the subsequent radial #ows are essentially no di!erent than those in isolated galaxies. The e!ects and evolution of galactic bars have been the object of intense study in recent years, and there is a large literature on it (see the proceedings edited by Sandquist and Lindblad, 1996; Buta et al., 1996). This literature provides several results that are very relevant to the topic of sustained in#ows. The "rst is that recent observational studies con"rm that bars do indeed enhance the global SFR in the central regions of galaxies containing them (see the review of Kennicutt (1994) and references therein, and more recently Huang et al. (1996)). However, the e!ect is more complicated than originally thought. In fact, centrally enhanced SF is only found in about 1/3}1/2 of the barred galaxies, and there is no di!erence between the global SFRs of barred and unbarred galaxies. There is also great diversity in the morphology and distribution of star-forming regions in barred galaxies. Yet there do appear to be systematic trends, especially with Hubble type according to Phillips (1993), as quoted in Kennicutt Jr. (1994). Late-type barred galaxies (SBc and later) `show exponential bars, rising rotation curves, relatively weak rings and star formation enhancements in the bara. While barred early-types `contain #at stellar surface brightness pro"les, #at rotation curves, strong rings, but no star formation in the bar regiona (Kennicutt Jr., 1994). In the latter case, SF is found in the circumnuclear region (often in rings), and in a normal distribution outside the barred region. Huang et al.'s recent study of the infrared properties of a large sample of barred galaxies is generally in accord with these generalizations, and adds the suggestion that `availability of fuela (HI gas) is especially important in the early types. Recent theoretical and numerical models help us understand these observations. A "ne summary is given in the review of Athanassoula (1994), also see Sellwood and Wilkinson (1993), Athanassoula (1996a,b) and Noguchi (1996b). In the "rst place Athanassoula makes it clear that the models have a comparable degree of complexity. The location, degree, and strength of bar-driven shocks, as well as the presence of central rings or spirals, all show considerable variation as a function of galaxy structural parameters. A second point that is especially relevant to collisionally induced bars, is that `the in#ow is high during or after violent events like the formation of the bar, and much less during quasi-steady slow evolution of the bara. Bar-driven circumnuclear activity should follow a similar history following a collision or merger. Athanassoula also summarizes current thinking on another important question, even if there is gas in#ow, how does this gas get down to the very small radii that characterize the SF regions and active nuclei? Overcoming the centrifugal barrier would be di$cult in itself, but there are additional dynamical barriers. Speci"cally, simulations show that the presence of an inner Lindblad resonance (ILR, see Section 3.5) is very e!ective at slowing in#ow, and retaining gas in a ring at the radius of the resonance. Some, but not all, galaxies have mass distributions that allow these resonances. Shlosman et al. (1989) suggested a clever way around these di$culties, the bars-within-bars scenario. The basic idea is that the driven in#ow raises the central surface density, which triggers
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gravitational instability on a small scale relative to the original bar. On this scale the potential is modi"ed, so an ILR can be removed (or changed to a corotation resonance), and gas can be funneled down to still smaller radii. The reality and e$ciency of this process are supported both by self-consistent simulations and observational evidence for small scale bars within a few `barreda galaxies (e.g., Athanassoula, 1994; Friedli, 1996). In principle, this process can repeat itself, and a nested set of bars can form and funnel gas down to arbitrarily small radii, though there is as yet little evidence for many multi-level bars (see Section 8.2). In conclusion, studies of isolated barred galaxies do indeed provide much insight into the question of what e!ects result from collisionally induced bars. However, they do not help much with the lingering questions of what are the requirements for such bar formation? Speci"cally, what combinations of structural and collisional parameters are needed? Since relatively little of the parameter space has been studied, these questions remain quite open. 5.6.3. Longevity, frequency and other matters Investigations of a number of related questions have begun recently. The "rst of these questions concerns the longevity of bars. While N-body simulations have demonstrated that bars can be long-lived, they are also subject to instabilities that can dissolve them. Hasan and Norman (1990) described how a central mass concentration, like a massive black hole, generate stochastic regions in the orbital phase space. As the central mass is increased, these regions grow, in association with the development and outward movement of an ILR. Ultimately this leads to the dissolution of the bar. Pfenniger and Norman (1990) studied the case where the central mass grows as a result of the in#ow of gas clouds on weakly dissipative orbits. This generalized the earlier work by including a realistic growth process in order to study the temporal development of the process. The basic results of the earlier study were con"rmed, the di!usion of stars from the disk into the bulge as the bar dissolved was also described. The authors suggested that a large fraction of all bulges might have formed by this mechanism, on a timescale of about a few billion years. The observational evidence for this idea was subsequently reviewed by Kormendy (1993), with a generally favorable conclusion, though the uncertainties are large. Later work with three dimensional simulations of accretion or the merging of a small satellite supported the earlier results and is described in the review of Pfenniger (1992) and Hasan et al. (1993). The case where the central concentration is a nuclear gas ring seems to be somewhat di!erent than the central point mass, according to the paper of Heller and Shlosman (1996). A related instability, the buckling or "rehose instability, was reviewed by Sellwood (1992), and simulations of how it generates `boxya or `peanuta shaped bulges were presented by Combes et al. (1990). Similar structures are formed from this instability following the merger of a small companion in the recent simulations of Mihos et al. (1995). Bar longevity is closely coupled to another important question, what is the relative fraction of induced vs. intrinisic bars? If bars are relatively short-lived compared to the age of the galaxies, then their high abundance suggests that they are constantly being regenerated, by collisions or other means. However, many of the barred galaxies may not have an ILR, and have relatively low in#ow rates, and so may have much longer lives than those studied in the papers above. Noguchi (1996a,b) has recently presented a uni"ed picture which incorporates these ideas, and a number of others above, with the disk formation process. The latter connection is made by associating the in#ow in isolated galaxies with the accretion process in the formation of the disk.
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Thence, 2 we propose that late-type barred galaxies, the disks of which are considered to have formed by slow accretion of the halo gas, have intrinsic origin, whereas the bars in early-type galaxies, whose disks are likely to have grown quickly, have been formed in tidal interactions with other galaxies. Numerical simulations2 show that the bars created by tidal perturbations tend to have a relatively #at density pro"le2 whereas spontaneous bars have a steeper pro"le. 2this numerical result can explain the observed dichotomy (between early and late-type barred galaxies) Noguchi (1996a). These results are extended in Miwa and Noguchi (1998), who "nd that the disk resonance structure and pattern speedsin isolated, `intrinsica barred models are di!erent than those of bars produced in strong tidal interactions. The latter were found to rotate more slowly, in part, simply because they tend to be derived from more stable, lower mass initial disks. A caveat is that weak tidal perturbations were found to produce bars whose structure depends mostly on internal properties, and so, are more `intrinsica. One of these complications is the "nal question for us to consider in this section, what e!ect do collisions have on pre-existing bars? It appears that this question was "rst addressed in the paper of Gerin et al. (1990), who noted that a tidal interaction could transiently increase or decrease the strength and angular velocity (pattern speed) of a pre-existing bar, depending on the relative phase at closest approach. In several recent papers Athanassoula (1996a,b) (Athanassoula et al., 1996) has reported the results of a much broader exploration of this question using self-consistent simulations. The main result is an enormous range of possible outcomes. Depending on the companion and orbital parameters the bar can be: essentially una!ected or destroyed, reformed in an o!-center con"guration, or turned into a lense, ring(s) or bulge on an intermediate timescale following collision or merger. Fig. 21 shows an example from Athanassoula et al. (1996). The companion can meet a similar range of fates as discussed above, though the simulations reported in these papers had not yet covered a large range of companion parameters. E.g., all companion masses are less than 10% of the primary mass. These models are especially relevant for the case of multiple collisions (on the way to merger), where the "rst collision might generate a bar, whose fate would depend on details of the subsequent collisions. So we end this chapter on a note of great potential complexity. 5.7. Intermediate summary The two most important processes that operate on intermediate timescales, but have long-term consequences, are dynamical friction and bar formation. Chandrasekhar's theory of dynamical friction operating on a massive object traveling through a uniform sea of low-mass stars played an extremely important role in helping to explain why galaxies merge. However, it is too idealized to account for the full range of `frictionala and collitive phenomena in galaxy collisions. That is, all #uctuation phenomena, and the collective e!ects of resonant orbit interactions. These are captured in the perturbative expansion formalism of Tremaine and Weinberg (1984), and the later papers cited above.
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Fig. 21. Collision involving a barred primary from Athanassoula et al. (1996). In this simulation the point of closest approach is near the edge of the barred galaxy. After the collision the bar is almost destroyed, but later an o!-centered oval structure develops.
At the same time that this theory was being developed the importance of collision-induced bar formation became recognized, largely due to the results of numerical simulations. Bar formation requires both a relatively strong tidal disturbance, and a susceptible region in the center of the galaxy disk, though there are additional complications. Induced bars drive enhance star formation on intermediate timescales, and may contribute signi"cantly to the net SF of the universe. Bars drive radial gas #ows, and thereby feed active nuclei, either directly or indirectly. Tidal tails and collisional shells, which can be drawn out to very large radii, are other examples of intermediate timescale e!ects. 6. Mergers: all the way back (Oxx) Despite much initial skepticism, the general notion that mergers are a major driving force in galaxy evolution has become widely, almost universally, accepted. In fact, it has become the dominant theme in the "eld in the last decade. Not only have there been numerous simulational
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and observational papers published, but there have been a number of major review articles published as well. For example, the interacting galaxies review of Barnes and Hernquist (1992) focusses heavily on mergers and the numerical techniques used to simulate them. Schweizer's (1983, 1986, 1990) reviews cover the observational study of merger remnants. Hernquist (1993) reviews the increasingly detailed e!orts to compare models to possible elliptical merger remnants. Barnes (1995) (also Barnes and Hernquist, 1996) summarizes simulational studies of gas dynamics in mergers. Recent updates can be found in the reviews of Barnes (1996), Bender (1996) and the articles in the Saas Fee lectures edited by Friedli et al. (1998). With so many excellent reviews available a detailed examination of this area does not seem necessary. Thus, the discussion here will be brief. (Similar comments apply to the following chapters on induced SF and nuclear activity.) As in the previous chapters I will try to focus on the important physical processes. Moreover, merging is in large part a repetition and elaboration of the processes described above, so we do not start from scratch. We will concentrate on a few key issues, including the following questions. How fast are mergers? What are their immediate e!ects on the galaxies involved? How do these e!ects di!er from those experienced in nonmerging collisions? What kind of remnants do they leave, and are there speci"c observable signatures of a past merger? How common are mergers? 6.1. Overview and historical highlights Toomre's (1977) paper, discussed in Section 1, would have been a landmark even without the mergers-make-ellipticals conjecture presented there. The extension of his earlier work to the conclusion that most collisions end in merger, because of the operation of dynamical friction, was pivotal in itself. The N-body and N-ring simulations he presented showed two galaxies combining with truly shocking rapidity, though `Na was very small by modern standards. This evidently led Toomre to search the galaxy atlases for examples, and good candidates were not hard to "nd. As discussed in Section 1.3 the merger elliptical idea initially seemed too speculative for many astronomers, and there was similar skepticism about the idea of rapid merging. The "eld of N-body modeling was still quite young, but within a few years after Toomre's paper quite a number of simulational studies had been published. In this "rst epoch of N-body merger simulations the galaxies were generally modeled by a single (spheroidal or halo) component, with typically a few thousand particles. Listings and brief descriptions of the original papers can be found in the reviews of Barnes and Hernquist (1992a) (Hernquist, 1993), and in the introduction to their recent paper (Barnes and Hernquist, 1996). In most cases the collision partners were assumed to be of equal or near-equal mass. The primary result of these models was con"rmation that galaxy collisions from initially nearly bound orbits do indeed generally result in rapid merger. This was hard to understand at "rst because two-body relaxation times are very long, and two-body collisions are the basis of Chandrasekhar's dynamical friction equation. But these models revealed that the transfer of orbital energy to internal motions was a global, not a local, phenomenon (see Section 5.1). Moreover, though the fraction of mass lost was found to be small, this material carries away signi"cant energy and angular momentum. This is also true of material which is not lost, but #ung out to large radii in tidal tails. These models also showed that the merger remnant relaxed rapidly to a quasi-steady state, which was at least qualitatively similar to galactic bulges or elliptical galaxies. This was interpreted
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to be the result of `violent relaxationa described below. The early simulation papers contain detailed discussions of mass loss, radial mixing, and rotational and structural properties of the remnants as a function of initial parameters. However, it was hard to know how far the comparison of these single component models to observation could be taken. The second epoch of simulational studies can be de"ned by the publication of two component (disk-halo) N-body models (e.g., Gerhard, 1981, 1983a,b; Farouki and Shapiro, 1982; Barnes, 1988, 1992), and the "rst attempts to simulate gaseous dissipation in collisions and mergers (e.g., Negroponte and White, 1983; Barnes and Hernquist, 1991). Since model galaxies with a dynamically cold disk are generally assumed to be stabilized against the bar instability by a hot halo, two components are needed in simulations with disks. Such simulations were required, in turn, to test the theory that an elliptical could be formed from the merger of two disk galaxies. Much larger numbers of particles are required for such simulations, and so, their development has been closely coupled to improvements in computer hardware and software. Because the technology developed so rapidly during this time, N-body simulations now contain four to "ve orders of magnitude more particles than in the early 1980s. Second epoch models revealed more complexity in both the merging process and in merger remnants. A "rst example is in the orbital deceleration. As described in Section 5.3 above, massive halos merge "rst, and the merging of the denser disks and bulges is facilitated by the binding of the large halo. As expected, the dynamical heating in this process forges remnants of much earlier Hubble type than the progenitors, i.e., elliptical-like. However, as will be discussed below, the memory of the initial structure and orbit are not immediately erased by the heating and relaxation processes. The gas dynamical models of this period produced another important result, that a large fraction of the interstellar gas can be funneled deep into the core of the merger remnant. Bar-driven #ows of the type described in Section 5.6 are often an important part of this process. There is no clear boundary between second epoch and current models, though we can arbitrarily mark it by the appearance of three component (bulge-disk-halo), gas-plus-star models (e.g., Barnes and Hernquist, 1996). While `third epocha models have not produced as many major new results as earlier ones, many new avenues are being explored. These include, for example, more extensive studies of collisional bars and their interaction with other components of the merging galaxies (see Section 5.6), and more sophisticated attempts to model the feedback e!ects of starbursts (see Section 7). Large particle numbers have also just simply allowed the production of better models for speci"c mergers, like Barne's (1998) model for the Antennae system. In conjunction with these developments in models and theory, there has been great progress in the observational studies of the merger process. These were pioneered by F. Schweizer, foremost among the observers who took up Toomre's challenge to "nd the evidence for mergers. Schweizer's (1983, 1998) review, already referred to in Section 5.5, summarizes observational indicators of merging and early work on merger remnants. This paper begins with a discussion of cD galaxies, the supergiant galaxies that are obvious merger candidates. This is because they have huge stellar envelopes, and generally reside in clusters, so the chance of a cluster galaxy being captured via dynamical friction are good (Ostriker and Tremaine, 1975). In fact, cD galaxies are probably made by a succession of such mergers (`galactic cannibalisma). Schweizer cites two other `promising pieces of observational evidence of mergers in cD(s)a } the presence of multiple cores and asymmetric envelopes. Both are nonequilibrium features and would be expected to disappear on
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timescales much less than the age of the galaxies. Moreover, neither feature is the exclusive property of cD galaxies. Schweizer not only provides examples of galaxies with multiple cores, but also of galaxies with gas disks that do not rotate at the same rate as the coextensive stars (indicating a relative tilt), and of galaxies apparently containing two gas disks. Again these are not equilibrium structures. Since that time many more multiple core, and `counter-rotatinga disk systems have been found. Schweizer also argued that the polar ring galaxies (discussed below) might be the result of accretion or merger. Finally, and most importantly, there is the presence of large scale `tidala features: ripples, shells, tails, etc., around many merger candidates (see Fig. 22). In many of the systems Schweizer reviewed there is no obvious alternative to a galaxy collision and merger origin for these morphologies. While Schweizer pursued detailed studies of individual systems, statistical studies were beginning to provide evidence for enhanced `activitya in colliding and merging galaxies, in the form of star formation and nuclear activity, especially in optical colors and radio continuum emission (also see references in the following chapters). However, the IRAS (Infrared Astronomical Satellite) mission in 1984 brought observational studies of merger remnants into a new era. IRAS's whole sky survey in the far-infrared (with passbands centered at 60 and 100 lm) revealed numerous ultra-luminous infrared galaxies (ULIRGs or ULIGs), many of which were soon determined to be merger remnants or other types of collisional galaxy (e.g., Aaronson and Olszewski, 1984; Houck et al., 1984, 1985; Lonsdale et al., 1984; Wright et al., 1984; Soifer et al., 1984a,b; Joseph and Wright, 1985). Given the results on central activity in merger remnants, and the "rst inklings of the process of gas funneling from simulations, discovering hot dust was not a surprise. However, the magnitude and the extent of the phenomenon, i.e., that this class contains the brightest galaxies known, was a shock. These discoveries enormously energized the "eld, generated many new observational and theoretical studies, and eventually suggested the possibility that the activity generated by mergers might be responsible for most of the star formation (and thus metal production), and nuclear (quasar) activity in the universe (e.g., Sanders et al. (1988), and the reviews of Soifer et al. (1987), Sanders and Mirabel (1996), and the conference proceedings edited by Persson (1987), Sulentic et al. (1990)). This proposition remains speculative since, even if most ordinary galaxies experienced `majora mergers in the past, the traces are hard to "nd now. However, the more limited proposition that almost all of the most infrared luminous objects (with ¸ '10 solar luminosi'0 ties) are active merger remnants has been quite "rmly established in the time since IRAS. This story is told in detail in the recent review of Sanders and Mirabel (1996). Reaching this point has been an arduous process of: optical identi"cation of IRAS sources, obtaining redshifts to determine distances, and obtaining observations in many other wavebands to understand the nature of the sources. Yet, by now many lines of evidence support the idea that these objects are generally, advanced mergers powered by a mixture of circumnuclear starburst(s) and active galactic (nuclei)2fueled by an enormous concentration of molecular gas that has been funneled into the merger nucleus. (Sanders and Mirabel, 1996) Also called FIRGs, far-infrared galaxies, ELFs, extremely luminous galaxies, and LIGs, luminous infrared galaxies, though this latter also includes somewhat less luminous objects.
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Fig. 22. Multiwavelength observations of four famous merger remnants: (a) NGC 4038/39 (Arp 244, `The Antennaea), (b) NGC 7252 (Arp 226, `Atoms for Peacea), (c) IRAS 19254-7245 (`The Super-Antennae), (d) IC 4553/54 (Arp 220). Contours show the surface density of neutral hydrogen gas superposed on optical images (greyscale). The insets show K band (2.2 lm) images of the central regions as greyscale, with white contours representing molecular gas (CO) intensities. The scale-bar represents 20 kpc in each case. See Sanders and Mirabel (1996) for details.
New high-resolution observational studies, like Scoville et al.'s (1997) recent study of molecular gas in Arp 220, are indicative of the type of advances we can expect in coming years. While much observational e!ort in the last decade has focussed on merger ULIRGs, optical and radio studies of nearby (non-ULIRG) merger systems have continued to advance. This includes the work of Schweizer and Seitzer (1992), who used high quality optical images of faint tidal features to estimate the relative ages of merger remnants in a large sample (see Section 2.2). Schweizer and his
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collaborators also presented an important series of papers detailing the star formation properties of several well-known merger remnants, using both ground-based and Hubble Space Telescope imaging (Whitmore et al., 1993a, b; Schweizer and Seitzer, 1993; Whitmore and Schweizer, 1995; Schweizer, 1996). This work will be described in the following sections. At the same time, Hibbard was studying an evolutionary sequence of "ve merger remnants for his thesis work (Hibbard, 1995; also see Hibbard et al., 1994; Scoville et al., 1994; Schiminovich et al., 1995; Hibbard and van Gorkom, 1996). The youngest two systems in this sample are yet not fully merged (Arp 295 and NGC 4676, vs. the older systems NGC 520, NGC 3921 and NGC 7252). He used the VLA to map the HI distribution, and also carried out H-alpha and R band imaging in all "ve systems. With these data he carried out the most complete analysis to date of the evolution the gas component in the merger process. In the systems he studied a signi"cant fraction of the gas remained in an atomic phase in long (but bound) tidal arms or tails through the merger process. Hibbard makes the interesting point that these systems and others like them are generally not ULIRGs. Thus, it appears that the ULIRG phase is either short-lived and under-represented among nearby mergers, or there are at least two classes of merger (see the contrasting examples in Hibbard (1997)). With the combined work of Hibbard and Schweizer (1996 and references therein), a great deal of high-resolution multi-waveband data is now published for the NGC 3921 and NGC 7252 systems, which provides much evidence in support of the Toomre's conjecture that both systems are well on their way to becoming elliptical galaxies. Another parallel strand of merger research is the study of shape and kinematic pro"les of elliptical galaxies, especially their cores, and comparison of these to merger simulations in the hope of "nding unique and long-lived signatures of the merger origin of ellipticals. `Boxya isophotes are a "rst example. These are de"ned as rather square-shaped surface brightness contours, that contrast with the usual spheroidal pro"les. Bender (1990, and references therein) has provided evidence that boxy ellipticals are intermediate age merger remnants. A second example is the presence of a `kinematically decoupled corea in an elliptical. That is, the spin axis of the core is at a large angle relative to the spin axis of the bulk of the galaxy. Counter-rotating cores are extreme cases. Bender (1996, and references therein) states that `more than 50% of all luminous ellipticals contain kinematically decoupled cores, and these galaxies are found in all environments. In general, the arguments that these cores result from some kind of merger are strong. It also seems that they can have a long lifetime, and so they may provide the best long-time signature of a merger yet discovered. More subtle kinematic indicators, like the misalignment between the core spin axis and the minor axis of a #attened (but slowly rotating) elliptical may also be useful in future (Barnes, 1992, 1998; Hernquist, 1993). Ultimately, the reason for the existence of such structures is the fact that violent relaxation, and radial mixing in particular, do not go to completion in the merger process. This concludes our general history and overview of mergers. In the following sections we will look at a few speci"c aspects of the process in more detail. 6.2. Major merger dynamics In this section we will consider some of the physical processes that characterize `major mergersa. This term, major merger, has been adopted in recent years to describe mergers between nearly equal progenitors, which have a major e!ect on both galaxies. `Minor mergersa involve a signi"cantly
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smaller companion, and so the primary galaxy is not highly disrupted. Note that major mergers do not necessarily involve large or massive galaxies, nor do minor mergers involve only small galaxies. 6.2.1. Violent relaxation Violent relaxation is a wonderful oxymoron! The basic idea is that large amplitude #uctuations in the gravitational "eld, as in galaxy formation or collisions, drive a relaxation process that is much faster than the two-body relaxation time due to star-star encounters with a galaxy. The statistical formulation and a derivation of the equilibrium stellar phase space distribution was originally given by Lynden-Bell (1967) also see BT Section 4.7. Speci"cally, he demonstrated that violent relaxation could lead to an equilibrium (e.g., Maxwell-Boltzmann) distribution function on a relatively short timescale. Tremaine et al. (1986) (also see Kull et al., 1997) later investigated constraints on mixing and the relaxation process using H-functions. They begin by noting that the equilibrium state that violent relaxation drives stellar systems towards, has in"nite mass. Since galaxies do not, violent relaxation cannot go to completion, and `potential variations die away before relaxation is completea. They "nd that the remnants of this process can only resemble real elliptical galaxies if the initial state is cold or clumpy, e.g., far enough from equilibrium in the intereaction environment to allow for considerable relaxation. This turns some of the old objections to forming ellipticals from dynamically cold spirals on their head. While `incomplete violent relaxationa is a bit di$cult to visualize, the theory is very helpful for understanding the merger simulations and observational properties discussed in the previous section. Speci"cally, it helps explain why even early simulations found rapid relaxation to an elliptical-like surface density pro"le, in conjunction with modest radial mixing. On the observational side, it explains why stellar surface density pro"les typical of quiescent ellipticals (e.g., King or de Vaucouleurs pro"les, see BT) are found in the presence of multiple cores and tidal structures. The discovery that dynamical friction is the result of a global response, and can be quite impulsive (see Section 5.1), suggests that it is closely related to violent relaxation. This in turn helps understand the short timescale of merging. Another aspect of the conjoint global response of violent relaxation and dynamical friction is a result emphasized by Barnes (1992), and which dates to Farouki and Shapiro (1982). This is that orbit shapes, and orbital decay are primarily determined by one parameter } the ratio of `pericentric separationa to the galactic half-mass radius. Note, however, that this result applies to mergers between equal-mass galaxies on initially parabolic orbits. Physically it is probably a result of the fact that in these cases halos merge promptly, and constrain subsequent evolution. In minor mergers with small companions other parameters are also important. To approach the topic from a slightly di!erent angle, consider the recent paper of Chavanis et al. (1996), which revives an old analogy between the equations of two dimensional turbulence (Euler eqs.) and stellar dynamics (Vlasov eq.), and compares them in detail. The authors reproduce Lynden-Bell's equilibrium distribution function using a mixing entropy function and the principle of maximum entropy production. They also derive a Fokker}Planck equation for the coarse-grained distribution function to describe violent relaxation. In both turbulence and stellar relaxation there is a tendency to `develop "ner and "ner "lamentsa, though in the stellar case these "laments lie in the six dimensional phase space. In the turbulence case the relaxation also proceeds in two stages, the "rst is rapid, like violent relaxation, and leads to the formation of large coherent structures. The "nal viscous relaxation to the true equilibrium takes much longer, like two-body relaxation.
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6.2.2. Gas funneling Because the gas disk is usually more extensive than the stellar disk in disk galaxies, and thus has higher mean speci"c angular momentum, it is reasonable to expect that more would be lost or swung out to large radii in tidal collisions. Thus, it is surprising how much gas can be funneled to the center of some merger remnants, and there provide fuel for high levels of activity. On the other hand, interstellar gas is highly dissipative. Large-scale shocks are not only good at radiating energy, but also can transport angular momentum e$ciently. In several sections above we have described the mechanisms that simulations have shown a!ect this transport and the gas funneling. Foremost among these are collisionally induced bars (Section 5.6) and tidal spirals, including both internal waves and extended tidal tails (Sections 3.5 and 5.4). While the numerical simulations demonstrate the e!ectiveness of these processes (see references in the previous section), to date no simple physical or analytical models have been produced which are capable of predicting the amount of funneled gas, or how it scales with collision parameters. This is understandable since very high-resolution, multi-component simulations with both gas and stars are required to study the phenomenon, and be con"dent of the accuracy of the results. Moreover, an extensive exploration of the high-dimensional parameter space will be required. Before leaving this topic we should note that the existence of the funneling process resolves an important problem in merger theory. That is, how to explain the very dense cores found in some ellipticals (see e.g., Faber et al., 1997). As Hernquist (1993) points out, mergers between purely stellar disk galaxies do not generally produce such dense cores, unless they have dense bulges to begin with. However, the dissipative gas is not bound by the same fundamental phase space constraints as collisionless stellar systems. Thus, as observed, funneling can build large central densities, and starbursts can convert the gas to a dense stellar core, though realistic simulations of this process have not yet been carried out. 6.3. Minor mergers: disk heating and aging Although, as discussed in Section 5.2, there has long been interest in the `sinking satellitea problem, there has been much less simulation work than on mergers between equal mass galaxies. The observational signatures of minor mergers are much weaker than those of major mergers, so observational comparisons are harder. Moreover, there are technical di$culties in the simulations, especially in adequately resolving a companion that is much smaller than the primary. On the other hand, minor mergers are likely to be much more common than major ones. Hernquist and Mihos (1995) summarize the evidence for this conclusion, and provide a listing of earlier works. At "rst this conclusion seems like a relatively simple matter. Galaxies have a wide range of masses (over 4}5 orders of magnitude), so it would seem unlikely that collision partners have nearly the same mass. However, the companion to primary mass ratio probably has to about 0.1 or less to make the merger `minora, and more than about 0.01 to have substantial consequences for the primary. We can make a simple estimate of the relative number of companions in these mass ranges based on the "eld galaxy luminosity function, but even this is dangerous. Collision partners are not selected randomly from the luminosity function. They are generally members of bound groups or of groups interacting with other groups or clusters (see Section 9). Thus, hierarchical clustering biases the statistics of collision partners.
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The work of Zaritsky et al. (1993b), discussed above in connection with the Holmberg e!ect, provides especially useful input on this subject. In this study of satellites around late-type spirals the authors "nd mean of 1.5 satellites per primary, and of course, there may have been more in the past, which have already merged. The mean mass of these satellites is estimated to be about 10%, exactly the right magnitude for minor mergers. The luminosity distribution of all the satellites is well "t by a Schechter form, like the "eld luminosity function, which shows that at least they are not an unusual population. The number of objects in the survey (69 satellites) is not large enough to de"nitively answer the questions above, but it does support the opinion that they are common. So too does the increasingly common discoveries of mild signatures of old collisions or mergers in otherwise normal galaxies. Despite the di$culties, several simulational studies of minor mergers have been published recently. These papers (Mihos and Hernquist, 1994a; Hernquist and Mihos, 1995; Walker et al., 1996), present the results of fully self-consistent, high particle number, multiple component simulations. These simulations use about a 10% mass companion, on a prograde orbit of modest inclination (about 303), with a companion density comparable to the mean of the primary disk. Thus, these models are a continuation and update of the sinking satellite studies described above. As in the earlier studies, the response is global, and the merger is prompt. Somewhat surprisingly, the results of these simulations are qualitatively similar to those of major merger models. The Hernquist and Mihos (1995) simulations, in particular, show strong gas funneling. Up to almost half of the primary gas mass can be deposited in a dense core, plausibly inducing starbursts and other nuclear activity. This is comparable to the funneling in major mergers, which is surprising because the collisional distortions are far less, and no strong bar is formed. Instead, another now familiar mechanism seems to be the cause. Strong spiral arms are induced by the interaction, and Hernquist and Mihos (1995) argue that gravitational torqueing by these waves drives the radial gas #ows. There is an interesting wrinkle however. The authors suggest that strong shocks form in the waves, and that the dissipation in these shocks yields a positional o!set between the gas and the stars in the wave. This o!set provides a lever arm with which of the stellar wave can torque the gas. Thus, the o!set is seen as the key to the strong radial #ow. For the present these new results should be viewed with caution, since they are based on an isothermal equation of state. For example, heating by star formation in the waves could stir the gas, yielding a smaller o!set, and less torque. Hernquist and Mihos (1995, and references therein) also con"rm a result hinted at in earlier works, the radial gas #ows can be delayed or inhibited by the presence of a compact bulge in the primary. They were not able to derive a speci"c mechanism for this e!ect, but they conjecture that it may be related to the presence of inner Lindblad resonances. On the basis of the studies of induced bars described in the previous chapter, this seems very reasonable. The paper of Walker et al. (1996) presents the most detailed study to date of another important consequence of sinking satellite minor mergers, dynamical heating of the stellar disk. The parameters of the 500 000 particle simulation presented there are essentially the same as those of Hernquist and Mihos. Over the 1.0 Gyr course of the merger the stellar disk thickens by 60%. Velocity dispersions are increased in all three directions. The net result is that the merger remnant is a disk galaxy of substantially earlier Hubble type. This result supports Schweizer's (1986, 1990) general conjecture, that minor mergers can push late-type progenitors along the Hubble sequence towards earlier types. This idea is a corollary of
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Toomre's mergers-make-ellipticals conjecture, and many of the same observational techniques are useful for investigating it. For example, in terms of tidal remnants, it has been clear from early on that many shell galaxies (Section 5.5) were S0, rather than elliptical galaxies. Similarly, many polar ring galaxies are of relatively early type, except for the ring (see Schweizer (1986) and the following section). Counter-rotating disks or cores are also found in objects of type S0 and Sa. The probable merger or accretion event which produced these had to be minor enough to preserve the disks. Another general point about minor mergers is that, since the outcome depends on both the structural and orbital parameters of the companion, they have the potential to amplify the diversity of galaxies. This raises the question, if minor mergers are common, then why haven't they disrupted structural relationships among the galaxies? This complex question may in fact have a fairly simple answer. Most of the increased diversity may be represented by relatively faint fossil collisional structures around the primary, while the e!ects of global or deeply penetrating perturbations rapidly relax to more generic forms. Processes like violent relaxation and dynamical friction evidently constrain evolution to approximate the fundamental relationships. In sum, though minor mergers have been somewhat neglected relative to their `majora siblings, there are many motivations for future study, and recent work demonstrates the feasibility of such studies. 6.4. New disks We have seen that major mergers can destroy galaxy disks, converting the progenitors into ellipticals, and that minor mergers can heat and age disks. But there is another side to this subject, minor mergers with companion disruption can lead to the formation of new, or reinvigorated disks. Since we have already considered both companion disruption and minor mergers, the goal of this section is simply to add a few missing pieces to the picture. 6.4.1. Disks in ellipticals Until about the mid-1970s there was little evidence for a cool gas, disk component in ellipticals, but with more sensitive instruments this situation has changed greatly (see Bregman et al. (1992); Macchetto et al. (1996) for survey statistics). Moreover, as noted above, kinematically decoupled cores have been found now in a high fraction of ellipticals. These core disks are discovered from their H emission or from dust obscuration (see the review of de Zeeuw (1994)). However, they are a not usually very blue, but rather normal photometrically relative to the cores of all early-type galaxies (Carollo, 1997; Carollo et al., 1997). There are other possible sources for cool gas in ellipticals, e.g., stellar mass loss or cooling #ows out of hot cluster gas. However, disks resulting from these sources would be expected to align with the kinematic axis of the galaxy. On the other hand, material from accretion or (minor) mergers is thought to settle promptly into a disk whose orientation depends on the spin of the progenitor and the orbital parameters. (See Barnes (1998) on the formation of inner and outer disks in major mergers, and Thakar and Ryden (1996), Struck (1997) on mass transfer.) These disks then evolve to a preferred plane of the host, but on a longer timescale (see de Zeeuw (1994) and references therein). Nonetheless, though the argument seems straightforward, it is still based on very circumstantial evidence in most cases. An HST study of the brightness pro"les in the cores of a sample of 61 early-type galaxies "nds that they can be divided into two distinct types: `corea galaxies with steep `power-lawa pro"les that
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break to a much #atter form within a core radius, and shallower power-laws with no resolvable changes into the center (Lauer et al., 1997; Faber et al., 1997). The latter type is most easily explained as the result of a merger of a small, gas-rich companion. However, disks were not detectable in most of the power-law objects in this sample, but many new searches for HI and molecular gas in elliptical galaxies are underway (van Gorkom and Schiminovich, 1997; Rupen, 1997). While kinematically decoupled cores are common in large, luminous ellipticals, a related phenomenon is at least as common in low luminosity ellipticals. This is the presence of #attened, pointed or `diskya isophotes. Galaxies containing them are called `disky ellipticalsa, and are usually also rotationally #attened, independent of the disky part. The basis for this statement is the fact that photometric and kinematic decompositions of disk and bulge parts, like those derived for disk galaxies, seem to work well on these objects (see Scorza and Bender, 1996; Bender, 1997). The derived disk-to-bulge luminosity ratios (D/B) overlap and extend the low end of the range for S0 galaxies, indicating continuity across the types. The disky ellipticals also extend the disk galaxy trends in plots of bulge or disk luminosity versus D/B. Thus, there are several indications that these galaxies represent an extension of the Hubble disk sequence. Do the disks and the high rotation rates of these galaxies result from mergers? Scorza and Bender point out that merger simulations with gas do lead to changes in stellar orbit families in such a way as to make a more oblate remnant, which is encouraging for the merger theory. However, the discovery and study of these objects is very recent, and there is far too little data for "rm conclusions. Ellipticals with extensive cool gas disks, extending beyond the optical disk, are still very rare (van Gorkom, 1992). Thus, if mergers of spirals make ellipticals, then they are indeed e$cient about consuming or heating and dispersing the cool gas of their progenitors. However, the nearby giant elliptical, NGC 5128"Cen A, with its large gas and dust disk, and evidence for a recent merger (Ebneter and Balick, 1983), stands as an apparent counter-example. More likely, however, the elliptical was formed long before the recent merger. (For other such examples see the discussion of Section 4.1, and the papers of Appleton (1983) and de Mello et al. (1995, 1996).) The work of Whitmore et al. (1997) on what may be dynamically young ellipticals provides especially interesting `archaeologicala support for these ideas. 6.4.2. Counter-rotating disks in S0 and Sa galaxies There are also numerous examples of counter-rotating disks among early-type spirals. E.g., Thakar and Ryden (1996) list a number of recent references for S0s. Bertola et al. (1992) estimated that about 40% of S0 galaxies contain ionized gas of external origin on the basis of a small survey. Very recently, Lovelace et al. (1997) have provided a list of (primarily) Sa type galaxies with extensive counter-rotating disks. For a review, history of the subject, and summary of several individual systems see Rubin (1994b). As in the case of the ellipticals, the most likely explanation for their existence is accretion or merger. What is most remarkable about these disks is that, in contrast to counter-rotating cores in luminous ellipticals, they are extensive, sometimes as large as the parent disk. They can also contain signi"cant mass, up to 20}50% of the disk. This makes it di$cult to account for them in any way except as the result of a merger. NGC 4550 in Virgo is the most famous S0 example, see Rubin (1994b). A later type (Sab) example is the so-called `Black-Eyea or `Evil-Eyea galaxy, NGC 4826, whose gas disk is nearly co-planar with the stars, but has very complicated
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kinematics. It rotates with the stars in the inner regions, but reverses direction in the outer disk (see Rubin, 1994a; Walterbos et al., 1994). The prominant dust lane (the black-eye) lies in the transition region. Another example, is NGC 4138, an Sa galaxy studied at multiple wavelengths by Jore et al. (1996). In this latter system 20% of the disk stars and all of the gas rotate counter to the majority of the stars. The counter-rotating HI gas extends to a radius of 2.5 times the outer radius of the stars. The smooth distribution of the components and Balmer absorption lines indicate that the SF stopped about 10 yr ago, giving some constraints on the age of the counter-disk. This raises the question of how stable and long-lived are these disks? Rubin (1994b) notes that this question has been considered for several decades, i.e., before the existence of oppositely rotating disks was demonstrated! The answers she records are quite varied. The latest input is the work of Lovelace et al. (1997), who summarize recent work and suggest that the oppositely rotating disks can drive density waves, especially the one-armed mode. These density waves increase the dissipation and e!ective viscosity in the gas leading to accretion. The other obvious question } how do these extensive counterotating disks form } is addressed by Thakar and Ryden (1996). They modeled counter-disk formation as a result of both continuous and episodic infall. The former could be infall out of a stream, like the Magellanic stream around the Milky Way. The latter could result from the merger of a gas-rich disk, for example. The key result of these simulations is that the infall rate must not be too rapid, or the sudden introduction of a large amount of mass leads to excessive heating of the pre-existing disk. This constraint becomes a real di$culty in producing massive counter-disks. Thus, the authors argue that it is unlikely that such disks are produced by the merger of a gas-rich dwarf. This is a surprising result, worthy of further study. 6.4.3. Polar ring galaxies In the present context, these objects are viewed as another kind of accreted disk. However, as their name implies they are generally oriented perpendicular to the main disk, rather than being contained within the same plane, e.g., Fig. 23. Moreover, it believed that they are usually annular disks, with empty middles, though in many cases this is hard to determine observationally, see Fig. 23 and the schematic examples of Fig. 24. This may be because they are polar and often contain gas; gas cloud collisions with clouds in the primary disk would remove the inner part of a complete polar disk. Moreover, if they do indeed result from accretion in most cases, the material may have a relatively large speci"c angular momentum. While they are the third type of `ringa galaxy we have considered they are clearly unrelated to either collisional ring waves or resonance rings within disks. The primary or host galaxy typically has the characteristics of an S0 type, with relatively little gas or dust (again in contrast to the other types of ring). This seems to con"rm the idea that polar rings generally orbit through the poles of an oblate galaxy, rather than around the equator of a prolate one. The polar ring is usually gas-rich and bluer than the host, with little or no old star component. Whitmore et al. (1990) have assembled a catalog with more than 100 examples. Review articles can be found in the book of Casertano et al. (1991), with a very recent one by Cox and Sparke (1996). Because this latter paper is so current, we will limit the discussion in this section. Polar rings probably form out of the total or partial disruption of a gas-rich companion (see e.g., Richter et al., 1994). The process is like that for forming disks in ellipticals described above.
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Fig. 23. A well-known polar ring galaxy, NGC 4650A. (Image courtesy European Southern Observatory.)
Speci"cally, the accreted gas settles promptly into a plane determined by the collision parameters, and evolves to a preferred plane on a timescale much longer than the orbital or azimuthal smearing time. The initial plane is likely the orbital plane of the companion if it is promptly destroyed (Katz and Rix, 1992; Thakar and Ryden, 1996). If the ring forms out of a bridge with gas from both galaxies, formed in a direct collision, its orbital plane may be perpendicular to the bridge (Struck, 1997). Recently, a couple of observational studies of forming polar rings have appeared. Cox et al. (1996) present HI mapping of the II Zw 70/71 system. Reshetnikov et al. (1996) present optical photometry of the Arp 87 and Arp 293 systems. In all three cases the two galaxies are still attached by a bridge, which may include gas from both galaxies as in the Struck (1997) models. Moreover, the polar ring is either the smaller galaxy (Arp and Madore, 1987), or the two galaxies are of comparable size. This is not consonant with the fact that most `maturea polar rings are isolated, so,
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Fig. 24. Schematic of a polar ring galaxy seen from a variety of di!erent orientations from Whitmore et al. (1990). According to the authors, less than half of these views are readily identi"able as a polar ring galaxy.
if these systems are future polar rings, a great deal of evolution remains, including the merger of the two galaxies. Since the discovery of most of these objects in the 1970s and 1980s much e!ort has focussed on studies of ring dynamics, with the goal of understanding why the rings are generally oriented around the poles, rather than at arbitrary angle. Speci"cally, the secondary evolution to a preferred plane is thought to be the result of di!erential precession. Orbital precession depends on both the inclination and the mean orbital radius of a gas cloud. Thus, annular rings at di!erent mean radii precess at di!erent rates (see the formula in Cox and Sparke, 1996). If these rings are elliptical rather than perfectly circular, which seems likely initially, di!erential precession will lead to ring intersections, gas cloud collisions, and dissipation. The most likely result is settling into one of the preferred planes, i.e., either the equatorial or a polar plane. One problem with this scenario is that there are some polar rings, which do not appear to be young, and in fact have red colors, but are still signi"cantly inclined. Katz and Rix (1992) have argued that this could be the result of viscous couplings that are strong enough to prevent the independent precession of di!erent rings. Thus, the whole polar disk would be viscously connected, and precess at some mean rate. However, another e!ect of the viscosity would be relatively rapid radial spreading in the disk, and presumably decreased lifetime. Nonetheless, Katz and Rix's simulations show that a quite long-lived and warped disk can develop. Recent HI (Richter et al., 1994) and CO (Galletta et al., 1997) studies of polar rings have shown that many contain as much gas as a typical late-type galaxy, rather than a dwarf. In this case the self-gravity of the gas can play a signi"cant role in holding the polar-ring together, again suggesting that they can be much longer-lived than originally thought. Because of their large radii and orthogonal orientation, polar rings can o!er a unique probe of the dark halos of galaxies. Cox and Sparke (1996) describe several techniques that have been used to analyse the halos of more than half a dozen galaxies. The particular advantage o!ered by the polar disk is information on the #attening of the host halo. Not many systems have been studied
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yet, and there are ambiguities or signi"cant uncertainties in all the methods. However, the derived #attenings range from round to quite #at, though the majority are quite round. Di!erent methods also give con#icting results, which is the case with NGC 4650A in particular, whose halo is either rounder than an E3 galaxy, or as #at as an E6-7, see Cox and Sparke. Nonetheless, these are early days in the use of this technique, and ultimately the information obtained should be of great importance. 6.5. Multiple mergers The topic of the simultaneous merger of more than two galaxies, or a rapid sequence of mergers, brings us to the edge of a nearly unexplored frontier. At "rst we might question the importance of this subject, since the universe is a low density gas of galaxies. However, a glance at the entwined galaxies in the compact groups illustrated in the Arp (1966) and Arp and Madore (1987) atlases, and the equivalent Vorontsov-Velyaminov (1959, 1977) galaxy `chainsa, assures us that multiple collisions do occur. Schneider and Gunn (1982) presented a photometric and spectroscopic study of a `nightmarisha example in the cluster V Zw 311, where the central object has at least 9 `nucleia wrapped in a common envelope. Each nucleus is roughly as bright as other cluster galaxies. As a virtual `poster-childa for mergers in progress it was adopted as the frontispiece for the book of Tinsley and Larson (1977). While this system illustrates the extremes of multiple merging, Ramella et al. (1994) present reasons why the process may be common. These authors studied the 38 compact galaxy groups from the catalog of Hickson (1982, 1993), and which also were included in published redshift surveys. The redshifts con"rm group membership and eliminate most chance superpositions. Ramella et al. "nd that 29 of these compact groups are embedded in larger, but looser systems. They report that these larger associations are similar to groups discovered in the Center for Astrophysics redshift survey, and that the latter also often contain compact subgroups. In addition they use N-body simulations to show that compact galaxy groups `form continually during the collapse of rich loose groupsa. In the introduction (Section 1.4) we reviewed how initial beliefs that galaxy collisions must be rare were overcome when it was realized that most collisions are between galaxies bound in groups. The environment makes a collision much more likely, if not inevitable. The results of the previous paragraph extend this conclusion to another level of complexity. Multiple mergers are also not accidental. They are apparently the result of the formation of rich substructures in the (hierarchical) evolution of rich groups. Unfortunately, this insight does not make multiple mergers easier to model. On the contrary, there are many di$culties, including the fact that there are myriad of initial conditions, The need for good spatial and particle resolution in each component of each galaxy strains computer capabilities. Nonetheless, some impressive simulational studies have been published recently. They include the paper of Governato et al. (1996), who studied the formation of Hickson compact groups in the collapse of loose groups. Weil and Hernquist (1996, 1994) followed the evolution of compact groups of 6 identical galaxies until all were merged, and they then studied the properties of the "nal remnant. Each galaxy in their half-dozen N-body simulations has bulge, disk and halo components. Some of their basic results are the same as in pairwise mergers, which is not surprising since the basic physics is the same. For example, the halos merge "rst, and the remnant is a dynamically hot elliptical-like object. On the
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other hand, there are a number of important di!erences in both morphological and kinematic properties. The multiple merger remnants are more commonly oblate spheroids that `appear round from many viewing anglesa. A large fraction (about half) of the substantial orbital angular momentum of the initial group is incorporated in the remnant. The spin of all components increases. However, the ratio v /p of the rotational velocity to the random velocity dispersion is small in the central regions, but increases to unity in the outer regions. At a given ellipticity, multiple mergers have somewhat higher (averaged) values of v /p, in agreement with the observa tions. Several other properties of the multiple merger models were also found to agree better with observation than pairwise merger remnants. This pioneering study has many limitations } only a few initial conditions are used, all galaxies in the group are identical, and gas is not included. However, the models are su$cient to support the idea that multiple mergers of early-type spirals might be the best way to make ellipticals. Support also comes from Statler et al.'s (1996) spectroscopic study of the elliptical NGC 1700, which found a variety of evidence for more than one merger, and also an oblate form and increasing rotation at large radii in accord with Weil and Hernquist's models. Presumably, the early-type spirals would themselves be made in earlier mergers. This suggests a modi"cation of Toomre's ellipticals-frommergers theory, to a multi-step or hierarchical buildup.
7. Induced star formation We have described a number of ways in which collisions and mergers profoundly a!ect the evolution of galaxies. However, one of the most important, and among the most brilliant, is induced SF. This is a daunting topic, because there is a vast literature. This ranges from detailed studies of the SF morphology in individual systems, to general theories of SF in galaxies, and how they might apply to interacting systems in particular. As to the former, we will not consider many speci"c systems beyond those already described. This chapter consists of a more general summary, and a look at some of the relevant mechanisms. Yet even this restricted area contains a large body of literature, and is a very #uid and rapidly changing "eld. Fortunately, there are a number of specialized review articles (e.g., Keel, 1991), and conference proceedings that contain reviews of many relevant aspects. These include the recent volumes edited by Franco et al. (1992), TenorioTagle et al. (1992), Tenorio-Tagle (1994), and Shlosman (1994). Note, we will not consider techniques for measuring SF in galaxies, and their relative advantages and disadvantages in detail. The interested reader is referred to the reviews of Kennicutt Jr. (1990, 1998a). 7.1. Color, H and other indicators of global enhancements a The early work of Spitzer and Baade (1951) on gas splashed out in collisions, and Zwicky's (e.g., 1959) emphasis on the importance of tidal distension, may have discouraged thinking on the possibility of gas compression and induced SF. If so, Toomre and Toomre's (1972) conjecture that collisional disequilibrium plus dissipation will lead to gas funneling into the central regions of interacting galaxies, changed the thinking. Larson and Tinsley (1978, henceforth LT) followed up on this suggestion with an extremely in#uential paper comparing the broad band (UBV) colors of
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(mostly normal) Hubble atlas galaxies to (collisional) Arp atlas galaxies. They found that the mean colors of the Arp atlas galaxies were somewhat bluer than the Hubble atlas galaxies, but that the color range of the Arp galaxies was much greater. They compared these samples to color evolution models consisting of two components: (1) an older stellar population with a continuous, but exponentially declining SFR, and (2) a relatively young population formed in a `bursta of short duration. Most of the Hubble galaxies could be accounted by models with little or no burst component, but a modest range of timescales for the declining SFR. To account for the color range in the Arp galaxies a substantial burst component was needed, as well as a range of times since the burst. This successful "t of models to optical observations, though not unique, gave rise to some durable ideas about collisionally induced SF. The "rst of these is that the nature of this SF is burst-like, rather than simply an enhanced level of continuous SF. The suggested timescale for these bursts was of order 10 yr; much shorter than the dynamical or merger timescale, implying that SF would not be greatly enhanced throughout the collision and merger process. (Note: that the possibility of multiple bursts was one of the variables not considered in LT.) A second conclusion is that because a range of burst strengths and ages would be represented, the primary characteristic of the colors of collisional (burst) galaxies would be a large range, rather than extreme blueness. At about the same time Huchra (1977) published a similar study of blue Haro, Zwicky and Markarian galaxies, which included similar color evolutionary modeling. He likewise concluded that most of these galaxies consisted of composite stellar populations, i.e., an old component as well as a substantial burst component. He found no `compelling evidencea for a population of purely young galaxies. Shortly thereafter Struck-Marcell and Tinsley (1978) published models for near-infrared colors. Though there was little data to compare to, these models predicted even greater color variations in composite populations over the longer infrared-to-optical baseline. In the time since these early studies, color evolutionary modeling has become a standard tool in the study of SF history in galaxies, despite the fact that the large number of possible variables often make conclusions based on it ambiguous. We will content ourselves here with citing a few relevant highlights. Many sophisticated models (e.g., the large grids of Arimoto and Yoshii (1986)) were developed more for application to long term galaxy evolution and the colors of early-type galaxies, rather than for studying color changes on the shorter timescales of galaxy collisions. The models of Arimoto and Bica (1989, and references therein) described rapid changes in colors that occur with the onset of the red supergiant and red giant evolutionary phases in young star clusters, or other coeval populations. These potentially important and observable changes were not adequately represented in earlier models. The models of Charlot and Bruzual (1991) (Bruzual and Charlot, 1993) also included a more sophisticated treatment of the giant phases, and a new interpolation technique for using stellar evolution tracks. An extensive compilation of current models, and model inputs has recently been published by Leitherer et al. (1996). One of the most important indicators of recent SF is the #ux of the H line. Kennicutt Jr. and a others investigated the range of H emission in normal galaxies, and interpreted it with the aid of a evolutionary models (see Gallagher et al. (1984) and the discussion and references in Kennicutt Jr. (1990, 1998a)). This provided a basis for comparison to collisional galaxies in the studies of Keel et al. (1985), Bushouse (1986, 1987) and Kennicutt Jr. et al. (1987). In addition to the H #uxes, the a
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"rst paper also provided nuclear spectrophotometry of a large sample, and the latter study included IRAS far-infrared #uxes for comparison (also see Bushouse et al., 1988). The basic results echo the earlier color studies } when compared to a control sample, the interacting galaxy sample had systematically higher emission #uxes in both H and the far-infrared. The dispersion in the a emission #uxes was also greater in the sample of interacting galaxies. And "nally, the H #uxes a could be accounted for with the addition of a short burst component in the SFR. One which produced a small (but bright) fraction of the total stellar mass of the galaxies. From this point on the case for global SF enhancements in galaxy collisions was demonstrated to the satisfaction of most. However, there are a number of complications. An early observational one was connected with searches for enhanced radio continuum emission, which can result from either SF or nuclear activity. Because of limited resolution, it was not clear initially whether or not radio continuum enhancements in strongly interacting systems were entirely due to active nuclei (e.g., Hummel, 1980, 1981). High surface brightness star-forming continuum disks were subsequently resolved, e.g., Hummel et al. (1987), and especially in the extensive work of Condon and collaborators (e.g., Condon and Broderick (1988), Condon et al. (1982, 1990, 1996). A number of works have also supported the idea that the strongest SF enhancements occur in the most `violentlya interacting systems (e.g., Kennicutt Jr. et al., 1987). This includes the multiwaveband studies of Bushouse (1986, 1987) (Bushouse et al., 1988; Bushouse and Werner, 1990), which were aimed speci"cally at such systems. Observations of molecular (CO) emission, which is another important indicator of SF activity (see review of Young, 1990; Young et al., 1996) have been used to estimate SF e$ciency, which is found to be extremely high in merger remnants. In fact, it is now believed that a signi"cant fraction of all stars ever created, and of the metals scattered in the intergalactic medium, may have been made in young merger remnants (see the previous chapter and refs. therein). While the spectacular results on merger remnants captured the spotlight, younger and nonmerging collisional systems received less attention. However, there have been a few studies examining the general question of the time dependence of SF in young to intermediate aged collisional systems, in addition to `case studiesa of individual systems too numerous to cite here. Bushouse's (1986, 1987) sample included a number of systems that seemed to be near the closest approach or impact point in a "rst collision. There are substantial enhancements in these young, violent systems. Another important work is Keel's (1993) study of `kinematic regulationa. In this work Keel found that the most important variable in determining SF enhancements is the relative amplitude of the velocity disturbance. His sample consisted of galaxy pairs with low relative velocities, and with what we can call young to intermediate ages. Sample galaxies also passed a geometric selection criteria, which required that at least one member be within 303 of edge-on, and that its companion be within 303 of the projected disk plane. These criteria allowed an optical rotation curve to be obtained, and a good estimate of whether the collision was prograde or retrograde for the galaxy under study. The sense of the collisional orbit, the projected separation, and a number of other variables were found not to correlate signi"cantly with the global SF. The velocity disturbance, which was found to be the important variable, depends on both the magnitude of the disturbance and the galaxy's susceptibility. In support of the latter dependence, enhanced SF was commonly found in galaxies with large regions of solid body rotation. Although a detailed comparison is probably not possible, these results seem in qualitative agreement with the dynamical models of induced waves and bars discussed in the preceding chapters.
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7.2. Spectral line diagnostics 7.2.1. SF histories The studies described in the previous section indicate that large amplitude disturbances resulting from direct collisions or strong tidal encounters are able to generate vigorous starbursts before the galaxies merge, and super-starbursts afterward. Beyond this, colors, H #uxed and many of the ? classical SF indicators do not have the sensitivity to elucidate the duration of bursts or the history of SF following a collision. Recently, attention has focussed on using optical and infrared spectral line #ux ratios to obtain additional information on these topics. A number of authors (e.g., Bica et al., 1990) have pointed out an age versus burst strength ambiguity in interpreting the broad band colors of old-plus-young mixed populations. The rapid evolution of the most massive stars means that very blue colors are very short-lived. Less blue colors can result from either a weak burst mixed with an old population, or a strong, but aging burst. Spectral indicators like the Balmer absorption lines, calcium H and K lines (Leonardi and Rose, 1996), or infrared CO bands (Bernlohr, 1992; Campbell and Terlevich, 1984) can be used to distinguish and date the starburst component. Except for a few nearby systems, the application of such methods to colliding galaxies is relatively recent. Bernlohr (1992) computed a set of evolutionary models for both broad band colors and a couple of dozen spectral lines, and then applied them to the interpretation of the prototypical starburst galaxy M82. We will return to discuss that object in a moment. Bernlohr (1993) applied these models to the study of a sample of about 30 interacting systems. The spectra he obtained for this sample were inevitably of lower resolution and sensitivity than nearby galaxies, and so, there were fewer lines to "t. Nonetheless, he obtained several interesting results concerning SF timescales in interacting systems. The "rst of these was that all the starbursts in his sample had ages of less than 2;10 yr, from which he concludes that starburst duration is generally less than this time. Recall that Larson and Tinsley (1978) (also see Kennicutt Jr. et al., 1987) had concluded that the colors of Arp atlas galaxies could be accounted for with an old population and an aging burst population, where the burst was of equally short duration and modes strength (410%). Published age estimates of individual starburst galaxies have also generally been short. Speci"cally, there has been no evidence for enhanced SFRs on intermediate timescales, despite the fact that dynamical and gas consumption timescales are typically intermediate (e.g., 10}10 yr). The implication is that starbursts are not ended by (global) starvation. Bernlohr notes an important caveat, however. He had no information on heavily obscured SF regions, such as those in merger remnants. In light of the discussion of ULIRGs above, we should not be surprised if the timescales and the nature of SF are very di!erent in merger remnants. Bernlohr also found that the minor or secondary galaxies in his sample pairs were much more likely to be in burst or post-burst phases than the primary galaxies. He estimated that time delays between bursts in the two partners range up to several times 10 yr. Bushouse (1986) found a similar result in his violently interacting sample, but Kennicutt Jr. et al. (1987) noted a tendency for galaxies in their pairs to have comparable H equivalent widths. Joseph et al. (1984) found a similar ? result, Lutz (1990) and Telesco et al. (1988) found the opposite result in samples selected by their far-infrared emission. There are many dynamical variables, so it is not surprising that the situation
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is very complex. Except for special subclasses of colliding galaxies, it seems likely that this result will not be strong in large samples. The art of spectral synthesis, and its use for deciphering SF histories, is advancing rapidly in many areas of extragalactic astronomy, but especially for young stellar populations. (The technique has also begun to be applied to data in other wavebands, e.g., PeH rez-Olea and Colina's (1995) recent evolutionary models of the ratio emission in starbursts.) Garcia-Vargas et al. (1995 and references therein) have produced a large set of model spectra for young star clusters (age (6;10 yr) and their surrounding ionized gas regions. The rapid evolution and high mass loss rates of massive young stars provide some distinctive color and spectral signatures. The rapid evolution is responsible for color shifts to the red as stars evolve to the red supergiant phase. The appearance of absorption lines, notably the near-infrared triplet of ionized calcium, is also associated with this change. The heavy mass loss is responsible for removing the hydrogen envelopes, and revealing layers enriched in heavier elements, which characterize Wolf}Rayet stars. These stars are identi"ed by helium lines, and emission lines of nitrogen and carbon in multiple ionization states (see the review of Massey, 1985). The Wolf}Rayet phase immediately proceeds the red supergiant phase. Finally the emission line intensities of the surrounding gas depend on the #ux of ionizing photons, and the ionizing #ux per unit mass of young stars is age dependent. Garcia-Vargas et al. have recently applied these models to the spectra of several regions in the colliding galaxy pair NGC 7714/15 (Arp 284). They argue that the spectra of at least one of these regions in the center of the primary galaxy is best "t as the combination of two mini-bursts of age 3}5 and 7}9 million years. (This system was also in Bernlohr's (1993) sample, where he classi"ed the primary as a starburst and the secondary as in a post-starburst phase.) Although high quality spectra are required, it seems likely that many more studies of this type will be undertaken in the coming years. 7.2.2. IMF variations and the example of M82 Since most of the luminosity of a young burst population is due to the massive stars, it is di$cult to determine the mass of the low mass stars produced in the burst, and thus, the total mass of stars formed and the importance of collisionally induced starbursts in galaxy evolution. This is especially true when optical/ultraviolet color and spectral line observations are used. In principle, infrared observations are more sensitive to the low mass stars, since not only are they brightest in the red parts of the spectrum, but in a burst population they will often be enshrouded in dusty `birth cacoonsa. However, in the near-infrared there is the potential for confusion with old stellar populations or red giants. More generally, these, and other problems, prevent us from determining the so-called `initial mass functiona (IMF), which describes the relative number of stars formed in di!erent mass ranges. This is a general problem in extragalactic astronomy, not peculiar to colliding galaxies. There has been a great deal of progress in the last couple of decades in determining the IMF of star clusters in the Milky and the Magellanic Clouds, see e.g., the reviews of Scalo (1986), and Garmany (1994). It is widely believed that the function has a universal, approximately power-law form, like that "rst suggested by Salpeter, except possibly at the high mass end. However, this very complicated topic is beyond the scope of this review. We refer interested readers to the recent review of Hunter et al. (1997), Leitherer (1998), and especially Scalo (1998) for a critical look at the observational foundations of the universal IMF idea.
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In particular, it has been suggested that IMFs may be di!erent, or at least extend over di!erent ranges, in galaxies with very high SFRs. That is, when the high SFRs in the objects we now call starburst galaxies were discovered, it was immediately recognized that with these SFRs the gas consumption timescales are much shorter than the age of the universe, assuming an IMF like that in the Milky Way (e.g., Sargent and Searle, 1970). Thus, either the high SFR is not maintained for very long, or the IMF is di!erent or both. The existence of a universal IMF would support the "rst possibility, hence `starburstsa. One example of the di$culty of observationally constraining the nature of individual starbursts is the nearby starburst and collisional galaxy M82, a member of the M81 group. In the late 1970s several papers suggested that the vigorous activity in the central regions of M82 was the result of massive, young stars recently produced at a very high rate (Solinger et al., 1977; O'Connell and Mangano, 1978; Rieke et al., 1980). The latter paper, in particular, included a detailed stellar population model of the starburst, despite the fact that M82 is nearly edge-on and highly obscured by dust. Based on a variety of near infrared observations Rieke et al. concluded that the extinction from the nucleus of M82 was as great as A +25 magnitudes (a factor of 10 in 4 the optical), though they suggested that it could be quite nonuniform across the central regions. Such large optical extinctions implied non-negligible extinctions in the near-infrared as well, and that the overall starburst luminosity is much greater than observed in those bands. The substantial UV and 2 micron #uxes constrain the duration of the burst to be more than 10 yr. The kinematically determined mass (and the gas mass), provided some of the tightest constraints on the mass of stars produced. To avoid producing too great a mass of stars, Rieke et al. argued that the burst age had to be 410 yr, and the IMF had to be very de"cient in stars of mass 43 solar masses relative to the `standarda (local group) IMF. The best-"t models predicted a high super- nova rate, whose young remnants were subsequently detected by radio continuum observations (Kronberg et al., 1985). Far-infrared observations (Telesco and Harper, 1980), also con"rmed the high luminosity. The tale of M82 took some twists with the papers of Lester et al. (1990) and Telesco et al. (1991). The "rst of these papers presented extensive near infrared spectroscopy of molecular and atomic hydrogen, and ionized iron. On the basis of these data the authors argued that the extinction of the infrared emitting region was much lower than previously thought (A +5 magnitudes), and 4 moreover, that emission from hot dust makes a substantial contribution at wavelengths near 2 lm. These conclusions considerably modify the constraints on population models, suggesting that the burst population is dominated by red supergiants (and thus in a very short-lived phase), and that a truncated IMF is not necessary. The paper of Telesco et al. presented near and mid-infrared mapping of M82. The primary result of this paper was the discovery of a 1 kpc bar, i.e., one which extends across the starburst and molecular gas region, and which might be responsible for funneling gas inwards. In addition, the authors concluded that hot dust emission is signi"cant at mid-infrared wavelengths, and supported the idea of a lower value for the visual extinction. Bernlohr (1992) and Rieke et al. (1993), also McLeod et al. (1993) have new data, and produced a new generation of starburst models. Bernlohr used new spectral constraints, and an extinction free estimate of the ionizing #ux from radio recombination lines (see Puxley et al., 1989; Puxley, 1991). His results are generally similar to Rieke et al. (1980), except that di!erent stellar evolutionary inputs yielded somewhat di!erent K band (2.2 lm) #uxes in the models. The extinction issue was not addressed, beyond a statement on it's patchy nature. The papers of Rieke et al. and McLeod et al. present models that include nonuniform dust `screensa as well as homogeneous
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ones, and conclude that the `visual extinctiona to the nucleus lies between A +12 and +27. (We 4 also note the recent infrared spectrophotometric study of Smith et al. (1996), which found an average extinction of 25 magnitudes in 20 luminous starburst galaxies.) They argue that with patchy extinction and a new value for an H O band strength the need for hot dust emission at 2 lm disappears. The parameters of their "nal best "t model are very similar to the Rieke et al. (1980) model, with an IMF truncated at the low end. More recently Satyapal et al. (1995) presented extensive, Fabry-Perot spectral-imaging observations of near-infrared hydrogen recombination lines. With many lines measurements at highresolution they were able to address the ambiguities of earlier studies. For example, they describe how previous results were based on #ux ratios derived by scaling large aperature data to a smaller aperature, in order to compare to the small aperature #ux measured in another line. On the basis of their own data they "nd that the visual extinction to the starburst region varies from 2 to 12 magnitudes, and that the 2.2 lm K-band luminosity is much lower than the previous estimates. Their results suggest little need for an IMF truncated at the low end, in accord with Lester et al. However, in agreement with McLeod et al. (1993), they "nd little hot dust emission in the K-band. The spatial variation of the Brackett c #ux, `along with other star formation diagnostics, suggests that the nucleus contains later-type stellar populations, and the starburst phenomena is propagating outwarda. This latter result is con"rmed with additional data in Satyapal et al. (1997). One moral of this rather long story is that, even in a nearby starburst galaxy with little or no nuclear activity, it can be very di$cult to decipher the stellar populations, SF history and IMF. It still seems likely that very low values of the mass-to-light ratio (e.g., M/¸;1.0, in units of the solar mass and luminosity) indicate an IMF with few low mass stars. The M/¸ constraint is one of the most fundamental in M82, and the debate hinges on the ¸, determined from the K-band. Generally, both M and ¸ can be very di$cult to determine. The frequent suggestion that interacting galaxies in general have an IMF enriched in high mass stars (e.g. Kennicutt Jr. et al., 1987; Rieke, 1991) remains uncertain. A related indicator has been studied by Young et al. (1996) (Young, 1993, and references therein). They have used the ratio of massive star luminosity to gas mass (¸/M , with M derived from CO observations) as a measure for high mass SF e$ciency, and found that while this factor varies little among di!erent Hubble (disk) types, it is signi"cantly elevated in interacting galaxies. How much of this enhancement is due to modi"ed IMFs is not yet clear. On a more optimistic note, we do seem to be converging on a good overall picture of the starburst in M82, which is the prototype. The primary starburst region is often described as a disk or ring of radius less than a 0.5 kpc at the inner edge of an annular disk of molecular gas. However, Telesco (1988) has suggested, on the basis of the kinematic observations, that the molecular gas may be located in spiral arms. He also provides us with a graphic description of a starburst region: If much of the apparent thickness of the IR ridge is due to projection of a thin tilted disk, then the surface brightness corresponds to an average separation of only a few parsecs between Orion-like complexes, each emitting +3;10 (solar luminosities)... Thus, the center of M82 is "lled with OB stars, supernova remnants, and a complex, dynamically advanced distribution of overlapping HII regions permeating a relatively warm, "lamentary medium bathed in UV radiation. The e!ect of these OB stars and supernova remnants is so profound that '10 (solar masses) of gas are being driven from the plane of M82 (in a galactic wind). (Telesco, 1988).
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The (stellar) bar is also found in this region, and not only helps explain the fueling of the starburst, but if it is an induced bar, provides a connection with the large-scale interactions in the M81/M82 group. The galaxies of this group are indeed strongly interacting, as is illustrated by the large scale HI maps of Appleton et al. (1981), Appleton and van der Hulst (1988), and Yun et al. (1994), see Fig. 25. In fact, it appears to be a complex group interaction, involving several galaxies or multiple encounters (e.g., Thomasson and Donner and references therein). 7.3. SF region morphologies In fact, we have already described many of the important morphologies of induced SF regions in Sections 2}6 above. However, most of those discussions were in the context of collision dynamics, with the SF serving primarily as an illuminator of characteristic morphologies and an indicator of gas compression zones. Nonetheless, to summarize, in Table 2 we list some general types of SF induced by galaxy collisions, give some examples, and provide an index to where they are discussed in this paper. The "rst three types can be induced in non-merging collisions. Mergers themselves can induce all three types, and potentially with much greater intensity, depending on the gas contents and relative sizes of the collision partners (see e.g., Schweizer, 1990, 1998; Barnes and Hernquist, 1992a). Table 2 also catalogs three general properties of these morphologies: location, timescale and relative intensity. The latter two are qualitative generalizations, since there are large variations between systems, and except for a few well-studied systems, our knowledge is very incomplete. The characterization of the intensities is based on gas consumption times or SFRs, which can be estimated from far infrared luminosities, albeit with substantial uncertainties. For example, the typical formation rates of stars of mass greater than 2.0 solar in normal spiral galaxies are about 3 solar masses per year or less (e.g., Solomon and Sage, 1988; Kennicutt, 1990; Young et al., 1996; and references therein). Collisional systems like the Cartwheel (Higdon, 1996) and M82 (McLeod et al., 1993) have moderately enhanced rates of a few to 10 solar masses per year. Merger remnants range upwards from a few tens of solar masses per year (Solomon and Sage, 1988). Thus, if SF continues for times of order 3;10}10 years in merger remnants, which is probable with continuing gas in#ow. The lower SFRs and shorter burst durations in collisional systems with central starbursts, like M82, suggest that the net SF in these systems is a small fraction of the whole. Collisionally driven waves also have relatively low net SFRs, but depending on the type of wave, have a wide range of durations. For example, ocular features are very short-lived (see Section 3.5), as are compression regions associated with high order caustic waves (Section 3.3). The circular waves in ring galaxies may take about 3;10 years to propagate through the disk, and simulations suggest two or three such waves may do so before the phenomenon disperses (Section 3.1). Collisionally induced bars and spirals may be longer lived (Section 5.6). Thus, waves may make a non-negligible contribution to disk SF, despite the fact that they are much less spectacular than mergers. Although star-forming tidal features in systems like the `Antennaea or `Super-antennaea have received a good deal of attention (Section 5.4), signi"cant amounts of SF are probably rare in tidal features, since gas is more generally dispersed rather than concentrated in such structures (see e.g., Schombert et al., 1990). Several sections above discussed systems with HI tidal features with few or no corresponding stellar features, and no on-going SF. It is unlikely that tidal features contribute
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Fig. 25. Maps of the distribution of atomic hydrogen in the M81/82 system. The top schematic (from Appleton et al. 1981) summarizes the large scale distribution. The large amount of gas between the bright optical galaxies emphasizes the magnitude of the disturbance in this system. The contour map at bottom shows the high resolution results of Yun et al. (1994) in the northeastern part of the system.
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Table 2 General types of induced star formation Type
Description
Activity timescale
Relative intensity
Example
Central starbursts
Nuclear, nuclear rings, or bars
Short to intermediate
Moderate
M82
Waves in disks
Rings, spirals or more complicated forms
Short to intermediate
Weak}moderate
M51, Cartwheel
Extended tidal features Tails, Bridges, etc.
Short to intermediate
Weak
Arp 295
Mergers
Intermediate to long
Weak to strong
Arp 220
Strong central SF (plus any of the above)
much to the net SF in the universe. However, there are a couple of situations where tidal SF is more common than usual. The "rst is in mergers between equal, gas-rich progenitors (see Section 6). The Antennae, and other objects on Toomre's classic list of merger candidates (1977) are examples (Schweizer, 1978; also Duc and Mirabel, 1997; Duc et al., 1998; Deeg et al., 1998). The second category, Hickson compact groups, has been pointed out more recently by Hunsberger et al. (1996) and Section 5.4. The Hickson groups have already been mentioned several times above. Hickson (1982, 1993), (Hickson et al., 1992) assembled his list of groups using Palomar survey prints. Membership was determined on the basis of morphological criteria, especially compactness and isolation. It has since been demonstrated that most of the groups are probably gravitationally bound. These groups typically have a few to half a dozen members, and given their compactness, the merging time must be relatively short. It was for this reason, and the fact that many of the groups contain obvious interactions, that Hunsberger et al. selected them for a study of SF in tidal structures. Using R-band images of 42 groups, Hunsberger et al. "nd 7 containing tidal arms or tails. Within these tidal features they discovered 47 knots or stellar concentrations, which they identify as newly formed dwarf galaxies, as noted in Section 5.4. This identi"cation is based on the luminosity of the knots, from which they estimate the knot mass. Hunsberger et al. interpret their "nding of an average of 3 dwarf galaxies per tidal arm as con"rmation of the models of Elmegreen et al. (1993). Followup studies, e.g., imagery in optical and infrared bands, are needed to determine the e!ects of reddening and obtain age estimates. If the ages of the stars in the knots matches the kinematic age of the tidal arms, we would have strong con"rmation of the tidal dwarf formation theory. The derivation of these kinematic timescales is independent of stellar evolutionary timescales, so the comparison between the two is potentially very interesting. In principle, this comparison between dynamical models, and stellar population models for local SF regions, could allow us to reconstruct the circumstances and the environment in which the SF occurred. Such studies would, provide powerful clues as to why speci"c forms of SF (e.g., formation of clusters, dwarf galaxies, etc.) occur at particular times and locations. We have considered several examples of this comparison in preceding sections, including the following. (1) The symmetric ring galaxies are among the simplest (Section 3.1). Theory and models predict that in these objects the collision has generated an outward propagating wave in the
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primary disk, and the wave compression triggers SF. Luminous young clusters in the Cartwheel and other rings, and strong color gradients behind the ring, indicative of an aging starburst population, strongly support these ideas. (2) Dynamical models also show the buildup of large clouds in tidal tails (e.g., Elmegreen et al., 1993), and help us understand how dwarf galaxies could be produced. (3) Models of induced bars and resonance rings help us understand nuclear starbursts, though as yet with less predictive power in terms of the probable SF history (see Section 5.6). These and other examples also show that the more detailed and unique the tidal structure in a given system, the more speci"c the modeling (both types) can be, and the `datinga will be correspondingly more precise. There has been and should continue to be much progress in this area, especially with a wealth of new Hubble Space Telescope observations, infrared, radio continuum and 21 cm line observations. The high resolution of HST, in particular, has allowed the study of individual star clusters in a number of colliding, merger remnant and starburst galaxies (see e.g., Miller et al., 1997; Whitmore et al., 1997; Kundu and Whitmore, 1998; Schweizer, 1998; and the recent text of Ashman and Zepf, 1998). In one spectacular case, the `Antennaea galaxies, over a thousand individual clusters have been identi"ed (Whitmore and Schweizer, 1995; Schweizer, 1998). With su$cient color data it is possible to estimate cluster ages, and learn about the history of starbursts in such systems. Thus, this work is beginning to teach us much about the modes and mechanisms of induced SF. One impediment to progress is the fact that, because multi-wavelength observations are required, the work is data intensive and time-consuming. Moreover, many individual systems must be studied to derive reliable generalizations. Mid-infrared observations provided by the Infrared Space Observatory (ISO) reveal dust heated by SF (or nuclear) activity, and are another source of exciting new information on SF morphologies, especially those buried in dust (see e.g., Genzel et al., 1998). A recent ISO study found that `the most intense starburst in this colliding system ... (is) in an o!-nuclear region that is inconspicuous at optical wavelengths'' (Mirabel et al., 1998). This result, and similar results on extinction in merger remnants, provide important cautionary notes on the dangers of over-generalizing results based on optical observations. These cautions are especially important for high-redshift studies. 7.4. Mechanisms and modes 7.4.1. Star formation enhancements Galaxy collisions drive starbursts and starburst waves, but what can studies of collisional systems teach us about the general mechanisms of SF? Before delving into these questions let me summarize current thinking about large-scale SF in isolated disk galaxies. It is widely believed that in isolated disk galaxies the sizes and masses of star-forming molecular clouds are determined by the scale of the gravitational instability in the gas disk (e.g., the reviews of Kennicutt Jr., 1990; Keel, 1991; Elmegreen, 1992). According to this theory, if the gas exceeds a threshold density (or surface density) in the disk, then gravity can overcome pressure and shear in some range of wavelengths (BT, Section 6.2). This leads to the formation of bound clouds on the scale of the most unstable wavelength. Kennicutt Jr. (1989) has presented evidence supporting the existence of the surface density threshold, and a number of recent studies support this picture (e.g., Caldwell et al., 1994; Skillman, 1996, 1997; Kenney and Jogee, 1997; Kennicutt Jr., 1998b, and references therein), though some recent studies of nearby galaxies also suggest di!erences in detail between observations and
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the theory. Above threshold, the SFR appears to scale as a power-law function of the gas surface density with an exponent of about 1}2. This is the Schmidt Law (Schmidt, 1959), which can be derived from cloud agglomeration models together with the assumption that SFR is proportional to cloud mass, for example. Kennicutt and others have argued that unperturbed galaxy disks are self-regulating, so that the gas surface density is maintained near the threshold value at all radii (also see e.g., Struck-Marcell, 1991; Dopita, 1990; Dopita and Ryder, 1994). In a gas disk the local threshold surface density is given by the expression (e.g., Kennicutt Jr., 1990, BT Section 6.2), iC , R "a ! nG
(16)
where a is a constant of order unity, c is the velocity dispersion of the gas clouds, and i is the radial epicyclic frequency (see Section 3.2). The gas surface density with declines radius as RJ1/r in most gas disks (e.g., Struck-Marcell, 1991), as does i when the rotation curve is #at (i.e., when the rotation velocity is constant). Moreover, c does not vary greatly with radius. Thus, based on these observations, it appears that R/R , the ratio of surface density to the local critical value does not ! vary greatly with radius. However, there is no a priori reason to believe that self-regulation or the Schmidt Law obtain in galaxies that are driven far from equilibrium by collisions. In many cases the induced SF seems too nonlinear to be described by a Schmidt Law, and appears to depend on additional variables, like that kinematic disturbance (Keel, 1993). A closely related idea is that of Elmegreen et al. (1993), (Elmegreen, 1994b) who suggest that the local turbulent velocity dispersion plays an important role in setting the mass scale of the star-forming clouds. The larger mass clouds, formed in more turbulent conditions, are more resistant to the disruptive e!ects of young stars, and so, it is argued, can sustain a higher SF e$ciency. High-resolution observations of the atomic and molecular gas distributions in many systems are needed to determine where the Schmidt law is applicable, and what is the role of other parameters. In collisions where the disk is highly disrupted, the role of the gravitational instability as the primary organizing force for induced SF can also be questioned. Yet, while suppression of SF is the likely short-term outcome in such extreme cases, the presently available evidence suggests that gravitational instability probably has a major role wherever SF is enhanced. For example, gravitational instabilities can build larger cloud complexes in large-scale spiral density waves (see the review of Elmegreen (1992)). Elmegreen et al. (1993) have further suggested that gravitational instabilities in collisionally driven waves could build superclouds, with masses comparable to dwarf galaxies, which in turn, produce more stars (also see Barnes and Hernquist (1992) and the discussion of the previous section). Gravitational instability theory has also been extended to describe strong central starbursts in ultraluminous infrared galaxies (ULIRGs, see Elmegreen, 1994b), and starburst rings at inner Lindblad resonances (Elmegreen, 1994a). Thus, with extensions, the process may play a role in essentially all types of interaction or merger induced star formation. However, other processes have also been suggested as the primary drivers of SF in more extreme cases of disequilibrium. These `violenta modes of SF include the direct triggering by cloud collisions or large-scale shocks. Perturbation analyses and cloud collision simulations in the 1970s inspired
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the idea that strong shocks can compress clouds and directly trigger (small-scale) gravitational collapse and SF. This idea has been studied in a number of simulational studies of galaxy collisions (e.g., Olson and Kwan, 1990a,b; Noguchi, 1990; Mihos et al., 1993; Mihos and Hernquist, 1994, 1996). However, observations of collisional systems suggest that in some cases this process does not play an important role in stimulating SF. The symmetric ring waves provide one class of examples. The vigorous SF observed within the outer ring of the Cartwheel galaxy is apparently not caused by this mechanism, since the velocity discontinuity across the ring is modest (Higdon, 1993, 1996). Moreover, in several ring galaxies with gas bridges connecting them to their collision partners, there is little or no young star population in the bridge (Appleton et al., 1996; Higdon, 1996; Smith et al., 1997). Yet, these bridges are most likely produced by direct, high velocity collisions, which would produce strong shocks. Jog and Das (1992) (also Jog and Solomon, 1992) have suggested another `violenta mechanism, that is, SF triggered by cloud crushing in regions where the intercloud medium reaches unusually high pressures. In this picture, even as massive clouds are broken down, continuing SF could be driven in the cloud remnants or in#owing clouds, increasing the net SF. This process is probably most important in the centers of galaxies with strong radial gas in#ows, and where young star activity and photoheating from AGN are additional sources of pressurization. Speci"cally, this idea was proposed to explain the high SF e$ciency in ULIRGs, but may be relevant in other starburst environments as well. We expect transient high pressures in vigorous SF regions in collisionally driven disk waves, but the pressure is relieved after a short time by blowing out of the disk. Pressurization and nonlinear enhancements from propagating SF seem to be `regulateda to modest amplitudes in unperturbed galaxy disks, but the existence of galactic winds in starburst galaxies (e.g., Heckman et al., 1993) provides one example of how they are not so tightly regulated in other environs. A recent symposium was held on the topic of `violent star formationa and its feedback e!ects in galaxies (Tenorio-Tagle, 1994; also see the review of Shore and Ferrini, 1995). Besides their direct, disruptive e!ects, heating and other feedbacks may drive transient nonlinear dynamical instabilities in collisional galaxies. Scalo and Struck-Marcell (e.g., Scalo and StruckMarcell, 1986; Struck-Marcell and Scalo, 1987) investigated cloud system dynamics with a model which included a cloud mass threshold for SF and young star feedbacks. The SF term in this `cloud #uida model behaves as expected from the Schmidt law in its more quiescent mode. However, it undergoes a transition to burst-like behavior, which is more nonlinear than the Schmidt law, above a critical density. This raises the possibility of a second density threshold above which the nature of SF is intrinsically time-dependent. The time between bursts and the burst duration in these models depend on intrinsic variables, independent of the gas consumption time (though subject to the e!ects of consumption). Observationally, it is hard to distinguish between vigorous (Schmidt law) SF driven by high gas densities (or pressures), and a qualitatively di!erent burst mode. What is needed is some measure of how nonlinear is the dependence of SF on gas density. Di!erent observational techniques are used to "nd the atomic and molecular components of the cool gas. Each method has its own ambiguities, and both generally have less resolution than we would like. Moreover, any region with strong SF does not become readily identi"able in the optical and near infrared until it has begun to disperse the gas, making it much more di$cult to assess the conditions at the onset of SF.
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Nonetheless, the study of young post-starburst environments, will provide useful insights. For example, in the region interior to the star-forming ring in M82, it appears that not all the gas has been blown out, but that the clouds have been broken down to very small sizes (e.g., Lord et al., 1996). 7.4.2. Star formation suppression Dynamical heating in galaxy collisions, and large-scale accretion processes can delay or inhibit SF in some regions via several mechanisms, though much less attention has been given to this than to enhancement mechanisms. For example, as already noted, strong bursts can trigger nuclear out#ows and galactic winds (e.g., Heckman, 1993; Lehnert and Heckman, 1996). Out#ows resulting from starbursts in the small companions in collisional systems may be particularly interesting, not only because they a!ect the SF history of the system, but also because, after escaping relatively modest gravitational potentials, enriched gas may be thrown to relatively large distances. There it could contribute to the numerous absorption line systems observed in the spectra of quasars. Within the galactic disks global compression waves are usually accompanied by comparable rarefaction regions. The region between the rings in ring galaxies provide striking examples (AS96). These waves can lower the gas density below relevant thresholds and terminate SF. Observations of the phenomenon can provide sensitive probes of threshold theories. To date, such observations have generally been consistent with the theory (e.g., in the Cartwheel, Lindsey-Shapley and Arp 10 ring galaxies, see Section 3.1). Collisions of high velocity clouds with the gas disk can provide strong local shock heating and push cool gas out of the disk (Tenorio-Tagle, 1981; Tenorio-Tagle et al., 1986, 1987). Both e!ects would tend to suppress cloud buildup and SF at least temporarily. Galaxy collisions involving two gas galaxy disks can push a great deal of gas out of the two galaxies. Most of this material will subsequently accrete back to the central regions of both of them, but at relatively high velocities. Struck et al. (1996) have suggested that the low level of SF within the inner ring of the Cartwheel galaxy may be partly the result of cloud disruption due to high velocity impacts of clouds infalling either radially from the disk or vertically from the bridge. Collisional galaxies are a unique environment for studying this process at a high #ow rate, where global consequences are observable. Models (Appleton et al., 1996; Struck et al., 1996; Struck, 1997) suggest that the companion gas disk will be highly disrupted in near-central collisions between two gas disks, and subsequently reform via accretion out of the disk (also see Thakar and Ryden, 1996). The heating involved in this process can delay the onset of SF while the gas mass builds up, and thus, setup a strong delayed starburst. While there are interesting hints in the available observations, the small companions in collisional systems have been relatively unexplored, and these predictions are largely untested. They potentially provide a valuable window on the process of galaxy disk formation. In sum, galaxy collisions can be considered as Nature's experiments in how SF is a!ected by major rearrangements in the distribution and kinematics of the gas in disks. Spectacular enhancements and suppressions can result; the enhancements may be su$cient to account for a large fraction of the stars formed in the universe. The collisional disturbance can temporarily destroy the self-regulated state that controls SF in isolated galaxies, providing unique opportunities for studying individual SF processes when they are not regulated by couplings to other processes. The list of such processes is rather long, and their separate and synergistic roles are not yet well understood. There is much work yet to be done.
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8. Active galactic nuclei in collisional galaxies 8.1. Phenomenology Toomre and Toomre's suggestion that collisional disruption together with gas dissipation could feed all kinds of nuclear activity (see Section 1.4), has inspired many searches for a connection between galaxy interactions and active galactic nuclei (AGN). As we will see these have turned up generally positive results, however, the correlations are not simple and direct, and many uncertainties and ambiguities remain. As in the case of the relation between SF and interactions, it seems that a number of processes are involved, each with its own parameter dependences and characteristic timescales. Given this fact, and the fact that there is a great deal of research underway, the results of which may change this situation enormously in coming years, we will not explore this subject very deeply here. The sets of active nucleus galaxies and interacting galaxies are not identical, but probably have considerable overlap. Both sets have been targeted in attempts to map out the degree of overlap, and discover the mechanisms responsible. First we will consider searches for an excess of AGNs in interacting galaxies, and secondly attempts to determine whether there is an excess of interactions among galaxies with AGN. Each has its own di$culties (see the reviews of Heckman, 1990; Stockton, 1990; Laurikainen and Salo, 1995; Peterson, 1997). There are two fundamental di$culties in trying to survey AGNs in interacting galaxies. The "rst is that even in isolated galaxies, and more so in collisionally disturbed galaxies, the active nucleus can be buried beneath thick layers of gas and dust. This problem can be overcome at long wavelengths (radio to far-infrared), but in doing so we usually encounter the second problem. This is that high-resolution is needed to uniquely identify the AGN. The AGN and its associated accretion disk are very small, i.e., sub-parsec scales. However, they are often also very luminous, so the resolution problem could be bypassed given unique spectral signatures. Unfortunately, the most commonly used spectral characteristic are broad emission lines in the near-infrared to ultraviolet, which can be buried. Many of the more easily observed broadband or continuum diagnostics are not unique, but rather shared with the spectra of starburst regions (see below). Most of the early surveys were hobbled by these di$culties. For example, radio (e.g., Hummel, 1981; Condon et al., 1982), far-infrared (e.g., Soifer et al., 1984a), and near-infrared (e.g., Joseph et al., 1984; Lonsdale et al., 1984; Cutri and McAlary, 1985) surveys have found excess emission in the nuclei of interacting systems relative to normal systems (also see the numerous articles in Sulentic, Keel and Telesco, 1990). However, generally the resolution in these studies was not su$cient to determine the source of the emission. In some cases radio and radio-to-infrared spectral indices can be used to distinguish starbursts from `monstersa (Condon and Broderick, 1988), but the nature of the monster isn't always clear. Keel et al. (1985) and Kennicutt Jr. et al. (1987) found optical line emission enhanced in their interacting sample, with evidence that Seyfert activity in particular was somewhat enhanced. They also found some evidence that Seyfert nuclei were more frequent in very close pairs. This may be a consequence of close pairs being farther along in the merger process, and the fact that Seyferts are over-represented in merger remnants (e.g., Keel, 1996). Now let us turn to the question of how often AGN galaxies are involved in collisions? The primary di$culty with approaching the problem from this direction is that (luminous) AGN are
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not as common as they used to be when the universe was younger. This means that most AGNs are at great distances from us, and di$cult to study. It was for this reason that the question of whether quasars generally are located in the centers of galaxies took a long time to answer. However, Hutchings (1995) notes that some 200 quasars had been resolved by 1995. These studies also provided insights into the relation between quasars and interactions. In the last few years high resolution HST studies of kinematics in the nuclei of nearby active galaxies have provided overwhelming evidence for the presence of supermassive black holes. There now seems to be little doubt that the AGN phenomenon is primarily as result of accretion onto these black holes (e.g., Maggorian et al. (1998) and references therein). The early study of nearby Seyfert galaxies, which contain low luminosity AGNs, by Simkin et al. (1980) found asymmetries or morphological disturbances in many of the host galaxies. They suggested that these asymmetries probably resulted either from internal causes, like bars, or from tidal interactions. Subsequent studies have con"rmed that tidal distortions are common (e.g., MacKenty, 1990 and the review of Heckman, 1990). However, recent HST observations of quasar hosts have provided the strongest evidence to date of an association between interactions and AGN. The six objects with the HST planetary camera by Hutchings and Morris (1995) (also Hutchings et al., 1994) all appeared to be in the `middle to late stages of merging with a smaller (M33 or LMC-size) companiona. (The advantages and disadvantages of HST for such studies were also detailed in Hutchings (1995).) The three quasars in the sample of Boyce et al. (1996) (also see Disney et al., 1995) also `appear embedded in spectacular interactions between two or more luminous galaxiesa. This latter sample consists of IRAS-bright (ULIRG) objects. The largest study published to date is that of Bahcall et al. (1997 and references therein), which contains 20 nearby, luminous quasars. Three of these objects appear to be in merging systems with extreme tidal distortions, and 13 have close companions (see Fig. 26). Since one of the merging systems does not have a close companion, at least 14 out of 20 systems may be involved in tidal interactions. Thus, it seems that quasar hosts involved in recent collisions are not only common, but perhaps ubiquitous. This latter result on quasar companions is only the most recent example of a sequence of studies on AGN companion frequencies and environments that extends back more than two decades. Early studies generally found a higher than average incidence of companions around galaxies with AGN, though the results of various studies were not always in complete agreement. See the summaries of Heckman (1990) and Peterson (1997). Some recent studies have worked with very large samples of Seyferts. With a sample of 104 Seyferts Laurikainen and Salo (1995) concluded that while Seyferts have more companions than average, these are concentrated in a small number of systems, and so, Seyferts are not involved in interactions any more than other galaxies. Rafanelli et al. (1995) found an interaction excess of a factor of a few in their complete, magnitude-limited sample, which contained 200 objects. In these studies there are many di$culties associated with sample selection, companion identi"cation and statistical analysis, whose consideration is beyond the scope of this paper. Presently, the only conclusion that does seem secure is that the possible triggering of Seyfert activity by interactions is not a direct, prompt, inevitable, and easily observable process! One factor that makes understanding the AGN-interactions connection di$cult is the likelihood that the connection is only one leg of a triangular relationship, in which starbursts are the third player. The previous chapter considered another leg of this triangle, the one connecting starbursts and interactions. The nature of the remaining leg, the connection between starbursts and AGN, is
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Fig. 26. Hubble Space Telescope image montage of QSO host galaxies, illustrating the frequency of disturbed or interacting hosts (from Bahcall et al. (1997), courtesy AURA/STSci.).
very unclear (despite a huge literature of studies of individual systems). At one extreme there is the possibility that the only connection is accidental or indirect, that both phenomena are triggered by similar conditions. Not far from this view is the proposal of Sanders et al. (1988) that the `tremendous reservoir of molecular gasa funneled into the centers of ULIRG merger remnants induces the formation of both massive starbursts and AGN (`buried quasarsa). Both forms of activity drive out the gas and dust, the starburst dies down and the quasar is revealed. One clear prediction of this model is that quasars have a signi"cantly longer lifetime than super-starbursts. A variation of this model, in which a supermassive black hole at the heart of the AGN is fueled by the `supera mass loss resulting from the super-starburst, was studied by Norman and Scoville (1988). In this model starburst and AGN can be successive, so that the AGN lifetime need not be as long. At the other extreme we have the idea promoted by Terlevich and collaborators that the black hole engine is not necessary, but rather (radio quiet) AGN might consist only of buried or unresolved starbursts and supernova remnants (e.g., Terlevich, 1994). AGN variability is accounted for by frequent supernova explosions and interactions of supernova remnants with the dense, turbulent environment.
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Heckman (1990) has pointed out the di$culty in determining which of these two extreme theories is correct. He begins his review article by pointing out that the energy output of AGN and uper-starbursts are comparable, and mischieviously concludes with the observation that 2it is amusing (though possibly irrelevant) that the spectra of BAL quasars bear a super"cial resemblence to the spectra of massive stars undergoing mass-loss! (Heckman, 1990) Nonetheless, in a later review, Heckman (1994) argues strongly against the Terlevich model, in part on the basis that the supernova energy requirements are not consistent with other properties of starbursts. Terlevich (1994) disagrees with Heckman's "gures. As in the previous chapter the nature of the IMF is crucial. New VLBI (very long baseline interferometry) results may resolve the problem. For example, in a study of 18 ULIRGs Smith et al. (1998) found that the compact radio cores in 7 out of the 11 that they modeled could be accounted for by starbursts and radio supernovae. The remaining 4 objects could not be "t with starburst models, and `almost certainly housea accreting black hole AGNs. Genzel et al.'s (1998) ISO (Infrared Space Observatory) mid-infrared spectroscopic study of a comparable sample reached similar conclusions about the relative roles of starbursts and AGN. In conclusion, understanding the nature of the three-way relationship between interactions, starbursts and AGN has proven extremely di$cult. Indeed, the Terlevich-Heckman dialogue shows the di$culty in determining whether we're looking at the results of accretion onto a supermassive black hole, or a pure starburst. However, VLBI observations are beginning to answer these questions. Self-consistent interaction simulations have taught us that the merger process usually involves multiple collisions of two or more galaxies, and that SF is probably induced by a variety of dynamical processes, whose characteristic timescales span orders of magnitude. Thus, we cannot expect that connection and the connections to interactions to be very obvious in most current statistical studies. On the other hand, Gunn's (1979) suggestion that the key aspect of this problem is how to get the gas fuel down to the very small scales of the accretion disk. If we accept this then it follows that theoretical and numerical studies of speci"c fueling processes are worthwhile, as are attempts to identify observational examples. There has, in fact, been a great deal of work along these lines, some of which we will consider in the next section. 8.2. Fueling mechanisms In their 1992 review Barnes and Hernquist pessimistically conclude: `theoretical studies on the relevance of mergers to quasar activity mainly comprise wishful thinkinga. (!) They cite two particular di$culties: (1) Gunn's feeding scale problem, (2) the possibility that the quasar hosts had little to do with the processes being modeled in merger simulations. In the "ve years since their review the intense research in this area has provided more grounds for optimism. The second problem is substantially alleviated by the HST observations described above, which provide direct evidence that some, and the implication that, most of the quasar hosts are collisional systems. The feeding scale problem remains tricky, however. Simulations of large-scale processes like mergers and collisionally induced bar information have improved with ever more particles and better resolution. This gives us more con"dence in their basic results, and allows more detailed studies of funneling mechanisms (see Hernquist
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and Barnes, 1994; and other reviews in Shlosman, 1994). The feeding scale problem begins to seem rather academic when these models show that 2/3 of the gas is funneled down to a scale of a few hundred parsecs in a major merger. While it is true that there are a couple of magnitudes left to get down to accretion disk sizes, this is still a very large mass of gas in a quite small volume! On the other hand, even if we accept this case as proven, we are still left with a number of hard questions. E.g., does it take a major merger to fuel a quasar? What about the other types of AGN, do they also require a major merger to initiate and sustain their activity? While there are no de"nite answers to these questions at present, other candidate fueling mechanisms are being investigated. Among the oldest idea is that dissipation and the nonaxisymmetric forces in the disks of barred galaxies can lead to gas funneling without the additional help of collisions or mergers. (Many barred galaxies are not presently interacting.) Unfortunately for this idea, observational studies to search for a correlation between the presence of a large-scale bar and an AGN have yielded con#icting results (see the references in Peterson (1997)). Most studies probably have not been sensitive enough to discover weak, small-scale bars. For example, the very recent near-infrared Seyfert imaging survey of Mulchaey and Regan (1997) found bars in 55% of the galaxies classi"ed as non-barred in the optical. However, they found the same percentage of bars in both the Seyfert and the control sample. In addition, 30% of their Seyfert sample showed no evidence for bars. The question of fueling by bars was also investigated by Ho et al. (1997) using data from their optical spectroscopic survey of ('300) galaxies. They found no signi"cant evidence of increased central SF or incidence of nuclear activity in the late-type (Sc-Sm) galaxies with bars versus those without. However, they did "nd a measureable increase in the SF of the early-type, barred galaxies of their sample. They conjecture that the reason for this is that inner Lindblad resonances are more common in the early-types, which have larger bulges. While the SF may be enhanced by gas accumulation in a ring at the radius of the resonance, it is prevented from fueling nuclear activity in the center. Moreover, barred galaxies may su!er from a more severe feeding scale problem than do merger remnants. That is, most bars may not be able to funnel enough gas down deep enough to feed the accretion disk without the help of an additional process. Several candidate for this process have been considered recently. For example, Noguchi (1994) and Shlosman and Heller (1994) emphasize the role of what might be called `bootstrappinga via the self-gravity in the in#owing gas. Heller and Shlosman (1994a,b) have also presented models showing much enhanced gas in#ow driven by enhanced viscosity and angular momentum transport resulting from a starburst triggered by the initial bar-induced in#ow. Finally, there is also the bars within bars mechanism discussed in Section 5.6. Mulchaey and Regan (1997) looked explicitly for evidence of this process. They did not "nd many candidate multiple bar systems, and most of the ones they did were in the non-Seyfert control sample. However, it is also believed that as the central black hole grows it can generate substantial dynamical heating, inhibiting or destroying (multiple) bars (e.g., Friedli, 1994; Shlosman and Heller, 1994). At this point the reader will recall that we began considering dynamical mechanisms in hopes of "nding a little more enlightment than was o!ered by the observational surveys. Instead, we seem to have found a great deal of dynamical complexity. Worse yet, we have not yet exhausted the list of dynamical processes; there are at least several important variations on the themes above. One is the process of mass transfer in non-merging collisions described in Section 4.1. When the transferred
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gas has low angular momentum along the host galaxy spin axis it will mix with ambient gas and accumulate in the central regions as in mergers. Most of these interactions are less extreme than mergers, except for small companions involved in direct collisions. Therefore, we would expect less direct feeding of the AGN, but indirect feeding via bar instabilities, induced starbursts, etc., seems more likely. Yet another possibility is that the active nucleus might contain more than one supermassive black hole. If most galaxies harbor a supermassive black hole, than massive binary or multiple black holes seem likely in the nucleus of a merger remnant (Begelman et al., 1980). Makino (1997) has recently presented self-consistent N-body simulations of this process. A supermassive binary black hole orbit decays as a result of scattering stars (two-body `thermala relaxation), resonant scattering, dynamical friction (collective scattering), and gravitational radiation if the orbit is small or highly eccentric. While the mean value of the average decay timescale is still being debated, it is generally thought to be longer than galaxy collision timescales. The (long-term) existence of such a binary would considerably ease the feeding scale problem. The accretion disks around the orbiting black holes will interact with gas in a much larger volume in the central disk than a single black hole, and will generate a non-axisymmetric gravitational disturbance much like a small-scale bar, funneling gas inward. By now it is clear that there are a variety of possible solutions to Gunn's feeding scale problem. The triangle of processes described above, with vertices labeled AGN, starburst and interactions, should be enlarged to a pentagon with the addition of vertices for bars and binary black holes, at least. On a more optimistic note, the high resolution observations of the near neighborhood of AGNs and their accretion disks or torii that are becoming available provide a much better understanding of the phenomena. This may be a prerequisite to a better understanding of the connections to large-scale phenomena. A "nal item on our wish list is more detailed and complete surveys. Keel's (1996) recent imaging and spectroscopic study of Seyfert galaxies with companions, noted above, provides a good example of what we can hope to "nd. Even negative results, like Keel's "nding that the presence of an AGN does not seem to correlate with the type of collisional interaction, are helpful. In fact, this result is a surprise, since possible fueling mechanisms like mass transfer and induced bar formation do depend strongly on the nature of the interaction. Are the time delays associated with fueling long enough to make it impossible to associate the e!ect with the initial cause, are these particular fueling processes simply not that important, or are many processes equally important? Keel's result that there does seem to a (complicated) dependence on the magnitude of the kinematic disturbance points the way to future studies.
9. Environments and redshift dependences In previous decades most research on galaxy collisions has focussed on questions like: how common are mergers, what fraction of galaxies are merger remnants and what traces remain of the merger process? These questions can be viewed as basic and fundamental, while environmental dependences may be seen as details to be "lled in later. Similarly, in the "eld of high-redshift studies the primary question has been how does the merger and interaction rate vary with time? Questions about the time dependences of environmental e!ects have been viewed as secondary.
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On the other hand, if the nature and frequency of interactions have strong environmental dependences, we will not be able to answer the basic questions without an understanding of some of the `detailsa. This situation is almost certainly qualitatively true in the present universe. For example, one important point is based on the galaxy density}morphology relation (e.g., Dressler, 1980; and more recently Whitmore and Gilmore, 1991; Whitmore et al., 1993a, b), which tells us that ellipticals and early-type spirals are preferentially found in dense environments (e.g., clusters), and late-type spirals and irregulars are found mostly in low density ("eld) environments. Since the former types are generally gas poor, we would expect gas dynamics, induced star formation, and perhaps, nuclear activity to be less important in collisions and mergers in the densest environments. Another factor of comparable importance is that the relative velocities of galaxies in dense environments are much higher than those in small groups. This means that interactions are generally more rapid, and generate smaller disturbances in the galaxies involved. The complete story of interactions within galaxy clusters is, of course, much more complicated, as we will see below. Currently we are entering a period of vigorous interest in the environmental dependences of collisions. On the theoretical side, large-scale N-body simulations of galaxy and cluster formation provide insights on not only the merger rate, but also on environmental and temporal dependences. Semianalytic hierarchical clustering models, based on the assumption that galaxies and large structures are built up by hierarchical mergers from small initial units, e.g., dwarf protogalaxies, have also become very popular recently. These models are a useful quantitative tool for analysing and describing the detailed numerical simulations, and are also useful aids to in interpreting high redshift observations. The discovery of large numbers of `Lyman-break galaxiesa at redshifts z'3 provides one example. Hierarchical models can readily account for the numbers and SF histories of these objects (Baugh et al., 1998). Representative `merger treesa derived from these models show that many mergers occur at early times, and that mergers continue to be orchestrated by the clustering environment down to the present time, much as envisioned by Toomre in the 1970s. We shall discuss these studies further in Section 9.3, after considering some speci"c environments in Sections 9.1 and 9.2. 9.1. Groups and compact groups There are many approaches to this subject, we will begin with the straightforward one of asking how many neighbors do collisional galaxies have? Unfortunately, there is not an equally straightforward answer, for several reasons. The "rst is that no very large-scale study of interacting systems has yet been undertaken. Optical searches of many systems for companions, down to low brightness levels, are very arduous. This should be clear from the discussion of the Holmberg e!ect in earlier chapters. On the other hand, anecdotal evidence suggests that such searches might be pro"table, especially HI searches, where discovering even small (gas-rich) dwarfs is relatively easy. For example, the HI study of the VII Zw 466 ring galaxy system by Appleton et al. (1996) discovered two gas-rich, dwarf galaxies at some distance from the compact central group, but with redshifts suggesting that they are part of the group. In fact, numerous dwarfs have been found in the vicinity of many interacting systems, see for example, the recent catalog of 12 systems Deeg et al. (1998). However, the goal of this study and others cited above is the discovery of dwarfs produced in the collision, a phenomenon which could greatly complicate the present question of neighbor
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numbers. Thus, the present question is best addressed in Y-type collisions, where there has not been time for the formation of new dwarfs. We also must keep in mind the more general context, illuminated by the work of Ramella et al. (1994, 1995, also see Palumbo et al., 1995) that compact groups, like the Hickson groups, are generally embedded in a larger loose groups which continually feed the compact center. Besides anecdotes and studies of groups that are di$cult to interpret, there are also some studies of the neighbor statistics of speci"c types of interacting galaxies. First of all, on the bright end of the spectrum we have the merging or merged ULIRGs. After reviewing a number of individual cases Sanders and Mirabel (1996) conclude that most of these objects `appear to involve strong interactions/mergers of molecular gas-rich spirals where the pairs are either isolated or part of small groupsa and are generally not in dense clusters. While this conclusion is not based on a rigorous statistical study, there are many examples. Few and Madore (1986) carried out a more formal study of the neighbors within two ring diameters of 69 ring galaxies, using southern sky survey plates. Rings are a readily identi"able morphology, and since collisional rings are a young Y morphology, the collision companion, at least, should be nearby. However, the disadvantage is that most rings are not collisional systems. Indeed, the goal of this study was to obtain some idea of what fraction might be collisional. The ring sample did indeed show a small excess of near neighbors over a control sample, though the most common number of neighbors was one in both cases. The authors took the analysis a step farther by classifying each ring as either P or O type, where the former have a knotty ring structure and usually a displaced nucleus or other morphological peculiarity, while O-types have a central nucleus, and a smooth ring. Thus, there is direct evidence for a collisional disturbance in the P-types, but little in the O-types. Most of the companion excess was found in the P subsample. We conclude that this special class of collisional system is generally found in small groups, from one to a few companions that are not much fainter than the primary. Another very interesting statistical study with a narrow focus is Odewahn's (1994) study of 170 Magellanic spirals. These are very late-type (beyond Sc, Sd) spirals, that are typically gas-rich. Odewahn "rst applied an arm strength classi"cation to each galaxy, and then noticed that virtually all of the 75 objects with `well classi"ed asymmetric armsa had a companion within a few diameters. Moreover, the companion distances peaked at small separations, indicating that most are probably not chance alignments, but physical associations, and in most cases, interactions. Thus, a very large fraction of one particular type of galaxy exists in small collisional groups, and most of the members of this class have unbalanced spiral waves or bars. The study of Fried (1988) constitutes one of the most general attempts to answer the question above. He assigned each of 517 nearby galaxies to an interaction class ranging from 0 (no evidence of disturbance) to 3 (strongly disturbed), using sky survey prints. His discussions of the di$culties in carrying out such a task are illuminating and explain why few others have attempted it. He found very few "eld galaxies were disturbed, i.e., outside of class 0. In groups, on the other hand, he found nearly 30% of the galaxies were disturbed. Most of these are spirals. Fried notes di$culties in detecting faint shells and other collisional debris around ellipticals in the survey prints, and in deriving this "gure adopts the fraction of disturbed ellipticals from the more sensitive study of ellipticals by Sadler and Gerhard (1985). Compact galaxy groups have received a great deal of attention in recent years, especially the 100 groups catalogued by Hickson (1982, 1994), and Hickson et al. (1992), see Fig. 27. The nature and
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Fig. 27. The "ve galaxies of Hickson compact group 40 ("Apr 321"VV116) are shown at the center of this image (Digital Sky Survey image courtesy of AURA/STScI.).
frequency of interactions in such provides valuable information on a very speci"c, but increasingly well-studied environment. Moreover, some important properties about this environment are reasonably well-established now, at least in the case of the Hickson groups. The "rst is that the groups are generally true physical associations, with a high density of galaxies and relatively low velocity dispersions, though they frequently contain one or more superposed background objects. This conclusion is based both on member redshifts (see e.g., Hickson et al., 1992; Ramella et al., 1994, 1995), and more indirect evidence. The latter includes the "nding of Palumbo et al. (1995) (also see Sulentic and Rabac7 a, 1994) that the spiral fraction of compact groups di!ers from that of "eld galaxies. The conclusion even seems to hold for `poora groups with fewer than 6 members (Zabludo! and Mulchaey, 1998). The very overdense environments of these groups are presumably very favorable for interactions. Another, more controversial property of these systems is that they may be commonly embedded in larger, loose groups, which continue to feed them galaxies. Speci"cally, Ramella et al. (1994, 1995) clearly "nd these extended groups in redshift space. However, in a study of sky survey prints, Palumbo et al. (1995) "nd that only about 18% of the groups have signi"cant concentrations of galaxies outside the core group. Sulentic and Rabac7 a argue from lack of compact group merger remnants that the groups must simply have a much longer lifetime than would be expected from merger simulations (see, e.g., Mamon, 1986, 1987; Barnes, 1989; Barnes and Hernquist, 1992a; Governato et al., 1996; Weil and Hernquist, 1996). Zabludo! and Mulchaey (1998) "nd that a large fraction of the dark matter in poor groups is located in a large common halo, rather than in the individual galaxy halos. Thus, as in large clusters, galaxy virial velocities within the common halos may be high enough to delay merging, without any need for continued feeding. However, these authors also "nd evidence for continuing accretion onto their groups, suggesting both phenomena play a role.
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The question of the interaction frequency in compact groups has been addressed with observational searches for enhanced activity in a variety of wavebands. Perhaps surprisingly, in a study of the optical luminosity function of compact groups, Sulentic and Rabac7 a (1994), found little evidence for an interaction-induced luminosity enhancement relative to control samples. This result does not necessarily contradict the conclusion that interactions are common in these systems, which is not only evident in the images, but con"rmed by more detailed studies. For example, Rubin et al. (1991) carried out detailed imaging and spectroscopic observations of 21 Hickson groups. In addition to the obvious morphological disturbances, they found a variety of kinematic peculiarities, including disk galaxies with asymmetric rotation curves and velocity patterns `too peculiar to form rotation curvesa. Speci"cally, they found that 10 out of 12 elliptical or S0 galaxies had ionized nitrogen emission from their nuclei, with resolved ionized gas disks in about half of them. They speculated that these gas disks are recent acquisitions. Pildis et al. (1995) undertook very deep photometry of about a dozen Hickson groups. With this high quality data they modeled the mean luminosity pro"les of the early-type galaxies and subtracted them out. In many cases shell systems remained, providing still more evidence for mergers and accretion events. Mendes de Oliveira and Hickson (1994) also used extensive surface photometry for a study of 202 galaxies in the Hickson groups. They found that 43% of the galaxies in their sample had disturbances indicative of interactions, and this was true of 75% of the galaxies in a subsample of groups with published kinematical data. Thus, the conclusion that interactions are common in compact groups remains secure. The unenhanced luminosity functions may be the result of a failure of interactions to induce star formation in these particular environments. We know from the discussion of the work of Hunsberger et al. (1996) in Sections 5.4 and 7.3 that SF does occur in tidal structures in some compact groups. Unfortunately, it appears that no large statistical study of the colors of galaxies in compact groups has been published yet (though many B-R colors are given in Hickson (1994)). Recently, Menon (1995, and references therein) presented the results of a radio continuum survey of 133 spirals in 68 Hickson groups. In this sample 56 galaxies were detected, and the continuum emission was generally found in `slightly extended nuclear regions suggestive of starburst activitya. On the other hand, the total continuum emission from Hickson spirals was found to be less on average than that of a comparison sample of isolated spirals. It appears that the nuclear emission in Hickson spirals is more than o!set by an emission de"cit in the other parts of the disks. Williams and Rood (1987) carried out an HI survey of 51 of the Hickson groups. They found that the amount of HI, the basic fuel of SF, in these groups was about half that in a control sample of loose groups with similar morphological and dynamical processes. Preliminary results of a survey of southern compact groups are similar (Oosterloo and Iovino, 1997). HI has been found outside the galaxies, and indeed, throughout the group in a few groups (e.g., Williams and van Gorkom, 1988; Williams et al., 1991), but this does not add a signi"cant amount of mass to the overall average. So where did the gas go? The far-infrared and CO study of Verdes-Montenegro et al. (1998) "nds that the molecular content of 80 galaxies in Hickson groups is very similar to that of a control sample of spirals. The far-infrared data also show little enhancement of SF or nuclear activity. These results are basically con"rmed in the study of Leon et al. (1998), which included 70 galaxies in 45 Hickson groups. The latter study did "nd some evidence of enhanced molecular and dust masses, which together with
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normal far-infrared #uxes may imply a lower than average SF e$ciency. Nonetheless, it appears that no great mass of gas has gone into the molecular phase. In some cases it appears that a signi"cant part of it has gone into a hot halo enveloping the group, which has been observed with the ROSAT X-ray satellite (e.g., Ebeling et al., 1994; Saracco and Ciliegi, 1995). Dynamical models suggest that it is plausible that this gas has been stripped from individual galaxies and heated in the collapse and subsequent evolution of the group, though there are di$culties in "tting the temperatures and emission pro"les of the X-ray gas (Diaferio et al., 1995; Pildis et al., 1996). Finally, in a detailed optical imaging and spectroscopic study of ellipticals in compact groups, Zepf and Whitmore (1993) "nd evidence for disturbances, but little evidence for young stellar populations. Thus, they conclude that these ellipticals are not the result of recent mergers between (gas-rich) spiral precursors. On the contrary, most of the stars in them must have been formed at a much earlier time. Thus, several lines of evidence suggest that compact groups of galaxies come together (collapse) rather late in the history of the universe, after forming most of their stars in the individual galaxies. The subsequent merger times can be increased by varying the galaxy mass distribution (Governato et al., 1991), or the fraction of dark matter distributed generally through the group (Bode et al., 1993). However, the continued input of galaxies from a larger, looser environment seems like the surest solution to the problem of the relatively high frequency, or longevity of compact groups. The special environments of these groups might be viewed as interaction generators. The overall conclusion of this section is that most vigorous galaxy interactions occur in groups containing a modest number of galaxies. Compact groups are special, and unusually well-studied, cases. Unfortunately, the statistical studies of groups are generally quite narrowly focused, and the collections of anecdotal results su!er even stronger selection e!ects. Thus, while the above conclusion appears to be quite "rm, it is di$cult to pursue further analyses of causes and connections at present. 9.2. Dense clusters Dense clusters of galaxies, the urban environment of the galaxy world (see Fig. 28), do not seem to be very conducive places for strong interactions. First of all, in the previous section we concluded that most strongly interacting systems are found in modest sized groups. A glance at the Arp or Arp and Madore atlases con"rms that most of the systems there do not lie within a dense cluster. Fried's (1988) study of interactions as a function of environment (discussed in the previous section), included the Virgo cluster, where he found signs of interaction in only 16% of the galaxies. This lack of obvious signs of galaxy collisions does not encourage studies of interactions in clusters, and until recently, relatively few studies, pursuing restricted subjects, have been published. However, the current evidence suggests that there is no lack of interaction in clusters, but rather that the interactions in this environment are qualitatively di!erent from other environs. Statistically, the best evidence for such di!erences comes from the study of SF in 15 749 (!) galaxies from the Las Campanas redshift survey undertaken by Hashimoto et al. (1998). They found that SF was reduced in all types of spirals in large clusters, and conjectured that this is the result of gas removal processes. On the other hand, they "nd a `prevalence of starbursts in
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Fig. 28. Broad band image of a distant (z"0.39) galaxy cluster, CL 0024#1654. Foreground stars are revealed by the white (saturated) dot in the center of the dark image, almost all remaining objects are galaxies, most in this representative cluster. (Unpublished image produced at the University of Hawaii 2.2 m telescope by R.J. Lavery and J.P. Henry, provided by R.J. Lavery.)
intermediate density environmentsa, such as groups and poor clusters, and they conlcude that this prevalence is the result of interactions. 9.2.1. cD galaxies The notion that strong interactions are rare in cluster environments is historically ironic, because the buildup of central cluster galaxies and the giant cD galaxies, in particular, by mergers and `galactic cannibalisma was an early area of study (e.g., Ostriker, 1977; Schneider and Gunn, 1982). A de"ning characteristic of cD galaxies is that while in their inner regions they have surface brightness pro"les like those of ellipticals, in the outer regions (which can be truly gigantic) they decline much more slowly. That is, cDs have giant luminous halos or envelopes, which can contain as much light as the rest of the galaxy. The studies of Schombert (1987, 1988, and references therein) of the surface brightness distributions of 342 bright cluster ellipticals substantiate and quantify these general statements. According to the galactic cannibalism theory, while the cD precursor may well have been the largest galaxy to form in the center of the cluster (or subcluster, see Merritt (1984)), the halo developed by the disruption and merger of numerous smaller galaxies. There is much numerical and observational support for this theory. First of all, the general structure of cD envelopes can be accounted for by homology calculations (Hausman and Ostriker, 1978), Monte Carlo calculations of the e!ects of collisions in clusters (Richstone, 1975, 1976), and N-body simulations (e.g., Farouki et al., 1983; Barnes, 1989). The accreted material ends up in an extended halo because it is dynamically hotter than the stars formed within the galaxy. The situation is like the case of the shell galaxies discussed in Section 5.5; the extended envelopes are essentially multiple, relaxed shell systems. (Interestingly, there has been relatively little modeling work on this subject in recent years, with much more e!ort going into details of the formation of
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giant ellipticals from major mergers, see above.) A second type of evidence is provided by velocity studies that establish that some cDs are surrounded by a population of galaxies that are bound to the cD itself, rather than just the general cluster potential (Bothun and Schombert, 1988, 1990). The e!ects of dynamical friction on these galaxies must be strong, and they should merge on a timescale much smaller than the age of the universe. However, the most convincing evidence is the discovery of examples of dense groups in which the galaxies are surrounded by a common envelope, all are contained within a region whose size is no more than a few diameters of the largest galaxy. Examples include V Zw 311 (Gunn and Schneider, 1982), discussed above, and Hickson 94 (Pildis, 1995). Cavaliere et al. (1991) use an analytic formalism to argue for the existence of a merging runaway in dense groups. Interestingly, these nascent cDs also seem to con"rm Merritt's suggestion that cDs form early, e.g., in subclusters, rather than at the core of large clusters of galaxies. On the other hand, Bothun and Schombert's work shows that their formation is a continuing process. In any case, the making of cDs is one example of how interactions in clusters, while not as spectacular as those in the "eld, can yield extreme products. 9.2.2. Collisions and harassment in clusters Why should the e!ects of collisions between comparable galaxies be weaker in clusters, and what are some of the qualitative di!erences? In fact, there are a number of ways in which the cluster environment can modify the e!ects of galaxy collisions. The "rst and most important of these e!ects results from the fact that the random velocities of the galaxies in large clusters are generally greater than the internal velocities of the galaxies. Thus, a typical collision between cluster members occurs at much higher relative velocities than we have generally considered, and so, the time the two galaxies are close together is correspondingly shorter. The chance that dynamical friction will leave them in a bound orbit is also much reduced. We have considered two estimates of collisional perturbation strength earlier in this article, and both contain terms that illustrate this e!ect. In Eq. (1) there is an inverse dependence on relative velocity, while in Eq. (10) there is a direct dependence on the duration of the collision. Richstone's (1975, 1976) early Monte Carlo studies of hyperbolic collisions between spheroidal galaxies in clusters showed that while the galaxy cores were relatively una!ected, the envelopes could be changed signi"cantly by tidal stripping and impulsive energy injection. Richstone also analysed how tidal cuto! radii of the galaxies evolve, and noted that halos can be stripped quite promptly. (However, Allen and Richstone (1988) revised the estimates of tidal radius evolution.) The result on halos has been con"rmed in more recent simulations, and can be viewed as an aspect of what is called the `overmerginga problem, i.e., the inability of cluster simulations to retain substructure. This problem, however, may be the result of limited numerical resolution and other technical details (Moore et al., 1996a; Frenk et al., 1996). Recently, the modest disturbances resulting from high-velocity encounters, which are frequent in the cluster environment, have been termed `galaxy harassmenta by Moore et al. (1996b), who present simulations of their e!ects. These e!ects appear qualitatively similar to the tidal disturbances of disk galaxies discussed in earlier chapters, but they are on the weak end of the disturbance continuum. Cluster ellipticals and other early types are relatively una!ected. They point out that the fraction of disturbed spirals is generally high in clusters, and the harassment explanation is very plausible. Icke's (1985) models of distant hyperbolic encounters produced
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similar results, and he emphasized that the e!ects on the gas would be greater. Moore et al. also suggest that the process plays a role in fueling quasars, and might account for the large fraction of blue galaxies in clusters at high redshift (the Butcher}Oemler e!ect). Oemler et al. (1997) "nd no evidence that the blue galaxies are merger remnants, and favor the harassment explanation. Joseph's (1996) review of the Moore et al. harassment paper is cautious, pointing out that in the simulations the haloes of the harrassing galaxies were assumed to be simple rigid, softened point-masses. This again raises the issues of how much halo cluster galaxies retain, and the structure of those haloes. These are complex issues, both observationally and theoretically, and we cannot digress into a proper consideration of them. We will have a little more to say about the results of cluster formation simulations, which provide some insights, below. Observationally, it is very di$cult to use traditional means, like HI rotation curves, because outer HI disks have often been stripped o!. Some rotation curve studies have been carried out, including a recent series of papers by Amram et al. (1993, 1994, 1995, 1996). These studies "nd generally #at rotation curves, in contrast to some earlier reports. However, in most case they do not extend far enough in galaxy radius to detect much halo dark matter, if it is present. Natarajan et al.'s (1998) HST study of gravitational lensing in a cluster with a redshift of z"0.31 provides evidence that cluster galaxies have smaller and less massive halos than their counterparts in the "eld. It will be exciting to see additional lensing studies in the coming years. 9.2.3. The cluster environment: stripping and cluster tidal ewects The "rst inkling of gas removal from cluster galaxies is Spitzer and Baade's (1951) suggestion that high velocity collisions in clusters could remove their gas and deposit it in an intracluster medium. Two decades later Gunn and Gott (1972) proposed an alternate mechanism that did not depend on such collisions, which, by that time, did not seem su$ciently frequent. Like Spitzer and Baade they reasoned that high galaxy random velocities would lead to the shock heating of any cool gas, so intracluster gas must be hot, millions of degrees Kelvin. Because of this they concluded that it must be smoothly distributed (a routine observation of X-ray telescopes now). The cluster galaxies would be constantly plowing through this medium, and experiencing ram pressure. Gunn and Gott estimated that this pressure could entirely strip the di!use gas from a spiral. Later, Valluri and Jog (1990) described some modi"cations that would result from the multi-phase structure of the interstellar gas, and speci"cally, concluded that it would be di$cult to strip dense molecular clouds. More recently, detailed numerical simulations indicate that only the outer parts of gas disks will be stripped (Kundic et al., 1992). This latter conclusion "nds some support in observation, especially in the case of the Virgo cluster, the nearest and best studied cluster. To begin with, Giovanelli and Haynes (1983) (Haynes et al., 1984; Haynes and Giovanelli, 1986) obtained HI observations of more than 160 Virgo Cluster spiral galaxies, and hundreds of "eld galaxies which could be used as a control sample. Using several di!erent measures, they found a signi"cant HI de"ciency for a subpopulation of Virgo spirals, though the remaining subpopulation was found to normal HI contents. They attributed the observed de"ciencies to ram pressure stripping, and noted that spirals within 53 of the Virgo center had lost 90% of their neutral hydrogen gas mass. They also noted that among the de"cient galaxies it appeared that SF had been `quencheda. Following up on this work, Kenney and Young (1986, 1988, 1989) surveyed the molecular gas masses and distributions of Virgo cluster galaxies, using observations of the CO molecule. They
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found that the molecular gas contents of the bright Virgo spirals are not greatly de"cient, or otherwise unusual, and that `the total gas de"ciency is manifested largely by a lack of HI in the outer diska (Kenney and Young, 1989). Inner disk regions appeared relatively une!ected. More recently, Koopman and Kenney (1993, 1994, 1995, 1996) have used H and R band imagery to ? study the star formation properties of a large sample of Virgo galaxies and an isolated control sample. They "nd a wide range in the H surface brightnesses in the inner disks of their sample, ? some are consistent with `fading diska models, e.g., with SF truncated by stripping. However, SF enhancements are evident in others, some of which are involved in an interaction, others may be accreting gas. Among their more interesting "ndings is a class `peculiar early-type spiralsa, which are `small bulge, gas-de"cient galaxies with featureless outer disks, but strong circumnuclear star formationa. They note the characteristics of these galaxies are what one might expect in stripped Sc type galaxies. From a broader point of view, these detailed results on SF in Virgo galaxies, provide an interesting complement to the global picture of the SF reduction in dense clusters provided by the Las Campanas survey cited above. The HI properties of Virgo cluster galaxies were surveyed again with higher resolution and sensitivity by Cayette et al. (1990, 1994), and also Warmels (1988a,b). The higher resolution of the radial pro"les of the galaxies they observed allowed Cayette et al. (1994) to draw somewhat more detailed conclusions than previously possible. E.g., they "nd that in `some galaxies ram-pressure stripping has done serious damage to the HI disks, while in other galaxies turbulent viscous stripping and thermal conductivity have caused a mild, but global HI de"ciency...a. Phookun and Mundy (1995) have also revived the idea of ram pressure `pushinga, rather than stripping, as a cause of the HI disk asymmetry in the Virgo galaxy NGC 4654, and possibly others. In sum, a number of di!erent authors "nd strong evidence that stripping has played an important role in shaping Virgo cluster disk galaxies. It appears that the data are becoming good enough to allow the study of details of the stripping process, and related processes. Thus, the prospects for learning more about these processes in Virgo appear very good. But what about other clusters, is Virgo representative? HI surveys of several other clusters have been reviewed recently by van Gorkom (1996). Like Virgo, `shrunken HI disksa seem to be common in cores of nearby large clusters, suggesting that stripping and related processes are equally common. The Ursa Major cluster studied by Verheijen (1996) provides a counterexample, since the HI disks of the spirals in the center of that cluster appear normal, not HI de"cient. However, the lack of X-ray gas in this cluster suggests that it may be much younger than other nearby clusters like Virgo, Coma and Abell 1367. The two clusters that have been mapped to date at somewhat higher redshift each show individual peculiarities (see van Gorkom, 1996). The "rst of these is the Hydra cluster at z"0.035. It has a bimodal velocity distribution function, no HI de"cit, and other morphological peculiarities all of which suggest that the spirals in the cluster core are either currently falling into the cluster, or are a chance superposition. In Abell 2670, at z"0.077, there is evidence for three distinct subsystems. It is not surprising that no general evolutionary trends have yet emerged from such studies, since the look-back times to these clusters are not a very large fraction of the age of the universe. Given optical observations of peculiarities in higher redshift clusters it will be interesting to see how such studies develop in the future. Another e!ect of the cluster environment, tidal disturbances on cluster galaxies due to the overall cluster potential, has been studied recently by Henriksen and Byrd (1996), also Byrd and Valtonen
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(1990). They conclude that under a fairly wide range of conditions, cluster tides can disturb galaxy disks enough to excite waves, global compressions, and enhanced star formation. They suggest that ram pressure stripping will later remove the gas in galaxies, guaranteeing a limited timescale for the SF enhancement, as in the observational Butcher}Oemler e!ect. The end results of the tidal disturbances seem very similar to those of the galaxy harrassment described above, although the processes are distinct. 9.2.4. Cluster}cluster collisions The HI observations described in the previous section provide evidence that substantial groups of galaxies continue to fall into large clusters. These observations can be viewed as a subset of a larger set of kinematic and morphological observations that suggest that unrelaxed substructure is common in galaxy clusters (see e.g., the review of Fitchett, 1988). A clumpy or asymmetric distribution of the hot X-ray gas in clusters also provides evidence of incomplete relaxation in nearby clusters, see e.g., the reviews of Jones and Forman (1991), and Henry and Briel (1993). More recent X-ray satellite (ROSAT) results are described in Henry and Briel (1996), and Knopp et al. (1996). Mahdavi et al. (1996), Mohr et al. (1996b) and Mohr et al. (1996a) have used optical redshifts, surface photometry and X-ray observations to demonstrate the existence of substructure and probable cluster}cluster mergers in several more clusters. While relaxation times are long in clusters, e.g., crossing times are of order 10 yr, these times are still much shorter than the age of the universe. Thus, they should have had time to relax since their formation. Moreover, the large fraction of highly evolved galaxies found in large clusters suggests that they were not formed recently, but in fact, probably formed quite early. This is in contrast to the poor clusters recently sampled by Ledlow et al. (1996), who "nd that the morphology, dynamics, and environments of these systems `are indicate of young, dynamically evolving clustersa. The solution to the paradox of highly evolved member galaxies observed in conjunction with nonequilibrium cluster dynamics favored by most of these authors is that these phenomena are the result of continuing hierarchical growth of large-scale structure. Clusters continue to merge with clusters, and substantial `cloudsa of galaxies continue to fall into large clusters, especially out of the sheets and "laments that make up the adjacent superclusters. The latter case (supercluster feeding) is very reminiscent of the discussion above of the continuing evolution of compact groups, and in a hierarchical growth picture we would expect such processes to co-exist on multiple scales. In the case of collisions between two large clusters of comparable mass, there can be dramatic and observable e!ects on the X-ray gas of the clusters, as recently shown in detail by the models of Roettiger et al. (1996). Research on the topics of cluster}cluster collisions and continuing infall are relatively new, and we will conclude this discussion by noting that collisions between larger entities may orchestrate some unique collisional processes for the galaxies contained within them. First, we note that since clusters have massive halos like galaxies, and since in collisions these halos will exert a strong dynamical friction force, we expect colliding clusters like galaxies to merge within a few crossing times. (A timescale we cannot really call short!) However, during this time the galaxies involved will feel a strongly #uctuating cluster gravitational "eld. This "eld should signi"cantly a!ect the orbits of dwarf satellite galaxies in the halos of large galaxies, and similarly a!ect the individual galaxies orbiting the center of mass in bound groups. In both cases, tidal torqueing may unbind
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some objects, releasing them into the general cluster "eld (Antunes, Wallin and Struck, in preparation). It may also remove orbital angular momentum from other objects, putting them on more radial orbits, and enhancing the chance of collision with the primary galaxy or other group members. This process is not the same as the galaxy harrassment described above, but if infall into clusters continues at a steady rate, it may be an important process in shaping clusters galaxies. 9.3. High redshift collisions The direct study of galaxies and galaxy collisions at high redshifts (and long lookback times) has become possible in this decade, especially because of the high sensitivity and resolution of instruments like the Keck telescopes and the Hubble Space Telescope. A large fraction of the evolutionary history of galaxies is now becoming directly observable. The amount of data obtained on high redshift objects, and our understanding of evolutionary processes will certainly grow rapidly in the coming decade as the exploration of this new frontier accelerates. We can expect new input on some of the oldest and most fundamental questions in this "eld, such as how much higher was the collision and merger rate in the past, and what is the average rate of change with redshift or lookback time? The time dependence of the merger rate has been a subject of increasing attention in the last decade after the discovery of numerous, faint `blue galaxiesa at a high redshift (e.g., Broadhurst et al. (1988), and more recently Driver et al. (1995a), and references therein), which may be the result of more frequent interactions and mergers at earlier times. The merger rate of galaxies is often parametrized as a power-law function of redshift, MRJ(1#z)K, and thus, it can be conveniently discussed in terms of the value of the exponent m. However, to date it has not proven easy to determine the value of this exponent. With current resolutions and sensitivities it is still di$cult to con"dently recognize ongoing mergers and merger remnants, and eliminate chance superpositions of galaxies. Many of the tidal structures discussed in earlier chapters are too faint to observe at high redshifts. Thus, it is perhaps not surprising that values of m reported in the literature range from 0 (no evolution in the merger rate) to m'3 (high rates of evolution). A sampling of values reported in the recent literature are given in Table 3. It seems premature to make any strong conclusions, but it Table 3 Merger rate exponent determinations Source
Exponent m
Mean survey redshift
Zepf and Koo (1989) Burkey et al. (1994) Carlberg et al. (1994) Woods et al. (1995) Yee and Ellingson (1995) Lavery et al. (1996) Neuschaefer et al. (1997) HST Medium Deep Survey Patton et al. (1997) Wu and Keel (1998)
4.0$2.5 2.5$0.5 3.4$1.0 +0 4.0$1.5 +4.5 1.2$0.4
0.25 0.4 0.4 50.4 0.38 +1.0 1}2
2.8$0.9 +2 (0}2)
0.33 2.4
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does appear that most studies "nd some evolution, and that the more extensive recent studies yield intermediate values of m. A recent study of Keel and Wu (1995) provides an estimate of the current merger rate, i.e., the constant in the power-law. Interestingly, they also "nd a large di!erence between the rate of spirals in pairs (4.2 per Hubble time), and the average for all spirals (0.33 per Hubble time). This suggests the possibility that the merger rate may depend sensitively on environment and the evolution of di!erent environments with time. Recently, there been a number of simulational studies that are relevant to the merger rate. Firstly, there are studies of e!ective merging cross sections and merger times, e.g., the recent work of Makino and Hut (1997). Secondly, there have been a number of recent papers on the merger rate in speci"c environments. These include studies of merging in compact groups (e.g., Athanassoula et al. (1997), and the references of Section 9.1), and in clusters and other large scale structures (e.g., Menci and Caldarini, 1994; Frenk et al., 1996), and in cosmological structure formation simulations (e.g., Lacey and Cole, 1994; Navarro et al., 1995; Baugh et al., 1996; Baugh et al., 1998). Some of the latter calculations have been able to derive merger rates, as well as giving a qualitative feel for what merger histories are typically like. Some of them give encouraging agreement with merger rates derived from analytical, hierarchical clustering models, and both would favor the larger values of m. Another fundamental question that will be clari"ed by high redshift observations is whether the nature of galaxy collisions was di!erent when the universe was appreciably younger? We have seen in previous sections that there are di!erences in collisions occurring in di!erent environments. Since these environments have themselves evolved considerably, we might expect di!erences at high redshifts. Average galaxy properties have also evolved continuously. For example, in the distant past galaxies were more gas-rich, and collisions and mergers between them may have generated spectacular star formation and nuclear activity more often than at present. One phenomena that is very relevant to this discussion is the Butcher}Oemler e!ect, the presence of unusually high numbers of blue galaxies in clusters at moderately high redshifts (Butcher and Oemler, 1978, 1984; Couch et al., 1994; Dressler et al., 1994; Barger et al., 1996). Spectral line analyses led to the discovery that many of the blue Butcher and Oemler galaxies are the so-called E#A galaxies (see Gunn and Dressler, 1988). These are galaxies with a relatively blue color, but strong hydrogen absorption lines rather than emission lines, plus absorption lines indicative of an old `elliptical-likea stellar population. These objects are now known to consist of two dominant stellar populations, an old one, together with an aging starburst, of order 1}2;10 yr old. Strong Balmer absorption line spectra (A-type star) are now used as essentially the de"ning characteristic of a `post-starbursta population, as these intermediate age populations are called. (Interestingly, Liu and Kennicutt Jr. (1995) also "nd E#A spectral features in a `considerable fractiona of the objects in their recent study of merging galaxies. However, it is not clear that there is any connection between these objects and the high redshift E#A galaxies.) There is as yet no consensus on the cause of this e!ect. Explanations range from galaxy harrassment or mergers at times when cluster galaxies had more gas, to higher rates of spiral galaxy infall into younger clusters. Caldwell and Rose's (1997) recent study of galaxies with Butcher}Oemler type spectral characteristics in nearby clusters provides some support for the latter explanation. They "nd evidence of substructure in the clusters containing these galaxies, and suggest that the infall of a subcluster about 10 yr ago, could have triggered the star formation in the galaxies that presently have post-starburst spectra. It also seems very reasonable to expect that
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interactions induce more vigorous star formation and other e!ects in young, gas-rich galaxies. There is already some evidence for stronger SF responses in interactions at high redshifts (e.g., Butcher and Oemler, 1978, 1984; Lavery and Henry, 1986; Lavery et al., 1992; Burkey et al., 1994; Driver et al., 1995a,b; Koo et al., 1996). The `faint blue excess galaxiesa seen at redshifts z(1, are a related observational phenomenon. These objects are not found in clusters, and may include many dwarfs and low surface brightness galaxies. This has led to much discussion of the idea of two epochs of `galaxy formationa, one associated with these objects, and the second forming giant galaxies at larger redshifts (e.g., at z'2.0, see Babul and Rees, 1992; Driver et al., 1995a, 1996; Lilly et al., 1995; Gwyn and Hartwick, 1996). The high redshift objects are becoming increasingly accessible to observation, e.g., in the Hubble Deep Field (e.g. Gwyn and Hartwick, 1996; Lowenthal et al., 1997; Madau et al., 1996; Steidel et al., 1996; Van den Bergh et al., 1996). Thus, we expect rapid progress in this area in the next few years, and hopefully, much clari"cation of these issues.
10. Conclusions We have come a long way from Shapley's `rare freaksa to the recognition that collisions are a primary force driving the continuing evolution of galaxies. Most of this progress is the result of the work of the last 25}30 years, beginning with the seminal work of Toomre and Toomre. There are two basic reasons that collisions are such an important process in galaxy evolution. The "rst is that collisions can strongly a!ect the structure, dynamics, and SFRs of the galaxies involved. This point was vigorously argued by the Toomres and Eneev, Kozlov, and Sunyaev, and their models triggered an ever-growing interest in the "eld which continued up to the present. Evidence for the strong e!ects of collisions can be found in almost every section of this paper. In particular, the discovery of massive halos around galaxies, and the realization that these halos would provide very strong frictional braking, virtually guaranteed that most galaxy collisions (outside of clusters) would result in mergers, with global consequences, as discussed above. The second basic reason for the importance of collisional processes is that collisions are not the result of rare, chance encounters. Rather they are the inevitable result of the continuing, (hierarchical) growth of large-scale structure in the universe. That is, collisions are written into the initial conditions. So why did Shapley and other galaxy pioneers mistake the importance and frequency of galaxy collisions? The vast distances between galaxies, and the consequent view of them as isolated, unevolving island universes certainly were important factors. With no knowledge of the dark halos it would have been di$cult to imagine that collisions were anything but rare, though Baade's recognition that galaxies rarely traveled alone was an early hint to the contrary. The early results of Hubble's classi"cation project, which found that most galaxies could be "t into a simple scheme with only a few basic categories provided further evidence that the freaks were rare. The relatively short time needed to complete a merger, and the rapid relaxation of the remnant into a surprisingly normal-looking state, were also not realized. With extensive help from numerical simulations we now understand not only these basic points, but also many details of the processes that create the enormous variety of collisional forms, and drive the evolution of the colliding galaxies. Following Shapley, Zwicky, Schweizer and others,
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I have presented descriptions and simple classi"cations of many of these collisional forms in hopes of highlighting the connections between di!erent forms. Despite the vast morphological variety, collisional phenomenology derives largely from a few basic dynamical processes. The most extreme morphologies are the result of tidal torqueing, centrifugal bounce, and hydrodynamic splash. The "rst two can be reasonably approximated as the combination of an impulsive disturbance, and subsequent ballistic kinematics, which provides a simple conceptual picture. Yet, this approximation is only applicable to transient disturbances. On longer timescales, globally #uctuating gravitational "elds and resonant couplings are responsible for dynamical friction and violent relaxation, which drive dynamical heating and the merger process. (Though we recall that dynamical friction also has an impulsive aspect.) These e!ects have been demonstrated in detail by self-consistent numerical models, and analytic theory provides a "rm foundation for understanding them, as described above. Our understanding of the process of mass transfer in nonmerging collisions is not so well developed. However, in the case of tidal torqueing in #yby collisions there are at least some good rules of thumb. Unfortunately, in the case of hydrodynamic `splashesa it appears that the amount of mass transfer is a very sensitive function of the relative orientation of the disks, and the orbital inclination. Thus, this case is more complex. Our understanding of interaction induced SF is even less complete. We have learned a great deal in the last decade about the processes that organize and compress the gas on large scales. The most dramatic of these is the funneling of gas into the central regions of merging galaxies as a result of the redistribution of angular momentum and the increased gravity. Funneling provides the fuel and processes like gravitational instability and cloud crushing provide the match to light the largest bon"res in the universe, the super-starbursts. A signi"cant fraction of the stars in the universe, especially in early-type galaxies may be formed in such events. Interaction induced bars also funnel gas and foster wave and resonant ring star formation. In this case, the "reworks are less spectacular, but also longer-lived, and so, probably contribute signi"cantly to the net SF of the universe as well. The intensity of SF induced by transient spiral and ring waves is comparable to the bar case, but more short-lived. However, waves e!ect the outer disk gas reservoirs of late-type galaxies, and so, have an e!ect by stimulating star formation and metal production in new regions. There is still a great deal to learn about the processes by which vigorous star formation is actually initiated in large gas clouds. Waves and other transient structures can serve as useful tools in advancing our understanding in this area. The fact that for some time after an impulsive collision, these structures largely follow ballistic kinematics (except for shocks in the gas), provides us with a good deal of understanding about the environment in which SF occurs. Probably the most universal SF process is the local, time-dependent, gravitational instability. The range of unstable wavelengths and the duration of the instability are predictable in transient compression waves. Observations are also more easily interpreted in these environments, which have not experienced multiple cycles of disturbance. Nevertheless, clear confrontations between theory and observation still require the highest available resolution in both multiwaveband observations and numerical simulations. As discussed above, a number of other processes may also play important roles in interaction induced SF. For example, pressure-induced SF (cloud crushing) may be important in very dense environments. Processes that limit or inhibit SF may be equally important. Young star heat and momentum inputs may heat and disperse cool, dense star-forming environments. The starburst
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phenomenon may be the result of such self-limiting behavior, together with the generation of galactic winds and fountains by the heating processes. Isolated galaxy disks appear to be strongly self-regulated. If so, collisional systems are one of the few places to study the nonlinear, nonequilibrium expression of these processes. The phenomena of gas disk disruption and reformation in the smaller partners in disk-disk collisions, and of accretional inhibition of SF in the central regions of the larger partner, provide special examples of collisions as laboratories for the study of nonlinear gas dynamical and SF processes. The study of such cases only begun, but they may help explain how starbursts can be delayed in interacting systems. All of these dynamical and SF processes are important aspects of galaxy evolution, and thus, astrophysical cosmology. For example, an understanding of galaxy-wide SF processes at lowredshift would be of great help in answering questions about the e$ciency and duration of SF in the epoch of galaxy formation. Analogous statements apply to AGN activity. These questions bear not only on the observability of young galaxies, but also on the hard photon #ux available for heating and reionizing gas on the largest scales, and also on the question of the degree to which intergalactic gas is enriched with heavy elements at early times. At present, only quite tentative answers are available for questions about how galaxy collisions depend on environment and redshift. This includes questions of the relative frequency of galaxy collisions of di!erent types as a function of redshift. However, the prospects for progress on these questions in the coming decade look very good. A vigorous, multi-faceted array of investigations is already underway. The most sensitive and highest resolution instruments in space and on the ground are being used to study galaxy collisions in unprecented detail, and at unprecented distances. Better instruments and more powerful computers are on the horizon. It seems likely that the overall picture of galaxy formation and evolution will be worked out (observed?) in the next few decades. It's an exciting time in this "eld!
Acknowledgements I am very grateful for helpful comments from colleagues who read all or parts of this manuscript, including Phil Appleton, Mark Bransford, Vassilis Charmandaris, John Scalo, Dave Schramm, and Bev Smith. Thank goodness for the many things you found missing or just plain wrong (and sorry for any I still missed), to the anonymous referee who really took it apart, providing dozens of detailed and expert suggestions. To many `collisional collaboratorsa and friends } including, C. Heisler, J. Higdon, R. Lavery, P. Marcum, M. Noguchi, E. Skillman, J. van Gorkom, J. Wallin, and the `ocularsa (E. Brinks, B. & D. Elmegreen, M. Kaufman, M. Klaric, and M. Thomasson) } for providing so much enlightenment on speci"c aspects of this "eld. Some support for this work was provided by NSF grant AST 93-19596. I am also grateful to the individuals and publishing companies who provided permission to use "gures, and in many cases digital versions of those "gures and preprints, including: P. Appleton, E. Athanassoula, J.L. Higdon, R.A. Gerber, R.J. Lavery, I.F. Mirabel, P.J. Quinn, D.B. Sanders, B.J. Smith, A. Toomre, I.R. Walker, B. Whitmore, and M. Yun. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Lab., Caltech, under contract with the National Aeronautics and
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Space Administration. Images from NOAO and STSci are the copyright of the Association of Universities for Research in Astronomy Inc. (AURA), all rights reserved. This research has made use of NASA's Astrophysics Data System Abstract Service. I am especially grateful to my longtime collaborator Phil Appleton for providing all of the above, and much encouragement besides. And to my wife, Megan Fairall, for a lot of support over the course of this project. This paper is dedicated to the memory of David N. Schramm, who conceived of the project a number of years ago, patiently endured the author's slow progress, and provided encouragement through the many interruptions. I am also grateful to Roberta L. Bernstein, David's editorial assistant, for her help, and to M. Kamionkowski for taking over the editorial tasks.
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MICROSCOPIC STUDIES ON TWO-PHONON GIANT RESONANCES
C.A. BERTULANI , V.Yu. PONOMAREV Instituto de FnHsica, Universidade Federal do Rio de Janeiro, 21945-970 Rio de Janeiro, R J, Brazil Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980, Dubna, Russia
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
Physics Reports 321 (1999) 139}251
Microscopic studies on two-phonon giant resonances C.A. Bertulani *, V.Yu. Ponomarev Instituto de Fn& sica, Universidade Federal do Rio de Janeiro, 21945-970 Rio de Janeiro, RJ, Brazil Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980, Dubna, Russia Received April 1999; editor: G.E. Brown Contents 1. Introduction 2. Heavy ion excitation of giant resonances 2.1. Coulomb excitation at relativistic energies 2.2. Coulomb excitation at intermediate energies 2.3. Comparison of Coulomb excitation of GRs at low energies and at relativistic energies 2.4. Quantum description of Coulomb excitation at high energies 2.5. Singles spectra in Coulomb excitation of GDR 2.6. Excitation and photon decay of the GDR 2.7. Nuclear excitation and strong absorption 2.8. Nucleon removal in peripheral relativistic heavy ion collisions 2.9. Excitation by a deformed nucleus 3. Heavy ion excitation of multiphonon resonances 3.1. Introduction 3.2. Perturbation theory and harmonic models 3.3. General arguments on the width of the double-phonon state
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3.4. Coupled-channel calculations with inclusion of the GR widths 4. Description of one- and multi-phonon excited states within the quasiparticle-phonon model 4.1. The model Hamiltonian and phonons 4.2. Mixing between simple and complex con"gurations in wave functions of excited states 4.3. Comparison with other approaches 5. Physical properties of the double-giant resonances 5.1. One-step excitation of two-phonon states in the energy region of giant resonances 5.2. 1> component of the DGDR 5.3. Position, width and cross section of excitation in RHIC of the DGDR in Xe and Pb 5.4. The role of transitions between complex con"gurations of the GDR and the DGDR 5.5. The DGDR in deformed nuclei Acknowledgements References
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Abstract A new class of giant resonances in nuclei, namely double-giant resonances, is discussed. They are giant resonances built on top of other giant resonances. Investigation on their properties, together with similar
* Corresponding author. E-mail addresses:
[email protected] (C.A. Bertulani),
[email protected] (V.Yu. Ponomarev) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 3 8 - 1
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studies on low-lying two-phonon states, should give an answer on how far the harmonic picture of boson-type excitations holds in the "nite fermion systems like atomic nuclei. The main attention in this review is paid to double-giant dipole resonances (DGDR) which are observed in relativistic heavy ion collisions with very large cross sections. A great experimental and theoretical e!ort is underway to understand the reaction mechanism which leads to the excitation of these states in nuclei, as well as the better microscopic understanding of their properties. The Coulomb mechanism of the excitation of single- and double-giant resonances in heavy ion collision at di!erent projectile energies is discussed in detail. A contribution of the nuclear excitation to the total cross section of the reaction is also considered. The Coulomb excitation of double resonances is described within both, the second-order perturbation theory approach and in coupled-channels calculation. The properties of single and double resonances are considered within the phenomenologic harmonic vibrator model and microscopic quasiparticle-RPA approach. For the last we use the quasiparticle-phonon model (QPM) the basic ideas and formalism of which are presented. The QPM predictions of the DGDR properties (energy centroids, widths, strength distributions, anharmonicities and excitation cross sections) are compared to predictions of harmonic vibrator model, results of other microscopic calculations and experimental data available. 1999 Elsevier Science B.V. All rights reserved. PACS: 24.30.Cz; 25.70.De; 21.60.!n Keywords: Multi-phonon resonances; Coulomb excitation
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1. Introduction The phenomenon of a Giant Resonance (GR) in a nucleus is now known for more than 60 years. The "rst article on this subject was published in 1937 by Bothe and Gentner [1] who observed an unexpectedly large absorption of 17.6 MeV photons (from the Li(p, c) reaction) in some targets. They noticed that the cross section for Cu was surprisingly high and they suggested that this might be due to a resonance phenomenon. These observations were later con"rmed by Baldwin and Klaiber (1947) with photons from a betatron. In 1948 Goldhaber and Teller [2] interpreted these resonances (named by isovector giant dipole resonances) with a hydrodynamical model in which rigid proton and neutron #uids vibrate against each other, the restoring force resulting from the surface energy. Steinwendel and Jensen [3] later developed the model, considering compressible neutron and proton #uids vibrating in opposite phase in a common "xed sphere, the restoring force resulting from the volume symmetry energy. The standard microscopic basis for the description of giant resonances is the random phase approximation (RPA) in which giant resonances appear as coherent superpositions of one-particle one-hole (1p1h) excitations in closed shell nuclei or two quasi-particle excitations in open shell nuclei (for a review of these techniques, see, e.g., Ref. [4]). The isoscalar quadrupole resonances were discovered in inelastic electron scattering by Pitthan and Walcher (1971) and in proton scattering by Lewis and Bertrand [5]. Giant monopole resonances were found later and their properties are closely related to the compression modulus of nuclear matter. Following these, other resonances of higher multipolarities and giant magnetic resonances were investigated. Typical probes for giant resonance studies are (a) c's and electrons for the excitation of GDR (isovector giant dipole resonance), (b) a-particles and electrons for the excitation of isoscalar GMR (giant monopole resonance) and GQR (giant quadrupole resonance), and (c) (p, n), or (He, t), for Gamow}Teller resonances, respectively. Relativistic coulomb excitation (RCE) is a well-established tool to unravel interesting aspects of nuclear structure [6]. Examples are the studies of multiphonon resonances in the SIS accelerator at the GSI facility, in Darmstadt, Germany [7}9]. Important properties of nuclei far from stability [10,11] have also been studied with this method. Inelastic scattering studies with heavy ion beams have opened new possibilities in the "eld (for a review the experimental developments, see Refs. [7,9]). A striking feature was observed when either the beam energy was increased, or heavier projectiles were used, or both [12]. This is displayed in Fig. 1, where the excitation of the GDR in Pb was observed in the inelastic scattering of O at 22A and 84A MeV, respectively, and Ar at 95A MeV [13,14]. What one clearly sees is that the `bumpa corresponding to the GDR at 13.5 MeV is appreciably enhanced. This feature is solely due to one agent: the electromagnetic interaction between the nuclei. This interaction is more e!ective at higher energies, and for increasing charge of the projectile. Baur and Bertulani showed in Ref. [15] that the excitation probabilities of the GDR in heavy ion collisions approach unity at grazing impact parameters. They also showed that, if double GDR resonance (i.e. a GDR excited on a GDR state) exists then the cross sections for their excitation in heavy ion collisions at relativistic energies are of order of a few hundred of milibarns. These calculations motivated experimentalists at the GSI [7,9] and elsewhere [13,14] to look for the signatures of the DGDR in the laboratory. This has by now become a very active "eld in nuclear physics with a great theoretical and experimental interest [7}9].
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Fig. 1. Experimental cross sections in arbitrary units for the excitation of Pb targets by O (22A and 84A MeV) and by Ar (95A MeV), as a function of the excitation energy.
In the "rst part of this review, we study the reaction mechanism in RCE of giant resonances in several collisions between heavy ions. In Section 2 we start with the description of the semiclassical theory for relativistic Coulomb excitation, then we consider the e!ects of recoil and later we describe the fully quantum mechanical approach. The role of nuclear excitation in relativistic heavy ion collisions is also discussed here. We demonstrate that the experimental data on the excitation and decay of single-giant resonances are well described by these formalisms. In Section 3 the process of the excitation of multi-phonon resonances in relativistic heavy ion collisions is considered within the second-order perturbation theory and in coupled-channels calculations. Giant resonances are treated in this section within the phenomenologic harmonic vibrator model. Some general arguments for the width of multi-phonon resonances are discussed here as well as an in#uence of giant resonances width on the total cross section of their excitation. A good part of this report (Sections 4 and 5) is dedicated to a review of the microscopic properties on the giant resonances in the quasiparticle-phonon model. In Section 4 we present the main ideas and formalism of this model. The particle}hole modes of nuclear excitation are projected into the space of quasi-bosons, phonons, and matrix elements of interaction between one- and multi-phonon con"gurations are calculated on microscopic footing within this approach. In Section 5, we use this model as a basis for a detailed investigation of the interplay between excitation mechanisms and the nuclear structure in the excitation of the DGDR. Di!erent aspects related to the physical properties of the DGDR in heavy nuclei (energy centroids, widths, strength distributions, anharmonicities and excitation cross sections) as predicted by microscopic studies are discussed in this section and compared to experimental data.
2. Heavy ion excitation of giant resonances 2.1. Coulomb excitation at relativistic energies In relativistic heavy ion collisions, the wavelength associated to the projectile-target separation is much smaller than the characteristic lengths of system. It is, therefore, a reasonable approximation
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to treat r as a classical variable r(t), given at each instant by the trajectory followed by the relative motion. At high energies, is also a good approximation to replace this trajectory by a straight line. The intrinsic dynamics can then be handled as a quantum mechanics problem with a timedependent Hamiltonian. This treatment is discussed in full details by Alder and Winther in Refs. [16}18]. We will describe next the formalism developed by Bertulani et al. [19], which explicitly gives the time dependence of the multipole "elds, useful for a coupled-channels calculation. The intrinsic state "t(t)2 satis"es the SchroK dinger equation [h#<(r(t))]"t(t)2"i R"t(t)2/Rt .
(1)
where h is the intrinsic Hamiltonian and < is the channel-coupling interaction. Expanding the wave function in the set +"m2; m"0, N, of eigenstates of h, where N is the number of excited states included in the coupled-channels problem, we obtain , "t(t)2" a (t)"m2 exp(!iE t/ ) , (2) K K K where E is the energy of the state "m2. Taking scalar product with each of the states 1n", we get the K set of coupled equations , (3) i a (t)" 1n"<"m2e #L\#KR a (t), n"0}N . K L K It should be remarked that the amplitudes depend also on the impact parameter b specifying the classical trajectory followed by the system. For the sake of keeping the notation simple, we do not indicate this dependence explicitly. We write, therefore, a (t) instead of a (b, t). Since the interaction L L < vanishes as tP$R, the amplitudes have as initial condition a (tP!R)"d(n, 0) and they L tend to constant values as tPR. Therefore, the excitation probability of an intrinsic state "n2 in a collision with impact parameter b is given as P (b)""a (R)" . (4) L L The total cross section for excitation of the state "n2 can be approximated by the classical expression
p "2p P (b)b db . L L
(5)
Since we are interested in the excitation of speci"c nuclear states, with good angular momentum and parity quantum numbers, it is appropriate to develop the time-dependent coupling interaction <(t) into multipoles. In Ref. [18], a multipole expansion of the electromagnetic excitation amplitudes in relativistic heavy ion collisions was carried out. This work used "rst-order perturbation theory and the semiclassical approximation. The time dependence of the multipole interactions was not explicitly given. This was accomplished in Ref. [19], which we describe next. We consider a nucleus 2 which is at rest and a relativistic nucleus 1 which moves along the z-axis and is excited from the initial state "I M 2 to the state "I M 2 by the electromagnetic "eld of nucG G D D leus 1. The nuclear states are speci"ed by the spin quantum numbers I , I and by the correspondG D ing magnetic quantum numbers M , M , respectively. We assume that the relativistic nucleus 1 G D
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moves along a straight-line trajectory with impact parameter b, which is therefore also the distance of the closest approach between the center of mass of the two nuclei at the time t"0. We shall consider the situation where b is larger than the sum of the two nuclear radii, such that the charge distributions of the two nuclei do not strongly overlap at any time. The electromagnetic "eld of the nucleus 1 in the reference frame of nucleus 2 is given by the usual Lorentz transformation [20] of the scalar potential (r)"Z e/"r", i.e.,
(r , t)"c [b !b, c(z !vt)] , A(r , t)"(*/c)c [b !b, c(z !vt)] . (6) Here b (impact parameter) and b are the components of the radius-vectors r and r transverse to *. The time-dependent matrix element for electromagnetic excitation is of the form < (t)"1I M "[o(r )!(*/c) ) J(r )] (r , t)"I M 2 , DG D D G G where o (J) is the nuclear transition density (current). A Taylor-series expansion of the LieH nard}Wiechert potential around r "0 yields
(r , t)"c [r(t)]#c
[r(t)] ) r #2 ,
(7)
(8)
where r"(b, cvt), and the following simplifying notation is used:
[r], (r , t)"r "! b (r)!(R/R(vt)) (r)z( "! b (r)!(*/v)(R/Rt) (r) . Y
(9)
Thus, < (t)"1I M "[o(r )!(*/c) ) J(r )] [c (r)#cr )
(r)]"I M 2 . DG D D G G Using the continuity equation
) J"!iuo ,
(11)
where u"(E !E )/ , and integrating by parts, D G
* ! [c (r)#cr )
(r)]"I M < (t)" I M " J(r) ) D D G G DG iu c
(10)
.
(12)
In spherical coordinates (4p a r >H , r )
" I I 3 I\ where a "e( )
, I I and e( are the spherical unit vectors I e( "G(1/(2)(e( $e( ), e( "e( . 6 7 8 ! We will use the relations and
*/c"(v/c)e( "(v/c)((4p/3) (r>H )
;L(rI> )"i(k#1) (rI> ) JK JK where L"!ir; .
(13)
(14)
(15) (16)
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Then, one can write J)
*
! [c #cr )
] iu c
v a I (r > )!
(r >H ) I c iu I\ The "rst term in the above equation can be rewritten as "!cJ )
* 4p (
) r )! c 3
.
(17)
J)
* v (r )
)" J ) [e( (r )
)#(r ) e( )
]#(v/2c)J ) [e( (r )
)!(r ) e( )
] . c 2c
(18)
The "rst term in this equation is symmetric under parity inversion, and contributes to the electric quadrupole (E2) excitation amplitudes, since (v/2c)J ) [e( (r )
)#(r ) e( )
]"(v/2c)J ) [z (r )
)] . (19) The second term in Eq. (18) is antisymmetric in J and r , and leads to magnetic dipole (M1) excitations. Indeed, using Eqs. (13)}(16), one "nds
4p a (!1)IL(r> ) . (20) I \I 3 I\ Thus, only the last two terms on the right-hand side of Eq. (17) contribute to the electric dipole (E1) excitations. Inserting them into Eq. (12), we get v v J ) [e( (r )
)!(r ) e( )
]" J) 2c 2c
<#(t)"c DG where
4p (!1)Ib 1I M "M(E1,!k)"I M 2 , I D D G G 3 I\
(21)
i M(E1,!k)" dr J(r) ) (r> )" d r o(r)r> (r) I I u and
(22)
b "!a "!(
) e( )"e( ) R /Rb, ! I I I b "!a !i(uv/c) . The derivatives of the potential are explicitly given by
(23)
Z e R
, bV "r "!x( b , Y V[b#cvt] Rb V Z e .
"r "!z( cvt X Y [b#cvt]
(24)
Using the results above, we get for the electric dipole potential
<#(t)" DG
2p uv c E (q)[M (E1,!1)!M (E1, 1)]#(2c qE (q)!i E (q) M (E1, 0) , DG DG DG 3 cc (25)
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where q"cv/b, and E (q)"Z e/b[1#q] and E (q)"Z e/b[1#q]
(26)
are proportional to the transverse and longitudinal electric "elds generated by the relativistic nucleus with charge Z e, respectively. From the de"nition
i M (M1, k)"! dr J(r) ) L(r> ) , DG I 2c
(27)
and Eq. (19), we "nd <+(t)"i DG
2p v cE (q)[M (M1, 1)#M (M1,!1)] . DG DG 3 c
(28)
The current J in Eq. (27) is made up of the usual convective part and a magnetization part, proportional to the intrinsic (Dirac and anomalous) magnetic moment of the nucleons. To obtain the electric quadrupole (E2) potential we use the third term in the Taylor expansion of Eq. (8). Using the continuity equation, a part of this term will contribute to E3 and M2 excitations, which we neglect. We then "nd that
p c 3E (q)[M (E2, 2)#M (E2,!2)] <#(q)"! DG DG DG 30
uv #c 6qE (q)!i E (q) [M (E2,!1)#M (E2, 1)] DG DG cc
uv #(6 c (2q!1)E (q)!i qE (q) M (E2, 0) , DG cc
(29)
where E (q) is the quadrupole electric "eld of nucleus 1, given by E (q)"Z e/b[1#q] .
(30)
The "elds E (q) peak around q"0, and decrease fastly within an interval *qK1. This corresponds to a collisional time *tKb/cv. This means that, numerically one needs to integrate the coupledchannels equations (Eq. (3)) only in a time interval within a range n;*q around q"0, with n equal to a small integer number. A computer code for coupled channels calculations of relativistic Coulomb excitation using the formalism presented in this section is available in the literature [121]. 2.1.1. First-order perturbation theory In most cases, the "rst-order perturbation theory is a good approximation to calculate the amplitudes for relativistic Coulomb excitation. It amounts to using a "d on the right-hand side I I
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of Eq. (3). The time integrals can be evaluated analytically for the < (t) perturbations, given by # Eqs. (25), (28), and (29). The result is a#"!i
8p Z e (2 m K (m)[M (E1,!1)!M (E1, 1)]#i K (m)M (E1, 0) , DG DG DG c 3 vb
(31)
where K (K ) is the modi"ed Bessel function of "rst (second) degree, and m"ub/cv. For the E2 and M1 multipolarities, we obtain respectively, a#"2i
p Z e m+K (m)[M (E2,2)#M (E2,!2)] DG DG 30 c vb
v #ic 2! K (m)[M (E2,!1)#M (E2,1)] DG DG c !(6K (m)M (E2,0), DG
(32)
(33)
and a+"
8p Z e mK (m)[M (M1, 1)!M (M1,!1)] . DG DG 3 cb
These expressions are the same as those obtained from the formulae deduced in Ref. [18]. We note that the multipole decomposition developed by those authors is accomplished by a di!erent approach, i.e., using recurrence relations for the Gegenbauer polynomials, after the integral on time is performed. Therefore, the above results present a good check for the time dependence of the multipole "elds deduced here. The formulas above have been derived under the assumption of the long-wavelength approximation. When this approximation is not valid the matrix elements given by Eqs. (22) and (27) are to be replaced by the non-approximated matrix elements for electromagnetic excitations [16], i.e.,
(2j#1)!! J(r) ) ;L[ j (ir)> (r( )] dr , M(Ej, k)" H HI iH>c(j#1)
(34)
(35)
M(Mj, k)"!i
(2j#1)!! J(r) ) L[ j (ir)> (r( )] dr H HI iHc(j#1)
for electric and magnetic excitations (i"u/c), respectively. However, the other factors do not change (see, e.g., [18]). 2.1.2. Excitation probabilities and virtual photon numbers The square modulus of Eqs. (31)}(33) gives the probability of exciting the target nucleus from the initial state "I M 2 to the "nal state "I M 2 in a collision with impact parameter b. If the orientation G G D D of the initial state is not speci"ed, the probability for exciting the nuclear state of energy E and spin D I is D 1 P " "a " . (36) G D 2I #1 DG G +G +D
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Integration of (36) over all energy transfers e" u, and summation over all possible "nal states of the projectile nucleus (making use of the Wigner}Eckart theorem and the orthogonality of the properties of the Clebsch}Gordan coe$cients) leads to the Coulomb excitation probability in a collision with impact parameter b:
P " P (b)o (e) de G D D ! D where o (e) is the density of "nal states of the target with energy E "E #e. D D G Inserting (31)}(33) into (37) one "nds
(37)
de n (b, e)ppH(e) , (38) P (b, e)" P (b, e)" A ! pH e pH pH pH where (2p)(j#1) o (e)iH\B(pj, I PI ) (39) ppH(e)" D G D A j[(2j#1)!!] D are the photonuclear absorption cross sections for a given multipolarity pj. The total photonuclear cross section is a sum of all these multipolarities, p " ppH(e) . A A pH The functions n (e) are called the virtual photon numbers, and are given by pH 1 Za m c K# K , n (b, e)" c # p b v
Za c 4 [K#mK K #mK]#m(2!v/c)K n (b, e)" # c pb v
(40)
(41) (42)
and n
Za m (b, e)" K , + p b
(43)
where all modi"ed Bessel functions K are functions of m(b)"ub/cv. I Since all nuclear excitation dynamics is contained in the photoabsorption cross section, the virtual photon numbers (41)}(43) do not depend on the nuclear structure. They are kinematical factors, depending on the orbital motion. They may be interpreted as the number of equivalent (virtual) photons that hit the target per unit area. These expressions show that Coulomb excitation probabilities are exactly directly proportional to the photonuclear cross sections, although the exchanged photons are o!-shell. This arises from the condition that the reaction is peripheral and the nuclear charge distributions of each nuclei do not overlap during the collision. This result can be proved from "rst principles, and has been shown in some textbooks (see, e.g., [21]). The usefulness of Coulomb excitation, even in "rst-order processes, is displayed in Eq. (38). The "eld of a real photon contains all multipolarities with the same weight and the photonuclear cross
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section (40) is a mixture of the contributions from all multipolarities, although only a few contribute in most processes. In the case of Coulomb excitation the total cross section is weighted by kinematical factors which are di!erent for each projectile or bombarding energy. This allows one to disentangle the multipolarities when several ones are involved in the excitation process, except for the very high bombarding energies c<1 for which all virtual photon numbers can be shown to be the same [22]. 2.1.3. Cross sections and total virtual photon numbers The cross section is obtained by the impact parameter integral of the excitation probabilities. Eq. (38) shows that we only need to integrate the number of virtual photons over impact parameter. One has to introduce a minimum impact parameter b in the integration. Impact parameters smaller than b are dominated by nuclear fragmentation processes. One "nds de dp " p " N (e)ppH(e) , (44) ! pH e pH A pH pH where the total virtual photon numbers N (e)"2pdb bn (b, e) are given analytically by pH LH vm 2Za c mK K ! (K!K) , (45) N (e)" # 2c p v
2Za c v v N (e)" K K 2 1! K#m 1! # p v c c mv # (K!K)#m(2!v/c)K 2c
(46)
and
m 2Za m mK K ! (K!K) , N (e)" + 2 p b
(47)
where all K 's are now functions of m(b)"ub /cv. I 2.2. Coulomb excitation at intermediate energies 2.2.1. Classical trajectory: recoil and retardation corrections The semiclassical theory of Coulomb excitation in low-energy collisions accounts for the Rutherford bending of the trajectory, but relativistic retardation e!ects are neglected [17]. On the other hand, in the theory of relativistic Coulomb excitation [18], recoil e!ects on the trajectory are neglected (one assumes straight-line motion) but retardation is handled correctly. In fact, the onset of retardation brings new important e!ects such as the steady increase of the excitation cross sections with bombarding energy. In a heavy ion collision around 100A MeV, the Lorentz factor c is about 1.1. Since this factor enters the excitation cross sections in many ways, like in the adiabaticity parameter m(R)"u R/cv , DG
(48)
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one expects that some sizable modi"cations in the theory of relativistic Coulomb excitation should occur [23]. Recoil corrections are not negligible either, and the relativistic calculations based on the straight-line parametrization should not be completely appropriate to describe the excitation probabilities and cross sections. The Coulomb recoil in a single collision is of the order of a "Z Z e/m v , (49) which is half-distance of closest approach in a head-on collision, with m equal to the reduced mass of the colliding nuclei. Although this recoil is small for intermediate energy collisions, the excitation probabilities are quite sensitive to it. This is important for example in the excitation of giant resonances because the adiabaticity parameter is of the order of one (see, Eq. (48)). When m(b);1, the excitation probabilities depends on b approximately like 1/b, while when m(b) becomes greater than one they decrease approximately as e\pK@/b. Therefore, when mK1, a slight change of b may vary appreciably the excitation probabilities. In the semiclassical theory of Coulomb excitation the nuclei are assumed to follow classical trajectories and the excitation probabilities are calculated in time-dependent perturbation theory. At low energies one assumes Rutherford trajectories for the relative motion while at relativistic energies one assumes straight-line motion. In intermediate energy collisions, where one wants to account for recoil and retardation simultaneously, one should solve the general classical problem of the motion of two relativistic charged particles. But, even if radiation is neglected, this problem can only be solved if one particle has in"nite mass [24]. This approximation should be su$cient if we take, e.g., the collision O#Pb as our system. An improved solution may be obtained by use of the reduced mass, as we show next, in a formalism developed by Aleixo and Bertulani [23]. In the classical one-body problem, one starts with the relativistic Lagrangian L"!m c+1!(1/c) (r #r Q ), !Z Z e/r , (50) where r and Q are the radial and the angular velocity of the particle, respectively (see Fig. 2). Using the Euler}Lagrange equations one "nds three kinds of solutions, depending on the sign of the charges and the angular momentum in the collision. In the case of our interest, the appropriate solution relating the collisional angle and the distance r between the nuclei is [24] 1/r"A[e cos (= )!1]
(51)
where ="[1!(Z Z e/c¸ )] , (52) A"Z Z eE/c¸= , (53) e"(c¸ /Z Z eE)[E!mc#(m cZ Z e/¸ )] . (54) E is the total bombarding energy in MeV, m is the mass of the particle and ¸ its angular momentum. In terms of the Lorentz factor c and of the impact parameter b, E"cm c and ¸ "cm vb. The above solution is valid if ¸ 'Z Z e/c. In heavy ion collisions at intermediate energies one has ¸
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Fig. 2. A nuclear target is Coulomb excited by a fast moving projectile. The coordinates used in text are shown.
also easy to show that, from the magnitudes of the parameters involved in heavy ion collisions at intermediate energies, the trajectory (51) can be very well described by approximating (55) A"a /cb, e"((bc/a)#1 , where a is half the distance of closest approach in a head on collision (if the nuclei were pointlike and if non-relativistic kinematics were used), and e is the eccentricity parameter. In the approximation (55) e is related to the de#ection angle 0 by e"(a /c) cot 0. The time dependence for a particle moving along the trajectory (51) may be directly obtained by solving the equation of angular momentum conservation. Introducing the parametrization ="1,
r(s)"(a /c)[e cosh s#1] we "nd
(56)
t"(a /cv)[s#e sinh s] . (57) Using the scattering plane perpendicular to the Z-axis, one "nds that the corresponding components of r may be written as x"a[cosh s#e] ,
(58)
y"a (e!1 sinh s ,
(59)
z"0 , (60) where a"a /c. This parametrization is of the same form as commonly used in the non-relativistic case [17], except that a is substituted by a /c,a. In the limit of straight-line motion eKb/a<1, and the equations above reduce to the simple parametrization y"vt, x"b and z"0 .
(61)
As we quoted before, the classical solution for the relative motion of two relativistic charges interacting electromagnetically can only be solved analytically if one of the particles has in"nite mass. Non-relativistically the two-body problem is solvable by introduction of center of mass and relative motion coordinates. Then, the result is equivalent to that of a particle with reduced mass m "m m /(m #m ) under the action of the same potential. The particle with reduced mass m is . 2 . 2 lighter than those with mass m and m , and this accounts for the simultaneous recoil of them. An . 2
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153
exact relativistic solution should reproduce this behavior as the relative motion energy is lowered. We shall use the reduced mass de"nition of m as usual in the parametrization of the classical trajectory of Coulomb excitation in intermediate energy collisions, as outlined above. In a O#Pb collision this is not a too serious approximation. For heavier systems like U#U it would be the simplest way to overcome this di$culty. But, as energy increases, this approximation is again unimportant since the trajectories will be straight lines parametrized by an impact parameter b. A more exact result can be obtained numerically using the Darwin Lagrangian to determine the classical trajectory in collisions at intermediate energies [25]. But, the parametrization of the classical trajectory as given by Eqs. (58)}(60) with a reduced mass particle, besides reproducing both the non-relativistic and the relativistic energies, gives a reasonable solution to the kind of collisions we want to study. 2.2.2. Excitation amplitudes Including retardation, the amplitude for Coulomb excitation of a target from the initial state "i2 to the "nal state " f 2 is given in "rst-order time-dependent perturbation theory by 1 a " DG i
1 o (r) (u, r)# J (r) ) A(u, r) dr DG c DG
(62)
where o (J ) is the nuclear transition density (current) and DG DG e G r!r R
(u, r)"Z e dt , (63) e SR "r!r (t)" \ e G r!r R Z e dt (64) * (t) e UR A(u, r)" "r!r (t)" c \ are the retarded potentials generated by a projectile with charge Z following a Coulomb trajectory, and i"u/c. When the magnitude of the amplitudes (62) is small compared to unity, the use of "rst-order perturbation theory is justi"ed. We now use the expansion
e G r!r "4pii j (ir )>H (r( )h (ir )> (r( ) , (65) H HI H HI "r!r " HI where j (h ) denotes the spherical Bessel (Hankel) functions (of "rst kind), r (r ) refers to H H whichever of r and r has the larger (smaller) magnitude. Assuming that the projectile does not penetrate the target, we use r (r ) for the projectile (target) coordinates. At collision energies above the Coulomb barrier this assumption only applies for impact parameters larger than a certain minimum, below which the nuclei penetrate each other. Using the continuity equation (11) for the nuclear transition current (we changed the notation: o,o , J,j ), we can show that the DG DG expansion (65) can be expressed in terms of spherical tensors (see, e.g., Ref. [21, Vol. II]) and Eq. (62) becomes 4p Z e (!1)I+S(Ej, k)M (Ej,!k)#S(Mj, k)M (Mj,!k), , (66) a " DG DG DG i 2j#1 HI where M(pj, k) are the matrix elements for electromagnetic transitions, as de"ned in (34) and (35).
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The orbital integrals S(pj, k) are given by
iiH> R S(Ej, k)"! +r (t)h [ir (t)],> [h (t), (t)] e SR dt H HI j(2j!1)!! Rr \ iH> * (t) ) r (t)h [ir (t)] > [h (t), (t)] e SR dt ! H HI cj(2j!1)!! \
(67)
and
iH> i
+h [ir (t)] > [h (t), (t)], e SR dt , S(Mj, k)"! L ) H HI cm c j(2j!1)!! \ where L is the angular momentum of relative motion, which is constant: ¸ "cam v cot 0/2 with 0 equal to the (center-of-mass) scattering angle. In non-relativistic collisions ir "ur /c"(v/c)(ur /v)((v/c);1
(68)
(69)
(70)
because when the relative distance r obeys the relations ur /v51 the interaction becomes adiabatic. Then one uses the limiting form of h for small values of its argument [26] to show that H S,0(Ej, k)K r \H\(t) > +h (t), (t), e SR dt (71) HI \ and 1 L S,0(Mj, k)K!
+r \H\(t) > [h (t), (t)], e SR dt (72) HI jm c \ which are the usual orbital integrals in the non-relativistic Coulomb excitation theory with hyperbolic trajectories (see Ref. [17, Eqs. (II.A.43)]). In the intermediate energy case the relation (69) is partially relaxed (of course, for relativistic energies, v&c, it is not valid) and one has to keep the more complex forms (67), (68) for the orbital integrals. Using the Z-axis perpendicular to the trajectory plane, the recursion relations for the spherical Hankel functions and the identity
* ) r"ds/dt dr/ds ) r"aev sinh s ,
(73)
we can rewrite the orbital integrals, in terms of the parametrization (58)}(60), as
iiHg (e#cosh s#i(e!1 sinhs)I C S(Ej, k)"! ds eGEC Q>Q (e cosh s#1)I\ cj(2j!1)!! HI \ v vg ge sinh s ) h , ; (j#1) h ! (e cosh s#1) h #i H H H> c c
(74)
C.A. Bertulani, V.Yu. Ponomarev / Physics Reports 321 (1999) 139}251
where
C " HI
155
2j#1 ((j!k)!(j#k)! (!1)H>I for j#k"even , 4p (j!k)!!(j#k)!!
(75)
for j#k"odd
0
and g"ua/v"ua /cv , and with all h 's as functions of (v/c)g (e cosh s#1). H For convenience, we de"ne
(76)
I(Ej, k)"(vaH/C )S(E , k) (77) HI H and we translate the path of integration by an amount ip/2 to avoid strong oscillations of the integral. We "nd,
vg H> 1 e\pE ds e\EC Q e EQ c j(2j!1)!! v (e#i sinh s!(e!1 cosh s)I (j#1)h !zh ! eg cosh s ) h , (78) ; H H> H (ie sinh s#1)I\ c
I(Ej, k)"!i
where all h 's are now functions of H z"(v/c)g(ie sinh s#1) .
(79)
In the case of magnetic excitations, one may explore the fact that L is perpendicular to the scattering plane to show that
av p 0 2j#1 1 L ) h (ir)> , "c cot ((j#1)!kC e I(h (ir) . H HI 2 H> I H r 2 2j#3 m The magnetic orbital integrals become S(Mj, k)"!ia
(80)
2j#1 v iH> ((j#1)!k c j(2j!1)!! 2j#3
0 1 h [ir (t)] e I(YRe SR dt . ;C cot H H> I r (t) 2 \
(81)
De"ning jcaHS(Mj, k) I(Mj, k)"! +[(2j#1)/(2j#3)][(j#1)!k],\ C cot 0/2 H> I we obtain, using the parametrization (58)}(60), and translating the integral path by ip/2,
i(vg/c)H> (e#i sinh s!(e!1 cosh s)I I(Mj, k)" ds h (z) e\EC Q e EQ e\pE . H (2j!1)!! (ie sinh s#1)I \
(82)
(83)
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Generally, the most important magnetic excitation has M1 multipolarity. The orbital integrals (78), (83) can only be solved numerically. 2.2.3. Cross sections and equivalent photon numbers In the high-energy limit the classical trajectory reduces to a straight line. One can show that using the approximation e"b/a<1, the orbital integrals (78) and (83) can be expressed in terms of simple analytical functions. However, it is instructive and useful to deduce the excitation amplitudes from the "rst principles again. The square modulus of Eq. (66) gives the probability of exciting the target nucleus from the initial state "I M 2 to the "nal state "I M 2 in a collision with c.m. scattering angle 0. If the G G D D orientation of the initial state is not speci"ed, the cross section for exciting the nuclear state of spin I is D 1 ae "a " dX , (84) dp " DG G D 4 2I #1 G +G +D where ae dX/4 is the elastic (Rutherford) cross section. Using the Wigner}Eckart theorem and the orthogonality properties of the Clebsch}Gordan coe$cients, one can show that dp 4pZe B(nj, I PI ) G D" ae G D "S(nj, k)" , (85) dX
(2j#1) HI where p"E or M stands for the electric or magnetic multipolarity, and the reduced transition probability is given by 1 B(nj; I PI )" "1I M "M(nj, k)" I M 2" G D G G D D 2I #1 G +G+D 1 " "1I ""M(nj)""I 2" . (86) D 2I #1 G G Integration of (85) over all energy transfers e" u, and summation over all possible "nal states of the projectile nucleus leads to
dp dp !" G Do (e) de , (87) dX dX D D where o (e) is the density of "nal states of the target with energy E "E #e. Inserting (85) into (87) D D G one "nds
dp dp de dn !" LH" LH (e)pLH(e) , (88) A dX dX e dX LH LH where pLH are the photonuclear absorption cross sections for a given multipolarity nj. The virtual A photon numbers, n (e), are given by LH dn Za j[(2j#1)!!] cae LH" "S(nj, k)" . (89) dX 2p (j#1)(2j#1) iH\ I
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In terms of the orbital integrals I(Ej, k), given by (79), and using Eq. (89), we "nd for the electric multipolarities
(j!k)!(j#k)! Za c H j[(2j#1)!!] dn #H" eg\H> "I(Ej, k)" . [(j!k)!!(j#k)!!] (j#1)(2j#1) 8p v dX I H>I In the case of magnetic excitations we "nd
(90)
dn Za c H\ [(2j#1)!!] +H" g\H>e(e!1) j(j#1)(2j#1) 8p v dX [(j#1)!k](j#1!k)!(j#1#k)! "I(Mj, k)" . (91) [(j#1!k)!!(j#1#k)!!] I H>I Only for the E1 multipolarity the integrals can be performed analytically and we get the closed expression ;
1 e!1 Za c dn #" efe\pD [K (ef)]#[K (ef)] , D D c e 4p v dX
(92)
where e"1/sin(h/2), a"1/137, f"ua /cv, a "Z Z e/2E , K is the modi"ed Bessel function * D with imaginary index, K is the derivative with respect to its argument. Since the impact parameter D is related to the scattering angle by b"a cot 0/2, we can also write 4 dn dn LH n (e, b), LH " LH 2pbdb ae dX
(93)
which are interpreted as the number of equivalent photons of energy e" u, incident on the target per unit area, in a collision with impact parameter b, in analogy with the results obtained in Section 2.1.2. Again we stress the usefulness of the concept of virtual photon numbers, especially in relativistic collisions. In these collisions the momentum and the energy transfer due to the Coulomb interaction are related by *p"*E/vK*E/c. This means that the virtual photons are almost real. One usually explores this fact to extract information about real photon processes from the reactions induced by relativistic charges, and vice versa. This is the basis of the WeizsaK cker} Williams method, commonly used to calculate cross sections for Coulomb excitation, particle production, Bremsstrahlung, etc. (see, e.g., Ref. [6]). In the case of Coulomb excitation, even at low energies, although the equivalent photon numbers should not be interpreted as (almost) real ones, the cross sections can still be written as a product of them and the cross sections induced by real photons, as we have shown above. 2.3. Comparison of Coulomb excitation of GRs at low energies and at relativistic energies Inserting the non-relativistic orbital integrals into Eq. (89), we get the following relation for the non-relativistic equivalent photon numbers (NR):
c H>Bdf j[(2j#1)!!] dn,0 LH(0, f) , LH "Za f\H> (2p)(j#1) v dX dX
(94)
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Fig. 3. Electric dipole number of equivalent photons per unit area db,2pb db, with energy of 10 MeV, incident on Pb in a collision with O at impact parameter b"15 fm, and as a function of the bombarding energy in MeV per nucleon. The dotted line and the dashed line correspond to calculations performed with the non-relativistic and with the relativistic approaches, respectively. The solid line represents a more correct calculation, as described in the text. Fig. 4. Same as Fig. 3, but for the E2 multipolarity.
where d"0 for electric, and d"!1 for magnetic multipolarities, and f"ua /v. The non relativistic Coulomb excitation functions f (0, f) are very well known and, e.g., are tabulated in LH Ref. [17], or maybe calculated numerically. Using Eqs. (90)}(92), we make an analysis of Coulomb excitation extending from low- to high-energy collisions. As an example, we study the excitations induced by O in O#Pb collisions. Since expression (89) is quite general, valid for all energies, under the assumption that the nuclei do not overlap, the equivalent photon numbers contain all information about the di!erences among the low- and the high-energy scattering. In Figs. 3}5 we show dn , for the E1 (Fig. 3), LH C E2 (Fig. 4), and M1 (Fig. 5) multipolarities, and for the collision O#Pb with an impact parameter b"15 fm. They are the equivalent photon numbers with frequency u"10 MeV/
incident on Pb. The dotted lines are obtained by using the non-relativistic equation (94), while the dashed lines correspond to the relativistic expressions (41)}(43). One observes that the relativistic expressions overestimate the equivalent photon numbers at low energies, while the non-relativistic expressions underestimate them at high energies. The most correct values are given by the solid lines, calculated according to Eqs. (90) and (91). They reproduce the low- and the high-energy limits, giving an improved interpolation between these limits at intermediate energies. These discrepancies are more apparent for the E1 and the E2 multipolarities. In the energy interval around 100A MeV neither the low-energy theory nor the high-energy one can reproduce well the correct values. This energy interval is indeed very sensitive to the e!ects of retardation and of Coulomb recoil. At these bombarding energies, the di!erences between the magnitude of the non-relativistic and the correct relativistic virtual photon numbers are kept at a constant value, of about 20%, for excitation energies e" u(10 MeV. However, they increase sharply when one reaches the
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Fig. 5. Same as Fig. 3, but for the M1 multipolarity. Fig. 6. Equivalent photon numbers per unit area incident on Pb, in a collision with O at 100A MeV and with impact parameter b"15 fm, as a function of the photon energy u. The curves for the E1, E2 and M1 multipolarities are shown.
excitation energy of e" u'10 MeV. The reason is that, for such excitation energies, the adiabaticity factor becomes greater than unity (m'1). This means that excitation energies of order of 10 MeV (like in the case of giant resonance excitation) are in the transition region from a constant behavior of the equivalent photon numbers to that of an exponential (&e\pK) decay. This is more transparent in Fig. 6 where we plot the equivalent photon numbers for E " 100 A MeV, b"15 fm, and as a function of u. One also observes from this "gure that the E2 multipolarity component of the electromagnetic "eld dominates at low frequencies. Nonetheless, over the range of u up to some tens of MeV, the E2 matrix elements of excitation are much smaller than the E1 elements for most nuclei, and the E2 e!ects become unimportant. However, such e!ects are relevant for the excitation of the isoscalar E2 giant resonance (GQR ) which have large matrix elements. As an application of the semiclassical approach to Coulomb excitation in intermediate energy collisions, we study the excitation of giant isovector dipole resonances (E1) and of giant isoscalar quadrupole resonances (E2) in Pb by means of the Coulomb interaction with a O projectile. At 100A MeV the maximum scattering angle which still leads to a pure Coulomb scattering (assuming a sharp cut-o! at an impact parameter b"R #R ) is 3.93. The cross sections are . 2 calculated by assuming a Lorentzian shape for the photonuclear cross sections: eC pLH"p A K(e!E )#eC K with p chosen to reproduce the Thomas}Reiche}Kuhn sum rule for E1 excitations, K NZ p#(e) deK60 MeV mb A A
(95)
(96)
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Fig. 7. Total cross sections for the excitation of giant electric dipole (E1) and quadrupole (E2) resonances in Pb by means of the Coulomb interaction with O, as a function of the laboratory energy.
and the energy-weighted sum rule for the quadrupole mode,
de p#(e) K0.22ZA lb/MeV . A e
(97)
The resonance energies are approximately given by E K77 ) A\ MeV and E K %"0 %/0 63 ) A\ MeV. We use the widths C "4 MeV and C "2.2 MeV for Pb. %"0 %/0 We will discuss the di!erential cross sections as a function of the scattering angle later, when we introduce the e!ects of strong absorption. To obtain the total cross sections, one has to integrate the equivalent photon numbers in (90) and (91) from 03 to a maximum scattering angle h , where
the nuclear absorption sets in, or equivalently, one can integrate over the impact parameter, from a minimum value b up to in"nity. Fig. 7 shows the total cross section for the excitation of giant
dipole and of giant quadrupole resonances in Pb in a collision with O as a function of the laboratory energy per nucleon. The same average behavior of the photonuclear cross sections, as assumed in Eqs. (95) and (96), is used. Only for the E1 multipolarity the angular integration can be performed analytically. One obtains
1 2 N " Za e\pD(c/v) !mK K ! (c/v)m D D 2 # p
RK RK i I I ;(f/m)K #K !K ! K !K GD GD GD e GD Rk GD Rk IGD IGD
,
(98)
We observe that the original formula for the dipole case appearing in [6] has a misprinted sign in one of its terms.
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where
1 for 2a'b ,
(99) e " R/a!1 for 2a(b ,
and m"e f"ub /cv.
It is easy to see that this equation reduces to Eq. (45) in the relativistic limit, when fP0, e PR. The cross sections increase very rapidly to large values, which are already attained at intermediate energies. A salient feature is that the cross section for the excitation of giant quadrupole modes is very large at low and intermediate energies, decreasing in importance (about 10% of the E1 cross section) as the energy increases above 1A GeV. This occurs because the equivalent photon number for the E2 multipolarity is much larger than that for the E1 multipolarity at low collision energies. That is, n
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2.4.1. Inelastic amplitudes and virtual photon numbers De"ning r as the separation between the centers of mass of the two nuclei and r to be the intrinsic coordinate of the target nucleus to "rst-order the inelastic scattering amplitude is given by
ik dr dr 1U\ f (h)" k (r) (r )"< (r, r )"U> k (r) (r )2 , Y D G 2p v
(100)
where U\ k (r) and U> k (r) are the incoming and outgoing distorted waves, respectively, for the Y scattering of the center of mass of the nuclei, and (r ) is the intrinsic nuclear wave function of the target nucleus. At intermediate energies, *E/E ;1, and forward angles, h;1, we can use eikonal wave functions for the distorted waves, i.e., (r)U> U\H k k (r)"exp+!iq . r#is(b), , Y where
(101)
i ;(z , b) dz #it (b) (102) s(b)" , !
v \ is the eikonal phase, q"k !k, ; is the nuclear optical potential, and t (b) is the Coulomb , ! eikonal phase. We have de"ned the impact parameter b by b""r;z( ". For light nuclei, one can assume Gaussian nuclear densities, and the Coulomb phase is given by
Z Z e b 1 t (b)"2 ln(kb)# E , (103) !
v R 2 % with RG equal to the size parameter of each Gaussian matter density, R "[R]#[R], and % % % % e\R E (x)" dt . (104) t V The "rst term in Eq. (103) is the contribution to the Coulomb phase of a point-like charge distribution. It reproduces the elastic Coulomb amplitude when introduced into the eikonal expression for the elastic scattering amplitude. The second term in Eq. (103) is a correction due to the extended Gaussian charge distribution. It eliminates the divergence of the Coulomb phase at b"0, so that
Z Z e t (0)"2 [ln (kR )!C] , ! %
v
(105)
where C"0.577 is the Euler constant. For heavy nuclei a `black-spherea absorption model is more appropriate. Assuming an absorption radius R , the Coulomb phase is given by s (b)"2(Z Z e/ v)+H(b!R )ln (kb)#H(R !b)[ln (kR ) ! ? #ln(1#(1!b/R) )!(1!b/R) !(1!b/R) ], . (106) Again, the "rst term inside the parentheses is the Coulomb eikonal phase for point-like charge distributions. The second term accounts for the "nite extension of the charge distributions.
C.A. Bertulani, V.Yu. Ponomarev / Physics Reports 321 (1999) 139}251 Table 1 Parameters [29] for the nucleon}nucleon amplitude, f
163
(h"03)"(k /4p) p (i#a ) ,, ,, ,, ,,
E (A MeV)
p (fm) ,,
a ,,
85 94 120 200 342.5 425 550 650 800 1000 2200
6.1 5.5 4.5 3.2 2.84 3.2 3.62 4.0 4.26 4.32 4.33
1 1.07 0.7 0.6 0.26 0.36 0.04 !0.095 !0.075 !0.275 !0.33
For high-energy collisions, the optical potential ;(r) can be constructed by using the t-oo approximation [28]. One gets
v ;(r)"! p (a #i) o (r )o (r!r ) dr , 2 ,, ,,
(107)
where p is the nucleon}nucleon cross section, and a is the real-to-imaginary ratio of the ,, ,, forward (h"03) nucleon}nucleon scattering amplitude. A set of the experimental values of these quantities, useful for our purposes, is given in Table 1. In Eq. (100) the interaction potential, assumed to be purely Coulomb, is given by e G r\rY vI , (108) < (r, r )" j (r ) c I "r!r " where vI"(c, *), with * equal to the projectile velocity, i"u/c, and j (r ) is the charge four-current I for the intrinsic excitation of nucleus 1 by an energy of u. Inserting (101), (102) and (108) in (100) and following the same steps as in Ref. [6], one "nds
u H c Z ek (2j#1 e\ K(X (q)G 1I M "M(nj,!m)"I M 2 , (109) f (h)"i iK K LHK v D D G G c c v LHK where njm denotes the multipolarity, G are the Winther}Alder relativistic functions [18], and LHK 1I M "M(nj,!m)"I M 2 is the matrix element for the electromagnetic transition of multipolarity D D G G njm from "I M 2 to "I M 2, with E !E " u. The function X (q) is given by G G D D D G K ub X (q)" db bJ (qb)K exp+is(b), , (110) K K K cv where q"2k sin(h/2) is the momentum transfer, h and are the polar and azimuthal scattering angles, respectively.
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For the E1, E2 and M1 multipolarity, the functions G (c/v) are given by [18] LHK G (x)"!G (x)"x(8p/3, G (x)"!i4(p(x!1)/3 , # # #\ (x)"0 , G (x)"G (x)"!i(8p/3, G + + +\ G (x)"G (x)"!2x(p(x!1)6/5 , # #\ G (x)"!G (x)"i2(p/6 (2x!1)/5, G (x)"2(p(x!1)/5 . (111) # #\ # Using the Wigner}Eckart theorem, one can calculate the inelastic di!erential cross section from (109), using techniques similar to those discussed in previous sections. One obtains dn dp 1 ! (E )" LHpLH(E ) (112) dX A A dX dE A E A A LH where pLH(E ) is the photonuclear cross section for the absorption of a real photon with energy A A E by nucleus 2, and dn /dX is the virtual photon number, which is given by [27] A LH uk j[(2j#1)!!] dn LH"Za "G ""X (q)" , (113) cv LHK K (2p)(j#1) dX K where a"e/ c. The total cross section for Coulomb excitation can be obtained from Eqs. (112) and (113), using the approximation dXK2pq dq/k, valid for small scattering angles and small energy losses. Using the closure relation for the Bessel functions, we obtain
1 dp ! (E )" n (E )pLH(E ) , LH A A A E dE A A LH A where j[(2j#1)!!] n (u)"Za "G "g (u) , LH (2p)(j#1) LHK K K
(114)
(115)
and
u ub db bK exp+!2s (b), , g (u)"2p K cv ' K cv
(116)
where s (b) is the imaginary part of s(b), which is obtained from Eq. (102) and the imaginary part of ' the optical potential. Before proceeding further, it is worthwhile to mention that the present calculations di!er from those of previous sections by the proper inclusion of absorption. To reproduce the angular distributions of the cross sections, it is essential to include the nuclear transparency. In the limit of a black-disk approximation, the above formulas reproduce the results presented in Ref. [6]. One also observes that the Coulomb phase in the distorted waves, which is necessary for the quantitative reproduction of the experimental angular distributions, is not important for the total cross section in high-energy collisions. This fact explains why semiclassical and quantum methods give the same result for the total cross section for Coulomb excitation at relativistic energies [6]. At
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intermediate energies, however, it is just this important phase which reproduces the semiclassical limit for the scattering of large-Z ions, as we shall see next. Using the semiclassical terminology, for E +100A MeV or less, the recoil in the Coulomb trajectory is relevant. At the distance of closest approach, when the Coulomb "eld is most e!ective at inducing the excitation, the ions are displaced farther from each other due to the Coulomb recoil. As we discussed before, this e!ect can be accounted for approximately by using the e!ective impact parameter b "b#pZ Z e/4E in the semiclassical calculations. This recoil approximation can also be used in Eq. (116), replacing b by b in the Bessel function and the nuclear phase, in order to obtain the total cross section. Since the modi"ed Bessel function is a rapidly decreasing function of its argument, this modi"cation leads to sizable modi"cations of the total cross section at intermediate energy collisions. Finally, we point out that for very light heavy ion partners, the distortion of the scattering wavefunctions caused by the nuclear "eld is not important. This distortion is manifested in the di!raction peaks of the angular distributions, characteristic of strong absorption processes. If Z Z a<1, one can neglect the di!raction peaks in the inelastic scattering cross sections and a purely Coulomb excitation process emerges. One can gain insight into the excitation mechanism by looking at how the semiclassical limit of the excitation amplitudes emerges from the general result (113). We do this next. 2.4.2. Semiclassical limit of the excitation amplitudes If we assume that Coulomb scattering is dominant and neglect the nuclear phase in Eq. (102), we get
ub exp+it (b), . db bJ (qb)K ! K K cv This integral can be done analytically by rewriting it as X (q)K K
(117)
ub , (118) db b> EJ (qb)K K K cv where we used the simple form t (b)"2g ln(kb), with g"Z Z e/ v. Using standard techniques ! found in Ref. [30], we "nd X (q)" K
cv > E 1 F(1#m#ig; 1#ig; 1#m;!K) , (119) X (q)"2 E C(1#m#ig)C(1#ig)KK K u m! where K"qcv/u ,
(120)
and F is the hypergeometric function [30]. The connection with the semiclassical results may be obtained by using the low-momentum transfer limit
pm p 1 2 cos qb! ! " +e O@e\ pK> #e\ O@e pK> , , J (qb)K K 2 4 pqb (2pqb
(121)
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and using the stationary phase method, i.e.,
G(x)e (V dxK
2pi G(x )e (V ,
(x )
(122)
where d (x )/dx"0 and (x )"d (x )/dx . (123) This result is valid for a slowly varying function G(x). Only the second term in brackets of Eq. (121) will have a positive (b"b '0) stationary point, and
p(m#1/2) 1 2pi ub exp i (b )#i , X (q)K (b K K K cv 2 (2pq (b)
(124)
where
(b)"!qb#2g ln(kb) .
(125)
The condition (b )"0 implies b "2g/q"a /sin(h/2) , (126) where a "Z Z e/kv is half the distance of closest approach in a classical head-on collision. We observe that the relation (126) is the same [with cot(h/2)&sin\(h/2)] as that between impact parameter and de#ection angle of a particle following a classical Rutherford trajectory. Also,
(b )"!2g/b"!q/2g , which implies that in the semiclassical limit
(127)
ua K . K cv sin(h/2) 0 Using the above results, Eq. (113) becomes 2ug 1 dp 4g " "X (q)" " K K K cvq k dX q
(128)
u j[(2j#1)!!] ua "G "K . (129) Za pHK K cv sin(h/2) cv (2p)(j#1) 0 K If strong absorption is not relevant, the above formula can be used to calculate the equivalent photon numbers. The stationary value given by Eq. (126) means that the important values of b which contribute to X (q) are those close to the classical impact parameter. Dropping the index K 0 from Eq. (126), we can also rewrite (129) as dn dp pH" dX dX
dn u j[(2j#1)!!] ub pH "Za "G "K , cv pHK K cv 2pb db (2p)(j#1) K which is equal to the semi-classical expression given in Ref. [23], Eq. (A.2).
(130)
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For very forward scattering angles, such that K;1, a further approximation can be made by setting the hypergeometric function in Eq. (119) equal to unity [30], and we obtain X (q)"2 E(1/m!)C(1#m#ig)C(1#ig)KK(cv/u)> E . K The main value of m in this case will be m"0, for which one gets X (q)K2 EC(1#ig)C(1#ig)(cv/u)> E"!g2 EC(ig)C(ig)(cv/u)> E
(131)
(132)
and "X (q)""g(cv/u) p/g sinh(pg) , which, for g<1, results in
(133)
"X (q)""4pg(cv/u)e\pE . (134) This result shows that in the absence of strong absorption and for g<1, Coulomb excitation is strongly suppressed at h"0. This also follows from semiclassical arguments, since hP0 means large impact parameters, b<1, for which the action of the Coulomb "eld is weak. 2.5. Singles spectra in Coulomb excitation of GDR In this section, we data of Ref. [31], in target nucleus Pb the elastic scattering form given by
apply the formalism developed in previous sections in the analysis of which a projectile of O with an energy of E "84A MeV excites to the GDR. We "rst seek parameters of the optical potential which data. We use the eikonal approximation for the elastic amplitude in
f (h)"ik J (qb)+1!exp[is(b)],b db ,
the the "ts the
(135)
where J is the Bessel function of zeroth order and the phase s(b) is given by Eq. (102). In Fig. 8 we compare the calculated elastic scattering angular distribution to the data from Ref. [12]. The calculation utilized Eq. (135), with s(b) obtained from an optical potential of a standard Woods}Saxon form with parameters < "50 MeV, = "58 MeV, R "R "8.5 fm and a "a "0.85 fm . (136) 4 5 4 5 The data are evidently very well reproduced by the eikonal approximation. In order to calculate the inelastic cross section for the excitation of the GDR, we use a Lorentzian parameterization for the photoabsorption cross section of Pb [32], assumed to be all E1, with E "13.5 MeV and C"4.0 MeV. Inserting this form into Eq. (114) and doing the calcu%"0 lations implicit in Eq. (113) for dn /dX, we calculate the angular distribution and compare it with # the data in Fig. 9. The agreement with the data is excellent, provided we adjust the overall normalization to a value corresponding to 93% of the energy weighted sum rule (EWSR) in the energy interval 7}18.9 MeV. Taking into account the $10% uncertainty in the absolute cross sections quoted in Ref. [12], this is consistent with photoabsorption cross section in that energy range, for which approximately 110% of the EWSR is exhausted.
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Fig. 8. Ratio to the Rutherford cross section of the elastic cross section for the O#Pb reaction at 84A MeV, as a function of the center-of-mass scattering angle. Data are from Ref. [12]. Fig. 9. Di!erential cross section for the excitation of the isovector giant dipole resonance in Pb by means of O projectiles at 84A MeV, as a function of the center-of-mass scattering angle. Data are from Ref. [12].
To unravel the e!ects of relativistic corrections, we repeat the previous calculations unplugging the factor c"(1!v/c)\ which appears in the expressions (115) and (116) and using the non-relativistic limit of the functions G , as given in Eq. (111). These modi"cations eliminate the #K relativistic corrections on the interaction potential. The result of this calculation is shown in Fig. 10 (dotted curve). For comparison, we also show the result of a full calculation, keeping the relativistic corrections (dashed curve). We observe that the two results have approximately the same pattern, except that the non-relativistic result is slightly smaller than the relativistic one. This fact may explain the discrepancy between the "t of Ref. [12] and ours as due to relativistic corrections not properly accounted for in the ECIS code [33]. In fact, if we repeat the calculation for the excitation of GDR using the non-relativistic limit of Eqs. (115) and (116), we "nd that the best "t to the data is obtained by exhausting 113% of the EWSR. This value is very close to the 110% obtained by Barrette et al. [12]. In Fig. 10 we also show the result of a semiclassical calculation (solid curve) for the GDR excitation in lead, using Eq. (129) for the virtual photon numbers. One observes that the semiclassical curve is not able to "t the experimental data. This is mainly because di!raction e!ects and strong absorption are not included. But the semiclassical calculation displays the region of relevance for Coulomb excitation. At small angles the scattering is dominated by large impact parameters, for which the Coulomb "eld is weak. Therefore, the Coulomb excitation is small and the semiclassical approximation fails. It also fails in describing the large angle data (dark side of the rainbow angle), since absorption is not treated properly. One sees that there is a `windowa in the inelastic scattering data near h"2}33 in which the semiclassical and full calculations give approximately the same cross section.
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Fig. 10. Virtual photon numbers for the electric dipole multipolarity generated by 84A MeV O projectiles incident on Pb, as a function of the center-of-mass scattering angle. The solid curve is a semiclassical calculation. The dashed and dotted curves are eikonal calculations with and without relativistic corrections, respectively. Fig. 11. Di!erential cross sections for the excitation of the giant dipole resonance (GDR), the isoscalar giant quadrupole resonance (GQR ), and the isovector giant quadrupole resonance (GQR ), in Pb for the collision Pb#Pb at G G 640A MeV. The solid (dotted) [dashed-dotted] line is the di!erential cross section for the excitation of the GDR (GQR ) G [GQR ]. The dashed line is the result of a semiclassical calculation. G
In Fig. 11 we perform the same calculation, but for the excitation of the GDR, the isoscalar giant quadrupole resonance (GQR ), and the isovector quadrupole resonance (GQR ), in Pb for the collision Pb#Pb at 640A MeV. The solid (dotted) (dashed-dotted) line is the di!erential cross section for the excitation of the GDR (GQR ) [GQR ]. The dashed line is the result of a semiclassical calculation. Here we see that a purely semiclassical calculation, using Eq. (92) is able to reproduce the quantum results up to a maximum scattering angle h , at which strong absorption
sets in. This justi"es the use of semiclassical calculations for heavy systems, even to calculate angular distributions. The cross sections increase rapidly with increasing scattering angle, up to an approximately constant value as the maximum Coulomb scattering angle is neared. This is explained as follows. Very forward angles correspond to large impact parameter collisions in which case ub/cv'1 and the excitation of giant resonances in the nuclei is not achieved. As the impact parameter decreases, increasing the scattering angle, this adiabaticity condition is ful"lled and excitation occurs. As discussed above, the semiclassical result works for large Z nuclei and for relativistic energies where the approximation of Eq. (117) is justi"ed. However, angular distributions are not useful at relativistic energies since the scattering is concentrated at extremely forward angles. The quantity of interest in this case is the total inelastic cross section. If we use a sharp-cuto! model for the strong absorption, so that s (b)"R for b(b and 0 otherwise, then Eqs. (115) and (116) yield '
the same result as an integration of the semiclassical expression, Eq. (130), from b to R. In fact,
this result has been obtained earlier in Ref. [6].
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2.6. Excitation and photon decay of the GDR We now consider the excitation of the target nucleus to the giant dipole resonance and the subsequent photon decay of that excited nucleus, leaving the target in the ground state. Experimentally, one detects the inelastically scattered projectile in coincidence with the decay photon and demands that the energy lost by the projectile is equal to the energy of the detected photon. To the extent that the excitation mechanism is dominated by Coulomb excitation, with the exchange of a single virtual photon, this reaction is very similar to the photon scattering reaction, except that in the present case the incident photon is virtual rather than real. In this section, we investigate whether the connection between these two reactions can be formalized. We "rst review the excitation mechanism. The physical situation is that of a heavy ion of energy E incident on a target. The projectile loses an energy *E while scattering through an angle h. We have shown that, under the conditions *E/E;1, the cross section for excitation of the target nucleus partitions into the following expression (we assume that the contribution of the E1multipolarity is dominant): 1 dn dp A (E )p (E ) , ! (E )" (137) E dX A A A dX dE A A A where p (E ) is the photonuclear cross section for the absorption of a real photon with energy A A E "*E by the target nucleus, and the remaining terms on the right-hand side are collectively the A number of virtual photons per unit energy with energy E . This latter quantity depends on the A kinematics of the scattered heavy ion and on the optical potential but is otherwise independent of the target degrees of freedom. This partitioning allows one to relate the excitation cross section to the photoabsorption cross section. Now, the usual way to write the cross section dp /dX dE for the excitation of the target !A A followed by photon decay to the ground state is simply to multiply the above expression by a branching ratio R , which represents the probability that the nucleus excited to an energy E will A A emit a photon leaving it in the ground state [13]: 1 dn dp A (E )p (E )R (E ) . !A (E )" E dX A A A A A dX dE A A A Instead, we propose the following expression, in complete analogy with Eq. (137):
(138)
1 dn dp A (E )p (E ) , !A (E )" (139) E dX A AA A dX dE A A A where p (E ) is the cross section for the elastic scattering of photons with energy E . Formally, AA A A these expressions are equivalent in that they simply de"ne the quantity R . However, if one treats A R literally as a branching ratio, then these expressions are equivalent only if it were true that the A photon scattering cross section is just product of the photoabsorption cross section and the branching ratio. In fact, it is well-known from the theory of photon scattering that the relationship between the photoabsorption cross section and the photon scattering cross section is more complicated [34]. In particular, it is not correct to think of photon scattering as a two-step process consisting of absorption, in which the target nucleus is excited to an intermediate state of energy E , A
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Fig. 12. Calculated cross section for the excitation followed by c-decay of Pb induced by O projectiles at 84A MeV. The photoabsorption cross section was parameterized by a simple Lorentzian representing the GDR, and the statistical component of the photon decay was neglected. The solid curve uses the formalism described in the text (Eq. (139)) while the dashed curve uses a constant branching ratio for photon decay (Eq. (138)). Fig. 13. Cross section for the excitation of the GDR without the detection of the decay photon. Data are from Ref. [13].
followed by emission, in which the emitted photon has the same energy E . Since the intermediate A state is not observable, one must sum over all possible intermediate states and not just those allowed by conservation of energy. Now, if the energy E happens to coincide with a narrow level, A then that level will completely dominate in the sum over intermediate states. In that case, it is proper to regard the scattering as a two-step process in the manner described above, and the two expressions for the cross section will be equal. However, for E in the nuclear continuum region A (e.g., in the region of the GDR), this will not be the case, as demonstrated in the following discussion. We again consider the inelastic scattering of O projectiles of energy E "84 MeV/nucleon from a Pb nucleus at an angle of 2.53. We use Eq. (113) to calculate the E1 virtual photon number and we use a Lorentzian parameterization of the GDR of Pb. We calculate R and A p according to the prescriptions of Refs. [13] and [34], respectively; in both cases we neglect the AA statistical contribution to the photon decay. The results are compared in Fig. 12, where it is very evident that they make very di!erent predictions for the cross section, especially in the wings of the GDR. We next use our expression to compare directly with the data of Ref. [13]. For this purpose, we again calculate p using the formalism of Ref. [34], which relates p to the total photoabsorption. AA AA For the latter, we use the numerically de"ned data set of Ref. [32] rather than a Lorentzian parameterization. The e!ect of the underlying compound nuclear levels (i.e., the statistical contribution to the photon scattering) is also included. The calculation is compared to the data in Figs. 13 and 14. Fig. 13 shows the cross section for the excitation of the GDR without the detection
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Fig. 14. Cross section for excitation followed by c-decay of Pb by O projectiles at 84A MeV. The solid (dashed) line includes (excludes) the Thomsom scattering amplitude. Data are from Ref. [13].
of the decay photon. The agreement with the data is excellent, giving us con"dence that our calculation of the virtual photon number as a function of E is correct. Fig. 14 shows the cross A section for the excitation-decay process as a function of E . Although the qualitative trend of A the data are well described, the calculation systematically overpredicts the cross section on the high-energy side of the GDR (solid curve). If the Thompson amplitude is not included in p , the AA calculation is in signi"cantly better agreement with the data (dashed curve). 2.7. Nuclear excitation and strong absorption Up to this point we have only considered the Coulomb excitation of the nuclei, without accounting for nuclear excitation. But, in peripheral collisions, the nuclear interaction between the ions can also induce excitations. This can be easily calculated in a vibrational model. The amplitude for the excitation of a vibrational mode by the nuclear interaction in relativistic heavy ion collisions can be obtained assuming that a residual interaction U between the projectile and the target exists, and that it is weak. According to the Bohr}Mottelson particle}vibrator coupling model, the matrix element for the transition iPf is given by (140) <,HI(r),1I M ";"I M 2"d /((2j#1)1I M "> "I M 2> (r( ); (r) , D D HI G G HI H DG D D G G H where d "b R is the vibrational amplitude, or deformation length, R is the nuclear radius, and H H ; (r) is the transition potential. H The deformation length d can be directly related to the reduced matrix elements for electromagnetic H transitions. Using well-known sum-rules for these matrix elements one "nds a relation between the deformation length and the nuclear masses and sizes. For isoscalar excitations one gets [35] 1
, d"2p m 1r2 AE V ,
2p 1 d " j(2j#1) , HY 3 m AE , V
(141)
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where A is the atomic number, 1r2 is the r.m.s. radius of the nucleus, and E is the excitation V energy. The transition potentials for nuclear excitations can be related to the optical potential in the elastic channel. The basic idea is that the interaction between the projectile and the target induces surface vibrations in the target. Only the contact region between the nuclei in grazing collisions is of relevance. One thus expects that the interaction potential is proportional to the derivatives of the optical potential in the elastic channel, which peak at the surface. This is discussed in detail in Ref. [35]. The transition potentials for isoscalar excitations are ; (r)"3; (r)#r d; (r)/dr , for monopole, and
(142)
; (r)"d; (r)/dr , for quadrupole modes. For dipole isovector excitations
(143)
p A 1 , d " 2 m NZ E , V where Z (N) is the charge (neutron) number. The transition potential in this case is [35]
; (r)"s
N!Z A
d; 1 d; # R , dr 3 dr
(144)
(145)
where the factor s depends on the di!erence between the proton and the neutron matter radii as R !R *R 2(N!Z) N " LN . " L (146) (R #R ) R 3A N L Thus, the strength of isovector excitations increases with the di!erence between the neutron and the proton matter radii. This di!erence is accentuated for neutron-rich nuclei and should be a good test for the quantity *R which enters the above equations. LN The time dependence of the matrix elements above can be obtained by making a Lorentz boost. Since the potentials ; [r(t)] peak strongly at t"0, we can safely approximate h(t)Kh(t"0)"p/2 H in the spherical harmonic of Eq. (140). One gets s
p d H 1I M "> "I M 2> h" ; [r(t)] , <,HI(r),1I M ";"I M 2"c D D HI G G HI DG D D G G 2 H (2j#1
(147)
where r(t)"(b#cvt. Using the Wigner}Eckart theorem, the matrix element of the spherical harmonics becomes
j I I j I (2I #1)(2j#1) ID G D G . (148) G 4p(2I #1) !M k M 0 0 0 D D For high-energy collisions, the optical potential ;(r) can be constructed by using the t-oo approximation [28], as given by Eq. (107). 1I M "> "I M 2"(!1)'D\+D D D HI G G
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Fig. 15. The GDR Coulomb excitation probabilities as functions of the impact parameter, for sharp and smooth absorptions. The system is Pb (640A MeV)#Pb. Fig. 16. Nuclear excitation probabilities, as functions of the impact parameter, of the isoscalar giant monopole resonance (GMR ), the GDR , and the GQR , in Pb for the collision Pb#Pb at 640A MeV. G G G
We are not interested here in di!raction and refraction e!ects in the scattering, but on the excitation probabilities for a given impact parameter. The strong absorption occurring in collisions with small impact parameters can be included. This can be done by using the eikonal approximation and the optical potential, given by Eq. (107). The practical result is that the excitation probabilities for a given impact parameter b, including the sum of the nuclear and the Coulomb contributions to the excitation, are given by
dz dr o (r )o (r!r ) , P (b)""a! (b)#a, (b)" exp !p ,, DG DG DG
(149)
where r"(b#z. The corresponding excitation cross sections are obtained by an integration of the above equation over impact parameters. 2.8. Nucleon removal in peripheral relativistic heavy ion collisions In Fig. 15 we plot the GDR excitation probability in Pb as a function of the impact parameter, for the system Pb#Pb at 640A MeV. We use 100% of the sum rule to calculate the B(E1)-value for the electromagnetic excitation of an isolated GDR state at 13.5 MeV. In the solid line, we consider absorption according to Eq. (149). In the construction of the optical potential we used the g.s. densities calculated from the droplet model of Myers and Swiatecki [36] in accordance with Shen et al. [37]. We will call it by soft-spheres model.
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As shown in Ref. [38], this parametrization yields the best agreement between experiment and theory. The dashed line does not include absorption. To simulate strong absorption at low impact parameters, we use b "15.1 fm as a lower limit in the impact parameter integration of Eq. (5).
This value was chosen such as to lead to the same cross section as that obtained from the solid line. However, a more detailed comparison of the soft-sphere model for strong absorption and a simple semiclassical calculation, based on a single parameter b , is described next [38].
In Fig. 16, we plot the nuclear contributions to the excitation probability as a function of the impact parameter. We study the excitation of the isoscalar giant monopole resonance (GMR ), the GDR , and the GQR , in lead for the collision Pb#Pb at 640A MeV. The GMR in Pb is located at 13.8 MeV. As discussed previously, isovector excitations are suppressed in nuclear excitation processes, due to the approximate charge independence of the nuclear interaction. We use the formalism of this section, with the deformation parameters such that 100% of the sum rule is exhausted. This corresponds to the monopole amplitude d "0.054. The GDR and GQR deformation parameters are d "0.31 fm and d "0.625 fm, respectively. The GQR excitation probability is much smaller than the other excitation probabilities and is, therefore, not shown. The nuclear excitation is peaked at the grazing impact parameter and is only relevant within an impact parameter range of &2 fm. Comparing to Fig. 15, we see that these excitation probabilities are orders of magnitude smaller than those for Coulomb excitation. Consequently, the corresponding cross sections are much smaller. We get 14.8 mb for the isovector GDR, 2.3 mb for the GQR , and 2.3 mb for the GDR . The interference between the nuclear and the Coulomb excitation is also small and can be neglected. Since they are high lying states above the continuum, giant resonances mostly decay by particle emission (mainly neutron emission in heavy nuclei). Therefore data on neutron removal in relativistic heavy ion collisions is an appropriate comparison between theory and experiment. As we have seen, above, nuclear excitation of GR's contribute very little to the cross section, as compared to Coulomb excitation. However, strong interactions at peripheral collisions also contribute to `directa knockout (or stripping) of neutrons, and also should be considered. It has been observed [6], however, that neutron removal cross sections induced by strong interactions scale with A #A , while the Coulomb excitation cross sections scale with the projectile's charge as Z, approximately. One can thus separate the nuclear contribution for the nucleon removal of a target (or projectile) by measuring the cross sections for di!erent projectiles (or targets). In the semiclassical approach, the total cross section for relativistic Coulomb excitation is obtained by integrating the excitation probabilities over impact parameter, starting from a minimum value b . It is assumed that below this minimum value the interaction is exclusively due to
the strong interaction (`sharp-cuto! a approximation). It has been found that with this approximation the Coulomb cross sections are very sensitive to the parameterization of the minimum impact parameter [39}42]. One commonly used parameterization at relativistic energies is that of Benesh et al. [43], "tted to Glauber-type calculations and reading b !4"1.35(A #A !0.75(A\ #A\ )) fm (150)
N R N R which we refer to hereafter as `BCVa. In Ref. [43] a detailed study has been made concerning the parametrization procedure of the minimum impact parameter. It was also found that the nuclear
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contribution to the neutron removal channels in peripheral collisions has a negligible interference with the Coulomb excitation mechanism. This is a very useful result since the Coulomb and nuclear part of the cross sections may be treated separately. The other parametrization is that of Kox et al. [44] which reproduced well measured total reaction cross sections of light and medium-mass systems:
A A R !1.9 fm . (151) b)"1.1 A #A #1.85 N N R
A #A N R This parametrization has been used previously [41] and a reasonable agreement with the measured data for 1n cross sections was found. It should be noted, however, that the Kox parametrization of total interaction cross sections has been derived mainly from experiments with light projectiles and that its application to heavy systems involves an extrapolation into a region where no data points are available. To achieve a good a comparison with experimental data on neutron-removal cross sections we will use the experimental photo-neutron emission cross sections from Refs. [32,45]. A Lorentzian "t to the (c, n)-data is used to parameterize the GDR in gold. The parameters are 13.72 MeV excitation energy, a width of 4.61 MeV, and a strength of 128% of the TRK sum rule. The Lorentz parameters for the isoscalar (isovector) GQR are taken as 10.8 (23.0) MeV for the excitation energy, 2.9 (7.0) MeV for the width, and we assume 95% exhaustion of the respective sum rules [5]. With these parameters we calculate the excitation cross sections dp(E)/dE for dipole and quadrupole excitations. The respective neutron emission cross sections are given by
p" L
dp(E) f (E) dE , dE L
(152)
where f (E) is the probability to evaporate one neutron at excitation energy E. f (E) is taken from L L the experimental (c, n)-data at low E and from a statistical decay calculation with the code HIVAP [46] for excitation energies above 20 MeV. Since the three-neutron emission threshold in gold is above the energy of the GDR state, this channel is fed mainly by the two-phonon excitation mechanism, while the 1n cross section is dominated by the excitation of the GDR. We expect that the BCV parametrization of b should yield similar results as the soft-spheres
calculation since it was derived in "tting the complementary process, the nuclear interaction, calculated also with Glauber theory. Fig. 17 shows that this expectation could be veri"ed: the soft-spheres calculation for 1n-removal from Au by ED processes (upper full curve) is almost indistinguishable from a sharp-cuto! calculation using b !4 (upper dotted curve). This remarkable
agreement tells us that for practical purposes we can avoid the extra numerical complication connected with the use of a soft spheres model and corroborates the use of b !4 in sharp-cuto!
calculations in earlier works [47,48]. We also think that the soft-spheres calculation (and the sharp-cuto! calculation using b !4) is physically better justi"ed than the Kox parametrization [44]
since the former is derived from realistic nuclear density distributions, whereas the latter is an extrapolation of measured total reaction cross sections into a region where no data points are available. We will return later to discuss the other data points of Fig. 17 when we treat the problem of the excitation of multiphonon states. However, it is worthwhile noticing that a point in the above
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Fig. 17. Experimental 1n- and 3n-removal cross sections for Au bombarded with relativistic projectiles (from Ref. [38]) in comparison with theoretical calculations from this work (solid curve: `soft-spheresa model; dotted curve: `sharp-cuto!a model with b !4 from Eq. (150)). For completeness, we also show a sharp-cuto! calculation with b)
(Eq. (151)) used in Ref. [41]. Fig. 18. A nuclear target is Coulomb excited by a fast moving deformed projectile. Besides the angle h, the orientation of the projectile also includes an azimuthal angle which can rotate its symmetry axis out the scattering plane. For simplicity, this is not shown. s is the angular position of the c.m. of the projectile with respect to the target.
curve, for uranium targets, is not well reproduced by the theory. In fact, this has been observed in other experiments [49], and deserves a special treatment. 2.9. Excitation by a deformed nucleus Either by using the soft-sphere model, or by means of a semiclassical calculation, Coulomb excitation by a relativistic projectile, or target, is well described theoretically if the charge distribution of the projectile is spherically symmetric [50]. However, there was found a discrepancy between theory and experiment with data with deformed projectiles, as measured by Justice et al. [49] for uranium projectiles. This problem was studied theoretically by Bertulani in [51], and we will brie#y discuss it here. To obtain a qualitative insight of the e!ects we shall consider a prolate deformed projectile with a variable deformation. In the frame of reference of the projectile the Coulomb "eld at a position r with respect to the center-of-charge of the distribution is given by 1 1 H > (h, )M(Ejk) ,
(r)"4p 2j#1 rH> HI H I
(153)
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where
M(Ejk)" o(r )r J> (r( ) dr , HI
(154)
with o(r) equal to the ground-state charge distribution of the projectile. For simplicity, we will consider a uniform spheroidal charge distribution with the z-axis along the symmetry axis. The charge distribution drops to zero for distances to the center greater than the angle-dependent radius R(h)"R (1#b> (h)) . In lowest order in the multipole expansion, Eq. (153) becomes
Z e p 1
(r)" # > (h)Q , r 5 r where Q is the quadrupole moment of the charge distribution, 3 Q" Z eRb(1#0.16b)#O(b) . (5p
(155)
(156)
(157)
To obtain the (time-dependent) "eld in the frame of reference of the target we perform a Lorentz transformation of Eq. (156). For a straight-line trajectory one "nds
p 1 cZ e > (h )Q
(r , t)" #c 5 r r
(158)
where r "(b#cvt, with b equal to the impact parameter, v the projectile velocity, and c"(1!v/c)\ . The "rst term in the above equation is the well-known LieH nard}Wiechert potential of a relativistic charge. It gives rise to monopole}multipole excitations of the target, which we have discussed so far. The second term accounts for quadrupole}multipole excitations of the target and is a correction due to the deformation of the projectile. This "eld will depend on the orientation of the projectile with respect to its trajectory (see Fig. 18). We can separate the orientation angles from the angular position of the projectile (along its trajectory) with respect to the target by using the identity
4p > (h, )> (s , 0) , (159) K K 5 K where (h, ) denotes the orientation of the projectile symmetry axis with respect to the bombarding axis and s "cos\ [cvt/r (t)]. The dipole excitation of the target is the most relevant and we shall restrict ourselves to this case only [6]. At a point r,(x, y, z) from the center of mass of the target the "eld is obtained by replacing r "(b, 0, cvt) by [b!x, y, c(vt!z)] in Eq. (158). The excitation amplitude to "rst order is given by Eq. (62). Using the continuity equation and expanding (62) to lowest order in r we "nd > (h )"
a "a#a DG DG DG
(160)
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where
1 2Z e m K (m)DV#i K (m)DX a"!i DG DG DG c
v b
(161)
and
p 1 4p Q m K (m)DV#i K (m)DX > (h, )> ,0 , (162) a"!i DG DG K K 2 DG c 5 v b K where m"ub/cv and K is the modi"ed Bessel function of order i. To simplify the notation, we have G used the Cartesian de"nition of the matrix elements. The dipole matrix elements for the nuclear excitation are given by DVX"1 f "x(z)"i2 . (163) DG In terms of the spherical coordinates, DV"(2p/3[M (E1,! 1)!M (E1, 1)], and DX" DG DG DG DG (4p/3 M (E1, 0). Thus, Eq. (161) is equal to Eq. (31). DG In expression (162) we have used the approximation > (s, 0)K> (p/2, 0) which is valid for K K high-energy collisions since the quadrupole "eld is strongly peaked at t"0, corresponding to the distance of closet approach. Eqs. (161) and (162) allow us to calculate the dipole excitation cross section by integrating their absolute squares over impact parameter, starting from a minimum impact parameter for which the strong interaction sets in. Neglecting the di!useness of the matter distribution of the nuclei we can write (see Fig. 18) b
(h)KR #R [1#b> (p/2#h)]
with the nuclear radii given by R "1.2A fm. The total cross section is G G
p"2p
db b1"a (b, X)"2 DG
(164)
(165)
@ F where the 122 sign means that an average over all the possible orientations of the projectile, i.e., over all angles X"(h, ), is done. We will apply the above formalism to the Coulomb excitation of Pb by U projectiles. We will give the U an arti"cial deformation in the range b"0}1 to check the dependence of the cross sections with this parameter. The cross section given above contains three terms: p"p #p #p . p is due to the monopole}dipole excitation amplitude, p is due to the quadrupole}dipole excitation amplitude, and p is the interference between them. In Fig. 19 we present the results for the numerical calculation of the quantity
D"100;(p !p@)/p@ (166) which is the percent correction of dipole excitations in Pb by a uranium projectile due to the average over the orientation of the projectile. p@ is the cross section for b"0. We present results for three bombarding energies, 10A GeV, 1A GeV and 100A MeV, and as a function of b. The quantity de"ned by Eq. (166) is independent of the nature of the state excited, since the dipole matrix elements cancel out. They depend on the energy of the state. In order to see how the e!ect depends qualitatively on the energy of the state we used three di!erent excitation energies E "1, DG
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Fig. 19. Percent increase of the Coulomb excitation cross section of dipole states in Pb due to the dependence of the minimum impact parameter on the deformation. The e!ect is shown for U projectiles at 100A MeV, 1A GeV and 10A GeV, respectively, and as a function of the deformation parameter b. The solid (dashed) [dotted] line corresponds to an excitation energy of 25 (10) [1] MeV. For the actual deformation of U, bK0.3, the e!ect is small. Fig. 20. Coulomb excitation cross section of a giant dipole resonance in Pb due to the quadrupole}dipole interaction with 100A MeV uranium projectiles, as a function of the deformation parameter b. These cross section are averaged over all possible orientations of the projectile.
10 and 25 MeV, respectively. These correspond to the dotted, dashed and solid lines in Fig. 19, respectively. One observes from Fig. 19 that the deformation e!ect accounted for by an average of the minimum impact parameter which enters Eq. (165) increases the magnitude of the cross section. Thus the average is equivalent to a smaller `e!ectivea impact parameter, since the cross sections increase with decreasing values of b . The e!ect is larger the greater the excitation energy is. This
e!ect also decreases with the bombarding energy. For very high bombarding energies it is very small even for the largest deformation. These results can be explained as follows. The Coulomb excitation cross section at very high bombarding energies, or very small excitation energies, is proportional to ln[ub (h)/cv)]. Averaging over orientation of the projectile means an average of
ln(b ) due to the additivity law of the logarithm. One can easily do this average and the net result
is a rescaling of b as f b , with f smaller, but very close to one.
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Table 2 Cross sections (in mb) for Coulomb excitation of the giant dipole resonance in Pb by U projectiles at 100A MeV. In the second (third) column the cross sections are due to the monopole (quadrupole)}dipole interaction. The last column is the total cross section. An average over the orientation of the projectile was done. A realistic value of the deformation of U corresponds to bK0.3. But, a variation of b is used to obtain an insight of the magnitude of the e!ect b
p (mb)
p (mb)
p (mb)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1171 1173 1179 1189 1202 1220 1241 1265 1294 1326 1362
0 0.179 0.748 1.773 3.34 5.57 8.61 12.6 17.9 24.7 33.3
1171 1174 1184 1200 1224 1242 1291 1335 1389 1446 1522
For high excitation energies, or small bombarding energies, the cross section is proportional to exp+!2ub (h)/cv, due to the adiabaticity condition [18]. Thus, in these situations, the cross
section is strongly dependent on the average over orientation due to the strong variation of the exponential function with the argument. Now we consider the e!ect of the second term of Eq. (158), namely of the quadrupole}dipole excitations. In Fig. 20 we show the excitation of a giant resonance dipole state in lead (E "13.5 MeV) due to the second term Eq. (158), as a function the deformation parameter b and DG for a bombarding energy of 100A MeV. We assume that the giant dipole state exhausts fully the TRK sum rule, Eq. (96), in lead. Now the average over orientation also includes the dependence of the quadrupole}dipole interaction on X"(h, ). As expected the cross section increases with b. But it is small as compared to the monopole}dipole excitations even for a large deformation. At this beam energy the monopole}dipole excitation is of order of 1 barn. The total cross section contains an interference between the amplitudes a and a. This is DG DG shown in Table 2 for 100A MeV for which the e!ect is larger. The second column gives the cross sections for monopole}dipole excitations of a giant resonance dipole state in lead. The e!ect of the orientation average can be seen as an increase of the cross section as compared to the value in the "rst row (zero deformation). For b"0.3 which is approximately the deformation parameter for U the correction to the cross section is negligible. In the third column the cross section for quadrupole}dipole excitation are given. They are also much smaller than those for the monopole}dipole excitations. The total cross sections, given in the last column, are also little dependent on the e!ect of the deformation. For b"0.3 it corresponds to an increase of 3% of the value of the original cross section ("rst row). This e!ect also decreases with the bombarding energy. For 1A GeV, p@"5922 mb, while p"5932 mb for b"0.3, with all e!ects included. In conclusion, the e!ect of excitation by a deformed projectile, which can be studied by averaging over the projectile orientation, is to increase slightly the cross sections. The inclusion of
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the quadrupole}dipole interaction increases the cross section, too. However, these corrections are small for realistic deformations. They cannot be responsible for the large deviations of the experimental values of the Coulomb fragmentation cross sections from the standard theory [6,18], as has been observed [41,49] for deformed projectiles.
3. Heavy ion excitation of multiphonon resonances 3.1. Introduction Much of the interest on multiphonon resonances relies on the possibility of looking at exotic particle decay of these states. For example, in Ref. [52], a hydrodynamical model was used to predict the proton and neutron dynamical densities in a multiphonon state of a nucleus. Large proton and neutron excesses at the surface are developed in a multiphonon state. Thus, the emission of exotic clusters from the decay of these states are a natural possibility. A more classical point of view is that the Lorentz contracted Coulomb "eld in a peripheral relativistic heavy ion collision acts as a hammer on the protons of the nuclei [6]. This (collective) motion of the protons seem only to be probed in relativistic Coulomb excitation. It is not well known how this classical view can be related to microscopic properties of the nuclei in a multiphonon state. Since there is more energy deposit in the nuclei, other decay channels are open for the multiphonon states. Generally, the GRs in heavy nuclei decay by neutron emission. One expects that the double, or triple, GDR decays mainly in the 2n and 3n decay channel. In fact, such a picture has been adopted in [38,41] with success to explain the total cross sections for the neutron removal in peripheral collisions. The method is the same that we used to explain the one-neutron removal cross sections, i.e., by replacing f by f , and f , in Eq. (152). L L L Although the perspectives for an experimental evidence of the DGDR via relativistic Coulomb excitation were good, on the basis of the large theoretical cross sections, it was "rst found in pion scattering at the Los Alamos Pion Facility [53]. In pion scattering o! nuclei the DGDR can be described as a two-step mechanism induced by the pion-nucleus interaction. Using the Axel}Brink hypotheses, the cross sections for the excitation of the DGDR with pions were shown to be well within the experimental possibilities [53]. Only about 5 years later, the "rst Coulomb excitation experiments for the excitation of the DGDR were performed at the GSI facility in Darmstadt/ Germany [39,40]. In Fig. 21 we show the result of one of these experiments, which looked for the neutron decay channels of giant resonances excited in relativistic projectiles. The excitation spectrum of relativistic Xe projectiles incident on Pb are compared with the spectrum obtained in C targets. A comparison of the two spectra immediately proofs that nuclear contribution to the excitation is very small. Another experiment [39] dealt with the photon decay of the double giant resonance. A clear bump in the spectra of coincident photon pairs was observed around the energy of two times the GDR centroid energy in Pb targets excited with relativistic Bi projectiles. The advantages of relativistic Coulomb excitation of heavy ions over other probes (pions, nuclear excitation, etc.) was clearly demonstrated in several GSI experiments [39}41,54]. A collection of the experimental data on the energy and width of the DGDR is shown in Fig. 22. The data points are from a compilation from pion (open symbols), and Coulomb excitation and nuclear excitation (full symbols) experiments [8].
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Fig. 21. Experimental results for Xe projectile excitation (at 690A MeV) on a Pb target (squares) and a C target (circles). The spectrum for the C target is multiplied by a factor 2 for better presentation. The resonance energies for oneand two-phonon giant resonances are indicated. The dashed curve re#ects the results of a "rst-order calculation for the Pb target. The "gure is taken from Ref. [40]. Fig. 22. Compilation of experimental "ndings with heavy ion (full symbols) and pion induced (open symbols) reactions for the energy, width, and cross sections of the double giant resonance. The data are compared to the energies and widths of the giant dipole resonance, respectively, and to the theoretical values of excitation cross sections.
The dashed lines are guide to the eyes. We see from Fig. 22(a) that the energy of the DGDR agrees reasonably with the expected harmonic prediction that the energy should be about twice the energy of the GDR, although small departures from this prediction are seen, especially in pion and nuclear excitation experiments. The width of the DGDR seems to agree with an average value of (2 times that of the GDR, although a factor 2 seems also to be possible, as we see from Fig. 22(b). Fig. 22(c) shows the ratio between the experimentally determined cross sections and the calculated ones. Here is where the data appear to be more dispersed. The largest values of p /p come from pion experiments, yielding up to a value of 5 for this quantity. We now discuss many features of the double GDR excitation theoretically and some attempts to solve the discrepancies between theory and experiment observed in Fig. 22.
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3.2. Perturbation theory and harmonic models 3.2.1. Sum rules for single and double resonances The simplest way to determine the matrix elements of excitation of giant resonances is by means of sum rules under the assumption that those sum rules are exhausted by collective states. We have done this when we used the sum rules (96), (97). Let us look at these with more details, since they will be useful for the determination of the matrix elements for multiphonon excitations. The conventional sum rules for the dipole and quadrupole transitions, derived without exchange and velocity-dependent corrections, are ( "1) 3 1 NZ e , (167) u "DK"" DG DG 4p 2m A , D Z/A, isoscalar excitations , 1 3R e; u "QK""2 (168) DG DG 2m 4p NZ/A, isovector excitations , , D where DK,M(E1m) and QK,M(E2m). We explain our procedure on the example of the dipole sum rule (167). The right-hand side S of " (167) being calculated for the "xed initial state "i2 in fact does not depend on the choice of "i2. (This dependence is rather weak even if the exchange terms are taken into account). Since S does not " depend on the projection m of the dipole operator DK as well, it is convenient to introduce in usual way the reduced matrix elements of multipole operators,
1 f; I M "OK"i; I M 2"1I M "I lM m2( f; I ""O ""i; I ) , (169) D D J G G D D G G D J G where f stands now for all quantum numbers except angular momentum ones, I and M, and to perform the additional summation of Eq. (167) over m. In such a way one obtains (170) u (2I #1)( f; I ""D""i; I )"3(2I #1)S . DG D D G G " D 'D Now let us take the ground state "02 of an even}even nucleus with angular momentum I "0 as an initial one "i; I 2. If we assume that the single GDR "12,"1; 12 is an isolated state saturating the G corresponding sum rule, we just divide the right-hand side of (170) by the excitation energy u to obtain the reduced matrix element (1""D""0)"S /u . (171) " In order to be able to calculate the cross section of excitation of the double GDR, we have to take the single GDR state "12 as an initial one. The corresponding sum in Eq. (170), according to our assumption, is saturated by (i) `downa transition to the ground state "02, which has negative transition energy !u and, due to the symmetry properties of the Clebsch}Gordan coe$cients, the strength which is 3 times larger than that of Eq. (171), and (ii) `upa transitions to the double GDR states "2; I "¸2 where ¸ can be equal to 0 and 2. The resulting sum rule for the up transitions is (2¸#1)u*(2; ¸""D""1)"12S , " *
(172)
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185
where u*,E !E is the energy of the second excitation. Actually, considering, instead of the _ * sum over m, the original dipole sum rule (167) for "xed m, one can separate the two contributions to the sum (172) and "nd (2; ¸""D""1)"2 S /u* . (173) " Obviously, it is consistent with the sum rule (172). Eqs. (171) and (173) imply the relation between the strengths of sequential excitation processes, (2; ¸""D""1)"2(u /u*)(1""D""0) . (174) For the equidistant vibrational spectrum this result is nothing but the standard Bose factor of stimulated radiation; our result is valid under more broad assumptions. The resulting enhancement factor includes, in addition, the ratio of transition frequencies which, according to the data, is slightly larger than 1. The generalization for the third and higher order excitation processes is straightforward. 3.2.2. Spreading widths of single and double resonances The above assumption of saturation certainly does not account for the fact that the resonances are wide. In fact, this might be also relevant for the calculation of total cross sections since the Coulomb excitation amplitudes given by may vary strongly with the excitation energy. Therefore, they might be sensitive to the shape of the strength function. The widths of the resonances can be taken into account in a simpli"ed approach, as we describe next. In a microscopic approach, the GDR is described by a coherent superposition of one-particle one-hole states. One of the many such states is pushed up by the residual interaction to the experimentally observed position of the GDR. This state carries practically all the E1 strength. This situation is simply realized in a model with a separable residual interaction. We write the GDR state as (one phonon with angular momentum 1M) "1, 1M2"AR "02 where AR is a proper + + superposition of particle}hole creation operators. Applying the quasi-boson approximation we can use the boson commutation relations and construct the multiphonon states (N-phonon states). A N-phonon state will be a coherent superposition of N-particle N-hole states. The width of the GDR in heavy nuclei is essentially due to the spreading width, i.e., to the coupling to more complex quasibound con"gurations. The escape width plays only a minor role. We are not interested in a detailed microscopic description of these states here. We use a simple model for the strength function [15]. We couple a state "a2 (i.e. a GDR state) by some mechanism to more complex states "a2, for simplicity we assume a constant coupling matrix element < "1a"<"a2"1a"<"a2"v. ?? With an equal spacing of D of the levels "a2 one obtains a width C"2p v/D ,
(175)
for the state "a2. We assume the same mechanism to be responsible for the width of the N-phonon state: one of the N-independent phonons decays into the more complex states "a2 while the other (N!1)-phonons remain spectators. We write the coupling interaction in terms of creation (destruction) operators cR(c ) of the complex states "a2 as ? ? <"v(AR c #A cR) . (176) + ? + ?
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For the coupling matrix elements v , which connects an N-phonon state "N2 to the state "N!1, a2 , (N!1 spectator phonons) one obtains v "1N!1, a"<"N2"v1N!1"A "N2"v.(N , , + i.e., one obtains for the width C of the N-phonon state , C"2pN(v/D)"NC ,
(177)
(178)
where C is given by Eq. (175). Thus, the factor N in (178) arises naturally from the bosonic character of the collective states. For the DGDR this would mean C "2C . The data points shown in Fig. 22(b) seem to favor a lower multiplicative factor. We will assume that the damping of the collective modes is mostly due to the coupling to the background of complex con"gurations in the vicinity of the resonance energy. Then the resonance state "j2 gets fragmented acquiring the spreading width C . The stationary "nal states " f 2 in the H region of the GR are superpositions (with the same exact quantum numbers as the collective mode) of the form " f 2"CD"j2# CD"l2 , (179) H J J where "j2 is a pure GR state and "l2 are complex many particle-many hole states. If the resonance component dominates in the excitation process as it should be for the one-body multipole operator, we "nd the "rst-order amplitude aH of the excitation of the individual state " f 2 in the DG fragmentation region aHK[CD]Ha(u ) . (180) DG H H DG Here a stands for the original "rst-order excitation amplitude. As a function of the transition H energy, the probability for the one-phonon excitation is P(u)" ["CD"d(u!u )]"a(u )",F (u)"a(u)" , (181) H H DG H DG H H D where we introduced the strength function F (u). H The traditional derivation of the strength function (see Ref. [55]) is based on the rough assumptions concerning mixing matrix elements and the equidistant spectrum of complex states. The matrix elements < which couple the collective mode to the background states are assumed to HJ be of the same average magnitude for all remote states "l2 from both sides of the resonance. Under those conditions the resulting strength function has the Breit}Wigner (BW) shape C 1 H , F (u)" H 2p (u!u )#C/4 H H where C is the spreading width of the collective resonance, H C "2p1< 2 /d , H HJ J
(182)
(183)
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187
d is the mean level spacing of complex states, coupling matrix elements are averaged over the states "l2 and u is the energy centroid. We will use in our numerical calculations the BW strength H function (182) with the empirical parameters u and C . However, the same procedure can be H H applied to any speci"c form of F (u). Later we come back to the question of justi"cation of the H model leading to Eqs. (182) and (183). The multiphonon states could also be reached by a direct excitation. Quite similarly, we can repeat the above arguments to calculate the probability for the direct excitation of a multiphonon state, with the appropriate spreading width and energy centroid of that state. The direct (or "rst-order) probabilities are then given by P(u)"F (u)"a(u)" . (184) Let us now treat the case of the two-step excitation of GR (double-phonon). For simplicity, we denote the single-phonon state by "12 and the double-phonon state by "22, the corresponding centroids being at u and u , respectively. The total probability to excite the double-phonon state is obtained by P(u)" "a#a"d(u!u ),P(u)#P(u)#P (u) , (185) DG DG DG D where P is the direct (or "rst-order) excitation of the double-phonon state, P is the two-step (or second-order) excitation term, and the last term in Eq. (185) is the interference between the two. 3.2.3. Second-order perturbation theory To second-order, the amplitude for a two-step excitation to a state "22 via intermediate states "12 is given by
R 1 dt e SR< (t) dt e SRY< (t ) , (186) a" (i ) \ \ where < (t) is a short notation for the interaction potential inside brackets of the integral of Eq. (186) for the transition "12P"22. Using the integral representation of the step function
1 if t't , 1 e\ OR\RY dq" H(t!t )"! lim 2pi q#id 0 if t(t , \ B > one "nds [16]
(187)
1 dq i a" a(u )a(u )# P a(u !q)a(u #q) , (188) 2 q 2p \ where P stands for the principal value of the integral. For numerical evaluation it is more appropriate to rewrite the principal value integral in Eq. (188) as
P
dq a(u !q)a(u #q) q \ dq " [a(u !q)a(u #q)!a(u #q) a(u !q)] . q
(189)
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To calculate a(u) for negative values of u, we note that the interaction potential can be written as a sum of an even and an odd part. This implies that a(!u)"![a(u)]H. For three-phonon excitation we use the third term of the time-dependent perturbation expansion, and the same procedure as above (Eqs. (187)}(189)). 3.2.4. Harmonic vibrator model A simpli"ed model, often used in connection with multiphonon excitations, is the harmonic vibrator model. In this model, the resonance widths are neglected and the Coupled-Channel equations can be solved exactly, in terms of the "rst-order excitation amplitudes [15]. The excitation amplitude of the nth harmonic oscillator state, for any time t, is given by (190) aL (t)"([a (t)]L/(n!) exp+!"a (t)"/2, , where a (t) is the excitation amplitude for the 0 (g.s.)P1 (one phonon) calculated with the "rst-order perturbation theory. For the excitation of giant resonances, n can be identi"ed with the state corresponding to a multiple n of the single giant resonance state. This procedure has been often used in order to calculate the cross sections for the excitation of multiphonon giant resonances. Since this result is exact in the harmonic vibrator model, it accounts for all coupling between the states. However, this result can be applied to studies of giant resonance excitation only if the same class of multipole states is involved. I.e., if one considers only electric dipole excitations, and use the harmonic oscillator model, one can calculate the excitation probabilities, and cross sections, of the GDR, double-GDR, triple-GDR, etc. Eq. (190) is not valid if the excitation of other multipolarities are involved, e.g., if the excitation of dipole states and quadrupole states are treated simultaneously. In Ref. [50] a hybrid harmonic oscillator model has been used. In this work, it is assumed that the di!erence between the amplitudes obtained with the harmonic oscillator model and with nth order perturbation theory is due to the appearance of the exponential term on the r.h.s. of Eq. (190). This exponential takes care of the decrease in the occupation amplitude of the ground state as a function of time. As argued in Ref. [50], the presence of other multipole states, e.g., of quadrupole states, together with dipole states, may be accounted for by adding the "rst-order excitation amplitudes for the quadrupole states to the exponent in Eq. (190). This would correct for the #ux from the ground state to the quadrupole states. In other words, Eq. (190) should be corrected to read
(191) aL (nj, t)"([a (nj, t)]L/(n!) exp ! "a (n j , t)"/2 . pYHY The harmonic oscillator model is not in complete agreement with the experimental "ndings. The double-GDR and -GQR states do not have exactly twice the energy of the respective GDR and GQR states [7}9]. Apparently, the matrix elements for the transition from the GDR (GQR) to the double-GDR (double-GQR) state does not follow the boson rule [42]. This is borne out by the discrepancy between the experimental cross sections for the excitation of the double-GDR and the -GQR with the perturbation theory, and with the harmonic oscillator model [7}9]. Thus, a Coupled-Channels calculation is useful to determine which matrix elements for the transitions among the giant resonance states reproduce the experimental data [121].
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Assuming that one has pLH(E) somehow (either from experiments, or from theory), a simple A harmonic model following the discussion above can be formulated to include the widths of the states. As we have mentioned, in the harmonic oscillator model the inclusion of the coupling between all multiphonon states can be performed analytically [15]. One of the basic changes is that the excitation probabilities calculated to "rst-order, P(E, b), are modi"ed to include the #ux of LH probability to the other states. That is, P (E, b)"P(E, b) exp+!P(b), , (192) LH LH LH where P(b) is the integral of over the excitation energy E. In general, the probability to reach LH a multiphonon state with the energy EL from the ground state, with energy E, is obtained by an integral over all intermediate energies
1 PLH H(EL, b)" exp+!P(b), dEL\ dEL\2dE LH LH n! ;P(EL!EL\, b)P(EL\!EL\, b)2P(E!E, b) . LH LH LH
(193)
3.2.5. Comparison with experiments The reactions Xe#Pb at 0.69A GeV and Bi#Pb at 1A GeV have been measured at GSI [39,40]. We apply the formalism developed in the preceding sections to calculate the excitation probabilities and cross sections for these systems. Cross sections (in mb) for the Coulomb excitation of the GDR , GQR and GQR in Xe incident on Pb at 0.69A GeV are given in Table 3. We have assumed that the GDR , GQR and the GQR are located at 15.3, 12.3 and 24 MeV, and that they exhaust 100%, 70% and 80% of the corresponding sum rules, respectively [56]. We used b "1.2(A #A ) fm "13.3 fm as
a lower limit guess and b "15.6 fm suggested by the parameterization [44] as an upper limit
(number inside parentheses). The parameterization [43] yields an intermediate value for this quantity. The contributions to various angular momentum projections of each state are shown separately. In the last column the total cross sections are calculated with the widths of the states taken into account. We use for the GDR , GQR and GQR the BW strength functions (182) with the resonance widths C"4.8, 4 and 7 MeV, respectively [56]. We see that states with higher angular momentum projections are more populated. The inclusion of the widths of the resonances
Table 3 Cross sections (in mb) for the Coulomb excitation of the GDR , GQR and GQR in Xe incident on Pb at 0.69A GeV. The cross sections in the last column are calculated with the widths of the states taken into account. The values outside (inside) parentheses use b "13.3 (15.6) fm
GDR GQR GQR
m"$2
m"$1
m"0
p
p
* 90 (64) 29.7 (25.6)
949 (712) 8.4 (6.09) 6.1 (5.46)
264 (201) 14.3 (10.6) 14 (12.4)
2162 (1630) 211 (150) 84.1 (74.5)
2482 (1820) 241 (169) 102 (93)
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in the calculation increases the cross sections by about 10}20%. The experimental value [40] 1110$80 mb for the GDR is much smaller which made the authors of [40] to claim that the GDR absorbs only 65% of the sum rule (this number apparently contradicts to the systematics of data for real monochromatic photons [56]). Using this value, our result reduces to 1613 (1183) mb which seems to prefer the upper value of b . The numbers in parentheses are also in rough agreement
with the data [40] for the GQR and GQR . Using the formalism developed in Section 2.8 we have also calculated the cross sections for the nuclear excitation of the GQR in the same reaction. The cross sections for the excitation of isovector modes are reduced by a factor [(N!Z)/A] since the isovector mode is excited due to the di!erence in strength of the nuclear interaction between the target and the protons and neutrons of the projectile [55]. This implies that the isovector excitations are strongly suppressed in nuclear excitations. Therefore, we do not consider them here. For the excitation of the GQR we "nd p,"5.3 mb, if we use the deformation parameter bR"0.7 fm for Xe. In the calculation of the nuclear potential we used Fermi density distributions with parameters o "0.17 fm\ and R"5.6 (6.5) fm, a"0.65 (0.65) fm for Xe (Pb). The nucleon}nucleon cross section used was 40 mb. Again we see that the nuclear contribution to the total cross section is very small. The double dipole phonon state can couple to total angular momentum 0 or 2. For the state with ¸"2 there is the possibility of a direct quadrupole Coulomb excitation (¸"0 states cannot be Coulomb excited [6]). For simplicity, we do not consider here the physics of the isospin coupling of the two GDR. We calculated the direct and the two-step probabilities for the excitation of the double-phonon state according to the approach discussed in the previous sections. The total cross sections obtained are shown in Table 4. We found that the principal value term in Eq. (188) contributes very little (less than 1%) to the GDR;GDR cross section via a two-step process. From Table 5 we see that the inclusion of the widths of the "nal (GDR;GDR) and the intermediate (GDR) state increase the cross sections by 10}20%. For the position and width of the GDR;GDR state we took E"28.3 MeV and C"7 MeV, respectively [40] which corresponds to u "15.3 MeV and u "13 MeV, both for ¸"2 and ¸"0. For the calculation of the direct excitation we assumed that the resonance would exhaust 20% of the GQR sum rule. It is based on the hypotheses that the missing strength of the low-lying GQR could be located at the double dipole phonon state as a consequence of the anharmonic phonon coupling of the (QDD)-type. Obviously, it should be considered as highly overestimated upper boundary of the direct excitation.
Table 4 Excitation cross sections (in mb) of the GDR , and of the [GDR]L states in the reaction Pb#Pb at 640A MeV. A comparison with "rst-order perturbation theory and the harmonic oscillator is made State GDR [GDR ] [GDR ] [GDR ]
1st pert. th. 3891 388 39.2 4.2
Harm. osc. 3235 281 27.3 2.4
c.c. 3210 280 32.7 3.2
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Table 5 Cross sections (in mb) for the Coulomb excitation of the double GDR in Xe incident on Pb at 0.69A GeV. The cross sections in the last column are calculated with the widths of the states taken into account. The values outside (inside) parentheses use b "13.3 (15.6) fm
DGDR state
m"$2
m"$1
m"0
p
p
L"0 (two-step) L"2 (two-step) L"2 (direct!20% of SR)
* 23.3 (11.2) 3.27 (2.85)
* 13.4 (6.6) 0.86 (0.77)
22.8 (10.7) 51.4 (26.8) 2.12 (1.88)
22.8 (10.7) 124.8 (62.4) 10.3 (9.12)
28.4 (13.3) 154 (77) 11.8 (10.8)
In Ref. [57] the reduced transition probability for the excitation of double-phonon states within the quasiparticle-phonon model have been calculated. The value B(2>, E2)"4.2e fm has been obtained. Using this value we get that the cross section for the direct excitation of the ¸"2 state is 12 lb, much smaller than what we quote above. We conclude that even in the more optimistic cases the contribution of the direct mechanism to the total cross section for Coulomb excitation of the double-phonon state is much less than that of the two-step process. Another conclusion drawn from the numbers of Table 5 is that the excitation of the ¸"2 double-phonon state is much stronger than for the ¸"0 state. Adding the two contributions we "nd that the total cross section for the excitation of the double-phonon state (excluding the direct mechanism) in the reaction above is equal to 182 (101) mb. The experimental value of Ref. [40] is about 215$50 mb. As stated above, the nuclear contribution to the (direct) excitation of the double-phonon state is not relevant. If we assume again that about 20% of the sum rule strength is exhausted by this state (using e.g. bR"0.1 fm), we get 1.1 mb for the nuclear excitation of the ¸"2 double-phonon state. Contrary to the single phonon case, the appropriate value of b for the
double GDR experiment [40] is b "13.3 fm.
We also compare our results with the experiment of Ritman et al. [39]. They measured the excitation of a Pb target by means of Bi projectiles at 1A GeV and obtained 770$220 mb for the excitation cross section of the double resonance. We calculate the cross sections for the same system, using E "13.5 MeV, C "4 MeV, E "27 MeV and C "6 MeV for the energy position and widths of the GDR and the GDR;GDR in Pb, respectively. Using the formalism developed in Sections 3.2.2 and 3.2.3 and including the e!ects of the widths of the states, we "nd p "5234 b for the excitation of the GDR and p "692 mb for the excitation of the GDR;GDR, using b "1.2(A #A ) fm"14.2 fm. Thus, while the cross section for the
. 2 excitation of single phonons is a factor 2.8 larger than that of the experiment of Ref. [40], the cross sections for the excitation of double phonons is larger by a factor 3.8. This is due to the larger value for the excitation probabilities caused by a larger B(E1) value for this reaction. The parameterization [44] with b "b "16.97 fm would lead to smaller cross sections p "4130 mb and
p "319 mb. We found the ratio of (P #P )/P "9.4 for the excitation of the GDR in the K> K\ K experiment of Ref. [39]. They quote the value 28 in their calculations and "t the gamma-ray angular distribution according to this value. We think that this result could somewhat change the extracted value of the GDR;GDR cross section which is quoted in Ref. [39].
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Using the formalism shown in Section 3.2.4 we "nd that the cross sections for the excitation of three-phonon states in the experiment of Schmidt et al. [40] is equal to 19.2 mb (with b "13.3 fm) while it is equal to 117 mb (with b "14.2 fm) for the experiment of Ritman et al.
[39]. The identi"cation of these resonances is therefore more di$cult, but possible with the present experimental techniques. Using the same arguments leading to Eq. (174) we "nd for the reduced matrix elements, in obvious notations, "D ""3(u /u )"D ", which we used in our calculation. We assumed that u /u Ku /u . These enhancement factors for the excitation of higher phonon states are very important to explain the magnitude of the cross sections. The anharmonic e!ects, suggested in [40] to explain the large excitation of double GDR, are expected to be small since the mixing of single- and double-phonon states is forbidden by the angular momentum and parity. The main anharmonic e!ect, apart from the weak coupling of the double GDR with ¸"2 to GQR, is the IBM-like scattering of dipole phonons which splits ¸"0 and ¸"2 states but hardly changes excitation and decay properties. Another important question is related to the expected width of the multiphonon states. Early estimates [15] presented in Section 3.2.2 indicated that these widths should scale as C "nC . L The experiments show however that a scaling as C "(nC is more appropriate, at least for the L double GDR. We next address in detail di!erent aspects of physics responsible for the width of the double-phonon state. 3.3. General arguments on the width of the double-phonon state Here we discuss in qualitative terms the problem of the width of a collective state which can be thought of as being created by the excitation of two quanta in a complex many-body system. We assume that the genuine decay to continuum is of minor importance at the given excitation energy. Therefore, we focus on the damping width which comes from the fact that the collective mode is a speci"c coherent superposition of simple con"gurations (for instance, of a particle-hole character) rather than a pure stationary state. In the actual excitation process the predominant mechanism is that of the sequential onephonon excitation. Under our assumption that the sum rule is saturated by the GR the intermediate states contribute to this process as far as they contain a signi"cant collective component. Therefore the interference of many incoherent paths can be neglected so that we are interested in the shape P(E) of the excitation function at a given energy E"E #E which can be obtained as a convolution of the single-phonon excitation functions,
P(E)" dE dE P (E )P (E )d(E!E !E ) .
(194)
The same shape should be revealed in the deexcitation process. In this formulation the problem is di!erent from what is usually looked at when one is interested, for example, in sound attenuation. In such classical problems the conventional exponential decrease of the wave intensity does not correspond to the decay of the state with a certain initial number of quanta. Contrary to that, here we have to compare the damping rates of individual quantum states with the "xed number of quanta, single- and double-phonon states in particular.
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193
We have to mention also that in the nuclear GR case quantum e!ects are more pronounced since the temperature corresponding to the relevant excitation energy is less than u, whereas in the measurements of the attenuation of the zero and "rst sound in the macroscopic Fermi liquid [58] the situation is always inverse and the quantum limit is hardly attainable. (In nuclear physics the classical case can be studied with low-lying quadrupole vibrations.) Independently of speci"c features of nuclear structure (level density, A-dependence, shell e!ects, "niteness of the system leading to the linear momentum nonconservation and, therefore, to the estimate of the available phase space which could be di!erent from that for in"nite matter, and so on) we can try to make several comments of general nature. If the anharmonic e!ects could be considered to be small we can assume that the phonons decay independently by what can be described, using the language of stationary quantum mechanics, as mixing to complex background states. The decay rate C (e) of an individual quasi particle (elementary excitation) with energy e depends on the background level density and, whence, on the excitation energy. The decay of a state with n quasiparticles occurs as far as one of the constituents decays. It implies the simple estimate of the width C of the n-quantum L state, C KnC (E/n). For the decay of typical many particle-many hole con"gurations [59}61] L one usually takes the Fermi-liquid estimate C (e)Je which leads to C J¹JE since L the average number of quasiparticles in a typical thermal con"guration at temperature ¹ is nJ¹. This estimate agrees with data. In the case of the pure n-phonon state E/n" u which results in the ratio r ,C /C Kn. L L Thus, the simplest line of reasoning favors the width of the double GR to be twice as big as the width of the single GR. At the "rst glance, this estimate is especially reasonable for the giant dipole since here the anharmonic e!ects, determining the whole pattern of low-lying vibrations, are expected to be very weak. Angular momentum and parity conservation forbids cubic anharmonicity which would mix single- and double-quantum states and in#uence both excitation cross sections and spreading widths. The main anharmonic term, apart from mentioned in Section 3.2.2 weak mixing of the giant quadrupole to the double dipole state with ¸"2, probably corresponds to the phonon scattering similar to that in the IBM. It results in the shift of the double-phonon state from 2 u and splitting of ¸"0 and ¸"2 states hardly changing the decay properties. Experimentally, the energy shift seems to be rather small. There are also other arguments for the width ratio r "2. In our calculation of cross sections we assumed the BW shape (182) of strength functions (181). If the sequential excitation is described by the BW functions P (E ) with the centroid at e and the width C, and P (E ) with corresponding parameters e and C , the convolution (194) restores the BW shape with the centroid at e#e and the total width C#C . For identical phonons it means that the width ratio r "2. As we mentioned in Section 3.2.2, the BW shape of the strength function is derived analytically within the simple model [55] of coupling between a phonon and complex background states. One diagonalizes "rst the Hamiltonian in the subspace of those complex states and get their energies e . J If the underlying dynamics is nearly chaotic, the resulting spectrum will show up level repulsion and rigid structure similar to that of the Gaussian Orthogonal Ensemble (GOE), with the mean level spacing d. Roughly speaking, one can assume the equidistant energy spectrum. The collective phonon "12 at energy E is coupled to those states and corresponding matrix elements < are J assumed to be of the same order of magnitude (much larger than the level spacing d) for all states "l2 in the large energy interval around the collective resonance. Then the energies of the stationary
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states ("nal states " f 2 in the notations of previous sections) are the roots E"E of the secular D equation < J "0 , F(E),E!E ! E!e J J and the distribution of the collective strength, Eq. (179),
(195)
< \ J (196) "CD""[dF/dE]\ D" 1# ## (E !e ) D J J reveals the BW shape (182) and the `golden rulea expression (183) for the width C . We can repeat the procedure for the double phonon state. Phonons of di!erent kind would couple to di!erent background states with di!erent level spacing and coupling matrix elements. It corresponds to independent decay leading as we discussed above to C "C#C . For the identical phonons, we should take into account that the double phonon state "22 is coupled to the states `single phonon#backgrounda and the background states here are the same as those determining the width of the single phonon state "12. This picture is in accordance with the famous Axel} Brink hypotheses. Therefore, the expression for the width, Eq. (183), contains the same level density whereas all coupling matrix elements for the transition to a complex state "l2 (plus a remaining phonon) have to be multiplied by the Bose factor, < "(2< . Thus, we come J J again to r "2. The approach of the proceeding paragraph can be slightly modi"ed by introducing explicitly coupling via a doorway state [62] or GOE internal dynamics [63]. In both cases the Bose factor (2 leads to the same result r "2. In addition, the collective resonance might be further broadened by the coupling to low-lying collective vibrational or rotational modes. For example, in the simplest model where the dipole phonon radiates and absorbs low-energy scalar quanta, it is easy to show that, in the stationary cloud of scalar quanta, their average number, which determines the fragmentation region of the dipole mode, is proportional to the squared number of dipole phonons. Hence it gives a large width ratio r "4. For the nuclei where actual data exist, this is not important since they are rather rigid spherical nuclei with no adiabatic collective modes. On the other hand, one can present some arguments in favor of the width ratio r "(2 which apparently is preferred by the existing data. First of all, this value follows from the convolution (194) of Gaussian distribution functions (instead of BW ones). Of course, this is the inconsistent approach since the experimentalists use BW or Lorentzian "t. But one can easily understand that the result r "(2 is not restricted to Gaussian "t. For an arbitrary sequence of two excitation processes we have 1E2"1E #E 2 and 1E2"1(E #E )2; for uncorrelated steps it results in the addition of #uctuations in quadrature (*E)"(*E )#(*E ). Identifying these #uctuations with the widths up to a common factor, we get for the identical phonons C"2C, or r "(2. The same conclusion will be valid for any distribution function which, as the Gaussian one, has a "nite second moment, contrary to the BW or Lorentzian ones with the second moment diverging. In some sense we may conclude that, in physical terms, the di!erence between r "2 and r "(2
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195
is due to the di!erent treatment of the wings of the distribution functions which re#ect small admixtures of far remote states. In the standard model of the strength function [55] all remote states are coupled to the collective mode equally strong. This is obviously an unrealistic assumption. The shell model (more generally, mean "eld) basis is the `naturala one [64] for estimating a degree of complexity of various states in a Fermi system at not very high excitation energy. In this representation matrix elements of residual interaction couple the collective state (coherent superposition of particle}hole excitations found for example in the framework of the RPA) only to the states of the next level of complexity (exciton class). Those states, in turn, become mixed with more complex con"gurations. This process proliferates and each simple state acquires its spreading, or fragmentation, width 2a"Nd where N stands for a typical number of stationary states carrying the noticeable weight of the ancestor state and the level spacing d is basically the same as in the mean "eld approximation. Inversely, N can be viewed as the localization length of a stationary complex state in the mean "eld basis. In the stochastic limit the local background dynamical properties can be modeled by those of the GOE with the semicircle radius a. This intrinsic spreading width a, which is expected to be of the order of magnitude of typical matrix elements of the original residual interaction between simple con"gurations, is the dynamical scale missed in the standard model which corresponds to the limit aPR. The existence of this intrinsic scale can be associated with the saturation [65] of the width of a single GR at high temperature. The standard model supposedly is valid for the spreading width C small in comparison with a. Because of the relatively weak interaction leading to the isospin impurity, this is the case for the isobaric analog states (IAS) [66,67] where typical spreading widths are less than 100 keV. This approach allows one to explain, at least qualitatively, small variations of the spreading widths of the IAS. The tunneling mixing of superdeformed states with the normal deformed background presents an extreme example of the small spreading width. However, in the case of GR the situation might be di!erent. To illustrate the new behavior in the opposite case of C5a, we can imagine the limit of the almost degenerate intrinsic states with very strong coupling to a collective mode. (The actual situation presumably is intermediate.) Assuming that the unperturbed phonon state has an energy in the same region, one can easily see from Eqs. (195) and (196) that the coupling results in the appearance of the two collective states sharing evenly the collective strength and shifted symmetrically from the unperturbed region by *E"$( <. The physical reason is evident: the interacJ J tion of the background states through the collective mode creates a speci"c coherent superposition which is hybridized with and repelled from the original collective state. The similar e!ect was discussed in di!erent context in [68] and observed in numerical simulations [69]. The well-known doubling of the resonance peak at the passage of a laser beam through a cavity containing a two-level atom is the simplest prototype of such a phenomenon. In this limit one gets the e!ective width of collective response 2*E"2(N1<2"2(aC /n Q J where C is the standard spreading width (183). This e!ective width is linearly proportional to the Q average coupling matrix element. Therefore it should increase by factor (n when applied to a n-phonon collective state. Thus, we anticipate in this limit r "(2. One may say that the phonons do not decay independently being correlated via common decay channels. In the literature the similar result, due to apparently the same physical reasons, was mentioned in [70] referring to the unpublished calculations in the framework of the second RPA.
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3.4. Coupled-channel calculations with inclusion of the GR widths We have seen that the excitation probabilities of excitation of single- and double-giant resonances are quite large. It is worthwhile to study the excitation process with a coupled channels calculation and compare to the other approximations. We will now study this e!ect by using the Coupled-Channels Born approximation. This approximation was used in Ref. [71] to describe the excitation of the double-giant resonance in relativistic heavy ion collisions. It is based on the idea that in such cases only the coupling between the ground state and the dominant giant dipole state has to be treated exactly. The reason is that the transitions to giant quadrupole and to the double-phonon states have low probability amplitudes, even for small impact parameters. However, an exact treatment of the back-and-forth transitions between the ground state and the giant dipole state is necessary. This leads to modi"cations of the transitions amplitudes to the remaining resonances, which are populated by the ground state and the GDR. In Ref. [71] the application of the method was limited to the use of an schematic interaction, and the magnetic substates were neglected. These de"ciencies are corrected here. The method allows the inclusion of the width of the giant resonances in a very simple and straightforward way. It will be useful for us to compare with the coupled-channels calculations with isolated states, as we described in the previous sections. Fig. 23 represents the procedure. The GDR is coupled to the ground state while the remaining resonances are fed by these two states according to "rst-order perturbation theory. The coupling matrix elements involve the ground state and a set of doorway states "DL2, where n speci"es the HI kind of resonance and jk are angular momentum quantum numbers. The amplitudes of these resonances in real continuum states are aL(e)"1 (e) " DL2 , (197) HI where (e) denotes the wave function of one of the numerous states which are responsible for the broad structure of the resonance. In this equation e"E !E , where E is the excitation energy V L V and E is the centroid of the resonance considered. L As we have stated above, in this approach we use the coupled-channels equations for the coupling between the ground state and the GDR. This results in the following coupled-channels equations:
i i a (t)" de1 (e) " D21D " < (t) " 02 exp ! (E #e)t a (t) I I # I C I
I i " de a(e)<(t) exp ! (E #e)t a (t) I C I
I
(198)
and i a (t)"[(a(e)<(t)]H exp+i(E #e)t/ ,a (t) . (199) C I I Above, n"1 stands for the GDR, a denotes the occupation amplitude of the ground state and a the occupation amplitude of a state located at an energy e away from the GDR centroid, C I and with magnetic quantum number k (k"!1, 0, 1). We used the short-hand notation <(t)"1D " < (t)" 02. I I # I
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197
Fig. 23. Schematic representation of the excitation of giant resonances, populated in heavy ion collisions.
Integrating Eq. (199) and inserting the result in Eq. (198), we get the integro-di!erential equation for the ground state occupation amplitude
R 1 aK (t)"! <(t) de"a(e)" dt [<(t )]Hexp+!i(E #e)(t!t )/ ,a (t ) , (200) I I
\ I where we used that a (t"!R)"0. To carry out the integration over e, we should use an C I appropriate parametrization for the doorway amplitude a(e). A convenient choice is the Breit}Wigner (BW) form
C 1 , (201) "a(e)"" 2p e#C/4 where C is chosen to "t the experimental width. In this case, this integral will be the simple exponential
(E #e)t (E !iC /2)t de"a(e)" exp !i "exp !i .
(202)
A better agreement with the experimental line shapes of the giant resonances is obtained by using a Lorentzian (L) parametrization for "a(e)", i.e.,
C E 2 V , (203) "a(e)"" p (E!E)#CE V V where E "E #e. The energy integral can still be performed exactly [72] but now it leads to the V more complicated result
(E #e)t C (E !iC /2)t de"a(e)" exp !i " 1!i exp !i #*C(t) , (204)
2E
where *C(t) is a non-exponential correction to the decay. For the energies and widths involved in the excitation of giant resonances, this correction can be shown numerically to be negligible. It will therefore be ignored in our subsequent calculations. After integration over e, Eq. (200) reduces to
R (E !iC /2)(t!t ) aK (t)"!S <(t) a (t ) , dt [<(t )]H exp !i I I
\ I where the factor S is S "1 for BW-shape and S "1!iC /2E for L-shape.
(205)
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We can take advantage of the exponential time dependence in the integral of the above equation, to reduce it to a set of second-order di!erential equations. Introducing the auxiliary amplitudes A (t), given by the relation I a (t)"1# A (t) , (206) I I with initial conditions A (t"!R)"0, and taking the derivative of Eq. (205), we get I
(208) P (b, E )""a(E !E )" dt exp+iE t ,[<(t )]Ha (t ) , V V V I I \ where "a(E !E )" is given by Eq. (201) or by Eq. (203), depending on the choice of the resonance V shape. To "rst order, DGDR continuum states can be populated through E2-coupling from the ground state or through E1-coupling from GDR states. The probability density arising from the former is given by Eq. (208), with the replacement of the line shape "a" by its DGDR counterpart "a" (de"ned in terms of parameters E and C ) and the use of the appropriate coupling-matrix elements <(t) with the E2 time dependence given by (29). On the other hand, the contribution from the I latter process is
P (b, E )""a(E !E )"S dt exp+iE t , (<(t ))H V V V JI J \ I (E !iC /2)(t!t ) RY a (t ) . dt (<(t )) exp !i (209) ; I
\ We should point out that Eq. (209) is not equivalent to second-order perturbation theory. This would be true only in the limit a (t)P1. In this approach, a (t)O1, since it is modi"ed by the time-dependent coupling to the GDR state. This coupling is treated exactly by means of the Coupled-Channels equations. We consider that this is the main e!ect on the calculation of the DGDR excitation probability. This approach is justi"ed due to the small excitation amplitude for the transition 1P2, since a (t);a (t). Equations similar to (208) can also be used to calculate the GQR and GQR excitation probabilities, with the proper choice of energies, widths, and transition potentials (e.g., < (t), or # < (t), or both). , In the next section we will apply the results of this section to analyze the e!ect of the widths of the GRs in a Coupled-Channels approach to relativistic Coulomb excitation.
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199
Fig. 24. Time dependence of the occupation probabilities "a " and "a ", in a collision with impact parameter b"15 fm. The time is measured in terms of the dimensionless variable q"(vc/b) t. The system is Pb (640A MeV)#Pb.
3.4.1. Zero-width calculations We consider the excitation of giant resonances in Pb projectiles, incident on Pb targets at 640A MeV, which has been studied at the GSI/SIS, Darmstadt [7,9]. For this system the excitation probabilities of the isovector giant dipole (GDR ) at 13.5 MeV are large and, consequently, high-order e!ects of channel coupling should be relevant. To assess the importance of these e!ects, we assume that the GDR state depletes 100% of the energy-weighted sum-rule and neglect the resonance width. As a "rst step, we study the time evolution of the excitation process, solving the CoupledChannels equations for a reduced set of states. We consider only the ground state (g.s.) and the GDR. The excitation probability is then compared with that obtained with "rst-order perturbation theory. This is done in Fig. 24, where we plot the occupation probabilities of the g.s., "a (t)", and of the GDR, "a (t)", as functions of time, for a collision with impact parameter b"15 fm. As discussed earlier, the Coulomb interaction is strongly peaked around t"0, with a width of the order *tKb/cv. Accordingly, the amplitudes are rapidly varying in this time range. A comparison between the CC-calculation (solid line) and "rst-order perturbation theory (dashed line) shows that the high-order processes contained in the former lead to an appreciable reduction of the GDR excitation probability. From this "gure we can also conclude that our numerical calculations can be restricted to the interval !10(q(10, where q"(cv/b) t is the time variable measured in natural units. Outside this range, the amplitudes reach asymptotic values. It is worthwhile to compare the predictions of "rst-order perturbation theory with those of the harmonic oscillator model and the CC calculations. In addition to the GDR, we include the following multiphonon states: a double-giant dipole state ([GDR ]) at 27 MeV, a triple-giant dipole state ([GDR ]) at 40.5 MeV, and a quadruple giant dipole state ([GDR ]) at 54 MeV. The
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coupling between the multiphonon states are determined by boson factors, i.e., for 0P1 and n!1Pn [42]: "1n!1""< ""n2""n"10""< ""12" . (210) # , # , Direct excitations of the multiphonon states from the g.s. are not considered. The angular momentum addition rules for bosons yields the following angular momentum states: ¸"0 and 2, for the [GDR] state; ¸"1, 2, and 3, for the [GDR] state; and ¸"0, 1, 2, 3, and 4, for the [GDR] state. We assume that states with the same number of phonons are degenerate. In Table 5, we show the resulting cross sections. The excitation probabilities and the cross section were calculated with the formalism of Section 3.4. The integration over impact parameter was carried out in the interval b (b(R. As we discuss below, the low-b cut-o! value [42] b "14.3 fm
mocks up absorption e!ects. We have checked that the CC results are not signi"cantly a!ected by the unknown phases of the transition matrix elements. Since the multiphonon spectrum is equally spaced, and the coupling matrix elements are related through boson factors (as in Eq. (210)), the harmonic oscillator and the CC cross sections should be equal. In fact the numerical results of these calculations given in the table are very close. We also see that the excitation cross sections of tripleand quartic-phonon states are much smaller than that for the [GDR]. Therefore, we shall concentrate our studies on the [GDR], neglecting other multiphonon states. Next, we include the remaining important giant resonances in Pb. Namely, the isoscalar giant quadrupole (GQR ) at 10.9 MeV and the isovector giant quadrupole (GDR ) at 22 MeV. Also in this case, we use 100% of the energy-weighted sum rules to deduce the strength matrix elements. In Table 6, we show the excitation probabilities in a grazing collision, with b"14.3 fm. We see that "rst-order perturbation theory yields a very large excitation probability for the GDR state. This is strongly reduced in a c.c. calculation, as we have already discussed in connection with Fig. 24. The excitations of the remaining states are also in#uenced. They are reduced due to the lowering of the occupation probabilities of the g.s. and of the GDR state in the c.c. calculation. As ex pected, perturbation theory and c.c. calculations agree at large impact parameters, when the transition probabilities are small. For the excitation of the [GDR ] state we used second-order perturbation theory to obtain the value in the second column. The presence of the GQR and the GQR in#uence the c.c. probabilities for the excitation of the GDR and the [GDR ], respectively.
Table 6 Transition probabilities at b"14.3 fm, for the reaction Pb#Pb at 640A MeV. A comparison with "rst order perturbation theory is made Trans.
1st pert. th.
c.c.
g.s.Pg.s. g.s.PGDR g.s.PGQR g.s.PGQR g.s.P[GDR ]
* 0.506 0.080 0.064 0.128
0.515 0.279 0.064 0.049 0.092
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201
Fig. 25. Excitation energy spectra of the main giant resonances for both Breit}Wigner and Lorentzian line shapes. The system is Pb (640A MeV)#Pb. Fig. 26. Ratio between the DGDR and the GDR cross sections in Pb#Pb collisions, as a function of the bombarding energy. Table 7 Centroid energies and widths of the main giant resonances in Pb
E (MeV) C (MeV)
GDR
DGDR
GQR GQ
GQR GT
13.5 4.0
27.0 5.7
10.9 4.8
20.2 5.5
3.4.2. Ewect of resonance widths We now turn to the in#uence of the giant resonance widths on the excitation dynamics. We had considered this in Section 3.2. But, now we show that the coupled-channels e!ects lead to important quantitative modi"cations of the results. We use the CCBA formalism developed in Section 3.4. Schematically, the CC problem is that represented Fig. 23. As we have seen above, the strongest coupling occurs between the g.s. and the GDR. In Fig. 25, we show the excitation energy spectrum for the GDR, the DGDR (a notation for the [GDR ]), GQR and GQR . The centroid energies and the widths of these resonances are listed in Table 7. The "gure shows excitation spectra obtained with both Breit-Wigner (BW) and Lorentzian (L) line shapes. One observes that the BW and L spectra have similar strengths at the resonance maxima. However, the low-energy parts (one or two widths below the centroid) of the spectra are more than one order of magnitude higher in the BW calculation. The reason for this behavior is that Coulomb excitation favors low energy transitions and the BW has a larger low
202
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Fig. 27. Dependence of p and p on the GDR width, treated as a free parameter. For details see the text. The %"0 "%"0 system is Pb (640A MeV)#Pb.
energy tail as compared with the Lorentzian line shape. The contribution from the DGDR leads to a pronounced bump in the total energy spectrum. This bump depends on the relative strength of the DGDR with respect to the GDR. In Fig. 26, we show the ratio p /p as a function of "%"0 %"0 the bombarding energy. We observe that this ratio is roughly constant in the energy range E /A"200}1000 MeV and it falls beyond these limits. This range corresponds to the SIS-energies at the GSI-Darmstadt facility. We now study the in#uence of the resonance widths and shapes on the GDR and DGDR cross sections. This study is similar to that presented in Ref. [71], except that we now have a realistic three-dimensional treatment of the states and consider di!erent line shapes. In the upper part of Fig. 27, denoted by (a), we show p as a function of C , treated as a free parameter. We note %"0 %"0 that the BW and ¸ parameterizations lead to di!erent trends. In the BW case the cross section grows with C while in the L case it decreases. The growing trend is also found in Ref. [71], %"0 which uses the BW line shape. The reason for this trend in the BW case is that an increase in the GDR width enhances the low energy tail of the line shape, picking up more contributions from the low energy transitions, favored in Coulomb excitation. On the other hand, an increase of the GDR width enhances the doorway amplitude to higher energies where Coulomb excitation is weaker. In Fig. 27(b) and (c), we study the dependence of p on C . In (b), the DGDR width is kept "xed at %"0 %"0 the value 5.7 MeV while in (c) it is kept proportional to p , "xing the ratio C /C "(2. %"0 "%"0 %"0 The "rst point to be noticed is that the BW results are systematically higher than the L ones. This is a consequence of the di!erent low-energy tails of these functions, as discussed above. One notices also that p decreases with C both in the BW and L cases. This trend can be understood in "%"0 %"0
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203
Table 8 Cross sections (in mb) for the excitation of giant resonances in lead, for the reaction Pb#Pb at 640A MeV. See text for details GDR
DGDR
GQR GQ
GQR GT
2704
184 (199) [198]
347
186
terms of the uncertainty principle. If the GDR width is increased, its lifetime is reduced. Since the DGDR is dominantly populated from the GDR, its short lifetime leads to decay before the transition to the DGDR. To assess the sensitivity of the DGDR cross section on the strength of the matrix elements and on the energy position of the resonance, we present in Table 8 the cross sections for the excitation of the GDR, DGDR, GQR and GQR , obtained with the CCBA approximation and 100% of the sum-rules for the respective modes. In this calculation we have included the strong absorption, as explained in Section 2.7. For comparison, the values inside parenthesis (and brackets) of the DGDR excitation cross section include a direct excitation of the ¸"2 DGDR state. We assumed that 20% of the E2 sum rule could be allocated for this excitation mode of the DGDR. The cross sections increase by less than 10% in this case. The value inside parentheses (brackets) assume a positive (negative) sign of the matrix element for the direct excitation. Since the excitation of the DGDR is weak, it is very well described by Eq. (209) and the DGDR population is approximately proportional to the squared strength of <. Therefore, to increase the DGDR cross section by a factor of 2, it is necessary to violate the relation E "2E by "%"0 %"0 the same factor. This would require a strongly anharmonic Hamiltonian for the nuclear collective modes, which would not be supported by traditional nuclear models [42]. Arguments supporting such anharmonicities have recently been presented in Refs. [73}75]. Another e!ect arising from anharmonicity would be the spin or isospin splitting of the DGDR. Since the Coulomb interaction favors lower energy excitations, it is clear that a decrease of the DGDR centroid would increase its cross section. A similar e!ect would occur if a strongly populated substate is splitted to lower energies. To study this point, we have varied the energy of the DGDR centroid in the range 20 MeV4E 427 MeV. The obtained DGDR cross sections (including direct excitations) are "%"0 equal to 620 mb, 299 mb and 199 mb, for the centroid energies of 20, 24 and 27 MeV, respectively. Although the experimental data on the DGDR excitation [7}9] seem to indicate that E &2E , a small deviation (in the range of 10}15%) of the centroid energy from this value "%"0 %"0 might be possible. However, the data are not conclusive, and more experiments are clearly necessary. We conclude, that from the arguments analyzed here, the magnitude of the DGDR cross section is more sensitive to the energy position of this state. The magnitude of the DGDR cross section would increase by a factor 2 if the energy position of the DGDR decreases by 20%, as found in Refs. [73}75], due to anharmonic e!ects. In Ref. [19] one obtained p "620, 299 and "%"0 199 mb for the centroid energies of E "20, 24 and 27 MeV, respectively. This shows that "%"0 anharmonic e!ects can play a big role in the Coulomb excitation cross sections of the DGDR, depending on the size of the shift of E . However, in Ref. [42] the source for anharmonic e!ects "%"0 were discussed and it was suggested that it should be very small, i.e., *E"E !2E K0. "%"0 %"0
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The anharmonic behavior of the giant resonances as a possibility to explain the increase of the Coulomb excitation cross sections has been studied by several authors (see also Ref. [76], and references therein). It was found that the e!ect is indeed negligible and it could be estimated [76] as *E(E /(50.A)&A\ MeV. %"0 One attempt to explain the larger experimental cross section is to include contributions from excitation of a single coherent phonon on `hota "ne structure states (Brink}Axel mechanism). Recently [77,78] this has been done through two di!erent approaches. In the "rst one [77], the nucleus is described as a collective harmonic oscillator interacting with a set of oscillators representing statistical degrees of freedom. In the second [78], a statistical approach along the lines proposed by Ko [79] (see also [15]) is used. These works indicate that the Brink}Axel mechanism should play an important role, being able to explain, in part, some discrepancies between theory and experimental cross sections. Further work along these lines were published by, Hussein and collaborators [80}82]. In a recent publication [83], the in#uence of the isospin structure of the double-giant resonance was studied in detail. It was shown that this structure also leads to an enhancement of the calculated cross sections. The calculations discussed so far are based on macroscopic properties of the nuclei, sum rules, etc. Now we show that, in order to obtain a better quantitative description of double-giant resonances it is necessary to include the internal degrees of freedom of the nuclei appropriately. We will discuss this next. But, we "rst describe the formalism that we will use for this purpose. 4. Description of one- and multi-phonon excited states within the quasiparticle-phonon model 4.1. The model Hamiltonian and phonons The Hamiltonian, H, of the quasiparticle-phonon model (QPM) (see Refs. [84}86] for more details) is introduced on the basis of physical ideas of nucleons moving in an average "eld and interacted among each other by means of a residual interaction. Schematically it can be written in the form H"H
#H #H . (211) We limit ourselves here only by the formalism for even}even spherical nuclei. The "rst term of Eq. (211), H , corresponds to the average "eld for neutrons (n) and protons (p). In the second quantized representation it can be written in terms of creation (annihilation) a> (a ) operators of HK HK particles on the level of the average "eld with quantum numbers j,[n, l, j] and m as follows: L N H " E a> a , (212) H HK HK O H K where E is the energy of the single-particle level degenerated in spherical nuclei by magnetic H quantum number m. The second term of Eq. (211), H , corresponds to residual interaction responsible for pairing in non-magic nuclei. In the QPM this interaction is described by monopole pairing with a constant matrix element G O L N a ] , (213) H " G ((2j#1)(2j #1)[a> a> ] [a HK H\K HY\KY HYKY O O H HY
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[a>a>] " CHI a> a> , (214) H HY HI HKHYKY HK HYKY K KY where CHI is the Clebsch}Gordan coe$cient. Since the QPM is usually applied for a description HKHYKY of properties of medium and heavy nuclei with a "lling of di!erent subshells for neutrons and protons, the neutron}proton monopole pairing is neglected. The residual interaction, H , is taken in the QPM in a separable form as a multipole decomposition. Its part in the particle}hole channel can be written as ! HN\F" (iH#oiH)M> (q)M (oq) , (215) HI HI HI OM where iH are the model parameters which determine the strength of isoscalar (isovector) residual interaction. The multipole operator M> (q) has the form HI M> (q)" 1 jm " iHf O(r)> (X) " j m 2a> a HI H HI HK HYKY H K HY KY for the natural parity states and the form
(216)
M> (q)" 1 jm " iJf O(r)[r ) Y (X)] " j m 2a> a (217) HI J JK HI HK HYKY H K HY KY JK for the unnatural parity states. The function f O(r) is a radial formfactor which in actual calculations H is taken either as rH or as a derivative of the central part of the average "eld: f O(r)"d;O(r)/dr. The H value q"!1(#1) corresponds to neutrons (protons). We will not consider here the residual interaction in the particle}particle channel which is the most important for the description of two-nucleon transfer reactions. The basic QPM equations are obtained by means of step-by-step diagonalization of the model Hamiltonian (211). In the "rst step its "rst two terms (212) and (213) are diagonalized. For that the Bogoliubov's canonical transformation from particle creation (annihilation) operators to quasiparticle creation (annihilation) operators a> (a ) is applied: HK HK a> "u a> #(!1)H\Kv a . (218) HK H HK H H\K The ground state of even}even nucleus, "2 , is assumed as a quasiparticle vacuum: a "2 ,0. Then O HK O the energy of the ground state is minimized:
(219) d 1"H #H "2 # k (u#v!1) "0 , O H H H H where k are Lagrange coe$cients. The result of this minimization are the well-known BCS H equations solving which one obtains correlation functions C "G u v and chemical potentials O O H H H j for neutron and proton systems. The coe$cients of the Bogoliubov transformation u and v can O H H be calculated from these values as follows:
1 E !j O , u"1!v , v" 1! H H 2 H H e H
(220)
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where e is the quasiparticle energy: H e "(C#[E !j ] . H O H O
(221)
In magic nuclei the BCS equations yield a zero value for the correlation function and the position of the chemical potential in the gap between particles and hole is uncertain. This results in vanishing of monopole pairing correlations and the Bogoliubov's coe$cients u (v ) equal to 0(1) for H H holes and to 1(0) for particles, respectively. After diagonalization of the "rst two terms of the model Hamiltonian (211) they can be written as: L N H #H " e a> a H HK HK O H K
(222)
and the multiple operator (216) in terms of quasiparticle operators has the form
u> O f H HHY ([a>a>] #(!1)H\I[a a ] )!v\B ( jj ; jk) , HHY M> (q)" H HY HI HY H H\I HHY O HI 2 (2j#1 HHY B ( jj ; jk)" (!1)HY>KYCHI a> a , O HKHYKY HK HY\KY KKY
(223)
(224)
where f H"1 j""iHf O(r)> (X)"" j 2 is the reduced matrix element of the multipole operator. We also HHY H H introduced the following combinations of the Bogoliubov's coe$cients: u!"u v $u v and HHY H HY HY H v8"u u Gv v to be used below. HHY H HY H HY We have determined the ground state of even}even nuclei as the quasiparticle vacuum. In this case, the simplest excited states of nucleus are two-quasiparticle states, a> a> "2 , which corresHK HYKY O pond to particle}hole transitions if monopole pairing vanishes. Two fermion quasiparticle operators couple to the total integer angular momentum corresponding to the Bose statistics. Thus, it is convenient to project the bi-fermion terms [a>a>] and [a a ] in Eq. (223) into the space of H HY HI HY H H\I quasi-boson operators. Following this boson mapping procedure, we introduce the phonon operators of the multipolarity j and projection k as 1L N Q> " +tHG [a>a>] !(!1)H\IuHG [a a ] , . HHY H HY HI HHY HY H H\I HIG 2 O HHY
(225)
The total number of di!erent phonons for the given multipolarity j should be equal to the sum of neutron and proton two-quasiparticle states coupled to the same angular momentum. The index i is used to number these di!erent phonons. One obtains the coe$cients tHG and uHG of the linear transformation (225) by diagonalization of HHY HHY the model Hamiltonian in the space of one-phonon states, Q> "2 . This can be done for example by HIG NF applying again the variation procedure
d 1"Q HQ> "2 !(u /2) +(tHG )!(uHG ),!2 HIG HIG NF HG HHY HHY HHY
"0 .
(226)
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It yields the well-known equations of the random-phase approximation (RPA) which for the case of the separable form of the residual interaction in ph-channel may be written as
(iH#iH)XH(u)!1 (iH!iH)XH(u) L L "0 , (iH!iH)XH(u) (iH#iH)XH(u)!1 N N
(227)
where the following notation have been used: 1 O ( f H u>)(e #e ) H HY . HHY HHY (228) XH(u)" O 2j#1 (e #e )!u HY HHY H The determinant equation (227) is a function of the nucleus excitation energy u. Solving this equation for each multipolarity jL, one obtains the spectrum of nuclei one-phonon excitation u . HG The index i in the de"nition of the phonon operator (225) gets the meaning of the order number of the solution of Eq. (227). The fermion structure of phonon excitation, i.e. the amplitudes t and u, corresponding to the contribution of di!erent two-quasiparticle components to the phonon operator, are obtained from the following equation:
t HG f H (q)u> 1 HHY HHY , (q)" e #e Gu u HHY (2YHG H HY HG O
(229)
where the value YHG is determined from normalization condition for phonon operators: O L N 1"Q Q> "2 " +(tHG )!(uHG ),"2 HIG HIG HHY HHY O HHY and one obtains
YHG">HG#>HG O O \O
(230)
1!(iH#iH)XH(u ) O HG , (iH!iH)XH (u ) \O HG
(231) 1 O ( f H u>)(e #e )u H HY HG . >HG" HHY HHY O [(e #e )!u] 2j#1 H HY HHY Equations (227), (229) and (231) correspond to natural parity phonons. Similar equations are valid for unnatural parity phonons by substituting the reduced spin-multipole matrix element f NJ H and HHY combination of coe$cients of Bogoliubov transformation u\ for f H and u>, respectively. Also, HHY HHY HHY amplitude uHG changes the sign in Eq. (229) for unnatural parity phonons. HHY The RPA equations have been obtained under the assumption that the nucleus ground state is the phonon vacuum, Q "2 ,0. This means that the ground-state correlations due to the last HIG NF term of the model Hamiltonian, H , are taken into account. If they are not accounted for and the ground state is still considered as a quasiparticle vacuum "2 , one obtains the so-called O Tamm}Dankov approximation (TDA). The TDA equations can be easily obtained from the RPA ones by neglecting backward going amplitudes in the de"nition of the phonon operator (225), i.e. applying uHG ,0. HHY
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Table 9 Parameters of Woods}Saxon potential, Eq. (233), for di!erent A-mass regions A
49 59 91 121 141 209
Neutrons
Protons
RL (fm)
aL (fm)
RN (fm)
aN (fm)
!41.35 !46.20 !44.70 !43.20 !45.95 !44.83
4.852 5.100 5.802 6.331 6.610 7.477
0.6200 0.6200 0.6200 0.6200 0.6200 0.6301
!9.655 !9.540 !9.231 !8.921 !9.489 !8.428
!58.65 !53.70 !56.86 !59.90 !57.70 !60.30
4.538 4.827 5.577 6.133 6.454 7.359
0.6301 0.6301 0.6301 0.6301 0.6301 0.6301
!9.506 !8.270 !9.609 !10.363 !10.069 !11.186
The relation between the wave functions of the phonon and quasiparticle vacuums is the following [84]:
1 1 "2 " exp ! (tHG )\uHG (!1)H\I[a> a> ] [a> a> ] , (232) NF N HH H H HI H H H\I 4 HH H H O H GI HH where N is a normalization factor. For actual numerical calculations one needs to determine the model parameters. The average "eld for neutrons and protons is described in the QPM by phenomenologic Woods}Saxon potential:
1 d
(233)
The parameters of this potential for di!erent A-mass regions are listed in Table 9 (see, also Ref. [87]). We usually use RO "RO , aO "aO , and R "RN . All single-particle levels from the JQ JQ ! bottom are included in calculation. The single-particle continuum is approximated by narrow quasibound states. This approximation gives a good description of the exhaust of the energy weighted sum rules (EWSR) for low values of j in medium and heavy nuclei. For the lead region we use the single-particle spectrum near the Fermi surface from Ref. [88] which was adjusted to achieve a correct description of low-lying states in neighboring odd nuclei. The parameters of the monopole G have been "tted to reproduce the pairing energies. O The parameters of the residual interaction are obtained the following way. The strength of the residual interaction for jL"2> and 3\ is adjusted to reproduce the properties (excitation energy and B(Ej) value, known from experiment) of the 2> and 3\ states. Usually it is not possible within one-phonon approximation, discussed in this subsection, if su$ciently large single-particle spectrum is used. When the energy of the lowest excitation is adjusted to the experimental value, the RPA equation yields an overestimated collectivity, B(Ej) value, for this state. And vice versa, if the collectivity of this state is reproduced, the excitation energy is too high as compared to the experimental value. The situation su$ciently improves when the coupling of one-phonon states to
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more complex con"gurations is taken into account as will be discussed in the next subsection. For the lowest excited state the coupling to complex con"gurations results in the energy shift downwards. Thus, for nuclei not very far from a closed shell it becomes possible to achieve a good description of both, the excitation energy and the B(Ej) value. The ratio between isoscalar and isovector strength of the residual interaction is usually "xed as iH/iH"!1.2 in calculation with the radial formfactor of the multipole operator as a derivative of the average "eld. With this ratio the best description of isovector multipole resonances with j'1 is achieved although the experimental information on these resonances is still sparse. For the dipole}dipole residual interaction the strength parameter are adjusted to exclude the spurious center of mass motion and to obtain a correct position of the GDR centroid. For the phonons with the multipolarity j54 the same procedure of adjusting the strength parameters as for jL"2> and 3\ cannot be applied. First, it is because the lowest states of high multipolarity are much less collective and their properties are more sensitive to description of single-particle levels near the Fermi surface than to the strength of the residual interaction. Second, in many cases the lowest states with j54 are either two-phonon states or the states with a large admixture of two-phonon con"gurations, thus, their properties are determined by phonons of another multipolarity. For these reasons we use iH "i> for even parity phonons and iH "i\ for odd parity in calculation with f O(r)"d;O(r)/dr. In fact, the di!erence between i> and i\ does not exceed H a few percent with this radial formfactor of residual force.
4.2. Mixing between simple and complex conxgurations in wave functions of excited states Diagonalization of the model Hamiltonian in the space of one-phonon states allows us to write it in the form H" u Q> Q #H , HG HIG HIG HIG
(234)
f H v\ 1 H "! [(!1)H\IQ> #Q ] HHY HHY B ( jj ; j!k)#h.c. , HIG H\IG O 2 HIG HHYO (2YHG O
(235)
where the origin of the second term in Eq. (234) can be traced back to the last term of multipole operator (223) which cannot be projected onto the space of the phonon operators. On the other hand, applying Marumori expansion technique [89], one may expand the operator B ( jj ; j!k)&a>a in an in"nite sum of even-number phonon operators. Keeping only the "rst O term of this expansion, the non-diagonal term of the model Hamiltonian, H , in the space of phonon operators may be re-written as H H " ;HGG(ji)Q> [Q Q ] #h.c. , HG HIG H I G H I G HI HIG HIG HIG
(236)
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where the matrix element of interaction between one- and two-phonon con"gurations, ;HG(ji), can HG be calculated by making use of the internal fermion structure of phonons, i.e. t and u coe$cients, and reduced matrix elements of the separable force formfactor, f H . It has the form HH ;HGG(ji)"1Q "H"[Q> Q> ] 2"(!1)H>H\H HG HG HG HG H
(2j #1)(2j #1) 2
f H v8 j j j L N ; HH HH (tHGuHG$tHGuHG) HH HH HH HH j j j (YHG O HHH O f H v8 j j j # HH HH (uHG uHG$tHG tHG) HH HH (YHG j j j HH HH O f H v8 j j j # HH HH (tHG tHG$uHG uHG) . (237) HH HH (YHG j j j HH HH O The upper (lower) sign in each of three terms in Eq. (237) correspond to multipole (spin-multipole) matrix element f H , f H or f H, respectively. HH HH HH Thus, we have completed a projection of the nuclear Hamiltonian into the space of phonon operators. Now we may assume that phonons obey boson statistics and work in the space of boson operators only. The presence of the term of interaction, H , in the model Hamiltonian means that the approximation, in which excited states of the nucleus are considered as pure one-, two-, multi-phonon states, is not su$cient. In fact, we have already mentioned above that it is not possible to describe the properties of the lowest collective vibrations in spherical nuclei in one-phonon approximation. It is also well-known that the coupling between one- and two-phonon con"gurations is the main mechanism for the damping of giant resonances. All this means that one needs to go beyond the approximation of independent phonons and take into account a coupling between them. To accomplish this task we write the wave function of excited states with angular momentum J and projection M in even}even nuclei in the most general form as a mixture of one-, two-, etc. phonon con"gurations:
DJ (J) ?@ [Q> Q>] WJ(JM)" SJ(J)Q> # ? ? ? @ (+ (1#d ? ?@ ? @
¹J (J) ?@A [Q> Q>Q> ] #2 # ? @ A (+ (1#d ?@A ? @ A d
,
(238)
NF
"d #d #d #2d d . (239) ? @ A ? @ ? A @ A ? @ ? A By greek characters we mean the phonon's identity, i.e. its multipolarity and order number, a,jLi, the index l ("1, 2, 3,2) labels whether a state J is the "rst, second, etc., state in the total energy spectrum of the system. It is assumed that any combination a, b, c of phonons appears only once. The second and the third terms in Eq. (238) include phonons of di!erent multipolarities and parities, they only must couple to the same total angular momentum J as the one-phonon term.
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Let us limit the wave function of excited states by three-phonon terms and diagonalize the model Hamiltonian of Eqs. (234) and (236) in the space of these states. We use for that a minimization procedure d+1WJ(JM)"H"WJ(JM)2!E(1WJ(JM) " WJ(JM)2,"0 , (240) V which yields a set of linear equations over unknown wave function coe$cients SJ(J), DJ (J) and ?@ ? ¹J (J): ?@A (u !E()SJ(J)# DJ (J);I ? "0 , V ? ?@ ? ?@ ? @ SJ(J);I ? #(u #u !E()DJ (J)# ¹J (J);I ?@ "0 , ? ? @ V ?@ ?@A ?@ ?@A ?@A ? (241) DJ (J);I ?@ #(u #u #u !E()¹J (J)"0 . ?@A ?@ ? @ A V ?@A ?@ Applying boson commutation relations for phonons, the matrix element of interaction between two- and three-phonon con"gurations, (242) ;I ?@ "(1#d (1#d 1[Q Q ] "H "[Q> Q>Q> ] 2 , ?@A ? @ ? @ A ? @ (+ ? @ A (+ can be expressed as a function of matrix elements of interaction between one- and two-phonon con"gurations, ;I ? "(1#d 1Q "H "[Q> Q>] 2"(1#d ;@(a ) , ? @ ? ? @ (+ ? @ ? ?@ as follows:
(243)
;I ?@ "(1#d [;I ? d #;I @ d ] ?@A @ A @ ? @ A ? ? @ A #(1#d [;I ? d #;I @ d ]#(1#d [;I ? d #;I @ d ] ? A @ @ ? A ? @ ? @ @ A ? @ ? A ? @ ? A (244) and the value ;I @(a ) is calculated according to Eq. (237). Since we have used pure boson ? commutation relations for phonons the two-phonon con"guration [a b ] couples only to those ( three-phonon con"gurations [a b c ] where either a , b or c are equal to a or b . This is ( governed by d-functions in Eq. (244). The number of linear equations (241) equals to the number of one-, two- and three-phonon con"gurations included in the wave function (238). Solving these equations we obtain the energy spectrum E( of excited states described by wave function (238) and the coe$cients of wave function J (241), S, D and ¹. It should be pointed out that within this approximation, in which phonons are considered as ideal bosons and nuclear Hamiltonian includes one-phonon exchange term, multi-phonon con"gurations of course possess no anharmonicity features. The strength of any one- or manyphonon con"guration included in the wave function (238) fragments over some energy interval due to the interaction with other con"gurations. But the centroid of the strength distribution remains at the unperturbed energy. Thus, the energy centroid of two-phonon con"guration
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[a b ] equals exactly to the sum of energies of a and b phonons for all values of J. To ( consider anharmonic properties of multi-phonon states, one needs to go beyond pure boson features of excitations in even}even nuclei and take into account their internal fermion structure. Another reason to return back to the fermion origin of phonon excitations is two main problems in considering multi-phonon states associated with the boson mapping procedure. The "rst problem is an admixture of spurious npnh con"gurations which violate Pauli principle in the wave function of n-phonon state. The second is related to the fact that the set of pure n-phonon states is mathematically non-orthonormal if the internal fermion structure of phonons is taken into account (see Refs. [90,91] for more details). To overcome these problems we will keep on using a phonon's imaging of nuclear excitation and use the same expression for the wave function of excited states (238) but in calculation of the norm of this wave function, 1WJ(JM) " WJ(JM)2, and the energy of this state, 1WJ(JM)"H"WJ(JM)2, we will use exact commutation relations between phonon operators: [Q , Q> ]}"d d d ! a> a H HY I IY G GY HK HYKY HIG HYIYGY HHYH KKYK CHI CHYIY !(!)H>HY>I>IYuHGuHYGYCH\I CHY\IY , ;+tHG tHYGY HH HYH HKH K HYKYH K HYH HH HYKYHK HKHK and exact commutation relations between phonon and quasiparticle operators:
(245)
[a , Q> ]}" tHG CHI a> , HHY HKHYKY HYKY HK HIG HYKY (246) [a> , Q> ]}"(!1)H\I uHG CH\I a . HHY HKHYKY HYKY HK HIG HYKY Also we will not expand the operator B ( jj ; jk) in Eq. (235) into a sum of phonon operators but use O its exact fermion structure. The "rst term of Eq. (245) corresponds to the ideal boson approximation while the second one is a correction due to the fermion structure of phonon operators. The overlap matrix elements between di!erent two-phonon con"gurations modify as 1[Q Q ] "[Q>Q>] 2"1[b b ] "[b>b>] 2#K((b a "ab) , (247) @Y ?Y ( ? @ ( @Y ?Y ( ? @ ( where b> is the ideal boson operator and the quantity K, ? K((b a "ab)"K((j i j i "j i j i )"((2j #1)(2j #1)(2j #1)(2j #1) j j j ;(!1)H>H (!1)H>H j j j (tHG tHG tHG tHG!uHG uHG uHG uHG) , HH HH HH HH HH HH HH HH HH HH j j J (248)
is the Pauli principle correction coe$cient. The experience of realistic calculations shows that usually "K((ba"ab)"<"K((b a "ab)" (where aOa and/or bOb ) and that the so-called diagonal Pauli principle approximation, K((b a "ab)"K((ab)d d , provides rather good accuracy and ? ?Y @ @Y su$ciently simpli"es the calculation. For these reasons we will use this diagonal Pauli principle approximation in what follows.
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A similar expression, as (247), is valid for the overlap matrix elements between di!erent three-phonon con"gurations. It can be used as a de"nition of the Pauli principle correction quantity K((c b a "abc) which we will also keep in diagonal approximation only. The relation ' between K((ab) and K((abc) quantities is the following [92]: '
(249) K((abc)"K'(ab) 3# ;(abJc; I, I )K'Y(bc) , ' 'Y where ; stands for the Jahn coe$cients [93]. When internal fermion structure of phonons is taken into account and exact commutation relations (245), (246) are applied the secular equation (241) transforms into (u !E()SJ(J)# DJ (J);I ? "0 , ?@ ? V ? ?@ ? @ SJ(J);I ? #(u #u #*u( !E()DJ (J)# ¹J (J);I ?@ "0 , ?@ ?@A ? ? @ ?@ V ?@ ?@A ?@A' ? DJ (J);I ?@ #(u #u #u #*u( !E()¹J (J)"0 . (250) ?@A ?@ ? @ A ?@A V ?@A ?@ The values *u( and *u( "*u( #*u( #*u( are anharmonicity shifts of two- and ?@ ?@A ?@ @A ?A three-phonon con"gurations, respectively, due to the Pauli principle corrections. In diagonal approximation they can be calculated according to
K((a b ) L N X? X@ O# O . *u( "! (251) ?@ Y? Y@ 4 O O O Another role of Pauli principle corrections is a somewhat renormalization of the interaction between n- and (n#1)-phonon con"gurations. We have used the same notations for these matrix elements ;I ? as in the case of the `ideal boson approximationa (see, Eqs. (242) and (243)). But ?@ calculating the matrix elements 1Q "H "[Q> Q>] 2 we take into account the fermion structure ? ? @ (+ of phonons and nuclear Hamiltonian and obtain ;I ? "(1#d ;@(a );[1#K((a b )] , (252) ? @ ? ?@ where the value ;@(a ) is calculated again according to Eq. (237). A similar additional factor ? [1#1/2;K((a b c )] receives the matrix element of interaction between two- and three-phonon ' con"gurations. The minimal value of the quantity K((ab) equals to !2. It corresponds to the case of the maximal Pauli principle violation, i.e. to a spurious multi-phonon con"guration. It happens only when a and b phonons are purely two-quasiparticle states. In such a case the matrix element of interaction ;I ? ,0 (see Eq. (252)) and the spurious state is completely separated from other ?@ states. While dealing with collective a and b phonons, when a possible admixture of the spurious four-quasiparticle con"gurations is small, the value of K((ab) is close to 0. Nevertheless, the value of the anharmonicity shift *u( is not vanishingly small for the later because of the relatively large ?@ value of the ratio X?/Y? in Eq. (251). This shift is the largest one for the collective low-lying O O
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multi-phonon con"gurations. For non-collective multi-phonon states the shift is small because of the small value of the above-mentioned ratio. Equations (250) have been obtained under two main assumptions. The "rst one is the already discussed diagonal Pauli principle approximation. The second assumption is the neglecting of the higher-order terms of the interaction part of the nuclear Hamiltonian as compared to the one in Eq. (236) which couples n- and (n$1)-phonon con"gurations. For Eqs. (250) it means that a direct coupling between one- and three-phonon con"gurations of the wave function (238) which is possible due to non-zero matrix element 1Q "a> a "[Q> Q>Q> ] 2, is neglected. In realistic ? HK HK ? @ A (+ calculation we will also use a selection of three-phonon con"gurations provided by Eq. (244) although now the matrix element ;I ?@ O0 even if one of a , b or c is not necessarily equal to ?@A a or b . These omitted matrix elements are orders of magnitude smaller as compared to the accounted for ones. Solving the system of linear equations (250) we obtain the spectrum of excited states, E(, J described by the wave function (238) and coe$cients SJ(J), DJ (J) and ¹J (J) re#ecting the ?@ ?@A ? phonon structure of excited states. Usually, in calculation of the properties of single giant resonances the three-phonon terms of the wave function (238) are omitted. Then it is possible to solve the system of linear equations (250) with the rank of the 10}10 order by a direct diagonalization. But while considering the damping properties of two-phonon resonances, threephonon con"gurations cannot be omitted. For this case instead of the diagonalization of the linear matrices of very high orders, an alternative solution is possible. We may substitute the "rst and last equations of (250) into the second equation and obtain the system of non-linear equations
;I ? ;I ? det (u #u #*u( !E()d ! ?@ ?@ ? @ ?@ V ? @ ? @ u !E( V ? ? ;I ?@ ;I ?@ ?@A ?@A ! "0 , (253) u #u #u #*u( !E( @ A ?@A V ?@A ? the rank of which equals to the number of two-phonon con"gurations included in the wave function (238). The solution of the system (253) by some iterative method yield again the spectrum of excited states E( and coe$cients DJ (J). Other coe$cients of the wave function (238) are related J ?@ to these coe$cients as follows:
DJ (J);I ? ?@ , SJ(J)"! ?@ ?@ ? u !EJ ? V DJ (J);I ?@ ?@A ?@ ?@ . (254) ¹J (J)"! ?@A u #u #u #*u( !E( ? @ A ?@A J It may be argued that the boson mapping with keeping the fermion information of the phonons' images at all stages of transformations gives no advantage as compared to npnh approach since, mathematically, a direct correspondence between two methods can be established only if the full basis of n-phonon states is used. However, many npnh con"gurations interact very weakly with other ones and as a result practically do not mix with them. It allows a su$cient truncation of multi-phonon con"gurations in the wave function (238) based on their physical properties with keeping a good accuracy for the components important for the subject of
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research. From the point of view of the Pauli principle violation the most dangerous multi-phonon con"gurations are the ones made of non-collective RPA states. On the other hand, these con"gurations interact with the other ones much weaker than the multi-phonon con"gurations including at least one collective phonon. For these reasons the "rst are not accounted for in the wave function (238) in realistic calculation. As the criteria `collective/non-collectivea we take the contribution of the main two-quasiparticle component to the wave function of the phonon operator. If the contribution exceeds 50}60% we will call the phonon non-collective. Let us consider now the electromagnetic excitation of pure one- and multi-phonon states from the ground state. The one-body operator of electromagnetic transition has the form 1 j""Ej"" j 2 L N CHI a> a , (255) M(Ejk)" eH (!1)HY>KY HKHYKY HK HY\KY O (2j#1 O HHY KKY where the single-particle transition matrix element 1 j""Ej"" j 2,1 j""iH> rH"" j 2 and eH are e!ective H O charges for neutrons and protons. In calculations we use the following values of e!ective charges: e"!Z/A and e"N/A to separate the center of mass motion and eH$"0 and eH$"1. L N L N Performing the transformation from particle operators to quasiparticle and phonon ones in Eq. (255), this equation transforms into
1 j""Ej"" j 2 u> L N HHY (tHG #uHG )(Q> #(!)H\IQ ) M(Ejk)" eH HHY HHY HIG H\IG O 2 (2j#1 G O HHY
#v\ CHI (!)HY>KYa> a , (256) HHY HKHYKY HYKY HY\KY KKY where the "rst term corresponds to one-phonon exchange between initial and "nal states and the second one is responsible for `boson-forbiddena electromagnetic transitions (see for details Ref. [94]). Then the reduced matrix element of the electromagnetic excitation of the one-phonon state ji from the ground state 0> in even}even nuclei may be calculated according to L N 1 1Q ""M(Ej)""0> 2" eH 1 j ""Ej"" j 2u> (tHG #uHG ) . (257) HG O HH HH HH 2 O HH Due to the ground-state correlations the direct excitation of pure two-phonon states [Q> ;Q> ] HG HG H from the ground state is also possible when we are dealing with the RPA phonons. The physical reason for that becomes clear if we remember that the ground state wave function includes a small admixture of four-, eight-, etc. quasiparticle con"gurations (see, Eq. (232)). The second term of Eq. (256) is responsible for these transitions and the reduced matrix element can be obtained by applying the commutation relations (246). It has the form L N 1[Q ;Q ] ""M(Ej)""0> 2"((2j #1)(2j #1) eH v\ O HG H HH HG O HHH j j j ;1 j ""Ej"" j 2 (tHGuHG#tHGuHG) . HH HH HH HH j j j
(258)
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Another type of boson-forbidden c-transitions which take place due to the internal fermion structure of phonons are the ones between one-phonon initial, Q> "2 , and "nal, Q> "2 , states. H G NF H G NF The reduced matrix element of such transitions can be calculated according to L N 1Q ""M(Ej)""Q> 2"(2j #1 eH v\ 1 j ""Ej"" j 2 O HG HG HH O HHH j j j (tHGtHG#uHGuHG) . (259) ; HH HH HH HH j j j The matrix element for transitions between the two-phonon states [Q> ;Q> ] "2 and H G HY NF HG [Q> ;Q> ] "2 is very complex and not presented here. Its "rst-order term is very similar H G H NF HG to the one for transitions between the one-phonon states Q> "2 and Q> "2 and may be H G NF H G NF obtained by assuming that the fermion structure of one phonon is `frozena, i.e., assuming that j i ,j i . When the coupling between one- and multi-phonon con"gurations is accounted for in the wave function of excited states, the reduced matrix element of the electromagnetic excitation of the states of Eq. (238) may be written as
1WJ(J)""M(Ej)""0> 2" SJ(J)1Q ""M(Ej)""0> 2 ? HG ? DJ (J) ?@ 1[Q ;Q ] ""M(Ej)""0> 2 , (260) # HG HG H (1#d ?@ ? @ where we have neglected the direct excitation of three-phonon con"gurations from the ground state. Since an admixture of multi-quasiparticle con"gurations in the ground-state wave function is very small, the reduced matrix element, Eq. (258), is typically about two orders of magnitude smaller as compared to the reduced matrix element, Eq. (257). For this reason, in most of the cases keeping only the "rst term in Eq. (260) and neglecting the second one together with interference e!ects provides very good accuracy in calculation. Nevertheless, there are a few exceptional cases. The "rst one is the excitation of the lowest 1\ state in spherical nuclei. It is well known that no collective one-phonon 1\ con"gurations appear in the low-energy region and the wave function of the 1\ state has the dominant two-phonon component [2>;3\] \. There are three main mechanisms to explain the E1-excitation of this state observed in the experiment [95]. The "rst is an in#uence of the GDR. In microscopic theories it appears in a natural way due to the coupling of one- and two-phonon con"gurations. Since the GDR is located about 10 MeV higher, this coupling yields only a very small portion of the observed strength. The second mechanism is the excitation of non- and weakly collective one-phonon 1\ con"gurations which have relatively small B(E1) values but are located in low-energy region. The last mechanism is the direct excitation of two-phonon con"gurations from the ground state. Although the direct excitation of two-phonon con"gurations from the ground state is a second-order e!ect, excitation of collective two-phonon con"gurations [2>;3\] \ play an essential role since the other two mechanisms yield much weaker E1 strengths. In this case, interference e!ects between the "rst and the third mechanisms are also important [94]. The second term of Eq. (260), although very weak as compared to the "rst one, may also play some role at the excitation energies above 20 MeV where the density two-phonon con"gurations is
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217
a few orders of magnitude higher as compared to the density of one-phonon con"gurations. It will be discussed below. Considering the two-step mechanism of the DGDR excitation in second-order perturbation theory we also need the reduced matrix element of the electromagnetic excitation of the twophonon DGDR state [1\;1\] from the one-phonon GDR state 1\. In ideal boson approximaG GY ( G tion this matrix element 1[1\;1\] ""M(E1)""1\2"((1#d )(2J#1)/311\""M(E1)""g.s.2 . GY GY G ( G G GY
(261)
4.3. Comparison with other approaches The properties of the double-giant resonances have been also microscopically studied with the Skyrme forces [74,96] and within the second-RPA approach [97,98]. The most close to the QPM approach is the one of the "rst group of papers. The main di!erence between these two approaches is that in calculations with the Skyrme forces the properties of the ground and 1p1h excited states are calculated self-consistently. As within the QPM, in calculations with the Skyrme forces the 1p1h basis is mapped into the phonon space. Multi-phonon states are obtained by folding of one-phonon states. The phonon basis in Refs. [74,96] is restricted by only a few, the most collective, phonons for each multipolarity. Calculations are performed with the wave function including one- and two-phonon terms. The main attention is paid to the e!ects of anharmonicity and non-linearity. The latter is an in#uence of taking into account the bosonforbidden transition matrix elements, Eqs. (258), (259), on the absolute value of the DGDR excitation in heavy ion collisions. In the second-RPA approach [99] the wave function of excited states is written as a mixture of 1p}1h and 2p}2h con"gurations: Q>"2 " (XJ a>a !>J a>a )# (XJ a>a>a a !>J a>a>a a )"2 . (262) J NF N F NF F N NNYFFY N NY FY F NNYFFY F FY NY N NF NNYFFY The operators Q> are assumed as bosons and the energy spectrum and coe$cients X and > are J obtained by diagonalization of the model Hamiltonian in the space of states described by the wave functions of Eq. (262).
5. Physical properties of the double-giant resonances In the present section we will consider the properties of the DGDR as predicted by the QPM mainly in Xe and Pb for which experimental data in relativistic heavy ion collision (RHIC) are available. Before proceeding with that let us brie#y check an accuracy of the description of the properties of low-lying states and single-giant resonances within this approach. It provides an estimate how good the phonon basis, to be used in the forthcoming calculation of the DGDR properties, is since no extra free parameters are used after this basis is "xed. The results of our calculations of the position and exhaust of the energy weighted sum rule (EWSR) of low-lying states and giant resonances as well as the width of resonances in Xe and Pb are presented in Table 10 in comparison with the experimental "ndings. The comparison indicate a rather good
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Table 10 Integral characteristics (position, E , exhaust of the energy weighted sum rule (EWSR) and width of resonances, C) of V low-lying excited states and one-phonon giant resonances in Xe and Pb Calculation Nucl.
jL
E V (MeV)
Xe
2> 3\ GDR GQR GQR
1.4 3.3 15.1 12.5 23.1
Pb
2> 3\ GDR GQR GQ GQR GT
4.2 2.4 13.35 10.6 21.9
Experiment
C (MeV)
EWSR (%)
E V (MeV)
4.0 3.2 3.6
2.6 5.6 107 75 80
1.31 3.28 15.2 12.3 22.1$0.7
3.5 3.1 5.0
16.4 21.3 94 67 81
4.09 2.61 13.4 10.5}10.9 22.6$0.4
C (MeV)
EWSR (%)
4.8 4.0 45.4
2.4 5.2 80}120 70 93$45
4.0 2.4}3.0 6$2
16.9 20 89}122 60}80 +50
Interpolation of experimental data [56]. Interpolation of experimental data [5].
correspondence between calculated characteristics and experimental data. The calculation somewhat underestimate the width of resonances, especially of the isovector GQR. The main reason is related to the necessity of truncating of complex con"gurations included in the wave function of excited states in actual calculation. The density of multi-phonon con"gurations is rapidly increasing with the excitation energy. That is why the e!ect of the basis truncation the most strongly in#uences on the width of the GQR located at higher energies. 5.1. One-step excitation of two-phonon states in the energy region of giant resonances Let us consider a direct photoexcitation of the two-phonon states in the energy region of giant resonances from the ground state of even}even nuclei (see, Refs. [57,100] for more details). Since in RHIC experiments the Coulomb mechanism of excitation plays the most essential role, the cross sections of photoexcitation can be easily recalculated into RHIC cross sections for di!erent energies and Z-values of target and projectile nuclei. In calculation of the B(Ej) values we use only the terms proportional to tu (see Eq. (258)). The complete set of diagrams corresponding to a direct transition to two-phonon states from the ground state is presented in Ref. [101]. As one can see from the analytical expressions the main part of the contributions from di!erent terms disappears due to the cancellation between particles and holes. The cross sections of the direct photoexcitation of the groups of two-phonon states made of phonons of de"nite multipolarities in Xe and Pb are presented in Fig. 28. E2-excitation of [1\ 1\] > states is plotted in the top part of the "gure. E1-excitation of the two-phonon states [1\ 2>] \ and [2> 3\] \ is shown in the middle and the bottom parts, respectively. The
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Fig. 28. Cross sections of the direct photoexcitation of two-phonon con"gurations [1\ 1\] >, [1\ 2>] \ and [2> 3\] \ from the ground state in Xe and Pb.
integral characteristics of two-phonon states which are a single giant resonances built on top of either a low-lying state or another single resonance in the same nuclei are given in Table 11. The main feature of the top part of Fig. 28 is that just all two-phonon states which form this double-phonon resonance are constructed of the one-phonon 1\ states belonging to the GDR in G the one-phonon approximation. The structure of the [1\ 2>] \ and [2> 3\] \ states is more
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Table 11 Integral characteristics (energy centroid, width and cross section of direct photoexcitation from the ground state) of some groups of two-phonon states which are a giant resonance built on top of either a low-lying state or another single-giant resonance in Xe and Pb Nucl.
Con"guration
Centroid (MeV)
Width (MeV)
p A (mb)
Xe
[1\ 2> ] \ %"0 %/0 [1\ 1\ ] > %"0 %"0 [2> 2> ] > %/0 %/0
24.0 30.2 21.3
2.9 4.0 0.5
4.3 0.33 0.1
[1\ 2>] \ %"0 [1\ 2>] \ %"0 [1\ 2>] \ %"0 [1\ 3\] > %"0 [1\ 2> ] \ %"0 %/0 [1\ 1\ ] > %"0 %"0
17.4 17.2 17.7 15.3 25.1 25.5
2.2 2.1 3.4 3.9 3.8 4.4
1.7 8.7;10\ 4.9;10\ 5.2;10\ 9.6 0.22
Pb
complex. For example, among [1\ 2>] \ states the substructure in the energy range from 15 to 20 MeV in Pb (right middle part of Fig. 28) is formed mainly by 1\ phonons from the GDR G region coupled to the 2> state. The small substructure above 32 MeV is due to the GDR 1\ phonons coupled to the 2> phonons of the isovector GQR. As for the broad structure between G GY 20 and 30 MeV not only [GDR GQR ] \ states but many other two-phonon states built of less GQ collective 1\ and 2> phonons, the role of which is marginal for properties of single resonances, play G GY an essential role. The same conclusions are valid for the direct photoexcitation of [1\ 2>] \ states in Xe. The cross section of the direct photoexcitation of the two-phonon 1\ states built of phonons of the higher multipolarities yield non-resonance feature. It is already seen for the case of [2> 3\] \ states (bottom part of Fig. 28), especially in Pb. While dealing with electromagnetic, or with Coulomb excitation from a 0> ground state, the priority attention has to be paid to the "nal states with the total angular momentum and parity JL"1\. For that we have calculated the cross section for the photoexcitation of two-phonon states [jL jL] \, where jL and jL are both natural jLL (nL"(!1)H) and unnatural jLS (nS"(!1)H>) parity phonons with multipolarity j from 0 to 9. The results of the calculation for Xe and Pb integrated over the energy interval from 20 to 35 MeV are presented in Table 12. Each con"guration [jL jL] in the table means a sum over a plenty of two-phonon states made of phonons with a given spin and parity jL, jL, but di!erent RPA root numbers i , i of its constituents p([jL jL])" p([jL(i ) jL(i )]) . G G
(263)
The total number of two-phonon 1\ states included in this calculation for each nucleus is about 10 and they exhaust 25% and 15% of the EWSR in Xe and Pb, respectively. The absolute
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Table 12 Cross sections for the direct photoexcitation of di!erent two-phonon con"gurations from the ground state integrated over the energy interval from 20 to 35 MeV in Xe and Pb. The GDR cross section integrated over the energy of its location is presented in the last line for a comparison p (mb) A Con"guration [0> 1\] \ [1\ 2>] \ [2> 3\] \ [3\ 4>] \ [4> 5\] \ [5\ 6>] \ [6> 7\] \ [7\ 8>] \ [8> 9\] \ [jLL jLS] \ H H [jLS jLS ] \ H H [jLL S jLL S] \ H H [GDR GDR] > [jLL jLL] > H H GDR
Xe
Pb
4.4 36.6 82.8 101.0 68.9 49.2 31.9 13.6 4.9
3.9 44.8 33.1 56.7 37.3 46.2 49.8 12.5 9.0
71.4
58.5
46.7
71.1
511.4
422.9
0.33 38.1 2006
0.22 21.7 2790
value of the photoexcitation of any two-phonon state under consideration is negligibly small but altogether they produce a sizable cross section. Table 12 demonstrates that di!erent two-phonon con"gurations give comparable contributions to the total cross section which decreases only for very high spins because of the lower densities of such states. As a rule, unnatural parity phonons play a less important role than natural parity ones. For these reasons we presented in the table only the sums for [natural unnatural] and [unnatural unnatural] two-phonon con"gurations. The cross section for the photoexcitation of all two-phonon 1\ states in the energy region 20}35 MeV from the ground state equals in our calculation to 511 and 423 mb for Xe and Pb, respectively. It is not surprising that we got a larger value for Xe than for Pb. This is because the phonon states in Xe are composed of a larger number of two-quasiparticle con"gurations due to the pairing. The same values for two-phonon states with angular momentum and parity JL"2> are an order of magnitude smaller. We point out that the direct excitation of [1\ 1\] > or [GDR GDR] con"gurations is negligibly weak (compare results in Tables 11 > and 12). The calculated values should be compared to the cross section for the photoexcitation of the single-phonon GDR which in our calculation equals to 2006 and 2790 mb, respectively.
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Fig. 29. Photoexcitation cross section of the GDR in Xe and Pb. Calculations are performed: (a) within one-phonon approximation and (b) with taking into account of the coupling between one- and two-phonon con"gurations. Continuous curves in the bottom part are the strength functions calculated with a smearing parameter D"1 MeV; dashed curve corresponds to electromagnetic transitions to one-phonon 1\ states, solid curve } to one- and two-phonon 1\ states.
A contribution of two-phonon 1\ states to the total cross section at GDR energies is weaker than at higher energies because of the lower density of two-phonon states and the lower excitation energy and can be neglected considering the GDR itself. It is clearly demonstrated in Fig. 29b. In this "gure the cross sections of the photoexcitation of 1\ states in Xe and Pb are presented. The top part of the "gure corresponds to a calculation performed in one-phonon approximation. The results of calculations with the wave function which includes a coupling between one- and two-phonon 1\ con"gurations are plotted in the bottom part of the "gure. For a visuality the last calculations are also presented as strength functions D 1 b(p, E)" p( J (E!E()#D/4 2p J J
(264)
with a smearing parameter D"1 MeV, where p( is a partial cross section for the state with the J excitation energy E( plotted also by a vertical line. The E1-transitions to one-phonon components J of the wave function of excited 1\ states are plotted by dashed curve. It should be compared with
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Fig. 30. Photoneutron cross sections in Pb. Solid curve is the result of calculation with the wave function including one- and two-phonon terms presented with a smearing parameter D"1 MeV; vertical lines (in arbitrary units) } within one-phonon approximation. Experimental data are plotted by experimental error bars.
Fig. 31. Photo-neutron cross section for Pb. Experimental data (dots with experimental errors) are from Ref. [103]. The long-dashed curve is the high energy tail of the GDR, the short-dashed curve is the GQR and the curve with squares G is their sum. The contribution of two-phonon states is plotted by a curve with triangles. The solid curve is the total calculated cross section.
the solid curve which is the sum of transitions to one- and two-phonon 1\ con"gurations in the GDR energy region. For Pb photoexcitation cross sections are known from experimental studies in (c, n) reactions up to the excitation energy about 25 MeV [102,103]. It was shown that QPM provides a very good description of the experimental data in the GDR region [102] (see, Fig. 30), while theoretical calculations at higher excitation energies which account for contributions from the single-phonon GDR and GQR essentially underestimated the experimental cross section [103]. The experi mental cross sections above 17 MeV are shown in Fig. 31 together with theoretical predictions. The results of the calculations are presented as strength functions obtained with averaging parameter
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equal to 1 MeV. The contribution to the total cross section of the GQR (short-dashed curve), the high-energy tail of GDR (long-dashed curve), and their sum (squared curve), are taken from Ref. [103]. The curve with triangles represents the contribution of the direct excitation of the twophonon states from our present studies. The two-phonon states form practically a #at background in the whole energy region under consideration. Summing together the photoexcitation cross sections of all one- and two-phonon states, we get a solid curve which is in a very good agreement with the experimental data. Thus, from our investigation of photoexcitation cross sections we conclude that in this reaction very many di!erent two-phonon states above the GDR contribute on a comparable level, forming altogether a #at physical background which should be taken into account in the description of experimental data. On the other hand, Coulomb excitation in relativistic heavy ion collisions provides a unique opportunity to excite a very selected number of two-phonon states by the absorption of two virtual c's in a single process of projectile}target interaction [6]. Theoretically, this process is described using the second-order perturbation theory of the semi-classical approach of Winther and Alder [6,18] and discussed in Section 3.2.3. Since excitation cross sections to second order are much weaker than to "rst order of the theory, two-phonon states connected to the ground states by two E1-transitions are predominantly excited. These two-phonon states have the structure [1\(i) 1\(i )] > and form the DGDR. ( 5.2. 1> component of the DGDR According to the rules of angular momentum coupling two one-phonon states with the spin and parity equal to 1\ may couple to the total angular momentum JL"0>, 1> and 2>. Thus, in principle, three components of the DGDR with these quantum numbers should exist. In phenomenological approaches describing the single GDR as one collective state, the [1\ 1\] > compon ent the DGDR is forbidden by symmetry properties. Taking into account the Landau damping this collective state splits into a set of di!erent 1\ states distributed over an energy interval. In G microscopic studies the Landau damping is taken into account by solving the RPA equations. Again, the diagonal components [1\ 1\] > are forbidden by the same symmetry properties but G G nondiagonal ones [1\ 1\] > exist and should be taken into consideration. Consequently, the role G GY of these nondiagonal components depends on how strong is the Landau damping. We produce here two-phonon DGDR states with quantum numbers JL"0>, 1> and 2> by coupling one-phonon RPA states with the wave function "1\2 , to each other. The index m stands G K for di!erent magnetic substates. The wave function of the two-phonon states has the form 1 (1m1m "JM)"1\2 "1\2 , "[1\ 1\] L > >2 " G K G KY G G ( + (2 K KY for two-phonon states made of two identical phonons while for other DGDR states it is
(265)
(266) "[1\ 1\] L > > >2 " (1m1m "JM)"1\2 "1\2 . G K GY KY G GY ( + K KY In the present calculation we do not include the interaction between DGDR states, of Eqs. (265) and (266), and we do not couple them to states with di!erent than two number of phonons (it will
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be considered below). Thus, our two-phonon states "[1\ 1\] L2 have excitation energy equal to G GY ( + the sum of one-phonon energies u #u and are degenerated for di!erent values of the total spin G GY JL and its projection M. Since the main mechanism of excitation in projectile ions at relativistic energies is the Coulomb part of interaction with a target, the nuclear part of interaction has been neglected in the present analysis. In a semi-classical approach [6], the two-phonon DGDR states can be excited in second-order perturbation theory via the two-step process g.s.PGDRPDGDR. The secondorder amplitude can be written as 1 a , a\G \GY " a\G K \G B\GY + \G K B + 2 K
(267)
where assuming the Coulomb mechanism of excitation the "rst-order amplitude aG D is pro( ( portional to the reduced matrix element of 1J ""E1""J 2. The reduced matrix element D G 1[1\ 1\] L""E1""1\2 of electromagnetic excitation of two-phonon states, Eqs. (265) and (266), G GY G ( from the one-phonon state "1\2 is related, in the boson picture of nuclear excitation, to the G K excitation of "1\2 from the ground state according to (261). It should be noted that although for G K the two-phonon states, Eq. (265), we have an extra factor (2, the states of Eq. (266) play a more important role in two-step excitations since they can be reached by two di!erent possibilities: g.s.P1\P[1\ 1\] and g.s.P1\P[1\ 1\]. First of all, we point out that in second-order G G GY GY G GY perturbation theory the amplitude for this process is identically zero in a semi-classical approach. This can be understood by looking at Fig. 32. The time-dependent "eld < carries angular # momentum with projections m"0, $1. Thus, to reach the 1> DGDR magnetic substates, many routes are possible. The lines represent transitions caused by the di!erent projections of < : (a) # dashed lines are for m"0, (b) dashed-dotted lines are for m"!1, and (c) solid lines are for m"#1. The relation < O< holds, so that not all routes yield the same excitation # K # K! amplitude. Since the phases of the wave functions of each set of magnetic substates are equal, the di!erence between the transition amplitudes to a "nal M, can also arise from di!erent values of the Clebsch}Gordan coe$cients (1m1m "1M). It is easy to see that, for any route to a "nal M, the second-order amplitude will be proportional to (001m"1m)(1m1m "1M) < < #(m m ). # KY # K
Fig. 32. The possible paths to the excitation of a given magnetic substate of the 1> component of the DGDR are displayed. The transitions caused by the di!erent projections of the operator < are shown by: (a) dashed lines for m"0, # (b) dashed-dotted lines for m"!1, and (c) solid lines are m"#1.
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The two amplitudes carry opposite signs from the value of the Clebsch}Gordan coe$cients. Since (001m"1m),1, the identically zero result for the excitation amplitude of the 1> DGDR state is therefore a consequence of (1m1m "1M)"0 . KKY
(268)
We have also performed a coupled-channels calculation [104] following the theory described in Ref. [19]. As shown in Ref. [19], the coupling of the electric quadrupole (isovector and isoscalar) and the electric dipole states is very weak and can be neglected. We therefore include in our space only one-phonon 1\ and two-phonon [1\ 1\] L (JL"0>, 1> and 2>) states. In coupledG GY ( channels calculation we take into account interference e!ects in the excitation of di!erent GDR and DGDR states and obtain the occupation amplitudes by solving the coupled-channels equations. By solving these equations we thus account for unitarity and for multi-step excitations, beyond the two-step processes of Eq. (267). The time-dependent electric dipole "eld is that of a straight-line moving particle with charge Ze, and impact parameter b (we use Eqs. (25) and (26) of Ref. [19]). Due to the large number of degenerate magnetic substates, to make our coupled-channels calculation feasible, we have chosen a limited set of GDR and DGDR states. We have taken six 1\ states which have the largest value of the reduced matrix element 11\""E1""g.s.2. These six states G exhaust 90.6% of the classical EWSR, while all 1\ states up to 25 MeV in our RPA calculation exhaust 94.3% of it. This value is somewhat smaller than the 122% reported in Ref. [32]. It is because the continuum in our RPA calculation was approximated by narrow quasibound states. From these six one-phonon 1\ states we construct two-phonon [1\ 1\] L states, Eqs. (265) and G GY ( (266), which also have the largest matrix element of excitation 1[1\ 1\] L""E1""1\2 for excitations G G GY ( starting from one-phonon states. The number of two-phonon states equals to 21 for JL"0> and 2>, and to 15 for JL"1>. The cross section for the DGDR excitation was obtained by summing over the "nal magnetic substates of the square of the occupation amplitudes and, "nally, by an integration over impact parameter. We have chosen the minimum impact parameter, b"15.54 fm, corresponding to the parameterization of Ref. [43], appropriate for lead}lead collisions. The electromagnetic excitation cross sections for the reaction Pb (640A MeV)#Pb with excitation of all our basic 63 states is shown in Fig. 33. The total cross sections for each multipolarity are presented in Table 13, together with the results of "rst-order (for one-phonon excitations) and second-order (for two-phonon excitations) perturbation theory. The coupledchannels calculation yields a non-zero cross section for the 1> DGDR state due to other possible
As demonstrated in the previous subsection the direct excitation of two-phonon con"gurations from the ground state is very weak. It allows us to exclude in our calculation matrix elements of the form 1[1\ 1\] > > ""E2(M1)""g.s.2 which G GY correspond to direct transitions and produce higher-order e!ects in comparison with accounted ones. These matrix elements give rise to DGDR excitation in "rst-order perturbation theory. Thus, to prove our approximation, we have calculated such cross sections and got total values equal to 0.11 and (0.01 mb for the 21 2> and the 15 1> basic two-phonon states, respectively. These values have to be compared to 244.9 mb for the total DGDR cross section in the second-order perturbation theory.
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Fig. 33. The electromagnetic excitation cross sections for the reaction Pb (640A MeV)#Pb calculated in coupled channels. It is shown the excitation of the GDR (top) and the three components JL"0>, 2> and 1> of the DGDR. The B(E1) strength distribution (in arbitrary units) over 1\ states is shown by dashed lines. For a visuality it is shifted up by 100 keV.
Table 13 Cross section (in mb) for the excitation of the GDR and the three components with JL"0>, 2>, 1> of the DGDR in Pb (640A MeV)#Pb collisions. Calculations are performed within coupled-channels (CC) and within "rst (PT-1) and second (PT-2) order perturbation theory, respectively
GDR DGDR > DGDR > DGDR > DGDR/GDR
CC
PT-1
PT-2
2830.
3275.
0.
0.0 0.11 (0.01
43.1 201.8 0.0
33.0 163.0 6.3 0.071
0.075
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routes (higher-order), not included in second-order perturbation theory. One observes a considerable reduction of the DGDR cross sections, as compared to the predictions of the second-order perturbation theory. The GDR cross sections are also reduced in magnitude. However, the population of the 1> DGDR states are not appreciable and cannot be the source of the missing excitation cross section needed to explain the experiments. In general, the coupled-channels calculation practically does not change the relative contribution of di!erent one-phonon 1\ and G two-phonon states [1\ 1\] L to the total cross section with given JL"1\, 0> and 2>. But since G GY ( the 1> component of the DGDR, with its zero value of excitation cross section in second-order perturbation theory, has a special place among the two other components, the main e!ect of coupled-channels is to redistribute the total cross section between the JL"0>, 2> and JL"1> components. The calculated cross section in coupled channels for both GDR and DGDR are somewhat lower than that reported in experimental "ndings [54,105]. This is not surprising since as mentioned above our chosen six 1\ states exhaust only 90.6% of EWSR while the photoneutron data [32] indicate that this value equals to 122%. Due to this underestimate of exhaust of the EWSR the cross section for the DGDR excitation reduces more strongly than the one for the single GDR. This is because the GDR cross section is roughly proportional to the total B(E1) value while for the DGDR it is proportional to the square of it. We will return back in more detail to the problem of absolute cross sections of the DGDR excitation in RHIC in the forthcoming subsection. 5.3. Position, width and cross section of excitation in RHIC of the DGDR in Xe and Pb To describe the width of two-phonon resonances it is necessary to take into account a coupling of two-phonon con"gurations, which form these resonances, with more complex ones. For this two types of calculations have been performed. In the "rst of them [106] the "ne structure of the GDR calculated with the wave function which includes one- and two-phonon con"gurations and presented in Fig. 29b has been used. The DGDR states have been constructed as a product of the GDR to itself. In other words, following the Axel}Brink hypotheses on top of each 1\ state in Fig. 29b we have built the full set of the same 1\ states. The calculation has been performed for the nucleus Xe. In the dipole case, jL"1\, the one-phonon states exhaust 107% of the classical oscillator strength and are displayed in the left part of Fig. 29a. Of these, 20 states have an oscillator strength which is at least 1% of the strongest strength and together exhaust 104% of the classical EWSR. We have used these states in the coupling to two-phonon states. We have included all the natural parity phonons jL"1\!8> with energy lower or equal to 21 MeV, obtaining 2632 two-phonon con"gurations. One obtains 1614 states described by the wave function which includes one- and two-phonon con"gurations, in the energy interval from 7 MeV to 19.5 MeV. Their photoexcitation cross sections are shown in Fig. 29b. The B(E1) value associated with each mixed state is calculated through its admixture with one-phonon states, as "1l""M(E1)""02""" SJ(1\)10""Q \ M(E1)""02". Also shown by dashed curve G G G in the left part of Fig. 29b is the result obtained adding an averaging parameter of 1.0 MeV. This parameter represents in some average way the coupling to increasingly more complicated states and eventually to the compound nuclear states. From the resulting smooth response it is easy to directly extract the centroid and the full width at half maximum of the GDR. The corresponding
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Fig. 34. The cross section for Coulomb excitation of the one-phonon GDR (continuous curve), of the isoscalar GQR (dash-dotted), of the isovector GQR (long dashed) as well as for the double-phonon GDR (short dashed) are shown. They have been calculated at E "681A MeV, taking into account the energy reduction of the beam in the target [40]. The one-phonon GDR cross section has been reduced in the "gure by a factor 10.
Table 14 Calculated (for two values of r ) and experimental cross section (in mb) for the excitation of giant resonances in Xe in M Xe (690A MeV)#Pb reaction. In the last row, the experimental cross sections for Coulomb excitation of one- and two-phonon states from Ref. [40] are shown. The value of the integrated cross section reported in Ref. [40] is 1.85$0.1 b. The nuclear contribution has been estimated in Ref. [40] to be about 100 mb, while about 3% (50 mb) of the cross section is found at higher energy. Subtracting these two contributions and the two-phonon cross section, leads to the value 1485$100 mb shown in the table
r "1, 2 fm M r "1, 5 fm M Experiment
GDR
GQR
GQR
GDR#GQR
DGDR
2180 1480 1024$100
170 110 }
120 60 }
2470 1650 1485$100
130 50 215$50
values are E "15.1 MeV and C "4 MeV. They can be compared with the values extracted %"0 %"0 from experiment, E "15.2 MeV and C "4.8 MeV. %"0 %"0 The isoscalar and the isovector giant quadrupole resonances (GQR) have also been calculated. The centroid, width and percentage of the EWSR associated with the isoscalar mode are 12.5 MeV, 3.2 MeV and 75%, respectively. The corresponding quantities associated with the isovector GQR are 23.1 MeV, 3.6 MeV and 80%. The di!erential Coulomb-excitation cross sections as a function of the energy associated with the one-phonon GDR and GQR resonances and the two-phonon DGDR in Xe (690A MeV) #Pb reaction are displayed in Fig. 34. It is seen that the centroid of the two-phonon dipole excitation falls at 30.6 MeV, about twice that of the one-phonon states, while the width is C+6 MeV, the ratio to that of the one-phonon excitation being 1.5. The associated integrated values are displayed in Table 14, in comparison with the experimental "ndings. The cross sections depend strongly on the choice of the value of b "r (A #A ).
M N R
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In keeping with the standard `safe distancea, that is, the distance beyond which nuclear excitation can be safely neglected, we have used r "1.5 fm. Because their values essentially do not M depend on the width of the GDR, we view the calculated cross section of 1650 mb as a rather accurate value and if anything an upper limit for the one-phonon Coulomb excitation cross section. It is satisfactory that the measured cross section is rather close to this value. Also shown in Table 14 are the predictions associated with the sequential excitation of the DGDR. This result is essentially not modi"ed evaluating the direct Coulomb excitation of the double GDR. In fact, the cross section associated with this process is a factor 10\ smaller than that associated with the two-step process. The calculated value of 50 mb is a factor of 0.25 smaller than that observed experimentally. Two other processes are possible within the sequential excitation of the giant modes which can lead to an excitation energy similar to that of the two-phonon GDR. They are the excitation of the isoscalar GQR mode followed by a GDR mode and vice versa. The resulting cross section is estimated to be an order of magnitude smaller, cf. Table 14, and does not change qualitatively this result. In order to make clearer the seriousness of this discrepancy, we have recalculated all the cross sections using r "1.2 fm, namely with a much smaller radius than that M prescribed in order to respect the safe Coulomb excitation distance of closest approach. The calculated value of 130 mb is still a factor of 0.6 smaller than the reported experimental cross section. At the same time the cross section of the one-phonon states has become a factor 1.7 larger than the empirical value. This factor becomes 1.5 when the coupling to higher multiphonon states is included according to the standard Poisson distribution for the excitation probabilities [17]. The main shortcoming of the above discussed theoretical scheme to treat the DGDR, when the DGDR states are obtained by folding of the "ne structure of two GDRs, is the fact that the DGDR states obtained this way are not eigenstates of the used microscopic Hamiltonian. To overcome this shortcoming other calculations have been performed in which two-phonon [1\ 1\] DGDR states are coupled directly to more complex ones [107}109]. From rather general arguments [60], the most important couplings leading to real transitions of the doublegiant resonances and thus to a damping width of these modes are to con"gurations built out by promoting three nucleon across the Fermi surface. That is, con"gurations containing three holes in the Fermi sea and three particles above the Fermi surface (3p3h con"gurations). We use the wave function (238) to describe the DGDR states and their coupling to 1p1h and to 3p3h doorway con"gurations. The spectrum of excited states which form the DGDR is obtained by solving the secular equation (253) and the wave function coe$cient S, D and ¹ are calculated from Eq. (254). Pauli principle corrections, the coe$cients KI (b a "a b ) and anharmonicity shifts *u( , were omitted ( ?@ in calculations presented in Ref. [107] and accounted for in Ref. [108]. While they are small, they produce shifts in the energy centroid of the double-giant resonance. Similar coe$cients appear also in connection with the term arising from the `doorway statesa containing three phonons in Eq. (238). We have neglected them because they again are small and furthermore act only in higher order as compared to the previous term, in de"ning the properties of the double-giant dipole resonance. Finally, the corresponding KI -coe$cient associated with the "rst term in Eq. (238) is proportional to the number of quasiparticles present in the ground state of the system, a quantity which is assumed to be zero within linear response theory.
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In keeping with the fact that the Q-value dependence of the Coulomb excitation amplitude is rather weak at relativistic energies [14], the cross section associated with the two-step excitation of the double-giant dipole resonance is proportional to [B(E1);B(E1)]"" 1WJ > > "M(E1)"WJ\2 ) 1WJ\"M(E1)"W 2" J M M ? @ # M M (1#d KI (b a "a b ) , " 2 DJ (J) ) @ ( ?@ ? @ ? (1#d ? @ ?@ ? @ (269)
where M "1Q ""M(E1)""0> 2 is the reduced matrix element of the E1-operator which acting on ? ? the ground state "2 excites the one-phonon state with quantum numbers a"(1\, i). NF Making use of the elements discussed above we calculated the distribution of the quantity Eq. (269) over the states Eq. (238) in Xe. We considered only JL"0> and 2> components of the two-phonon giant dipole resonance. As already discussed above its JL"1> component cannot be excited in the second-order perturbation theory and is su$ciently quenched in coupled-channels calculation. The 15 con"gurations +1\i, 1\i ,"+a , b , displaying the largest [B(E1);B(E1)] values were used in the calculation. They are built up out of the "ve most collective RPA roots associated with the one-phonon giant dipole resonance carrying the largest B(E1) values and exhausting 77% of energy weighted sum rule (EWSR). Two-phonon states of collective character and with quantum numbers di!erent from 1\ lie, as a rule, at energies few MeV away from the double-giant dipole states and were not included in the calculations. The three-phonon states +a b c , were built out of phonons with angular momentum and parity 1\, 2>, 3\ and 4>. Only those con"gurations where either a , b or c were equal to a or b were chosen. This is because other con"gurations lead to matrix elements ;?@ (J) of the interaction, which are orders of ?@A magnitude smaller than those associated with the above-mentioned three-phonon con"gurations, and which contain in the present calculation 5742 states up to an excitation energy 38 MeV. The single-particle continuum has been approximated in the present calculation by quasibound states. This approximation provides rather good description of the single GDR properties in Xe. This means that our (2p2h) \ \ spectrum is also rather complete for the description of the DGDR "
properties although it is located at higher energies. If one assumes a pure boson picture to describe the phonons, without taking into account their fermion structure, the three-phonon con"gurations omitted in the present calculation do not couple to two-phonon states under consideration. Furthermore, although the density of 3p3h con"gurations is quite high in the energy region corresponding to the DGDR, a selection of the important doorway con"gurations in terms of the e$ciency with which con"gurations couple to the DGDR, can be done rather easily. The above considerations testify to the advantage of employing a microscopic phonon picture in describing the nuclear excitation spectrum, instead of a particle}hole representation. One can more readily identify the regularities typical of the collective picture of the vibrational spectrum, and still deal with the fermion structure of these excitations. As far as the one-phonon term appearing in Eq. (238) is concerned, essentially all phonons with angular momentum and parity 0> and 2> were taken into account within the energy interval 20}40 MeV.
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Fig. 35. Fragmentation of the most collective (a) one-phonon 1\ and (b) two-phonon [1\ 1\] con"gurations in Xe due to the coupling to more complex con"gurations. The results are presented with a smearing parameter D"0.2 MeV.
A rather general feature displayed by the results of the present calculation is that all two-phonon con"gurations of the type +1\i, 1\i , building the DGDR in the `harmonica picture are fragmented over a few MeV due to the coupling to 3p3h `doorway statesa. Fragmentation of the most collective one is presented in the bottom part of Fig. 35. For a comparison the fragmentation of the most collective one-phonon 1\ con"guration due to the coupling to 2p2h `doorway statesa is plotted in the top part of the same "gure. The results have been averaged with the aid of a Breit}Wigner distribution of width 0.2 MeV. The maximum amplitude with which each twophonon con"guration enters in the wave function (238) does not exceed a few percent. Two-phonon con"gurations made out of two di!erent 1\ phonons are fragmented stronger than two-phonon con"gurations made out of two identical 1\ phonons. This in keeping with the fact that, as a rule, states of the type +1\i, 1\i , with iOi are less harmonic than states with i"i and consequently are coupled to a larger number of three-phonon con"gurations. In Figs. 36b and c, the [B(E1);B(E1)] quantity of Eq. (269) associated with Coulomb excitation of the almost degenerate JL"0> and JL"2> components of double-giant dipole resonance are shown. For comparison, the B(E1) quantity associated with the Coulomb excitation of the one-phonon giant dipole resonance is also shown in Fig. 36a. The reason why the two angular momentum components of the DGDR are almost degenerate can be traced back to the fact that
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Fig. 36. (a) B(E1) values for the GDR and (b, c) [B(E1);B(E1)] values Eq. (269) for the DGDR associated with Coulomb excitation in Xe in relativistic heavy ion collision. (b) and (c) correspond to J"0> and J"2> components of the DGDR, respectively. A smooth curve is a result of averaging over all states with a smearing parameter D"0.5 MeV. See text for details.
the density of one-phonon con"gurations to which the DGDR couple and which are di!erent for JL"0> and JL"2> type states is much lower than the density of states associated with 3p3h `doorway statesa, density of states which is the same in the present calculation for the two di!erent angular momentum and parity. E!ects associated with the J-dependence of the KI and D coe$( ( cients are not able to remove the mentioned degeneracy, because of the small size of these coe$cients. These coe$cients can also a!ect the excitation probability with which the JL"0> and JL"2> states are excited (cf. Eq. (269)). The e!ect however is rather small, leading to a decrease of the order of 2}3% in both cases. The J-degeneracy would be probably somehow broken if one goes beyond a one-boson exchange picture in the present approach of interaction between di!erent nuclear modes. The next order term of interaction would couple the DGDR states to many other 3p3h con"gurations, not included in the present studies, some of these 3p3h con"gurations would
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Table 15 Position, width and the ratio values R, Eq. (270), for of J"0> and J"2> components of the DGDR with respect to the ones of the single GDR in Xe. The third row corresponds to pure harmonic picture J
1E 2!2 ) 1E 2 (keV) "%"0 %"0
C /C "%"0 %"0
R
0> 2>
!120 !90 0
1.44 1.45 (2
1.94 1.96 2
be di!erent for di!erent JL values. Unfortunately, such calculation is not possible at the present moment. The calculated excitation functions displayed in Figs. 36b and c yield the following values for the centroid and width of the DGDR in Xe: 1E >2"30.68 MeV and C >"6.82 MeV for the 0> component of the DGDR and 1E >2"30.71 MeV and C >"6.84 MeV for the 2> component. These values have to be compared to 1E \2"15.40 MeV and C \"4.72 MeV for the single GDR in this nucleus from our calculation. The correspondence between these values is presented in Table 15 in comparison with the prediction of the harmonic model. Also shown is the ratio 1WJ > > "M(E1)"WJ\2 ) 1WJ\"M(E1)"W 2" R" J J " 1WJ\"M(E1)"W 2" J
(270)
between the two-step excitation probability of the DGDR normalized to the summed excitation probability of the one-phonon GDR. The numerical results lie quite close to the predictions of the harmonical model (see also a discussion of this problem in Ref. [110]). While the on-the-energyshell transitions are easier to identify and calculate properly, o!-the-energy shell corrections are considerably more elusive. In fact, it may be argued that the calculated shift of the energy centroid of the DGDR with respect to that expected in the harmonic picture is somewhat underestimated, because of the limitations used in selecting two-phonon basis states used in the calculation. Our calculated value *E"21E 2!1E 2 shown in Table 15 can be compared to the ones in %"0 "%"0 Ca [111] and Pb [74] in calculations with Skyrme forces. One of the purposes of the last calculations was to consider the anharmonic properties of the DGDR with the wave function which includes collective 1p1h and 2p2h states. Thus, an interaction not only between the two-phonon DGDR states, [1\;1\] among themselves, but with other two-phonon states made up of collective 2> and 3\ phonons was taken into account. The reported value of *E in these studies is of the order of !200 keV in consistency with our results. It should be pointed out that the calculation with Skyrme forces also yield somewhat larger anharmonicity shifts for low-lying two-phonon states as compared to the QPM calculations [112]. The most complete basis of the 2p2h con"gurations has been used in the second-RPA calculations of the DGDR properties in Ca [97] and Pb [98] which includes not only `collective phononsa but non-collective as well. The authors of Refs. [97,98] obtained the values of *E equal to !670 (!40) and !960 (!470) keV for 0> and 2> components of the DGDR, respectively, in Ca (Pb). Recently, the problem of anharmonicity for the DGDR has been also studied within macroscopic approaches in
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Fig. 37. (a) B(E1) values for the GDR and (b) [B(E1);B(E1)] values, Eq. (269), for the DGDR (J"0>#2>) associated with Coulomb excitation in Pb in relativistic heavy ion collision. A smooth curve is a result of averaging over all states with a smearing parameter D"0.5 MeV.
Refs. [76,113]. In Ref. [76] it has been concluded that the A-dependence of it should be as A\ while in Ref. [113] it is A\ in consistency with Ref. [55]. The fragmentation of the DGDR due to the coupling to three-phonon con"gurations has been also calculated in Pb [109]. The "ne structure of the GDR as a result of interaction with two-phonon 1\ con"gurations in this nucleus is presented in Fig. 37a. The [B(E1);B(E1)] values for the DGDR states described by the wave function which includes two- and three-phonon con"gurations are plotted in Fig. 37b. In this calculation we have used the same basis of six the most collective one-phonon 1\ states for the GDR and 21 the most collective two-phonon [1\;1\] states for 0> and 2> components of the DGDR as in the coupled-channels calculation in the previous subsection (see Fig. 33). For a description of the GDR width a coupling to 1161 two-phonon 1\ con"gurations was taken into account. In calculation of the DGDR strength distribution we have neglected the interaction with one-phonon con"gurations and Pauli principle corrections since these e!ects are weaker in double-magic nucleus Pb as compared to the ones in Xe. As a result, the 0> and 2> components of the DGDR are completely degenerated in this calculation. The DGDR width is determined by the coupling of the selected 21 two-phonon con"gurations with 6972 three-phonon ones and is very close to (2 times the width of the GDR. This is a natural result for a folding of two independent phonons in microscopic treatment of the
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problem. As already discussed in Section 3.3, when damping width of giant resonances is described phenomenologically by Breit}Wigner strength distribution one obtains the value 2 for the quantity r"C /C . On the other hand, when the Gaussian strength distribution is used, it yields the "%"0 %"0 value r"(2. This is due to the di!erent behavior of the wings of the above-mentioned strength functions at in"nity. In a microscopic picture, collective resonance state(s) couples to some "nite number of doorway con"gurations and the strength distribution, as a result of this coupling, is always concentrated in a de"nite energy region. It results in r"(2. The square of the amplitude aD D G G(E) of the Coulomb excitation of one-phonon resonances ( + _( + in RHIC in the "rst-order perturbation theory has a smooth exponential energy dependence. This rather simpli"es a calculation of the excitation cross section in RHIC of the states of Eq. (238) which form single-giant resonances. Although a large number of the states of Eq. (238), the giant resonance excitation cross section in this reaction can be easily calculated as a product of the B(E1) values of each state, presented in Figs. 36a and 37a, and an interpolated value of the tabulated function "aD D G G(E)" at E"E(D. The cross sections of the GDR and GQR excitation in Xe J ( + _( + (see Fig. 34) have been calculated this way. A similar procedure may be applied for calculation of the cross section of the DGDR(l > >) states excitation via the GDR(l \) states. In the second-order perturbation theory it equals to (271) p > >"" A(E \, E > >)11\(l \)""E1""0> 21[1\ 1\](l > >)""E1""1\(l \)2" , J J J \ J where A(E , E ) is the reaction amplitude which has a very smooth dependence on both arguments. This function was tabulated and used in the "nal calculation of the DGDR Coulomb excitation cross section in relativistic heavy ion collisions. Let us consider the excitation of the DGDR in the projectile for a Pb (640A MeV)#Pb collision, according to the experiment in Ref. [54], and use the minimum value of the impact parameter, b"15.54 fm, corresponding to the parameterization of Ref. [43]. The cross section for
Fig. 38. The contribution for the excitation of two-phonon 1\ states (long-dashed curve) in "rst-order perturbation theory, and for two-phonon 0> and 2> DGDR states in second order (short-dashed curve). The total cross section (for Pb (640A MeV)#Pb) is shown by the solid curve.
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Coulomb excitation of the DGDR is presented in Fig. 38 by the short-dashed curve as a strength function calculated with an averaging parameter equal to 1 MeV. The contribution of the background of the two-phonon 1\ states to the total cross section is shown by a long-dashed curve in the same "gure. It was calculated in "rst-order perturbation theory. The role of the background in this reaction is much less important than in photoexcitation studies. First, it is because in heavy ion collisions we have a special mechanism to excite selected two-phonon states in the two-step process. Second, the Coulomb excitation amplitude is exponentially decreasing with the excitation energy, while the E1-photoexcitation amplitude is linearly increasing. Nonetheless, Fig. 38 shows that the direct excitation of two-phonon 1\ states cannot be completely excluded from consideration of this reaction. Integrated over the energy interval from 20 to 35 MeV these states give a cross section of 50.3 mb which should be compared with the experimental cross section in the DGDR region for the Pb (640A MeV)#Pb reaction which is equal to 380 mb [54]. The solid line in Fig. 38 is the sum of DGDR and two-phonon background excitations in relativistic heavy ion collisions. The "rst and second moments of excitation functions, displayed by the short-dashed and solid curves in Fig. 38, indicate that the centroid of the total strength is 200 keV lower and the width is 16% larger than the same quantities for the pure DGDR. We point out that this 200 keV shift is even somewhat larger than the one due to the anharmonicities studied in Pb [74]. Direct excitation of the two-phonon 1\ states in Pb (640A MeV)#Pb reaction was also investigated in Ref. [74] in calculation with Skyrme forces. The reported e!ect (a di!erence between 5.07 and 3.55 mb for 22(E (28 MeV) is much weaker than in our calculation because of a rather V limited two-phonon space. Another source of the DGDR enhancement in [74] is due to anharmonicity e!ects. We also checked the last by coupling one-phonon GDR states to (the most important) 1200 two-phonon 1\ states in the DGDR region. Due to the constructive interference between one- and two-phonon states at DGDR energies we got an additional enhancement of 24 mb, which is again larger than the di!erence between 6.42 and 3.55 mb obtained in Ref. [74] for the same reason. The absolute value of the total cross section of the DGDR excitation in RHIC in Pb is somewhat small in our calculation (cf. Table 13) as compared with the experimental "ndings. For example, the experimental value of the total DGDR excitation in the reaction Pb (640A MeV) #Pb, for which the calculations have been performed, equals to 0,38(4) b. As mentioned above, our chosen six 1\ states exhaust only 90.6% of EWSR while the photo-neutron data [32] indicate that this value equals to 122%. Due to this underestimate of exhaust of the EWSR the cross section for the DGDR excitation reduces more strongly than the one for the single GDR. This is because the GDR cross section is roughly proportional to the total B(E1) value while for the DGDR it is proportional to the square of it. If we apply a primitive scaling to obtain the experimental value 122% of EWSR the ratio R"p /p , the last line of Table 13, changes into 0.096 and 0.101 for the coupled-channels "%"0 %"0 calculation and for the perturbation theory, respectively. The experimental "ndings [54] yield the value R "0.116$0.014. The reported [54] disagreement R /R "1.33$0.16 is the result of a comparison with R obtained within a folding model, assuming 122% of the EWSR. We get a somewhat larger value of R (taking into account our scaling procedure) because the B(E1) strength distribution over our six 1\ states is not symmetrical with respect to the centroid energy, E : the lower part is enhanced. A weak energy dependence in the excitation amplitude, which is %"0
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also squared for the DGDR, enhances the DGDR cross section for a non-symmetrical distribution with respect to the symmetrical one, or when the GDR is treated as a single state. The e!ect of the energy dependence is demonstrated for a single GDR in the top part of Fig. 33 where the excitation cross sections are compared to the B(E1) strength distribution. It produces a shift to lower energies of the centroid of the GDR and the DGDR cross sections with respect to the centroid of the B(E1) and the [B(E1);B(E1)] strength distribution, respectively. In our calculation this shift equals to 0.26 MeV for the GDR and to 0.33, 0.28 MeV for the DGDR within coupled channels and perturbation theory, respectively. Of course, this scaling procedure has no deep physical meaning but we have included this discussion to indicate that the disagreement between experiment and theory for the DGDR excitation cross sections in Pb reached the stage when theoretical calculations have to provide a very precise description of both the GDR and the DGDR to draw up "nal conclusions. The situation with the absolute values of cross sections of the DGDR excitation in Xe in RHIC is much less clear than in Pb. The experimental value for the reaction Xe (700A MeV) #Pb is reported to be equal to 215$50 mb [40]. This value is su$ciently larger as compared to any theoretical predictions available (cf. Table 14). But it should be pointed out that a comparison between experimental data for xenon [40] and lead [54] reveal some essential contradiction. While for Pb the above discussed quantity of the ratio between the total cross sections of the DGDR and GDR excitation R ( Pb)"0.116$0.014, its value for Xe: R ( Xe)"0.21$0.05 [40]. Taking into account that experiments for both nuclei have been performed at close projectile energies (per nucleon) and the cross section of the GDR excitation in Xe is about three times less as compared to the one in Pb, the ratio R ( Xe) should be su$ciently smaller than R ( Pb) and not vice versa. Probably, the problem with the absolute value of the DGDR excitation in Xe is related to uncertainties in separating of the contribution of single resonances, the characteristics of which are unknown experimentally for this nucleus and the results of interpolation have been used in evaluating the experimental data. Recently, the experiment for Xe has been repeated by LAND collaboration [114]. The analysis of the new data in the nearest future should clear up the situation. 5.4. The role of transitions between complex conxgurations of the GDR and the DGDR In the previous subsection considering the excitation properties of the DGDR, [B(E1);B(E1)] values or excitation cross sections in RHIC in second-order perturbation theory, we have taken into account only the transition matrix elements between simple one-phonon 1\ GDR and two-phonon 0> or 2> DGDR con"gurations for the second step of the excitation process g.s.PGDRPDGDR. In fact, as already discussed above these con"gurations couple to more complex ones to produce the widths of single and double resonances and, in principle, additional transitions between complex con"gurations of the GDR and the DGDR, together with interference e!ects, may alter the predicted values of excitation probabilities. This problem will be considered in the present subsection (see, also Ref. [115]). It will be concluded that their role is marginal in the process under consideration although a huge amount of the E1-strength is hidden in the GDRPDGDR transition. This negative result ensures that calculations, in which only transitions between simple components of the GDR and DGDR are taken into account and which are much easier to carry out, require no further corrections.
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In microscopic approaches the strength of the GDR is split among several one-phonon 1\ states ? (due to the Landau damping). The wave function "1\2 couples to complex con"gurations "1\2 ? @ yielding the GDR width. We use the index a for simple con"gurations and the index b for complex ones, respectively. Thus, the wave function of the ith 1\ state in the GDR energy region can be schematically written as "1\2" S%"0(a)"1\2# C%"0(b)"1\2 , (272) G G ? G @ ? @ where the coe$cients S%"0(a) and C%"0(b) may be obtained by diagonalizing the nuclear model G G Hamiltonian on the set of wave functions (272). The total E1-strength of the GDR excitation from the ground state, B (E1)" "11\""E1""0> 2" , %"0 G G remains practically the same as in the one-phonon RPA calculation because the direct excitation of complex con"gurations from the ground state is a few orders of magnitude weaker as compared to excitation of one-phonon states. However these complex con"gurations play a fundamental role for the width of the GDR. The wave function of the 2> component of the DGDR states can be written in the similar fashion: "2>2" S"%"0(a)"[1\ ;1\ ] >2# SI "%"0(a )"2> 2# C"%"0(b )"2>2 . (273) D D ? ? D ? D @Y ? +?"?, ? @Y In this equation, we have separated in the "rst term the [1\;1\] DGDR con"gurations from other two-phonon con"gurations (second term) and complex con"gurations (the last term). The same equation as (273) is valid for the 0> DGDR states. The total E1-transition strength between the GDR and DGDR, "12>(0>) ""E1""1\2" , D G D G is much larger as compared to that for the GDR excitation, "11\""E1""0> 2", from the ground G G state. This is because the former includes transitions not only between simple GDR and DGDR states but also between complex con"gurations as well. The enhancement factor should be the ratio between the density of simple and complex con"guration in the GDR energy region. But in the two-step excitation process the sum over intermediate GDR states in Eq. (274) reduces the total transition strength for g.s.PGDRPDGDR to &2 ) "B (E1)" (the factor 2 appears due to the %"0 bosonic character of the two phonons which also holds if Landau damping is taken into account). To prove this let us consider the excitation probability of the DGDR
1 (E !E , b) , (274) P (E , b)" a#I \G G (E , b);a#IY \ \ \ G D D G + " + D G "%"0 D + G 4 D G + G + where the index i labels intermediate states belonging to the GDR, and a#I is the (+ (+ "rst-order E1 excitation amplitude for the transition J (M )PJ (M ) in a collision with impact
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parameter b. For each state, J and M denote the total angular momentum and the magnetic projection, respectively. is given by The amplitude a#I (+ (+ (E, b)"(J M 1k"J M );1J ""E1""J 2f (E, b) . (275) a#I #I (+ (+ It is a product of the reduced matrix element 1J ""E1""J 2 for the E1-transition between the states J (M ) and J (M ) which carries nuclear structure information and the reaction function f (E, b). The latter depends on the excitation energy, charge of the target, beam energy, and is #I calculated according to Ref. [18]. Except for the dependence on the excitation energy, it does not carry any nuclear structure information. The cross section for the DGDR is obtained from Eq. (274) by integration over impact parameters, starting from a minimal value b to in"nity. This
minimal value is chosen according to Ref. [19]. Now we substitute the wave functions of the GDR and DGDR states given by Eqs. (272) and (273) in expression (274). We obtain two terms. The "rst one corresponds to transitions between simple GDR and DGDR states (after the GDR is excited from the ground state through its simple component): (E , b)11\""E1""0> 2 A " S%"0(a) f G #I G ? IIY G ??Y? (276) ;S%"0(a )S"%"0(a) f (E !E , b)1[1\ ;1\ ] ""E1""1\2d ? D ?Y ? ?Y G D #IY D G ? and the second one accounts transitions between complex con"gurations in the wave functions of Eqs. (272) and (273): B " S%"0(a) f (E , b)11\""E1""0> 2 IIY G #I G ? G ??Y@@Y ;C%"0(b)C"%"0(b ) f (E !E , b)1[1\;1\] ""E1""1\2d . (277) G D #IY D G ?Y @ D @ @Y ?Y"@
The second reduced matrix element in the above equations is proportional to the reduced matrix element between the ground state and the simple one-phonon con"guration (see Eq. (261)). For a given impact parameter b, the function f (E, b) can be approximated by a constant value #I f [6] for the relevant values of the excitation energies. Then the energy dependence can be taken #I out of summations and orthogonality relations between di!erent components of the GDR wave functions can be applied [107]. The orthogonality relations between the wave functions imply that S%"0(a)C%"0(b),0 . (278) G G G This means that the term B vanishes. The term A summed over projections and all "nal states IIY IIY yields a transition probability to the DGDR, P (E , b), which is proportional to 2 ) "B (E1)" "%"0 D %"0 in second-order perturbation theory. This argument was the reason for neglecting the term B in IIY previous calculations of DGDR excitation where the coupling of simple GDR and DGDR states to complex con"gurations was taken into account. In Fig. 39 we plot the value of s (E)"2pdb b I " f#I(E, b)" as a function of energy calculated # for the Pb (640A MeV)#Pb reaction. This value corresponds to p%"0 if B%"0(E1)"1. The square in this "gure indicates the location of the GDR in Pb. This "gure demonstrates that the
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Fig. 39. The energy dependence of the Pb (640A MeV)#Pb reaction function calculated within "rst-order perturbation theory. The square indicates the location of the GDR in Pb.
function s#(E) changes by 60% in the GDR energy region. The role of this energy dependence for other e!ects has been considered in Refs. [19,74]. Taking into account that one-phonon 1\ ? con"gurations are fragmented over a few MeV [108], when a su$ciently large two-phonon basis is included in the wave function given by Eq. (272), the role of the BIIY term in the excitation of the DGDR should be studied in more detail. To accomplish this task we have performed "rstly a simpli"ed calculation in which we used the boson-type Hamiltonian: H" u?QR? Q?# u@QI R@QI @# ;?@(QR? QI @#h.c.) , (279) ? @ ? @ where QR? is the phonon creation operator and u? is the energy of this one-phonon con"guration; QI R@ is the operator for creation of a complex con"guration with energy u@ and ;?@ is the matrix element for the interaction between these con"gurations. We have assumed that the energy di!erence between two neighboring one-phonon con"gurations is constant and equals to *u. An equidistant spacing with the energy *u was assumed for the complex con"gurations. We also have used a constant value ; for the matrix elements of the interaction. The B%"0(E1) value was distributed symmetrically over one-phonon con"gurations. Thus, the free parameters of this model are: *u, *u, ;, the number of one-phonon and complex con"gurations, and the distribution of the B%"0(E1) value over the simple con"gurations. The only condition we want to be satis"ed is that the energy spectrum for the GDR photoexcitation is the same as the one known from experiment. After all parameters are "xed we diagonalize the model Hamiltonian of Eq. (279) on the set of wave functions of Eq. (272) for the GDR and on the set of Eq. (273) for the DGDR. The diagonalization procedure yields the information on eigenenergies of the 1\ G GDR states and on the coe$cients S%"0 (a) and C%"0 (b), respectively. One also obtains information on eigenenergies G G of the 2> or 0> DGDR states and the coe$cients S"%"0 (a ) and C"%"0 (b ), respectively. With this D D D D
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Fig. 40. The cross section for the excitation of the 2> component of the DGDR in the reaction Pb (640A MeV)# Pb, calculated within second-order perturbation theory. The dashed curve shows the contribution of the E1-transition between simple GDR and DGDR con"gurations only. The solid curve is a sum of the above result and the contribution of the E1-transitions between complex GDR and DGDR con"gurations. See text for details.
information we are able to study the role of the BIIY term in the excitation of the DGDR in RHIC. The big number of free parameters allows an in"nite number of suitable choices. In fact, not all of the parameters are really independent. For example, the increase in the number of simple or complex con"gurations goes together with the decreasing of the value of ;. This is necessary for a correct description of the GDR photoabsorption cross section. This makes it possible to investigate the role of the BIIY term in di!erent conditions of weak and strong Landau damping and for di!erent density of complex con"gurations. In our calculations we vary the number of collective simple states from one to seven and the number of complex con"gurations from 50 to 500. The value of ; then changes from about 100}500 keV. The results of one of these calculations for the excitation of the 2> component of the DGDR in Pb (640A MeV)# Pb collisions are presented in Fig. 40. For a better visual appearance the results are averaged with a smearing parameter equal to 1 MeV. The dashed curve shows the results of a calculation in which p "%"0(E),p"%"0(E)&db b"AIIY" and the results of another one in which p > "%"0(E), p"%"0(E)&db b"AIIY#BIIY" are represented by a solid curve. Our calculation within this simple model indicates that the role of the BIIY term in second-order perturbation theory is negligibly small, although the total B(E1) strength for transitions between complex GDR and DGDR con"gurations, considered separately, is more than two orders of magnitude larger than the ones between simple GDR and DGDR con"gurations. The value *p"(p > "%"0!p "%"0)/p "%"0, where p > "%"0 "p > "%"0 (E) dE, changes in these calculations from 1% to 2.5%. The results practically do not depend on the number of complex con"gurations accounted for. The maximum value of *p is achieved in a calculation with a single one-phonon GDR state (no Landau damping). This is because the value of ; is larger in this case and the fragmentation of the one-phonon state is stronger. Thus, in such a situation, the energy dependence of the reaction amplitude modi"es appreciably the orthogonality relations. But in general the e!ect is marginal. We also performed a calculation with more realistic wave functions for the GDR and DGDR states taken from our studies presented in the previous subsection. These wave functions include
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6 and 21 simple states for the GDR and DGDR, respectively. The complex con"gurations are two-phonon states for the GDR and three-phonon states for the DGDR. The value *p equals in this realistic calculation to 0.5%. This result is not surprising because realistic calculations with only two-phonon complex con"gurations, and a limited number of them, somewhat underestimate the GDR width which is crucial for the modi"cation of the orthogonality relations. We have proved that the transitions between complex GDR and DGDR con"gurations within second-order perturbation theory for the DGDR excitation in RHI collisions play a marginal role in the process under consideration and it is su$cient to take into account only transitions between the ground-state and one-phonon GDR and two-phonon DGDR con"gurations. 5.5. The DGDR in deformed nuclei The possibility to observe two-phonon giant resonances in deformed nuclei with the present state-of-art experimental techniques is still questionable. This is mainly due to the fact that one has to expect a larger width of these resonances as compared to spherical nuclei. Also, the situation with the low-lying two-phonon states in deformed nuclei is much less clear than in spherical ones. The "rst experiment with the aim to observe the double-giant dipole resonance (DGDR) in U in relativistic heavy ion collisions (RHIC) was performed recently at the GSI/SIS facility by the LAND collaboration [114]. It will take some time to analyze the experimental data and to present the "rst experimental evidence of the DGDR in deformed nuclei, if any. Thus, we present here the "rst theoretical predictions of the properties of the DGDR in deformed nuclei based on microscopic study [116]. The main attention will be paid to the width of the DGDR and its shape. In a phenomenological approach the GDR is considered as a collective vibration of protons against neutrons. In spherical nuclei this state is degenerate in energy for di!erent values of the spin J"1\ projection M"0,$1. The same is true for the 2> component of the DGDR with projection M"0,$1,$2. In deformed nuclei with an axial symmetry like U, the GDR is spit into two components Ip(K)"1\(0) and Ip(K)"1\($1) corresponding to vibrations against two di!erent axes. In this approach one expects a three-bump structure for the DGDR with the value K"0, K"$1 and K"0,$2, respectively (see Fig. 41). Actually, the GDR possesses a width and the main mechanism responsible for it in deformed nuclei is the Landau damping. Thus, the conclusion on how three bumps overlap and what is the real shape of the DGDR in these nuclei, i.e., either a three-bump or a #at broad structure, can be drawn out only from some consistent microscopic studies. We use in our calculations for U the parameters of Woods}Saxon potential for the average "eld and monopole pairing from Ref. [117]. They were adjusted to reproduce the properties of the ground-state and low-lying excited states. The average "eld has a static deformation with the deformation parameters b"0.22 and b"0.08. To construct the phonon basis for the K"0 and K"$1 components of the GDR we use the dipole-dipole residual interaction (for more details of the QPM application to deformed nuclei, see e.g. Ref. [84]). The strength parameters of this interaction are taken from Ref. [118] where they have been "tted to obtain the centroid of the B(E1, 0> P1\(K"0,$1)) strength distribution at the value known from experiment [119] and to exclude the center of mass motion. In this approach, the information on the phonon basis (i.e. the excitation energies of phonons and their internal fermion structure) is obtained by solving
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Fig. 41. The possible paths to the excitation of a given magnetic substate of the 0> and 2> components of the DGDR in spherical and deformed nuclei. The notations are the same as in Fig. 32.
Fig. 42. The B(E1) strength distribution over K"0 (short-dashed curve) and K"$1 (long-dashed curve) 1\ states in U. The solid curve is their sum. The strongest one-phonon 1\ states are shown by vertical lines, the ones with K"0 are marked by a triangle on top. Experimental data are from Ref. [119].
the RPA equations. For electromagnetic E1-transitions we use the values of the e!ective charges, e8, "eN(!Z)/A to separate the center of mass motion. The results of our calculation of the B(E1) strength distribution over "1\ )(i)2 and "1\ )!(i )2 GDR states are presented in Fig. 42, together with experimental data. The index i in the wave function stands for the di!erent RPA states. All one-phonon states with the energy lower than 20 MeV and with the B(E1) value larger than 10\ e fm are accounted for. Their total number equals to 447 and 835 for the K"0 and K"$1 components, respectively. Only the strongest of them with B(E1)50.2 e fm are shown in the "gure by vertical lines. Our phonon basis exhausts
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32.6% and 76.3% of the energy weighted sum rules, 14.8 ) NZ/A e fm MeV, by the K"0 and K"$1 components, respectively. For a better visual appearance we also present in the same "gure the strength functions averaged with a smearing parameter, which we take as 1 MeV. The short (long) dashed-curve represent the K"0 (K"$1) components of the GDR. The solid curve is their sum. The calculation reproduces well the two-bump structure of the GDR and the larger width of its K"$1 component. The last is consistent with the experiment [119] which is best "tted by two Lorentzians with widths equal to C "2.99 MeV and C "5.10 MeV, respectively. The amplitudes of both maxima in the calculation are somewhat overestimated as compared to the experimental data. This happens because the coupling of one-phonon states to complex con"gurations is not taken into account which can be more relevant for the K"$1 peak at higher energies. But in general the coupling matrix elements are much weaker in deformed nuclei as compared to spherical ones and the Landau damping describes the GDR width on a reasonable level. The wave functions of the 0> and 2> states belonging to the DGDR are constructed by the folding of two 1\ phonons from the previous calculation. When a two-phonon state is constructed as the product of two identical phonons its wave function gets an additional factor 1/(2. The 1> component of the DGDR is not considered here for the same reasons as in spherical nuclei. The anharmonicity e!ects which arise from interactions between di!erent two-phonon states are also not included in the present study. The folding procedure yields three groups of the DGDR states: (280a) "[1\ (i ) 1\ (i )] >) >)2 , ) ) "[1\ (i) 1\ (i )] 2, (280b) ) )! >)! "[1\ (i ) 1\ (i )] 2. (280c) )! )! >) >) ! The total number of non-degenerate two-phonon states equals to about 1.5;10. The energy centroid of the "rst group is the lowest and of the last group is the highest among them. So, we also obtain the three-bump structure of the DGDR. But the total strength of each bump is fragmented over a wide energy region and they strongly overlap. Making use of the nuclear structure elements discussed above, we have calculated the excitation of the DGDR in U projectiles (0.5 GeV ) A) incident on Sn and Pb targets, following the conditions of the experiment in Ref. [114]. These calculations have been performed in secondorder perturbation theory [6], in which the DGDR states of Eqs. (280a), (280b) and (280c) are excited within a two-step process: g.s.PGDRPDGDR. As intermediate states, the full set of one-phonon "1\ (i)2 and "1\ (i )2 states was used. We have also calculated the GDR excitation ) )! to "rst order for the same systems. The minimal value of the impact parameter, which is very essential for the absolute values of excitation cross section has been taken according to b "1.28 ) (A #A ).
R N The results of our calculations are summarized in Fig. 43 and Table 16. In Fig. 43 we present the cross sections of the GDR (part a) and the DGDR (part b) excitation in the U (0.5 GeV ) A)#Pb reaction. We plot only the smeared strength functions of the energy distributions because the number of two-phonon states involved is numerous. The results for U (0.5 GeV ) A)#Sn reaction look very similar and di!er only by the absolute value of cross
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Fig. 43. The strength functions for the excitation: (a) of the GDR, and (b) of the DGDR in U in the U (0.5A GeV)#Pb reaction. In (a), the short-dashed curve corresponds to the GDR (K"0) and the long-dashed curve to the GDR (K"$1). In (b) the dashed curve corresponds to the DGDR > (K"0), the curve with circles to the DGDR >(K"0), the curve with squares to the DGDR > (K"$1), and the curve with triangles to the DGDR > (K"$2). The solid curve is the sum of all components. The strength functions are calculated with the smearing parameter equal to 1 MeV.
Table 16 The properties of the di!erent components of the GDR and the DGDR in U. The energy centroid E , the second moment of the strength distribution m in RHIC, and the cross sections p for the excitation of the projectile are presented for: (a) U (0.5A GeV)#Sn, and (b) U (0.5A GeV)#Pb p (mb)
E
m
(MeV)
(MeV)
(a)
(b)
GDR(K"0) GDR(K"$1) GDR(total)
11.0 12.3 12.0
2.1 2.6 2.6
431.2 1560.2 1991.4
1035.4 3579.1 4614.5
DGDR >(K"0) DGDR >(K"0) DGDR >(K"$1) DGDR >(K"$2) DGDR(total)
25.0 24.4 23.9 25.3 24.8
3.4 3.5 3.2 3.4 3.4
18.3 11.8 22.7 49.7 102.5
88.9 58.7 115.4 238.3 501.3
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sections. In Table 16 the properties of the GDR and the DGDR, and their di!erent K components are given. The energy centroid E and the second moment, m "( p (E !E )/ p , of the I I I I I distributions are averaged values for the two reactions under consideration. The two-bump structure can still be seen in the curve representing the cross section of the GDR excitation in U in RHIC as a function of the excitation energy. But its shape di!ers appreciably from the B(E1) strength distribution (see Fig. 43a in comparison with Fig. 42). The reason for this is the role of the virtual photon spectra. First, for the given value of the excitation energy and impact parameter it is larger for the K"$1 component than that for the K"0 one (see also the "rst two lines in Table 16). Second, for both components it has a decreasing tendency with an increase of the excitation energy [6]. As a result, the energy centroid of the GDR excitation in RHIC shifts by the value 0.7 MeV to lower energies as compared to the same value for the B(E1) strength distribution. The second moment m increases by 0.2 MeV. The curves representing the cross sections of the excitation of the K"$1 and K"$2 components of the DGDR in U in RHIC have typically a one-bump structure (see the curves with squares and triangles in Fig. 43b, respectively). It is because they are made of two-phonon 2> states of one type: the states of Eqs. (280b) and (280c), respectively. Their centroids should be separated by an energy approximately equal to the di!erence between the energy centroids of the K"0 and K"$1 components of the GDR. They correspond to the second and the third bumps in a phenomenological treatment of the DGDR. The K"0 components of the DGDR include two group of states: the states represented by Eq. (280a) and those of Eq. (280c). Its strength distribution has two bumps (see the curve with circles for the 2>(K"0) and the dashed curve for the 0>(K"0) components of the DGDR, respectively). The excitation of the states given by Eq. (280a) in RHIC is enhanced due to their lower energies, while the enhancement of the excitation of the states given by Eq. (280c) is related to the strongest response of the K"$1 components to the external E1 Coulomb "eld in both stages of the two-step process. Summing together all components of the DGDR yields a broad one-bump distribution for the cross section for the excitation of the DGDR in U, as a function of excitation energy. It is presented by the solid curve in Fig. 43b. Another interesting result of our calculations is related to the position of the DGDR energy centroid and to the second moment of the DGDR cross section. The centroid of the DGDR in RHIC is shifted to the higher energies by about 0.8 MeV from the expected value of two times the energy of the GDR centroid. The origin for this shift is in the energy dependence of the virtual photon spectra and it has nothing to do with anharmonicities of the two-phonon DGDR states. In fact, the energy centroid of the B(E1, g.s.P1\); G B(E1, 1\PDGDR ) strength function appears exactly at twice the energy of the centroid of the G D B(E1, g.s.PGDR) strength distribution because the coupling between di!erent two-phonon DGDR states are not accounted for in the present calculation. The same shift of the DGDR from twice the energy position of the GDR in RHIC also takes place in spherical nuclei. But the value of the shift is smaller there because in spherical nuclei the GDR and the DGDR strength is less fragmented over their simple con"gurations due to the Landau damping. But the larger value of the shift under consideration in deformed nuclei should somehow simplify the separation of the DGDR from the total cross section in RHIC. Another e!ect which also works in favor of the extraction of the DGDR from RHIC excitation studies with deformed nuclei is its smaller width than (2 times the width of the GDR, as observed with spherical nuclei. Our calculation yields the value 1.33 for the ratio C /C in this "%"0 %"0
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reaction. The origin for this e!ect is in the di!erent contributions of the GDR K"0 and K"$1 components to the total cross section, due to the reaction mechanism. It should be remembered that only the Landau damping is accounted for the width of both the GDR and the DGDR. But since the e!ect of narrowing of the DGDR width is due to the selectivity of the reaction mechanism it will still hold if the coupling to complex con"gurations is included in the calculation. It may be argued that the procedure of independent excitations of two RPA phonons applied here is not su$cient for a consistent description of the properties of the two-phonon giant resonances. This is true for the case of spherical nuclei where only the coupling of two GDR phonons to more complex, 3p3h, con"gurations allows one to describe the DGDR width as discussed above. But the typical matrix element of this coupling in deformed nuclei does not exceed the value of 200 keV [120] while in spherical nuclei it is an order of magnitude larger. It means that due to the coupling, the strength of each GDR RPA-phonon will fragment within the energy interval of 100}200 keV in deformed nuclei. The last value should be compared to the second moment, m , presented in Table 16 which is the result of the Landau damping accounted for in our calculation. Taking into account that the reaction amplitude has very weak energy dependence and that mixing of di!erent RPA phonons in the GDR wave function does not change the total strength [115], the total cross sections of the GDR and DGDR excitation in RHIC will be also conserved.
Acknowledgements We thank our colleagues G. Baur, P.F. Bortignon, R.A. Broglia, L.F. Canto, M. Hussein, A.F.N. de Toledo Piza, A.V. Sushkov, V.V. Voronov for fruitful collaboration. We also thank H. Emling for many fruitful and stimulative discussions. This work was supported in part by the Brazilian agencies CNPq, FINEP/PRONEX, FAPERJ and FUJB, the Russian Fund for Basic Research (grant no. 96-15-96729) and the Research Council of the University of Gent.
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T. Elsaesser, M. Woerner / Physics Reports 321 (1999) 253}305
FEMTOSECOND INFRARED SPECTROSCOPY OF SEMICONDUCTORS AND SEMICONDUCTOR NANOSTRUCTURES
Thomas ELSAESSER, Michael WOERNER Max-Born-Institut fu( r Nichtlineare Optik und Kurzzeitspektroskopie, Rudower Chaussee 6, D-12489 Berlin, Germany
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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Physics Reports 321 (1999) 253}305
Femtosecond infrared spectroscopy of semiconductors and semiconductor nanostructures Thomas Elsaesser*, Michael Woerner Max-Born-Institut fu( r Nichtlineare Optik und Kurzzeitspektroskopie, Rudower Chaussee 6, D-12489 Berlin, Germany Received February 1999; editor: J. Eichler Contents 1. Introduction 2. Below-bandgap excitations in semiconductors 2.1. Elementary excitations of the electronic system below the fundamental bandgap 2.2. Hierarchy of ultrafast processes 3. Experimental techniques 3.1. Free electron lasers 3.2. Modelocked solid state lasers for the infrared 3.3. Nonlinear frequency conversion 3.4. Techniques of time-resolved spectroscopy
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257 265 272 272 272 274 278
4. Ultrafast dynamics in bulk semiconductors 4.1. Relaxation processes of holes 4.2. Intraband excitations 5. Ultrafast dynamics of intersubband excitations in quasi-two-dimensional semiconductor structures 5.1. Coherent intersubband polarizations 5.2. Intersubband scattering and thermalization 5.3. Carrier cooling 6. Conclusions and outlook Acknowledgements References
281 282 286
287 287 291 295 299 300 300
Abstract Infrared spectroscopy on ultrafast time scales represents a powerful technique to investigate the nonequilibrium dynamics of elementary excitations in bulk and nanostructured semiconductors. In this article, recent progress in this "eld is reviewed. After a brief introduction into electronic excitations below the fundamental bandgap and ultrafast processes in semiconductors, infrared pulse generation and the methodology of time-resolved infrared spectroscopy are reviewed. The main part of this paper is devoted to coherent optical polarizations and nonequilibrium excitations of the electronic system in the spectral range below the fundamental band gap. The focus is on the physics of single component plasmas, i.e. electrons or holes. In particular, intraband, inter-valence and intersubband transitions are considered. Processes of phase
* Corresponding author. Tel.: #49-30-6392-1400; fax: #49-30-6392-1409. E-mail address:
[email protected] (T. Elsaesser) 0370-1573/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 3 4 - 4
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relaxation, carrier and energy redistribution are analyzed. The potential of ultrafast infrared technology and spectroscopy for future applications is discussed in the "nal part. ( 1999 Elsevier Science B.V. All rights reserved. PACS: 78.47.#p; 42.65.Re; 73.20.Dx Keywords: Ultrafast phenomena; Infrared spectroscopy; Semiconductors; Below bandgap excitations
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1. Introduction Elementary excitations in solids show a complex nonequilibrium behavior with a dynamics governed by their mutual coupling. The fastest nonequilibrium processes occur on ultrafast time scales and strongly in#uence both optical properties and carrier transport. Semiconductors represent an interesting class of model systems in which the physical parameters relevant for such processes, e.g. electronic bandstructure, optical transition energies, carrier concentration, and phonon frequencies, vary over a broad range, leading to a variety of ultrafast phenomena. Moreover, the quantum con"nement of wavefunctions in low-dimensional semiconductor nanostructures allows a systematic variation of material properties. Such aspects of fundamental physics are complemented by the relevance of nonequilibrium carrier dynamics in high speed semiconductor devices. Optical spectroscopy with ultrashort pulses provides direct insight into these processes occurring on a time scale between about 10~14 and 10~10 s. Most experimental studies have concentrated on interband excitation at photon energies close to or higher than the fundamental bandgap of the semiconductor, i.e. the properties of excitons and/or electron}hole pairs in the band-to-band continuum have been investigated. Such studies have provided much information on the dynamics of coherent interband polarizations and relaxation processes of electron}hole plasmas, both from an experimental and a theoretical point of view. A number of excellent reviews of these topics are available in the literature [1,2]. A principal drawback of such studies, however, consists in the presence of two types of carriers, electrons and holes. The transient optical properties measured in such ultrafast experiments are determined by the distribution functions of both electrons and holes which are di$cult to separate. In most cases, the distribution of holes in the nonparabolic warped valence bands is not well characterized. Furthermore, the correlation of electrons and holes via their Coulomb interaction a!ects the optical interband spectra, in particular in the range close to the bandgap. Electron}hole scattering represents another consequence of this type of Coulomb interaction. These e!ects make a quantitative analysis of such experiments quite di$cult. A variety of elementary excitations occurs in the energy range below the fundamental bandgap. Intraband (free carrier), inter-valence band, and inter-subband transitions are determined by a single type of carrier, i.e. electrons or holes and can serve as a probe of the dynamics of single component plasmas. In many cases, a time-resolved study of these excitations gives much more speci"c information than experiments with electron}hole plasmas. Moreover, the theoretical analysis may be easier to perform than for electron}hole excitations. Below-bandgap excitations occur mainly in the infrared spectral range beyond a wavelength of 1 lm and investigation of their dynamics requires ultrashort pulses in that range. During the last ten years, the generation of stable and widely tunable femtosecond pulses in the infrared has made substantial progress and a variety of femtosecond infrared sources covering the wavelength range between 1 and about 100 lm are available now. Concomitantly, an increasing number of ultrafast infrared studies of semiconductors and applications in optoelectronics have been reported. In this article, we present a review of femtosecond infrared spectroscopy of semiconductors and semiconductor nanostructures. We concentrate on studies of below-bandgap excitations in the infrared wavelength range from 1.5 to about 20 lm. The main emphasis is on experiments providing insight into the nonequilibrium dynamics of optical polarizations and photoexcited carriers. The article is organized in the following way: Section 2 gives a tutorial discussion of
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elementary excitations below the bandgap which is followed by an introduction into ultrafast processes. This part includes a discussion of the relevant literature. Techniques of generation of ultrashort infrared pulses and methods of time-resolved infrared spectroscopy are reviewed in Section 3. In Sections 4 and 5, some recent experimental results are discussed in more detail, together with their theoretical analysis. Section 4 is devoted to hole dynamics and intraband excitations in bulk semiconductors, Section 5 to nonequilibrium phenomena in quasi-two-dimensional semiconductor structures occurring after intersubband excitation of electrons. In both Sections 4 and 5, we focus on the behavior of single component plasmas of electrons or holes. Conclusions and an outlook are presented in the "nal Section 6.
2. Below-bandgap excitations in semiconductors 2.1. Elementary excitations of the electronic system below the fundamental bandgap There are four di!erent types of optical excitations below the fundamental absorption edge of doped bulk and nanostructured semiconductors which are depicted schematically in Fig. 1: (a) Transitions between electronic states of impurity atoms or vacancies, and from impurity levels into the valence or conduction band continuum, (b) indirect intraband excitations, the so-called free carrier absorption, (c) inter-valence band transitions of holes, and (d) intersubband transitions between valence and conduction subbands in low-dimensional semiconductor nanostructures, e.g. quantum wells, wires and dots. In order to classify experimental infrared spectra of doped semiconductors according to these categories, carrier statistics has to be taken into account. In general, there are carriers bound to impurity atoms, and free carriers populating the valence or
Fig. 1. Optical excitations of the electronic system below the fundamental absorption edge of doped bulk (left-hand side) and quasi-two-dimensional (right-hand side) semiconductors: (a) impurity related transitions, (b) indirect intraband excitations (free carrier absorption), (c) intervalence band transitions, and (d) intersubband transitions between di!erent conduction subbands in quantum wells.
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conduction band continuum. According to the law of mass action [3], the fraction of free carriers is determined by the binding energy and concentration of the dopant and by the lattice temperature. In the following, we brie#y review the basic physics of the excitations (b)}(d), for a review of impurity related transitions, the reader is referred to Ref. [4]. 2.1.1. Indirect intraband excitations The presence of free carriers in the conduction or valence band leads to free carrier absorption (FCA) in the mid- to far-infrared spectral range. FCA represents an intraband absorption process in which an electron or hole is promoted to a state of higher energy in the same band. Both energy and k-vector have to be conserved in this process. As the optical excitation is vertical in k-space, k-conservation requires coupling to a third particle, e.g. a phonon or an impurity, of "nite momentum. In the classical formulation, FCA is described by the Drude model which is also applied for high frequency transport properties of the material. This treatment starts from the frequency dependent conductivity of a collision broadened carrier plasma: Ne2q 1 p(u)" . mH 1#iuq
(1)
Here, N is the density of carriers, mH their e!ective mass, and q is the momentum relaxation time of the carriers (e: elementary charge). In the simplest approximation, this relaxation is independent of the carrier energy. Momentum relaxation of the carriers is caused by elastic and inelastic collisions with third particles, e.g. impurities or phonons. For a single component plasma populating a spherical and parabolic band, electron-electron scattering does not lead to any momentum relaxation and, thus, does not a!ect FCA. In the Drude model, the frequency dependent absorption coe$cient a (u) for infrared frequencies u with uq<1 is approximately given by: FCA 2nNe2 1 a (u)+ Jj2 , (2) FCA cnmHq u2 where n is the refractive index at the frequency u. Experimentally, the spectra of FCA have been investigated for silicon, germanium, and numerous III}V semiconductors. The measurements demonstrate } if at all } a j2-dependence only over limited spectral intervals. For instance, the FCA of n-type InAs measured in Ref. [5] exhibits an increase with j3, in contrast to Eq. (2). Part of this discrepancy has been attributed to an energy dependent momentum relaxation time q, however, without providing a clear picture of the underlying physics. Quantum mechanical models in which the microscopic three particle interaction underlying FCA is treated in second order perturbation theory, give a more appropriate description [6]. In n-type III}V semiconductors like InAs or GaAs coupling to longitudinal optical (LO) phonons via the polar}optical interaction represents an important mechanism of momentum conservation. The four relevant coupling schemes are depicted in Fig. 2: absorption of a photon in combination with absorption or emission of a LO phonon and stimulated emission of a photon in combination with absorption or emission of a LO phonon. In second order perturbation theory, the transition rate of electrons from initial states at energies e(k) into "nal states at e@(k@) by absorption of an infrared photon is given by R "(R #R ) with !"4 ` ~ 2< 2n R " d3k d3k@ (DHkkD2DHk@kD2 )d(e@(k@)!e(k)!+u$+u ) f (k)[1!f (k@)] . (3) B (2n)6 B q +3u2
P
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Fig. 2. Schematic of indirect intraband photon absorption and emission processes involving longitudinal optical (LO) phonons via the polar interaction for k-vector conservation. Four indirect processes are depicted for a parabolic band: absorption of a photon in combination with absorption (emission) of a LO phonon and stimulated emission of a photon in combination with absorption (emission) of a LO phonon.
Here DHkkD is the matrix element for the interaction of an electron with a photon of energy +u which is described in dipole approximation, DHk@kD describes the coupling to LO phonons where #and B } signs stand for phonon emission and absorption, respectively. Expressions for both matrix elements can be found, e.g., in Refs. [3,7]. The distribution functions f (k) and f (k@) give the occupation probability of the initial and "nal states. The delta function assures energy conservation (+u: photon energy, +u : phonon energy, +"h/2n with h: Planck's constant, < volume of the q crystal). A corresponding rate expression holds for the intraband emission rate of photons R of %. an energy +u. The net absorption coe$cient a (+u) is proportional to the di!erence of the two FCA rates, i.e. a (+u)J(R !R ). FCA !"4 %. It is important to note that there is no one-to-one relation between the carrier energy and the photon energy in FCA. However, there is a close relation between the photon energy and the LO phonon momentum required for k-conservation. FCA at high/low frequencies is predominantly connected with LO phonons of large/small momentum. In time-resolved experiments, this fact can be exploited to monitor transient phonon populations created by carrier cooling as will be discussed in more detail in Section 4.2. The absolute value of the absorption coe$cient and the wavelength dependence are in#uenced by the variation of the matrix element DHk@kD with the phonon wavevector q"k@!k. For low carrier concentration, the relation DHk@kD2J1/q2 holds, for higher carrier density screening of this long-range interaction by the carrier plasma leads to changes of the q-dependence. In the quantum-mechanical treatment outlined above, excitation of individual electrons is considered, corresponding to a single particle picture of the carriers. This approach neglects many-body e!ects arising from the Coulomb interaction among the carriers which are important at higher carrier concentration. Many-body aspects of FCA have been discussed in Ref. [8]. In addition to many-body e!ects, the coherent character of the intraband excitation process and the dephasing of the macroscopic intraband polarization are neglected in the present discussion. On
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time scales shorter than the LO phonon period which is on the order of 120 fs in GaAs, the picture of incoherent phonon scattering based on the Boltzmann equation breaks down and a quantum kinetic description of phonon coupling has to be applied [9]. To our knowledge, a quantum kinetic treatment of intraband absorption has not been presented so far. In addition, the coherence properties of intraband excitations have not been explored in ultrafast experiments. 2.1.2. Inter-valence band absorption Inter-valence band absorption is due to dipole-allowed transitions of free holes from states in one valence band to states of higher energy in another valence band. For bulk semiconductors with a diamond-like, e.g. silicon and germanium, or zinc-blende lattice like most III}V semiconductors, inter-valence band absorption is dominated by transitions between the heavy hole (HH), light hole (LH), and split-o! (SO) valence band. Early experimental work concentrated on those materials, in particular p-type germanium [10] and p-type GaAs [11]. In Fig. 3, the infrared absorption spectrum of a p-type Ge crystal with an acceptor concentration of N "7]1016 cm~3 is shown on A a logarithmic scale [12]. There are three absorption bands which are due to intervalence band transitions from the heavy hole to the light hole valence band (1P2), from the light hole to the split-o! band (2P3), and from the heavy hole to the split-o! band (1P3). The strength and the shape of the three bands depend strongly on temperature and on the concentration of free holes. The theoretical analysis of inter-valence band absorption started with the early work of Refs. [13,14]. In this approach, the inter-valence band absorption coe$cient is given by a
C (u)" + DM (k)D2 ) [ f (k)!f (k)] ) d(E (k)!E (k)!+u) . IVA iaj i j j i uk
(4)
Fig. 3. Mid-infrared absorption of a p-type germanium sample (impurity concentration N "7]1016 cm~3) for four A di!erent lattice temperatures. The absorption coe$cient a is plotted on a logarithmic scale as a function of the photon energy. In this spectral range transitions between the di!erent valence-bands result in three absorption bands. 1P2: from the HH to the LH valence band, 2P3: from the LH to SO band, and 1P3: from the HH to SO band.
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M (k) is the k-dependent optical dipole matrix element of the particular inter-valence band iaj transition; E (k) and E (k) represent the energy dispersion of the involved valence bands. In i j Ref. [14], the nonspherical and nonparabolic valence band structure and the optical matrix elements were derived from a k ) p bandstructure calculation. This approach which was re"ned later [15], accounts quite well for the experimental spectra of p-type germanium, whereas approximations based on a spherical valence band structure are in disagreement with the experiments. The calculations demonstrate that holes in di!erent areas of the warped valence band structure contribute to absorption at a particular photon energy. The dipole matrix elements and the valence band dispersion show a very weak dependence on temperature. Thus, the temperature dependent shape of the inter-valence band absorption spectra directly re#ects the di!erence between the temperature dependent hole distributions [ f (k)!f (k)] i j in the respective valence bands. This makes inter-valence band absorption a powerful probe to monitor transient changes of hole distributions induced in ultrafast experiments. In Section 4.1, such studies of hole thermalization and cooling will be discussed in detail. It should be noted that expression (4) neglects the homogeneous broadening of the inter-valence band spectra which originates from the phase relaxation of the coherent inter-valence band polarization created in the excitation process. In a crude approximation, this can be included by replacing the delta function in Eq. (4) by a lineshape function of "nite width. For a microscopic understanding, however, the role of hole scattering for dephasing has to be analyzed, e.g. on the basis of the semiconductor Bloch equations. There are "rst experiments giving insight into coherent inter-valence band polarizations and their subpicosecond dephasing dynamics [16]. Similar to inter-valence band absorption, dipole-allowed infrared transitions occur from speci"c conduction band minima to higher conduction bands. For instance, the III}V semiconductors GaAs and GaP exhibit strong absorption between the X and X conduction band minima, 6 7 occurring in the wavelength range between 3 and 4 lm. Transient changes of these absorption bands on the subpicosecond time scale have been studied to monitor electron dynamics, in particular intervalley scattering in bulk GaAs and GaP [17,18]. 2.1.3. Intersubband transitions in quasi-two-dimensional nanostructures Epitaxial growth techniques like molecular beam epitaxy (MBE) [19,20] or metal-organic vapor phase epitaxy (MOVPE) [21] allow the controlled growth of semiconductor nanostructures on an atomic scale. Quantum wells or superlattices in which carrier motion is restricted to a quasi-twodimensional semiconductor layer, have reached a very high degree of perfection using material systems like GaAs/AlGaAs, GaInAs/InP, GaInAs/AlInAs, and Ge/Si. In quantum wells (Fig. 4), the depth of the potential is determined by the bandgap discontinuity between the well and the barrier material. Quantum con"nement occurs in a situation where the length scale of the potential structure, i.e. the well width, is on the order of the de-Broglie wavelength of the carriers j"h/J2mHE (mH, E: e!ective mass and energy of the particles). This leads to a quantized motion in the z-direction. Parallel to the well layer, there is free motion with a continuum of carrier energies. A series of quasi-two-dimensional subbands [Fig. 4(b)] with di!erent e!ective masses and nonparabolicities develops according to the details of the bandstructure. The energy positions and } thus } the separation of the subbands can be tailored by varying the width of the quantum wells. The density of electronic states in such quasi-two-dimensional structures exhibits a step-like shape
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Fig. 4. Basic properties of semiconductor quantum wells. (a) Variation of the conduction and valence band potentials as a function of the spatial coordinate in z-direction, i.e. along the stack axis. Energy levels of the con"ned wavefunctions in the wells are labeled with subsequent numbers. (b) Schematic energy dispersion of various subbands as a function of k according to the free motion of carriers parallel to the layers. (c) Steplike density of states D(E) in the conduction and @@ valence subbands of the quasi-two-dimensional structure.
as a function of the carrier energy, di!erent from the square-root dependence in bulk material [Fig. 4(c)]. The quasi-two-dimensional con"nement of carriers in quantum wells leads to a variety of new phenomena, both with respect to the optical properties and the dynamics of carriers. Due to the splitting of the bandstructure into various subbands both intersubband and intrasubband processes are relevant. Optical transitions between subbands of di!erent parity are dipole-allowed and result in intersubband absorption and emission, occurring in the spectral range below the fundamental interband absorption edge of the material. In contrast to the optical dipole moment of interband transitions between valence and conduction subbands which relies on a change of the fast oscillating, cellperiodic component of the Bloch wavefunction, the optical dipole moment of intersubband transitions is due to a change of the slowly varying e!ective mass component of the Bloch wavefunction. Consequently, the orientation of the intersubband dipole is perpendicular to the layers parallel to the z axis requiring an electric "eld component in z-direction for optical interaction with this dipole. Intersubband absorption has mainly been studied in n- or p-doped multiple quantum well structures. Modulation-doped structures contain a layer of shallow impurities in the center of the barriers, e.g. Si-delta-doping in AlInAs barriers between GaInAs wells, spatially separated from the quantum wells. Those donors provide free electrons forming a quasi-two-dimensional electron plasma in the lowest (n"1) subband of the quantum wells. After carrier transfer to the wells, the spatial separation of electrons and ionized impurities modi"es the potential energy pro"le of the free carriers. For instance, additional potential minima for electrons occur in the barriers. In most cases, the well potential for the electrons is much deeper than the barrier minima, resulting in a complete carrier transfer to the wells and in a strong con"nement of the carriers to the wells in the lowest (n"1) subband. A di!erent situation exists for higher subbands if the states in the wells come energetically close to states in the barrier minimum. Using structures with a large discontinuity in the conduction band and small built-in potentials due to charge separation this problem can
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be avoided and even carriers excited to the (n"2) subband are strongly con"ned to the quantum. For Ga In As/Al In As quantum wells with a conduction band o!set of 0.5 eV, such 0.47 0.53 0.58 0.52 a situation can be realized quite easily [22]. In the literature, mainly transitions from the lowest (n"1) to the (n"2) conduction subband have been studied and high oscillator strengths of f+20 have been found [23,24]. Numerous measurements of linear absorption have been reported with samples of di!erent quantum well width, lattice temperature and carrier concentration. These experimental studies were complemented by extensive theoretical studies of intersubband absorption pro"les. In the following, we concentrate on quantum well systems in which the energy separation between the (n"1) and (n"2) conduction subband is larger than the energy of optical phonons. Observation of intersubband absorption requires a z-component of the incident electric "eld. In the simplest geometry, the sample is put under Brewster angle in the incident infrared beam, resulting in a z-component of several percent of the total electric "eld. Larger z-components of the electric "eld can be realized in waveguide geometries [25,26] and with metal gratings on top of the samples [27]. In contrast to the broad spectra of free carrier or inter-valence band absorption, the identical sign of the subband curvatures (cf. Fig. 4) and the similar e!ective masses of subsequent subbands lead to a concentration of the intersubband absorption in narrow lines. As an example, the intersubband absorption spectrum of an n-type modulation-doped GaInAs/AlInAs multiple quantum well structure (well width 8 nm) is displayed in Fig. 5. The peak position of this line at 198 meV is determined by the quantum well width of 8 nm, the linewidth (FWHM) has a value of 14 meV. A detailed experimental study of intersubband absorption lines in the GaInAs/AlInAs system has been reported in Ref. [28]. Particularly large intersubband splittings of up to 300 meV have been realized [29] and absorption lines as narrow as 7 meV (FWHM) centered at a photon energy of 148 meV were observed in an early high-quality structure [24]. The narrowest intersubband absorption lines for subband spacings larger than the LO phonon energy are observed in n-type modulation-doped GaAs/AlGaAs quantum wells. Early measurements
Fig. 5. Steady state intersubband absorption between the (n"1) and (n"2) conduction subband of a n-type modulation doped GaInAs/AlInAs multiple quantum well structure (50 wells, well width 8 nm, electron concentration 5]1011 cm~2). The spectrum was measured with the sample under Brewster angle in the infrared beam at a lattice temperature of 10 K. Inset: Schematic of the intersubband transition (E : Fermi level of the electrons). &
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with multiple-quantum well samples gave typical linewidths between 5 and 10 meV (FWHM) for carrier concentrations of several 1011 cm~2 [23,30]. Recent improvements of the structural quality allow the growth of samples with an intersubband linewidth as narrow as 2.3 meV [31]. In a coupled quantum well structure an even narrower line of only 1.3 meV FWHM was reported for a (n"1) to (n"3) transition [32]. In the following, we brie#y discuss the di!erent broadening mechanisms which determine the linewidth of intersubband absorption. (i) Phase breaking scattering processes: Each scattering event which changes the relative phase between the electron waves of the two involved subbands will also lead to a damping of the coherent macroscopic intersubband polarization. The relevant phase-breaking scattering processes are electron}electron scattering, electron}phonon scattering and scattering from disorder potentials. In the Markov limit, i.e. neglecting all memory e!ects, the scattering processes lead to a homogeneous broadening of the intersubband absorption line which is usually characterized by a dephasing time ¹ . A complete theoretical modelling of intersubband phase relaxation including 2 all microscopic scattering processes has not been performed so far. In such a treatment not only the rate of a given scattering process but also its in#uence on the intersubband phase has to be considered. In addition, the already mentioned memory e!ects, i.e. quantum kinetics, may be relevant for the coherent dynamics and for phase relaxation. Keeping this in mind, the frequently used dephasing time ¹ just represents a phenomenological parameter for the time scale on which 2 the macroscopic intersubband polarization decays. Although homogeneous broadening makes a signi"cant contribution to the intersubband absorption line, a direct, quantitative, experimental determination of the dephasing time ¹ is rather di$cult and has not been possible until very 2 recently [33]. The essential problem in measuring ¹ is a clear separation of the homogeneous 2 broadening from other mechanisms. Only nonlinear spectroscopic methods allow for a separation of the various contributions. (ii) Inhomogeneous broadening: Inhomogeneous broadening, i.e. a distribution of intersubband transition energies in the optically coupled range, leads to a fast decay of the macroscopic polarization caused by the destructive interference between the individual components. There is a variety of inhomogeneous broadening mechanisms: In most cases, the dispersion of subband energies with the in-plane k-vector [Fig. 4(b)] is not parallel in the optically coupled range of k-space and } thus } the intersubband transition energy depends on the in-plane k-vector. Whenever the initial carrier distribution covers a "nite interval in k-space, this variation of intersubband transition energies results in an inhomogeneous broadening of the intersubband absorption line. In a single particle picture of noninteracting, independent two-level systems, this broadening has been estimated from the calculated subband dispersions, i.e. the e!ective masses and nonparabolicities [34,35], and from the width of the carrier distribution function in k-space. This picture is supported by experiments with hot electron plasmas of moderate density in GaInAs/AlInAs quantum wells where a single-particle description is appropriate to account for the transient lineshapes of intersubband absorption [36,37]. In addition to the non-parallel k-dispersion, the di!erence between the e!ective mass in the quantum well and the barrier can a!ect the linewidth [38]. In a description including many-body e!ects due to the Coulomb interaction in the carrier plasma, the single particle dispersion is modi"ed and the plane-wave part of the carrier wavefunction is altered, leading to an additional broadening of the states in k-space. However, coupling of the di!erent transition dipoles in the many-body system a!ects the shape of the absorption band, reducing the linewidth and shifting the position of the line to higher photon
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energies. This behavior which has been found in many absorption measurements, will be discussed in paragraph (iii) below. A second important contribution to inhomogeneous broadening stems from disorder e!ects like #uctuations of the quantum well width or the alloy composition. These #uctuations occur within one well as a function of the spatial coordinates parallel to the layer as well as between di!erent wells in a multiple quantum well structure. Concerning disorder inhomogeneities, the parameters of the growth process play an important role. Here the GaAs/AlGaAs system is superior to the GaInAs/AlInAs system, partly because alloy #uctuations are absent in the binary GaAs. In addition, the randomly distributed ionized impurities in the barriers of modulation-doped quantum well structures lead to a weak #uctuating potential for the carriers in the wells. The in#uence of this e!ect on the intersubband absorption line depends strongly on the separation between quantum well center and impurity layer in the barriers [39]. Those #uctuating in-plane potentials are partly compensated by screening in a dense electron plasma [40,41]. (iii) Many body ewects: The single particle picture of noninteracting, independent two-level systems breaks down at higher carrier concentrations, typically on the order of several 1012 cm~2. In addition to the already mentioned single particle excitations, there are collective charge-density and spin density oscillations in a dense two-dimensional plasma [42}53]. All three components were identi"ed in light scattering (Raman) experiments on quantum wells with small subband spacings below the energy of optical phonons and were analyzed theoretically [45,48]. For any carrier distribution in the (n"1) subband, each occupied (n"1) state represents a separate intersubband transition to its corresponding state in the (n"2) subband with the same in-plane wave vector k . The Coulomb interaction between carriers occupying states in both @@ subbands will introduce coupling between intersubband transitions at di!erent k-vectors. In a long series of papers such many-body e!ects have been discussed [54}74]. In particular, the shape, width, and spectral position of the intersubband absorption line are drastically changed by the Coulomb interaction. The most prominent feature is the depolarization shift [74]. A collective charge oscillation of the plasma in z-direction is caused by a phase coherent excitation of the ensemble of intersubband oscillators. During oscillation, charge is separated leading to additional restoring forces which corresponds to a more rigid oscillator. This has been observed in inversion layers of heterostructures [55,58] as a blue shift of the intersubband transition. In addition to the blue shift of the intersubband absorption line, a strong narrowing of the line is observed, as discussed in Ref. [35]. Correlations between di!erent intersubband dipoles are very important in the case of wide wells (i.e. large intersubband dipoles) and high carrier concentrations. In this report we will concentrate on the opposite case, i.e. narrow wells (6}8 nm) with subband spacings *E "100}300 meV, much 12 larger than the optical phonon energy and moderate carrier concentrations (+5]1011 cm~2). Here, many-body e!ects are less important. We point out that a detailed quantitative understanding of the di!erent broadening processes and of their relative contribution to the overall line pro"le has not been reached so far. 2.2. Hierarchy of ultrafast processes Resonant excitation of a semiconductor or a semiconductor nanostructure by an ultrashort laser pulse initiates a complex relaxation scenario which occurs on ultrafast time scales. Both processes
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of phase relaxation, i.e. the decay of macroscopic coherent optical polarizations, and carrier redistribution, i.e. population relaxation, play an important role. Di!erent steps during the excitation and the subsequent relaxation can be characterized by quasi-equilibrium situations in the various subsystems of the crystal [1]. Generally, the carrier relaxation in single component plasmas can be divided into four regimes depicted in Fig. 6. Those regimes are not strictly sequential but overlap in time. In the following, we discuss the di!erent relaxation stages with emphasis on intraband, inter-valence band and intersubband excitation. 2.2.1. Excitation and dephasing of coherent polarizations Ultrashort pulses are optical wavepackets consisting of a coherent superposition of wavetrains in a well-de"ned interval of photon energies. Resonant interaction of a coherent ultrashort pulse with a particular transition in the semiconductor creates both a coherent optical polarization between the optically coupled states and promotes carriers (electrons or holes) from energetically lower to higher states in the same or a di!erent band. In a quantum mechanical density matrix description, the coherent polarization is described by the o!-diagonal elements of the density matrix, whereas the population changes are described by the diagonal elements. The coherent polarization is based on a nonstationary superposition of the quantum mechanical ground and excited state of the optical transition with an initially well-de"ned phase relation to the electric "eld of the excitation pulse. As time elapses, this well-de"ned phase relation is destroyed by a variety of scattering processes which change the relative phase of the wavefunction between the ground and excited state. This phase relaxation or dephasing process is synonymous with a fast decay of the macroscopic polarization P(t) and results in a homogeneous broadening of the particular optical
Fig. 6. Schematic of di!erent relaxation processes initiated by ultrafast excitation of inter-valence band transitions in p-type bulk semiconductors or intersubband transitions in n-type modulation-doped quantum wells. The lateral extensions of the di!erent boxes indicate the relevant time intervals of the relaxation regimes on a logarithmic time scale. Note that the di!erent regimes overlap in time.
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transition. In the simplest picture, one describes the phase relaxation by an exponential decay of P(t) which is characterized by a time constant ¹ , the phase relaxation time. 2 Most ultrafast experiments have concentrated on the dynamics of coherent interband polarizations, both resonant to excitonic lines and in the interband continuum (for a review see e.g. [1]). In contrast, the phase relaxation of below-bandgap excitations has been much less explored. For instance, coherent intraband polarizations in the wavelength range up to 20 lm corresponding to a frequency of 15 THz have not been investigated yet. A number of experiments on the coherent dynamics of plasmons, i.e. coherent plasma oscillations, have been reported in which the generation of electromagnetic radiation in the frequency range of a few THz was studied [75,76]. The dephasing dynamics of coherent inter-valence band polarizations has been investigated only very recently [16]. In such experiments, simultaneous excitation of interband transitions from the heavy and the light hole valence band to the conduction band of bulk GaAs by a 20 fs near-infrared pulse created a coherent heavy}light hole polarization. This polarization gave rise to an oscillatory component, i.e. quantum beats, in the nonlinear transmission change of the sample which was monitored in temporally and spectrally resolved pump-probe experiments. The beat frequency was determined by the heavy}light hole splitting in the optically coupled range. The theoretical analysis of these data on the basis of the semiconductor Bloch equations gave evidence that the phase relaxation of coherent inter-valence band polarizations occurs on a time scale of several hundreds of femtoseconds, distinctly slower than the dephasing of transitions from the valence to the conduction band continuum. For intersubband excitations in quantum wells, both the time scale and the mechanisms of phase relaxation are not well understood. The time-dependent many-body e!ects during phase relaxation and the resulting homogeneous broadening of the absorption line are barely characterized. In a recent experiment carried out at room temperature, a coherent intersubband polarization was created by femtosecond interband excitation of asymmetric quantum wells. The decay of the macroscopic polarization was monitored via the electric "eld transients emitted by the sample giving time constants of 110}180 fs [77] for the decay of the macroscopic intersubband polarization. In this experiment, both the destructive interference between di!erent components of the inhomogeneously broadened ensemble and the irreversible phase relaxation due to scattering mechanisms are responsible for the decay and cannot be separated. In addition, a complex relaxation scenario is expected after broadband interband excitation due to a variety of phase breaking scattering processes, namely electron}electron, electron}hole and carrier}phonon scattering. Because of this complex situation, an analysis of the relevant relaxation mechanisms is almost impossible. In Section 5, we discuss the results of the "rst ultrafast four-wave-mixing experiments on coherent intersubband polarizations. These measurements were performed with 130 fs pulses in the mid-infrared and allowed a real-time observation of the decay of the coherent polarization resonant to the transition between the (n"1) and (n"2) conduction subbands in GaInAs/AlInAs quantum wells. The homogeneous and the inhomogeneous contributions to the linewidth of intersubband absorption were separated. Coherent dynamics involving intersubband excitations have been studied for much smaller energy separation of the subbands, corresponding to frequencies in the THz range. Coherent oscillations of wavepackets in coupled quantum wells and Bloch oscillations in superlattices
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represent such phenomena which are beyond the scope of this article. For detailed information on such experiments the reader is referred to Refs. [1,79,80]. 2.2.2. Population relaxation involving diwerent bands Inter-valence and intersubband excitations create carrier distributions populating di!erent valence bands and subbands, respectively. Population transfer between those bands is part of the relaxation of such nonequilibrium populations towards a quasi-equilibrium carrier distribution and occurs on times scales between about 30 fs and several tens of picoseconds, substantially faster than electron}hole recombination after interband excitation. Theoretical calculations [81] show that inter-valence band scattering of holes from the split-o! band or from high-lying states in the light hole band (E '100 meV) into heavy hole states of bulk LH semiconductors like Ge or GaAs is an extremely fast process occurring on a sub-100 fs time scale. Emission of optical phonons via the optical deformation potential and } in polar semiconductors } the polar}optical interaction represents the dominant scattering mechanism. So far, a timeresolved observation of inter-valence band scattering has not been reported and most experiments on hole dynamics gave only upper limits for inter-valence band scattering times [82,83]. Intersubband scattering in quantum well structures has been studied in much more detail. Most experiments and theoretical calculations have concentrated on intersubband scattering of electrons, mainly from the (n"2) back to the (n"1) conduction subband. Two cases have been distinguished, namely energy spacings of the subbands smaller [84}91] or larger than the energy of the longitudinal optical phonons in the system [26,92}109]. In the "rst case, spontaneous emission of LO phonons (E "30}40 meV) by carriers is not possible and } consequently } intersubband LO scattering should be governed by carrier}carrier scattering and by interaction of the carriers with acoustic phonons. For such small subband spacings, the literature gives quite contradictory information. Intersubband relaxation times of hundreds of picoseconds were derived from early Raman experiments with GaAs/AlGaAs quantum wells [84]. Those long scattering times were attributed to acoustic phonon scattering. A later Monte Carlo simulation of this experiment has shown that a fraction of (n"2) electrons has enough energy to undergo intersubband relaxation by emitting LO phonons, resulting in a non-equilibrium LO phonon population [91]. In this picture, the long relaxation time is essentially due to carrier cooling which is slowed down by the hot phonon e!ect. Other quasi-stationary and time resolved measurements of nonlinear intersubband absorption [89] gave intersubband scattering times of several tens of picoseconds. A recent study of interband luminescence in wide GaAs/AlGaAs suggests fast intersubband scattering of electrons via carrier}carrier interaction, leading to subpicosecond lifetimes in higher subbands at carrier densities of approximately 1011 cm~2 [90]. In this article, we concentrate on quantum wells with subband spacings substantially larger than the energy of LO phonons. The corresponding optical intersubband transitions lie in the midinfrared wavelength range from 3 to 20 lm. For this case, intersubband relaxation times between about 200 fs and 10 ps have been derived from experiments [92}109], whereas theoretical calculations suggest intersubband relaxation by LO phonon emission with time constants of about 1 ps [110}115]. The long relaxation times on the order of 10 ps were found in picosecond pump-probe studies of nonlinear intersubband absorption in n-type modulation-doped GaAs/AlGaAs multiple quantum wells [26]. The slow relaxation is mainly due to a real space transfer of (n"2) carriers from the quantum well into the pronounced potential minima in the barriers of the quantum well
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structure (barrier thickness 40 nm, doping concentrations 1012 cm~2) and the slow return of those electrons to the (n"1) subband. The experiments of Refs. [94,99] give intersubband relaxation rates of about 1 ps, in agreement with theoretical estimates. Very short scattering times of 200 fs were derived from femtosecond luminescence studies [100,108] where both electron and hole dynamics contribute to the measured response, making an unambiguous interpretation of the data very di$cult. In Section 5, we discuss a novel femtosecond technique to measure the lifetime of electrons in the (n"2) conduction subband which is free of the problems indicated above. This technique was applied to study intersubband relaxation of strongly con"ned electrons in GaInAs/AlInAs quantum wells. 2.2.3. Equilibration of athermal carrier distributions Inter-valence band excitation in p-type bulk semiconductors or intersubband excitation in n-type modulation-doped quantum wells generate carrier distributions in k-space which are far from thermal equilibrium, i.e. markedly di!erent from hot Fermi distributions. The overall relaxation scenario thus involves thermalization, a carrier redistribution process transforming the athermal into a quasi-equilibrium distribution which is characterized by a carrier temperature. Here, the most relevant scattering processes are carrier}carrier and carrier-optical phonon scattering which occur with very high rates of 1013}1014 s~1. As a result, thermalization processes overlap in time with phase relaxation of coherently excited polarizations and with inter-valence band or intersubband scattering. The time scale of thermalization depends crucially on the excitation conditions, the speci"c character of the initial nonequilibrium distribution and/or the presence of a cold background plasma. In the literature, carrier thermalization was mainly investigated after interband excitation of electron}hole plasmas in bulk or quasi-two-dimensional semiconductors. Transient carrier distribution functions were derived from femtosecond pump-probe studies of nonlinear interband absorption [116,117] or from time resolved luminescence measurements [118]. In contrast, carrier thermalization after intraband excitation has not been studied until now. There is only limited information on the ultrafast thermalization of holes. In femtosecond luminescence experiments with n-doped bulk GaAs and GaAs/AlGaAs quantum wells, i.e. with a quasi-stationary electron distribution, the time evolution of the emission is mainly determined by hole dynamics. Thermalization times of nonequilibrium holes of up to several hundreds of femtoseconds were derived from those experiments [119}121]. For lattice temperatures below about 100 K, electron}hole scattering was considered the main equilibration mechanism [119,120] whereas the substantially faster thermalization at 300 K was attributed to scattering with optical phonons [121]. More recently, the intraband thermalization of heavy holes after interband excitation of an electron}hole plasma was studied in a specially designed pump-probe experiment with GaAs at room temperature [122]. The thermalization times on the order of 150 fs were attributed to intraband scattering with optical phonons. Another femtosecond study of GaAs at 300 K which made use of mid-infrared probing of transient inter-valence band absorption after interband excitation of an electron}hole plasma, gave redistribution times of heavy and light holes of less than 100 fs [82]. In all those experiments, the photoexcited holes scatter with electrons which are present by doping or by interband excitation. The quantitative analysis of electron}hole scattering for a realistic valence bandstructure and of its contribution to hole thermalization is very complicated and the di!erent models used in the literature gave di!erent, partly con#icting
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results. The study of pure hole plasmas in p-type materials avoids the problem of hole}electron interaction and gives direct access to ultrafast hole relaxation. Such an experiment will be discussed in Section 4.1. The thermalization of carrier distributions in quantum wells depends strongly on the initial excitation conditions. In Fig. 7, three di!erent initial situations are sketched which allow for an investigation of the in#uence of a cold electron plasma at the bottom of the (n"1) conduction subband. Thermalization of photo-excited electron}hole plasmas has been investigated in timeresolved pump-probe experiments in nominally undoped [Fig. 7(a)] and in n-type [Fig. 7(b)] and p-type modulation-doped GaAs/AlGaAs quantum wells at room temperature [117]. The nonequilibrium carrier distributions were monitored via the transmission changes in the spectral range of (n"1) valence to conduction band transitions with a time-resolution of 100 fs for excitation densities n "p "5]1011 cm~2 which are comparable to the doping concentrations %9 %9 in the modulation-doped structures. In the undoped sample the authors of Ref. [117] found an equilibration within the time resolution into a hot Maxwellian distribution. A thermalization time of 60 fs was estimated. An even faster thermalization was reported for the n-type modulation-doped quantum wells. At all times during and after excitation of the electron-hole plasma, no deviation from thermal carrier distributions was observed and a thermalization time of 10 fs was estimated [117]. Such fast scattering rates on the order of several 1013 s~1 were con"rmed in transient four-wave-mixing experiments on interband transitions in n-type modulation-doped GaAs quantum wells [123] and in stationary emission experiments monitoring LO phonon replicas in band-to-acceptor (BA) luminescence as a function of the plasma density [124]. The ultrafast equilibration rates reported in the experiments are in good agreement with theoretical treatments of carrier}carrier scattering [125}129]. Much less is known about thermalization of a quasi-two-dimensional electron plasma after resonant intersubband excitation (Fig. 7c). Here the dispersion of the optically coupled subbands in
Fig. 7. Thermalization scenarios in quantum wells for di!erent excitation conditions. (a) Nonequilibrium photo-excited electron hole plasma in an undoped quantum well structure. (b) In#uence of a cold electron plasma at the bottom of the (n"1) conduction subband on the thermalization of a photo-excited electron}hole plasma. (c) Thermalization of a pure electron plasma after ultrafast intersubband excitation in n-type modulation-doped quantum wells.
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k-space is more or less parallel leading to a completely di!erent initial distribution of nonequilibrium carriers after intersubband excitation of electrons and to a fundamentally di!erent scenario of electron relaxation. In Section 5, we discuss the "rst femtosecond experiments on electron thermalization after intersubband excitation. Ensemble Monte Carlo simulations demonstrate that Pauli blocking and screening in the cold plasma of unexcited electrons in the (n"1) subband can signi"cantly slow down the thermalization of a two-dimensional electron plasma after intersubband excitation. 2.2.4. Cooling of hot carrier distributions Thermalization of carriers leads to a hot Fermi distribution which is described by two parameters: the quasi-Fermi level k and the carrier temperature ¹ which is initially higher than the C lattice temperature ¹ '¹ . In the subsequent stages of relaxation, the carriers obeying a hot C L Fermi distribution exchange energy with the lattice by emission and absorption of phonons, resulting in a net #ow of energy from the carriers to the lattice. At low lattice temperatures (¹ "10 K), there will be "rst a fast cooling process (few picoseconds) due to emission of optical L phonons which is followed by a slower cooling due to emission of acoustic phonons (¹ (50 K) on C a time scale up to 1 ns [130,131]. In some cases, the strong coupling of hot carriers to a small subsystem of the phonon modes } usually LO phonons with small q-vector } via the polar optical interaction, i.e. the high phonon emission rate, and the substantially lower phonon decay rate result in pronounced athermal populations of such LO phonon modes. These nonequilibrium phonons emitted in the early stage of carrier cooling will partly be reabsorbed by the carriers leading to a slowing down of the overall cooling process of the carrier gas. A detailed theoretical analysis of the `hot phonona phenomenon has been given in Ref. [132]. In Section 4.2, the in#uence of hot phonons on the strength and dynamics of the transient intraband absorption of electrons will be discussed. The cooling dynamics of hot holes in p-type germanium was recently studied by time-resolved Raman experiments [133] and } in more detail } in a series of mid-infrared pump-probe experiments using picosecond pulses for excitation and probing the inter-valence band transitions [12,134]. It turned out that the `hot phonona e!ect discussed above is not relevant for the cooling of hot holes in valence band of bulk Ge. This is due to the facts that (i) the holes couple exclusively via the deformation potential interaction to optical phonons which does not favour small phonon q vectors in contrast to the polar interaction, and (ii) phonons are emitted over a rather broad range of q-vectors. Cooling of hot electron}hole plasmas in quantum wells and superlattices has been studied in many time-resolved luminescence experiments (for a review, see e.g., Ref. [1]). For bulk Ga In As and Ga In As/Al In As quantum wells, cooling times of tens of 0.47 0.53 0.47 0.53 0.48 0.52 picoseconds have been reported [130,131]. The energy transfer to the lattice is dominated by the emission of longitudinal optical (LO) phonons with high scattering rates on the order of 7]1012 s~1 [127]. Here, the buildup of nonequilibrium LO phonons leads to a signi"cant reduction of the cooling rate by factors of 10 and more [132]. Much less information exists on the cooling of pure electron plasmas after intersubband excitation. In time-resolved mid-infrared experiments with n-type modulation-doped GaInAs/AlInAs quantum wells, overall cooling times on the order of 50}100 ps were found [92]. In Section 5, experiments with femtosecond time resolution are described in which electron cooling within the "rst 20 ps after intersubband excitation is characterized in detail. The results give evidence of hot phonon populations slowing down the overall
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cooling process. The reduction of cooling rates, however, is substantially less pronounced than suggested by picosecond experiments with photoexcited electron}hole plasmas [131].
3. Experimental techniques Progress in ultrafast spectroscopy is closely related to the development of femtosecond laser sources. Femtosecond pulses are required for inducing elementary excitations and for probing their nonequilibrium dynamics. The advent of mode-locked solid state lasers and ampli"cation techniques based on solid state materials has led to a dramatic improvement of pulse parameters, e.g. with respect to pulse energy, stability and reduction of the pulse duration in broad spectral ranges. In particular, the generation of femtosecond infrared pulses in the wavelength range beyond 1 lm has made substantial progress. Using di!erent techniques, the wavelength range between 1 and 300 lm (corresponding range of optical frequencies 1}300 THz) has been covered continuously. There are essentially three classes of ultrafast infrared sources, (i) free electron lasers, (ii) modelocked solid state lasers, and (iii) sources based on nonlinear optical frequency conversion of ultrashort laser pulses, in particular parametric frequency mixing. In the following, these methods of infrared generation are brie#y reviewed and techniques of ultrafast infrared spectroscopy are outlined. 3.1. Free electron lasers In recent years, infrared generation in free electron lasers (FELs) has received increasing interest. There are a number of facilities around the world providing coherent widely tunable radiation in the wavelength range between about 1 and 2000 lm [135]. At wavelengths between 5 and 10 lm, macropulses consisting of a sequence of 0.5 ps micropulses and microjoule energies per micropulse have been produced using short electron bunches from a linear electron accelerator [136}138]. Within each macropulse of 10 ls duration, the temporal separation of the subpicosecond pulses is between 1 and 40 ns. The wide tuning range and the microjoule energies per micropulse make FELs attractive for spectroscopic studies and there are "rst experiments on semiconductors in the subpicosecond domain [139,140]. For some applications, however, averaging of the signals over the pulses within a macropulse and heating of the samples due to repetitive excitation represent major experimental limitations. In addition, synchronization of FEL pulses with pulses from an external laser source, e.g. for two-color pump-probe experiments, represents a major issue. First results demonstrating synchronization with a timing jitter of several picoseconds have been reported in Ref. [141]. This has been improved substantially in very recent experiments demonstrating synchronization of a FEL and a mode-locked Ti : sapphire laser with a jitter of about 400 fs [142]. 3.2. Modelocked solid state lasers for the infrared In the wavelength range between 1 and 3 lm, modelocked solid state lasers are available for femtosecond pulse generation for which the following requirements have to be ful"lled: f The gain bandwidth of the laser medium must be large enough to allow ampli"cation of the spectral components necessary for a well-de"ned optical wavepacket of femtosecond duration.
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For pulses of q"50 fs duration, a bandwidth *lK1/q"2]1013 s~1 is required which is equivalent to *l/lK0.1 for a center frequency of l"2]1014 s~1 (j"1500 nm). f A phase-coherent superposition of laser oscillation on many longitudinal modes of the laser resonator, i.e. modelocking, has to be accomplished in order to form the wavepacket. Di!erent techniques of modelocking are available for this purpose, among them additive pulse modelocking (APM), passive modelocking with semiconductor nonlinear absorbers, and Kerr lens modelocking. APM requires a laser consisting of two coupled cavities, a main cavity containing the active medium and an auxiliary cavity with a nonlinear medium serving for self-phasemodulation of the pulses [143]. Phase-coherent superposition of the pulse in the main cavity with a chirped pulse coupled back from the auxiliary cavity results in a pulse shortening down to the femtosecond regime. Passive modelocking is based on a saturable loss in an absorber medium placed in the laser resonator which } together with gain in the active medium } shapes the pulse envelope. For Kerr lens mode-locking [144], self-focusing in the active medium in combination with a hard or soft aperture serves as a fast saturable loss. f Dispersion control, in particular compensation of group velocity dispersion (GVD) represents an important issue for preserving short pulses and high time resolution in the experiments. In general, the positive GVD in the gain medium and optical components has to be balanced by introducing negative GVD with prism pairs [145], grating arrangements or specially designed (`chirpeda) mirrors [146]. In color center lasers, generation of stable femtosecond pulses started with the soliton laser consisting of two coupled cavities [147,148]. The "rst cavity contains the active medium, Tl0(1) color centers in a KCl crystal pumped by a continuous wave Nd : YAG laser, and is coupled through one end mirror to the second cavity containing a single-mode polarization preserving optical "ber as the nonlinear optical element. The combined action of self phase modulation by the "ber index nonlinearity n , negative GVD in the "ber, and ampli"cation in the main cavity leads to 2 the formation of a stable (N"2) soliton in the system. The laser output, a pulse train with a repetition rate of about 100 MHz, was tunable in the wavelength range between 1.4 and 1.6 lm. Pulse durations between 130 fs and } after external recompression } 80 fs were reported. Soon afterwards, other coupled cavity techniques like additive pulse modelocking (APM) were invented, leading to femtosecond pulse generation with color centers emitting in the wavelength range up to 1.7 lm [143,149]. Passive modelocking of color center lasers by saturable semiconductor absorbers provided pulses of about 100 fs duration up to wavelengths of 2.8 lm [150,151]. Later work concentrated on the generation of ampli"ed femtosecond pulses around 1.5 lm, resulting in single pulse energies of several tens of lJ at kilohertz repetition rate [152]. Although 100 fs pulses tunable over a substantial wavelength range are available from color center lasers, their application in experiments has been quite limited. This is partly due to the limited lifetime of color center crystals under such operation conditions and the need for cryogenic temperatures of the active medium. Femtosecond pulses at wavelengths between 1.2 and 1.6 lm have been generated by passive modelocking of Cr4` : YAG and Cr4` : forsterite lasers [153}157]. Optical "bers doped with rare earth ions like Er3` represent a particularly compact gain medium for femtosecond pulse generation. Er-doped silica "bers have received much interest for the generation and ampli"cation of femtosecond pulses in the wavelength range around 1.55 lm which is important for optical communication technologies. The gain bandwidth of Er3` in silica supports ampli"cation of pulses
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in the sub-100 fs range. For pumping, continuous wave diode lasers emitting at 980 nm are available. Di!erent techniques of passive modelocking have been developed for Er-doped "ber lasers [158}161]. For instance, the nonlinear polarization rotation in the "ber in conjunction with a polarizer was used to generate soliton-like femtosecond pulses in a "ber laser with an external cavity [159]. Nonlinear polarization rotation was also used in an all-"ber ring laser combining an Er-doped "ber with positive GVD and a standard silica "ber with negative GVD [160,161]. Ampli"cation of femtosecond pulses in highly compact, diode pumped "ber systems [162] represents an important technique to increase the peak intensity of the pulses. A comment should be made on the generation of ultrashort infrared pulses with semiconductor lasers. Infrared pulses of picosecond duration have been generated using techniques of gain switching, i.e. injection of short electrical pump pulses, and modelocking. In some cases, subpicosecond pulse generation has been successful and even colliding pulse modelocking has been implemented in monolithic quantum well lasers [163]. So far, however, the low average power, the very high repetition rates and the di$culties of chirp control in such lasers have prevented spectroscopic applications. 3.3. Nonlinear frequency conversion In most experiments, nonlinear optical frequency conversion of laser pulses at relatively short wavelengths is used to generate pulses tunable in a wide infrared wavelength range. Techniques of parametric ampli"cation, di!erence frequency mixing, optical recti"cation, and } to a minor extent } stimulated scattering [164] in di!erent solid, liquid and gaseous nonlinear media have been applied successfully. Parametric processes in materials with a nonvanishing second order nonlinearity s(2) have been exploited for infrared generation. In such a process, three pulses at center frequencies u , u , and u (p: pump, s: signal, i: idler) interact with each other in the nonlinear 1 4 * medium. Energy conservation requires u "u #u with u (u (u . In parametric generation 1 4 * * 4 1 and ampli"cation, an intense input pulse at u generates new pulses at u and u . In di!erence 1 4 * frequency mixing, pulses at a low frequency u "u !u are generated from two input pulses at * 1 4 u and u . Optical recti"cation represents a special case of di!erence frequency mixing where 1 4 u Ku and u is on the order of *u , the spectral bandwidth of the input pulses. Optical 1 4 * 1,4 recti"cation is particularly simple if the u and u waves are taken from a single pulse of 1 4 a bandwidth *u. A well-de"ned phase relationship between the electric "elds of the three interacting pulses is maintained by ful"lling the phasematching condition k "k #k where 1 4 * k "n (u /c) are the respective wavevectors. This phasematching condition can be ful"lled 1,4,* 1,4,* 1,4,* by adjusting the refractive indices n in birefringent nonlinear media. For this purpose, di!erent 1,4,* linear polarizations for the pulses are selected and the orientation and/or temperature of the crystal are adjusted. For the di!erent phasematching schemes, the reader is referred to Ref. [165]. Frequency tuning of the generated pulses is achieved by changing the phase-matching angle through rotation of the nonlinear crystal or variation of its temperature. Even for phasematched parametric interaction, there is a substantial mismatch of the group velocities of the three interacting pulses. The di!erent group velocities limit the e!ective interaction length in the nonlinear medium and determine the minimum pulse duration achieved [166]. The energy conversion e$ciency of a phase-matched parametric process is determined by the second order nonlinear susceptibility of the material, the e!ective interaction length, and the intensity of the
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input pulses [165]. In most practical sources for femtosecond infrared pulses, very high peak intensities between 1 GW/cm2 and 1 TW/cm2 are used. For standard nonlinear materials and interaction lengths between 100 lm and several mm, this results in an energy conversion into the infrared between several 10~5 and several percent. There is a limited number of birefringent nonlinear materials suitable for phase-matched parametric frequency mixing in the mid-infrared, the main issues being a su$ciently broad range of infrared transparency and high crystal quality, e.g. optical homogeneity and the absence of scattering centers [167]. For idler wavelengths of up to 5 lm, LiNbO , LiIO , KNbO , beta3 3 3 barium borate (BBO) and KTiOPO (KTP) have been used. AgGaS represents a standard 4 2 material for wavelengths up to 12 lm, AgGaSe and GaSe allow parametric mixing at even longer 2 wavelengths up to 18 lm. A summary of crystal data can be found in Ref. [167]. Di!erent experimental schemes have been developed for the reliable and e$cient generation of femtosecond pulses in the mid-infrared [164,168]. In recent years, modelocked solid state lasers and/or ampli"ers have been used to provide stable input pulses for the respective parametric process whereas earlier work was based on dye laser systems [169}171]. Two types of generation schemes are to be distinguished, namely (i) sources for quasi-continuous trains of mid-infrared pulses with high MHz repetition rates and low average power and (ii) systems based on ampli"ed pulses of low kHz repetition rates which provide mid-infrared pulses of up to microjoule energies per pulse. Femtosecond mid-infrared pulses at high repetition rates of several tens of MHz have been derived from near-infrared pulses which were generated in self-modelocked Ti : sapphire oscillators or in optical parametric oscillators (OPOs) synchronously pumped by modelocked Ti : sapphire lasers. In Ref. [172], two synchronous pulse trains at di!erent wavelengths were generated in a Ti : sapphire laser and subsequently fed into a AgGaS crystal to generate pulses at 2 9 lm by di!erence frequency mixing. In a much simpler and more reliable approach, two frequency components from a single broadband near-infrared pulse can serve for di!erence frequency mixing. Pulses of 10}20 fs duration corresponding to a spectral bandwidth of up to *j"100 nm (FWHM) have been used for such generation schemes. In Refs. [78,173], the second order nonlinearity occurring at the surface of bulk GaAs was used to generate a broadband infrared pulse by optical recti"cation of 15 fs pulses. As this nonlinear process is not phase-matched, all pairs of frequency components within the pulse spectrum contribute to di!erence frequency mixing and } thus } a very broad infrared spectrum is generated. Part of this spectrum between 7 and 18 lm has been analyzed in Ref. [78], a full characterization of the temporal properties of such mid-infrared pulses has not been reported so far. Furthermore, the very short interaction length at the surface of the absorbing nonlinear medium results in a very low e$ciency of the nonlinear process. Substantially higher conversion e$ciencies leading to mid-infrared average powers in the lW range for an input power of 100 mW have been achieved by phase-matched di!erence frequency mixing within the spectrum of 20 fs near-infrared pulses using a GaSe crystal [174]. The "nite acceptance bandwidth of the phase-matching process results in a selection of input wavelengths contributing to the nonlinear process and } consequently } a well-de"ned spectral envelope of the generated midinfrared pulse. Such pulses are continuously tunable by changing the phase-matching angle of the nonlinear crystal. This is shown in Fig. 8 displaying the spectra of mid-infrared pulses generated in a 1 mm thick GaSe crystal. The corresponding tuning curve between 9 and 18 lm is shown in the inset. A pulse duration of 100 fs was measured for a center wavelength of 13 lm, demonstrating that the time-bandwidth product of such pulses is close to the Fourier limit. Very recently, the same
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Fig. 8. Spectra of femtosecond mid-infrared pulses generated by di!erence frequency mixing in GaSe. The pulses at a repetition rate of 88 MHz are continuously tunable between 9 and 18 lm. The spectra were derived from linear electric "eld autocorrelations of the pulses. Inset: Tuning curve of phase-matched (type I) di!erence frequency mixing in GaSe. The measured (external) phase-matching angle is plotted versus the center wavelength of the pulses (circles). The dashed curve represents the calculated phase-matching curve.
concept was applied for infrared generation with a cavity-dumped modelocked Ti : sapphire oscillator [175]. This oscillator works at a lower repetition rate around 2 MHz and provides single pulse energies in the near-infrared which are 20 times higher than from the 88 MHz system. The mid-infrared pulses generated with the input from this laser show a single pulse energy which is approximately 100 times higher than with the 88 MHz laser [176]. The spectral and temporal properties of the mid-infrared pulses at 1 MHz are similar to those at high repetition rate. Optical parametric oscillators (OPOs) provide femtosecond pulse trains with average powers of up to several hundreds of milliwatts and a tuning range of the signal pulses between about 1 and 2 lm, depending on the speci"c nonlinear material [177}179]. For the corresponding idler pulses, tunability between about 2 and 5 lm has been demonstrated with lower average powers. The duration of signal and idler pulses was between 100 and 500 fs, strongly depending on the wavelength position. Extension of the tuning range to wavelengths beyond 5 lm has been achieved by di!erence frequency mixing of the signal and idler pulses from the OPO in an additional nonlinear crystal outside the resonator of the OPO [180]. This technique has been applied recently to generate pulses of several hundreds of femtoseconds duration up to wavelengths of 18 lm [181,182]. Intense near-infrared pulses generated in regenerative Ti : sapphire ampli"ers at kHz repetition rates have been used for pumping a variety of parametric conversion schemes [168]. In the
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Fig. 9. Experimental setup for the generation of intense femtosecond mid-infrared pulses at a 1 kHz repetition rate. A regeneratively ampli"ed Ti : sapphire laser system provides intense pulses of 100 fs duration at a wavelength of 810 nm. Pulses with 300 lJ energy pump a parametric device working in the near-infrared. Signal and idler pulses tunable between 1.2 and 2.6 lm were obtained by parametric generation in a temperature-tuned lithium triborate (LBO) generator crystal followed by a beta-barium-borate (BBO) ampli"er crystal. In a subsequent AgGaS crystal, signal and 2 idler pulses generate mid-infrared pulses by di!erence frequency mixing in the spectral range between 3 and 10 lm with pulse energies of several tens of nJ. Fig. 10. Autocorrelation trace of mid-infrared pulses at a wavelength of j"5 lm (repetition rate 1 kHz). The second harmonic signal generated in a thin AgGaS crystal is plotted versus the delay time between the incident pulses (symbols). 2 Solid line: autocorrelation signal calculated for sech2 shaped pulses of 72 fs duration.
following, we brie#y discuss the mid-infrared source applied in some of the experiments discussed in Section 5 [183]. The experimental setup is shown in Fig. 9. Intense 800 nm pulses of 100 fs duration and up to 800 lJ energy/pulse were generated in an ampli"ed Ti : sapphire laser. Signal and idler pulses tunable between 1.2 and 2.6 lm were obtained with microjoule energies by parametric ampli"cation in a setup consisting of a lithium triborate (LBO) generator crystal and a beta-barium-borate (BBO) ampli"er crystal. The duration of signal and idler pulses was as low as 50 fs. In a subsequent AgGaS crystal, signal and idler pulses generate mid-infrared pulses by 2 di!erence frequency mixing. Changing the near-infrared input wavelengths allows for a continuous tuning of the mid-infrared output between 4 and 12 lm with pulse energies of several tens of nJ. In Fig. 10, an autocorrelation trace of such pulses is shown which was recorded at a center wavelength of 5 lm (symbols). The temporal pulse envelope is close to a sech2 dependence (solid line: autocorrelation trace calculated for sech2 shaped pulses). A pulse duration of about 70 fs corresponding to a few optical cycles of the infrared electric "eld is derived from the autocorrelation trace. For the measurements reported in the following, somewhat longer pulses of 130 fs duration were used.
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3.4. Techniques of time-resolved spectroscopy Ultrafast spectroscopy aims at monitoring the dynamics of elementary excitations in real-time. In addition to ultrashort pulses, this requires speci"c spectroscopic techniques for studying the relaxation stages which were introduced in Section 2.2 and extend over several orders of magnitude in time. Most of the studies discussed here imply excitation of the semiconductor by infrared pulses resonant to the relevant optical transition. In cases of small transition energies beyond the range covered by infrared pulses, i.e. excitations at very long wavelengths, impulsive excitation schemes by a pulse at shorter wavelengths represents an alternative technique. For instance, coherent polarizations between the heavy and light hole valence bands can be generated by simultaneously exciting the heavy hole and the light hole to conduction band transitions by broadband near-infrared pulses [16]. A related technique are Raman-type excitations of the electronic system or the lattice. In the coherent regime, the optical polarization of the semiconductor exhibits a well-de"ned phase relationship to the electric "eld of the optical pulse and initially shows a phase-coherent time evolution. A variety of experimental schemes like coherent emission measurements including interferometric detection [77], degenerate and quasi-two-color pump-probe techniques and } in particular } four-wave-mixing techniques based on the third order nonlinearity of the semiconductor have been applied to investigate coherent dynamics. A standard scheme for degenerate four-wave-mixing, i.e. with pulses of identical photon energy, is depicted schematically in Fig. 11. Two pulses incident onto the sample with wavevector k and k and a mutual delay *t create 1 2 a transient grating from which coherent third-order nonlinear signals are generated by selfdi!raction of the pulses into the directions 2k !k and 2k !k . The intensity of the di!racted 2 1 1 2 beams is proportional to the square of the third-order polarization generated in the sample and } thus } the change of the di!racted signal with time re#ects the dynamics of the nonlinear polarization. There are di!erent methods to analyze the di!racted signal: (i) In the simplest case, the di!racted intensity is recorded with a time-integrating detector as a function of the delay time *t between the two pulses generating the transient grating. The signals di!racted into the directions 2k !k and 2k !k are symmetric in time with respect to zero delay between the incident pulses. 1 2 2 1 Such a measurement gives information on the build-up and decay of the macroscopic nonlinear polarization in the sample. The decay of the signal is a measure of the loss of phase coherence and provides information on the dephasing processes. In the simplest theoretical approximation, this behavior has been analyzed with the optical Bloch equations for an ensemble of independent two-level systems. This approach gives a single exponential decay of the FWM signal proportional to exp (!*t/q). The decay time q has a value of ¹ /2 and ¹ /4 for a homogeneously and an 2 2 inhomogeneously broadened set of two level systems, respectively (¹ : dephasing time). For 2 semiconductors, however, this frequently applied model does not represent an adequate description, as it neglects the Coulomb interaction among the carriers and } thus } does not account for many-body e!ects strongly in#uencing the nonlinear response of a semiconductor. Here, more sophisticated models, e.g. based on the semiconductor Bloch equations, have to be used [184]. (ii) Spectrally resolved time integrated detection gives additional information on the nature of the nonlinear polarization. For instance, excitonic and free carrier contributions showing di!erent spectral signatures can be separated by this technique. (iii) Information on the inherent time dependence of the di!racted signal can be obtained in time-resolved detection schemes. For instance, gating of the di!racted signal by sum-frequency mixing with a third pulse gives the time
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Fig. 11. Schematic of an experimental setup of degenerate four-wave-mixing. Two ultrashort pulses propagating along directions k and k create a transient excitation grating in the sample. In dependence of the delay between the pulses *t, 1 2 light will be di!racted of this grating into the directions 2k !k and 2k !k . 1 2 2 1
envelope of the di!racted intensity (convoluted with the third pulse). Such measurements provided detailed insight into many-body and/or local "eld e!ects in#uencing the nonlinear response, as is discussed in detail elsewhere [1]. Three pulse photon echo measurements and interferometric techniques of signal detection [185] represent other methods to analyze the temporal structure of nonlinear polarization. The incoherent redistribution of carriers after ultrafast intersubband excitation has mainly been studied by pump-probe methods. A "rst pulse creates a nonequilibrium excitation which is monitored via nonlinear changes of absorption or re#ection by a second, in general much weaker probe pulse. Measurements with a tunable wavelength of the probe pulse provide transient spectra from which detailed information on the carrier distribution functions can be derived. In the following, we concentrate on resonant excitation of a mid-infrared transition in doped samples. In Fig. 12, di!erent pump-probe schemes for bulk and quasi-two-dimensional semiconductors are depicted schematically. Intraband excitation of electrons (Fig. 12a) is a three-particle interaction involving photon absorption and coupling to a phonon or impurity for k-conservation. Intraband excitation initiates intraband redistribution of carriers and subsequent carrier cooling by emission of phonons. Both processes can be monitored via transient changes of free-carrier absorption extending over a very wide wavelength range. It should be noted that intraband absorption is much weaker than dipole allowed interband transitions, leading to rather small absorption changes to be observed in pump-probe experiments. Fig. 12(b) shows a pump-probe scheme for studying the dynamics of holes in p-doped bulk materials. The strong dipole-allowed inter-valence band absorption is used for excitation of holes from the heavy-hole to the light-hole or split-o! band. For such an excitation, hole redistribution involves both inter- and intra-valence band scattering which can be followed by changes of inter-valence band absorption on the same or a di!erent inter-valence band transition. The non-parallel dispersion of the di!erent valence bands and warping of the valence band structure in k-space results in the rather complex relationship between the optical transition energy and the k-vector of the holes given in Eq. (4) (Section 2.1.2). For an analysis of time resolved data, the transient hole distribution functions f (k, t) and f (k, t) have to be derived from the momentary * j inter-valence band absorption a (u, t). This requires a detailed modelling of the k-dependent IVA dipole matrix elements. In the lower part of Fig. 12, we present pump-probe schemes for studying intersubband excitations in quasi-two-dimensional semiconductors. In Fig. 12(c), the pump pulse promotes electrons from the (n"1) to the (n"2) subband by resonant absorption on the intersubband
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Fig. 12. Di!erent pump-probe schemes for investigating ultrafast excitations below the fundamental absorption edge in doped semiconductors. The pumping processes are marked by solid arrows whereas dashed arrows indicate the respective probing processes. (a) Indirect free carrier absorption in the conduction band. (b) Inter-valence band excitation and probing in p-type semiconductors. (c) Intersubband excitation of (n"1) electrons to the (n"2) subband and subsequent probing of the same transition. (d) Left-hand side: intersubband excitation and subsequent probing of the (n"2) population via the corresponding valence- to conduction subband transition. Right-hand side: monitoring the transient (n"1) distribution function after ultrafast intersubband excitation via transient spectra of the corresponding (n"1) valence to conduction subband transition.
transition. The transient carrier distribution is probed by a second pulse in the mid-infrared, monitoring nonlinear changes of the mid-infrared absorption of the sample. The second pulse is either at the same spectral position as the pump or tunable in the range of the intersubband absorption line. In the simplest approximation of single particle excitations, the change of intersubband absorption *a(u) is given by *a(u)"!a ( f (u, t)#f (u, t)) where a is the steady 0 * & 0 state intersubband absorption coe$cient and f and f are the distribution functions in the initial * & and "nal subband optically coupled at a transition frequency u. This expression shows that the nonlinear absorption signals observed in such experiments cannot fully be interpreted within a simple two-level picture of the intersubband transition. Such a simple model predicts a bleaching of intersubband absorption due to the reduced population in the initial state and an enhanced population in the "nal state that decays with the intersubband scattering time. Because of the in-plane dispersion of the two subbands, however, both pump and probe couple optically to states
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in a broad range of in-plane k-vectors, i.e. the signal is sensitive to population transfer between the subbands and intrasubband carrier redistribution. Depending on the spectral position of the probe, both bleaching and enhanced absorption can occur in di!erent time windows. Moreover, at high carrier densities ('1012 cm~2), the in#uence of many-body e!ects on the intersubband absorption spectrum is strong, making an interpretation of the signals in terms of carrier distribution functions quite di$cult. Changes of the carrier distribution by intersubband excitation in the mid-infrared result in changes of the near-infrared interband absorption of the quasi-2D system [Fig. 12(d)]. In particular, one expects an absorption decrease by state-"lling on interband transitions the "nal k-states of which are populated after intersubband excitation. Correspondingly, initially blocked interband transitions with "nal states depopulated by intersubband excitation exhibit an increase of absorption. This fact and the selection rules for interband transitions allow to selectively monitor (n"1) and (n"2) carrier distributions with probe pulses in the near-infrared. In addition to absorption changes caused by population e!ects, the signal measured in this scheme might be in#uenced by many-body e!ects. In particular, transient absorption features related to the Coulomb enhancement of the interband absorption and the Fermi edge singularity of the plasma could play a role [186,187]. A recent experimental study [188], however, demonstrates that such signals are limited to a narrow spectral range of a few meV and show a small amplitude for moderate carrier densities at low temperatures. For probe pulses of about 100 fs duration, the spectral bandwidth of the probe is much larger than this narrow energy interval and } thus } population e!ects dominate the signals [188]. It is important to note that the total carrier concentration is constant in the pump-probe schemes discussed. Excitation results exclusively in a redistribution of carriers present by doping, making an analysis of the data much simpler than in the case of interband excitation where both electrons and holes contribute to the optical signal. Selective observation of dynamics after below bandgap excitation is also possible using spontaneous anti-Stokes Raman scattering (SASRS) as a probe for the population of higher bands [84,94,133]. In experiments on intersubband dynamics, population of the (n"2) conduction subband is created by exciting an electron}hole plasma via the corresponding interband transitions and probed via the anti-Stokes Raman signal generated with a probe pulse of the same [94] or di!erent [84] photon energy. The strength of the anti-Stokes signal shifted by the intersubband transition energy relative to the center of the probe pulse is measured as a function of time delay to the pump and re#ects the population dynamics of the upper subband. The time resolution of pump-probe experiments depends on the duration of pump and probe pulses, which should be short compared to the time scale of relaxation in the sample, and on geometry and interaction length in the sample. For studies in the femtosecond regime, group velocity dispersion and/or reshaping of the pulses by interaction with strong absorption resonances have to be minimized in order to achieve optimum time resolution. This requires optically thin samples with typical geometrical thickness between 0.5 and 100 lm. 4. Ultrafast dynamics in bulk semiconductors In the following, we review experiments with bulk semiconductors in which ultrafast carrier dynamics was monitored after optical excitation below the bandgap. Particular emphasis is on
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relaxation processes of holes, including hole thermalization and cooling after inter-valence band excitation. Such phenomena are discussed in Section 4.1. Section 4.2 is devoted to transient intraband absorption processes of electrons. 4.1. Relaxation processes of holes In most ultrafast pump-probe studies of photoexcited electron}hole plasmas, electrons give rise to stronger changes of transmission or re#ection than holes and } consequently } the measured signals mainly re#ect the nonequilibrium dynamics of electrons. The study of pure hole plasmas in p-type materials avoids this problem and gives direct access to ultrafast hole relaxation. The experiments presented here address both ultrafast dynamics and picosecond cooling of holes in p-type germanium which has a valence band structure representative for a wider class of materials, including GaAs and other III}V compounds. A gallium-doped germanium crystal with an acceptor concentration of 3]1017 cm~3 (sample thickness 0.02 cm) was studied in two-color pump-probe experiments with femtosecond pulses in the mid-infrared. The pump pulse excites holes from the heavy hole (HH) to the split-o! (SO) band (spin-orbit splitting E "290 meV) as depicted schematically in Fig. 13(a). The resulting change of SO the HH to SO absorption is monitored by weak delayed pulses in spectrally and temporally resolved measurements. Synchronized pump and probe pulses of 250 fs duration were used which were independently tunable in the wavelength range from 2.7 lm (460 meV) to 5.0 lm (250 meV). The excitation density estimated from the incoming #ux of infrared photons and from the absorption of the sample lies between 3]1015 and 1016 cm~3, i.e. a small fraction of 1}3% of the total hole concentration was excited to the SO band. The spectrum of inter-valence band absorption from the HH to the SO band of the p-type germanium sample is plotted in Fig. 13(b) for lattice temperatures of ¹ "10 K and 80 K [83]. In L the femtosecond experiments, transient changes of this absorption are studied with excitation (E ) %9 at 430 meV and probe (E ) pulses between 400 and 445 meV as indicated by the arrows in 13 Fig. 13(b). Time-resolved data for the four probe energies are presented in Figs. 14(b)}(e), where the change of absorption *A is plotted versus delay time (points; lattice temperature ¹ "10 K) [83]. L At all probe energies, a transient increase of absorption (*A'0) is found that rises with a distinct delay relative to the instantaneous response in Fig. 14(a). A numerical analysis assuming monoexponential kinetics gives a common rise time of 700$150 fs for the di!erent spectral positions (solid lines). It is important to note that a transient decrease of absorption is not detected. Thus, spectral hole burning is absent even around delay zero where pump and probe pulses coincide. In Fig. 15, we present time-resolved data taken at di!erent lattice temperatures ¹ of (a) 10 K, L (b) 40 K and (c) 60 K with identical photon energies E "E "390 meV. On the left-hand side, %9 13 the change of absorption is plotted for the "rst two picoseconds. The data show an increase of absorption with a rise time of 700 fs for the di!erent ¹ values. The induced absorption decays with L a picosecond kinetics depending upon the speci"c lattice temperature (right-hand side of Fig. 15). For ¹ "10 K (Fig. 15a), one observes a decay time of several hundreds of picoseconds whereas L a considerably faster relaxation is found at higher lattice temperatures [Figs. 15(b), (c)]. For a qualitative discussion, the di!erent processes occurring during and after inter-valence band excitation are depicted schematically in Fig. 13(a), showing the hole distribution functions f (E) (i)}(iv). H
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Fig. 13. (a) Left-hand side: Schematic of heavy hole (HH), light hole (LH), and split o! (SO) valence bands in germanium. HH to SO excitation by infrared photons and inter-valence band scattering by emission of optical phonons are indicated by arrows. Right-hand side: Hole distribution functions f (E) are plotted on a logarithmic scale versus the hole energy H E for a sequence of processes in the femtosecond experiment: (i) excitation, (ii) intervalence band scattering, (iii) H thermalization, and (iv) cooling. (b) Stationary HH to SO inter-valence band absorption of p-type germanium (acceptor concentration 3]1017 cm~3) for lattice temperatures ¹ of 10 and 80 K. The spectral positions of the femtosecond L excitation (E ) and probe (E ) pulses are indicated. %9 13 Fig. 14. (a) Time-integrated cross correlation function of the femtosecond mid-infrared pump and probe pulses, determining time zero and the pulse durations of t K250 fs. (b)}(e) Transient increase of HH to SO inter-valence band P absorption of p-type germanium after femtosecond excitation at E "430 meV (2.9 lm, ¹ "10 K). The change of %9 L absorption *A"!ln(¹/¹ ) is plotted versus the delay time between pump and probe pulses for four probe energies 0 E of (b) 445 meV, (c) 430 meV, (d) 415 meV, and (e) 400 meV (¹ ,¹: transmission before and after excitation). 13 0
(i) Excitation: Absorption of pump photons leads to a transient depletion of HH states and to an excess population of SO levels that are optically coupled by the femtosecond excitation pulse. These changes of the hole distribution are expected to cause a decrease of inter-valence band absorption by state-"lling which manifests itself in spectral hole burning. In contrast, the data show an increase of absorption for all spectral positions and delay times. This "nding demonstrates that within the time resolution of 100 fs the holes excited to the SO band scatter rapidly out of the optically coupled range in k-space and that the HH states depleted by the pump pulse are quickly repopulated. The latter quasi-equilibrium is established within about 100 fs by scattering among unexcited holes with small changes of the k-vectors [189].
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Fig. 15. Time dependent increase of absorption after excitation and probing at 390 meV for lattice temperatures ¹ of L (a) 10 K, (b) 40 K, and (c) 60 K. The signal is normalized to its peak value. Left-hand side: Femtosecond rise of the signal. Right-hand side: Picosecond decay of the transient absorption due to cooling of the hole plasma.
(ii) Inter-valence band scattering and (iii) thermalization: In the present experiments, the total carrier concentration is constant. Thus, population of SO states corresponds to a reduced density of heavy holes and a decrease of absorption at any spectral position in the HH to SO band (cf. Fig. 13b). The holes excited to the SO band undergo two major relaxation processes, inter-valence band scattering by emission of optical phonons and thermalization. For the present experimental conditions, inter-valence band scattering is fast compared to HH thermalization, as has been discussed in detail in Ref. [83]. The holes excited to SO states scatter back to the HH band within the time resolution of the experiments. This initial step is followed by thermalization within the HH band, a process in which a quasi-equilibrium distribution of heavy holes is formed out of the backscattered and the unexcited holes. Both the scattering of heavy holes with optical phonons and Coulomb scattering among the holes contribute to thermalization. Coulomb scattering transfers part of the excess energy from the back-scattered holes to the cold plasma of unexcited holes, resulting in a strong heating of those holes. This heating enhances the population of the HH states monitored by the probe pulses and leads to the observed delayed increase of HH to SO absorption. Thus, our data directly show that the thermalization of heavy holes proceeds with a time constant of 700 fs. This interpretation is supported by theoretical calculations which have been presented in Ref. [81]. The rates of inter-valence band scattering via the optical deformation potential were calculated in "rst order perturbation theory taking into account the nonparabolic valence band structure and the k-dependent overlap of the corresponding hole wavefunctions [14,190]. A value of D "6.3]108 eV/cm determined in earlier ps measurements was used for the deformation 0 potential [12]. A very short lifetime of q "100 fs was estimated for carriers in the SO band. SO A similar formalism was used to calculate the rates of intra-valence band scattering of heavy holes by optical phonons.
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Fig. 16. (a) Transient energy loss of holes excited to HH states with E "500 meV in p-type Ge with a hole density of HH p "3]1017 cm~3 at ¹ "¹ "0 K. The transient energy per carrier (solid line) is plotted vs time in picoseconds. The 0 C L transferred energies to the cold hole plasma of unexcited holes and to the lattice via optical phonon emission are shown as long and short dashed lines, respectively.
Inelastic hole}hole scattering was treated in a model including the full dynamical screening of the Coulomb interaction as well as intra- and inter-valence band transitions [191,192]. Here, a single carrier looses energy by scattering with the #uctuations of the longitudinal macroscopic electric "eld in the plasma of unexcited holes at wavevectors q and frequencies u. The longitudinal dielectric reponse function e(q, u) was calculated in random phase approximation (RPA) [193]. Scattering rates for both intraband optical phonon and inelastic Coulomb scattering have been presented in Ref. [81]. In Fig. 16, the energy of holes which initially populate heavy hole states 500 meV above the bottom of the heavy hole band and subsequently thermalize, is plotted as a function of time (solid line). The dashed lines give the amounts of excess energy transferred to the cold hole plasma and to the lattice (via phonon emission). The calculation gives an increase of the energy in the hole plasma that is close to an exponential rise with a time constant of approximately 800 fs. This value is in good agreement with the rise time of the absorption changes plotted in Fig. 14 which was 700 fs. This con"rms that the transient inter-valence band absorption re#ects the heating of the cold hole plasma by intraband thermalization of the excited holes. The carrier energy of 340 meV which is reached after about 2 ps corresponds to a plasma temperature of 80 K. A more detailed inspection of the scattering scenario shows that excitation of HH to LH inter-valence band transitions in the cold plasma represents the main mechanism by which hot holes loose their excess energy. The weak dynamical screening of the Coulomb interaction among the carriers is essential to account for the observed high scattering rates [189] and for the large fraction of the total excess energy which is transferred to the hole plasma. (iv) Cooling: The transient absorption reaches its maximum *A (E ) at a delay time of .!9 13 t K3 ps when a hot quasi-equilibrium distribution of holes has been formed. Within the experiD mental accuracy, *A (E ) corresponds to the di!erence between the stationary absorption .!9 13 spectra for ¹ "10 K and 80 K (cf. Fig. 13b). This temperature rise derived from the experiments is L
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in good agreement with the maximum carrier temperature calculated from the theoretical model. Cooling of the hot holes down to lattice temperature ¹ represents the "nal relaxation process [83] L occurring in the picosecond regime. For ¹ "60 K [Fig. 15(c)], emission of optical phonons via L the deformation potential leads to carrier cooling within 50 ps. For lower (carrier and lattice) temperatures, the fraction of holes emitting optical phonons is reduced and acoustic phonon scattering determines the substantially slower relaxation kinetics [Figs. 15(a), (b)]. The cooling dynamics of hot holes in p-type germanium and the concomitant changes of inter-valence band absorption have been studied by time-resolved Raman scattering [133] and } in more detail } in a series of mid-infrared pump-probe experiments with picosecond pulses [12,134]. Such experiments have demonstrated that changes of inter-valence band absorption in the picosecond regime are dominated by the transient temperature of a thermalized hole plasma, in contrast to earlier claims in the literature. The results of Refs. [12,134] are in quantitative agreement with the cooling data presented here. In conclusion, femtosecond infrared studies of p-type germanium reveal that holes excited to the split-o! band scatter back to high lying heavy and light hole states within 100 fs. The subsequent thermalization of the excited carriers proceeds on a time scale of 700 fs, predominantly by inelastic scattering with unexcited holes. The high energy loss rates are due to the weak dynamical screening of the Coulomb interaction in the carrier system. This scheme of hole thermalization is relevant for other semiconductors like GaAs which shows similar valence band parameters, optical deformation potentials and } consequently } scattering rates of holes. 4.2. Intraband excitations Intraband transitions of free carriers give rise to weak, in general structureless absorption spectra extending over a broad range in the infrared. Steady-state free carrier absorption has been studied in many experiments and detailed theoretical descriptions of intraband absorption representing a three-particle interaction (cf. Section 2.1) have been developed. In contrast, information on transient free carrier absorption, e.g. of hot carriers, has remained scarce. Investigation of transient free carrier absorption is interesting for the following reasons: (i) The strength and the shape of intraband absorption spectra depend strongly on the carrier distribution. Thus, transient carrier distributions, in particular nonequilibrium distributions, can be monitored via changes of the intraband absorption spectra. (ii) Conservation of k-vector in the intraband absorption process requires coupling to a lattice excitation, i.e. phonon or an impurity. At mid-infrared wavelengths up to about 30 lm, coupling to optical phonons represents the dominant mechanism of k-conservation in III}V semiconductors. The strength of intraband absorption is proportional to the phonon population at the wavevector required for k-conservation. As a result, transient changes of phonon populations, in particular excess (`hota) optical phonons created by carrier cooling can be studied via changes of the free carrier absorption. For di!erent spectral positions, phonons at di!erent wavevectors couple in the absorption process and } consequently } the distribution of excess phonons in k-space can be mapped in spectrally resolved experiments. Pioneering experiments based on these ideas have been reported in Refs. [194,195]. The electron plasma in an n-doped InAs crystal was excited at a wavelength of 7 lm by mid-infrared pulses of 8 ps duration and the resulting change of intraband absorption was monitored at di!erent probe wavelengths. The measurements demonstrated a transient enhancement of free carrier absorption
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decaying on a time scale of 70 ps, parallel to electron cooling. The increase of free carrier absorption was due to both the change of the electron distribution at high carrier temperatures and to the coupling of hot phonons which were created in the cooling process. Recently, the role of nonequilibrium phonon populations was con"rmed in an experiment with n-doped GaAs at longer wavelengths around 17 lm [139]. All those studies dealt with heated quasi-equilibrium distributions of electrons. Future studies of transient free carrier absorption with higher time resolution could provide new insight into the ultrafast thermalization dynamics of electron plasmas.
5. Ultrafast dynamics of intersubband excitations in quasi-two-dimensional semiconductor structures In this section, we discuss a series of experiments and theoretical calculations in which the ultrafast dynamics of intersubband excitations of a pure electron plasma were studied in Ga In As/Al In As quantum wells. This material system is of high technological rel0.47 0.53 0.48 0.52 evance and has been used, e.g., in the "rst quantum cascade lasers [196]. Both coherent dynamics of intersubband polarizations and incoherent carrier dynamics, i.e. intersubband relaxation, carrier thermalization and cooling, are addressed. The results give insight into the relevant carrier}carrier and carrier}phonon scattering mechanisms. The experiments were performed with a set of n-type modulation-doped Ga In As/Al In As quantum well samples grown by molecular 0.47 0.53 0.48 0.52 beam epitaxy on InP substrates. Because of the large conduction band o!set of 0.5 eV in this material system, both the (n"1) and (n"2) conduction subbands are strongly con"ned to the quantum wells and real-space transfer of carriers into the barriers can be neglected. The di!erent samples consist of 50 Ga In As quantum wells of a respective width of 6 or 8 nm. For such 0.47 0.53 quantum well widths, the optical intersubband transition between the (n"1) and (n"2) conduction subbands is located at mid-infrared photon energies which are much higher than the energy of LO phonons. The quantum wells are separated by 14 nm wide Al In As barriers the center of 0.48 0.52 which is d-doped with Si donors, resulting in an electron concentration per quantum well between 1.5]1011 and 1.5]1012 cm~2. As an example, the intersubband absorption spectra of three samples consisting of 6 nm wide quantum wells with electron concentration of 1.5]1011, 5]1011, and 1.5]1012 cm~2 are displayed in Fig. 17. The absorption bands were measured with the samples under Brewster angle in the infrared beam. In all time-resolved experiments, resonant intersubband excitation by 130 fs pulses (spectral bandwidth 15 meV) in the mid-infrared was applied. The excitation densities were between 10% and 20% of the total electron concentration. All measurements were performed at a lattice temperature of ¹ "10 K if not indicated otherwise. L 5.1. Coherent intersubband polarizations In the following, we discuss results of the "rst femtosecond four-wave-mixing (FWM) study of coherent intersubband polarizations. FWM gives direct information on the decay of the macroscopic coherent intersubband polarizations. For experimental details, the reader is referred to Section 3.4 and Ref. [33].
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Fig. 17. Steady-state intersubband absorption spectra between the (n"1) and (n"2) conduction subbands of Ga In As/Al In As multiple quantum well structures of di!erent doping density (well width 6 nm: electron 0.47 0.53 0.48 0.52 concentrations 1.5]1011 cm~2 (dotted line), 5]1011 cm~2 (dashed line), 1.5]1012 cm~2 (solid line), lattice temperature 10 K). The spectra were recorded with the sample under Brewster angle in the incident infrared beam. Fig. 18. Four-wave-mixing signals recorded with femtosecond excitation resonant to the intersubband transition in the mid-infrared. The intensity di!racted from the transient grating in the sample is plotted as a function of delay time between the two pulses generating the grating. (a) Data for an electron concentration of 1.5]1011 cm~2 (symbols, lattice temperature 10 K). The dashed line gives the cross-correlation of the two femtosecond pulses. Inset: Intersubband absorption spectrum (solid line) and spectrum of the FWM signal at delay zero (symbols). (b), (c) Data for electron concentrations of 5]1011 and 1.5]1012 cm~2.
In Fig. 18, FWM signals from samples with 6 nm wide quantum wells and electron densities of 1.5]1011, 5]1011, and 1.5]1012 cm~2 are presented. The respective Fermi energies of electrons are 7, 18, and 50 meV. The spectrally integrated intensity di!racted into the direction 2k !k is 2 1 plotted on a logarithmic scale as a function of delay time between the two pulses generating the transient grating. The dashed line gives the cross-correlation of the two pulses. At the lowest density of 1.5]1011 cm~2 [Fig. 18(a)], the signal rises within the time resolution of the experiment, reaches a maximum after 100 fs and decays within several hundreds of femtoseconds. A monoexponential "t gives a decay time of 80$15 fs. With increasing density [Figs. 18(b), (c)], the maximum of the signal shifts to earlier delay times and the decay becomes substantially faster. For electron densities of 5]1011 and 1.5]1012 cm~2, the respective decay times are K65 and (50 fs, in the latter case close to the time resolution of the experiment. In all samples, one "nds a strong
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resonant enhancement of the FWM signal at the position of the intersubband absorption line. In the inset of Fig. 18(a) (symbols), this is shown for the sample with the lowest carrier concentration. The FWM signals in Fig. 18 exhibit essentially monoexponential decays with time constants between 450 and 80 fs. Such dynamics are much faster than the picosecond intersubband relaxation of photoexcited electrons from the (n"2) back to the (n"1) subband which occurs with a time constant of 1.3 ps [37,197] and is addressed in detail in Section 5.2. Thus, intersubband population relaxation plays a minor role for dephasing. Instead, electron}electron, intraband electron}LO phonon scattering and } to a minor extent } electron}impurity scattering are potential mechanisms which determine the phase relaxation dynamics and lead to a homogeneous broadening of the intersubband transition. In addition, quantum well thickness and alloy #uctuations and the di!erent dispersion of the two subbands with the in-plane k-vector result in inhomogeneous broadening [35,37]. The excitation bandwidth in our experiments is close to the spectral width of intersubband absorption, i.e. the whole distribution of intersubband transition frequencies is excited phase-coherently and contributes to the overall polarization. In the simplest theoretical description based on independent two-level systems, the FWM signal from a homogeneously and inhomogeneously broadened ensemble decays with time constants of ¹ /2 and ¹ /4, respectively. For intersubband excitations, however, the nonparabolic dispersion of 2 2 the subbands with the in-plane k-vector and the coupling of the transition dipoles by many-body e!ects results in a more complex behavior requiring a more sophisticated theoretical treatment. To get insight into such phenomena, calculations based on the time dependent Hartree}Fock equations (TDHF) were performed [33] which account for many-body e!ects on a mean-"eld level and allow for a simultaneous study of inhomogeneous broadening [35,48,65]. In the case of interband dynamics in an intrinsic semiconductor, the TDHF approach is equivalent to the semiconductor Bloch equations. To analyze the di!erent factors in#uencing the signal, the following four cases were investigated: (i) Two-band model with equal e!ective electron masses m "m "0.05m in 1 2 0 the two subbands (m : free electron mass), i.e. a k-independent energy separation of the two 0 subbands. The system was assumed to be exclusively homogeneously broadened with a constant, k-independent dephasing time ¹ and the Coulomb interaction was switched o!. The other cases 2 correspond to adding successively (ii) the Coulomb interaction (many-body e!ects), (iii) the e!ect of di!erent masses in the subbands (m "0.05m , m "0.065m ), and (iv) the in#uence of (Gaussian) 1 0 2 0 inhomogeneous broadening *u due to quantum well thickness and alloy #uctuations G (*u "12 meV). For the di!erent electron concentrations, "rst the correct Hartree}Fock ground G state was determined and then the FWM signals in the directions 2k !k and 2k !k were 2 1 1 2 calculated as a function of delay time. Fig. 19 shows the results for the lowest and highest doping density with respective ¹ -values of 2 310 and 100 fs. In the noninteracting homogeneously broadened case (i), a decay rate of ¹ /2 is 2 found. With increasing complexity, the decay becomes faster and nonexponential. The importance of the di!erent contributions varies strongly from the low to the high density. Coulomb interaction leads to a nonparabolic dispersion in the Hartree}Fock ground state below the Fermi energy [48]. While in the low density case Coulomb interaction and di!erent masses enhance the decay rate by about 50%, at high density they already lead to a decay rate close to the time resolution of our experiment. Here, additional inhomogeneous broadening does not further enhance the decay. A comparison of case (i) with (ii) demonstrates that the in#uence of many body e!ects (Coulomb interaction) on the decay rates of the time-integrated FWM signals is rather limited.
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Fig. 19. Theoretical results for the spectrally integrated four-wave-mixing signals obtained from the TDHF equations for di!erent model cases (lattice temperature 10 K). (a) Electron density 1.5]1011 cm~2, dephasing time ¹ "310 fs. 2 (b) Electron density 1.5]1012 cm~2, ¹ "100 fs. 2
For a comparison with the experimental results, both the time-resolved FWM data (Fig. 18) and the width of the intersubband absorption spectra (Fig. 17) have to be accounted for in a consistent way. Considering the spectral widths and the decay rates, one concludes that there is a substantial inhomogeneous contribution to the overall linewidth in the absorption spectra for all electron densities. The theoretical results show that the intersubband absorption line of the sample with the lowest electron density (1.5]1011 cm~2) is dominantly inhomogeneously broadened and correspondingly the FWM signal decays essentially with 4/¹ . This gives a dephasing time of 2 ¹ K310 fs. The corresponding homogeneous linewidth has a value of 4.3 meV, representing 2 about 30 percent of the total linewidth. For the sample with the high doping concentration (1.5]1012 cm~2), the decay of the FWM signal is considerably faster and close to the timeresolution of the experiment. In this case, an upper limit of ¹ (200 fs is estimated from the FWM 2 decay assuming predominant inhomogeneous broadening. A lower limit ¹ '50 fs is given by the 2 inverse linewidth of the intersubband absorption band assuming predominant homogeneous broadening. Next, the scattering processes relevant for intersubband dephasing are discussed. In the modulation-doped samples, the electron gas in the quantum wells is spatially separated from the ionized donors in the barriers. This reduces ionized impurity scattering of electrons and results in characteristic scattering times of 1}2 ps [198], much longer than the observed dephasing times. For ¹ "10 K, the thermal LO phonon population and } thus } LO phonon absorption by electrons L are negligible. For an electron density of 1.5]1011 cm~2, the Fermi energy of E "7 meV is much F
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smaller than the LO phonon energy which is on the order of 30 meV. The resonant intersubband excitation applied in the experiments promotes electrons to states close to the bottom of the (n"2) subband, well below the LO phonon energy. Consequently, intraband LO phonon emission is suppressed in both subbands and its contribution to the fast dephasing is negligible. For an electron density of 5]1011 cm~2 (Fermi energy E "18 meV), LO phonon emission is still F inhibited. For the highest electron concentration of 1.5]1012 cm~2 (E "50 meV), intersubband F excitation creates a carrier distribution in which about 30% of the electrons can emit LO phonons. Under such conditions, electron}electron scattering represents the main dephasing mechanism. The increase of the dephasing rate with carrier density (Fig. 18) is mainly due to the rise of electron}electron scattering rates. Intraband scattering in the (n"1) and (n"2) subbands and scattering events where an electron in one subband interacts with an electron in the other subband and each carrier is scattered to a "nal state in its own subband, represent the main contributions to the overall scattering rate. Because of the di!erent symmetry of the (n"1) and (n"2) electron wavefunctions, the rates of such processes are much higher than for scattering involving intersubband transfer of electrons [110]. A quantitative calculation of electron}electron scattering rates is di$cult for the nonequilibrium electron distribution created by intersubband excitation. The results strongly depend on the type of screening of the Coulomb interaction used in the calculation. Furthermore, inclusion of quantum kinetic e!ects changes the results substantially compared to calculations based on solving the Boltzmann equation for in"nitely short collision times. In the literature, rates have been calculated for di!erent limiting cases of electron}electron scattering [189}199]. In Ref. [200], quasi-twodimensional electron densities around 1011 cm~2 and carrier energies of up to 10 meV have been considered. The calculation including dynamic screening of the Coulomb interaction among the electrons gives dephasing times of several hundred femtoseconds for valence to conduction band transitions, a time scale similar to that of intersubband dephasing reported here. It should be noted, however, that a theoretical analysis of intersubband dephasing by electron}electron scattering has not been performed until now. In summary, the decay of coherent intersubband polarizations in GaInAs/AlInAs quantum wells occurs on a time scale of several hundreds of femtoseconds with dephasing rates increasing with carrier concentration. Theoretical calculations based on the TDHF equations allow to relate the decay rate of the signal with the intersubband dephasing rate and thus to determine the homogeneous contribution to the linewidth of intersubband absorption spectra. The results clearly demonstrate that electron}electron scattering represents the dominant dephasing mechanism. In the samples studied here, there is a strong inhomogeneous contribution to the overall width of the intersubband transitions. Future experiments with samples showing narrower absorption lines with predominant homogeneous broadening should provide even better insight into the manybody e!ects relevant for the coherent response. 5.2. Intersubband scattering and thermalization Electrons excited to a higher subband undergo a complex relaxation scenario, involving di!erent types of scattering processes until the initial electron distribution is restored. In the following, we discuss the relaxation of electrons excited to the (n"2) subband and their interaction both with the plasma of unexcited (n"1) electrons via Coulomb scattering and with the lattice via phonon
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Fig. 20. Schematic of intersubband scattering and thermalization after resonant intersubband excitation. (a) intersubband excitation of (n"1) electrons to the (n"2) subband and subsequent backscattering to energetic states in the (n"1) subband by LO phonon emission. (b) Non-thermal (n"1) carrier distribution during and after intersubband excitation and intersubband relaxation. The initial cold electron distribution before excitation is shown as dashed line. (c) Hot Fermi distribution of (n"1) electrons after the thermalization process. (d) Transient spectra of (n"1) interband absorption reveal the di!erence between the transient (n"1) electron distribution minus the initial cold distribution.
scattering. The principal relaxation scheme is depicted in Fig. 20: Resonant intersubband excitation creates a nonequilibrium population in the (n"2) subband and leads to a distinct distortion of the initial Fermi distribution of the (n"1) electron plasma. Due to the similar dispersion of the two subbands with the in-plane k-vector, a broad range of initially populated (n"1) states is depopulated by resonant excitation, leaving behind a nonequilibrium distribution of (n"1) electrons. In the cases considered here, the total electron concentration has a value of several 1011 cm~2 and } consequently } the distribution of unexcited (n"1) electrons populates an energy interval much narrower than the intersubband energy spacing. The electrons promoted to the (n"2) subband subsequently undergo intersubband scattering back to the (n"1) subband. As the energy separation of the two subbands is much larger than the amount of energy exchanged in intersubband Coulomb or phonon scattering, high lying (n"1) states at large values of the in-plane wavevector are populated by intersubband scattering. Thermalization of this nonequilibrium distribution of electrons involves a transfer of carriers from such high lying states to the bottom of the (n"1) subband and the formation of a quasi-equilibrium (quasi-Fermi) population. As the excess energy provided by optical intersubband excitation partly remains in the carrier system, this quasi-equilibrium is characterized by an elevated electron temperature. Monitoring this relaxation scheme requires an experimental technique providing information on the transient electron distributions. As shown in Section 3.4, changes of the interband absorption probed by tunable near-infrared pulses after resonant intersubband excitation in the mid-infrared give insight into the (n"1) and (n"2) electron distributions. In the following, such experimental results are discussed for a sample consisting of 8 nm wide quantum wells with an electron density of 5]1011 cm~2. In Fig. 21, time-resolved data (symbols) are shown which were taken at the onset of the (n"2) interband absorption, at photon energies of the probe between 1.130 and 1.150 eV. The change of transmission *¹/¹ "(¹!¹ )/¹ is plotted versus delay time between the mid-infrared pump 0 0 0
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Fig. 21. Time-resolved transmission change of the (n"2) interband transition after excitation of (n"1) electrons to the (n"2) subband. The change of transmission *¹/¹ "(¹!¹ )/¹ (solid circles) is plotted vs. the delay time between 0 0 0 mid-infrared excitation pulses (E "0.2 eV) and probe pulses at E "1.141 eV (¹,¹ : transmission with and without %9 13 0 excitation). The Monte Carlo simulation gives a decay time of 1 ps (solid line). Dashed line: cross correlation of pump and probe pulses. Inset: Pump-probe scheme of the experiment. Fig. 22. Left-hand side: Transient spectra of an n-type modulation-doped GaInAs/AlInAs MQW structure (8 nm wells, electron density 5]1011 cm~2) in the spectral range of the (n"1) interband transition after femtosecond intersubband excitation. The transmission change *¹/¹ is plotted as a function of the photon energy of the probe pulses for delay 0 times of (a) 0.2 ps, (b) 0.4 ps, (c) 0.8 ps, (d) 1.1 ps, and (e) 2 ps (circles), as well as (b) 11 ps (squares) for comparison. The bandwidth of the probe pulses is 13 meV (FWHM). Solid lines: Results of the Monte Carlo simulation. Right-hand side: Transient distribution functions of (n"1) electrons (solid lines) as calculated by the ensemble Monte Carlo simulation. Dashed line: Initial cold Fermi distribution.
pulses and the probe pulses (¹ , ¹: transmission of the sample before and after excitation). The 0 bandwidth of the probe pulses was 13 meV. We observe a transient increase of transmission, i.e. a bleaching, which rises within the time resolution of the experiment and decays with a time constant q "1 ps which is clearly resolved in this measurement. The bleaching is caused by IS the transient electron population of the (n"2) conduction subband, leading to a blocking of the corresponding interband transitions. The decay of the signal with q is due to the depopulation of IS the (n"2) subband by intersubband scattering of electrons back to the (n"1) subband. Transient absorption spectra recorded in the range of the stationary (n"1) absorption edge around 0.913 eV give insight into the dynamics of (n"1) electrons. In this range, one probes states at the bottom of the (n"1) subband which are initially populated by the steady-state electron distribution. In Fig. 22, spectra are displayed for various delay times after intersubband excitation
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(solid circles). At 0.2 ps [Fig. 22(a)], a decrease of transmission in the range of the initially populated electron states is found which is due to the depletion by the excitation pulse. At later times [Fig. 22(b)}(e)], the amplitude of this signal rises and } in addition } a delayed bleaching occurs at higher photon energies. For delay times between 0.4 and 1.1 ps, the maximum amplitude of this bleaching signal is substantially smaller than that of the absorption increase occurring below the initial Fermi level, whereas similar amplitudes are approached after about 2 ps [see data for a delay of 11 ps in Fig. 22(b)]. At even later times, the signals decay by carrier cooling which will be discussed in Section 5.3. The absorption changes studied here occur on a time scale which is substantially slower than the decay of intersubband phase coherence. Many-body e!ects make a minor contribution to the signals observed under the present experimental conditions as was discussed in Section 3.4 [188]. Thus, the transient spectra are determined by the di!erence of the transient and the initial equilibrium electron distribution functions and their time evolution re#ects the redistribution processes of electrons involving both inter- and intrasubband scattering. In such a case, Ensemble Monte Carlo (EMC) techniques to solve the Boltzmann equations for the electron distribution functions are an appropriate tool for analyzing the relaxation scenario. The present experiments were simulated by the EMC approach described in Refs. [120,201]. Inter- and intrasubband electron}electron scattering, Pauli exclusion principle and nonequilibrium phonons are taken into account. The speci"c properties of the phonon modes in the GaInAs/AlInAs quantum wells were considered within the dielectric continuum model [111,201]. Intersubband excitation was simulated by promoting about 15% of the electrons from the (n"1) to the (n"2) subband by a 130 fs pulse that was resonant to the intersubband transition between the (n"1) and (n"2) conduction subbands of parallel in-plane k-dispersion. For comparison with the measured absorption changes, the di!erence between the transient and the initial electron distribution in each subband was multiplied with the step-like absorption coe$cient of the respective valence to conduction band transition and convoluted with the temporal and spectral envelope of the 100 fs probe pulses (spectral bandwidth 13 meV). Intersubband relaxation of (n"2) electrons occurs with a characteristic time constant of 1 ps, as is evident from the decay of the signal in Fig. 21. This decay is very well reproduced by the EMC simulation (solid line), indicating that emission of optical phonons via the polar interaction represents the dominant scattering mechanism. In this way, nonequilibrium electrons are transferred to high lying (n"1) states. No bleaching due to accumulation of backscattered carriers in those high-lying states is detected at the corresponding probe energies around 1.08 eV (not shown in Fig. 22). This shows that spreading of the backscattered electrons over a broad energy range occurs much faster than the supply of nonequilibrium carriers with the intersubband scattering time of 1 ps. The redistribution of nonequilibrium electrons of high energy is part of the thermalization process within the (n"1) subband. Thermalization requires a transfer of such electrons towards the bottom of the band where a quasi-equilibrium distribution of all electrons is formed. This relaxation involves both emission of optical phonons transferring excess energy to the lattice and Coulomb scattering between high and low-energy carriers, leading to a heating of the cold electron plasma. We "rst concentrate on the cold plasma. Just after intersubband excitation [delay time 0.2 ps, Fig. 22(a)], one observes a transmission decrease from the onset of (n"1) interband absorption up to the Fermi edge of the initially
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distribution [indicated by the arrow in Fig. 22(a)]. The broad range of enhanced absorption demonstrates a monotoneous depletion of (n"1) states over the entire range of in-plane k-vectors from the (n"1) bandgap up to the initial Fermi level. This "nding is due to the nearly constant energy separation and dipole moment between the (n"1) and (n"2) subbands. The EMC simulation for 0.2 ps (solid lines) gives a constant depletion of the initial Fermi distribution (dashed line) by 15% of the carriers consistent with the excitation conditions. The EMC simulation shows that the rate of electron}electron scattering into states below the initial Fermi level is strongly reduced by the small fraction of unoccupied states to which electrons can be transferred, i.e. the Pauli blocking of those states, and by screening of the Coulomb interaction among the carriers. As a result, the carrier depletion between the bandgap and the initial Fermi level persists for hundreds of femtoseconds, as is directly evident from the data in Fig. 22(b). The slow redistribution of cold electrons at early times demonstrates that intersubband electron}electron scattering makes a minor contribution to the thermalization of the cold plasma. In addition to scattering within the sea of cold electrons, thermalization involves the redistribution of the backscattered nonequilibrium carriers of high energy. The inelastic Coulomb scattering between cold and backscattered carriers during this redistribution leads to a transfer of cold electrons to states above the initial Fermi level E , corresponding to the formation of a high F0 energy tail which includes both formerly cold and excited electrons. In Fig. 22, this is evident from the enhancement of induced absorption below E and the bleaching, i.e. population, above E . F0 F0 Between 0.2 and 1 ps, the very broad high-energy tail results in an amplitude of bleaching much smaller than that of induced absorption. After a time delay of about 2 ps, this distinctly nonthermal distribution has evolved into a hot Fermi distribution with similar amplitudes for enhanced absorption and bleaching. It is interesting to note that the main energy transfer to the cold plasma occurs at times between 500 fs and 2 ps, due to the delayed supply of energetic carriers from the (n"2) subband. Much faster intrasubband thermalization of (n"1) electrons was observed after interband excitation of additional electron}hole pairs in GaAs/AlGaAs quantum wells [117]. This di!erence is mainly due to the completely di!erent carrier distribution created by optical excitation. For interband excitation, the excess electrons populate states above the Fermi sea where Pauli blocking and screening are much weaker than in the distribution created by intersubband excitation. Thus, thermalization after interband excitation is much faster. In summary, the redistribution of an electron plasma during and after intersubband excitation was monitored by probing changes of interband absorption. An intersubband relaxation time of 1 ps was measured which is determined by emission of longitudinal optical phonons. After intersubband scattering, athermal electron distributions were observed up to delay times of 2 ps. The data were analyzed by ensemble Monte Carlo simulations of the carrier dynamics. The calculations demonstrate that this slow electron thermalization is due to electron}electron scattering with rates strongly reduced by Pauli blocking and screening and to the picosecond supply of hot electrons from the (n"2) subband. 5.3. Carrier cooling The thermalization processes discussed in Section 5.2 create a hot plasma of (n"1) electrons which is characterized by a carrier temperature substantially higher than that of the lattice
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Fig. 23. Transient spectra of an n-type modulation-doped GaInAs/AlInAs multiple quantum well structure (well width 6 nm, electron concentration n"1.5]1012 cm~2) in the spectral range of the (n"1) interband transition after femtosecond intersubband excitation. Solid lines: calculated spectra for hot Fermi distributions with temperatures of 270 K (2 ps), 220 K (2.5 ps), 170 K (4 ps), 115 K (6 ps), 82 K (10 ps), and 50 K (20 ps). Inset: Schematic of the pump-probe experiment.
(¹ "10 K). In the following, we focus on the heating of this electron plasma, a process which is L part of thermalization, and on the subsequent cooling by transferring the excess energy from the carriers to the lattice [197]. Cooling involves optical and } to lesser extent } acoustic phonon scattering and represents the "nal relaxation stage by which the initial electron distribution in the (n"1) subband is restored. In the experiments, resonant intersubband excitation by a 130 fs mid-infrared pulse was applied, promoting 10}20% of the total electron concentration to the (n"2) subband. The subsequent changes of the electron distribution were monitored by probing transient changes of the (n"1) interband absorption with widely tunable near-infrared pulses, as was discussed in Section 5.2. Two n-type modulation-doped samples consisting of 50 quantum wells of 6 nm width were investigated, the respective electron concentrations were 1.5]1011 and 1.5]1012 cm~2. In Fig. 23, we present transient absorption spectra for higher doping concentration which were recorded in the range of the (n"1) absorption edge around 0.99 eV. At 0.2 ps, a decrease of transmission occurs below the Fermi level E which is due to depopulation by the mid-infrared excitation pulse. At later times F0 (0.6 and 1.2 ps), the amplitude of this signal rises and an enhanced transmission is found above E . F0
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The signal amplitude reaches its maximum at delay times around 2 ps and, eventually, it decays on a 40 ps time scale. During the decay the transmission decrease below and the bleaching above the Fermi edge E are of similar amplitude. F0 The dynamics at early delay times was discussed in Section 5.2 and re#ects the thermalization of the electron distribution during and after intersubband relaxation. After about 2 ps, a hot quasiequilibrium distribution of (n"1) electrons is formed. The transient spectra on the right hand side of Fig. 23 are well reproduced by hot Fermi distributions (solid lines), calculated from the di!erence of the transient hot minus the initially cold distribution which was "nally convoluted with a gaussian pro"le of a width of 30 meV (FWHM). The convolution accounts for the bandwidth of the probe pulses and the "nite slope of the low temperature absorption edge of the samples which is due to inhomogeneities, collision broadening and many-body e!ects like the Fermi edge singularity. The average amount of excess energy per (n"1) electron is obtained by spectrally integrating the carrier energy times the transient distribution function underlying the spectra of Fig. 23. Similar experiments [197] and analysis were carried out for the 6 nm sample with the lower plasma density of 1.5]1011 cm~2. Results for both samples are plotted in Figs. 24 and 25(a). The time evolution of both heating and cooling depends on the plasma density. For delay times up to 0.2 ps we observe, for both structures, no changes of the mean (n"1) carrier energy. As discussed earlier, intersubband excitation leads to a monotonous depletion of (n"1) states over the entire range of in-plane k-vectors from the (n"1) bandgap up to the initial Fermi level E . Such an energy independent F0 extraction of carriers does not change the mean energy of the remaining (n"1) electrons. Intersubband scattering and thermalization by electron}electron scattering lead to the delayed rise of excess energy in Fig. 24. One estimates that in both samples approximately 50% of the total excess energy are transferred to the plasma, the remainder 50% going directly to the lattice via LO phonon emission. For delay times around 0.35 ps, the highest rates of plasma heating with 20 and 10 meV/ps occur for the high and the low electron concentration, respectively. This energy #ux is exclusively supplied by the small fraction of hot electrons in the system (about 3}4% of the total density) which have already undergone intersubband scattering, as estimated from the intersubband scattering rate and the excitation density. Thus, the energy loss rate per energetic (n"1) electron to the cold plasma is 30 times higher and lies in the order of 0.6 eV/ps for n "1.5]1012 cm~2 and 0.3 eV/ps for n "1.5]1011 cm~2. It should be noted that the loss rate in S S the "rst case is twice as high as in the second, whereas the total electron densities di!er by a factor of 10. The rate of inelastic Coulomb scattering depends on the total electron concentration and is in#uenced by screening of the Coulomb interaction among the carriers. The increase of the loss rate with density is partly compensated by the stronger screening at higher electron density. The absolute values of the loss rates are well reproduced by theoretical calculations considering the density dependence of carrier}plasma scattering on the basis of full dynamical screening of the Coulomb interaction [191]. Cooling of the hot (n"1) electron plasma down to the lattice temperature (¹ "8 K) occurs on L a longer picosecond time scale. In Fig. 25(a), the excess energy per (n"1) carrier is plotted for delay times up to 25 ps. For a constant carrier concentration like in our case, the energy content of the thermalized plasma is exclusively a function of the carrier temperature ¹ . The transient plasma C temperature plotted in Figs. 25(b), (c) was calculated from the energy content for the two samples. For comparison, we show the plasma temperature (open circles and solid diamonds) gained by
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Fig. 24. Transient mean excess energy of carriers in the (n"1) subband as calculated for two samples with 6 nm wells and di!erent electron concentrations from the transient interband absorption spectra (cf. Fig. 23). Fig. 25. Time resolved excess energy per (n"1) carrier on a picosecond time scale as calculated from transient (n"1) interband spectra (circles: n"1.5]1012 cm~2, diamonds: n"1.5]1011 cm~2). (b), (c) Transient temperature of the (n"1) electron plasma during the cooling period as calculated from the mean excess energy. Symbols with error bars: Plasma temperatures gained from a best "t to the transient spectra in Fig. 23. Solid lines: calculated carrier cooling due to LO phonon emission using a hot phonon model. Dashed lines: carrier cooling without hot phonon e!ects.
a best "t procedure directly from the transient spectra in Fig. 23 and in Fig. 1 in Ref. [37]. The estimated error bars account for both the uncertainty of the absolute amplitude of the signal and the noise of the transient spectra. For plasma temperatures lower than 170 K the plots for the di!erent samples are almost identical. This means that the cooling dynamics, i.e. (n"1) carrier temperature as a function of time, does not depend signi"cantly on the plasma density in the presented density range. The apparent di!erences in the dynamics of the mean excess energies shown in Fig. 25(a) are due to the di!erent heat capacities of the Fermi gases. At higher density, the electron gas is more degenerate and shows a smaller speci"c heat. The cooling dynamics was simulated in a theoretical model (similar to Ref. [132]) including hot phonon e!ects for the electron}LO phonon interaction. The electrons are treated strictly twodimensional and the LO phonons are approximated by three-dimensional bulk modes. We use a common LO phonon energy of 32 meV, a value between InAs-like (29 meV) and GaAs-like
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(35 meV) LO phonon energies. For delay times longer than 2 ps the thermalization of the carrier system is completed and, thus, the electron plasma is described by a hot Fermi distribution. In contrast, the LO phonon distribution still shows a pronounced athermal character requiring a separate equation of motion for each phonon mode. The occupancy in the individual modes is determined by the emission or reabsorption of LO phonons by electrons and by the anharmonic decay of LO phonons into acoustic phonons with a time constant of q "5 ps. To our knowledge, LO the LO phonon lifetime q in InGaAs has not been determined experimentally. For this reason, we LO approximate q by using the average between the values of GaAs and InAs, see also Refs. LO [202,209]. The supply of excess energy due to intersubband scattering and thermalization was accounted for by using a time dependent source term of excess energy decaying single-exponentially with q "1.3 ps. The results of the theoretical simulation are shown in Figs. 25(b) and (c) as IS solid lines. Good agreement with the experiment is found for both electron concentrations studied. For comparison, calculations not taking into account the reabsorption of phonons are shown [dashed lines in Figs. 25(b), (c)]. The distinct di!erence between the experimental data and these dashed lines shows that the carrier cooling rate is slowed down by the hot phonon e!ect. The insensitivity of the cooling dynamics to the plasma density in the presented density range is well explained by the model, showing that the density dependence of the energy-loss reduction saturates as the electron plasma becomes degenerate (k¹ (E ). All carriers which are below the smeared C F Fermi edge do not contribute to the energy-loss and consequently not to the hot phonon e!ect. From our calculation we expect that plasmas with densities smaller than 1011 cm~2 will eventually show cooling with a smaller, if not vanishing hot phonon e!ect. In summary, heating and cooling of a pure electron plasma after resonant intersubband excitation were analyzed. Excited electrons undergo intersubband scattering with a time constant of 1.3 ps and thermalize subsequently with the unexcited (n"1) electrons within 2 ps. Concomitantly, the plasma is strongly heated by electron}electron scattering with the excited carriers. The energy transfer rate increases with plasma density. The resulting hot Fermi distribution cools down to 50 K within 25 ps by emission of LO phonons. Cooling is found to be independent on the plasma density in a range of electron concentrations between 1.5]1011 and 1.5]1012 cm~2.
6. Conclusions and outlook In conclusion, femtosecond infrared spectroscopy of below-bandgap excitations in semiconductors has provided new and highly speci"c information on the physics of coherent optical polarizations and of nonequilibrium carriers. So far, most experiments have concentrated on carrier dynamics in bulk and nanostructured materials in order to understand the fundamental physical processes underlying ultrafast processes. Concomitant theoretical work based, e.g., on ensemble Monte Carlo techniques, plays an important role for the quantitative analysis of such data. In contrast to incoherent carrier dynamics, coherent optical polarizations at photon energies below the fundamental bandgap have been investigated in much less detail. There are interesting topics to be studied in future experiments like the physics of coherent intraband polarizations, the role of carrier quantum kinetics for intraband excitations, and coherent intersubband excitations between valence bands. Here, an extension of the spectral range covered with femtosecond pulses towards the far-infrared could be helpful.
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In addition to these aspects of fundamental physics, there is an increasing interest in using below-bandgap excitations for device applications. Recent examples are photodetectors [203] and light-emitting devices for the mid-infrared making use of intersubband transitions, among them the quantum-cascade laser [204]. Ultrafast studies of carrier dynamics in such devices is relevant for clarifying the operation principles and optimizing device performance and } thus } is a topic of current research. The work discussed in this review concentrated on the analysis of ultrafast phenomena and the high time resolution of the experimental techniques served mainly for monitoring nonequilibrium dynamics in real-time. Beyond this, there are "rst concepts and "rst experimental demonstrations of all-optical control of ultrafast processes in semiconductors [205}208]. In such schemes, interaction of the semiconductor with taylored pulses, multiple pulses at di!erent spectral positions or pulse sequences induces a time evolution of the system which is di!erent from the ultrafast relaxation dynamics outlined in Section 2.2. The coherent regime of material response, i.e. the time interval in which the optical polarization and the electric "eld of the ultrashort pulses show a well-de"ned phase relationship, is particularly interesting for such control strategies which could lead to new all-optical devices for switching and data processing.
Acknowledgements We would like to thank a number of colleagues who were involved in the studies discussed here. The work on bulk semiconductors was performed at the Physics Department E11 of the Technical University of Munich, together with W. Kaiser, R.J. BaK uerle, W. Frey, C. Ludwig, and M.T. Portella. At the Max-Born-Institute, S. Lutgen and R. Kaindl performed most of the experimental work on intersubband dynamics. The experiments on coherent intersubband polarizations were analyzed in close collaboration with T. Kuhn, B. Nottelmann, and M. Axt who are with the Institute for Theoretical Physics of the University of Muenster, Germany. P. Lugli and his coworkers, II. University of Rome Tor Vergata, Italy performed Monte Carlo simulations of intersubband scattering and electron thermalization. The GaInAs/AlInAs samples studied in our experiments were grown by H. Kuenzel and A. Hase from the Heinrich Hertz Institut fuK r Nachrichtentechnik GmbH, Berlin, Germany. We gratefully acknowledge "nancial support by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 296 and by the European Union through the ULTRAFAST Network.
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CONTENTS VOLUME 321 C. Struck. Galaxy collisions
1
C.A. Bertulani, V.Yu. Ponomarev. Microscopic studies on two-phonon giant resonances
139
T. Elsaesser, M. Woerner. Femtosecond infrared spectroscopy of semiconductors and semiconductor nanostructures
253
